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Organic Chemistry/Cover. Welcome to the world's foremost open content<br>Organic Chemistry Textbook<br>on the web! The Study of Organic Chemistry. Organic chemistry is primarily devoted to the unique properties of the carbon atom and its compounds. These compounds play a critical role in biology and ecology, Earth sciences and geology, physics, industry, medicine and — of course — chemistry. At first glance, the new material that organic chemistry brings to the table may seem complicated and daunting, but all it takes is concentration and perseverance. Millions of students before you have successfully passed this course and you can too! This field of chemistry is based less on formulas and more on reactions between various molecules under different conditions. Whereas a typical general chemistry question may ask a student to compute an answer with an equation from the chapter that they memorized, a more typical organic chemistry question is along the lines of "what product will form when substance X is treated with solution Y and bombarded by light". The key to learning organic chemistry is to "understand" it rather than cram it in the night before a test. It is all well and good to memorize the mechanism of Michael addition, but a superior accomplishment would be the ability to explain "why" such a reaction would take place. As in all things, it is easier to build up a body of new knowledge on a foundation of solid prior knowledge. Students will be well served by much of the knowledge brought to this subject from the subject of General Chemistry. Concepts with particular importance to organic chemists are covalent bonding, Molecular Orbit theory, VSEPR Modeling, understanding acid/base chemistry vis-a-vis pKa values, and even trends of the periodic table. This is by no means a comprehensive list of the knowledge you should have gained already in order to fully understand the subject of organic chemistry, but it should give you some idea of the things you need to know to succeed in an organic chemistry test or course. Organic Chemistry is one of the subjects which are very useful and close to our daily life. We always try to figure out some of the unknown mysteries of our daily life through our factious thinking habit, which generates superstitions. Through the help of chemistry we can help ourselves to get out of this kind of superstition. We always try to find the ultimate truth through our own convenience. In the ancient past we had struggled to make things to go as per our need. In that context we have found fire, house, food, transportation, etc... Now the burning question is: "how can chemistry help our daily life?" To find the answer of this questions, we have to know the subject thoroughly. Let us start it from now.

Organic Chemistry/Foundational concepts of organic chemistry. The purpose of this section is to review topics from freshman chemistry and build the foundation for further studies in organic chemistry. | Alkanes »

Organic Chemistry/Introduction to reactions. > Introduction to reactions > Introduction to reactions

Organic Chemistry/Alkenes. « Haloalkanes |Alkenes| Alkynes » Alkenes are aliphatic hydrocarbons containing carbon-carbon double bonds and general formula CnH2n. =Naming Alkenes= Alkenes are named as if they were alkanes, but the "-ane" suffix is changed to "-ene". If the alkene contains only one double bond and that double bond is terminal (the double bond is at one end of the molecule or another) then it is not necessary to place any number in front of the name. butane: C4H10 (CH3CH2CH2CH3)<br> butene: C4H8 (CH2=CHCH2CH3) If the double bond is not terminal (if it is on a carbon somewhere in the center of the chain) then the carbons should be numbered in such a way as to give the first of the two double-bonded carbons the lowest possible number, and that number should precede the "ene" suffix with a dash, as shown below. correct: pent-2-ene (CH3CH=CHCH2CH3)<br> incorrect: pent-3-ene (CH3CH2CH=CHCH3)<br> "The second one is incorrect because flipping the formula horizontally results in a lower number for the alkene." If there is more than one double bond in an alkene, all of the bonds should be numbered in the name of the molecule - even terminal double bonds. The numbers should go from lowest to highest, and be separated from one another by a comma. The IUPAC numerical prefixes are used to indicate the number of double bonds. octa-2,4-diene: CH3CH=CHCH=CHCH2CH2CH3<br> deca-1,5-diene: CH2=CHCH2CH2CH=CHCH2CH2CH2CH3 Note that the numbering of "2-4" above yields a molecule with two double bonds separated by just one single bond. Double bonds in such a condition are called "conjugated", and they represent an enhanced stability of conformation, so they are energetically favored as reactants in many situations and combinations. EZ Notation. Earlier in stereochemistry, we discussed cis/trans notation where cis- means same side and trans- means opposite side. Alkenes can present a unique problem, however in that the cis/trans notation sometimes breaks down. The first thing to keep in mind is that alkenes are planar and there's no rotation of the bonds, as we'll discuss later. So when a substituent is on one side of the double-bond, it stays on that side. The above example is pretty straight-forward. On the left, we have two methyl groups on the same side, so it's cis-but-2-ene. And on the right, we have them on opposite sides, so we have trans-but-2-ene. So in this situation, the cis/trans notation works and, in fact, these are the correct names. From the example above, how would you use cis and trans? Which is the same side and which is the opposite side? Whenever an alkene has 3 or 4 differing substituents, one must use the what's called the EZ nomenclature, coming from the German words, Entgegen (opposite) and Zusammen (same). Let's begin with (Z)-3-methylpent-2-ene. We begin by dividing our alkene into left and right halves. On each side, we assign a substituent as being either a high priority or low priority substituent. The priority is based on the atomic number of the substituents. So on the left side, hydrogen is the lowest priority because its atomic number is 1 and carbon is higher because its atomic number is 6. On the right side, we have carbon substituents on both the top and bottom, so we go out to the next bond. On to the top, there's another carbon, but on the bottom, a hydrogen. So the top gets high priority and the bottom gets low priority. Because the high priorities from both sides are on the same side, they are Zusammen (as a mnemonic, think 'Zame Zide'). Now let's look at (E)-3-methylpent-2-ene. On the left, we have the same substituents on the same sides, so the priorities are the same as in the Zusammen version. However, the substituents are reversed on the right side with the high priority substituent on the bottom and the low priority substituent on the top. Because the High and Low priorities are opposite on the left and right, these are Entgegen, or opposite. The system takes a little getting used to and it's usually easier to name an alkene than it is to write one out given its name. But with a little practice, you'll find that it's quite easy. Comparison of E-Z with cis-trans. To a certain extent, the Z configuration can be regarded as the "cis-" isomer and the E as the "trans-" isomers. This correspondence is exact only if the two carbon atoms are identically substituted. In general, cis-trans should only be used if each double-bonded carbon atom has a hydrogen atom (i.e. R-CH=CH-R'). IUPAC Gold book on cis-trans notation. IUPAC Gold book on E-Z notation. =Properties= Alkenes are molecules with carbons bonded to hydrogens which contain at least two sp2 hybridized carbon atoms. That is, to say, at least one carbon-to-carbon double bond, where the carbon atoms, in addition to an electron pair shared in a "sigma" (σ) bond, share one pair of electrons in a "pi" (π) bond between them. The general formula for an aliphatic alkene is: CnH2n -- "e.g." C2H4 "or" C3H6 Diastereomerism. Restricted rotation. Because of the characteristics of pi-bonds, alkenes have very limited rotation around the double bonds between two atoms. In order for the alkene structure to rotate the pi-bond would first have to be broken - which would require about 60 or 70 kcal of energy per mol. For this reason alkenes have different chemical properties based on which side of the bond each atom is located. For example, but-2-ene exists as two diastereomers: =Relative stability= Observing the reaction of the addition of hydrogen to 1-butene, (Z)-2-butene, and (E)-2-butene, we can see that all of the products are butane. The difference between the reactions is that each reaction has a different energy: -30.3 kcal/mol for 1-butene, -28.6 kcal/mol for (Z)-2-butene and -27.6 kcal/mol for (E)-2-butene. This illustrates that there are differences in the stabilities of the three species of butene isomers, due to the difference in how much energy can be released by reducing them. The relative stability of alkenes may be estimated based on the following concepts: Internal alkenes are more stable than terminal alkenes because they are connected to more carbons on the chain. Since a terminal alkene is located at the end of the chain, the double bond is only connected to one carbon, and is called primary (1°). Primary carbons are the least stable. In the middle of a chain, a double bond could be connected to two carbons. This is called secondary (2°). The most stable would be quaternary (4°). =Reactions= Preparation. There are several methods for creating alkenes. Some of these methods, such as the Wittig reaction, we'll only describe briefly in this chapter and instead, cover them in more detail later in the book. For now, it's enough to know that they are ways of creating alkenes. Dehydrohalogenation of Haloalkanes. Alkyl halides are converted into alkenes by dehydrohalogenation: elimination of the elements of hydrogen halide. Dehydrohalogenation involves removal of the halogen atom together with a hydrogen atom from a carbon adjacent to the one bearing the halogen. It uses the E2 elimination mechanism that we'll discuss in detail at the end of this chapter The haloalkane must have a hydrogen and halide 180° from each other on neighboring carbons. If there is no hydrogen 180° from the halogen on a neighboring carbon, the reaction will not take place. It is not surprising that the reagent required for the elimination of what amounts to a molecule of acid is a strong base for example: alcoholic KOH. In some cases this reaction yields a single alkene. and in other cases yield a mixture. n-Butyl chloride, for example, can eliminate hydrogen only from C-2 and hence yields only 1-butene. sec-Butyl chloride, on the other hand, can eliminate hydrogen from either C-l or C-3 and hence yields both 1-butene and 2-butene. Where the two alkenes can be formed, 2-butene is the chief product. Dehalogenation of Vicinal Dihalides. The dehalogenation of vicinal dihalides (halides on two neighboring carbons, think "vicinity") is another method for synthesizing alkenes. The reaction can take place using either sodium iodide in a solution of acetone, or it can be performed using zinc dust in a solution of either heated ethanol or acetic acid. This reaction can also be performed with magnesium in ether, though the mechanism is different as this actually produces, as an intermediate, a Grignard reagent that reacts with itself and causes an elimination, resulting in the alkene. Dehydration of alcohols. An alcohol is converted into an alkene by dehydration: elimination of a molecule of water. Dehydration requires the presence of an acid and the application of heat. It is generally carried out in either of two ways, heating the alcohol with sulfuric or phosphoric acid to temperatures as high as 200, or passing the alcohol vapor over alumina, Al2O3 , at 350-400, alumina here serving as a Lewis acid. Ease of dehydration of alcohols : 3° > 2° > 1° Where isomeric alkenes can be formed, we again find the tendency for one isomer to predominate. Thus, sec-butyl alcohol, which might yield both 2-butene and 1-butene, actually yields almost exclusively the 2-isomer The formation of 2-butene from n-butyl alcohol illustrates a characteristic of dehydration that is not shared by dehydrohalogenalion: the double bond can be formed at a position remote from the carbon originally holding the -OH group. This characteristic is accounted for later. It is chiefly because of the greater certainty as to where the double bond will appear that dehydrohalogeation is often preferred over dehydration as a method of making alkenes. Reduction of Alkynes. Reduction of an alkyne to the double-bond stage can yield either a cis-alkene or a trans-alkene, unless the triple bond is at the end of a chain. Just which isomer predominates depends upon the choice of reducing agent. Predominantly trans-alkene is obtained by reduction of alkynes with sodium or lithium in liquid ammonia. Almost entirely cis-alkene (as high as 98%) is obtained by hydrogenation of alkynes with several different catalysts : a specially prepared palladium called Lindlar's catalyst; or a nickel boride called P-2 catalyst. Each of these reactions is, then, highly stereoselective. The stereoselectivity in the cis-reduction of alkynes is attributed, in a general way, to the attachment of two hydrogens to the same side of an alkyne sitting on the catalyst surface; presumably this same stereochemistry holds for the hydrogenation of terminal alkynes which cannot yield cis- and trans-alkenes. Markovnikov's Rule. Before we continue discussing reactions, we need to take a detour and discuss a subject that's very important in Alkene reactions, "Markovnikov's Rule." This is a simple rule stated by the Russian Vladmir Markovnikov in 1869, as he was showing the orientation of addition of HBr to alkenes. His rule states:"When an unsymmetrical alkene reacts with a hydrogen halide to give an alkyl halide, the hydrogen adds to the carbon of the alkene that has the greater number of hydrogen substituents, and the halogen to the carbon of the alkene with the fewer number of hydrogen substituents" (This rule is often compared to the phrase: "The rich get richer and the poor get poorer." Aka, the Carbon with the most Hydrogens gets another Hydrogen and the one with the least Hydrogens gets the halogen) This means that the nucleophile of the electrophile-nucleophile pair is bonded to the position most stable for a carbocation, or partial positive charge in the case of a transition state. Examples. formula_1 Here the Br attaches to the middle carbon over the terminal carbon, because of Markovnikov's rule, and this is called a Markovnikov product. Markovnikov product. The product of a reaction that follows Markovnikov's rule is called a Markovnikov product. Markovnikov addition. Markovnikov addition is an addition reaction which follows Markovnikov's rule, producing a Markovnikov product. Anti-Markovnikov addition. Certain reactions produce the opposite of the Markovnikov product, yielding what is called anti-Markovnikov product. That is, hydrogen ends up on the more substituted carbon of the double bond. The hydroboration/oxidation reaction that we'll discuss shortly, is an example of this, as are reactions that are conducted in peroxides. A modernized version of Markovnikov's rule often explains the "anti-Markovnikov" behavior. The original Markovnikov rule predicts that the hydrogen (an electrophile) being added across a double bond will end up on the carbon with more hydrogens. Generalizing to all electrophiles, it is really the electrophile which ends up on the carbon with the greatest number of hydrogens. Usually hydrogen plays the role of the electrophile; however, hydrogen can also act as an nucleophile in some reactions. The following expansion of Markovnikov's rule is more versatile: "When an alkene undergoes electrophilic addition, the electrophile adds to the carbon with the greatest number of hydrogen substituents. The nucleophile adds to the more highly substituated carbon." Or more simply: "The species that adds first adds to the carbon with the greatest number of hydrogens." The fact that some reactions reliably produce anti-Markovnikov products is actually a powerful tool in organic chemistry. For example, in the reactions we discuss below, we'll show two different ways of creating alcohols from alkenes: Oxymercuration-Reduction and Hydroboration/Oxidation. Oxymercuration produces a Markovnikov product while Hydroboration produces an anti-Markovnikov product. This gives the organic chemist a choice in products without having to be stuck with a single product that might not be the most desired. Why it works. Markovnikov's rule works because of the stability of carbocation intermediates. Experiments tend to reveal that carbocations are planar molecules, with a carbon that has three substituents at 120° to each other and a vacant p orbital that is perpendicular to it in the 3rd plane. The p orbital extends above and below the trisubstituent plane. This leads to a stabilizing effect called hyperconjugation. Hyperconjugation is what happens when there is an unfilled (antibonding or vacant) C-C π orbital and a filled C-H σ bond orbital next to each other. The result is that the filled C-H σ orbital interacts with the unfilled C-C π orbital and stabilizes the molecule. The more highly substituted the molecule, the more chances there are for hyperconjugation and thus the more stable the molecule is. Another stabilizing effect is an inductive effect. Exceptions to the Rule. There are a few exceptions to the Markovnikov rule, and these are of tremendous importance to organic synthesis. Addition reactions. Hydroboration. Hydroboration is a very useful reaction in Alkenes, not as an end product so much as an intermediate product for further reactions. The primary one we'll discuss below is the Hydroboration/Oxidation reaction which is actually a hydroboration reaction followed by a completely separate oxidation reaction. The addition of BH3 is a concerted reaction in that several bonds are broken and formed at the same time. Hydroboration happens in what's called syn-addition because the boron and one of its hydrogens attach to the same side of the alkene at the same time. As you can see from the transition state in the center of the image, this produces a sort of box between the two alkene carbons and the boron and its hydrogen. In the final step, the boron, along with its other two hydrogens, remains attached to one carbon and the other hydrogen attaches to the adjacent carbon. This description is fairly adequate, however, the reaction actually continues to happen and the -BH2 continue to react with other alkenes giving an R2BH and then again, until you end up with a complex of the boron atom attached to 3 alkyl groups, or R3B. This trialkyl-boron complex is then used in other reactions to produce various products. Borane, in reality, is not stable as BH3. Boron, in this configuration has only 6 electrons and wants 8, so in its natural state it actually creates the B2H6 complex shown on the left. Furthermore, instead of using B2H6 itself, BH3 is often used in a complex with tetrahydrofuran (THF) as shown in the image on the right.In either situation, the result of the reactions are the same. Hydroboration/Oxidation. With the reagent diborane, (BH3)2, alkenes undergo hydroboration to yield alkylboranes, R3B, which on oxidation give alcohols.The reaction procedure is simple and convenient, the yields are exceedingly high, and the products are ones difficult to obtain from alkenes in anyother way. Diborane is the dimer of the hypothetical BH3 (borane) and, in the reactions that concern us, acts much as though it were BH3 . Indeed, in tetrahydrofuran, one of the solvents used for hydroboration, the reagent exists as the monomer, in the form of an acid-base complex with the solvent. Hydroboration involves addition to the double bond of BH3 (or, in following stages, BH2R and BHR2), with hydrogen becoming attached to one doubly-bonded carbon, and boron to the other. The alkylborane can then undergo oxidation, in which the boron is replaced by -OH. Thus, the two-stage reaction process of hydroboration-oxidation permits, in effect, the addition to the carbon-carbon double bond of the elements of H-OH. Reaction is carried out in an ether, commonly tetrahydrofuran or "diglyme" (diethylene glycol methyl ether, CH3OCH2CH2OCH2CH2OCH3). Diborane is commercially available in tetrahydrofuran solution. The alkylboranes are not isolated, but are simply treated in situ with alkaline hydrogen peroxide. Stereochemistry and Orientation. Hydroboration-oxidation, then, converts alkenes into alcohols. Addition is highly regiospecific; the preferred product here, however, is exactly opposite to the one formed by oxymercuration-demercuration or by direct acid-catalyzed hydration. The hydroboration-oxidation process gives products corresponding to anti-Markovnikov addition of water to the carbon-carbon double bond. The reaction of 3,3-dimethyl-l -butene illustrates a particular advantage of the method. Rearrangement does not occur in hydroboration evidently because carbonium ions are not intermediates and hence the method can be used without the complications that often accompany other addition reactions. The reaction of 1,2-dimethylcyclopentene illustrates the stereochemistry of the synthesis: hydroboration-oxidation involves overall syn addition. Oxymercuration/Reduction. Alkenes react with mercuric acetate in the presence of water to give hydroxymercurial compounds which on reduction yield alcohols. The first stage, oxymercuration, involves addition to the carbon-carbon double bond of -OH and -HgOAc. Then, in reduction, the -HgOAc is replaced by -H. The reaction sequence amounts to hydration of the alkene, but is much more widely applicable than direct hydration. The two-stage process of oxymercuration/reduction is fast and convenient, takes place under mild conditions, and gives excellent yields often over 90%. The alkene is added at room temperature to an aqueous solution of mercuric acetate diluted with the solvent tetrahydrofuran. Reaction is generally complete within minutes. The organomercurial compound is not isolated but is simply reduced in situ by sodium borohydride, NaBH4. (The mercury is recovered as a ball of elemental mercury.) Oxymercuration/reduction is highly regiospecific, and gives alcohols corresponding to Markovnikov addition of water to the carbon-carbon double bond. Oxymercuration involves electrophilic addition to the carbon-carbon double bond, with the mercuric ion acting as electrophile. The absence of rearrangement and the high degree of stereospecificity (typically anti) in the oxymercuration step argues against an open carbonium ion as intermediate. Instead, it has been proposed, there is formed a cyclic mercurinium ion, analogous to the bromonium and chloronium ions involved in the addition of halogens. In 1971, Olah reported spectroscopic evidence for the preparation of stable solutions of such mercurinium ions. The mercurinium ion is attacked by the nucleophilic solvent water, in the present case to yield the addition product. This attack is back-side (unless prevented by some structural feature) and the net result is anti addition, as in the addition of halogens. Attack is thus of the SN2 type; yet the orientation of addition shows that the nucleophile becomes attached to the more highly substituted carbon as though there were a free carbonium ion intermediate. As we shall see, the transition state in reactions of such unstable threemembered rings has much SN1 character. Reduction is generally not stereospecific and can, in certain special cases, be accompanied by rearrangement. Despite the stereospecificity of the first stage, then, the overall process is not,in general, stereospecific. Rearrangements can occur, but are not common. The reaction of 3,3-dimethyl-1-butene illustrates the absence of the rearrangements that are typical of intermediate carbonium ions. Diels-Alder Reaction. The Diels–Alder reaction is a reaction (specifically, a cycloaddition) between a conjugated diene and a substituted alkene, commonly termed the dienophile, to form a substituted cyclohexene system. The reaction can proceed even if some of the atoms in the newly formed ring are not carbon. Some of the Diels–Alder reactions are reversible; the decomposition reaction of the cyclic system is then called the retro-Diels–Alder. The Diels–Alder reaction is generally considered one of the more useful reactions in organic chemistry since it requires very little energy to create a cyclohexene ring, which is useful in many other organic reactions A concerted, single-step mechanism is almost certainly involved; both new carbon-carbon bonds are partly formed in the same transition state, although not necessarily to the same extent. The Diels-Alder reaction is the most important example of cycloaddition. Since reaction involves a system of 4 π electrons (the diene) and a system of 2 π it electrons (the dienophile), it is known as a [4 + 2] cycloaddition. Catalytic addition of hydrogen. Catalytic hydrogenation of alkenes produce the corresponding alkanes. The reaction is carried out under pressure in the presence of a metallic catalyst. Common industrial catalysts are based on platinum, nickel or palladium, but for laboratory syntheses, Raney nickel (formed from an alloy of nickel and aluminium) is often employed. The catalytic hydrogenation of ethylene to yield ethane proceeds thusly: Electrophilic addition. Most addition reactions to alkenes follow the mechanism of electrophilic addition. An example is the Prins reaction, where the electrophile is a carbonyl group. Halogenation. Addition of elementary bromine or chlorine in the presence of an organic solvent to alkenes yield vicinal dibromo- and dichloroalkanes, respectively. The decoloration of a solution of bromine in water is an analytical test for the presence of alkenes: CH2=CH2 + Br2 → BrCH2-CH2Br The reaction works because the high electron density at the double bond causes a temporary shift of electrons in the Br-Br bond causing a temporary induced dipole. This makes the Br closest to the double bond slightly positive and therefore an electrophile. Hydrohalogenation. Addition of hydrohalic acids like HCl or HBr to alkenes yield the corresponding haloalkanes. If the two carbon atoms at the double bond are linked to a different number of hydrogen atoms, the halogen is found preferentially at the carbon with less hydrogen substituents (Markovnikov's rule). Addition of a carbene or carbenoid yields the corresponding cyclopropane Oxidation. Alkenes are oxidized with a large number of oxidizing agents. In the presence of oxygen, alkenes burn with a bright flame to form carbon dioxide and water. Catalytic oxidation with oxygen or the reaction with percarboxylic acids yields epoxides. Reaction with ozone in ozonolysis leads to the breaking of the double bond, yielding two aldehydes or ketones: R1-CH=CH-R2 + O3 → R1-CHO + R2-CHO + H2O This reaction can be used to determine the position of a double bond in an unknown alkene. Polymerization. Polymerization of alkenes is an economically important reaction which yields polymers of high industrial value, such as the plastics polyethylene and polypropylene. Polymerization can either proceed via a free-radical or an ionic mechanism. =Substitution and Elimination Reaction Mechanisms= Nucleophilic Substitution Reactions. Nucleophilic substitution reactions (SN1 and SN2) are very closely related to the E1 and E2 elimination reactions, discussed later in this section, and it is generally a good idea to learn the reactions together, as there are parallels in reaction mechanism, preferred substrates, and the reactions sometimes compete with each other. It's important to understand that substitution and elimination reactions are not associated with a specific compound or mixture so much as they're a representation of how certain reactions take place. At times, combinations of these mechanisms may occur together in the same reaction or may compete against each other, with influences such as solvent or nucleophile choice being the determining factor as to which reaction will dominate. In nucleophilic substitution, a nucleophile attacks a molecule and takes the place of another nucleophile, which then leaves. The nucleophile that leaves is called the leaving group. Nucleophilic substitutions "require " A leaving group is a charged or neutral moiety (group) which breaks free. SN1 vs SN2. One of the main differences between SN1 and SN2 is that the SN1 reaction is a 2-step reaction, initiated by disassociation of the leaving group. The SN2 reaction, on the other hand, is a 1-step reaction where the attacking nucleophile, because of its higher affinity for and stronger bonding with the carbon, forces the leaving group to leave. These two things happen in a single step. These two different mechanisms explain the difference in reaction rates between SN1 and SN2 reactions. SN1 reactions are dependent on the leaving group disassociating itself from the carbon. It is the rate-limiting step and thus, the reaction rate is a first-order reaction whose rate depends solely on that step. Alternatively, in SN2 reactions, the single step of the nucleophile coming together with the reactant from the opposite side of the leaving group, is the key to its rate. Because of this, the rate is dependent on both the concentration of the nucleophile as well as the concentration of the reactant. The higher these two concentrations, the more frequent the collisions. Thus the reaction rate is a second-order reaction: SN2 Reactions. There are primarily 3 things that affect whether an SN2 reaction will take place or not. The most important is structure. That is whether the alkyl halide is on a methyl, primary, secondary, or tertiary carbon. The other two components that determine whether an SN2 reaction will take place or not, are the nucleophilicity of the nucleophile and the solvent used in the reaction. The structure of the alkyl halide has a great effect on mechanism. CH3X & RCH2X are the preferred structures for SN2. R2CHX can undergo the SN2 under the proper conditions (see below), and R3CX rarely, if ever, is involved in SN2 reactions. The reaction takes place by the nucleophile attacking from the opposite side of the bromine atom. Notice that the other 3 bonds are all pointed away from the bromine and towards the attacking nucleophile. When these 3 bonds are hydrogen bonds, there's very little steric hinderance of the approaching nucleophile. However, as the number of R groups increases, so does the steric hinderance, making it more difficult for the nucleophile to get close enough to the α-carbon, to expel the bromine atom. In fact, tertiary carbons (R3CX) are so sterically hindered as to prevent the SN2 mechanism from taking place at all. In the case of this example, a secondary α-carbon, there is still a great deal of steric hinderance and whether the SN2 mechanism will happen will depend entirely on what the nucleophile and solvent are. SN2 reactions are preferred for methyl halides and primary halides. Another important point to keep in mind, and this can be seen clearly in the example above, during an SN2 reaction, the molecule undergoes an inversion. The bonds attached to the α-carbon are pushed away as the nucleophile approaches. During the transition state, these bonds become planar with the carbon and, as the bromine leaves and the nucleophile bonds to the α-carbon, the other bonds fold back away from the nucleophile. This is particularly important in chiral or pro-chiral molecules, where an R configuration will be converted into an S configuration and vice versa. As you'll see below, this is in contrast to the results of SN1 reactions. Examples: OH- is the nucleophile, Cl is the electrophile, HOCH3 is the product, and Cl- is the leaving group. or, The above reaction, taking place in acetone as the solvent, sodium and iodide disassociate almost completely in the acetone, leaving the iodide ions free to attack the CH-Br molecules. The negatively charged iodide ion, a nucleophile, attacks the methyl bromide molecule, forcing off the negatively charged bromide ion and taking its place. The bromide ion is the leaving group. Nucleophilicity. Nucleophilicity is the rate at which a nucleophile displaces the leaving group in a reaction. Generally, nucleophilicity is stronger, the larger, more polarizable, and/or the less stable the nucleophile. No specific number or unit of measure is used. All other things being equal, nucleophiles are generally compared to each other in terms of relative reactivity. For example, a particular strong nucleophile might have a relative reactivity of 10,000 that of a particular weak nucleophile. These relationships are generalities as things like solvent and substrate can affect the relative rates, but they are generally good guidelines for which species make the best nucleophiles. All nucleophiles are Lewis bases. In SN2 reactions, the preferred nucleophile is a strong nucleophile that is a weak base. Examples of these are N3-, RS-, I-, Br-, and CN-. Alternatively, a strong nucleophile that's also a strong base can also work. However, as mentioned earlier in the text, sometimes reaction mechanisms compete and in the case of a strong nucleophile that's a strong base, the SN2 mechanism will compete with the E2 mechanism. Examples of strong nucleophiles that are also strong bases, include RO- and OH-. Leaving Group. Leaving group is the group on the substrate that leaves. In the case of an alkyl halide, this is the halide ion that leaves the carbon atom when the nucleophile attacks. The tendency of the nucleophile to leave is Fluoride ions are very poor leaving groups because they bond very strongly and are very rarely used in alkyl halide substitution reactions. Reactivity of a leaving group is related to its basicity with stronger bases being poorer leaving groups. Solvent. The solvent can play an important role in SN2 reactions, particularly in SN2 involving secondary alkyl halide substrates, where it can be the determining factor in mechanism. Solvent can also have a great effect on reaction rate of SN2 reactions. The SN2 mechanism is preferred when the solvent is an aprotic, polar solvent. That is, a solvent that is polar, but without a polar hydrogen. Polar, protic solvents would include water, alcohols, and generally, solvents with polar NH or OH bonds. Good aprotic, polar solvents are HMPA, CH3CN, DMSO, and DMF. A polar solvent is preferred because it better allows the dissociation of the halide from the alkyl group. A protic solvent with a polar hydrogen, however, forms a 'cage' of hydrogen-bonded solvent around the nucleophile, hindering its approach to the substrate. SN1 Reactions. The SN1 mechanism is very different from the SN2 mechanism. In some of its preferences, its exactly the opposite and, in some cases, the results of the reaction can be significantly different. Like the SN2 mechanism, structure plays an important role in the SN1 mechanism. The role of structure in the SN1 mechanism, however, is quite different and because of this, the reactivity of structures is more or less reversed. The SN1 mechanism is preferred for tertiary alkyl halides and, depending on the solvent, may be preferred in secondary alkyl halides. The SN1 mechanism does not operate on primary alkyl halides or methyl halides. To understand why this is so, let's take a look at how the SN1 mechanism works. At the top of the diagram, the first step is the spontaneous dissociation of the halide from the alkyl halide. Unlike the SN2 mechanism, where the attacking nucleophile causes the halide to leave, the SN1 mechanism depends on the ability of the halide to leave on its own. This requires certain conditions. In particular, the stability of the carbocation is crucial to the ability of the halide to leave. Since we know tertiary carbocations are the most stable, they are the best candidates for the SN1 mechanism. And with appropriate conditions, secondary carbocations will also operate by the SN1 mechanism. Primary and methyl carbocations however, are not stable enough to allow this mechanism to happen. Once the halide has dissociated, the water acts as a nucleophile to bond to the carbocation. In theSN2 reactions, there is an inversion caused by the nucleophile attacking from the opposite side while the halide is still bonded to the carbon. In the SN1 mechanism, since the halide has left, and the bonds off of the α-carbon have become planar, the water molecule is free to attack from either side. This results in, primarily, a racemic mixture. In the final step, one of the hydrogens of the bonded water molecule is attacked by another water molecule, leaving an alcohol. "Note: Racemic mixtures imply entirely equal amounts of mixture, however this is rarely the case in SN1. There is a slight tendency towards attack from the opposite side of the halide. This is the result some steric hinderence from the leaving halide which is sometimes close enough to the leaving side to block the nucleophile's approach from that side." Solvent. Like the SN2 mechanism, the SN1 is affected by solvent as well. As with structure, however, the reasons differ. In the SN1 mechanism, a polar, protic solvent is used. The polarity of the solvent is associated with the dielectric constant of the solvent and solutions with high dielectric constants are better able to support separated ions in solution. In SN2 reactions, we were concerned about polar hydrogen atoms "caging" our nucleophile. This still happens with a polar protic solvent in SN1 reactions, so why don't we worry about it? You have to keep in mind the mechanism of the reaction. The first step, and more importantly, the rate-limiting step, of the SN1 reaction, is the ability to create a stable carbocation by getting the halide anion to leave. With a polar protic solvent, just as with a polar aprotic solvent,we're creating a stable cation, however it's the polar hydrogens that stabilize the halide anion and make it better able to leave. Improving the rate-limiting step is always the goal. The "caging" of the nucleophile is unrelated to the rate-limiting step and even in its "caged" state, the second step, the attack of the nucleophile, is so much faster than the first step, that the "caging" can simply be ignored. Summary. SN1, SN2, E1, and E2, are all reaction mechanisms, not reactions themselves. They are mechanisms used by a number of different reactions. Usually in organic chemistry, the goal is to synthesize a product. In cases where you have possibly competing mechanisms, and this is particularly the case where an SN1 and an E1 reaction are competing, the dominating mechanism is going to decide what your product is, so knowing the mechanisms and which conditions favor one over the other, will determine your product. In other cases, knowing the mechanism allows you to set up an environment favorable to that mechanism. This can mean the difference between having your product in a few minutes, or sometime around the next ice age. So when you're designing a synthesis for a product, you need to consider, I want to get product Y, so what are my options to get to Y? Once you know your options and you've decided on a reaction, then you need to consider the mechanism of the reaction and ask yourself, how do I create conditions that are going to make this happen correctly and happen quickly? Elimination Reactions. Nucleophilic substitution reactions and Elimination reactions share a lot of common characteristics, on top of which, the E1 and SN1 as well as E2 and SN2 reactions can sometimes compete and, since their products are different, it's important to understand them both. Without understanding both kinds of mechanisms, it would be difficult to get the product you desire from a reaction. In addition, the SN1 and SN2 reactions will be referenced quite a bit by way of comparison and contrast, so it's probably best to read that section first and then continue here. Elimination reactions are the mechanisms for creating alkene products from haloalkane reactants. E1 and E2 elimination, unlike SN1 and SN2 substitution, mechanisms do not occur with methyl halides because the reaction creates a double bond between two carbon atoms and methylhalides have only one carbon. E1 vs E2. Reaction rates. E1 and E2 are two different pathways to creating alkenes from haloalkanes. As with SN1 and SN2 reactions, one of the key differences is in the reaction rate, as it provides great insight into the mechanisms. E1 reactions, like SN1 reactions are 2-step reactions. Also like SN1 reactions, the rate-limiting step is the dissociation of the halide from its alkane, making it a first-order reaction, depending on the concentration of the haloalkane, with a reaction rate of: On the other hand, E2 reactions, like SN2 reactions are 1-step reactions. And again, as with SN2 reactions, the rate limiting step is the ability of a nucleophile to attach to the alkane and displace the halide. Thus it is a second-order reaction that depends on the concentrations of both the nucleophile and haloalkane, with a reaction rate of: Zaitsev's Rule. Zaitsev's rule (sometimes spelled "Saytzeff") states that in an elimination reaction, when multiple products are possible, the most stable alkene is the major product. That is to say, the most highly substituted alkene (the alkene with the most non-hydrogen substituents) is the major product. Both E1 and E2 reactions produce a mixture of products, when possible, but generally follow Zaitsev's rule. We'll see below why E1 reactions follow Zaitsev's rule more reliably and tend to produce a purer product. The above image represents two possible pathways for the dehydrohalogenation of (S)-2-bromo-3-methylbutane. The two potential products are 2-methylbut-2-ene and 3-methylbut-1-ene. The images on the right are simplified drawings of the molecular product shown in the images in the center. As you can see on the left, the bromine is on the second carbon and in an E1 or E2 reaction, the hydrogen could be removed from either the 1st or the 3rd carbon. Zaitsev's rule says that the hydrogen will be removed predominantly from the 3rd carbon. In reality, there will be a mixture, but most of the product will be 2-methylbut-2-ene by the E1 mechanism. By the E2 reaction, as we'll see later, this might not necessarily be the case. E2. The E2 mechanism is concerted and highly stereospecific, because it can occur only when the H and the leaving group X are in an anti-coplanar position. That is, in a Newman projection, the H and X must be 180°, or in the anti-configuration. This behaviour stems from the best overlap of the 2"p" orbitals of the adjacent carbons when the pi bond has to be formed. If the H and the leaving group cannot be brought into this position due to the structure of the molecule, the "E2" mechanism will not take place. Therefore, only molecules having accessible H-X anti-coplanar conformations can react via this route. Furthermore, the E2 mechanism will operate contrary to Zaitsev's rule if the only anti-coplanar hydrogen from the leaving group results in the least stable alkene. A good example of how this can happen is by looking at how cyclohexane and cyclohexene derivatives might operate in E2 conditions. Let's look at the example above. The reactant we're using is 1-chloro-2-isopropylcyclohexane. The drawing at the top left is one conformation and the drawing below is after a ring flip. In the center are Newman projections of both conformations and the drawings on the right, the products. If we assume we're treating the 1-chloro-2-isopropylcyclohexane with a strong base, for example CH3CH2O- (ethanolate), the mechanism that dominates is E2. There are 3 hydrogens off of the carbons adjacent to our chlorinated carbon. The red and the green ones are two of them. The third would be hard to show but is attached to the same carbon as the red hydrogen, angled a little down from the plane and towards the viewer. The red hydrogen is the only hydrogen that's 180° from the chlorine atom, so it's the only one eligible for the E2 mechanism. Because of this, the product is going to be only 3-isopropylcylcohex-1-ene. Notice how this is contrary to Zaitsev's rule which says the most substituted alkene is preferred. By his rule, 1-isopropylcyclohexene should be our primary product, as that would leave the most substituted alkene. However it simply can't be produced because of the steric hindrance. The images below shows the molecule after a ring flip. In this conformation, no product is possible. As you can see from the Newman projection, there are no hydrogens 180° from the chlorine atom. So it's important, when considering the E2 mechanism, to understand the geometry of the molecule. Sometimes the geometry can be used to your advantage to preferentially get a single product. Other times it will prevent you from getting the product you want, and you'll need to consider a different mechanism to get your product. "Note: Often the word periplanar is used instead of coplanar. Coplanar implies precisely 180 degree separation and "peri-", from Greek for "near", implies near 180 degrees. Periplanar may actually be more accurate. In the case of the 1-chloro-3-isopropylcyclohexane example, because of molecular forces, the chlorine atom is actually slightly less than 180 degrees from both the hydrogen and the isopropyl group, so in this case, periplanar might be a more correct term." E1. The E1 mechanism begins with the dissociation of the leaving group from an alkyl, producing a carbocation on the alkyl group and a leaving anion. This is the same way the SN1 reaction begins, so the same thing that helps initiate that step in SN1 reactions, help initiate the step in E1 reactions. More specifically, secondary and tertiary carbocations are preferred because they're more stable than primary carbocations. The choice of solvent is the same as SN1 as well; a polar protic solvent is preferred because the polar aspect stabilizes the carbocation and the protic aspect stabilizes the anion. What makes the difference between whether the reaction takes the SN1 or E1 pathway then, must depend on the second step; the action of the nucleophile. In SN1 reactions, a strong nucleophile that's a weak base is preferred. The nucleophile will then attack and bond to the carbocation. In E1 reactions, a strong nucleophile is still preferred. The difference is that a strong nucleophile that's also a strong base, causes the nucleophile to attack the hydrogen at the β-carbon instead of the α-carbocation. The nucleophile/base then extracts the hydrogen causing the bonding electrons to fall in and produce a pi bond with the carbocation. Because the hydrogen and the leaving group are lost in two separate steps and the fact that it has no requirements as to geometry, the E1 mechanism more reliably produces products that follow Zaitsev's rule. =References= « Haloalkanes |Alkenes| Alkynes »

Organic Chemistry/Chirality. Introduction. Chirality (pronounced kie-RAL-it-tee) is the property of "handedness". If you attempt to superimpose your right hand on top of your left, the two do not match up in the sense that your right hand's thumb overlays your left hand's pinky finger. Your two hands cannot be superimposed identically, despite the fact that your fingers of each hand are connected in the same way. Any object can have this property, including molecules. An object that is chiral is an object that can not be superimposed on its mirror image. Chiral objects don't have a "plane of symmetry". An achiral object has a plane of symmetry or a rotation-reflection axis, i.e. reflection gives a rotated version. Optical isomers or enantiomers are stereoisomers which exhibit chirality. Optical isomerism is of interest because of its application in inorganic chemistry, organic chemistry, physical chemistry, pharmacology and biochemistry. They are often formed when asymmetric centers are present, for example, a carbon with four different groups bonded to it. Every stereocenter in one enantiomer has the opposite configuration in the other. When a molecule has more than one source of asymmetry, two optical isomers may be neither perfect reflections of each other nor superimposeable: some but not all stereocenters are inverted. These molecules are an example of diastereomers: They are not enantiomers. Diastereomers seldom have the same physical properties. Sometimes, the stereocentres are themselves symmetrical. This causes the counterintuitive situation where two chiral centres may be present but no isomers result. Such compounds are called meso compounds. A mixture of equal amounts of both enantiomers is said to be a racemic mixture. It is the symmetry of a molecule (or any other object) that determines whether it is chiral or not. Technically, a molecule is achiral (not chiral) if and only if it has an axis of improper rotation; that is, an n-fold rotation (rotation by 360°/n) followed by a reflection in the plane perpendicular to this axis which maps the molecule onto itself. A chiral molecule is not necessarily dissymmetric (completely devoid of symmetry) as it can have, e.g., rotational symmetry. A simplified rule applies to tetrahedrally-bonded carbon, as shown in the illustration: if all four substituents are different, the molecule is chiral. It is important to keep in mind that molecules which are dissolved in solution or are in the gas phase usually have considerable flexibility and thus may adopt a variety of different conformations. These various conformations are themselves almost always chiral. However, when assessing chirality, one must use a structural picture of the molecule which corresponds to just one chemical conformation – the one of lowest energy. Chiral Compounds With Stereocenters. Most commonly, chiral molecules have point chirality, centering around a single atom, usually carbon, which has four different substituents. The two enantiomers of such compounds are said to have different absolute configurations at this center. This center is thus stereogenic (i.e., a grouping within a molecular entity that may be considered a focus of stereoisomerism), and is exemplified by the α-carbon of amino acids. The special nature of carbon, its ability to form four bonds to different substituents, means that a mirror image of the carbon with four different bonds will not be the same as the original compound, no matter how you try to rotate it. Understanding this is vital because the goal of organic chemistry is understanding how to use tools to synthesize a compound with the desired chirality, because a different arrangement may have no effect, or even an undesired one. A carbon atom is chiral if it has four different items bonded to it at the same time. Most often this refers to a carbon with three heteroatoms and a hydrogen, or two heteroatoms plus a bond to another carbon plus a bond to a hydrogen atom. It can also refer to a nitrogen atom bonded to four different types of molecules, if the nitrogen atom is utilizing its lone pair as a nucleophile. If the nitrogen has only three bonds it is not chiral, because the lone pair of electrons can flip from one side of the atom to the other spontaneously. "Any atom in an organic molecule that is bonded to four different types of atoms or chains of atoms can be considered "chiral"." If a carbon atom (or other type of atom) has four different substituents, that carbon atom forms a "chiral center" (also known as a "stereocenter"). Chiral molecules often have one or more stereocenters. When drawing molecules, stereocenters are usually indicated with an asterisk near the carbon. Example: Left: The carbon atom has a Cl, a Br, and 2 CH3. That's only 3 different substituents, which means this is not a stereocenter. Center: The carbon atom has one ethyl group (CH2CH3), one methyl group (CH3) and 2 H. This is not a stereocenter. Right: The carbon atom has a Cl and 1 H. Then you must look around the ring. Since one side has a double bond and the other doesn't, it means the substituents off that carbon are different. The 4 different substituents make this carbon a stereocenter and makes the molecule chiral. A molecule can have multiple chiral centers without being chiral overall: It is then called a meso compound. This occurs if there is a symmetry element (a mirror plane or inversion center) which relates the chiral centers. Fischer projections. Fischer projections (after the German chemist ) are an ingenious means for representing configurations of carbon atoms. Considering the carbon atom as the center, the bonds which extend towards the viewer are placed horizontally. Those extending away from the viewer are drawn vertically. This process, when using the common dash and wedge representations of bonds, yields what is sometimes referred to as the "bowtie" drawing due to its characteristic shape. This representation is then further shorthanded as two lines: the horizontal (forward) and the vertical (back), as showed in the figure below: Naming conventions. There are three main systems for describing configuration: the oldest, the "relative" whose use is now deprecated, and the current, or "absolute". The relative configuration description is still used mainly in glycochemistry. Configuration can also be assigned on the purely empirical basis of the optical activity. By optical activity: (+)- and (-)-. An optical isomer can be named by the direction in which it rotates the plane of polarized light. If an isomer rotates the plane clockwise as seen by a viewer towards whom the light is traveling, that isomer is labeled (+). Its counterpart is labeled (-). The (+) and (-) isomers have also been termed d- and l-, respectively (for dextrorotatory and levorotatory). This labeling is easy to confuse with D- and L- and is therefore not encouraged by IUPAC. The fact that an enantiomer can rotate polarised light clockwise ("d"- or "+"- enantiomer) does not relate with the relative configuration (D- or L-) of it. By relative configuration: D- and L-. Fischer, whose research interest was in carbohydrate chemistry, took glyceraldehyde (the simplest sugar, systematic name 2,3-dihydroxypropanal) as a template chiral molecule and denoted the two possible configurations with D- and L-, which rotated polarised light clockwise and counterclockwise, respectively. All other molecules are assigned the D- or L- configuration if the chiral centre can be formally obtained from glyceraldehyde by substitution. For this reason the D- or L- naming scheme is called "relative configuration". An optical isomer can be named by the spatial configuration of its atoms. The D/L system does this by relating the molecule to glyceraldehyde. Glyceraldehyde is chiral itself, and its two isomers are labeled D and L. Certain chemical manipulations can be performed on glyceraldehyde without affecting its configuration, and its historical use for this purpose (possibly combined with its convenience as one of the smallest commonly-used chiral molecules) has resulted in its use for nomenclature. In this system, compounds are named by analogy to glyceraldehyde, which generally produces unambiguous designations, but is easiest to see in the small biomolecules similar to glyceraldehyde. One example is the amino acid alanine: alanine has two optical isomers, and they are labeled according to which isomer of glyceraldehyde they come from. Glycine, the amino acid derived from glyceraldehyde, incidentally, does not retain its optical activity, since its central carbon is not chiral. Alanine, however, is essentially methylated glycine and shows optical activity. The D/L labeling is unrelated to (+)/(-); it does not indicate which enantiomer is dextrorotatory and which is levorotatory. Rather, it says that the compound's stereochemistry is related to that of the dextrorotatory or levorotatory enantiomer of glyceraldehyde. Nine of the nineteen L-amino acids commonly found in proteins are dextrorotatory (at a wavelength of 589 nm), and D-fructose is also referred to as levulose because it is levorotatory. The dextrorotatory isomer of glyceraldehyde is in fact the D isomer, but this was a lucky guess. At the time this system was established, there was no way to tell which configuration was dextrorotatory. (If the guess had turned out wrong, the labeling situation would now be even more confusing.) A rule of thumb for determining the D/L isomeric form of an amino acid is the "CORN" rule. The groups: are arranged around the chiral center carbon atom. If these groups are arranged clockwise around the carbon atom, then it is the L-form. If counter-clockwise, it is the D-form.This rule only holds when the hydrogen atom is pointing out of the page. By absolute configuration: R- and S-. Main article: R-S System The absolute configuration system stems from the , which allow a precise description of a stereocenter without using any reference compound. In fact the basis is now the atomic number of the stereocenter substituents. The R/S system is another way to name an optical isomer by its configuration, without involving a reference molecule such as glyceraldehyde. It labels each chiral center R or S according to a system by which its ligands are each assigned a priority, according to the Cahn Ingold Prelog priority rules, based on atomic number. This system labels each chiral center in a molecule (and also has an extension to chiral molecules not involving chiral centers). It thus has greater generality than the D/L system, and can label, for example, an (R,R) isomer versus an (R,S) — diastereomers. The R/S system has no fixed relation to the (+)/(-) system. An R isomer can be either dextrorotatory or levorotatory, depending on its exact ligands. The R/S system also has no fixed relation to the D/L system. For example, one of glyceraldehyde's ligands is a hydroxy group, -OH. If a thiol group, -SH, were swapped in for it, the D/L labeling would, by its definition, not be affected by the substitution. But this substitution would invert the molecule's R/S labeling, due to the fact that sulfur's atomic number is higher than carbon's, whereas oxygen's is lower. [Note: This seems incorrect. Oxygen has a higher atomic number than carbon. Sulfur has a higher atomic number than oxygen. The reason the assignment priorities change in this example is because the CH2SH group gets a higher priority than the CHO, whereas in glyceraldehyde the CHO takes priority over the CH2OH.] For this reason, the D/L system remains in common use in certain areas, such as amino acid and carbohydrate chemistry. It is convenient to have all of the common amino acids of higher organisms labeled the same way. In D/L, they are all L. In R/S, they are not, conversely, all S — most are, but cysteine, for example, is R, again because of sulfur's higher atomic number. The word “racemic” is derived from the Latin word for grape; the term having its origins in the work of Louis Pasteur who isolated racemic tartaric acid from wine. Chiral Compounds Without Stereocenters. It is also possible for a molecule to be chiral without having actual point chirality (stereocenters). Commonly encountered examples include 1,1'-bi-2-naphthol (BINOL) and 1,3-dichloro-allene which have axial chirality, and (E)-cyclooctene which has planar chirality. For example, the isomers which are shown by the following figure are different. The two isomers cannot convert from one to another spontaneously because of restriction of rotation of double bonds. Other types of chiral compounds without stereocenters (like restriction of rotation of a single bond because of steric hindrance) also exist. Consider the following example of the R and S binol molecules: "The biphenyl C-C bond cannot rotate if the X and Y groups cause steric hindrance." "This compound exhibits spiral chirality." Properties of optical isomers. Enantiomers have – "when present in a symmetric environment" – identical chemical and physical properties except for their ability to rotate plane-polarized light by equal amounts but in opposite directions. A solution of equal parts of an optically-active isomer and its enantiomer is known as a racemic solution and has a net rotation of plane-polarized light of zero. Enantiomers differ in how they interact with different optical isomers of other compounds. In nature, most biological compounds (such as amino acids) occur as single enantiomers. As a result, different enantiomers of a compound may have substantially different biological effects. Different enantiomers of the same chiral drug can have very different pharmological effects, mainly because the proteins they bind to are also chiral. For example, spearmint leaves and caraway seeds respectively contain L-carvone and D-carvone – enantiomers of carvone. These smell different to most people because our taste receptors also contain chiral molecules which behave differently in the presence of different enantiomers. D-form Amino acids tend to taste sweet, whereas L-forms are usually tasteless. This is again due to our chiral taste molecules. The smells of oranges and lemons are examples of the D and L enantiomers. Penicillin's activity is stereoselective. The antibiotic only works on peptide links of D-alanine which occur in the cell walls of bacteria – but not in humans. The antibiotic can kill only the bacteria, and not us, because we don't have these D-amino acids. The electric and magnetic fields of polarized light oscillate in a geometric plane. An axis normal to this plane gives the direction of energy propagation. Optically active isomers rotate the plane that the fields oscillate in. The polarized light is actually rotated in a racemic mixture as well, but it is rotated to the left by one of the two enantiomers, and to the right by the other, which cancel out to zero net rotation. Chirality in biology. Many biologically-active molecules are chiral, including the naturally-occurring amino acids (the building blocks of proteins), and sugars. Interestingly, in biological systems most of these compounds are of the same chirality: most amino acids are L and sugars are D. The origin of this homochirality in biology is the subject of much debate. Chiral objects have different interactions with the two enantiomers of other chiral objects. Enzymes, which are chiral, often distinguish between the two enantiomers of a chiral substrate. Imagine an enzyme as having a glove-like cavity which binds a substrate. If this glove is right handed, then one enantiomer will fit inside and be bound while the other enantiomer will have a poor fit and is unlikely to bind. Chirality in inorganic chemistry. Many coordination compounds are chiral; for example the well-known [Ru(2,2'-bipyridine)3]2+ complex in which the three bipyridine ligands adopt a chiral propeller-like arrangement [7]. In this case, the Ru atom may be regarded as a stereogenic centre, with the complex having point chirality. The two enantiomers of complexes such as [Ru(2,2'-bipyridine)3]2+ may be designated as Λ (left-handed twist of the propeller described by the ligands) and Δ (right-handed twist). Hexol is a chiral cobalt compound. Enantiopure preparations. Several strategies exist for the preparation of enantiopure compounds. The first method is the separation of a racemic mixture into its isomers. Louis Pasteur in his pioneering work was able to isolate the isomers of tartaric acid because they crystallize from solution as crystals with differing symmetry. A less common and more recently discovered method is by enantiomer self-disproportionation, which is an advanced technique involving the separation of a primarily racemic fraction from a nearly enantiopure fraction via column chromatography. In a non-symmetric environment (such as a biological environment) enantiomers may react at different speeds with other substances. This is the basis for "chiral synthesis", which preserves a molecule's desired chirality by reacting it with or catalyzing it with chiral molecules capable of maintaining the product's chirality in the desired conformation (using certain chiral molecules to help it keep its configuration). Other methods also exist and are used by organic chemists to synthesize only (or maybe only "mostly") the desired enantiomer in a given reaction. Enantiopure medications. Advances in industrial chemical processes have allowed pharmaceutical manufacturers to take drugs that were originally marketed in racemic form and divide them into individual enantiomers, each of which may have unique properties. For some drugs, such as zopiclone, only one enantiomer (eszopiclone) is active; the FDA has allowed such once-generic drugs to be patented and marketed under another name. In other cases, such as ibuprofen, both enantiomers produce the same effects. Steroid receptor sites also show stereoisomer specificity. Examples of racemic mixtures and enantiomers that have been marketed include: Many chiral drugs must be made with high enantiomeric purity due to potential side-effects of the other enantiomer. (The other enantiomer may also merely be inactive.) Consider a racemic sample of thalidomide. One enantiomer was thought to be effective against morning sickness while the other is now known to be teratogenic. Unfortunately, in this case administering just one of the enantiomers to a pregnant patient would still be very dangerous as the two enantiomers are readily interconverted "in vivo". Thus, if a person is given either enantiomer, both the D and L isomers will eventually be present in the patient's serum and so chemical processes may not be used to mitigate its toxicity. See also. Optical activity

Organic Chemistry/Dienes. In alkene chemistry, we demonstrated that allylic carbon could maintain a cation charge because the double bond could de-localize to support the charge. What of having two double bonds separated by a single bond? What of having a compound that alternates between double bond and single bond? In addition to other concepts, this chapter will explore what a having a conjugated system means in terms of stability and reaction. Dienes are simply hydrocarbons which contain two double bonds. Dienes are intermediate between alkenes and polyenes. Dienes can divided into three classes:

Organic Chemistry/External links. =Resources= =Other online textbooks=

Organic Chemistry/Foundational concepts of organic chemistry/History of organic chemistry. Brief History. Jöns Jacob Berzelius, a physician by trade, first coined the term "organic chemistry" in 1806 for the study of compounds derived from biological sources. Up through the early 19th century, naturalists and scientists observed critical differences between compounds that were derived from living things and those that were not. Chemists of the period noted that there seemed to be an essential yet inexplicable difference between the properties of the two different types of compounds. The vital force theory, sometimes called "vitalism" (vital means "life force"), was therefore proposed, and widely accepted, as a way to explain these differences, that a "vital force" existed within organic material but did not exist in any inorganic materials. Synthesis of Urea. Friedrich Wöhler is widely regarded as a pioneer in organic chemistry as a result of his synthesizing of the biological compound urea (a component of urine in many animals) utilizing what is now called "the Wöhler synthesis." Wöhler mixed silver or lead cyanate with ammonium nitrate; this was supposed to yield ammonium cyanate as a result of an exchange reaction, according to Berzelius's dualism theory. Wöhler, however, discovered that the end product of this reaction is "not" ammonium cyanate (NH4OCN), an inorganic salt, but urea ((NH2)2CO), a biological compound. (Furthermore, heating ammonium cyanate turns it into urea.) Faced with this result, Berzelius had to concede that (NH2)2CO and NH4OCN were "isomers". Until this discovery in the year 1828, it was widely believed by chemists that organic substances could only be formed under the influence of the "vital force" in the bodies of animals and plants. Wöhler's synthesis dramatically proved that view to be false. Urea synthesis was a critical discovery for biochemists because it showed that a compound known to be produced in nature only by biological organisms could be produced in a laboratory under controlled conditions from inanimate matter. This "in vitro" synthesis of organic matter disproved the common theory (vitalism) about the vis vitalis, a transcendent "life force" needed for producing organic compounds. Organic vs Inorganic Chemistry. Although originally defined as the chemistry of biological molecules, organic chemistry has since been redefined to refer specifically to carbon compounds — even those with non-biological origin. Some carbon molecules are not considered organic, with carbon dioxide being the most well known and most common inorganic carbon compound, but such molecules are the exception and not the rule. Organic chemistry focuses on carbon and following movement of the electrons in carbon chains and rings, and also how electrons are shared with other carbon atoms and heteroatoms. Organic chemistry is primarily concerned with the properties of covalent bonds and non-metallic elements, though ions and metals do play critical roles in some reactions. The applications of organic chemistry are myriad, and include all sorts of plastics, dyes, flavorings, scents, detergents, explosives, fuels and many, many other products. Read the ingredient list for almost any kind of food that you eat — or even your shampoo bottle — and you will see the handiwork of organic chemists listed there. Major Advances in the Field of Organic Chemistry. Of course a chemistry text should at least mention Antoine Laurent Lavoisier. The French chemist is often called the "Father of Modern Chemistry" and his place is first in any pantheon of great chemistry figures. Your general chemistry textbook should contain information on the specific work and discoveries of Lavoisier — they will not be repeated here because his discoveries did not relate directly to organic chemistry in particular. Berzelius and Wöhler are discussed above, and their work was foundational to the specific field of organic chemistry. After those two, three more scientists are famed for independently proposing the elements of structural theory. Those chemists were August Kekulé, Archibald Couper, and Alexander Butlerov. Kekulé was a German, an architect by training, and he was perhaps the first to propose that isomerism was due to carbon's proclivity towards forming four bonds. Its ability to bond with up to four other atoms made it ideal for forming long chains of atoms in a single molecule, and also made it possible for the same number of atoms to be connected in an enormous variety of ways. Couper, a Scot, and Butlerov, a Russian, came to many of the same conclusions at the same time or just a short time after. Through the nineteenth century and into the twentieth, experimental results brought to light much new knowledge about atoms, molecules, and molecular bonding. In 1916 it was Gilbert Lewis of U.C. Berkeley who described covalent bonding largely as we know it today (electron-sharing). Nobel laureate Linus Pauling further developed Lewis' concepts by proposing resonance while he was at the California Institute of Technology. At about the same time, Sir Robert Robinson of Oxford University focused primarily on the electrons of atoms as the engines of molecular change. Sir Christopher Ingold of University College, London, organized what was known of organic chemical reactions by arranging them in schemes we now know as mechanisms, in order to better understand the sequence of changes in a synthesis or reaction. The field of organic chemistry is probably the most active and important field of chemistry at the moment, due to its extreme applicability to both biochemistry (especially in the pharmaceutical industry) and petrochemistry (especially in the energy industry). Organic chemistry has a relatively recent history, but it will have an enormously important future, affecting the lives of everyone around the world for many, many years to come. <noinclude> « Foundational concepts| History of Organic Chemistry | Atomic Structure > | Alkanes » </alkynes>

Computer Programming/Authors. List of Authors. "You always need someone to blame."

Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure. Atomic Structure. Atoms are made up of a nucleus and electrons that orbit the nucleus. The nucleus consists of protons and neutrons. An atom in its natural, uncharged state has the same number of electrons as protons. The nucleus. The nucleus is made up of protons, which are positively charged, and neutrons, which have no charge. Neutrons and protons have about the same mass, and together account for most of the mass of the atom. Electrons. The electrons are negatively charged particles. The mass of an electron is about 2000 times smaller than that of a proton or neutron at 0.00055 amu. Electrons circle so fast that it cannot be determined where electrons are at any point in time. The image depicts the old Bohr model of the atom, in which the electrons inhabit discrete "orbitals" around the nucleus much like planets orbit the sun. This model is outdated. Current models of the atomic structure hold that electrons occupy fuzzy clouds around the nucleus of specific shapes, some spherical, some dumbbell shaped, some with even more complex shapes. Shells and Orbitals. Electron shells. Electrons orbit atoms in clouds of distinct shapes and sizes. The electron clouds are layered one inside the other into units called shells, with the electrons occupying the simplest orbitals in the innermost shell having the lowest energy state and the electrons in the most complex orbitals in the outermost shell having the highest energy state. The higher the energy state, the more energy the electron has, just like a rock at the top of a hill has more potential energy than a rock at the bottom of a valley. The main reason why electrons exist in higher energy orbitals is because only two electrons can exist in any orbital. So electrons fill up orbitals, always taking the lowest energy orbitals available. An electron can also be pushed to a higher energy orbital, for example by a photon. Typically this is not a stable state and after a while the electron descends to lower energy states by emitting a photon spontaneously. These concepts will be important in understanding later concepts like optical activity of chiral compounds as well as many interesting phenomena outside the realm of organic chemistry (for example, how lasers work). Electron orbitals. Each different shell is subdivided into one or more orbitals, which also have different energy levels, although the energy difference between orbitals is less than the energy difference between shells. Longer wavelengths have less energy; the s orbital has the longest wavelength allowed for an electron orbiting a nucleus and this orbital is observed to have the lowest energy. Each orbital has a characteristic shape which shows where electrons most often exist. The orbitals are named using letters of the alphabet. In order of increasing energy the orbitals are: s, p, d, and f orbitals. As one progresses up through the shells (represented by the principal quantum number n) more types of orbitals become possible. The shells are designated by numbers. So the 2s orbital refers to the s orbital in the second shell. S orbital. The s orbital is the orbital lowest in energy and is spherical in shape. Electrons in this orbital are in their fundamental frequency. This orbital can hold a maximum of two electrons. P orbital. The next lowest-energy orbital is the p orbital. Its shape is often described as like that of a dumbbell. There are three p-orbitals each oriented along one of the 3-dimensional coordinates x, y or z. Each of these three "p" orbitals can hold a maximum of two electrons. These three different p orbitals can be referred to as the px, py, and pz. The s and p orbitals are important for understanding most of organic chemistry as these are the orbitals that are occupied in the type of atoms that are most common in organic compounds. D and F orbitals. There are also D and F orbitals. D orbitals are present in transition metals. Sulphur and phosphorus have empty D orbitals. Compounds involving atoms with D orbitals do come into play, but are rarely part of an organic molecule. F are present in the elements of the lanthanide and actinide series. Lanthanides and actinides are mostly irrelevant to organic chemistry. Filling electron shells. When an atom or ion receives electrons into its orbitals, the orbitals and shells fill up in a particular manner. There are three principles that govern this process: Pauli exclusion principle. No two electrons in an atom can have all four quantum numbers the same. What this translates to in terms of our pictures of orbitals is that each orbital can only hold two electrons, one "spin up" and one "spin down". Hund's rule. This states that filled and half-filled shells tend to have additional stability. In some instances, then, for example, the 4s orbitals will be filled before the 3d orbitals. This rule is applicable only for those elements that have d electrons, and so is less important in organic chemistry (though it is important in organometallic chemistry). Octet rule. The octet rule states that atoms tend to prefer to have eight electrons in their valence shell, so will tend to "combine" in such a way that each atom can have eight electrons in its valence shell, similar to the electronic configuration of a noble gas. In simple terms, molecules are more stable when the outer shells of their constituent atoms are empty, full, or have eight electrons in the outer shell. The main exception to the rule is helium, which is at lowest energy when it has two electrons in its valence shell. Other notable exceptions are aluminum and boron, which can function well with six valence electrons; and some atoms beyond group three on the periodic table that can have over eight electrons, such as sulphur. Additionally, some noble gasses can form compounds when expanding their valence shell. Hybridization. Hybridization refers to the combining of the orbitals of two or more covalently bonded atoms. Depending on how many free electrons a given atom has and how many bonds it is forming, the electrons in the s and the p orbitals will combine in certain manners to form the bonds. It is easy to determine the hybridization of an atom given a Lewis structure. First, you count the number of pairs of free electrons and the number of "sigma" bonds (single bonds). Do not count double bonds, since they do not affect the hybridization of the atom. Once the total of these two is determined, the hybridization pattern is as follows: Sigma Bonds + Electron Pairs Hybridization 2 sp 3 sp2 4 sp3 The pattern here is the same as that for the electron orbitals, which serves as a memory guide.

Organic Chemistry/Foundational concepts of organic chemistry/Bonding. Ionic Bonding. Ionic bonding is when positively and negatively charged ions stick to each other through electrostatic force. These bonds are " slightly weaker than covalent bonds" and stronger than Van der Waals bonding or hydrogen bonding. In ionic bonds the electronegativity of the negative ion is so much stronger than the electronegativity of the positive ion that the two ions do not share electrons. Rather, the more electronegative ion assumes full ownership of the electron(s). Perhaps the most common example of an ionically bonded substance is NaCl, or table salt. In this, the sodium (Na) atom gives up an electron to the much more electronegative chlorine (Cl) atom, and the two atoms become ions, Na+ and Cl-.The electrostatic bonding force between the two oppositely charged ions extends outside the local area attracting other ions to form giant crystal structures. For this reason most ionically bonded materials are solid at room temperature. Sodium chloride forms crystals with cubic symmetry. In these, the larger chloride ions are arranged in a cubic close-packing, while the smaller sodium ions fill the octahedral gaps between them. Each ion is surrounded by six of the other kind. This same basic structure is found in many other minerals, and is known as the halite structure. Covalent Bonding. Covalent bonding is close to the heart of organic chemistry. This is where two atoms share electrons in a bond. The goal of each atom is to "fill its octet" as well as have a "formal charge of zero". To do this, atomic nuclei share electrons in the space between them. This sharing also allows the atoms to reach a lower energy state, which stabilizes the molecule. Most reactions in chemistry are due to molecules achieving a lower energy state. Covalent bonds are most frequently seen between atoms with similar electronegativity. In molecules that only have one type of atom, e.g. H2 or O2 , the electronegativity of the atoms is essentially identical, so they cannot form ionic bonds. They always form covalent bonds. Carbon is especially good at covalent bonding because its electronegativity is intermediate relative to other atoms. That means it can give as well as take electrons as needs warrant. Covalently bonded compounds have strong internal bonds but weak attractive forces between molecules. Because of these weak attractive forces, the melting and boiling points of these compounds are much lower than compounds with ionic bonds. Therefore, such compounds are much more likely to be liquids or gases at room temperature than ionically bonded compounds. In molecules formed from two atoms of the same element, there is no difference in the electronegativity of the bonded atoms, so the electrons in the covalent bond are shared equally, resulting in a completely non-polar covalent bond. In covalent bonds where the bonded atoms are different elements, there is a difference in electronegativities between the two atoms. The atom that is more electronegative will attract the bonding electrons more toward itself than the less electronegative atom. The difference in charge on the two atoms because of the electrons causes the covalent bond to be polar. Greater differences in electronegativity result in more polar bonds. Depending on the difference in electronegativities, the polarity of a bond can range from non-polar covalent to ionic with varying degrees of polar covalent in between. An overall imbalance in charge from one side of a molecule to the other side is called a dipole moment. Such molecules are said to be polar. For a completely symmetrical covalently bonded molecule, the overall dipole moment of the molecule is zero. Molecules with larger dipole moments are more polar. The most common polar molecule is water. Bond Polarity and Dipole Moment. The ideas of bond polarity and dipole moment play important roles in organic chemistry. If you look at the image of methane on the right, the single most important aspect of it in terms of bond polarity is that it is a symmetric molecule. It has 4 hydrogens, all bonded at 109.5° from the other, and all with precisely the same bond angle. Each carbon-hydrogen bond is slightly polar (hydrogen has an electronegativity of 2.1, carbon 2.5), but because of this symmetry, the polarities cancel each other out and overall, methane is a non-polar molecule. The distinction is between Bond Polarity and Molecular polarity. The total polarity of a molecule is measured as Dipole Moment. The actual calculation of dipole moment isn't really necessary so much as an understanding of what it means. Frequently, a guesstimate of dipole moment is pretty easy once you understand the concept and until you get into the more advanced organic chemistry, exact values are of little value. Basically, the molecular polarity is, essentially, the summation of the vectors of all of the bond polarities in a molecule. Van der Waals Bonding. Van der Waals bonding is the collective name for three types of interactions: A Dipole is caused by an atom or molecule fragment having a higher electronegativity (this is a measure of its effective nuclear charge, and thus the attraction of the nucleus by electrons) than one to which it is attached. This means that it pulls electrons closer to it, and has a higher share of the electrons in the bond. Dipoles can cancel out by symmetry, eg: Carbon dioxide (O=C=O) is linear so there is no dipole, but the charge distribution is asymmetric causing a quadrupole moment (this acts similarly to a dipole, but is much weaker). Organometallic Compounds and Bonding. Organometallic chemistry combines aspects of inorganic chemistry and organic chemistry, because organometallic compounds are chemical compounds containing bonds between carbon and a metal or metalloid element. Organometallic bonds are different from other bonds in that they are not either truly covalent or truly ionic, but each type of metal has individual bond character. Cuprate (copper) compounds, for example, behave quite differently than Grignard reagents (magnesium), and so beginning organic chemists should concentrate on how to use the most basic compounds mechanistically, while leaving the explanation of exactly what occurs at the molecular level until later and more in-depth studies in the subject. Basic organometallic interactions are discussed fully in a later chapter.

Organic Chemistry/Foundational concepts of organic chemistry/Resonance. Resonance. Resonance refers to structures that are not easily represented by a single electron dot structure but that are intermediates between two or more drawn structures. Resonance is easily misunderstood in part because of the way certain chemistry textbooks attempt to explain the concept. In science, analogies can provide an aid to understanding, but analogies should not be taken too literally. It is sometimes best to use analogies to introduce a topic, but then explain the differences and inevitable complications as further details on a complicated subject. This is the case for resonance. Just as entropic principles cannot be applied to individual molecules, it is impossible to say whether or not any given individual molecule with a resonance structure is literally in one configuration or another. The actual situation on the molecular scale is that each configuration of the molecule contributes a percentage to the possible configurations, resulting in a "blend" of the possible structures. Changes in molecular shape occur so rapidly, and on such a tiny scale, that the actual physical locations of individual electrons cannot be precisely known (due to Heisenberg's Uncertainty Principle). The result of all that complexity is simply this: molecules with resonance structures are treated as mixtures of their multiple forms, with a greater percentage of probability given to the most stable configurations. The nuclei of the atoms are not moving when they are represented by resonance structure drawings. Rather, the electrons are portrayed as if they were moving instead. The true situation is that no one can say for certain exactly where any individual electron is at any specific moment, but rather electron location can be expressed as a probability only. What a dot structure is actually showing is where electrons almost certainly are located, therefore resonance structures indicate a split in those same probabilities. Chemists are absolutely certain where electrons are located when one carbon bonds four hydrogens (methane), but it is less certain where precisely any given electron is located when six carbons bond six hydrogens in a ring structrue (benzene). Resonance is an expression of this uncertainty, and is therefore the average of probable locations. Resonance structures are stabilizing in molecules because they allow electrons to lengthen their wavelengths and thereby lower their energy. This is the reason that benzene (C6H6) has a lower heat of formation than organic chemists would predict, not accounting for resonance. Other aromatic molecules have a similar stability, which leads to an overall entropic preference for aromaticity (a subject that will be covered fully in a later chapter). Resonance stability plays a major role in organic chemistry due to resonant molecules' lower energy of formation, so students of organic chemistry should understand this effect and practice spotting molecules stabilized by resonant forms. <br> In the Lewis structures above, carbonate (CO32-) has a resonance structure. Using laboratory procedures to measure the bond length of each bond, we do not find that one bond is shorter than the two others (remember, double bonds are shorter than single bonds), but instead that all bonds are of the same length somewhere between the length of typical double and single bonds. Resonance Structures. Resonance structures are diagrammatic tools used predominately in organic chemistry to symbolize resonant bonds between atoms in molecules. The electron density of these bonds is spread over the molecule, also known as the delocalization of electrons. Resonance contributors for the same molecule all have the same chemical formula and same sigma framework, but the pi electrons will be distributed differently among the atoms. Because Lewis dot diagrams often cannot represent the true electronic structure of a molecule, resonance structures are often employed to approximate the true electronic structure. Resonance structures of the same molecule are connected with a double-headed arrow. While organic chemists use resonance structures frequently, they are also used in inorganic structures, with nitrate as an example. Key characteristics. The key elements of resonance are: What resonance is not. Significantly, resonance structures do not represent different, isolatable structures or compounds. In the case of benzene, for example, there are two important resonance structures - which can be thought of as cyclohexa-1,3,5-trienes. There are other resonance forms possible, but because they are higher in energy than the triene structures (due to charge separation or other effects) they are less important and contribute less to the "real" electronic structure (average hybrid). However, this does not mean there are two different, interconvertable forms of benzene; rather, the true electronic structure of benzene is an average of the two structures. The six carbon-carbon bond lengths are identical when measured, which would be invalid for the cyclic triene. Resonance should also not be confused with a chemical equilibrium or tautomerism which are equilibria between compounds that have different sigma bonding patterns. Hyperconjugation is a special case of resonance. History. The concept of resonance was introduced by Linus Pauling in 1928. He was inspired by the quantum mechanical treatment of the H2+ ion in which an electron is located between two hydrogen nuclei. The alternative term mesomerism popular in German and French publications with the same meaning was introduced by Christopher Ingold in 1938 but did not catch on in the English literature. The current concept of Mesomeric effect has taken on a related but different meaning. The double headed arrow was introduced by the German chemist Arndt (also responsible for the Arndt-Eistert synthesis) who preferred the German phrase "zwischenstufe" or "intermediate phase". Due to confusion with the physical meaning of the word "resonance", as no elements do actually appear to be resonating, it is suggested to abandon the term "resonance" in favor of "delocalization" . Resonance energy would become delocalization energy and a "resonance structure" becomes contributing structure. The double headed arrows would get replaced by commas. Examples. The ozone molecule is represented by two resonance structures in the top of "scheme 2". In reality the two terminal oxygen atoms are equivalent and the hybrid structure is drawn on the right with a charge of -1/2 on both oxygen atoms and partial double bonds. The concept of benzene as a hybrid of two conventional structures (middle "scheme 2") was a major breakthrough in chemistry made by Kekule, and the two forms of the ring which together represent the total resonance of the system are called "Kekule structures". In the hybrid structure on the right the circle replaces three double bonds. The allyl cation (bottom "scheme 2") has two resonance forms and in the hybrid structure the positive charge is delocalized over the terminal methylene groups.

Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases. Arrhenius Definition: Hydroxide and Hydronium Ions. The first and earliest definition of acids and bases was proposed in the 1800s by Swedish scientist Svante Arrhenius, who said that an acid was anything that dissolved in water to yield H+ ions (like stomach acid HCl, hydrochloric acid), and a base was anything that dissolved in water to give up OH- ions (like soda lye NaOH, sodium hydroxide). Acids and bases were already widely used in various occupations and activities of the time, so Arrhenius' definition merely attempted to explained well-known and long-observed phenomenon. Although simple, at the time this definition of the two types of substances was significant. It allowed chemists to explain certain reactions as ion chemistry, and it also expanded the ability of scientists of the time to predict certain chemical reactions. The definition left a great deal wanting, however, in that many types of reactions that did not involve hydroxide or hydronium ions directly remained unexplained. Many general chemistry classes (especially in the lower grades or introductory levels) still use this simple definition of acids and bases today, but modern organic chemists make further distinctions between acids and bases than the distinctions provided under Arrhenius's definition. Brønsted-Lowry Acids and Bases: Proton donors and acceptors. A new definition for acids and bases, building upon the one already proposed by Arrhenius, was brought forth independently by Johannes Nicolaus Brønsted and Thomas Martin Lowry in 1923. The new definition did not depend on a substance's dissolution in water for definition, but instead suggested that a substance was acidic if it readily donated a proton (H+) to a reaction and a substance was basic if it accepted a proton in a reaction. The major advantage of the updated definition was that it was not limited to aqueous solution. This definition of acids and bases allowed chemists to explain a great number of reactions that took place in protic or aprotic solvents that were not water, and it also allowed for gaseous and solid phase reactions (although those reactions are more rare). For example, the hypothetical acid HA will disassociate into H+ and A-: The Brønsted-Lowry definition of acids and bases is one of two definitions still in common use by modern chemists. Lewis Acids and Bases: Electron donors and acceptors. The second definition in widespread use deals not with a molecule's propensity for accepting or donating protons but rather with accepting or donating electrons, thereby demonstrating a slightly different emphasis and further broadening the explanatory and predictive powers of acid-base chemistry. Probably the most important aspect of Lewis acids and bases is which types of atoms can donate electrons, and which types of atoms can receive them. Essentially atoms with lone pairs, i.e. unshared pairs of electrons in an outer shell, have the capability of using those lone pairs to attract electron-deficient atoms or ions. This is why ammonia can bond a fourth hydrogen ion to create the ammonium ion; its lone pair of electrons can attract and bond to a free H+ ion in solution and hold on to it. For the same reason, methane cannot become methanium ion under ordinary circumstances, because the carbon in methane does not have any unshared pairs of electrons orbiting its nucleus. Generally speaking, Lewis acid are in the nitrogen, oxygen or halogen groups of the periodic table. Nucleophiles and Electrophiles. Whether or not an atom can donate or accept electrons it can be called a nucleophile or electrophile, respectively. Electrophiles (literally, "lovers of electrons") are attracted to electrons. Electrophiles therefore seek to pair with unshared electrons of other atoms. Nucleophiles, or "nucleus lovers", seek positively charged nuclei such as those available in acidic solutions as hydronium ions. It is important to note that electrophiles and nucleophiles are often ions, but sometimes they are not. Understanding electrophiles and nucleophiles goes beyond simply ideas of acids and bases. They are, in a majority of cases, the major players in organic reactions. As we will, over and over again, find reactions that are the result of nucleophiles "attacking" electrophiles. Keep in mind that the idea of nucleophiles and electrophiles is very related to the ideas of acids and bases in the Lewis context. But it is also important to understand that, while they are related, they are not exactly the same thing either. An ion or molecule can be a strong nucleophile and a weak base (e.g. N3-, RS-, I-, Br- and CN-). Another ion can be a poor nucleophile and a strong base ((CH3)3CO-, R2N-). And yet others are strong nucleophiles and strong bases (R3C-, RO-, HO-) and poor nucleophiles and poor bases (RCO2-, ROH, NH3). This will all be discussed in greater detail as the topics of specific reactions and reaction mechanisms are covered. In the meantime, try to bear in mind that nucleophiles are basic and electrophiles are acidic. pKa and Acidity. The acid dissociation constant of a substance is commonly called its pKa, and it is a measure of the negative log of the K value of an acid dissociation reaction. (The K value refers to the equilibrium calculations you learned how to perform in general chemistry—if you have forgotten your K's and Q's, now would be a good time to refresh your memory on the topic.) The lower the pKa value is, the more acidic (and consequently, less basic) a substance is. There is also a pKb value for all relevant substances, but it is common in organic chemistry to use pKa exclusively, even when discussing bases. This is because extremely high pKa values correlate exactly to extremely low pKb values, so there is no need to use both kinds of measurements. Any pKa value higher than seven means that a substance is not acidic when placed in water, but it does not mean that substance cannot be an acid. Alcohols are a good example of this: they can donate a hydrogen ion in chemical reactions but they do not do so readily, which makes them acidic but only very weakly so. Many of the acids in organic chemistry are considerably weaker than acids used for inorganic chemistry, so discussion of acid-base chemistry in organic reactions may not necessarily relate well to your previous understanding of the topic.

Organic Chemistry/Foundational concepts of organic chemistry/Electron dot structures. Electron Dot Structures. Electron dot structures, also called "Lewis structures", give a representation of the valence electrons surrounding an atom. Each valence electron is represented by one dot, thus, a lone atom of hydrogen would be drawn as an "H" with one dot, whereas a lone atom of Helium would be drawn as an "He" with two dots, and so forth. Representing two atoms joined by a covalent bond is done by drawing the atomic symbols near to each other, and drawing a single line to represent a shared pair of electrons. It is important to note: a single valence electron is represented by a dot, whereas a pair of electrons is represented by a line. The covalent compound hydrogen fluoride, for example, would be represented by the symbol "H" joined to the symbol "F" by a single line, with three pairs (six more dots) surrounding the symbol "F". The line represents the two electrons shared by both hydrogen and fluorine, whereas the six paired dots represent fluorine's remaining six valence electrons. Dot structures are useful in illustrating simple covalent molecules, but the limitations of dot structures become obvious when diagramming even relatively simple organic molecules. The dot structures have no ability to represent the actual physical orientation of molecules, and they become overly cumbersome when more than three or four atoms are represented. Lewis dot structures are useful for introducing the idea of covalence and bonding in small molecules, but other model types have much more capability to communicate chemistry concepts. Drawing electron dot structures. <br> Some examples of electron dot structures for a few commonly encountered molecules from inorganic chemistry. A note about Gilbert N. Lewis. Lewis was born in Weymouth, Massachusetts as the son of a Dartmouth-graduated lawyer/broker. He attended the University of Nebraska at age 14, then three years later transferred to Harvard. After showing an initial interest in Economics, Gilbert Newton Lewis earned first a B.A. in Chemistry, and then a Ph.D. in Chemistry in 1899. For a few years after obtaining his doctorate, Lewis worked and studied both in the United States and abroad (including Germany and the Philippines) and he was even a professor at M.I.T. from 1907 until 1911. He then went on to U.C. Berkeley in order to be Dean of the College of Chemistry in 1912. In 1916 Dr. Lewis formulated the idea that a covalent bond consisted of a shared pair of electrons. His ideas on chemical bonding were expanded upon by Irving Langmuir and became the inspiration for the studies on the nature of the chemical bond by Linus Pauling. In 1923, he formulated the electron-pair theory of acid-base reactions. In the so-called Lewis theory of acids and bases, a "Lewis acid" is an electron-pair acceptor and a "Lewis base" is an electron-pair donor. In 1926, he coined the term "photon" for the smallest unit of radiant energy. Lewis was also the first to produce a pure sample of deuterium oxide (heavy water) in 1933. By accelerating deuterons (deuterium nuclei) in Ernest O. Lawrence's cyclotron, he was able to study many of the properties of atomic nuclei. During his career he published on many other subjects, and he died at age 70 of a heart attack while working in his laboratory in Berkeley. He had one daughter and two sons; both of his sons became chemistry professors themselves. Formal Charge. The formal charge of an atom is the charge that it would have if every bond were 100% covalent (non-polar). Formal charges are computed by using a set of rules and are useful for accounting for the electrons when writing a reaction mechanism, but they don't have any intrinsic physical meaning. They may also be used for qualitative comparisons between different resonance structures (see below) of the same molecule, and often have the same sign as the partial charge of the atom, but there are exceptions. The formal charge of an atom is computed as the difference between the number of valence electrons that a neutral atom would have and the number of electrons that "belong" to it in the Lewis structure when one counts lone pair electrons as belonging fully to the atom, while electrons in covalent bonds are split equally between the atoms involved in the bond. The total of the formal charges on an ion should be equal to the charge on the ion, and the total of the formal charges on a neutral molecule should be equal to zero. For example, in the hydronium ion, H3O+, the oxygen atom has 5 electrons for the purpose of computing the formal charge—2 from one lone pair, and 3 from the covalent bonds with the hydrogen atoms. The other 3 electrons in the covalent bonds are counted as belonging to the hydrogen atoms (one each). A neutral oxygen atom has 6 valence electrons (due to its position in group 16 of the periodic table); therefore the formal charge on the oxygen atom is 6 – 5 = +1. A neutral hydrogen atom has one electron. Since each of the hydrogen atoms in the hydronium atom has one electron from a covalent bond, the formal charge on the hydrogen atoms is zero. The sum of the formal charges is +1, which matches the total charge of the ion. In chemistry, a formal charge (FC) on an atom in a molecule is defined as: When determining the correct Lewis structure (or predominant resonance structure) for a molecule, the structure is chosen such that the formal charge on each of the atoms is minimized.

Organic Chemistry/Introduction to reactions/Overview of addition, elimination, substitution and rearrangement reactions. The real heart of organic chemistry is the reactions. Everything that you study is geared to prepare you for organic syntheses and other chemical transformations performed in the lab. This chapter gives you the basic tools to begin looking at these reactions. Some basic reaction types. One way to organize organic reactions places them into a few basic categories: Other categories include: Sometimes one reaction can fall into more than one category. These classifications are just a tool and are not rigid. Addition reaction. Something is added to something else to produce a third thing. "Note: the letters A, B and C here represent any atomic, ionic or molecular species which can undergo this type of reaction." Elimination reaction. Something comes off of a molecule, resulting in two products. Substitution reactions. This involves the exchange of one group for another. Common reaction types include Rearrangement reactions. A molecule shifts or otherwise rearranges to form a different molecule. This typically happens when one molecule changes into an isomer of itself.

Organic Chemistry/Introduction to reactions/Polar and radical reactions. Homolytic vs heterolytic cleavage. Two bonded atoms can come apart from each other in one of two ways. Either In homolytic cleavage, each atom leaves with one-half of the shared electrons (one electron for a single bond, or two for double bonds). A—B → A* + B* A* and B* represent uncharged radicals. The "*" represents an unbonded, unpaired valence electron. In heterolytic cleavage, one atom leaves with all of the previously shared electrons and the other atom gets none of them. A—B → A− + B+ "Homo" (from the Greek for same) indicates that each atom leaves with the same number of electrons from the bond. "Hetero" (from the Greek for different) refers to the fact that one atom gets all of the bonding electrons, while the other gets none. Polar reactions. Polar reactions occur when two bonded atoms come apart, one taking more of the shared electrons than the other. They involve heterolytic cleavage. The result is two charged species—one cation and one anion. Radical reactions. Radical reactions don't deal with charged particles but with radicals. Radicals are uncharged atoms or molecules with an incomplete octet of valence electrons. When a molecule comes apart by homolytic cleavage the result is two radicals. Although uncharged, radicals are usually very reactive because the unfilled octet is unstable and the radical can lower its energy by forming a bond in a way that allows it to fill its valence shell while avoiding any electrostatic charge..

Organic Chemistry/Introduction to functional groups. « Introduction to reactions | Introduction to functional groups | Overview of Functional Groups » | Alkenes »

Organic Chemistry/Foundational concepts of organic chemistry/History of organic chemistry/Vital force theory. « Foundational concepts | Synthesis of urea » Long ago, people observed the differences between compounds that were derived from living things and those that were not. There seemed to be an impassable gap between the properties of the two groups. Someone proposed the vital force theory to explain the difference. The theory said that there was a something called a vital force that dwelled within the organic material that did not exist in the nonorganic materials. However, to echo the words of President Ronald Reagan, it was "only" a theory. Synthesis of urea »

Organic Chemistry/Foundational concepts of organic chemistry/History of organic chemistry/Synthesis of urea. « Vital force theory | Organic vs inorganic chemistry »

Organic Chemistry/Foundational concepts of organic chemistry/History of organic chemistry/Organic vs inorganic chemistry. « Synthesis of urea | Atomic structure »

Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Nucleus and electrons. « Atomic structure | Shells and orbitals » Atoms are made up of a nucleus and electrons that orbit the nucleus. An atom in its natural, uncharged state has the same number of electrons as protons. If it gains or loses electrons, the atom is then referred to as an ion. The nucleus. The nucleus is made up of protons, which each have a positive charge, and neutrons, which have no charge. Neutrons and protons have about the same mass, and together account for most of the mass of the atom. Each of these particles is made up of even smaller particles, though the existence of these particles do not come into play at the energies and time spans in which most chemical reactions occur. Electrons. The electrons are negatively charged and fly around the nucleus of an atom at something like light speed. We cannot determine exactly where electrons are at any point in time, rather, we can only guess at the probability of finding an electron at a point in space relative to a nucleus at any point in time. The image depicts the Bohr model of the atom, in which the electrons inhabit discrete "orbitals" around the nucleus much like planets orbit the sun. Current models of the atomic structure hold that electrons occupy fuzzy clouds around the nucleus of specific shapes, some spherical, some dumbbell shaped, some with even more complex shapes. Even though the simpler Bohr model of atomic structure has been superseded, we still refer to these electron clouds as "orbitals". The number of electrons and the nature of the orbitals they occupy greatly influence the reactivity of atoms in organic chemistry. « Atomic structure | Nucleus and electrons | Shells and orbitals »

Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Shells and orbitals. « Nucleus and electrons | Filling electron shells » Electron orbitals. Electrons orbit atoms in clouds of distinct shapes and sizes. The electron clouds are layered one inside the other into units called shells (think nested Russian dolls), with the electrons occupying the smallest, innermost shell having the lowest energy state and the electrons in the largest, outermost shell having the highest energy state. The higher the energy state, the more potential energy the electron has, just like a rock at the top of a hill has more potential energy than a rock at the bottom of a valley. These concepts will be important in understanding later concepts like optical activity of chiral compounds as well as many interesting things outside the realm of organic chemistry (like how lasers work). Wave nature of electrons. Electrons behave as particles but also as waves. (Work by Albert Einstein and others revealed that in fact, light and all matter behaves with this dual nature, and it is most clearly observed in the tiniest particles.) One of the results of this observation is that electrons are not just in simple orbit around the nucleus as we imagine the moon to circle the earth, but instead occupy space as if they were a wave on the surface of a sphere. If you jump a jumprope you could imagine that the wave in the rope is in its fundamental frequency. The high and low points fall right in the middle, and the places where the rope doesn't move much (the nodes) occur only at the two ends. If you shake the rope fast enough in a rythmic way, using more energy than you would just jumping rope, you might be able to make the rope vibrate with a wavelength shorter than the fundamental. You them might see that the rope has more than one place along its length where it vibrates from its highest spot to its lowest spot. Furthermore, you'll see that there is one or more places (or nodes) along its length where the rope seems to move very little, if at all. Or consider stringed musical instruments. The sound made by these instruments comes from the different ways, or modes the strings can vibrate. We can refer to these different patterns or modes of vibrations as linear harmonics. Going from there, we can recognize that a drum makes sound by vibrations that occur across the 2-dimensional surface of the drumhead. Extending this now into three dimensions, we think of the electron as vibrating across a 3-dimensional sphere, and the patterns or modes of vibration are referred to as spherical harmonics. The mathematical analysis of spherical harmonics were worked out by the French mathematician Legendre long before anyone started to think about the shapes of electron orbitals. The algebraic expressions he developed, known as Legendre polynomials, describe the three dimension shapes of electron orbitals in much the same way that the expression x2+y2 = z describes a circle (or, for that matter, a drumhead). Many organic chemists need never actually work with these equations, but it helps to understand where the pictures we use to think about the shapes of these orbitals come from. Electron shells. Each different shell is subdivided into one or more orbitals, which also have different energy levels, although the energy difference between orbitals is less than the energy difference between shells. Longer wavelengths have less energy; the s orbital has the longest wavelength allowed for an electron orbiting a nucleus and this orbital is observed to have the lowest energy. Each sub-shell in the main electron shell has a characteristic shape, and are named by a letter. The sub-shells are: s, p, d, and f. As one progresses up through the shells (represented by the principle quantum number n) more types of orbitals become possible. S orbital. The s orbital is the orbital lowest in energy and is spherical in shape. Electrons in this orbital are in their fundamental frequency. P orbital. The next lowest-energy orbital is the p orbital. Its shape is often described as like that of a dumbbell. There are three p-orbitals each oriented along one of the 3-dimensional coordinates x, y or z. These three different p orbitals can be referred to as the px, py, and pz. The s and p orbitals are important for understanding most of organic chemistry as these are the orbitals that are occupied by the type of atoms that are most common in organic compounds. D orbital. There are 5 types of d orbitals. Three of them are roughly X-shaped, as shown here, and might be viewed as being shaped like a crossed pair of dumbbells . They are referred to as dxy, dxz</sub u>, and dyz. Like the p-orbitals, these three d orbitals have a node at the origin of the coordinate system where the three axes all come together. Unlike the p orbitals, however, these three d orbitals are not oriented along the x, y, or z axes, but instead are oriented in between them. The dxy orbital, for instance, lies in the xy plane, but the lobes of the orbital point out in between the x and y axes. F orbital and beyond. There are 7 kinds of F orbitals, but we will not discuss their shapes here. F orbitals are filled in the elements of the lanthanide and actinide series, although electrons in these orbitals rarely come into play in organometallic reactions involving these elements.

Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Filling electron shells. « Shells and orbitals | Octet rule and exceptions » When an atom or ion receives electrons into its orbitals, the orbitals and shells fill up in a particular manner. There are three principles that govern this process: 1) the Aufbau (build-up) principle, 2) the Pauli exclusion principle, and 3) Hund's rule. Exclusion principle. No more than one electron can have all four quantum numbers the same. What this translates to in terms of our pictures of orbitals is that each orbital can only hold two electrons, one "spin up" and one "spin down". Build-up principle. You may consider an atom as being "built up" from a naked nucleus by gradually adding to it one electron after another, until all the electrons it will hold have been added. Much as one fills up a container with liquid from the bottom up, so also are the orbitals of an atom filled from the lowest energy orbitals to the highest energy orbitals. However, the three p orbitals of a given shell all occur at the same energy level. So, how are they filled up? Is one of them filled full with the two electrons it can hold first? Or do each of the three orbitals receive one electron apiece before the any single orbital is double occupied. As it turns out, the latter situation occurs. Hund's rule. This rule is applicable only for those elements that have d electrons, and so is less important in organic chemistry (though it is important in organometallic chemistry). It says that filled and half-filled shells tend to have additional stability. In some instances, then, for example, the 4s orbitals will be filled before the 3d orbitals. Building atoms with quantum mechanics (advanced topic). The equations (like the Legendre polynomials that describe spherical harmonics, and thus the shapes of orbitals) of quantum mechanics are distinguished by four types of numbers. The first of these quantum numbers is referred to as the principal quantum number, and is indicated by n. This merely represents which shell electrons occupy, and shows up in the periodic chart as the rows of the periodic chart. It has integral values, n=1,2,3 . . . . The highest energy electrons of the atoms in the first row all have n=1. Those in the atoms in the second row all have n=2. As one gets into n=3, Hund's rule mixes it up a little bit, but when one gets to the end of the third row, at least, the electrons with the highest energy have n=3. The next quantum number is indicated by the letter m and indicates how many different types of shells an atom can have. Those elements in the first row can have just one, the s orbital. The elements in the second row can have two, the s and the p orbitals. The elements in the third row can have three, the s, p, and d orbitals. And so on. It may be funny to think of s, p and d as "numbers", but these are used as an historical and geometrical convenience. The third kind of quantum number ml specifies, for those kinds of orbitals that can have different shapes, which of the possible shapes one is referring to. So, for example, a 2pz orbital indicates three quantum numbers, represented respectively by the 2, the p and the z. Finally, the fourth quantum number is the spin of the electron. It has only two possible values, +1/2 or -1/2. Pretty much only computational chemists have to treat quantum numbers as numbers per se is equations. But it helps to know that the wide variety of elements of the periodic table and the different shapes and other properties of electron orbitals have a unifying principle--the proliferation of different shapes is not completely arbitrary, but is instead bounded by very specific rules. Afbau Principle (it means 'building up'):- It states that the orbitals should be filled according to their increasing energies Thus the lowest energy orbital which is available is filled first. The increasing order of energies of the various orbitals is:- 1s,2s,2p,3s,3p,4s,3d,4p,5s,4d,5p,6s,4f,5d,6p,7s,5f... The order of increasing of energy of orbitals can be calc. from(n+l) rule or 'Bohr bury rule' According to this rule, the value of n+l is the energy of the orbital and such on orbital will be filled up first. e.g. 4s orbital having lower value of(n+l) has lower energy than 3d orbital and hence 4s orbital is filled up first. For 4s orbital, n+l=4+0=4 For 3d orbital, n+l=3+2=5,therefore 4s orbital will be filled first.

Linear Algebra. This book discusses proof-based linear algebra. The book was designed specifically for students who have not previously been exposed to mathematics as mathematicians view it: that is, as a subject whose goal is to "rigorously" prove theorems starting from clear consistent definitions. This book attempts to build students up from a background where mathematics is simply a tool that provides useful calculations to the point where the students have a grasp of the clear and precise nature of mathematics. A more detailed discussion of the prerequisites and goal of this book is given in the introduction. Because of the proof-based nature of this book, readers are recommended to be familiar with mathematical proof before reading this book (although this is not a prerequisite, strictly speaking), so that their reading experiences can be smoother. To gain familiarity with mathematical proof and also some basic mathematical concepts, readers may read the wikibook Mathematical Proof. For a milder introduction to linear algebra that is not too proof-based, see the wikibook Introductory Linear Algebra. Table of Contents. Appendix. The following is a brief overview of some basic concepts in mathematics. For more details, reader can read the wikibook Mathematical Proof.

Calculus/Introduction. What is calculus? Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behavior of a function at points ever closer to a particular point, but without ever actually reaching that point. As a typical application of the methods of calculus, consider a moving car. It is possible to create a function describing the displacement of the car (where it is located in relation to a reference point) at any point in time as well as a function describing the velocity (speed and direction of movement) of the car at any point in time. If the car were traveling at a constant velocity, then algebra would be sufficient to determine the position of the car at any time; if the velocity is unknown but still constant, the position of the car could be used (along with the time) to find the velocity. However, the velocity of a car cannot jump from zero to 35 miles per hour at the beginning of a trip, stay constant throughout, and then jump back to zero at the end. As the accelerator is pressed down, the velocity rises gradually, and usually not at a constant "rate" (i.e., the driver may push on the gas pedal harder at the beginning, in order to speed up). Describing such motion and finding velocities and distances at particular times cannot be done using methods taught in pre-calculus, whereas it is not only possible but straightforward with calculus. Calculus has two basic applications: "differential calculus" and "integral calculus". The simplest introduction to differential calculus involves an explicit series of numbers. Given the series (42, 43, 3, 18, 34), the differential of this series would be (1, -40, 15, 16). The new series is derived from the difference of successive numbers which gives rise to its name "differential". Rarely, if ever, are differentials used on an explicit series of numbers as done here. Instead, they are derived from a continuous function in a manner which is described later. Integral calculus, like differential calculus, can also be introduced via series of numbers. Notice that in the previous example, the original series can almost be derived solely from its differential. Instead of taking the difference, however, integration involves taking the sum. Given the first number of the original series, 42 in this case, the rest of the original series can be derived by adding each successive number in its differential (42+1, 43-40, 3+15, 18+16). Note that knowledge of the first number in the original series is crucial in deriving the integral. As with differentials, integration is performed on continuous functions rather than explicit series of numbers, but the concept is still the same. Integral calculus allows us to calculate the area under a curve of almost any shape; in the car example, this enables you to find the displacement of the car based on the velocity curve. This is because the area under the curve is the total distance moved, as we will soon see. Why learn calculus? Calculus is essential for many areas of science and engineering. Both make heavy use of mathematical functions to describe and predict physical phenomena that are subject to continuous change, and this requires the use of calculus. Take our car example: if you want to design cars, you need to know how to calculate forces, velocities, accelerations, and positions. All require calculus. Calculus is also necessary to study the motion of gases and particles, the interaction of forces, and the transfer of energy. It is also useful in business whenever rates are involved. For example, equations involving interest or supply and demand curves are grounded in the language of calculus. Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and non-euclidean geometry. Notwithstanding calculus' "functional" utility (pun intended), many non-scientists and non-engineers have chosen to study calculus just for the challenge of doing so. A smaller number of persons undertake such a challenge and then discover that calculus is beautiful in and of itself. What is involved in learning calculus? Learning calculus, like much of mathematics, involves two parts: What you should know before using this text. There are some basic skills that you need before you can use this text. Continuing with our example of a moving car: Scope. The first four chapters of this textbook cover the topics taught in a typical high school or first year college course. The first chapter, ../Precalculus/, reviews those aspects of functions most essential to the mastery of calculus. The second, ../Limits/, introduces the concept of the limit process. It also discusses some applications of limits and proposes using limits to examine slope and area of functions. The next two chapters, ../Differentiation/ and ../Integration/, apply limits to calculate derivatives and integrals. The Fundamental Theorem of Calculus is used, as are the essential formulas for computation of derivatives and integrals without resorting to the limit process. The third and fourth chapters include articles that apply the concepts previously learned to calculating volumes, and as other important formulas. The remainder of the central calculus chapters cover topics taught in higher-level calculus topics: parametric and polar equations, sequences and series, multivariable calculus, and differential equations. The final chapters cover the same material, using formal notation. They introduce the material at a much faster pace, and cover many more theorems than the other two sections. They assume knowledge of some set theory and set notation.

Calculus/Functions. Functions are everywhere, from a simple correlation between distance and time to complex heat waves. This chapter focuses on the fundamentals of functions: the definition, basic concepts, and other defining aspects. It is very concept-heavy, and expect a lot of reading and understanding. However, this is simply a review and an introduction on what is to come in future chapters. Introduction. Whenever one quantity uniquely determines the value of another quantity, we have a function. That is, the set formula_1 uniquely determines the set formula_2. You can think of a "function" as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product. Think about dropping a ball from a bridge. At each moment in time, the ball is a height above the ground. The height of the ball is a function of time. It was the job of physicists to come up with a formula for this function. This type of function is called "real-valued" since the "finished product" is a number (or, more specifically, a real number). Think about a wind storm. At different places, the wind can be blowing in different directions with different intensities. The direction and intensity of the wind can be thought of as a function of position. This is a function of two real variables (a location is described by two values - an formula_3 and a formula_4) which results in a vector (which is something that can be used to hold a direction and an intensity). These functions are studied in multivariable calculus (which is usually studied after a one year college level calculus course). This a vector-valued function of two real variables. We will be looking at real-valued functions until studying multivariable calculus. Think of a real-valued function as an "input-output machine"; you give the function an input, and it gives you an output which is a number (more specifically, a real number). For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input -1 and gives the output value 1. Notation. Functions are used so much that there is a special notation for them. The notation is somewhat ambiguous, so familiarity with it is important in order to understand the intention of an equation or formula. Though there are no strict rules for naming a function, it is standard practice to use the letters formula_5 , formula_6 , and formula_7 to denote functions, and the variable formula_3 to denote an independent variable. formula_4 is used for both dependent and independent variables. When discussing or working with a function formula_5 , it's important to know not only the function, but also its independent variable formula_3 . Thus, when referring to a function formula_5, you usually do not write formula_5, but instead formula_14 . The function is now referred to as "formula_5 of formula_3". The name of the function is adjacent to the independent variable (in parentheses). This is useful for indicating the value of the function at a particular value of the independent variable. For instance, if and if we want to use the value of formula_5 for formula_3 equal to formula_20 , then we would substitute 2 for formula_3 on both sides of the definition above and write This notation is more informative than leaving off the independent variable and writing simply 'formula_5' , but can be ambiguous since the parentheses next to formula_5 can be misinterpreted as multiplication, formula_25. To make sure nobody is too confused, follow this procedure: Description. There are many ways which people describe functions. In the examples above, a verbal description is given (the height of the ball above the earth as a function of time). Here is a list of ways to describe functions. The top three listed approaches to describing functions are the most popular. When a function is given a name (like in number 1 above), the name of the function is usually a single letter of the alphabet (such as formula_5 or formula_6). Some functions whose names are multiple letters (like the sine function formula_34) If we write formula_30 , then we know that How would we know the value of the function formula_5 at 3? We would have the following three thoughts: and we would write formula_46. The value of formula_5 at 3 is 11. Note that formula_48 means the value of the dependent variable when formula_3 takes on the value of 3. So we see that the number "11" is the output of the function when we give the number "3" as the input. People often summarize the work above by writing "the value of formula_5 at three is eleven", or simply "formula_5 of three equals eleven". Basic concepts of functions. The formal definition of a function states that a function is actually a "mapping" that associates the elements of one set called the domain of the function, formula_52, with the elements of another set called the "range" of the function, formula_53. For each value we select from the domain of the function, there exists "exactly one" corresponding element in the range of the function. The definition of the function tells us which element in the range corresponds to the element we picked from the domain. An example of how is given below. In mathematics, it is important to notice any repetition. If something seems to repeat, keep a note of that in the back of your mind somewhere. Here, notice that formula_54 and formula_55. Because formula_14 is equal to two different things, it must be the case that the other side of the equal sign to formula_14 is equal to the other. This property is known as the transitive property and could thus make the following equation below true:<br> Next, simplify — make your life easier rather than harder. In this instance, since formula_59 has formula_3 as a like-term, and the two terms are fractions added to the other, put it over a common denominator and simplify. Similar, since formula_61 is a mixed fraction, formula_62.<br> Multiply both sides by the reciprocal of the coefficient of formula_3, formula_68 from both sides by multiplying by it.<br> The value of formula_3 that makes formula_55 is formula_71.formula_75. Classically, the element picked from the domain is pictured as something that is fed into the function and the corresponding element in the range is pictured as the output. Since we "pick" the element in the domain whose corresponding element in the range we want to find, we have control over what element we pick and hence this element is also known as the "independent variable". The element mapped in the range is beyond our control and is "mapped to" by the function. This element is hence also known as the "dependent variable", for it depends on which independent variable we pick. Since the elementary idea of functions is better understood from the classical viewpoint, we shall use it hereafter. However, it is still important to remember the correct definition of functions at all times. Basic types of transformation. To make it simple, for the function formula_14, all of the possible formula_3 values constitute the domain, and all of the values formula_14 (formula_4 on the x-y plane) constitute the range. To put it in more formal terms, a function formula_5 is a mapping of some element formula_81, called the domain, to exactly one element formula_82, called the range, such that formula_83. The image below should help explain the modern definition of a function: The modern definition describes a function sufficiently such that it helps us determine whether some new type of "function" is indeed one. Now that each case is defined above, you can now prove whether functions are one of these function cases. Here is an example problem: Notice that the only changing element in the function formula_5 is formula_3. To prove a function is one-to-one by applying the definition may be impossible because although two random elements of domain set formula_52 may not be many-to-one, there may be some elements formula_99 that may make the function many-to-one. To account for this possibility, we must prove that it is impossible to have some result like that. Assume there exists two distinct elements formula_115 that will result in formula_116. This would make the function many-to-one. In consequence,<br> formula_117 Subtract formula_89 from both sides of the equation. Subtract formula_120 from both sides of the equation. Factor formula_91 from both terms to the left of the equation. Multiply formula_124 to both sides of the equation. Add formula_126 to both sides of the equation. Notice that formula_128. However, this is impossible because formula_129 and formula_126 are distinct. Ergo, formula_131. No two distinct inputs can give the same output. Therefore, the function must be one-to-one. Basic concepts. There are a few more important ideas to learn from the modern definition of the function, and it comes from knowing the difference between domain, range, and codomain. We already discussed some of these, yet knowing about sets adds a new definition for each of the following ideas. Therefore, let us discuss these based on these new ideas. Let formula_52 and formula_53 be a set. If we were to combine these two sets, it would be defined as the "cartesian cross product" formula_134. The subset of this product is the function. The below definitions are a little confusing. The best way to interpret these is to draw an image. To the right of these definitions is the image that corresponds to it. Note that the codomain is not as important as the other two concepts. Take formula_135 for example:Because of the square root, the content in the square root should be strictly not smaller than 0.formula_136formula_137Thus the domain isformula_138 Correspondingly, the range will beformula_139 Other types of transformation. There is one more final aspect that needs to be defined. We already have a good idea of what makes a mapping a function (e.g. it cannot be one-to-many). However, three more definitions that you will often hear will be necessary to talk about: "injective", "surjective", "bijective". Again, the above definitions are often very confusing. Again, another image is provided to the right of the bullet points. Along with that, another example is also provided. Let us analyze the function formula_142. Notice that the only changing element in the function formula_6 is formula_3. Let us check to see the conditions of this function. Is it injective? Let formula_145, and solve for formula_3. First, divide by formula_91. Then take the square root of formula_150. By definition, formula_151, so Notice, however, what we learned from the above manipulation. As a result of solving for formula_3, we found that there are two solutions for formula_3. However, this resulted in two different values from formula_156. This means that for some individual formula_3 that gives formula_4, there are two different inputs that result in the same value. Because formula_159 when formula_160, this is by definition non-injective. Is it surjective? As a natural consequence of what we learned about inputs, let us determine the range of the function. After all, the only way to determine if something is surjective is to see if the range applies to all real numbers. A good way to determine this is by finding a pattern involving our domains. Let us say we input a negative number for the domain: formula_161. This seemingly trivial exercise tells us that negative numbers give us non-negative numbers for our range. This is huge information! For formula_162, the function always results formula_163 for our range. For the set formula_53, we have elements in that set that have no mappings from the set formula_52. As such, formula_166 is the codomain of set formula_53. Therefore, this function is non-surjective! Tests for equations. The vertical line test. The vertical line test is a systematic test to find out if an equation involving formula_3 and formula_4 can serve as a function (with formula_3 the independent variable and formula_4 the dependent variable). Simply graph the equation and draw a vertical line through each point of the formula_3-axis. If any vertical line ever touches the graph at more than one point, then the equation is not a function; if the line always touches at most one point of the graph, then the equation is a function. The circle, on the right, is not a function because the vertical line intercepts two points on the graph when formula_173. The horizontal line and the algebraic 1-1 test. Similarly, the horizontal line test, though does not test if an equation is a function, tests if a function is injective (one-to-one). If any horizontal line ever touches the graph at more than one point, then the function is not one-to-one; if the line always touches at most one point on the graph, then the function is one-to-one. The algebraic 1-1 test is the non-geometric way to see if a function is one-to-one. The basic concept is that: Assume there is a function formula_5. If:formula_175, and formula_176, thenfunction formula_5 is one-to-one. Here is an example: prove that formula_178 is injective. Since the notation is the notation for a function, the equation is a function. So we only need to prove that it is injective. Let formula_91 and formula_89 be the inputs of the function and that formula_175. Thus, So, the result is formula_176, proving that the function is injective. Another example is proving that formula_190 is not injective. Using the same method, one can find that formula_191, which is not formula_176. So, the function is not injective. Remarks. The following arise as a direct consequence of the definition of functions: Functions are an important foundation of mathematics. This modern interpretation is a result of the hard work of the mathematicians of history. It was not until recently that the definition of the relation was introduced by René Descartes in "Geometry" (1637). Nearly a century later, the standard notation (formula_197) was first introduced by Leonhard Euler in "Introductio in Analysin Infinitorum" and "Institutiones Calculi Differentialis". The term function was also a new innovation during Euler's time as well, being introduced Gottfried Wilhelm Leibniz in a 1673 letter "to describe a quantity related to points of a curve, such as a coordinate or curve's slope." Finally, the modern definition of the function being the relation among sets was first introduced in 1908 by Godfrey Harold Hardy where there is a relation between two variables formula_3 and formula_4 such that "to some values of formula_3 at any rate correspond values of formula_4." For the person that wants to learn more about the history of the function, go to History of functions. Important functions. The functions listed below are essential to calculus. This section only serves as a review and scratches the surface of those functions. If there are any questions about those functions, please review the materials related to those functions before continuing. More about graphing will be explained in Chapter Polynomials. Polynomial functions are the most common and most convenient functions in the world of calculus. Their frequent appearances and their applications on computer calculations have made them important. Constant. When formula_202, the polynomial can be rewritten into the following function:formula_203, where formula_204 is a constant.The graph of this function is a horizontal line passing the point formula_205. Linear. When formula_206, the polynomial can be rewritten intoformula_207, where formula_208 are constants.The graph of this function is a straight line passing the point formula_209 and formula_210, and the slope of this function is formula_211. Quadratic. When formula_212, the polynomial can be rewritten intoformula_213, where formula_214 are constants.The graph of this function is a parabola, like the trajectory of a basketball thrown into the hoop. If there are questions about the quadratic formula and other basic polynomial concepts, please review the respective chapters in Algebra. Trigonometric. Trigonometric functions are also very important because it can connect algebra and geometry. Trigonometric functions are explained in detail here due to its importance and difficulty. Exponential and Logarithmic. Exponential and logarithmic functions have a unique identity when calculating the derivatives, so this is a great time to review those functions. A special number will be frequently seen in those functions: the Euler's constant, also known as the base of the natural logarithm. Notated as formula_215, it is defined as formula_216. Signum. The Signum (sign) function is simply defined asformula_217 Properties of functions. Sometimes, a lot of function manipulations cannot be achieved without some help from basic properties of functions. Domain and range. This concept is discussed above. Growth. Although it seems obvious to spot a function increasing or decreasing, without the help of graphing software, we need a mathematical way to spot whether the function is increasing or decreasing, or else our human minds cannot immediately comprehend the huge amount of information. Assume a function formula_14 with inputs formula_219, and that formula_220, formula_221, and formula_222 at all times.If for all formula_223 and formula_224, formula_225, then formula_14 is increasing in formula_227 If for all formula_223 and formula_224, formula_230, then formula_14 is decreasing in formula_227Example: In which intervals is formula_233 increasing? Firstly, the domain is important. Because the denominator cannot be 0, the domain for this function is formula_234 In formula_235, the growth of the function is:Let formula_236 and formula_222 Thus,formula_238formula_239 both formula_236 formula_241 formula_242 formula_239 formula_222 and formula_245 formula_241 formula_247So, formula_248formula_14 is decreasing in formula_235In formula_251Let formula_252 and formula_222 Thus,formula_238formula_239 both formula_236 formula_241formula_242 However, the sign of formula_259 in formula_251 cannot be determined. It can only be determined in formula_261.In formula_262 formula_239 formula_222 and formula_245 formula_241 formula_247 In formula_268 formula_269 formula_270As a result, formula_14 is decreasing in formula_262 and increasing in formula_268.In formula_274Let formula_275 and formula_222 Thus,formula_238formula_239 both formula_236 formula_241formula_242 formula_269 formula_270So, formula_284formula_14 is increasing in formula_274.Therefore, the intervals in which the function is increasing are formula_287. formula_75 After learning derivatives, there will be more ways to determine the growth of a function. Parity. The properties odd and even are associated with symmetry. While even functions have a symmetry about the formula_4-axis, odd functions are symmetric about the origin. In mathematical terms:A function is even when formula_290 A function is odd when formula_291Example: Prove that formula_233 is an even function. formula_293 formula_294 is an even function formula_295 Manipulating functions. Addition, Subtraction, Multiplication and Division of functions. For two real-valued functions, we can add the functions, multiply the functions, raised to a power, etc. If we add the functions formula_31 and formula_297 , we obtain formula_298 . If we subtract formula_31 from formula_297 , we obtain formula_301 . We can also write this as formula_302 . If we multiply the function formula_31 and the function formula_297 , we obtain formula_305 . We can also write this as formula_306 . If we divide the function formula_31 by the function formula_297 , we obtain formula_309 . If a math problem wants you to add two functions formula_5 and formula_6 , there are two ways that the problem will likely be worded: Similar statements can be made for subtraction, multiplication and division. Let formula_30 and: formula_190 . Let's add, subtract, multiply and divide. Composition of functions. We begin with a fun (and not too complicated) application of composition of functions before we talk about what composition of functions is. If we drop a ball from a bridge which is 20 meters above the ground, then the height of our ball above the earth is a function of time. The physicists tell us that if we measure time in seconds and distance in meters, then the formula for height in terms of time is formula_330 . Suppose we are tracking the ball with a camera and always want the ball to be in the center of our picture. Suppose we have formula_331 The angle will depend upon the height of the ball above the ground and the height above the ground depends upon time. So the angle will depend upon time. This can be written as formula_332 . We replace formula_7 with what it is equal to. This is the essence of composition. Composition of functions is another way to combine functions which is different from addition, subtraction, multiplication or division. The value of a function formula_5 depends upon the value of another variable formula_3 ; however, that variable could be equal to another function formula_6 , so its value depends on the value of a third variable. If this is the case, then the first variable is a function formula_7 of the third variable; this function (formula_7) is called the composition of the other two functions (formula_5 and formula_6). Let formula_30 and: formula_190 . The composition of formula_5 with formula_6 is read as either "f composed with g" or "f of g of x." Let Then Sometimes a math problem asks you compute formula_347 when they want you to compute formula_348 , Here, formula_7 is the composition of formula_5 and formula_6 and we write formula_352 . Note that composition is not commutative: Composition of functions is very common, mainly because functions themselves are common. For instance, squaring and sine are both functions: Thus, the expression formula_358 is a composition of functions: Since the function sine equals formula_360 if formula_361 , Since the function square equals formula_363 if formula_361 , Transformations. Transformations are a type of function manipulation that are very common. They consist of multiplying, dividing, adding or subtracting constants to either the input or the output. Multiplying by a constant is called dilation and adding a constant is called translation. Here are a few examples: Translations and dilations can be either horizontal or vertical. Examples of both vertical and horizontal translations can be seen at right. The red graphs represent functions in their 'original' state, the solid blue graphs have been translated (shifted) horizontally, and the dashed graphs have been translated vertically. Dilations are demonstrated in a similar fashion. The function has had its input doubled. One way to think about this is that now any change in the input will be doubled. If I add one to formula_3, I add two to the input of formula_5, so it will now change twice as quickly. Thus, this is a horizontal dilation by formula_373 because the distance to the formula_4-axis has been halved. A vertical dilation, such as is slightly more straightforward. In this case, you double the output of the function. The output represents the distance from the formula_3-axis, so in effect, you have made the graph of the function 'taller'. Here are a few basic examples where formula_91 is any positive constant: Inverse functions. We call formula_378 the inverse function of formula_14 if, for all formula_3 : A function formula_14 has an inverse function if and only if formula_14 is one-to-one. For example, the inverse of formula_384 is formula_385 . The function formula_135 has no inverse because it is not injective. Similarly, the inverse functions of trigonometric functions have to undergo transformations and limitations to be considered as valid functions. Notation. The inverse function of formula_5 is denoted as formula_388 . Thus, formula_388 is defined as the function that follows this rule To determine formula_388 when given a function formula_5 , substitute formula_388 for formula_3 and substitute formula_3 for formula_14 . Then solve for formula_388 , provided that it is also a function. Example: Given formula_398 , find formula_388 . Substitute formula_388 for formula_3 and substitute formula_3 for formula_14 . Then solve for formula_388 : To check your work, confirm that formula_409 :formula_410formula_411formula_412If formula_5 isn't one-to-one, then, as we said before, it doesn't have an inverse. Then this method will fail. Example: Given formula_414 , find formula_388. Substitute formula_388 for formula_3 and substitute formula_3 for formula_14 . Then solve for formula_388 : Since there are two possibilities for formula_388 , it's not a function. Thus formula_414 doesn't have an inverse. Of course, we could also have found this out from the graph by applying the Horizontal Line Test. It's useful, though, to have lots of ways to solve a problem, since in a specific case some of them might be very difficult while others might be easy. For example, we might only know an algebraic expression for formula_14 but not a graph. =External links=

Waves/Vectors. Math Tutorial -- Vectors. Figure 1: Displacement vectors in a plane. Vector formula_1 represents the displacement of George from Mary, while vector formula_2 represents the displacement of Paul from George. Vector formula_3 represents the displacement of Paul from Mary and formula_4. The quantities formula_5, formula_6, etc., represent the Cartesian components of the vectors. Before we can proceed further we need to explore the idea of a "vector". A vector is a quantity which expresses both magnitude and direction. Graphically we represent a vector as an arrow. In typeset notation a vector is represented by a boldface character, while in handwriting an arrow is drawn over the character representing the vector. Figure 1 shows some examples of "displacement vectors", i. e., vectors which represent the displacement of one object from another, and introduces the idea of vector addition. The tail of vector formula_2 is collocated with the head of vector formula_1, and the vector which stretches from the tail of formula_1 to the head of formula_2 is the sum of formula_1 and formula_2, called formula_3 in figure 1. Figure 2: Definition sketch for the angle formula_14 representing the orientation of a two dimensional vector. The quantities formula_5, formula_6, etc., represent the Cartesian components of the vectors in figure 2. A vector can be represented either by its Cartesian components, which are just the projections of the vector onto the Cartesian coordinate axes, or by its direction and magnitude. The direction of a vector in two dimensions is generally represented by the counterclockwise angle of the vector relative to the formula_17 axis, as shown in figure 2. Conversion from one form to the other is given by the equations where formula_20 is the magnitude of the vector. A vector magnitude is sometimes represented by absolute value notation: formula_21. Notice that the inverse tangent gives a result which is ambiguous relative to adding or subtracting integer multiples of formula_22. Thus the quadrant in which the angle lies must be resolved by independently examining the signs of formula_5 and formula_6 and choosing the appropriate value of formula_14. To add two vectors, formula_1 and formula_2, it is easiest to convert them to Cartesian component form. The components of the sum formula_4 are then just the sums of the components: Subtraction of vectors is done similarly, e. g., if formula_30, then A unit vector is a vector of unit length. One can always construct a unit vector from an ordinary vector by dividing the vector by its length: formula_32. This division operation is carried out by dividing each of the vector components by the number in the denominator. Alternatively, if the vector is expressed in terms of length and direction, the magnitude of the vector is divided by the denominator and the direction is unchanged. Unit vectors can be used to define a Cartesian coordinate system. Conventionally, formula_33, formula_34, and formula_35 indicate the formula_17, formula_37, and formula_38 axes of such a system. Note that formula_33, formula_34, and formula_35 are mutually perpendicular. Any vector can be represented in terms of unit vectors and its Cartesian components: formula_42. An alternate way to represent a vector is as list of components: formula_43. We tend to use the latter representation since it is somewhat more economical notation. There are two ways to multiply two vectors, yielding respectively what are known as the dot product and the cross product. The cross product yields another vector while the dot product yields a number. Here we will discuss only the dot product. Figure 3: Definition sketch for dot product. Given vectors formula_1 and formula_2, the dot product of the two is defined where formula_14 is the angle between the two vectors. An alternate expression for the dot product exists in terms of the Cartesian components of the vectors: It is easy to show that this is equivalent to the cosine form of the dot product when the formula_17 axis lies along one of the vectors, as in figure 3. Notice in particular that formula_50, while formula_51 and formula_52. Thus, formula_53 in this case, which is identical to the form given above. By the law of cosines we can also see that which is an alternate coordinate-free expression for the dot product. Figure 4: Definition figure for rotated coordinate system. The vector formula_55 has components formula_56 and formula_57 in the unprimed coordinate system and components formula_58 and formula_59 in the primed coordinate system. All that remains to be proven for equation (2.6) to hold in general is to show that it yields the same answer regardless of how the Cartesian coordinate system is oriented relative to the vectors. To do this, we must show that formula_60, where the primes indicate components in a coordinate system rotated from the original coordinate system. This can be shown nearly instantly by applying the pythagorean theorem. Due to the fact that R is invariant and represents the hypotenuse for both triangles (X, X', Y and Y') we can conclude: formula_61 Since the dot product can be written solely in terms of magnitudes, as we did above, if the magnitude of a vector is invariant the dot product of two vectors must also be invariant. To deduce a general formula for X' and Y' you will have to do a bit more thinking: Figure 2.4 shows the vector formula_55 resolved in two coordinate systems rotated with respect to each other. From this figure it is clear that formula_63. Focusing on the shaded triangles, we see that formula_64 and formula_65. Thus, we find formula_66. Similar reasoning shows that formula_67 (Just imagine to rotate the constructs in the image further 90° without changing the axis-names. You will instantly notice that in the second quadrant X is negative while Y positive). Thus, the new and old coordinates are related by This is true of the position vector. We can use it to extend the notion of vector to concepts other than position by stating that a pair of numbers is a vector "if and only if" its values change in exactly this way under rotation. Substituting this relation into our earlier expression for the dot product and using the trigonometric identity formula_69 results in which proves the complete equivalence of the two forms of the dot product quoted above. (Multiply out the above expression to verify this.) A numerical quantity which doesn't depend on which coordinate system is being used is called a scalar. The dot product of two vectors is a scalar. However, the components of a vector, taken individually, are not scalars, since the components change as the coordinate system changes. Since the laws of physics cannot depend on the choice of coordinate system being used, we insist that physical laws be expressed in terms of scalars and vectors, but not in terms of the components of vectors. In three dimensions the cosine form of the dot product remains the same, while the component form is

Waves/Reflection and Refraction. Reflection and Refraction. Most of what we need to know about geometrical optics can be summarized in two rules, the laws of reflection and refraction. These rules may both be inferred by considering what happens when a plane wave segment impinges on a flat surface. If the surface is polished metal, the wave is "reflected", whereas if the surface is an interface between two transparent media with differing indices of refraction, the wave is partially reflected and partially "refracted". Reflection means that the wave is turned back into the half-space from which it came, while refraction means that it passes through the interface, acquiring a different direction of motion from that which it had before reaching the interface. <br> Figure 3.1 shows the wave vector and wave front of a wave being reflected from a plane mirror. The angles of incidence, formula_1, and reflection, formula_2, are defined to be the angles between the incoming and outgoing wave vectors respectively and the line normal to the mirror. The law of reflection states that formula_3. This is a consequence of the need for the incoming and outgoing wave fronts to be in phase with each other all along the mirror surface. This plus the equality of the incoming and outgoing wavelengths is sufficient to insure the above result. <br> Refraction, as illustrated in figure 3.2, is slightly more complicated. Since formula_4, the speed of light in the right-hand medium is less than in the left-hand medium. (Recall that the speed of light in a medium with refractive index formula_5 is formula_6.) The frequency of the wave packet doesn't change as it passes through the interface, so the wavelength of the light on the right side is less than the wavelength on the left side. Let us examine the triangle ABC in figure 3.2. The side AC is equal to the side BC times formula_7. However, AC is also equal to formula_8, or twice the wavelength of the wave to the left of the interface. Similar reasoning shows that formula_9, twice the wavelength to the right of the interface, equals BC times formula_10. Since the interval BC is common to both triangles, we easily see that Since formula_12 and formula_13 where formula_14 and formula_15 are the wave speeds to the left and right of the interface, formula_16 is the speed of light in a vacuum, and formula_17 is the (common) period, we can easily recast the above equation in the form This is called "Snell's law", and it governs how a ray of light bends as it passes through a discontinuity in the index of refraction. The angle formula_1 is called the incident angle and formula_2 is called the refracted angle. Notice that these angles are measured from the normal to the surface, not the tangent. Derivation for Law of Reflection. The derivation of Law of Reflection using Fermat's principle is straightforward. The Law of Reflection can be derived using elementary Calculus and Trigonometry. The generalization of the Law of Reflection is Snell's law, which is derived bellow using the same principle. The medium that light travels through doesn't change. In order to minimize the time for light travel between to points, we should minimize the path taken. 1. Total path length of the light is given by 2. Using Pythagorean theorem from Euclidean Geometry we see that 3. When we substitute both values of d1 and d2 for above, we get 4. In order to minimize the path traveled by light, we take the first derivative of L with respect to x. 5. Set both sides equal to each other. 6. We can now tell that the left side is nothing but formula_27 and the right side formula_28 means 7. Taking the inverse sine of both sides we see that the angle of incidence equals the angle of reflection Derivation for Snell's Law. The derivation of Snell's Law using Fermat's Priciple is straightforward. Snell's Law can be derived using elementary calculus and trigonometry. Snell's Law is the generalization of the above in that it does not require the medium to be the same everywhere. To mark the speed of light in different media refractive indices named n1 and n2 are used. Here formula_33 is the speed of light in the vacuum and formula_34 because all materials slow down light as it travels through them. 1. Time for the trip equals distance traveled divided by the speed. 2. Using the Pythagorean theorem from Euclidean Geometry we see that 3. Substituting this result into equation (1) we get 4. Differentiating and setting the derivative equal to zero gives 5. After careful examination the above equation we see that it is nothing but 6. Thus 7. Multiplying both sides by formula_42 we get 8. Substituting formula_44 for v1 and formula_45 for formula_46 we get 9. Simplifying both sides we get our final result

Special Relativity/Spacetime. The modern approach to relativity. Although the special theory of relativity was first proposed by Einstein in 1905, the modern approach to the theory depends upon the concept of a four-dimensional universe, that was first proposed by Hermann Minkowski in 1908. Minkowski's contribution appears complicated but is simply an extension of Pythagoras' Theorem: In 2 dimensions: formula_1 In 3 dimensions: formula_2 in 4 dimensions: formula_3 The modern approach uses the concept of invariance to explore the types of coordinate systems that are required to provide a full physical description of the location and extent of things. The modern theory of special relativity begins with the concept of "length". In everyday experience, it seems that the length of objects remains the same no matter how they are rotated or moved from place to place. We think that the simple length of a thing is "invariant". However, as is shown in the illustrations below, what we are actually suggesting is that length seems to be invariant in a three-dimensional coordinate system. The length of a thing in a two-dimensional coordinate system is given by Pythagoras's theorem: This two-dimensional length is not invariant if the thing is tilted out of the two-dimensional plane. In everyday life, a three-dimensional coordinate system seems to describe the length fully. The length is given by the three-dimensional version of Pythagoras's theorem: The derivation of this formula is shown in the illustration below. It seems that, provided all the directions in which a thing can be tilted or arranged are represented within a coordinate system, then the coordinate system can fully represent the length of a thing. However, it is clear that things may also be changed over a period of time. Time is another direction in which things can be arranged. This is shown in the following diagram: The length of a straight line between two events in space and time is called a "space-time interval". In 1908 Hermann Minkowski pointed out that if things could be rearranged in time, then the universe might be four-dimensional. He boldly suggested that Einstein's recently-discovered theory of Special Relativity was a consequence of this four-dimensional universe. He proposed that the space-time interval might be related to space and time by Pythagoras' theorem in four dimensions: Where "i" is the imaginary unit (sometimes imprecisely called formula_8), "c" is a constant, and "t" is the time interval spanned by the space-time interval, "s". The symbols "x", "y" and "z" represent displacements in space along the corresponding axes. In this equation, the 'second' becomes just another unit of length. In the same way as centimetres and inches are both units of length related by centimetres = 'conversion constant' times inches, metres and seconds are related by metres = 'conversion constant' times seconds. The conversion constant, "c" has a value of about 300,000,000 meters per second. Now formula_9 is equal to minus one, so the space-time interval is given by: Minkowski's use of the imaginary unit has been superseded by the use of advanced geometry that uses a tool known as the "metric tensor". The metric tensor permits the existence of "real" time and the negative sign in the expression for the square of the space-time interval originates in the way that distance changes with time when the curvature of spacetime is analysed (see advanced text). We now use real time but Minkowski's original equation for the square of the interval survives so that the space-time interval is still given by: Space-time intervals are difficult to imagine; they extend between one place and time and another place and time, so the velocity of the thing that travels along the interval is already determined for a given observer. If the universe is four-dimensional, then the space-time interval (rather than the spatial length) will be invariant. Whoever measures a particular space-time interval will get the same value, no matter how fast they are travelling. In physical terminology the invariance of the spacetime interval is a type of Lorentz Invariance. The invariance of the spacetime interval has some dramatic consequences. The first consequence is the prediction that if a thing is travelling at a velocity of "c" metres per second, then all observers, no matter how fast they are travelling, will measure the same velocity for the thing. The velocity "c" will be a universal constant. This is explained below. When an object is travelling at "c", the space time interval is zero, this is shown below: A space-time interval of zero only occurs when the velocity is "c" (if x>0). All observers observe the same space-time interval so when observers observe something with a space-time interval of zero, they all observe it to have a velocity of "c", no matter how fast they are moving themselves. The universal constant, "c", is known for historical reasons as the "speed of light in a vacuum". In the first decade or two after the formulation of Minkowski's approach many physicists, although supporting Special Relativity, expected that light might not travel at exactly "c", but might travel at very nearly "c". There are now few physicists who believe that light in a vacuum does not propagate at "c". The second consequence of the invariance of the space-time interval is that clocks will appear to go slower on objects that are moving relative to you. Suppose there are two people, Bill and John, on separate planets that are moving away from each other. John draws a graph of Bill's motion through space and time. This is shown in the illustration below: Being on planets, both Bill and John think they are stationary, and just moving through time. John spots that Bill is moving through what John calls space, as well as time, when Bill thinks he is moving through time alone. Bill would also draw the same conclusion about John's motion. To John, it is as if Bill's time axis is leaning over in the direction of travel and to Bill, it is as if John's time axis leans over. The space-time interval, formula_19, is invariant. It has the same value for all observers, no matter who measures it or how they are moving in a straight line. Bill's formula_20 equals John's formula_20 so: So, if John sees Bill measure a time interval of 1 second (formula_25) between two ticks of a clock that is at rest in Bill's frame, John will find that his own clock measures between these same ticks an interval formula_26, called coordinate time, which is greater than one second. It is said that clocks in motion slow down, relative to those on observers at rest. This is known as "relativistic time dilation of a moving clock". The time that is measured in the rest frame of the clock (in Bill's frame) is called the proper time of the clock. John will also observe measuring rods at rest on Bill's planet to be shorter than his own measuring rods, in the direction of motion. This is a prediction known as "relativistic length contraction of a moving rod". If the length of a rod at rest on Bill's planet is formula_27, then we call this quantity the proper length of the rod. The length formula_28 of that same rod as measured from John's planet, is called coordinate length, and given by This equation can be derived directly and validly from the time dilation result with the assumption that the speed of light is constant. The last consequence is that clocks will appear to be out of phase with each other along the length of a moving object. This means that if one observer sets up a line of clocks that are all synchronised so they all read the same time, then another observer who is moving along the line at high speed will see the clocks all reading different times. In other words observers who are moving relative to each other see different events as simultaneous. This effect is known as Relativistic Phase or the Relativity of Simultaneity. Relativistic phase is often overlooked by students of Special Relativity, but if it is understood then phenomena such as the twin paradox are easier to understand. The way that clocks go out of phase along the line of travel can be calculated from the concepts of the invariance of the space-time interval and length contraction. In the diagram above John is conventionally stationary. Distances between two points according to Bill are simple lengths in space (x) all at t=0 whereas John sees Bill's measurement of distance as a combination of a distance (X) and a time interval (T): Notice that the quantities represented by capital letters are proper lengths and times and in this example refer to John's measurements. Bill's distance, x, is the length that he would obtain for things that John believes to be X metres in length. For Bill it is John who has rods that contract in the direction of motion so Bill's determination "x" of John's distance "X" is given from: This relationship between proper and coordinate lengths was seen above to relate Bill's proper lengths to John's measurements. It also applies to how Bill observes John's proper lengths. Clocks that are synchronised for one observer go out of phase along the line of travel for another observer moving at formula_37 metres per second by :formula_38 seconds for every metre. This is one of the most important results of Special Relativity and should be thoroughly understood by students. The net effect of the four-dimensional universe is that observers who are in motion relative to you seem to have time coordinates that lean over in the direction of motion and consider things to be simultaneous that are not simultaneous for you. Spatial lengths in the direction of travel are shortened, because they tip upwards and downwards, relative to the time axis in the direction of travel, akin to a rotation out of three-dimensional space. Interpreting space-time diagrams. Great care is needed when interpreting space-time diagrams. Diagrams present data in two dimensions, and cannot show faithfully how, for instance, a zero length space-time interval appears. When diagrams are used to show both space and time it is important to be alert to space and time being related by Minkowski's equation and not by simple Euclidean geometry. The diagrams are only aids to understanding the approximate relation between space and time and it must not be assumed, for instance, that simple trigonometric relationships can be used to relate lines that represent spatial displacements and lines that represent temporal displacements. It is sometimes mistakenly held that the time dilation and length contraction results only apply for observers at x=0 and t=0. This is untrue. An inertial frame of reference is defined so that length and time comparisons can be made anywhere within a given reference frame. Time differences in one inertial reference frame can be compared with time differences anywhere in another inertial reference frame provided it is remembered that these differences apply to corresponding pairs of lines or pairs of planes of simultaneous events. Spacetime. In order to gain an understanding of both Galilean and Special Relativity it is important to begin thinking of space and time as being different dimensions of a four-dimensional vector space called spacetime. Actually, since we can't visualize four dimensions very well, it is easiest to start with only one space dimension and the time dimension. The figure shows a graph with time plotted on the vertical axis and the one space dimension plotted on the horizontal axis. An "event" is something that occurs at a particular time and a particular point in space. ("Julius X. wrecks his car in Lemitar, NM on 21 June at 6:17 PM.") A "world line" is a plot of the position of some object as a function of time (more properly, the time of the object as a function of position) on a spacetime diagram. Thus, a world line is really a line in spacetime, while an event is a point in spacetime. A horizontal line parallel to the position axis (x-axis) is a "line of simultaneity"; in Galilean Relativity all events on this line occur simultaneously for all observers. It will be seen that the line of simultaneity differs between Galilean and Special Relativity; in Special Relativity the line of simultaneity depends on the state of motion of the observer. In a spacetime diagram the slope of a world line has a special meaning. Notice that a vertical world line means that the object it represents does not move -- the velocity is zero. If the object moves to the right, then the world line tilts to the right, and the faster it moves, the more the world line tilts. Quantitatively, we say that Notice that this works for negative slopes and velocities as well as positive ones. If the object changes its velocity with time, then the world line is curved, and the instantaneous velocity at any time is the inverse of the slope of the tangent to the world line at that time. The hardest thing to realize about spacetime diagrams is that they represent the past, present, and future all in one diagram. Thus, spacetime diagrams don't change with time -- the evolution of physical systems is represented by looking at successive horizontal slices in the diagram at successive times. Spacetime diagrams represent the evolution of events, but they don't evolve themselves. The lightcone. Things that move at the speed of light in our four dimensional universe have surprising properties. If something travels at the speed of light along the x-axis and covers x meters from the origin in t seconds the space-time interval of its path is zero. formula_42 but formula_43 so: formula_44 Extending this result to the general case, if something travels at the speed of light in any direction into or out from the origin it has a space-time interval of 0: formula_45 This equation is known as the Minkowski Light Cone Equation. If light were travelling towards the origin then the Light Cone Equation would describe the position and time of emission of all those photons that could be at the origin at a particular instant. If light were travelling away from the origin the equation would describe the position of the photons emitted at a particular instant at any future time 't'. At the superficial level the light cone is easy to interpret. Its backward surface represents the path of light rays that strike a point observer at an instant and its forward surface represents the possible paths of rays emitted from the point observer. Things that travel along the surface of the light cone are said to be light- like and the path taken by such things is known as a null geodesic. Events that lie outside the cones are said to be space-like or, better still space separated because their space time interval from the observer has the same sign as space (positive according to the convention used here). Events that lie within the cones are said to be time-like or time separated because their space-time interval has the same sign as time. However, there is more to the light cone than the propagation of light. If the added assumption is made that the speed of light is the maximum possible velocity then events that are space separated cannot affect the observer directly. Events within the backward cone can have affected the observer so the backward cone is known as the "affective past" and the observer can affect events in the forward cone hence the forward cone is known as the "affective future". The assumption that the speed of light is the maximum velocity for all communications is neither inherent in nor required by four dimensional geometry although the speed of light is indeed the maximum velocity for objects if the principle of causality is to be preserved by physical theories (ie: that causes precede effects). The Lorentz transformation equations. The discussion so far has involved the comparison of interval measurements (time intervals and space intervals) between two observers. The observers might also want to compare more general sorts of measurement such as the time and position of a single event that is recorded by both of them. The equations that describe how each observer describes the other's recordings in this circumstance are known as the Lorentz Transformation Equations. (Note that the symbols below signify coordinates.) The table below shows the Lorentz Transformation Equations. See mathematical derivation of Lorentz transformation. Notice how the phase ( (v/c2)x ) is important and how these formulae for absolute time and position of a joint event differ from the formulae for intervals. A spacetime representation of the Lorentz Transformation. Bill and John are moving at a relative velocity, v, and synchronise clocks when they pass each other. Both Bill and John observe an event along Bill's direction of motion. What times will Bill and John assign to the event? It was shown above that the relativistic phase was given by: formula_46. This means that Bill will observe an extra amount of time elapsing on John's time axis due to the position of the event. Taking phase into account and using the time dilation equation Bill is going to observe that the amount of time his own clocks measure can be compared with John's clocks using: formula_47. This relationship between the times of a common event between reference frames is known as the Lorentz Transformation Equation for time. Continue

Waves/Transverse, Longitudinal and Torsional waves. Transverse and Longitudinal Waves. With the exception of light, waves are undulations in some material medium. For instance, waves on a slinky are either "transverse", in that the motion of the material of the slinky is perpendicular to the orientation of the slinky, if you vibrate the slinky like a rope, or they are "longitudinal", with material motion in the direction of the stretched slinky, if you treat it like a spring. (See image on right) Ocean waves are simultaneously transverse and longitudinal, the net effect being (nearly) circular undulations in the position of water parcels. The oscillations in neighboring parcels are phased such that a "pattern" moves across the ocean surface. Some media support only longitudinal waves, others support only transverse waves, while yet others support both types. Sound waves are purely longitudinal in gases and liquids, but can be either type in solids. Mechanical transverse waves require a material medium and propogate by means of vibrations of the medium perpendicular to the direction of travel. Examples are water waves, ripples, seismic shear waves, and waves in stretched strings as above. Electromagnetic (EM) waves (such as light) are also transverse waves but they do not require a medium and thus can pass through a vacuum (see intro). They consist of oscillating electric (E) and magnetic (B) fields which are perpendicular to the direction of propagation while also being mutually perpendicular. EM waves are a disturbance of space itself, which can be thought of as being stretched and therefore being elastic and having a tension. The B and E fields are in phase as shown on the left. The fundamental S.I. unit of length (meter) is defined in terms of the speed of light in vacuum and the definition of the unit of time, the second. The previous definition was in terms of the wavelength of a particular color of light in the line spectrum of Krypton 86. The modern definition is more accurate as well as being the same to all observers regardless of their relative velocity. Longitudinal waves propogate by means of vibrations or disturbances in the medium that are in the same direction that the wave travels. Examples are sound (as above), seismic shock waves, slinky springs and part of the motion in ocean waves. Sound waves are a series of high-pressure compressions and low-pressure rarefaction (for this reason, sound is sometimes called a "pressure wave"). While it might be convenient to think of individual molecules vibrating back and forth around an equilibrium position (producing areas of high and low pressure), the individual molecules in a gas generally move randomly, and it is only in large numbers that this pattern is visible. In longitudinal waves, the convention is to describe displacement in the direction the wave is going (the direction of propagation) as positive, and displacement against that direction as negative. The pressure and displacement of a molecule at a point are formula_1 out of phase, so that, for example, when a molecule is farthest displaced it is in a normal pressure area, while molecules in compressions or rarefactions have displacement close to zero. Torsional waves consist of a twisting disturbance moving through a medium such as a wire or a rod.

Waves/Sine Waves. Sine Waves. A particularly simple kind of wave, the sine wave, is illustrated in figure 1.2. This has the mathematical form: where Figure 1.2: Definition sketch for a sine wave, showing the wavelength λ and the amplitude formula_2 and the phase φ at various points. So far we have only considered a sine wave as it appears at a particular time. All interesting waves move with time. The movement of a sine wave to the right, a distance formula_4 may be accounted for by replacing formula_5 in the above formula by formula_6. If this movement occurs in time formula_7, then the wave moves at velocity formula_8. Solving this for formula_4 and substituting yields a formula for the displacement of a sine wave as a function of both distance formula_5 and time formula_7: The time for a wave to move one wavelength is called the period of the wave: formula_13. Thus, we can also write Physicists actually like to write the equation for a sine wave in a slightly simpler form. Defining the wavenumber as formula_15 and the angular frequency as formula_16, we write We normally think of the frequency of oscillatory motion as the number of cycles completed per second and is given by formula_18. It is related to the angular frequency omega by formula_19. The angular frequency is used because it is directly analogous to the wavenumber, see above. Converting between the two is not difficult. Frequency is measured in units of hertz, abbreviated Hz; formula_20 and angular frequency formula_21 is in units of radians per second. The argument of the sine function is by definition an angle. We refer to this angle as the phase of the wave, formula_22. The difference in the phase of a wave at fixed time over a distance of one wavelength is formula_23, as is the difference in phase at fixed position over a time interval of one wave period. As previously noted, we call formula_2, the maximum displacement of the wave, the amplitude. Often we are interested in the intensity of a wave, which is defined as the square of the amplitude, formula_25. The wave speed we have defined above, formula_26, is actually called the phase speed. Since formula_27 and formula_28, we can write the phase speed in terms of the angular frequency and the wavenumber:

Waves/Types of Waves. Types of Waves. In order to make the above material more concrete, we now examine the characteristics of various types of waves which may be observed in the real world. Ocean Surface Waves. <br> Figure 1.3: Wave on an ocean of depth formula_1. The wave is moving to the right and the particles of water at the surface move up and down as shown by the small vertical arrows. The particles move up, and gravity is the 'restoring' force. Water waves are also longitudinal, the water particles moving forward, then back. The restoring forces are more complex, but involve the inertia of the mass of water surrounding. Think of water waves as superimposed transverse and longitudinal waves. The net particle paths are nearly circular. This is typical of waves that travel along the boundary (interface) between two substances, in this case water and air. These waves are manifested as undulations of the ocean surface as seen in figure 1.3. The speed of ocean waves is given by the formula where formula_3 is a constant related to the strength of the Earth's gravity, formula_1 is the depth of the ocean, and the hyperbolic tangent is defined as 2.7 <br> Figure 1.4: Plot of the function formula_6. The dashed line shows our approximation formula_7 for formula_8. As figure 1.4 shows, for very small x, we can approximate the hyperbolic tangent by formula_7, while for very large x it is positive 1 for positive x and negative 1 for negative x. This leads to two limits: Since formula_10, the "shallow water" limit, which occurs when formula_11, yields a wave speed of while the "deep water" limit, which occurs when formula_13, yields Notice that the speed of shallow water waves depends only on the depth of the water and on formula_15. In other words, all shallow water waves move at the same speed. On the other hand, deep water waves of longer wavelength (and hence smaller wavenumber) move more rapidly than those with shorter wavelength. Waves for which the wave speed varies with wavelength are called "dispersive". Thus, deep water waves are dispersive, while shallow water waves are non-dispersive. For water waves with wavelengths of a few centimeters or less, surface tension becomes important to the dynamics of the waves. In the deep water case the wave speed at short wavelengths is actually given by the formula where the constant formula_17 is related to surface tension and depends on the surfaces involved. For an air-water interface near room temperature, formula_18.

Waves/Sound Waves. Sound Waves. Sound is a longitudinal compression-expansion wave through matter. The wave speed is where formula_2 and formula_3 are constants and formula_4 is the "absolute temperature". The absolute temperature is measured in Kelvins and is numerically given by where formula_6 is the temperature in Celsius degrees. The angular frequency of sound waves is thus given by The speed of sound in air at normal temperatures is about formula_8.

Waves/Light. Light. Light moves in a vacuum at a speed of formula_1. In transparent materials it moves at a speed less than formula_2 by a factor formula_3 which is called the "refractive index" of the material: Often the refractive index takes the form where formula_6 is the wavenumber and formula_7 and formula_8 are constants characteristic of the material. The angular frequency of light in a transparent medium is thus so the frequency increases slightly with increasing "k". Typically, when "k" is near "k""R", the material becomes opaque. Ultimately, this is due to resonance between the light and the atoms of the materials.

Waves/Superposition. Superposition Principle. It is found empirically that as long as the amplitudes of waves in most media are small, two waves in the same physical location don't interact with each other. Thus, for example, two waves moving in the opposite direction simply pass through each other without their shapes or amplitudes being changed. When superimposed, the total wave displacement is just the sum of the displacements of the individual waves. This is called the "superposition principle". At sufficiently large amplitude the superposition principle often breaks down -- interacting waves may scatter off of each other, lose amplitude, or change their form. "Interference" is a consequence of the superposition principle. When two or more waves are superimposed, the net wave displacement is just the algebraic sum of the displacements of the individual waves. Since these displacements can be positive or negative, the net displacement can either be greater or less than the individual wave displacements. The former case is called "constructive interference", while the latter is called "destructive interference". <br> Figure 1.5: Superposition (lower panel) of two sine waves (shown individually in the upper panel) with equal amplitudes and wavenumbers formula_1 and formula_2 Let us see what happens when we superimpose two sine waves with different wavenumbers. Figure 1.5 shows the superposition of two waves with wavenumbers formula_1 and formula_2. Notice that the result is a wave with about the same wavelength as the two initial waves, but which varies in amplitude depending on whether the two sine waves are in or out of phase. When the waves are in phase, constructive interference is occurring, while destructive interference occurs where the waves are out of phase. <br> Figure 1.6: Superposition of two sine waves with equal amplitudes and wavenumbers formula_5 and formula_6 What happens when the wavenumbers of the two sine waves are changed? Figure 1.6 shows the result when formula_5 and formula_6. Notice that though the wavelength of the resultant wave is decreased, the locations where the amplitude is maximum have the same separation in formula_9 as in figure 1.5. <br> Figure 1.7: Superposition of two sine waves with equal amplitudes and wavenumbers formula_5 and formula_11. If we superimpose waves with formula_5 and formula_11, as is shown in figure 1.7, we see that the formula_9 spacing of the regions of maximum amplitude has decreased by a factor of two. Thus, while the wavenumber of the resultant wave seems to be related to something like the "average" of the wavenumbers of the component waves, the spacing between regions of maximum wave amplitude appears to go inversely with the "difference" of the wavenumbers of the component waves. In other words, if formula_15 and formula_16 are close together, the amplitude maxima are far apart and vice versa. <br> Figure 1.8: Representation of the wavenumbers and amplitudes of two superimposed sine waves. We can symbolically represent the sine waves that make up figures 1.5, 1.6, and 1.7 by a plot such as that shown in figure 1.8. The amplitudes and wavenumbers of each of the sine waves are indicated by vertical lines in this figure. The regions of large wave amplitude are called wave packets. Wave packets will play a central role in what is to follow, so it is important that we acquire a good understanding of them. The wave packets produced by only two sine waves are not well separated along the formula_9-axis. However, if we superimpose many waves, we can produce an isolated wave packet. For example, figure 1.9 shows the results of superimposing formula_18 sine waves with wavenumbers formula_19, formula_20, where the amplitudes of the waves are largest for wavenumbers near formula_21. <br> Figure 1.9: Superposition of twenty sine waves with formula_22 and formula_23. In particular, we assume that the amplitude of each sine wave is proportional to formula_24, where formula_22 and formula_23. The amplitudes of each of the sine waves making up the wave packet in figure 1.9 are shown schematically in figure 1.10. <br> Figure 1.10: Representation of the wavenumbers and amplitudes of 20 superimposed sine waves with formula_22 and formula_23. The quantity formula_29 controls the distribution of the sine waves being superimposed -- only those waves with a wavenumber formula_30 within approximately formula_29 of the central wavenumber formula_32 of the wave packet, i. e., for formula_33 in this case, contribute significantly to the sum. If formula_29 is changed to formula_35, so that wavenumbers in the range formula_36 contribute significantly, the wavepacket becomes narrower, as is shown in figures 1.11 and 1.12. <br> Figure 1.11: Superposition of twenty sine waves with formula_22 and formula_38. <br> Figure 1.12: Representation of the wavenumbers and amplitudes of 20 superimposed sine waves with formula_22 and formula_38. formula_29 is called the wavenumber spread of the wave packet, and it evidently plays a role similar to the difference in wavenumbers in the superposition of two sine waves -- the larger the wavenumber spread, the smaller the physical size of the wave packet. Furthermore, the wavenumber of the oscillations within the wave packet is given approximately by the central wavenumber. We can better understand how wave packets work by mathematically analyzing the simple case of the superposition of two sine waves. Let us define formula_42 where formula_15 and formula_16 are the wavenumbers of the component waves. Furthermore let us set formula_45. The quantities formula_32 and formula_29 are graphically illustrated in figure 1.8. We can write formula_48 and formula_49 and use the trigonometric identity formula_50 to find formula_51 formula_52 formula_53 (2.17) The sine factor on the bottom line of the above equation produces the oscillations within the wave packet, and as speculated earlier, this oscillation has a wavenumber formula_32 equal to the average of the wavenumbers of the component waves. The cosine factor modulates this wave with a spacing between regions of maximum amplitude of Thus, as we observed in the earlier examples, the length of the wave packet formula_56 is inversely related to the spread of the wavenumbers formula_29 (which in this case is just the difference between the two wavenumbers) of the component waves. This relationship is central to the uncertainty principle of quantum mechanics.

Waves/Beats. Beats. Suppose two sound waves of slightly different frequencies impinge on your ear at the same time. The displacement perceived by your ear is the superposition of these two waves, with time dependence formula_1 (2.19) where we now have formula_2 and formula_3. What you actually hear is a tone with angular frequency formula_4 which fades in and out with period formula_5 (2.20) The "beat frequency" is simply formula_6 (2.21) Note how beats are the time analog of wave packets -- the mathematics are the same except that frequency replaces wavenumber and time replaces space.

Waves/Interferometers. Interferometers. An interferometer is a device which splits a beam of light into two sub-beams, shifts the phase of one sub-beam with respect to the other, and then superimposes the sub-beams so that they interfere constructively or destructively, depending on the magnitude of the phase shift between them. In this section we study the Michelson interferometer and interferometric effects in thin films. The Michelson Interferometer. <br> Figure 1.13: Sketch of a Michelson interferometer. The American physicist Albert Michelson invented the optical interferometer illustrated in figure 1.13. The incoming beam is split into two beams by the half-silvered mirror. Each sub-beam reflects off of another mirror which returns it to the half-silvered mirror, where the two sub-beams recombine as shown. One of the reflecting mirrors is movable by a sensitive micrometer device, allowing the path length of the corresponding sub-beam, and hence the phase relationship between the two sub-beams, to be altered. As figure 1.13 shows, the difference in path length between the two sub-beams is formula_1 because the horizontal sub-beam traverses the path twice. Thus, constructive interference occurs when this path difference is an integral number of wavelengths, i. e., where formula_3 is the wavelength of the light and formula_4 is an integer. Note that formula_4 is the number of wavelengths that fits evenly into the distance formula_1.

Waves/Thin Films. Thin Films. <br> Figure 1.14: Plane light wave normally incident on a transparent thin film of thickness formula_1 and index of refraction formula_2. Partial reflection occurs at the front surface of the film, resulting in beam A, and at the rear surface, resulting in beam B. Much of the wave passes completely through the film, as with C. One of the most revealing examples of interference occurs when light interacts with a thin film of transparent material such as a soap bubble. Figure 1.14 shows how a plane wave normally incident on the film is partially reflected by the front and rear surfaces. The waves reflected off the front and rear surfaces of the film interfere with each other. The interference can be either constructive or destructive depending on the phase difference between the two reflected waves. If the wavelength of the incoming wave is formula_3, one would naively expect constructive interference to occur between the A and B beams if formula_4 were an integral multiple of formula_3. Two factors complicate this picture. First, the wavelength inside the film is not formula_3, but formula_7, where formula_8 is the index of refraction of the film. Constructive interference would then occur if formula_9. Second, it turns out that an additional phase shift of half a wavelength occurs upon reflection when the wave is incident on material with a higher index of refraction than the medium in which the incident beam is immersed. This phase shift doesn't occur when light is reflected from a region with lower index of refraction than felt by the incident beam. Thus beam B doesn't acquire any additional phase shift upon reflection. As a consequence, constructive interference actually occurs when while destructive interference results when When we look at a soap bubble, we see bands of colors reflected back from a light source. What is the origin of these bands? Light from ordinary sources is generally a mixture of wavelengths ranging from roughly formula_12 (violet light) to formula_13 (red light). In between violet and red we also have blue, green, and yellow light, in that order. Because of the different wavelengths associated with different colors, it is clear that for a mixed light source we will have some colors interfering constructively while others interfere destructively. Those undergoing constructive interference will be visible in reflection, while those undergoing destructive interference will not. Another factor enters as well. If the light is not normally incident on the film, the difference in the distances traveled between beams reflected off of the front and rear faces of the film will not be just twice the thickness of the film. To understand this case quantitatively, we need the concept of refraction, which will be developed later in the context of geometrical optics. However, it should be clear that different wavelengths will undergo constructive interference for different angles of incidence of the incoming light. Different portions of the thin film will in general be viewed at different angles, and will therefore exhibit different colors under reflection, resulting in the colorful patterns normally seen in soap bubbles.

Waves/Derivatives. Math Tutorial -- Derivatives. <br> Figure 1.15: Estimation of the derivative, which is the slope of the tangent line. When point B approaches point A, the slope of the line AB approaches the slope of the tangent to the curve at point A. This section provides a quick introduction to the idea of the derivative. For a more detailed discussion and exploration of the differentiation and of Calculus, see Calculus and Differentiation. Often we are interested in the slope of a line tangent to a function formula_1 at some value of formula_2. This slope is called the derivative and is denoted formula_3. Since a tangent line to the function can be defined at any point formula_2, the derivative itself is a function of formula_2: As figure 1.15 illustrates, the slope of the tangent line at some point on the function may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve: As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact. Table of Derivatives. Derivatives of some common functions are now given. In each case formula_8 is a constant. The product and chain rules are used to compute the derivatives of complex functions. For instance, and

Waves/Group Velocity. Group Velocity. We now ask the following question: How fast do wave packets move? Surprisingly, we often find that wave packets move at a speed very different from the phase speed, formula_1, of the wave composing the wave packet. We shall find that the speed of motion of wave packets, referred to as the group velocity, is given by Dispersive waves are waves in which the phase speed varies with wavenumber. It is easy to show that dispersive waves have unequal phase and group velocities, while these velocities are equal for non-dispersive waves. Derivation of Group Velocity Formula. We now derive equation (1.36). It is easiest to do this for the simplest wave packets, namely those constructed out of the superposition of just two sine waves. We will proceed by adding two waves with full space and time dependence: After algebraic and trigonometric manipulations familiar from earlier sections, we find where as before we have formula_11, formula_12, formula_13, and formula_14. Again think of this as a sine wave of frequency formula_15 and wavenumber formula_16 modulated by a cosine function. In this case the modulation pattern moves with a speed so as to keep the argument of the cosine function constant: Differentiating this with respect to formula_18 while holding formula_19 and formula_20 constant yields In the limit in which the deltas become very small, this reduces to the derivative which is the desired result.

Waves/1D Examples. Examples. We now illustrate some examples of phase speed and group velocity by showing the displacement resulting from the superposition of two sine waves, as given by equation (1.38), in the formula_1-formula_2 plane. This is an example of a spacetime diagram, of which we will see many examples later on. <br> Figure 1.16: Net displacement of the sum of two traveling sine waves plotted in the formula_3 plane. The short vertical lines indicate where the displacement is large and positive, while the short horizontal lines indicate where it is large and negative. One wave has formula_4 and formula_5, while the other has formula_6 and formula_7. Thus, formula_8 and formula_9 and we have formula_10. Notice that the phase speed for the first sine wave is formula_11 and for the second wave is formula_12. Thus, formula_13 in this case. Figure 1.16 shows a non-dispersive case in which the phase speed equals the group velocity. The regions with vertical and horizontal hatching (short vertical or horizontal lines) indicate where the wave displacement is large and positive or large and negative. Large displacements indicate the location of wave packets. The positions of waves and wave packets at any given time may therefore be determined by drawing a horizontal line across the graph at the desired time and examining the variations in wave displacement along this line. The crests of the waves are indicated by regions of short vertical lines. Notice that as time increases, the crests move to the right. This corresponds to the motion of the waves within the wave packets. Note also that the wave packets, i. e., the broad regions of large positive and negative amplitudes, move to the right with increasing time as well. Since velocity is distance moved formula_14 divided by elapsed time formula_15, the slope of a line in figure 1.16, formula_16, is one over the velocity of whatever that line represents. The slopes of lines representing crests (the slanted lines, not the short horizontal and vertical lines) are the same as the slopes of lines representing wave packets in this case, which indicates that the two move at the same velocity. Since the speed of movement of wave crests is the phase speed and the speed of movement of wave packets is the group velocity, the two velocities are equal and the non-dispersive nature of this case is confirmed. <br> Figure 1.17: Net displacement of the sum of two traveling sine waves plotted in the formula_1-formula_2 plane. One wave has formula_19 and formula_5, while the other has formula_21 and formula_22. In this case formula_23 while formula_24, so the group velocity is formula_25. However, the phase speeds for the two waves are formula_26 and formula_27. The average of the two phase speeds is about formula_28, so the group velocity is about twice the average phase speed in this case. <br> Figure 1.18: Net displacement of the sum of two traveling sine waves plotted in the formula_1-formula_2 plane. One wave has formula_4 and formula_7, while the other has formula_6 and formula_5. Can you figure out the group velocity and the average phase speed in this case? Do these velocities match the apparent phase and group speeds in the figure? Figure 1.17 shows a dispersive wave in which the group velocity is twice the phase speed, while figure 1.18 shows a case in which the group velocity is actually opposite in sign to the phase speed. See if you can confirm that the phase and group velocities seen in each figure correspond to the values for these quantities calculated from the specified frequencies and wavenumbers.

Waves/1D Problems. Problems. <br> Figure 1.19: Sketch of a police radar. <br> Figure 1.20: Sketch of a Fabry-Perot interferometer. <br> Figure 1.21: Sketch of a weird dispersion relation.

Waves. The wave is a physical phenomenon that is found in a variety of contexts, a perturbation in the medium. You undoubtedly know about ocean waves and have probably played with a stretched slinky toy, producing undulations which move rapidly along the slinky. Other examples of waves are sound, vibrations in solids, and light (photons have the particularity of sharing characteristics of both waves and particles). Despite the vast differences in these different types of waves, they all consist of a fluctuation around an equilibrium position caused by a restoring force, and can be described by the same types of mathematical equations. Understanding these equations is a powerful tool because it will let you understand the basics of a wide variety of seemingly unrelated phenomena. The purpose of this section is to describe the "kinematics" of waves, i.e., to provide tools for describing the form and motion of all waves irrespective of their underlying physical mechanisms. In this chapter we learn first about the basic properties of waves and introduce the most common type of wave, called the sinusoidal (sine) wave. In fact just about any type of wave can be expressed as a combination of sine functions using a technique proposed by Joseph Fourier. Examples of waves seen in the real world are presented. We then learn about the superposition principle, which allows us to construct complex wave patterns by superimposing sine waves. Using these ideas, we discuss the related ideas of beats and interferometry. Finally, the ideas of wave packets and group velocity are introduced.

Organic Chemistry/Alkanes/Methane. =Methane= « Foundational concepts | Ethane » Methane, CH4 is the simplest organic molecule. It is a gas at standard temperature and pressure and is a carbon atom with four hydrogen atoms bonded to it in a tetrahedral shape. This is a flattened, two-dimensional representation of methane that you will see commonly. The true three-dimensional form of methane does not have any 90 degree angles between bonded hydrogens. The bonds point to the four corners of a tetrahedron, forming cos−1(−1/3) ≈ 109.5 degree bond angles. « Foundational concepts | « Alkanes | Methane | Ethane » | Introduction to reactions »

Organic Chemistry/Alkanes/Ethane. =Ethane= « Methane | Propane thru decane » Ethane (/ˈɛθeɪn/ or /ˈiːθeɪn/) is a chemical compound, or hydrocarbon with a chemical formula of C2H6. At standard temperature and pressure, ethane is a colorless, odorless gas. Ethane is isolated on an industrial scale from natural gas, and as a byproduct of petroleum refining. Its chief use is as petrochemical feedstock for ethylene production. It could be described as two methane molecules attached to each other, less two hydrogens. Number of hydrogens to carbons. This equation describes the relationship between the number of hydrogen and carbon atoms in alkanes: where "C" and "H" are used to represent the number of carbon and hydrogen atoms present in one molecule. If C = 2, then H = 6. Many textbooks put this in the following format: where "CN" and "H2N+2" represent the number of carbon and hydrogen atoms present in one molecule. If CN = 3, then H2N+2 = 2(3) + 2 = 8. (For this formula look to the "N" for the number, the "C" and the "H" letters themselves do not change.) « Organic chemistry | « Alkanes | « Methane | Ethane | Propane thru decane » | Introduction to reactions »

Optics. All help is welcome. Optics. "about this book" Contents. X-ray optics /Authors/ External Resources. __NOEDITSECTION__

Computer Programming/Tools. Wikify. Wikify is a program written by Josh Ferguson specifically designed to turn a standard text file into a html/wiki format. It is being used to do the color syntax highlighting for the C++ code in the C++ section of this textbook. Author's Statement: Wikify can not yet wikify itself, due to the fact that it has html codes in it, which confuses itself. I plan to rewrite it soon so it can wikify itself. Right now though, it works good with simple code. And it can be modified to work with other languages by changing the variables in it. Later on I'll probably move all the important variables into text files so it will be more easy to change things for other languages (i.e. by changing the text files, or extension checks), etc... The code was written really fast, so the code isn't perfect, even though the author stubbornly maintains that it is close. Code Snapshot as of 7-22-03 The escape sequences in the code give the wikiscript problems, so here is a link to it. Click on it, and it will take you to a page, where you can click on it, and it will come up. This should be compilable by any standard C++ compiler.

BASIC Programming. = List of Pages =

Pascal Programming. Pascal is an influential computer programming language named after the mathematician . It was invented by in 1968 as a research project into the nascent field of compiler theory. The backronym PASCAL standing for "primary algorithmic scientific commercial application language" highlights its suitability for computing tasks in science, making it certainly usable for general programming as well. Contents. Standard Pascal Extensions Appendix Alternative resources. Tutorials, Textbooks, and the like: References, Articles on certain topics:

Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Octet rule and exceptions. « Filling electron shells | Molecular orbitals » The octet rule refers to the tendency of atoms to prefer to have eight electrons in their valence shell. The main exception to the rule is hydrogen, which is at lowest energy when it has two electrons in its valence shell. Other notable exceptions are aluminum and boron, which can function well with six valence electrons; and some atoms beyond group three on the periodic table that can have over 8 electrons, including sulfur. Additionally, some noble gasses can form compounds when expanding their valence shell. The other tendency of atoms with regard to their electrons is to maintain a neutral charge. Only the noble gasses have zero charge with filled valence octets. All of the other elements have a charge when they have eight electrons all to themselves. The result of these two guiding principles is the explanation for much of the reactivity and bonding that is observed within atoms; atoms seeking to share electrons in a way that minimizes charge while fulfilling an octet in the valence shell. « Foundational concepts | « Filling electron shells | Molecular orbitals »

Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/Proton donors and acceptors. « Acids and bases | Electron donors and acceptors » The first and earliest definition of acids and bases was proposed in the 1800s by Swedish scientist Svante Arrhenius, who said that an acid was anything that dissolved in water to give up H+ ions. Such as hydrochloric acid and a base was anything that dissolved in water to give up OH- ions such a sodium hydroxide. The Brønsted-Lowry definition of 1923 broadened this idea a bit: The focus of this definition is on donating and accepting protons, and is not limited to aqueous solution. The Brønsted-Lowry definition of acids and bases is one of two definitions we commonly use. The second definition deals not with protons but with electrons, and has a slightly different emphasis. « Foundational concepts | « Resonance | « Acids and bases | Proton donors and acceptors | Alkanes »

Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Molecular orbitals. In organic chemistry we look at the hybridization of electron orbitals in to something called molecular orbitals. The s and p orbitals in a carbon atom combine into four hybridized orbitals that repel each other in a shape much like that of four balloons tied together. Remember what a tetrahedron looks like ? I knew you did ...

D Programming. The goal for this book is to create a complete, free, open-content, well-organized online book for the D programming language. D is a programming language being designed as a successor to C++. Until this page gets better written and more informative, the D home can be found here. Introduction. This book aims at beginners learning D language. It will cover all the language basics and some design aspects. In addition it will introduce topics like multi-threading, GUI programming and standard library to get you started with real-world applications. Overview. To quote Walter Bright, the author of the D Programming Language: D is a general purpose systems and applications programming language. It is a higher level language than C++, but retains the ability to write high performance code and interface directly with the operating system API's and with hardware. D is well suited to writing medium to large scale million line programs with teams of developers. D is easy to learn, provides many capabilities to aid the programmer, and is well suited to aggressive compiler optimization technology. D is not a scripting language, nor an interpreted language. It doesn't come with a VM, a religion, or an overriding philosophy. It's a practical language for practical programmers who need to get the job done quickly, reliably, and leave behind maintainable, easy to understand code. D is the culmination of decades of experience implementing compilers for many diverse languages, and attempting to construct large projects using those languages. D draws inspiration from those other languages (most especially C++) and tempers it with experience and real world practicality. D is a statically-typed, multi-paradigm language supporting imperative programming, object-oriented programming, and template metaprogramming. It also supports generics and design by contract. Main features of D. D has many features not seen in C++, implementing design by contract, unit testing, true modules, automatic memory management (garbage collection), first class arrays, associative arrays, dynamic arrays, array slicing, nested functions, inner classes, closures (anonymous functions), and has a reengineered template syntax. D retains C++'s ability to do low-level coding, and adds to it with support for an integrated inline assembler. C++ multiple inheritance is replaced by single inheritance with interfaces and mixins. D's declaration, statement and expression syntax closely matches that of C++. The inline assembler typifies the differentiation between D and application languages like Java and C#. An inline assembler allows a programmer to enter machine-specific assembly code alongside standard D code—a technique often used by systems programmers to access the low-level features of the processor needed to run programs that interface directly with the underlying hardware, such as operating systems and device drivers. Built into the language is a documentation generator called Ddoc. Memory management. Memory is usually managed with garbage collection, but specific objects can be finalized immediately when they go out of scope. Explicit memory management is possible using the overloaded operators new and delete, as well as simply calling C's malloc and free directly. It is also possible to disable garbage collection for individual objects, or even for the entire program if more control over memory management is desired. Interaction with other systems. C's ABI (Application Binary Interface) is supported as well as all of C's fundamental and derived types, enabling direct access to existing C code and libraries. C's standard library is part of standard D. In D 1.0, C++'s ABI is not supported, although it can access C++ code that is written to the C ABI, and can access C++ COM (Component Object Model) code. D 2 already supports some interaction with the C++ ABI. Implementation. Current D implementations compile directly into native code for efficient execution. Getting and installing D. The Digital Mars D compiler can be obtained from the digital mars website. http://www.digitalmars.com/d/download.html You will need the two files dmd.zip and dmc.zip. According to the manual both files should be extracted into a root directory or one without spaces or other special characters. The location of link.exe should then be added to the path. D programs can now be compiled by calling 'dmd'. Win32: Example of configuration 1 (dmd). Create a batch file "dmd_vars.bat" and move it to a directory that is included in the path: @echo off echo Setting up a dmd environment... set PATH=c:\dm\bin;c:\dmd\bin rem ;%SystemRoot%\System32 set LIB=c:\dmd\lib;c:\dm\lib echo PATH set to %PATH% Then: Note: This works well, even if you have other compilers/tools installed (which might also have link.exe/make.exe etc.) Your first D programs. /First Program Examples/ With Tango library the classic hello world program is: import tango.io.Console; void main() Cout("Hello, World").newline; /Alternate Program Examples/ With the Phobos library the classic hello world program is: import std.stdio; void main() writefln("Hello, World"); Compilation. Compiling hello world: dmd. dmd hello.d -ofhello gdc. gdc hello.d -o hello All D features. This is an incomplete list of all D features. It is specifically created to show and teach the D programming language with lots of examples.

Organic Chemistry/Alkanes/Propane thru decane. « Ethane | Stereoisomers » Many more linear alkanes can be formed by adding one additional carbon to the end of a chain of carbons. Ethane is the shortest chain with two carbons but DNA is known to have carbon chains containing millions of linked carbons. Alkanes are named based on the number of carbons in their longest chain: Naming carbon chains up to ten. The general equation for Alkanes is CnH2n+2.(Substute the n with no.of carbons, eg:-Propane 3 C 'C3H2*3+2=C3H8') The suffixis on the first four are from an obscure system but you should be familiar with the rest. Propane and butane are gases at standard temperature and pressure and are used commonly in lighters. Pentane on down the list are liquids at STP. Octane is the same as the octane in your gas tank. You may also want to name hydrocarbon chains of more than ten carbons. Here is a list for your reference: Naming longer carbon chains. etc... As the carbon chains get longer, molecules get relatively heavier and tend to move from being gases at Standard Temperature and Pressure to liquids to waxy solids. Foundational concepts » | « Alkanes | « Ethane | Propane thru decane | Drawing alkanes » | Introduction to reactions »

Organic Chemistry/Alkanes. Alkanes are the simplest organic molecules, consisting solely of singly-bonded carbon and hydrogen atoms. Alkanes are used as the basis for naming the majority of organic compounds (their nomenclature). Alkanes have the general formula CnH2n+2. Although their reactivities are often rather uninteresting, they provide an excellent basis for understanding bonding, conformation, and other important concepts which can be generalized to more "useful" molecules. =Introduction= Alkanes are the simplest and the least reactive hydrocarbon species containing only carbons and hydrogens. They are commercially very important, for being the principal constituent of gasoline and lubricating oils and are extensively employed in organic chemistry; though the role of pure alkanes (such as hexanes) is delegated mostly to solvents. The distinguishing feature of an alkane, making it distinct from other compounds that also exclusively contain carbon and hydrogen, is its lack of unsaturation. That is to say, it contains no double or triple bonds, which are highly reactive in organic chemistry. Though not totally devoid of reactivity, their lack of reactivity under most laboratory conditions makes them a relatively uninteresting, though very important component of organic chemistry. As you will learn about later, the energy confined within the carbon-carbon bond and the carbon-hydrogen bond is quite high and their rapid oxidation produces a large amount of heat, typically in the form of fire. As said it is important, not considered very important component in the chemistry. Introductory Definitions. Organic compounds contain carbon and hydrogen by definition and usually other elements (e.g. nitrogen and oxygen) as well. (CO2 is not an organic compound because it has no hydrogen). Hydrocarbons are organic compounds that contain carbon and hydrogen only. Alkanes are hydrocarbons or organic compounds made up of only carbon-carbon single bonds.Hence they are saturated. (as opposed to double and triple bonds). The simplest alkane is methane. Methane. Methane, (CH4, one carbon bonded to four hydrogens) is the simplest organic molecule. It is a gas at standard temperature and pressure (STP). This is a flattened, two-dimensional representation of methane that you will see commonly. The true three-dimensional form of methane does not have any 90 degree angles between bonded hydrogens. The bonds point to the four corners of a tetrahedron, forming cos-1(-1/3) ≈ 109.5 degree bond angles. Ethane. Two carbons singly bonded to each other with six hydrogens is called ethane. Ethane is the second simplest hydrocarbon molecule. It can be thought of as two methane molecules attached to each other, but with two fewer hydrogen atoms. Note that, if we were simply to create a new bond between the carbon centers of two methane molecules, this would violate the octet rule for the involved atoms. There are several common methods to draw organic molecules. They are often used interchangeably, although some methods work better for one situation or another. It is important to be familiar with the common methods, as these are the "languages" organic chemists can use to discuss structure with one another. =Drawing alkanes= When writing out the alkane structures, you can use different levels of the shorthand depending on the needs at hand in hand. For example, pentane can be written out. Its formula is C5H12. or CH3–CH2–CH2–CH2–CH3, or CH3(CH2)3CH3, or minimized to Line drawing shorthand. Although non-cyclic alkanes are called straight-chain alkanes they are technically made of linked chains. This is reflected in the line-drawing method. Each ending point and bend in the line represents one carbon atom and each short line represents one single carbon-carbon bond. Every carbon is assumed to be surrounded with a maximum number of hydrogen atoms unless shown otherwise. Structures drawn without explicitly showing all carbon atoms are often called "skeletal" structures, since they represent the skeleton or the backbone of the molecule. In organic chemistry, carbon is very frequently used, so chemists know that there is a carbon atom at the endpoints of every line that is not specifically labeled. =Conformations= Conformers, also called conformational isomers, or rotational isomers,or rotomers are arrangements of the same molecule made transiently different by the rotation in space about one or more single bonds. Other types of isomer can only be converted from one form to another by "breaking" bonds, but conformational isomers can be made simply by "rotating" bonds. Newman projections. Newman projections are drawings used to represent different positions of parts of molecules relative to each other in space. Remember that single bonds can rotate in space if not impeded. Newman projections represent different positions of rotating molecule parts. Conformations and energy. Different conformations have different potential energies. The staggered conformation is at a lower potential energy than the eclipsed conformation, and is favored. In ethane, the barrier to rotation is approximately 25 kJ/mol, indicating that each pair of eclipsed hydrogens raises the energy by about 8 kJ/mol. This number also applies to other organic compounds which have hydrogen atoms at similar distances from each other. At very low temperatures all conformations revert to the stabler( due to minimized vibration of atoms at it's mean position) , lower energy staggered conformation. Steric effects. Steric effects have to do with size. Two bulky objects run into each other and invade each others space. If we replace one or more hydrogen atoms on the above Newman projections with a methyl or other group, the potential energy goes up especially for the eclipsed conformations. Lets look at a Newman projection of butane as it rotates counterclockwise around its axes. When the larger groups overlap they repel each other more strongly than do hydrogen, and the potential energy goes up. Entropy. Entropy, represented as a ΔS, is a mathematical construct that represents disorder or probability. Natural systems want to find the lowest energy or organization possible, which translates to the highest entropy. "A note about potential energy: If you are rusty on this, remember the analogy of a big rock pushed to the top of a hill. At the top it has a maximum of potential energy. When you push it and allow it to roll down the hill the potential energy stored in it is transformed into kinetic energy that can be used to generate heat or smash something. " Notice that statistically, the ethane molecule has twice as many opportunities to be in the gauche conformation as in the anti conformation. However, because the Gauche configuration brings the methyl groups closer together in space, this generates high energy steric interactions and do not occur without the input of energy. Thus, the butane molecules shown will almost never be found in such unfavorable conformations. = Preparation of Alkanes = Wurtz reaction. Wurtz reaction is coupling of haloalkanes using sodium metal in solvent like dry ether 2R-X + 2Na → R-R + 2Na+X− Mechanism. The reaction consists of a halogen-metal exchange involving the free radical species R• (in a similar fashion to the formation of a Grignard reagent and then the carbon-carbon bond formation in a nucleophilic substitution reaction.) One electron from the metal is transferred to the halogen to produce a metal halide and an alkyl radical. The alkyl radical then accepts an electron from another metal atom to form an alkyl anion and the metal becomes cationic. This intermediate has been isolated in a several cases. The nucleophilic carbon of the alkyl anion then displaces the halide in an SN2 reaction, forming a new carbon-carbon covalent bond. Clemmensen reduction. Clemmensen reduction is a reduction of ketones (or aldehydes) to alkanes using zinc amalgam and hydrochloric acid The Clemmensen reduction is particularly effective at reducing aryl-alkyl ketones. With aliphatic or cyclic ketones, zinc metal reduction is much more effective The substrate must be stable in the strongly acidic conditions of the Clemmensen reduction. Acid sensitive substrates should be reacted in the Wolff-Kishner reduction, which utilizes strongly basic conditions; a further, milder method is the Mozingo reduction. As a result of Clemmensen Reduction, the carbon of the carbonyl group involved is converted from sp2 hybridisation to sp3 hybridisation. The oxygen atom is lost in the form of one molecule of water. Wolff-Kishner reduction. The Wolff–Kishner reduction is a chemical reaction that fully reduces a ketone (or aldehyde) to an alkane. Condensation of the carbonyl compound with hydrazine forms the hydrazone, and treatment with base induces the reduction of the carbon coupled with oxidation of the hydrazine to gaseous nitrogen, to yield the corresponding alkane. Mechanism. The mechanism first involves the formation of the hydrazone in a mechanism that is probably analogous to the formation of an imine. Successive deprotonations eventually result in the evolution of nitrogen. The mechanism can be justified by the evolution of nitrogen as the thermodynamic driving force. This reaction is also used to distinguish between aldehydes and ketones. Mozingo Reduction. A thioketal is first produced by reaction of the ketone with an appropriate thiol. The product is then hydrogenolyzed to the alkane, using Raney nickel = Properties of Alkanes = Alkanes are not very reactive when compared with other chemical species. This is because the backbone carbon atoms in alkanes have attained their octet of electrons through forming four covalent bonds (the maximum allowed number of bonds under the octet rule; which is why carbon's valence number is 4). These four bonds formed by carbon in alkanes are sigma bonds, which are more stable than other types of bond because of the greater overlap of carbon's atomic orbitals with neighboring atoms' atomic orbitals. To make alkanes react, the input of additional energy is needed; either through heat or radiation. Gasoline is a mixture of the alkanes and unlike many chemicals, can be stored for long periods and transported without problem. It is only when ignited that it has enough energy to continue reacting. This property makes it difficult for alkanes to be converted into other types of organic molecules. (There are only a few ways to do this). Alkanes are also less dense than water, as one can observe, oil, an alkane, floats on water. Alkanes are non-polar solvents. Since only C and H atoms are present, alkanes are nonpolar. Alkanes are immiscible in water but freely miscible in other non-polar solvents. Alkanes consisting of weak dipole dipole bonds can not break the strong hydrogen bond between water molecules hence it is not miscible in water. The same character is also shown by alkenes. Because alkanes contain only carbon and hydrogen, combustion produces compounds that contain only carbon, hydrogen, and/or oxygen. Like other hydrocarbons, combustion under most circumstances produces mainly carbon dioxide and water. However, alkanes require more heat to combust and do not release as much heat when they combust as other classes of hydrocarbons. Therefore, combustion of alkanes produces higher concentrations of organic compounds containing oxygen, such as aldehydes and ketones, when combusting at the same temperature as other hydrocarbons. The general formula for alkanes is CNH2N+2; the simplest possible alkane is therefore methane, CH4. The next simplest is ethane, C2H6; the series continues indefinitely. Each carbon atom in an alkane has sp³ hybridization. Alkanes are also known as paraffins, or collectively as the paraffin series. These terms are also used for alkanes whose carbon atoms form a single, unbranched chain. Branched-chain alkanes are called isoparaffins. Methane through Butane are very flammable gases at standard temperature and pressure (STP). Pentane is an extremely flammable liquid boiling at 36 °C and boiling points and melting points steadily increase from there; octadecane is the first alkane which is solid at room temperature. Longer alkanes are waxy solids; candle wax generally has between C20 and C25 chains. As chain length increases ultimately we reach polyethylene, which consists of carbon chains of indefinite length, which is generally a hard white solid. Chemical properties. Alkanes react only very poorly with ionic or other polar substances. The pKa values of all alkanes are above 50, and so they are practically inert to acids and bases. This inertness is the source of the term paraffins (Latin para + affinis, with the meaning here of "lacking affinity"). In crude oil the alkane molecules have remained chemically unchanged for millions of years. However redox reactions of alkanes, in particular with oxygen and the halogens, are possible as the carbon atoms are in a strongly reduced condition; in the case of methane, the lowest possible oxidation state for carbon (−4) is reached. Reaction with oxygen leads to combustion without any smoke; with halogens, substitution. In addition, alkanes have been shown to interact with, and bind to, certain transition metal complexes. Free radicals, molecules with unpaired electrons, play a large role in most reactions of alkanes, such as cracking and reformation where long-chain alkanes are converted into shorter-chain alkanes and straight-chain alkanes into branched-chain isomers. In highly branched alkanes and cycloalkanes, the bond angles may differ significantly from the optimal value (109.5°) in order to allow the different groups sufficient space. This causes a tension in the molecule, known as steric hinderance, and can substantially increase the reactivity. The same is preferred for alkenes too. =Introduction to Nomenclature= Before we can understand reactions in organic chemistry, we must begin with a basic knowledge of naming the compounds. The IUPAC nomenclature is a system on which most organic chemists have agreed to provide guidelines to allow them to learn from each others' works. Nomenclature, in other words, provides a foundation of language for organic chemistry. The names of all alkanes end with "-ane". Whether or not the carbons are linked together end-to-end in a ring (called "cyclic alkanes" or "cycloalkanes") or whether they contain side chains and branches, the name of every carbon-hydrogen chain that lacks any double bonds or functional groups will end with the suffix "-ane". Alkanes with unbranched carbon chains are simply named by the number of carbons in the chain. The first four members of the series (in terms of number of carbon atoms) are named as follows: Alkanes with five or more carbon atoms are named by adding the suffix "-ane" to the appropriate numerical multiplier, except the terminal "-a" is removed from the basic numerical term. Hence, C5H12 is called "pentane", C6H14 is called "hexane", C7H16 is called "heptane" and so forth. Straight-chain alkanes are sometimes indicated by the prefix "n-" (for normal) to distinguish them from branched-chain alkanes having the same number of carbon atoms. Although this is not strictly necessary, the usage is still common in cases where there is an important difference in properties between the straight-chain and branched-chain isomers: e.g. "n-hexane" is a neurotoxin while its branched-chain isomers are not. Number of hydrogens to carbons. This equation describes the relationship between the number of hydrogen and carbon atoms in alkanes: where "C" and "H" are used to represent the number of carbon and hydrogen atoms present in one molecule. If C = 2, then H = 6. Many textbooks put this in the following format: where "Cn" and "H2n+2" represent the number of carbon and hydrogen atoms present in one molecule. If Cn = 3, then H2n+2 = 2(3) + 2 = 8. (For this formula look to the "n" for the number, the "C" and the "H" letters themselves do not change.) Progressively longer hydrocarbon chains can be made and are named systematically, depending on the number of carbons in the longest chain. Naming carbon chains up to twelve. The prefixes of the first three are the contribution of a German Chemist, August Wilhelm Hoffman, who also suggested the name quartane for 4 carbons in 1866. However, the but- prefix had already been in use since the 1820s and the name quartane never caught on. He also recommended the endings to use the vowels, a, e, i (or y), o, and u, or -ane, -ene, -ine or -yne, -one, and -une. Again, only the first three caught on for single, double, and triple bonds and -one was already in use for ketones. Pent, hex, hept, oct, and dec all come from the ancient Greek numbers (penta, hex, hepta, octa, deka) and oddly, non, from the Latin novem. For longer-chained alkanes we use the special IUPAC multiplying affixes. For example, pentadecane signifies an alkane with 5+10 = 15 carbon atoms. For chains of length 30, 40, 50, and so on the basic prefix is added to -contane. For example, C57H116 is named as heptapentacontane. When the chain contains 20-29 atoms we have an exception. C20H42 is known as icosane, and then we have, e.g. tetracosane (eliding the "i" when necessary). For the length 100 we have "hecta" but for 200, 300 ... 900 we have "dicta", "tricta", and so on, eliding the "i" on "icta" when necessary; for 1000 we have "kilia" but for 2000 and so on, "dilia", "trilia", and so on, eliding the "i" on "ilia" when necessary. We then put all of the prefixes together in reverse order. The alkane with 9236 carbon atoms is then hexatridinoniliane. Isomerism. The atoms in alkanes with more than three carbon atoms can be arranged in many ways, leading to a large number of potential different configurations (isomers). So-called "normal" alkanes have a linear, unbranched configuration, but the "n-" isomer of any given alkane is only one of potentially hundreds or even possibly millions of configurations for that number of carbon and hydrogen atoms in some sort of chain arrangement.<br>Isomerism is defined as the compound having same moleculer formula the formula which present the different moleculer formula arrangement are called as Isomerism.<br>e.g.- The molecular formula for butane is C4H10. The number of isomers increases rapidly with the number of carbon atoms in a given alkane molecule; for alkanes with as few as 12 carbon atoms, there are over three hundred and fifty-five possible forms the molecule can take! Branched chains. Carbon is able to bond in all four directions and easily forms strong bonds with other carbon atoms. When one carbon is bonded to more than two other carbons it forms a branch. Above you see a carbon bonded to three and four other carbons. The common system has naming convention for carbon chains as they relate to branching. "Note: "R" in organic chemistry is a placeholder that can represent any carbon group." Constitutional isomers. One of the most important characteristics of carbon is its ability to form several relatively strong bonds per atom. It is for this reason that many scientists believe that carbon is the only element that could be the basis for the many complicated molecules needed to support a living being. One carbon atom can have attached to it not just the one or two other carbons needed to form a single chain but can bond to up to four other carbons. It is this ability to bond multiply that allows isomerism. Isomers are two molecules with the same molecular formula but different physical arrangements. Constitutional isomers have their atoms arranged in a different order. A constitutional isomer of butane has a main chain that is forked at the end and one carbon shorter in its main chain than butane. Naming Alkanes. There are several ways or systems for the nomenclature, or naming, of organic molecules, but just two main ones. The IUPAC system is necessary for complicated organic compounds. It gives a series of unified rules for naming a large compound by conceptually dividing it up into smaller, more manageable nameable units. Many traditional (non-IUPAC) names are still commonly used in industry, especially for simpler and more common chemicals, as the traditional names were already entrenched. IUPAC naming rules. Substituents are named like a parent, and replacing the "-ane" ending with "-yl". Numbering. The above molecule is numbered as follows: 2,3,7-Trimethyloctane Not 2,6,7-Trimethyloctane. Remember, number so as to give the smallest numbers to the substituents. Alphabetizing. 3-Ethyl-3-methylpentane "Ethyl" is listed before "methyl" for alphabetizing purposes. Branched Substituents. Naming branched substituents. 3-(1-methylethyl)-2,4-dimethylpentane The main chain in the drawing is numbered 1-5. The main part of the branched substituent, an ethyl group, is numbered 1' and 2'. The methyl substituent off of the ethyl substituent is not numbered in the drawing. To name the compound, put the whole branched substituent name in parentheses and then number and alphabetize as if a simple substituent. Common system. Some prefixes from the common system are accepted in the IUPAC system. For alphabetization purposes, iso- and neo- are considered part of the name, and alphabetized. Sec- and tert- are not considered an alphabetizable part of the name. Iso-. Iso- can be used for substituents that branch at the second-to-last carbon and end with two methyls. An isobutyl has four carbons total: "Isobutyl" Sec-. Sec- can be used for substituents that branch at the first carbon . Neo-. Neo- refers to a substituent whose second-to-last carbon of the chain is trisubstituted (has three methyl groups attached to it). A neo-pentyl has five carbons total. "Neopentyl" Tert-. Tert- is short for tertiary and refers to a substituent whose first carbon has three other carbon groups attached to it. =See also=

Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/Electron donors and acceptors. Proton donors and acceptors » | Electrophiles and nucleophiles » The Lewis definition of acids and bases describes an acid as an electron acceptor and a base as an electron donator. « Foundational concepts | « Acids and bases | Proton donors and acceptors » | Electron donors and acceptors | Electrophiles and nucleophiles » | Alkanes »

Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/Electrophiles and nucleophiles. « Electron donors and acceptors | pKa and acidity » Electrophiles are "electron-lovers". (The suffix -phile means "lover of", as "bibliophile" means "lover of books"). Electrophiles seek electrons. Nucleophiles, or "nucleus lovers", seek positively charged nuclei. Electrophiles and nucleophiles are often ions, but sometimes not. « Foundational concepts | « Acids and bases | Electron donors and acceptors » | Electrophiles and nucleophiles | pKa and acidity » | Alkanes »

Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/PKa and acidity. « Electrophiles and nucleophiles | Alkanes » To measure acid strength of a compound, scientist typically use a quantity called pKa. It is defined as formula_1 where Ka is the acid dissociation constant. Stronger acids have small pKa values, where as weaker acids have higher pKa values. « Foundational concepts | « Acids and bases | Electrophiles and nucleophiles » | pKa and acidity | Alkanes »

Organic Chemistry/Alkanes/Stereoisomers. Stereoisomer of 2-bromo-3-hexanol </noinclude> = Stereoisomers = Stereoisomers are a type of isomer where the order of the atoms in the two molecules is the same but their arrangement in space is different. To understand this we need to take a look at the ways that organic molecules can and cannot move. Again, usingthree-dimensional models is a great tool to visualize this and almost essential for most people to grasp these concepts clearly. With cyclo-alkanes, we observe that a group placed on one side of a ring stays on that same side. Except in very large rings (13+ carbons) the carbons are not free to rotate all of the way around their axes. This means that a group that is axial will not move into an equatorial position, and vice versa. Stereoisomerism is the arrangement of atoms in molecules whose connectivity remains the same but their arrangement in space is different in each isomer. The two main types of stereoisomerism are: Cis-trans Isomerism. "Main article:" Diastereomers Cis/trans isomerism occurs when a double bond is present, because the pi bond involved prevents that bond from being "twisted" the same way that a single bond can be. A good example is 1,2-dichloroethene: C2H2Cl2. Consider the two examples below: The two molecules shown above are "cis"-1,2-dichloroethene and "trans"-1,2-dichloroethene. This is more specifically an example of diastereomerism. These two molecules are stereoisomers because the two carbon atoms cannot be rotated relative to each other, due to the rigidity caused by the pi bond between them. Therefore, they are not "superimposeable" - they are not identical, and cannot take each other's place. However, the isomers are not mirror images of one another, so they are not enantiomers; therefore they must be diastereomers. Diastereomers usually have different chemical and physical properties and can exhibit dramatically different biological activity. There are two forms of these isomers; the "cis" and "trans" versions. The form in which the substituent hydrogen atoms are on the same side of the bond that doesn't allow rotation is called "cis"; the form in which the hydrogens are on opposite sides of the bond is called "trans". An example of a small hydrocarbon displaying cis-trans isomerism is 2-butene. Alicyclic compounds can also display cis-trans isomerism. As an example of a geometric isomer due to a ring structure, consider 1,2-dichlorocyclohexane: Optical Isomerism. "Main article:" Chirality Optical isomers are stereoisomers formed when asymmetric centers are present, for example, a carbon with four different groups bonded to it. Enantiomers are two optical isomers (i.e. isomers that are reflections of each other). Every stereocenter in one isomer has the opposite configuration in the other. Compounds that are enantiomers of each other have the same physical properties, except for the direction in which they rotate polarized light and how they interact with different optical isomers of other compounds. In nature, most biological compounds, such as amino acids, occur as single enantiomers. As a result, different enantiomers of a compound may have substantially different biological effects. When a molecule has more than one source of asymmetry, two optical isomers may be neither perfect reflections of each other nor superimposeable: some but not all stereocenters are inverted. These molecules are diastereomers, not enantiomers. Diastereomers seldom have the same physical properties. Optical isomerism is a form of isomerism (specifically stereoisomerism) where the two different isomers are the same in every way except being non-superposable [1] mirror images of each other. Optical isomers are known as chiral molecules. « Foundational concepts | « Alkanes | « Naming cycloalkanes | Stereoisomers | Introduction to reactions » </noinclude>

Organic Chemistry/Places to buy organic chemistry models. The links below will search on the named sites for the terms "organic chemistry model kit":

Organic Chemistry/Alkanes/Cycloalkanes. Cycloalkanes are hydrocarbons containing one or more rings. (Alkanes without rings are referred to as aliphatic.) → + H2 Under certain reaction conditions, propane can be transformed into cyclopropane. (H2 comes off as a sideproduct.) Cyclopropane (unstable, lots of ring strain) Cyclobutane (ring strain) Cyclopentane (little ring strain) Cyclohexane (Next to no ring strain) Cyclodecane Rings with thirteen or more carbons have virtually no ring strain. Naming cycloalkanes. Cycloalkanes are named similarly to their straight-chain counterparts. Simply add the root "cyclo-" before the alkane part of the name. Example: Propane » Cyclopropane When naming cycloalkanes, the cyclo prefix is used for alphabetization. Substituents. If a cycloalkane has only one substituent, it is not necessary to assign that substituent a number. If there is more than one substituent, then it is necessary to number the carbons and specify which substituent is on which carbon. Methylcyclopentane 1,1-dimethylcyclopentane 1,2-dimethylcyclopentane 1,3-dimethylcyclopentane The organic compound could be named and numbered 1-cyclopropyl-5-ethyl-2-methylcyclohexane but instead should be named 2-cyclopropyl-4-ethyl-1-methylcyclohexane because it produces a lower numbered name (1+5+2=8 vs. 2+4+1=7). In the following examples, notice that the longer chain is the parent and the cycloalkane is the substituent. 2-Cyclopropylbutane 1,3-dicyclopropylpropane Multicyclic alkanes. Multicyclic alkanes are hydrocarbons that have "more than one bonded cyclic ring". These abound in biology as all kinds of hormones, steroids, cholesterol,carbohydrates, etc. They are named as bicycloalkanes, tricycloalkanes, etc. They are named slightly differently than singularly cyclic alkanes. Bicyclo[2.1.0]pentane Multicyclic alkanes are found frequently in living beings: Part of Cholesterol We will get to some of the most interesting multicyclic rings later on when we study benzene and aromaticity. Stereochemistry. Because the C-C bonds in cycles cannot rotate through 360 degrees, substituted cycloalkanes and similar compounds can exhibit diastereomerism. This is comparable to alkenes which show cis/trans (or E/Z) isomerism. The isomers can be named using cis/trans notation, or more rigorously using R-S notation. Conformers, or conformational isomers, are different arrangements of the same molecule in space. Do not confuse them with any kind of true isomer as they are in every way the same molecule. The difference is in how the molecule is bent or twisted is space in any one instant of time. Cyclohexane. The first molecule that is generally presented in a discussion of cycloalkane conformers is cyclohexane. It comes in several flavors; the main ones are the chair conformation and the boat conformation. "Note: In the above models, the straight lines represents single bonds, the lumps represent carbon atoms, and the open ends represent hydrogen atoms." Consider getting a good molecular model set if you do not yet have one. They are not as inexpensive as you would hope but they help most people immensely to understand the way molecules look in three dimensions. Follow this link to places you can buy a molecular model kit. The chair conformation (can you see how it looks like a chair?) is lower in energy than the boat conformation. This is because the two ends of the molecule are farther apart and avoid steric hindrance. Hydrogen atoms in a cyclohexane can be divided into two types: When hydrogens are replaced with other, bulkier groups, it becomes apparent that the axial positions are less energetically favored than the equatorial positions. That means that, if given a choice, bulkier groups will tend to bond to cyclohexane in equatorial positions, as this reduces their steric hinderance and potential energy. Other cycloalkanes. Cyclopentane flips between slightly different conformers as well.

Organic Chemistry/Chirality/Enantiomers. Enantiomers. Main article: Chirality In chemistry, two stereoisomers are said to be enantiomers if they are mirror images of each other. Much as a left and right hand are different but one is the mirror image of the other, enantiomers are stereoisomers whose molecules are nonsuperposable mirror images of each other.

Investing. Investing involves using your money (or borrowed money that you control) to earn more money. Before proceeding, make sure that you understand the concepts of Personal Finance. Topics covered here are: Many excellent resources available on the web:

Organic Chemistry/Periodic table. Some of the good news about organic chemistry is that it focuses on a subset of the periodic table, so there are fewer elements to worry about. These are the main elements with which we must concern ourselves, although occasionally a few others will be used in reactions.

Calculus/Limits/An Introduction to Limits. Limits, the first step into calculus, explain the complex nature of the subject. It is used to define the process of derivation and integration. It is also used in other circumstances to intuitively demonstrate the process of "approaching". Introduction. Intuitive Look into Limits. The limit is one of the greatest tools in the hands of any mathematician. We will give the limit an approach. Because mathematics came only due to approaches... remember?! We designate limit in the form: This is read as "The limit of formula_2 of formula_3 as formula_3 approaches formula_5". This is an important thing to remember, it is basic notation which is accepted by the world. We'll take up later the question of how we can determine whether a limit exists for formula_6 at formula_5 and, if so, what it is. For now, we'll look at it from an intuitive standpoint. Let's say that the function that we're interested in is formula_8 , and that we're interested in its limit as formula_3 approaches formula_10. Using the above notation, we can write the limit that we're interested in as follows: One way to try to evaluate what this limit is would be to choose values near 2, compute formula_6 for each, and see what happens as they get closer to 2. There are two ways to approach values near 2. One is to approach from below, and the other is to approach from above: The table above is the case from below. The table above is the case from above. We can see from the tables that as formula_3 grows closer and closer to 2, formula_6 seems to get closer and closer to 4, regardless of whether formula_3 approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of formula_16 as formula_3 approaches 2 is 4, or, written in limit notation, We could have also just substituted 2 into formula_16 and evaluated: formula_20. However, this will not work with all limits. Now let's look at another example. Suppose we're interested in the behavior of the function formula_21 as formula_3 approaches 2. Here's the limit in limit notation: Just as before, we can compute function values as formula_3 approaches 2 from below and from above. Here's a table, approaching from below: And here from above: In this case, the function doesn't seem to be approaching a single value as formula_3 approaches 2, but instead becomes an extremely large positive or negative number (depending on the direction of approach). Well, one says such a limit does not exist because no finite number is approached. This arises the concept of infinity: an undefined quantity and the limit is also called infinite limit or limit without a bound. Note that we cannot just substitute 2 into formula_26 and evaluate as we could with the first example, since we would be dividing by 0. Both of these examples may seem trivial, but consider the following function: This function is the same as Note that these functions are really completely identical; not just "almost the same," but actually, in terms of the definition of a function, completely the same; they give exactly the same output for every input. In elementary algebra, a typical approach is to simply say that we can cancel the term formula_29 , and then we have the function formula_8. However, that would be inaccurate; the function that we have now is not really the same as the one we started with, because it is defined when formula_31 , and our original function was specifically not defined when formula_31. This may seem like a minor point, but from making this kind of assumptions we can easily derive absurd results, such that formula_33 (see Mathematical Fallacy § All numbers equal all other numbers in Wikipedia for a complete example). Even without calculus we can avoid this error by stating that: In calculus, we can introduce a more intuitive and also correct way of looking at this type of function. What we want is to be able to say that, although the function isn't defined when formula_31, it works almost as if it was. It may not get there, but it gets really, really close. For instance, formula_36. The only question that we have is: what do we mean by "close"? Informal Definition of a Limit. As the precise definition of a limit is a bit technical, it is easier to start with an informal definition; we'll explain the formal definition later. We suppose that a function formula_2 is defined for formula_3 near formula_39 (but we do not require that it be defined when formula_40). Notice that the definition of a limit is not concerned with the value of formula_6 when formula_40 (which may exist or may not). All we care about are the values of formula_6 when formula_3 is close to formula_39 , on either the left or the right (i.e. less or greater). Limit can also be understood as: formula_3 is infinitely approaching to formula_39 but never equals to formula_39, just like the function formula_49, which infinitely approaches to formula_50 but never equals formula_50. Basics. Rules and Identities. Now that we have defined, informally, what a limit is, we will list some rules that are useful for working with and computing limits. You will be able to prove all these once we formally define the fundamental concept of the limit of a function. First, the constant rule states that if formula_52 (that is, formula_2 is constant for all formula_3) then the limit as formula_3 approaches formula_39 must be equal to formula_57. In other words Second, the identity rule states that if formula_59 (that is, formula_2 just gives back whatever number you put in) then the limit of formula_2 as formula_3 approaches formula_39 is equal to formula_39. That is, The next few rules tell us how, given the values of some limits, to compute others. Notice that in the last rule we need to require that formula_66 is not equal to 0 (otherwise we would be dividing by zero which is an undefined operation). These rules are known as identities; they are the scalar product, sum, difference, product, and quotient rules for limits. (A scalar is a constant, and, when you multiply a function by a constant, we say that you are performing scalar multiplication.) Using these rules we can deduce another. Namely, using the rule for products many times we get that This is called the power rule. As a result, we can safely say that all limits for polynomial functions can be deduced into several limits that satisfy the identity rule and thus easier to compute. Find the limit formula_69. We need to simplify the problem, since we have no rules about this expression by itself. We know from the identity rule above that formula_70. By the power rule, formula_71. Lastly, by the scalar multiplication rule, we get formula_72. formula_73 Find the limit formula_74. To do this informally, we split up the expression, once again, into its components. As above, formula_75. Also formula_76 and formula_77. Adding these together gives Find the limit formula_80. From the previous example the limit of the numerator is formula_81. The limit of the denominator is As the limit of the denominator is not equal to zero we can divide. This gives Find the limit formula_85. We apply the same process here as we did in the previous set of examples; We can evaluate each of these; formula_87 Thus, the answer is formula_88. Find the limit formula_90. In this example, evaluating the result directly will result in a division by 0. While you can determine the answer experimentally, a mathematical solution is possible as well. First, the numerator is a polynomial that may be factored: formula_91 Now, you can divide both the numerator and denominator by formula_29: formula_93 Remember that the limit is a method to determine the approaching value of a function instead of the value of the function itself. So, though the function is undefined at formula_31, the value of the function is approaching to formula_95 when formula_96 Find the limit formula_98. To evaluate this seemingly complex limit, we will need to recall some sine and cosine identities (see Chapter ). We will also have to use two new facts. First, if formula_6 is a trigonometric function (that is, one of sine, cosine, tangent, cotangent, secant and cosecant functions), and is defined at formula_5 , then formula_101. Second, formula_102. This can be proved using squeeze theorem. Note that L'Hospital's rule is not allowed to be used to evaluate this limit because it causes circular reasoning, in the sense that differentiating formula_103.requires this limit to equal one, which is exactly the equation that is being proven. Method 1 (before learning L'Hôpital's rule): To evaluate the limit, recognize that formula_104 can be multiplied by formula_105 to obtain formula_106 which, by our trig identities, is formula_107. So, multiply the top and bottom by formula_105. (This is allowed because it is identical to multiplying by one.) This is a standard trick for evaluating limits of fractions; multiply the numerator and the denominator by a carefully chosen expression which will make the expression simplify somehow. In this case, we should end up with: Our next step should be to break this up into formula_109 by the product rule. As mentioned above, formula_110. Next, formula_111. Thus, by multiplying these two results, we obtain 0. formula_73 Note that we also cannot apply L'Hospital's rule to evaluate this limit because it causes circular reasoning. We will now present an amazingly useful result, even though we cannot prove it yet. We can find the limit at formula_39 of any polynomial or rational function, as long as that rational function is defined at formula_39 (so we are not dividing by 0). That is, formula_39 must be in the domain of the function. We already learned this for trigonometric functions, so we see that it is easy to find limits of polynomial, rational or trigonometric functions wherever they are defined. In fact, this is true even for combinations of these functions; thus, for example, formula_116. The Squeeze Theorem. The Squeeze Theorem is very important in calculus, where it is typically used to find the limit of a function by comparison with two other functions whose limits are known. It is called the Squeeze Theorem because it refers to a function formula_2 whose values are squeezed between the values of two other functions formula_118 and formula_119 , both of which have the same limit formula_120. If the value of formula_2 is trapped between the values of the two functions formula_118 and formula_119 , the values of formula_2 must also approach formula_120. Expressed more precisely: Example: Compute formula_126. Since we know thatformula_127Multiplying formula_3 into the inequality yieldsformula_129Now we apply the squeeze theoremformula_130Since formula_131, formula_132 formula_73 Finding Limits. Now, we will discuss how, in practice, to find limits. First, if the function can be built out of rational, trigonometric, logarithmic, or exponential functions, then if a number formula_39 is in the domain of the function, then the limit at formula_39 is simply the value of the function at formula_39:formula_137 when formula_6 can be built out of rational, trigonometric, logarithmic, or exponential functions and formula_139 the Domain of formula_6If formula_39 is not in the domain of the function, then in many cases (as with rational functions) the domain of the function includes all the points near formula_39, but not formula_39 itself. An example would be if we wanted to find formula_144 , where the domain includes all numbers besides 0. In that case, in order to find formula_145 we want to find a function formula_146 similar to formula_6 , except with the hole at formula_39 filled in. The limits of formula_2 and formula_118 will be the same, as can be seen from the definition of a limit. By definition, the limit depends on formula_6 only at the points where formula_3 is close to formula_39 but not equal to it, so the limit at formula_39 does not depend on the value of the function at formula_39. Therefore, if formula_156 , formula_157 also. And since the domain of our new function formula_118 includes formula_39 , we can now (assuming formula_118 is still built out of rational, trigonometric, logarithmic and exponential functions) just evaluate it at formula_39 as before. Thus we have formula_162. In our example, this is easy; canceling the formula_3's gives formula_164, which equals formula_165 at all points except 0. Thus, we have formula_166. In general, when computing limits of rational functions, it's a good idea to look for common factors in the numerator and denominator. Specific DNE Situations. Note that the limit might not exist at all (DNE means "does not exist"). There are a number of ways in which this can occur: "Gap" Determining Limits. The formal way to determine whether a limit exists is to find out whether the value of the limit is the same when approaching from below and above (see at the top of this chapter). The notation for the limit approaching from below (in increasing order) isformula_209, notice the negative sign on the constant formula_39The notation for the limit approaching from above (from decreasing order) isformula_211, notice the positive sign on the constant formula_39For example, let us find the limits of formula_49 when formula_3 is approaching formula_50 in both directions. In other words, find formula_216 and formula_217.Recall the table we made when we are trying to intuitively feel the limit. We can use that to help us. However, if familiar enough with reciprocal functions, we can simply determine the two values by imagining the graph of the function. The following table is when formula_3 is approaching from below. Thus, we found that when formula_3 is approaching from below to formula_50, the function approaches negative infinity. In mathematical terms: formula_221 Now let's talk about the approach from above. We found that formula_222formula_73 The method of determining if limits exist is relatively intuitive. It only requires some practice to be familiar with the process. Let's use the same example: find formula_224.Since we already found that formula_221 and formula_222, and obviously, formula_227 We can say that formula_224 does not exist.formula_73 Infinity Situations. Now consider the function What is the limit as formula_3 approaches zero? The value of formula_232 does not exist; it is not defined. Notice, also, that we can make formula_146 as large as we like, by choosing a small formula_3 , as long as formula_235. For example, to make formula_146 equal to formula_237 , we choose formula_3 to be formula_239. Thus, formula_240 does not exist. However, we "do" know something about what happens to formula_146 when formula_3 gets close to 0 without reaching it. We want to say we can make formula_146 arbitrarily large (as large as we like) by taking formula_3 to be sufficiently close to 0, but not equal to 0. We express this symbolically as follows: Note that the limit does not exist at formula_50 ; for a limit, being formula_247 is a special kind of not existing. In general, we make the following definition. An example of the second half of the definition would be that formula_248. Applications of Limits. To see the power of the concept of the limit, let's consider a moving car. Suppose we have a car whose position is linear with respect to time (that is, a graph plotting the position with respect to time will show a straight line). We want to find the velocity. This is easy to do from algebra; we just take the slope, and that's our velocity. But unfortunately, things in the real world don't always travel in nice straight lines. Cars speed up, slow down, and generally behave in ways that make it difficult to calculate their velocities. Now what we really want to do is to find the velocity at a given moment (the instantaneous velocity). The trouble is that in order to find the velocity we need two points, while at any given time, we only have one point. We can, of course, always find the average speed of the car, given two points in time, but we want to find the speed of the car at one precise moment. This is the basic trick of differential calculus, the first of the two main subjects of this book. We take the average speed at two moments in time, and then make those two moments in time closer and closer together. We then see what the limit of the slope is as these two moments in time are closer and closer, and say that this limit is the slope at a single instant. We will study this process in much greater depth later in the book. First, however, we will need to study limits more carefully.

Artificial Intelligence. Welcome to the Wikibook about Artificial Intelligence. Book Contents. The following is a first proposal for a basic layout. This is not yet complete, ideas are welcome. Discuss on the talk page or just add them here. The book is laid out into 5 sections, with increasing detail and complexity. Each section contains a number of chapters. In addition to regular chapters, there are case-study chapters that investigate full and complex AI systems using several techniques from the regular chapters (as well as perhaps some new ones). Introduction. Overview

Spanish/Cover. ¡Bienvenidos! "Welcome!" Vaya a los contenidos» "Go to the contents»"

Spanish/Lesson 1. Introduction. This is the very first lesson in learning a second language, the Spanish language! This lesson begins with simple greetings, and covers important ideas of the Spanish language. Throughout education, methods of teaching Spanish have changed greatly. Years ago, the Spanish language was taught simply by memory. Today, however, the Spanish Language is taught by moving more slowly and covering grammar and spelling rules. Again, this is an introduction. If this is the first time you are attempting to learn Spanish, do not become discouraged if you cannot understand, pronounce, or memorize some of the things discussed here. In addition, learning a second language requires a basic understanding of your own language. You may find, as you study Spanish, that you learn a lot about English as well. At their core, all languages share some simple components like verbs, nouns, adjectives, and plurals. English, as your first language, comes naturally to you and you don't think about things like subject-verb agreement, verb conjugation, or usage of the various tenses; yet you use these concepts on a daily basis. While English is described as a very complicated language to learn, many of the distinguishing grammar structures have been simplified over the years. This is not true for many other languages. Following the grammatical conventions of Spanish will be very important, and can actually change the meaning of phrases. You'll see what is meant by this as you learn your first verbs ser and estar. Do not become discouraged! You can do it. Dialogue 1. Two good friends - Carmen and Roberto - are meeting: (139KB) Dialogue 2. Two people - Señor González and Señora Pérez - are meeting for the first time: Nice to meet you. Listen to the Dialogue. Vocabulary. Exercise: Greetings Grammar: Subject Pronouns. </br> A few things to keep in mind: Grammar: Verbs ser and estar. Spanish has two different words that can be translated with ""to be". Ser is used more for more permanent characteristics ("Soy Luis") whereas estar is used for more temporary or changeable conditions, such as location ("La papelera está al lado del escritorio"", "The trash can is beside the desk") and feeling (""Estoy bien""). A good way to remember when to use "estar" is by using the rhyme, "To tell how you feel or where you are, always use the verb estar." In future lessons we will come back to the uses of ser and estar. Here we will look at the conjugations in the "present indicative". <br> Ejemplos de los verbos ser y estar (Examples of the verbs ser and estar). Note: *This use of estar is the Spanish "present progressive" which is used for actions in progress. More about the "present progressive" in Lesson 4 Dialect Note: Spanish which uses the "vos" form conjugates ser with the following irregular form: "sos". Exercise: Verbs ser and estar Hay. Spanish uses a different verb (haber) to express ""there is " and "there are". The form of haber used for this purpose is hay, for both singular ("there is") and plural ("there are"). Spanish alphabet. Here is the normal Spanish alphabet. However words aren't alphabetized by it. Please read the notes and sections below. (Blue letters are a part of the normal English alphabet.) Audio: (646KB) Although the above will help you understand, proper pronunciation of Spanish consonants is a bit more complicated: Most of the consonants are pronounced as they are in American English with these exceptions: Vowel pronunciation. The pronunciation of vowels is as follows: The "u" is always silent after q (as in "qué" pronounced kā). Spanish also uses the ¨ (diaeresis) diacritic mark over the vowel u to indicate that it is pronounced separately in places where it would normally be silent. For example, in words such as "vergüenza" ("shame") or "pingüino" ("penguin"), the "u" (sounds the same as the "u" in "ultra") is pronounced similarly but with more strength to the English "w" forming a diphthong with the following vowel: [we] and [wi] respectively. It is also used to preserve sound in stem changes and in commands. Semi-Vowels. At the end of a word or when it means "and" ("y") it is pronounced like i. Acute accents. Spanish uses the ´ (Acute) diacritic mark over vowels to indicate a vocal stress on a word that would normally be stressed on another syllable; Stress is contrastive. For example, the word "ánimo" is normally stressed on "a", meaning "mood, spirit." While "animo" is stressed on "ni" meaning "I cheer." And "animó" is stressed on "mó" meaning "he cheered." Additionally the acute mark is used to disambiguate certain words which would otherwise be homographs. It's used in various question word or relative pronoun pairs such as "cómo" (how?)& "como" (as), "dónde"(where?) & "donde" (where), and some other words such as "tú" (you) & "tu" (your), "él" (he/him) & "el" (the). Emphasis. The rules of stress in Spanish are: 1. When the word ends in a vowel or in "n" or "s" the emphasis falls on the second to last syllable. Eg: Mañana, Como, Dedos, Hablan. 2. When the word ends in a consonant other than "n" or "s", the emphasis falls on the last syllable. Eg: Ciudad, Comer, Reptil. 3. If the above two rules don't apply, there will be an accent to show the stress. Eg: Fíjate, Inglés, Teléfono. 4. SPECIAL CASE: Adverbs ending in "-mente", which are derived from adjectives, have two stresses. The first stress occurs in the "adjective part" of the adverb, on the syllable where the adjective would normally be stressed. The second stress occurs on the "-men-" syllable. Eg: Solamente, Felizmente, Cortésmente.

Spanish/Lesson 2. Díalogo 1: El salón de clase. "La profesora entra en el salón." "Profesora": Buenos días alumnos. Hoy estudiamos los objetos en el salón de clase. ¿Cuántos libros hay? "Carlos": Hay catorce estudiantes. Hay catorce libros. "Profesora": Bien Carlos. Marianela, ¿Hay un mapa en el salón? "Marianela": Sí, señora. Hay un mapa a la izquierda de la pizarra. Está en la pared. "Profesora": Bien, Marianela. ¿De qué color es el mapa? "Carlos": El mapa es verde y azul. Es el mapa de México. "Marianela": ¿Profesora, porqué está el mapa en la pared y no está en la pizarra? "Profesora": La pizarra es para escribir con la tiza. Tengo un borrador aquí. ¿Quieres escribir la capital de México en la pizarra? "Marianela": No sé la capital de México. "Carlos": Marianela, mira el mapa. "Marianela": Ay, gracias Carlos. La capital es Ciudad de México. "Profesora": Bien. Vocabulario: El salón de clase. All vocabulary will be given with the appropriate definite article; however, the article will not be translated. In addition, note that all nouns have a gender: they are either masculine or feminine. This will be important in later lessons when you begin to learn adjectives. Grammar: The Definite Article. Like in English, Spanish has definite articles that serve to identify the location of the noun in the sentence. These are commonly called "noun markers". These articles have a gender that is equivalent to the gender of the noun they modify. However, sometimes pronunciation requires the article to be different than the gender of the word. In these situations, the gender of the noun will always be indicated. Otherwise, you can use the article to determine the gender of the noun. <br> Examples: The following examples come from the vocabulary of the first dialogue. Note that no matter whether the word is singular or plural, the meaning of the definite article does not change. <br> The Indefinite Article. In English the indefinite articles are "a" and "an" (singular) or "some" (plural). In Spanish there are different forms for masculine-gender, feminine-gender, singular or plural. <br> Examples: <br> <br> For phonetic reasons some words beginning with accented a may have the article un: un ave blanca "(a white bird)," las aves blancas "(the white birds)". This is basically the same idea as el ave blanca "(the white bird)". Remember, do not confuse uno "(one)" with un "(a or an)".<br> Also, do not confuse una "(a or an)" with uña "(fingernail)". Exercise: Spanish/Exercises/Articles The colors. As in English, colors in Spanish are adjectives. As adjectives, their endings will vary according to the nouns they modify. The following adjectives are in the masculine singular form. Most adjectives will be presented to you in this form. <br> Examples: <br> <br> Los números (Numbers). The numbers 16 through 29 can be formed in two ways: (1) using the conjunction "y" like veinte y uno - 21, or (2) as one complete word, changing the spelling of the word like dieciocho (18) or veintiséis (26).

Algebra/Arithmetic/Numerical Axioms. Numerical Axioms. It is possible to define a regular set of numbers in a formal fashion. The set of "Peano axioms" define the series of numbers known as the natural numbers. They are as follows: Let us attempt to motivate these axioms. We want these axioms to eliminate any set which is not the natural numbers. E.g., any set fulfilling the above should at least be infinite. The first two are obvious properties of the natural numbers (and of integers) as we know them. Note that some prefer to use 1 as the lowest number. The reason for choosing zero has root in [set theory], in which the first natural number is chosen as the empty set formula_1. The 3rd axiom prevents circularity. If this axiom was not included, defining formula_2 would trivially fulfill the remaining axioms --- prove this for yourself by considering each remaining axiom! The 4th prevents a partial loop. Consider a the set formula_3 and set formula_4 and formula_5. This set fulfills every axiom but the 4th --- prove this for yourself. The 5th is sometimes called the induction axiom. It ensures that the set is "connected", i.e. that we can reach any number by using the 2nd axiom repeatedly on 0. An example of a set that fulfills every axiom but the 5th is formula_6 with the usual meaning of +1. From this we can deduce the existence of a series of quantities like this: where '0' is a constant and the first natural number and '1' is a constant natural number equivalent to the difference in value between two consecutive natural numbers. This set is sufficient for counting. However, it is inconvenient to refer to a large natural number as '0' followed by the requisite large number of '+ 1' expressions. Due to this, each of the natural numbers is given a label, and to make the labelling easier another axiom is introduced: '0 + 1' is equivalent to '1'. Thus the series of natural numbers may be written so for some brevity: Once this is done, giving each quantity its own label is trivial. And so the series of natural numbers can then be written:

Discrete Mathematics. formula_1 formula_2 Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Contents. __NOEDITSECTION__

Discrete Mathematics/Introduction. The subject of mathematics is commited to rigorous reasoning. This book aims to raise your confidence in the manipulation and interpretation of formal notations, as well as to train you to solve problems that are fundamentally discrete: problems like chess, in which the moves you make are exact; problems where fields like calculus fail because there's no continuity; problems that appear frequently in games, puzzles, and computer science. We hope you'll enjoy discovering discrete mathematics here, and we hope you'll find this a good reference for quickly picking up the details you may forget.

Lojban. "This is currently a pre-draft textbook. Therefore some information is inaccurate." Coi! This book will help you learn Lojban. Old Table Of Contents. Appendices

Lojban/Sounds and alphabet. Lojban has a 24-letter alphabet (23 of which come from the Roman alphabet):<br> <br> ' a b c d e f g i j k l m n o p r s t u v x y z<br> <br> And two special symbols:<br> <br> , .<br> <br> This includes most of the traditional English alphabet except for three letters:<br> <br> h q w<br> <br> Lojban has a phonetic spelling and audio-visual isomorphism, meaning that each letter has exactly one sound and each sound has exactly one letter.<br> <br> ' only appears in Lojban words between vowels. "," only appears in Lojban names. Lojbanists can optionally not include the "." in their writing. The stress is on a word's penultimate syllable. The exception is that a syllable whose nucleus is a 'y' cannot be stressed; in that case, stress the syllable immediately preceding it. Names may indicate different from penultimate stress by capitalizing the entire stressed syllable or the vowel of the stressed syllable. Some Lojbanists have unofficially proposed other characters to mark stress like "^".<br> <br> Lojban has eleven diphthongs written by composed vowels. The following transliterations are assuming that the reader has an American English accent.<br> A pair of adjacent vowels which are capable of forming a diphthong "will" form a diphthong unless a comma is placed between them. That is the purpose of the comma in Lojban; to act as a separator between syllables whenever necessary for pronunciation purposes. (It is not for pausing between words or phrases like in English.) To make pronouncing some words easier, you may put unwritten buffer vowels in your speech between consonants. Any non-Lojban vowel can act as a buffer vowel, but i as in h"i"t has the most popularity. For example, you could pronounce (but not write) a buffer vowel between the "s" and "f" of "sfani".<br> <br> Exercises. Write/Stress Answer True or False Answers to exercises. Pronounce Write/Stress Answer True or False

Lojban/Cmene. In Lojban names are called "cmene" (pronouced shmen-eh, IPA /ʃme.ne/), and are transliterated to use only Lojban sounds. In Lojban all words end in a vowel, and so to avoid any possible confusion all cmene end in a consonant; if a transliterated name would end in a vowel, it is usual to add an s to the end. All cmene are followed by a full stop, so confusion to where the name stops and the next word starts it avoided; for this reason, any names beginning in a vowel have a full stop in front of them. To indicate an accent somewhere other than the second-to-last syllable, the entire syllable to be accented must be capitalised. A comma should be used to stop two vowels next to each other that should not be pronouced as a dipthong. All names must have the word 'la', meaning 'the', placed before them. A few examples are below: Exercises. English names to Cmene. Translate these names to Lojban cmene. Cmene to English names. Translate these Lojban cmene to English names. Answers to Exercises. No one pronouciation of a name is the right one, so use common sense whe deciding if you got these right or not.

Spanish/Lesson 4. Stem Changing Verbs. In Spanish, some verbs change their stems when they are conjugated. These verbs are known as "stem-changing verbs". Many of these verbs are important and often used. There are three different types of stem changing verbs in Spanish: The stem changes for all conjugations, excepting nosotros/as and vosotros/as. The endings are the same as for regular verbs (-o for yo, -as/-es for tú, ...). <br> Entender<br>To Understand Note that the stem change is done for the second 'e' (not the first one) - in general the stem always changes for the last vowel before the -ar/-er/-ir ending. Example: pedir (e->i) "to ask for, to order" Pedir<br>To Ask For, Order Note: all e->i stem changing verbs are -ir verbs. Dormir<br>To Sleep Here is a list of some other common stem changing verbs: Exercise: Stem Changing Verbs Present Participle. The present participle in Spanish is used either for continuous tenses (with "estar", e.g. "I am running.") and can also be used as an adjective. The Spanish present participle corresponds to the English -ing form of the verb.To form the present participle for regular -ar verbs, add -ando to the stem. For -er and -ir verbs add -iendo: <br> However, not all present participles are that regular. Some verbs add a "y," or change the spelling, to adhere to Spanish orthography (spelling) rules. Here is a list of some common verbs that have an irregular present participle: Present Progressive. Like in English, the Spanish present progressive is used to describe an "action in progress". It is formed by conjugating the verb estar and then adding the present participle:

Spanish/Lesson 3. Text. Here are a couple of sentences and short dialogs about people planning/doing leisure activities. Besides the new vocabulary you should also have a look at how the verbs are conjugated depending on the subject of the sentence. As you may see, each verb is bolded. These verbs are "conjugated", that is, changed by the person(s) to which they are referring. Notice that subject pronouns are "not" necessary. Regular Verbs. Spanish has three different types of regular verbs: -ar, -er, and -ir verbs. The subject pronoun is not necessary and in conversational Spanish it is only used for emphasis. For this lesson, we will omit it. One can still use pronouns, however. The conjugation pattern is the following: </br> As one can see, the endings for each person are different. This is similar to other Romance languages, such as Portuguese and Italian (the notable exception is French). This is the reason why we may omit the pronouns while we speak. Remember that sometimes it is best to clarify whether él, ella, or usted is speaking, because they share the same form. However, the context of the rest of the sentence sometimes clarifies this. There are a few steps involved with conjugating a verb. Here are the steps involved: Notice that there are only two differences between the conjugations of -er and -ir verbs. The "nosotros" (4) and "vosotros" (5) forms are the only differences. Those forms have an "i" in the stem instead of an "e." Exercise: Regular Verbs "G" Verbs. The verb "hacer" means to do or to make. Hacer is irregular in the first person singular form "(I)" of the present tense only. The irregular form is hago. Hacer is one of the many verbs in Spanish which add a "g" in the first person singular of the verb. This is the present indicative of the verb hacer. Note that the verb hacer is translated as "to do" and "to make" when referring to activities. But it can also be used to talk about some weather conditions: But: When speaking about the weather using hacer, the Ud. form (third singular form) is always used. El vocabulario (Vocabulary) - Los días (Days). Audio: (157KB) Una fiesta. Una fiesta entre amigos. Nosotros bailamos y lo pasamos bien en el jardín de esta casa.<br> "A party among friends. We dance and enjoy ourselves in the patio (garden) of this house."

Lojban/Sapir-Whorf Hypothesis. The Sapir-Whorf Hypothesis (SWH) is a hypothesis in linguistics, stating that there are notable differences in thought patterns of speakers of different languages, and that the way people's brains function is strongly affected by their native languages. It's a very controversial theory, championed by linguist Edward Sapir and his student Benjamin Whorf. First discussed by Sapir in 1929, the hypothesis became popular in the 1950s following posthumous publication of Whorf's writings on the subject. In 1955, Dr. James Cooke Brown created the Loglan language (which led to the offshoot Lojban) in order to test the hypothesis. After vigorous attack from followers of Noam Chomsky in the following decades, the hypothesis is now only believed by linguists with a grain of salt; that thought processes are somewhat affected by language, but that differences aren't that notable. Central to the Sapir-Whorf hypothesis is the idea of linguistic relativity: distinctions of meaning between related terms in a language are often arbitrary and particular to that language. Sapir and Whorf took this one step further by arguing that a person's whole world view is determined by the vocabulary and syntax available in his or her language. The extreme ("Weltanschauung") version of this idea, that all mental function is constrained by language, can be disproved through personal experience: people in every language occasionally struggle to express their exact thoughts, feeling constrained by the language. It's common to say or write something, only to correct one's self or further clarify meaning, especially to someone being explained to. These show that ideas are not merely words, because one can imagine something without being able to express it in words. The opposite extreme — that language does not influence thought at all — is also widely considered to be false. For example, it has been shown in studies that people's discrimination of similar colors can be influenced by their vocabulary for distinguishing said colors. Another study showed that deaf children of hearing parents are more likely to fail on some cognitive tasks unrelated to hearing, while deaf children of deaf parents succeed, due to parents being able to more extensively communicate. Computer programmers who know different programming languages often see the same problem in completely different ways. The Neuro-Linguistic Programming (NLP) analysis of the problem is direct: most people do notable thinking by talking to themselves and by imagining images and other sensory phantasms. To the extent that people think by talking to themselves, they are limited by their vocabulary and the structure of their language and linguistic habits. (However it should also be noted that everyone have idiolects, mental language patterns individual to them.) John Grinder, a founder of NLP, was a linguistics professor who perhaps unconsciously combined the ideas of Chomsky with the Sapir-Whorf hypothesis. A seminal NLP insight came from a challenge he gave to his students: coin a neologism to describe an idea for which you have no words. Student Robert Dilts gave an example by coining a word for the way people stare into space when they are thinking, and for the different directions they stare. These new words enabled users to describe patterns in the ways people stare into space, which led directly to NLP results — as notable a validation of the weak hypothesis as one could ask. Lojban is thus designed to test the Sapir-Whorf hypothesis, by attempting to expand the speakers' minds and express thoughts as conveniently as possible to see if there's any notable effects in the speakers' thought patterns/worldview. The only complication with this is that the third factor — the speakers all wanting to learn Lojban, an obscure language — could skew the results somewhat, but the only way to fix that is to get more Lojban speakers. External links. "This article was originally taken from Wikipedia and is licensed under the GNU FDL. Update as needed."

Lojban/Introduction to Lojban. Loglan. In 1955, sociologist from Philippines Dr. James Cooke Brown began work on Loglan, a constructed language designed for linguistic research, particularly investigation of the Sapir-Whorf hypothesis, a theory stating that linguistic structures affect people's thoughts. The object was to make a language so powerfully expressive for logic and calculation that people learning it would become measurably smarter if the hypothesis was true. He intended it to be as culturally neutral as possible, logically and linguistically powerful, incorporating all known expressive features of any language (e.g., compounded location tenses), metaphysically parsimonious (e.g., you are not required to express any feature of reality, as you are in English time-tenses of verbs), and totally regular and unambiguous. He even used maximally stable phonemes. The formal grammar was disambiguated mechanically (at first). The language's grammar is based on predicate logic (the name Loglan is short for "logical language"), which also makes it particularly suitable for human-computer communication, an application that led Robert Heinlein to mention the language in his novel The Moon is a Harsh Mistress. Dr. Brown founded The Loglan Institute to develop the language and other applications of it. He always considered the language a research project that was never complete, so although he released many papers about its design he never "released" it to become a usable language. A group of his followers later formed The Logical Language Group to create the language Lojban along the same principles, but with the intention to make it freely available and encourage its use as a real language. This latter group has a small but active community of speakers. Loglan allows very strange speech, because one can say things that are simply not meaningful. For example, you can literally say that John, a person, is a short word. Or one can directly and precisely say any of the many possible meanings of the English phrase "a pretty little girls school." In natural languages, the ambiguity of the grammar hides these odd meanings; but in Loglan they are all available and add expressive freedom to the language. This feature is so pronounced that people fluent in Loglan say impossible things as a sort of joke — a type of humour simply not supported by the linguistic machinery of natural languages. The oddest, most difficult thing for a speaker of an Indo-European language like English is that Loglan has no distinction between nouns or verbs, objects, direct objects, indirect objects, possessive forms or tenses. There are only predicates, with places for variables: for example, "botso": X buys item Y from seller P for price Q. There are prefixes to reorder predicates; for example, price would be "botso" with a little word to make price the first variable. With different reorderings, one speaks of buyer, bought-thing, or seller. Tenses for time, location, actor, type of action, etc. are provided by "little words" which are optional. Predicates compound, so a predicate can fit in the variable of another predicate. Every feature of the language has standard, regular forms for acting in compounds. For example, time-travel tenses are available trivially in Loglan (I did X from time Y to P in time Q.) using compounding forms normally used for location tenses. After long use of Loglan, the world gains a sort of timeless, objectless, actorless flavor. Time words and location words fall away except when needed to make a point, usually with emotional emphasis. It is rather easy to avoid blame for responsibilities in Loglan, and scheduling may be creatively ambiguous because the tenses are optional. The language is designed so that the patterns of phonemes always parse into words. Thus, one cannot babble Loglan, because even when run together, the language is still parsable. Lojban. The constructed language Lojban (SAMPA ['loZban]) was created by the Logical Language Group in 1987 based on the earlier Loglan, with the intent to make the language more complete, usable, and freely available. The language itself shares many of the features and goals of Loglan; in particular: While the initial goal of the Loglan project was to investigate the Sapir-Whorf hypothesis, the active Lojban community has additional goals for the language, including: Lojban Grammar. All words in Lojban belong to one of three overall categories: "brivla", for both common nouns and verbs; "cmene", for proper nouns; "cmavo", for structural particles: articles, numerals, tense indicators and other such modifiers. The "cmavo" are further subdivided into "selma'o", which are closer to the notion of parts of speech (e.g., UI includes interjections and discursives). There is no distinct class of words for adjectives or adverbs, unlike most Indo-European languages. Most types of noun phrases are always preceded by an article; there are different articles to indicate whether it is being treated as an individual, mass, set, or typical element. "Brivla" do not inflect for tense, person, or number; tense may optionally be indicated by separate "cmavo", and number may optionally be indicated in various ways. As befits a logical language, there is a large assortment of conjunctions. Logical conjunctions take different forms depending on whether they connect "sumti" (the equivalent of noun phrases), "selbri" (phrases that can serve as verbs; all "brivla" are "selbri"), parts of a "tanru" (a construct whose closest English equivalent is a string of nouns), or clauses in a sentence. The typology is Subject Verb Object, with Subject Object Verb also common. Word formation is polysynthetic; many "brivla" (all of which, except for a handful of borrowings such as "alga", have at least five letters) have one to three three-letter forms called "rafsi" which are used in making longer words. For example, "gasnu" means "to make something happen"; its "rafsi" form of "-gau" regularly forms compounds meaning "to cause...x", in which the agent is in the subject place of the new predicate. Lojban has a positional case system, though this can be overridden by marking predicate arguments with explicit case particles. For instance, "bramau" means "is bigger than"; the bigger thing is in first position, and the smaller is second, and the measured property in the third. So "mi bramau do le ka clani" means "I am bigger than you in the property of height" or "I am taller than you"; but this could also be expressed as something like "fi le ka clani fe do fa mi bramau", "In height, you are exceeded by me". What a particular place means depends entirely on the "brivla". For animals and plants the second place is the species, variety, breed, or other taxon; for verbs of measurement it is the numerical measurement, and a further place is the standard; for "klama" ("go" / "come") it is the destination. There may be up to five places for some "brivla". Something of the flavor of Lojban (and Loglan) can be imparted by this lightbulb joke: Q: How many Lojbanists does it take to change a broken light bulb?<br> A: Two: one to decide what to change it into, and one to figure out what kind of bulb emits broken light. This makes use of two features of the language; first, the language attempts to eliminate polysemy, that is, having a word with more than one meaning. So while the English word "change" can mean "to transform into a different state", or "to replace", or even "small-denomination currency", Lojban has different words for each. In particular, the use of a "brivla" such as the word for "change" ("binxo") implies that all of its predicate places exist, so there must be something for it to change into. Another feature of the language is that it has no grammatical ambiguities such as appear in English phrases like "big dog catcher", which can mean either a big person who catches dogs or a person who catches big dogs. In Lojban, unless you clearly specify otherwise with "cmavo", such modifiers always group left-to-right, so "big dog catcher" is a catcher of big dogs, and a "broken light bulb" is a bulb that emits broken light (you can also avoid the ambiguity by creating a new word, so "broken lightbulb" has the intended meaning).

Lojban/Attitudinals. Attitudinals are markers of attitude or emotion, modifying the word (brivla, cmavo, cmene, sumti, or tanru) directly before it. For example, the simple « "mlatu" » for "cat" can become « "mlatu .ui" » for "cat (happiness!/:D)". At the beginning of a sentence (after the place of .i) they modify the entire sentence (bridi), and after the attitudinal beginning bracket « "fu'e" », they continue the emotion for all the statement until the closing bracket « "fu'o" ». Attitudinals can also affect vocatives, the same way a change in vocal tone can greatly affect a simple "hello". For example, « "coi .ui" » means not a simple "Hello.", but something like "Hello! :D". Combined with the attitudinal question-maker "pei", you can easily create more complex « "coi .uipei" » may be thought of as "Hello - are you glad to see me?", while « "co'o .uipei" » may be viewed as "Good-bye - are you happy that we part?". Beyond mere emotion, they can mark prepositional ideas such as desire ("I want this to happen"), obligation ("This needs to happen"), and many other complex functions we take for granted - for example, « "mi citka" » means "I eat", while « ".au mi citka" » means "I want to eat", and « ".e'e mi citka" » means "I am able to eat". Attitudinal indicators and suffixes with examples of modifier use:. Miscellaneous indicators:. See also: Emotions

Lojban/Vocatives. Since {ki'e} means approximately "Thank you," it might seem appropriate to respond by saying {fi'i}, seemingly corresponding to the English "You're welcome." However, {fi'i} means "welcome" in the sense of an invitation/acceptance only. In order to respond, {je'e} should be used. "ju'i" can be used as an attention-grabber such as the first word in each of the following utterances (if they started a conversation): "Attention, all shoppers", "So, I just got back from my date with Ryan", "Lo! Listen to the news which I bring". This is the function of the very first word of "Beowulf". It can also return an audience's attention to the speaker/discussion, such as with some usages of "ahem", clearing the throat, or wrapping on a desk. Exercises. In Exercises 1-7, give the most suitable Lojban word(s) to say in each situation. 01. The beginning of a conversation.<br> 02. Begging.<br> 03. During an oath.<br> 04. Inviting a guest into your house.<br> 05. Getting a group of people to listen to your announcement.<br> 06. You desire the floor while someone else has it.<br> 07. Leaving a party. In Exercises 8-43, give the closest Lojban translation of the English sentences. 08. Attention!<br> 09. Listen to me, Victoria.<br> 10. I swear, Benjamin.<br> 11. I don't understand, Christian.<br> 12. I am Pete.<br> 13. Goodbye, Jonathan.<br> 14. Roger, Logan.<br> 15. Wilco.<br> 16. Hello, Rodrigo.<br> 17. Hi, Bertand.<br> 18. Wait! We're not done discussing this!<br> 19. Hark!<br> 20. Agreed.<br> 21. Can I say something, Jose?<br> 22. Just a minute, Jennifer.<br> 23. I'm not done talking, Morgan.<br> 24. Please, Sydney.<br> 25. Over and out.<br> 26. I appreciate that, Chloe.<br> 27. Make yourself at home, Rachel.<br> 28. May I speak, Jasmine?<br> 29. Sorry for interrupting, Sophia, but...<br> 30. I promise, Megan.<br> 31. Thanks, Samuel.<br> 32. Hold on, Nathan.<br> 33. What did you say, Natalie?<br> 34. I interrupt.<br> 35. Greetings, Justin.<br> 36. Uh-huh.<br> 37. Thank you, Dylan.<br> 38. At your service, Austin.<br> 39. I'm listening, Kevin.<br> 40. I understand, Julia.<br> 41. "Houston, we have a problem."<br> 42. Hello! Susan has just left Harold.<br> 43. Hello, Susan! Harold has just left. Answers to exercises. 01. coi<br> 02. pe'u<br> 03. nu'e<br> 04. fi'i<br> 05. ju'i<br> 06. ta'a<br> 07. co'o 08. ju'i<br> 09. ju'i. vektarias.<br> 10. nu'e. bendjamen.<br> 11. je'enai. krestien.<br> 12. mi'e. pit.<br> 13. co'o. janaten.<br> 14. je'e. logen.<br> 15. vi'o<br> 16. coi. radrigos.<br> 17. coi. bertrend.<br> 18. fe'onai<br> 19. ju'i<br> 20. vi'o<br> 21. be'e. xozeis.<br> 22. re'inai. djenefer.<br> 23. mu'onai. morgen.<br> 24. pe'u. sednis.<br> 25. fe'o<br> 26. ki'e. clos.<br> 27. fi'i. reitcel.<br> 28. be'e. djezmen.<br> 29. ta'a. sofias.<br> 30. nu'e. meigen.<br> 31. ki'e. semiul.<br> 32. re'inai. neitan.<br> 33. ke'o. netalis.<br> 34. ta'a<br> 35. coi. jesten.<br> 36. je'e<br> 37. ki'e. delen.<br> 38. fi'i. .asten.<br> 39. re'i. keven.<br> 40. je'e. djulias.<br> 41. doi xustyn. mi'a se nabmi<br> 42. coi do'u la suzyn. la xeirold. puzi cliva<br> 43. coi la suzyn. la xeirold. puzi cliva

Discrete Mathematics/Set theory. Introduction. Set Theory starts very simply: it examines whether an object "belongs", or does "not belong", to a "set" of objects which has been described in some non-ambiguous way. From this simple beginning, an increasingly complex (and useful!) series of ideas can be developed, which lead to notations and techniques with many varied applications. Definition: Set. The present definition of a set may sound very vague. A set can be defined as an unordered collection of entities that are related because they obey a certain rule. 'Entities' may be anything, "literally": numbers, people, shapes, cities, bits of text, ... etc The key fact about the 'rule' they all obey is that it must be "well-defined". In other words, it must describe clearly what the entities obey. If the entities we're talking about are words, for example, a well-defined rule is: X is English A rule which is not well-defined (and therefore couldn't be used to define a set) might be: X is hard to spell Elements. An entity that belongs to a given set is called an element of that set. For example: Set Notation. formula_1 formula_2 formula_4 formula_5 formula_8 formula_9 formula_10 Note that the use of ellipses may cause ambiguities, the set above may be taken as the set of integers individible by 4, for example. Special Sets. The universal set. The set of all the entities in the current context is called the universal set, or simply the universe. It is denoted by formula_15. The context may be a homework exercise, for example, where the Universal set is limited to the particular entities under its consideration. Also, it may be any arbitrary problem, where we clearly know where it is applied. The empty set. The set containing no elements at all is called the null set, or empty set. It is denoted by a pair of empty braces: formula_16 or by the symbol formula_17. It may seem odd to define a set that contains no elements. Bear in mind, however, that one may be looking for solutions to a problem where it isn't clear at the outset whether or not such solutions even exist. If it turns out that there isn't a solution, then the set of solutions is empty. For example: Operations on the empty set. Operations performed on the empty set (as a set of things to be operated upon) can also be confusing. (Such operations are nullary operations.) For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). This may seem odd, since there are no elements of the empty set, so how could it matter whether they are added or multiplied (since “they” do not exist)? Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, notice that zero is the identity element for addition, and one is the identity element for multiplication. Special numerical sets. Several sets are used so often, they are given special symbols. Natural numbers. Note that, when we write this set by hand, we can't write in bold type so we write an N in blackboard bold font: formula_22 Integers. In blackboard bold, it looks like this: formula_23 Real numbers. If we expand the set of integers to include all decimal numbers, we form the set of real numbers. The set of reals is sometimes denoted by R. A real number may have a "finite" number of digits after the decimal point (e.g. 3.625), or an "infinite" number of decimal digits. In the case of an infinite number of digits, these digits may: In blackboard bold: formula_24 Rational numbers. Those real numbers whose decimal digits are finite in number, or which recur, are called rational numbers. The set of rationals is sometimes denoted by the letter Q. A rational number can always be written as exact fraction "p"/"q"; where "p" and "q" are integers. If "q" equals 1, the fraction is just the integer "p". Note that "q" may NOT equal zero as the value is then undefined. In blackboard bold: formula_25 Irrational numbers. If a number "can't" be represented exactly by a fraction "p"/"q", it is said to be irrational. Set Theory Exercise 1. Click the link for Set Theory Exercise 1 Relationships between Sets. We’ll now look at various ways in which sets may be related to one another. Equality. Two sets formula_11 and formula_12 are said to be equal if and only if they have exactly the same elements. In this case, we simply write: formula_28 Note two further facts about equal sets: So, for example, the following sets are all equal: formula_29 Subsets. If all the elements of a set formula_11 are also elements of a set formula_12, then we say that formula_11 is a subset of formula_12 and we write: formula_35 For example: In the examples below: If formula_36 and formula_37, then formula_38 If formula_39 and formula_40, then formula_41 If formula_42 and formula_43, then formula_44 Notice that formula_35 does not imply that formula_12 must necessarily contain extra elements that are not in formula_11; the two sets could be equal – as indeed formula_48 and formula_49 are above. However, if, in addition, formula_12 does contain at least one element that isn’t in formula_11, then we say that formula_11 is a proper subset of formula_12. In such a case we would write: formula_54 In the examples above: formula_55 contains ... -4, -2, 0, 2, 4, 6, 8, 10, 12, 14, ... , so formula_56 formula_57 contains $, ;, &, ..., so formula_58 But formula_48 and formula_49 are just different ways of saying the same thing, so formula_61 The use of formula_62 and formula_63; is clearly analogous to the use of < and ≤ when comparing two numbers. Notice also that "every" set is a subset of the "universal set", and the "empty set" is a subset of "every" set. Finally, note that if formula_68 and formula_12 must contain exactly the same elements, and are therefore equal. In other words: formula_70 Disjoint. Two sets are said to be disjoint if they have no elements in common. For example: If formula_71 and formula_72, then formula_11 and formula_12 are disjoint sets Venn Diagrams. A "Venn diagram" can be a useful way of illustrating relationships between sets. In a Venn diagram: On the left, the sets "A" and "B" are disjoint, because the loops don't overlap. On the right "A" is a subset of "B", because the loop representing set "A" is entirely enclosed by loop "B". <br clear="all"> Venn diagrams: Worked Examples. "Example 1" "Fig. 3" represents a Venn diagram showing two sets "A" and "B", in the general case where nothing is known about any relationships between the sets. Note that the rectangle representing the universal set is divided into four regions, labelled "i", "ii", "iii" and "iv". What can be said about the sets "A" and "B" if it turns out that: <br clear="all"> (a) If region "ii" is empty, then "A" contains no elements that are not in "B". So "A" is a subset of "B", and the diagram should be re-drawn like "Fig 2" above. (b) If region "iii" is empty, then "A" and "B" have no elements in common and are therefore disjoint. The diagram should then be re-drawn like "Fig 1" above. "Example 2" (a) The diagram in "Fig. 4" shows the general case of three sets where nothing is known about any possible relationships between them. (b) The rectangle representing U is now divided into 8 regions, indicated by the Roman numerals "i" to "viii". (c) Various combinations of empty regions are possible. In each case, the Venn diagram can be re-drawn so that empty regions are no longer included. For example: "Example 3" The following sets are defined: Using the two-stage technique described below, draw a Venn diagram to represent these sets, marking all the elements in the appropriate regions. The technique is as follows: Don't begin by entering the elements of set "A", then set "B", then "C" – you'll risk missing elements out or including them twice! "Solution" After drawing the three empty loops in a diagram looking like "Fig. 4" (but without the Roman numerals!), go through each of the ten elements in U - the numbers 1 to 10 - asking each one three questions; like this: First element: 1 A 'no' to all three questions means that the number 1 is outside all three loops. So write it in the appropriate region (region number "i" in "Fig. 4"). Second element: 2 Yes, yes, no: so the number 2 is inside "A" and "B" but outside "C". Goes in region "iii" then. ...and so on, with elements 3 to 10. The resulting diagram looks like "Fig. 5". <br clear="all"> The final stage is to examine the diagram for empty regions - in this case the regions we called "iv", "vi" and "vii" in "Fig. 4" - and then re-draw the diagram to eliminate these regions. When we've done so, we shall clearly see the relationships between the three sets. So we need to: The finished result is shown in "Fig. 6". The regions in a Venn Diagram and Truth Tables. Perhaps you've realized that adding an additional set to a Venn diagram "doubles" the number of regions into which the rectangle representing the universal set is divided. This gives us a very simple pattern, as follows: It's not hard to see why this should be so. Each new loop we add to the diagram divides each existing region into two, thus doubling the number of regions altogether. But there's another way of looking at this, and it's this. In the solution to "Example 3" above, we asked three questions of each element: "Are you in A? Are you in B?" and "Are you in C?" Now there are obviously two possible answers to each of these questions: "yes" and "no". When we "combine" the answers to three questions like this, one after the other, there are then 23 = 8 possible sets of answers altogether. Each of these eight possible combinations of answers corresponds to a different region on the Venn diagram. The complete set of answers resembles very closely a "Truth Table" - an important concept in "Logic", which deals with statements which may be "true" or "false". The table on the right shows the eight possible combinations of answers for 3 sets "A", "B" and "C". You'll find it helpful to study the patterns of Y's and N's in each column. Set Theory Exercise 2. Click link for Set Theory Exercise 2 Operations on Sets. Just as we can combine two numbers to form a third number, with operations like 'add', 'subtract', 'multiply' and 'divide', so we can combine two sets to form a third set in various ways. We'll begin by looking again at the Venn diagram which shows two sets "A" and "B" in a general position, where we don't have any information about how they may be related. <br clear="all"> The first two columns in the table on the right show the four sets of possible answers to the questions "Are you in A?" and "Are you in B?" for two sets "A" and "B"; the Roman numerals in the third column show the corresponding region in the Venn diagram in "Fig. 7". Intersection. Region "iii", where the two loops overlap (the region corresponding to 'Y' followed by 'Y'), is called the "intersection" of the sets "A" and "B". It is denoted by "A" ∩ "B". So we can define intersection as follows: For example, if "A" = {1, 2, 3, 4} and "B" = {2, 4, 6, 8}, then "A" ∩ "B" = {2, 4}. We can say, then, that we have combined two sets to form a third set using the "operation of intersection". Union. In a similar way we can define the "union" of two sets as follows: The union, then, is represented by regions "ii", "iii" and "iv" in "Fig. 7". You'll see, then, that in order to get into the intersection, an element must answer 'Yes' to "both" questions, whereas to get into the union, "either" answer may be 'Yes'. The ∪ symbol looks like the first letter of 'Union' and like a cup that will hold a lot of items. The ∩ symbol looks like a spilled cup that won't hold a lot of items, or possibly the letter 'n', for i'n'tersection. Take care not to confuse the two. Difference. This is written "A" - "B", or sometimes "A" \ "B". The elements in the difference, then, are the ones that answer 'Yes' to the first question "Are you in A?", but 'No' to the second "Are you in B?". This combination of answers is on row 2 of the above table, and corresponds to region "ii" in "Fig.7". Complement. So far, we have considered operations in which "two" sets combine to form a third: "binary" operations. Now we look at a "unary" operation - one that involves just "one" set. Clearly, this is the set of elements that answer 'No' to the question "Are you in A?". Notice the spelling of the word "complement": its literal meaning is 'a complementary item or items'; in other words, 'that which completes'. So if we already have the elements of "A", the complement of "A" is the set that "completes" the universal set. Summary. <br clear="all"> Cardinality. Finally, in this section on "Set Operations" we look at an operation on a set that yields not another set, but an integer. Generalized set operations. If we want to denote the intersection or union of "n" sets, "A"1, "A"2, ..., "A""n" (where we may not know the value of "n") then the following "generalized set notation" may be useful: In the symbol formula_90 "A""i", then, "i" is a variable that takes values from 1 to "n", to indicate the repeated intersection of all the sets "A"1 to "A""n". Set Theory Exercise 3. Click link for Set Theory Exercise 3

Biochemistry/Introduction. Intro: What Is Biochemistry? Biochemistry is the study of the chemistry of, and relating to, biological organisms. It forms a bridge between biology and chemistry by studying how complex chemical reactions and chemical structures give rise to life and life's processes. Biochemistry is sometimes viewed as a hybrid branch of organic chemistry which specializes in the chemical processes and chemical transformations that take place inside of living organisms, but the truth is that the study of biochemistry should generally be considered neither fully "biology" nor fully "chemistry" in nature. Biochemistry incorporates everything in size between a molecule and a cell and all the interactions between them. The aim of biochemists is to describe in molecular terms the structures, mechanisms and chemical processes shared by all organisms, providing organizing principles that underlie life in all its diverse forms. Biochemistry essentially remains the study of the structure and function of cellular components (such as enzymes and cellular organelles) and the processes carried out both on and by organic macromolecules - especially proteins, but also carbohydrates, lipids, nucleic acids, and other biomolecules. All life forms alive today are generally believed to have descended from a single proto-biotic ancestor, which could explain why all known living things naturally have similar biochemistries. Even when it comes to matters which could appear to be arbitrary - such as the genetic code and meanings of codons, or the "handedness" of various biomolecules - it is irrefutable fact that all marine and terrestrial living things demonstrate certain unchanging patterns throughout every level of organization, from family and phylum to kingdom and clade. Biochemistry is, most simply put, the chemistry of life.

Biochemistry/Catalysis. Catalysis refers to the acceleration of the rate of a chemical reaction by a substance, called a catalyst, that is itself unchanged by the overall reaction. Catalysis is crucial for any known form of life, as it makes chemical reactions happen much faster than they would "by themselves", sometimes by a factor of several million times. A common misunderstanding is that catalysis "makes the reaction happen", that the reaction would not otherwise proceed without the presence of the catalyst. However, a catalyst cannot make a thermodynamically unfavorable reaction proceed. Rather, it can only speed up a reaction that is already thermodynamically favorable. Such a reaction in the absence of a catalyst would proceed, even without the catalyst, although perhaps too slowly to be observed or to be useful in a given context. Catalysts accelerate the chemical reaction by providing a lower energy pathway between the reactants and the products. This usually involves the formation of an intermediate, which cannot be formed without the catalyst. The formation of this intermediate and subsequent reaction generally has a much lower activation energy barrier than is required for the direct reaction of reactants to products. Catalysis is a very important process from an industrial point of view since the production of most industrially important chemicals involve catalysis. Research into catalysis is a major field in applied science, and involves many fields of chemistry and physics. Two types of catalysis are generally distinguished. In homogeneous catalysis the reactants and catalyst are in the same phase. For example acids (H+ ion donors) are common catalysts in many aqueous reactions. In this case both the reactants and the catalysts are in the aqueous phase. In heterogeneous catalysis the catalyst is in a different phase than the reactants and products. Usually, the catalyst is a solid and the reactants and products are gases or liquids. In order for the reaction to occur one or more of the reactants must diffuse to the catalyst surface and adsorb onto it. After reaction, the products must desorb from the surface and diffuse away from the solid surface. Frequently, this transport of reactants and products from one phase to another plays a dominant role in limiting the rate of reaction. Understanding these transport phenomena is an important area of heterogeneous catalyst research. Enzymes. Enzyme (from Greek, "in ferment") are special protein molecules whose function is to facilitate or otherwise accelerate most chemical reactions in cells. They are simply biological catalysts. Most enzymes are proteins, although a few are catalytic RNA molecules called ribozymes. Many chemical reactions occur within biological cells, but without catalysts most of them happen too slowly in the test tube to be biologically relevant. Enzymes can also serve to couple two or more reactions together, so that a thermodynamically favorable reaction can be used to "drive" a thermodynamically unfavorable one. One of the most common examples is enzymes which use the dephosphorylation of ATP to drive some otherwise unrelated chemical reaction. Chemical reactions need a certain amount of activation energy to take place. Enzymes can increase the reaction speed by favoring or enabling a different reaction path with a lower activation energy (Fig. 1), making it easier for the reaction to occur. Enzymes are large globular proteins that catalyze (accelerate) chemical reactions. They are essential for the function of cells. Enzymes are very specific as to the reactions they catalyze and the chemicals (substrates) that are involved in the reactions. Substrates fit their enzymes like a key fits its lock (Fig. 2). Many enzymes are composed of several proteins that act together as a unit. Most parts of an enzyme have regulatory or structural purposes. The catalyzed reaction takes place in only a small part of the enzyme called the active site, which is made up of approximately 2 - 20 amino acids. The substrates (A and B) need a large amount of energy ("E"1) to reach the intermediate state A...B, which then reacts to form the end product (AB). The enzyme (E) creates a microenvironment in which A and B can reach the intermediate state (A...E...B) more easily, reducing the amount of energy needed ("E"2). As a result, the reaction is more likely to take place, thus improving the reaction speed. Enzymes can perform up to several million catalytic reactions per second. To determine the maximum speed of an enzymatic reaction, the substrate concentration is increased until a constant rate of product formation is achieved (Fig. 3). This is the maximum velocity ("V"max) of the enzyme. In this state, all enzyme active sites are saturated with substrate. This was proposed in 1913 by Leonor Michaelis and Maud Menten. Since the substrate concentration at Vmax cannot be measured exactly, enzymes are characterized by the substrate concentration at which the rate of reaction is half its maximum. This substrate concentration is called the Michaelis-Menten constant ("K"M). Many enzymes obey Michaelis-Menten kinetics. The speed "V" means the number of reactions per second that are catalyzed by an enzyme. With increasing substrate concentration [S], the enzyme is asymptotically approaching its maximum speed "V"max, but never actually reaching it. Because of that, no [S] for "V"max can be given. Instead, the characteristic value for the enzyme is defined by the substrate concentration at its half-maximum speed ("V"max"/2"). This KM value is also called Michaelis-Menten constant. Several factors can influence the reaction speed, catalytic activity, and specificity of an enzyme. Besides de novo synthesis (the production of more enzyme molecules to increase catalysis rates), properties such as pH or temperature can denature an enzyme (alter its shape) so that it can no longer function. More specific regulation is possible by posttranslational modification (e.g., phosphorylation) of the enzyme or by adding cofactors like metal ions or organic molecules (e.g., NAD+, FAD, CoA, or vitamins) that interact with the enzyme. Allosteric enzymes are composed of several subunits (proteins) that interact with each other and thus influence each other's catalytic activity. Enzymes can also be regulated by competitive inhibitors (Fig. 4) and non-competitive inhibitors and activators (Fig. 5). Inhibitors and activators are often used as medicines, but they can also be poisonous. A competitive inhibitor fits the enzyme as well as its real substrate, sometimes even better. The inhibitor takes the place of the substrate in the active center, but cannot undergo the catalytic reaction, thus inhibiting the enzyme from binding with a substrate molecule. Some inhibitors form covalent bonds with the enzyme, deactivating it permanently (suicide inhibitors). In terms of the kinetics of a competitive inhibitor, it will increase Km but leave Vmax unchanged. Non-competitive inhibitors/activators (I) do not bind to the active center, but to other parts of the enzyme (E) that can be far away from the substrate (S) binding site. By changing the conformation (the three-dimensional structure) of the enzyme (E), they disable or enable the ability of the enzyme (E) to bind its substrate (S) and catalyze the desired reaction. The noncompetitive inhibitor will lower Vmax but leave Km unchanged. An "uncompetitive" inhibitor will only bind to the enzyme-substrate complex forming an enzyme-substrate-inhibitor (ESI) complex and cannot be overcome by additional substrate. Since the ESI is nonreactive, Vmax is effectively lowered. The uncompetitive inhibitor will in turn lower the Km due to a lower concentration of substrate needed to achieve half the maximum concentration of ES. Several enzymes can work together in a specific order, creating metabolic pathways (e.g., the citric acid cycle, a series of enzymatic reactions in the cells of aerobic organisms, important in cellular respiration). In a metabolic pathway, one enzyme takes the product of another enzyme as a substrate. After the catalytic reaction, the product is then passed on to another enzyme. The end product(s) of such a pathway are often non-competitive inhibitors (Fig. 5) for one of the first enzymes of the pathway (usually the first irreversible step, called "committed step"), thus regulating the amount of end product made by the pathway (Fig. 6). Enzymes are essential to living organisms, and a malfunction of even a single enzyme out of approximately 2,000 present in our bodies can lead to severe or lethal illness. An example of a disease caused by an enzyme malfunction in humans is phenylketonuria (PKU). The enzyme phenylalanine hydroxylase, which usually converts the essential amino acid phenylalanine into tyrosine does not work, resulting in a buildup of phenylalanine that leads to mental retardation. Enzymes in the human body can also be influenced by inhibitors in good or bad ways. Aspirin, for example, inhibits an enzyme that produces prostaglandins (inflammation messengers), thus suppressing pain. But not all enzymes are in living things. Enzymes are also used in everyday products such as biological washing detergents where they speed up chemical reactions, (to get your clothes clean). Digestive and Metabolic Enzymes. In the previous section we have been talking about the digestive enzymes, both the ones produced by the body, such as salivary amylase, and the food enzymes. Their primary role is for the digestion of food. Another class of enzymes is called metabolic enzymes. Their role is to catalyze chemical reactions involving every process in the body, including the absorption of oxygen. Our cells would literally starve for oxygen even with an abundance of oxygen without the action of the enzyme, cytochrome oxidase. Enzymes are also necessary for muscle contraction and relaxation. The fact is, without both of these classes of enzymes, (digestive and metabolic,) life could not exist. Digestive enzymes function as biological catalysts in which it helps to breakdown carbohydrates, proteins, and fats. On the other hand, metabolic enzymes function as a remodel of cells. Digestion of food has a high priority and demand for enzymes; digestive enzymes get priority over metabolic enzymes. Any deficiency in metabolic enzyme can lead to over work, which could lead to enlarge organs in order to perform the increased workload. The result is unhealthy and could cause enlarged heart or pancreas. The deficiencies of metabolic enzymes can have a tremendous impact on health. As we grow older enzyme level decline and the efficiency in the body decline. Enzyme naming conventions. By common convention, an enzyme's name consists of a description of what it does, with the word ending in "-ase". Examples are alcohol dehydrogenase and DNA polymerase. Kinases are enzymes that transfer phosphate groups. The International Union of Biochemistry and Molecular Biology has developed a nomenclature for enzymes, the EC numbers; each enzyme is described by a sequence of four numbers, preceded by "EC". The first number broadly classifies the enzyme based on its mechanism: Some other important enzymes are: Protease: breaks the protein into amino acids in high acidity environments such as stomach, pancreatic and intestinal juices. Act on bacteria, viruses and some cancerous cells. Amylase: Break complex carbohydrates such as starch into simpler sugars (dextrin and maltose). It found in the intestines, pancreas and also in salivary glands. Lipase: breaks down fats and some fat soluble vitamins (A,E,K, and D). helpful in treating cardiovascular diseases. Cellulase: break down cellulose that found in fruits, grains, and vegetables. It increases the nutritional values of vegetables, and fruits. Pectinase: break down pectin that found in citrus fruits, carrots, beets, tomatoes, and apple. Antioxidants: protect from free radical negative effect that can damage cell in the body. Cathepsin: break animal protein down. Lactase: break down lactose that found in milk products. the production of lactase decrease with age. Invertase: assimilate sucrose that can contribute to digestive stress if not digested properly. Papain: break down protein and help the body in digestion. Bromelain: Break proteins that found in plants and animals. it could help the body to fight cancer and treat inflammation. Glucoamylase: break down maltose that found in all grains in to two glucose molecules.

HyperText Markup Language. This is a guide to HTML, a standard for web pages. A text editor and a web browser is all you need to create web pages, view your handiwork, and share information with others all over the world. This book covers simple HTML syntax. For dynamic behavior in websites, see the JavaScript wikibook. Another separate book covers Cascading Style Sheets (CSS) which handle overall look and styling, but the present book addresses CSS briefly. Additionally, XHTML has its own textbook. Other Wikibooks.

World History/California Content Standard. http://www.cde.ca.gov/re/pn/fd/documents/hist-social-sci-frame.pdf http://www.cde.ca.gov/be/st/ss/hstgrades9through12.asp

Spanish/How to study Spanish. To study Spanish or any non-native language you should determine your goals. Many people study a language in school to fulfill a requirement. Others do it generally as a "broadening" experience. Still others do it so that they can speak with others in their own language. Number one tip: study every day. To speak any language fluently and be able to survive in the language among native speakers, it is probably necessary to immerse yourself in the language for an extended period such as 9 months to a year. Short of that it is best to study intensively every day for an extended period. Studying every day is much more effective than studying every other day. Studying only once a week is a sure way to forget everything by the end of the week and be eternally relearning the same little bit of vocabulary. The problem is that your mind will forget much of what you learn in the course of one day and by the end of two days it is even more. If you study every day, even for a little while, it will mean that you can focus more of your time learning new material and waste less time relearning old material. This is one of the secrets to learning a language: study it every day. Tip for studying new words: Have your own vocabulary. It can be some note, every page is e.g. 10 words. And every day study 10 new words, 10 words from the yesterday, 10 words from the day before yesterday, 10 words from the week before, 10 from 2 week before, 10 from 3 week before and 10 from 4 week before. It's overall 70 words. It want about half or 1 hour studying. 10 new words want more time than then repeat 60 others and you have confidence that you will know for a long time. Spaced learning software such as anki does a similar thing by scheduling reviews depending on how well you do. Using web resources. Learning via the web offers a lot of benefits that a standard textbook can not provide. One example are audio files which we already include in some of our lessons. Another benefit are online dictionaries that allow you to quickly look up vocabulary. Check out our Web Resources page to find out more. Using media resources. Once you have learned the basics of Spanish, try reading or viewing material in spanish. A newspaper, a magazine or a children's book is a great place to start. Online retailers, such as Amazon.com, have international storefronts that sell books in foreign languages. Watching Spanish television or Spanish movies will help you practice listening. Playing your favorite DVD or television show with Spanish close captioning on or with the Spanish soundtrack will help you learn new vocabulary and will assist in putting items into context. Areas where a significant number of people speak Spanish should have reading material available at local supermarkets and bookstores, and may have some local television or radio stations in Spanish. If you cannot find Spanish media where you live, the Internet has online editions of many Spanish newspapers and magazines available. Keep a dictionary nearby if you have trouble with unfamiliar vocabulary. The dictionary can be a very effective study tool. Each time you look up a new word, place a dot or symbol near the word. If you forget a word and need to look it up again, add another dot near the word. Pretty soon, you will have a collection of words with two, three and four dots. This marking system helps you figure out which words are either important to remember (because you use them frequently) or which ones are difficult to remember. During your vocabulary review, you can flip through your dictionary and find important words to practice. Regular use of a dictionary has the added benefit of putting things into context. Many dictionaries have tidbits and trivia added to the definitions, to help you see how components of language fit together. In addition, every time you open your dictionary, you will see other Spanish words, while searching for your target word and this has the effect of reminding you of other words. Use it. The best way to learn Spanish is by using it with other speakers. Do not be discouraged by your bad accent and choppy flow; these are things that come with being a beginner. People will understand that you are learning and will appreciate you applying the skills that you have acquired. So go ahead—say something. Ask your neighbor "¿Dónde está el baño?" Look around your local community. Often times coffee shops, libraries, or other local organization sponsor a "Spanish club" for adults. Both native and non-native speakers attend. This is a great way to use Spanish by speaking with others (especially native speakers). Another great way to practice your new language skills is to travel to the country where the language is spoken. Most major cities in Spanish speaking countries host a number of Spanish language schools for foreigners. These schools offer language classes at all levels by hour, week, or month of classes. Also, they normally will arrange for their students to live with a local host family, which provides another source for practicing the Spanish language with native speakers.

Spanish/Lesson 5. Grammar - Questions. Unlike English, yes/no questions in Spanish are not usually formed by switching the position of subject and verb (if the subject is explicit). To recognize a sentence as affirmative or as a question one must pay attention to the intonation pattern. Unlike English, Spanish uses a reversed question mark (¿) at the beginning of a question: become For other type of questions Spanish uses the following question words (note that all of them have an accent in the word): Here are some Spanish sentences where specific question words are used: Questions can also be posed within a sentence: Exercise: Questions Grammar - Possessive Adjectives. Like English, the Spanish possessive adjectives differ depending on the person they are referring to. Unlike English, the possessive article also changes depending on the number of items that one possesses (for example: mi libro = "my book", mis libros = "my books"). It can also change depending on the gender of the item (for example: nuestro perro = "our dog", nuestra casa = "our house"). The following table summarizes all Spanish possessive adjectives: Exercise: Possessive Adjectives Grammar - Comparisons. Equality. Spanish uses three slightly different constructions for comparisons of equality. One for comparing verbs, one for comparing nouns and one for comparing adjectives/adverbs. The following examples show the three different possibilities: When comparing nouns, the ending of tanto will be modified to tanta, tantos, or tantas in order to match gender and quantity of the noun. The general pattern for comparisons of equality is the following: Inequality. For comparisons of inequality, Spanish uses the same form for both nouns and adjectives/adverbs. There are two types of inequalities: más ... que ("more than") and menos ... que ("less than"): In general: Superlatives. Superlatives in Spanish are similar to comparisons of inequality: They use más for "the most", menos for "the least". Then follows the adjective and finally there is a preposition (de): Note that in some cases (la más inteligente) you can just write the article and omit the noun. The general pattern for Spanish superlatives is: Exercise: Comparisons

General Biology/Classification of Living Things/Eukaryotes/Animals/Chordates. The phylum Chordata includes three subphyla. These include vertebrates and invertebrate chordates. Shared characteristics. Non-synapomorphic characteristics (not limited to chordates): Subphylum Urochordata. The tunicates are located in this subphylum. Along with the subphylum Cephalochordata, these two subphyla make up the invertebrate chordates. Only the tunicate larvae have notochords, nerve cords, and postanal tails. Most adult tunicates are sessile, filter-feeders which retain their pharyngeal slits. Adult tunicates also develop a sac, called a tunic, which gives tunicates their name. Cilia beating within the tunicate cause water to enter the incurrent siphon. The water enters the body, passes through the pharyngeal slits, and leaves the body through the excurrent siphon. Undigested food is removed through the anus. Tunicates are hermaphrodites and can reproduce asexually through budding. In urochordates notochord is confined to larval tail. Subphylum Cephalochordata. The lancelets are located in this subphylum. Along with the subphylum Urochordata, these two subphyla make up the invertebrate chordates. Lancelets receive their name from their blade-like shape. They resemble fish, but they are actually scale-less chordates only a few centimeters long. They spend most of their time buried in the sand with their mouths protruding. Fossils of lancelets have been found to be over 550 million years old. Dropped out sessile stage, what was the larval stage is now sexually reproductive. Includes Branchiostoma (“amphioxus”). Subphylum Vertebrata. This group gets its name from the Latin verb "vertere", meaning "to turn." They are characterized by separate bones or cartilage blocks firmly joined as a backbone. The backbone supports and protects a dorsal nerve cord. Vertebrates have tissues which are organized into organs which in turn are organized into organ systems. Characteristics. All vertebrates share the following characteristics: The vertebrate organ systems are broadly categorized into: The vertebral column is not present in higher vertebrate organisms that have reached maturity. (In humans, the gel-like, spongy core of the vertebral column is the only remainder; ruptured or herniated discs are an injury to this.) The cranium is a composite structure of bone and cartilage that serves two primary functions: What evolutionary relationship could we imagine between sessile echinoderms and the higher chordate animals? Paedomorphic (child-form) hypothesis: basically, evolution of sexual reproduction in what had previously been a larval life stage, or the retention of at least one juvenile characteristic into the adult (capable of sexual reproduction) stage. Some scientists believe that this occurred in a proto-chordate animal lineage. Maybe chordates (and vertebrates) arose from sessile (attached) ancestors. Selection in these proto-chordates maybe began to favor more time in the larval stage, as feeding was more successful or mortality lower in this stage. As larvae got bigger physics shows that the cilia become less efficient for locomotion, favoring the undulating motion allowed by a notochord. Is this hypothesis crazy? A similar example of this today is Ephemeroptera, the mayfly, which has almost abandoned its adult stage. Its one-year lifespan is mostly larval with just a brief day of reproduce-and-die as an adult, which doesn’t even have usable mouthparts. Tunicate (sea squirt) larva has all four chordate characteristics, although adult sessile (“attached”). Class Agnatha. "jawless fish" Appeared approximately 500 million years ago and dominated the oceans for about 100 million years. The first group of fish to appear. They had neither jaws, paired fins, nor scales, but they were the first organisms with a backbone. Class Acanthodia. "spiny fish" Appeared about 430 million years ago. An extinct class of fish that developed jaws with bony edges. They had internal skeletons made of cartilage and some bone. Class Placodermi. Appeared about 410 million years ago, dominated the sea for about 50 million years. An extinct class of fissive heads. Class Chondrichthyes. "cartilaginous fish" Appeared about 400 million years ago with bony fish. Includes sharks, skates and rays, and chimaeras. Their skeletons are made of cartilage strengthened by the mineral calcium carbonate. The main characteristics and distinguishing features of this class: - gills - single-loop blood circulation - vertebral column - presence of placoid scales on their bodies - internal skeleton of cartilage - paired, fleshy pectoral and pelvic fins - asymmetrical tail fin prevents sinking - fatty liver provides neutral buoyancy - visceral clefts present as separate and distinct gills - no external ear - oviparous - internal fertilization - ectoderms - cold blooded Class Osteichthyes. "bony fish" Appeared about 400 million years ago with cartilaginous fish. Includes about 95% of today's fish species. Subclass Sarcopterygii. fleshy-finned fishes. Fins have bones and muscles, homologous to our limbs. Order Dipnoi. lung fishes, two groups isolated when continents separated Order Crossopterygii. includes coelacanths and rhipodistians, gave rise to amphibians, had lungs which evolved into a swim bladder in bony fishes, and labyrinthodont teeth, characterized by complex folding of enamel. • Skeleton made of bone, jaws, fins, most with scales, two-chambered heart. Class Amphibia. means “both lives”, aquatic larvae, terrestrial adult Amphibians: - Legs - Lungs - Double-Loop Circulation - Partially Divided Heart - Cutaneous Respiration (Breathes through Skin) Order Salientia. frogs (jumping) (aka Anura) Order Urodela. salamanders (tailed) Labyrinthodont amphibians: oldest known amphibians, inherited characteristic teeth from crossopterygii ancestor, had stocky, aquatic larvae. Amphibians have limbs instead of fins. Girdles and vertebral column now more substantial and connected, support body on legs. Lisamphybia: no scales, “smooth”, eggs with no shell, laid in water (water-reliant). Amphibians gave rise to cotylosaurs, from which arose dinosaurs, turtles, lizards, and therapsids. Class Reptilia. amniotic egg allowed freedom from water, shelled egg. (Amnion: protection). Reptiles have four extra-embryonic membranes: Reptiles are cold-blooded, or ectothermic, meaning that their heat come from their environment. Sometimes defined as all amniotes that are not birds or mammals. Reptiles can be classified by skull structure into four groups: Refers to number of holes in the skull. Cotylosaurs had Anapsid skull Dermatocranium: from bony outer skull structure, precursor to human cranium. Subclass Testudinata. turtles, terrapins Subclass Diapsida. dinosaurs, snakes, most stuff Subclass Diapsida. includes Ichthyosaurs, marine reptiles convergent on dolphins; Plesiosaurs, ancient sea monsters; Squamates, including lizards and snakes; and Thecodonts, which gave rise to Dinosaurs: broken into two groups, based on hip structure Crocodilians: come from archosaurs, the only extant (still living today) archosaur descendant. Ancestrally bipedal, secondarily quadripedal. Synapsids: refers to joined (Greek syn-, together with) parts of skull. Led eventually to mammals. Synapsid pelycosaur » therapsid » mammals Pelycosaur: Sail-backed dinosaur, legs not spread out like lizard but more pillar-like and under body, allowing greater activity and competence in motion, pendulum like rather than constant push-up. Teeth differentiated into different types, for pre-processing of food needed by higher metabolism. Skull changes, bone histology, suggestions of warm-bloodedness. Class Aves. arose late Jurassic, early Cretaceous. Feathers, skeleton modified for flight. Feathers: epidermal derivative, made of keratin (like fingernails). Carpometacarpis: bears primary flight feathers, parallel to hand parts. Keeled sternum: breastbone, powerful one needed to support flight muscles. Strong, light, occasionally hollow bones. All birds lay eggs (as contrasted to reptiles, which have developed live birthing over 100 independent times.) Why are there no live-bearing birds? Early birds had teeth, lost them. With mammals, only exothermic animals. Archaeopteryx: “ancient wing”, Jurassic bird-reptile, very dinosaur-like. Good fossils found in Zolenhoffen, German sandstone mine with fine sand, shows feathers clearly, found shortly after Darwin’s publication and used to support his hypothesis. Thick, heavy bones and no sternum, bony tail, not a good flyer but did have primary flight feathers. Archaeornithes: includes archaeopteryx. Paleognathae: gave rise to Australian flightless birds. Neognathae: remaining live birds. Class Mammalia. Two unique characteristics, or synapomorphies: Three skeletal characteristics (fossilize) Mammals basically have a synapsid skull design inherited from ancestor Non diagnostic characteristics (not unique to mammals): Subclass Protheria. monotremes (Greek mon-, one; and trema, hole), or egg-laying mammals, have one opening for excretion and urination. Subclass Theria. Metatheria: Marsupials (opossum, kangaroo…) Eutheria: Placental mammals (all common mammals) Marsupium: (from Greek marsypion, purse or pouch). Gestation period much shorter than in Eutherian mammals, but after leaving the uterus the tiny offspring crawls into a pouch where it completes development latched onto a teat. Recent molecular (read: genetic) evidence suggests that two different mammal groups may have developed live-bearing ability separately. Instead of being a “rough draft” for placental-style live bearing, perhaps the marsupial pouch approach is another solution to the same problem. Advantage: in tough times the parent can pitch out the offspring and increase its own chance of survival.

General Biology/Tissues. General Biology =Vertebrate tissues = Definition of tissue: an aggregation of cells, usually of the same kind, organized to perform a common function. A group of similar cells organized into a structural and functional unit (Raven/Johnson). Primary categories of adult tissues (non-dogmatic categories of convenience):

General Biology/Evolution. <br> <br>

General Biology/Classification of Living Things/Eukaryotes/Animals/Phyla. Introduction to animal phyla. There currently are almost 40 recognized phyla. Phylum — Number of Species — Common Name Phylum Porifera. The name Porifera means "pore-bearing". This phylum is commonly called sponges. The number of species is estimated to be between 5,000 and 10,000. All are aquatic and almost all are marine. Animals in this phyla have no true tissues, which means, for example, that they have no nervous system or sense organs. Although sponges are multicellular, they are described as being essentially at a cellular level of organization. They are sessile as adults, but have a free swimming larva. Their bodies are porous. They are filter feeders; water flows in through many small openings (ostia), and out through fewer, large openings (osculum). They have inner and outer cell layers, and a variable middle layer. The middle layer often is gelatinous with spiny skeletal elements (called spicules) of silica or calcium carbonate, and fibres made of spongin (a form of collagen). Choanocytes are flagellated cells lining the inside of the body that generate a current, and trap and phagocytize food particles. Their cells remain totipotent, or developmentally flexible: they can become any type of cell at any point in the sponge's development. This allows for the great regenerative power sponges have. Sponges are an ancient group, with fossils from the early Cambrian (ca. 540 mya) and possibly from the Precambrian. Sponges often are abundant in reef ecosystems. They somehow are protected from predators (spicules? bad taste?). Many organisms are commensals of sponges, living inside them. Some sponges harbor endosymbiotic cyanobacteria or algae (dinoflagellates, a.k.a. "zooxanthellae"). Phylum Cnidaria. Name comes from the Greek knide- meaning "nettle". This phylum formerly known as phylum coelenterata consists of the jellyfish, hydra, sea anemones, corals, sea pens, sea wasps, sea whips and box jellyfish. There are about 9,000 species. Almost all are marine. This is another ancient group, with fossils perhaps reaching back to 700 mya. Cnidarians exhibit radial symmetry. Their basic body plan is a sac with a gastrovascular cavity, or a central digestive system. They have one opening, which serves as both mouth and anus. The body wall has an outer ectoderm, an inner endoderm, and a variable undifferentiated middle layer called mesoglea or mesenchyme that may be jelly-like. The mesoglea is NOT considered to be true mesoderm and so the Cnidaria are described as diploblastic. Tentacles usually extend from the body wall around the mouth/anus. There are two basic body plans: the polyp and the medusa. The polyp is sessile and attaches to substrate by the aboral end (i.e., the end away from the mouth). The medusa ("jellyfish") is a floating form, and looks like an upside-down version of the polyp. Some cnidarians only have the polyp stage, some have only the medusa stage, and others have both. The typical life cycle of a cnidarian involves what is called "alternation of generations": an alternation between an asexual polyp stage and a sexual medusa stage. The tentacles are armed with cnidae (or nematocysts), small intracellular "harpoons" that function in defense and prey capture. When fired, the cnidae deliver a powerful toxin that in some cases is dangerous to humans. The phylum is named after the cnidae. Cnidarians have no head, no centralized nervous system, and no specialized organs for gas exchange, excretion, or circulation. They do have a "nerve net" and nerve rings (in jellyfish). Many cnidarians have intracellular algae living within them in a mutualistic symbiotic relationship (Dinoflagellates = zooxanthellae). This combination is responsible for much of the primary productivity of coral reefs. There are three main classes in the phylum Phylum Platyhelminthes. Name means "flat worm" Most members of this phylum are parasitic (flukes and tapeworms), but some are free living (e.g., planaria). There are about 20,000 species. They are dorsoventrally compressed (i.e., "flat"). Animals in this phylum are acoelomate, triploblastic, bilaterally symmetrical, and unsegmented. Platyhelminths have a simple anterior "brain" and a simple ladder-like nervous system. Their gut has only one opening. Flatworms have NO circulatory or gas exchange systems. They do have simple excretory/osmoregulatory structures (protonephridia or "flame cells"). Platyhelminths are hermaphroditic, and the parasitic species often have complex reproductive (life) cycles. There are four main classes of platyhelminths: Phylum Rotifera. The Rotifers. The name means "wheel bearing," a reference to the corona, a feeding structure (see below). They are triploblastic, bilaterally symmetrical, and unsegmented. They are considered pseudocoelomates. Most less than 2 mm, some as large as 2 – 3 mm. Rotifers have a three part body: head, trunk, and foot. The head has a ciliary organ called the corona that, when beating, looks like wheels turning, hence the name of the phylum. The corona is a feeding structure that surrounds the animal's jaws. The gut is complete (i.e., mouth & anus), and regionally specialized. They have protonephridia but no specialized circulatory or gas-exchange structures. Most live in fresh water, a very few are marine or live in damp terrestrial habitats. They typically are very abundant. There are about 2,000 species. Parthenogenesis, where females produce more females from unfertilized but diploid eggs, is common. Males may be absent (as in bdelloid rotifers) or reduced. When males are present, sexual and asexual life cycles alternate. Males develop from unfertilized haploid eggs and are haploid. Males produce sperm by mitosis which can fertilize haploid eggs, yielding a diploid zygote that develops into a diploid female. Sexual reproduction occurs primarily when living conditions are unfavorable. Most structures in rotifers are syncytial ("a mulitnucleate mass of protoplasm not divided into separate cells," or "a multinucleated cell") and show eutely (here, "constant or near-constant number of nuclei"). Phylum Nematoda. Name from the Greek for "thread". This phylum consists of the round worms. There are about 12,000 named species but the true number probably is 10 - 100 times this! These animals are triploblastic, bilaterally symmetrical, unsegmented pseudocoelomates. They are vermiform, or wormlike. In cross-section, they are round, and covered by a layered cuticle (remember this cuticle !!). Probably due to this cuticle, juveniles in this phylum grow by molting. The gut is complete. They have a unique excretory system but they lack special circulatory or gas-exchange structures. The body has only longitudinal muscle fibers. The sexes are separate. Nematodes can be incredibly common, widespread, and of great medical and economic importance. They are parasites of humans and our crops. They can live pretty much anywhere. Nematodes can be free living or important parasites of our crops, or of humans and other animals. They have become very important in development studies, especially the species Caenorhabditis elegans, presumably due to its small size and constancy of cell number (eutely - 959 cells in C. elegans). Phylum Annelida. Name means "ringed", from the Greek annulatus. This phylum consists of earthworms, leeches, and various marine worms given many different names (e.g., sand worms, tube worms). There are about 12,000 - 15,000 species. Animals in this phylum are triploblastic, bilaterally symmetrical, segmented coelomates. Development is typically protostomous. They have a complete circulatory system, and a well-developed nervous system. Typically, each segment has paired epidermal "bristles" (setae or chaetae). Most are marine but they are successful occupants of almost anywhere sufficient water is available. They can be free living, parasitic, mutualistic, or commensalistic. Major advances of this phylum include the true coelom, segmentation, both longitundinal and circular muscles, a closed circulatory system and, for most, a more advanced excretory system (metanephridia). There are three main classes of Annelids Phylum Arthropoda. Name means "jointed feet". This phylum consists of spiders, ticks, mites, insects, lobsters, crabs, and shrimp, and is the largest of all the phyla. So far, over 1 million species have been named, and it is likely that the true number out there is 10 - 100 times greater. These animals are triploblastic, bilaterally symmetrical, segmented, protostome coelomates. The coelom is generally reduced to portions of the reproductive and excretory systems. They have an open circulatory system. The most notable advancement of this phylum is a rigid exoskeleton. It has major implications in these organisms' locomotion, flexibility, circulatory systems, gas exchange systems, and growth. It also was partially responsible for the ability of the arthropods to move on to land. There are several major groupings of arthropods: Phylum Mollusca. Name means "soft". This phylum consists of snails, slugs, bivalves, chitons, squids, octopus, and many others. About 110,000 species All molluscs have a similar body plan: Molluscs are bilaterally symmetrical, or secondarily asymmetrical. They are coelomates, but the coelom generally has been greatly reduced; the main body cavity is a hemocoel. Development is typically protostomous. The gut is complete with marked regional specialization. Large, complex, metanephridia (excretion). Many molluscan life cycles include a trochophore larva. This stage also is characteristic of annelids. There are several major classes of molluscs: Phylum Echinodermata. Name means "spiny skin" This phylum consists of sea stars, brittle stars, sea urchins, and sea cucumbers. Echinoderms are mostly sessile or very slow moving animals. As adults, they are radially symmetrical, but in the larval stage, they are bilaterally symmetrical. They are considered deuterostomes. Echinoderms are unique in that they have a water vascular system composed of a system of fluid-filled canals. These canals branch into the tube feet, which function in feeding, locomotion, and gas exchange. There are six major classes of echinoderms: Phylum Chordata. Name means "the chordates", i.e., these animals have a notochord at some stage in their lifecycle. This phylum consists of tunicates, lancelets, and the vertebrates. There are four major features that characterize the phylum Chordata. Chordates have a segmented body plan, at least in development. This segmentation evolved independently from the segmentation of annelids. Three subphyla make up the phylum Chordata: Formally, the phyla Urochordata and Cephalochordata are considered invertebrates. Subphylum Vertebrata. Vertebrata refers to the presence of vertebrae and a vertebral column. This subphylum includes most of the animals with which most people are familiar. The notochord generally is replaced by the cranium & vertebral column in adults. Neural Crest Cells. Later in development, these give rise to many cells of the body, including some cartilage cells, pigment cells, neurons & glial cells of the peripheral nervous systems, much of the cranium, and some of the cells of the endocrine system. Some scientists would like to classify the neural crest as the fourth germ layer. Neural crest cells come from the dorsal edge of the neural plate, thus ectoderm.

General Biology/Classification of Living Things/Eukaryotes/Animals. Key Terms. synapomorphy Introduction. What makes an animal an animal? If animals are a monophyletic taxon, then animals should be able to be defined by synapomorphies, (shared, derived characteristics). Ideally, we would NOT define this or any taxon using symplesiomorphies (shared ancestral or primitive characteristics) or homoplastic characters (the independent evolution of similarity, or "convergent evolution"). See pages 654 - 656 and Fig. 32.6 in your text to review these concepts. As you consider the characteristics listed below, ask yourself whether or not each is a synapomorphy. Characteristics of an Animal. Animals are multicellular heterotrophic eukaryotes Animals share unique characteristics Animals share certain reproductive characteristics Other commonly used definitions or characterizations What kinds of animals are there? "This text is based on notes very generously donated by Ralph Gibson, Ph.D. of the Cleveland State University."

General Biology/Animal Evolution. The Evolutionary Tree in Animals. There are many competing hypotheses for the form of the evolutionary tree of animals. A traditional hypothesis is that the tree resembles a tuning fork: it has a short base and two main branches. However, there is recent molecular evidence that challenges part of this traditional hypothesis. Under the tuning fork model, the "base" of the tree includes structurally simple animals like sponges, corals, and their relatives. One main branch includes arthropods, molluscs, annelids, and nematodes. This branch, or a large part of it, usually is called the protostomes. The second main branch includes vertebrates (phylum Chordata), and starfish, sea urchins, and their relatives (phylum Echinodermata). This branch usually is called the deuterostomes. Flatworms (phylum Platyhelminthes), which include free living planarians as well as parasitic flukes and tapeworms may be placed very low on the protostome branch, or high on the trunk just below the protostome - deuterostome branching. Support for the "Tuning Fork" Model. The features of animals that have been interpreted as suggesting a tuning fork model are extremely basic characteristics of body organization and early embryonic development. Body Symmetry. Asymmetry. Lack of any symmetry. Many sponges are asymmetric. Bilateral Symmetry. There is only one plane of symmetry, and it is anterior-to-posterior, dorsal-to-ventral, through the midline. Characteristic of most protostomes and the higher deuterostomes. Presence of True Tissues. Tissues are defined as an integrated group of cells that share a common structure and a common function (for example, nervous tissue or muscle tissue). Sponges are described as lacking true tissues. True tissues are present in Cnidaria, flatworms, and all higher animals. Number of Embryonic Germ Layers. Germ layers are defined as the basic tissue layers in the early embryo which give rise developmentally to the organs and tissues of the adult (e.g., ectoderm, mesoderm, endoderm). This is a concept that is applied "only" to organisms considered to have true tissues. Two Germ Layers. Organisms with two germ layers are said to be diploblastic. This is characteristic of Cnidarians. Three Germ Layers. Such organisms are said to be triploblastic. This is characteristic of flatworms and all higher organisms. Four Germ Layers (?). Some developmental biologists consider the neural crest tissue of vertebrate embryos to be a fourth germ layer. Nature of the Main Body Cavity. Most triploblastic animals have a fluid-filled space somewhere between the body wall and the gut. Such a cavity can provide numerous functional advantages. For example, peristalsis of the gut need not affect the body wall, and movements of the body wall during locomotion need not distort the internal organs. We will consider three conditions with respect to the body cavity: Acoelomates. These animals lack an enclosed body cavity; the only "body cavity" is the lumen of the digestive tube. The space between the gut and the body wall is filled with a more or less solid mass of mesodermal tissue. The major example of this is the phylum Platyhelminthes, the flatworms. Minor examples: Phylum Nemertea (Rhynchocoela) and Phylum Gnathostomulida (not responsible for these "minor" examples). Pseudocoelomates. Pseudocoelomates have a fluid-filled cavity between the body wall and the gut, but it does not form within mesoderm, nor does it end up fully enclosed by mesoderm. This cavity often is interpreted as being a developmental remnant of the blastocoel, the fluid-filled cavity of the blastula stage of the embryo. To distinguish it from the next grade, this type of cavity is called a pseudocoelom. Major examples of pseudocoelomates include the phyla Nematoda (round worms) and Rotifera (rotifers). Other phyla listed in the table that are considered to be pseudocoelomates are flagged with an asterisk. Note that recent molecular data in particular have challenged the "naturalness" of the pseudocoelomates as a possible taxon. Coelomates. Coelomate animals also have a fluid-filled cavity between the body wall and the gut. In this grade, however, the cavity is completely enclosed by mesoderm. Major examples of coelomates include molluscs, arthropods, echinoderms, and chordates. The protostome-deuterostome distinction. The distinction is based on several fundamental characteristics of early development. Characteristic: Determinate vs. indeterminate cleavage Spiral vs. radial cleavage Fate of the blastopore Source of mesoderm Formation of coelom Phyla Echinodermata, Hemichordata, Chordata "This text is based on notes very generously donated by Ralph Gibson, Ph.D. of the Cleveland State University."

Structured Query Language. __NOEDITSECTION__ Structured Query Language (SQL) is a widely-used programming language for working with relational databases. The name of the language is generally pronounced as the three letters of its abbreviation or, in some people's usage, as . This Wikibook provides a short description of SQL, its origins, basic concepts and components, and many examples. The book follows the specifications of the SQL:2011 standard developed by a common committee of ISO and IEC. Their publications are not freely available but can be ordered online. Or you may want to refer to a working draft that you can download from Whitemarsh Information Systems Corporation.

Internet Technologies. Scope. This work is aimed at people already familiar with using the Internet, who want to know how and why it works. When we say technology we don't just mean the software and hardware, but also the human components which are an integral part of the overall system of the Internet.

Spanish/Slang. Spanish slang is more localized than English slang and sometimes people from one Spanish-speaking country get confused talking to people from other Spanish-speaking countries.

World History. " The World History Project" Welcome to the World History Project. This organization is dedicated to making a free, open-content, standardized textbook on World History based on the /AP World History Standard/. The goal is to create a standard of quality which will suffice for a secondary and post-secondary environment. The World History Project is the "brains" behind the organization. We are a set of regular contributors who organize and give the major guidance to the World History page. We welcome contribution of any who wish to help (whether as part of the World History Project or no), as well as collaboration with other projects - contact us at our main discussion page or here at our Authors page. Standards | Our Golden Rule | The Authors | Prologue - Other. __NOEDITSECTION__

World History/World Religions. This page is for an elementary explanation of the origins and nature of the religions. It is not meant to be an in depth study by any means. Animism. Animism (from Latin animus, -i "soul, life") is the worldview that non-human entities (animals, plants, and inanimate objects or phenomena), possess a spiritual essence. Baha'i Faith. The "Bahá'í Faith" is the youngest of the world's independent religions. Its founder, Bahá'u'lláh (1817–1892), is regarded by Bahá'ís as the most recent in the line of Messengers of God that stretches back beyond recorded time and that includes Abraham, Moses, Buddha, Zoroaster, Christ and Muhammad. The central theme of Bahá'u'lláh's message is that humanity is one single race and that the day has come for its unification in one global society. God, Bahá'u'lláh said, has set in motion historical forces that are breaking down traditional barriers of race, class, creed, and nation and that will, in time, give birth to a universal civilization. The principal challenge facing the peoples of the earth is to accept the fact of their oneness and to assist the processes of unification. Wikipedia article: Bahá'í Faith Buddhism. Buddhism is a religion based largely around the teachings of the Siddhārtha Gautama (although his exact status is still controversial and changes by sect) and is now the central religion of most of Southeast Asia, Mongolia, Sri Lanka, and parts of China, Nepal, India, and a small part of Russia. It is also a major religion in Korea and Japan and has a growing influence in the west. Gautama was born in ancient Nepal around the 6th century BC, son of a King and relieved of all tasks. One day, while being brought around by his charioteer, he saw the four passing sights (an old man, a sickly man, a decaying corpse, and a holy man), which brought him to the realization that birth, old age, sickness, and death happen to all people over countless lives. He left his wife, children, rank, and his entire life to solve that problem. Gautama tried everything to achieve inner peace (he nearly killed himself numerous times) but found nothing that worked. He then tried sitting peacefully under a Bohdi tree and meditating. This proved very successful, and he soon achieved the inner peace he wanted. He then traveled the lands, preaching his new faith. (Postscript-Buddhism is largely based on Jainism, and shares many beliefs with Jainism.) Within the context of postclassical China, dominations such as pure land and Zen Buddhism appealed to both aristocratic elites and the mass peasantry. A commonality of religion in the global context: the induction of fervent belief system in the presence of societal corruption and lack of intellectual synthesis. Modern Buddhism still follows the ideals of Gautama-peace, kindness to man, and love of nature (including vegetarianism). There are three modern sects of Buddhism - Theravada, Mahāyāna, and Vajrayāna (practiced in southeast Asia, East Asia, and scattered parts of Asia, respectively). For information on the sects of Buddhism, or some tenets of Buddhism, go to the Wikipedia articles on: Buddhism Theravada Mahayana Vajrayana Christianity. Wikipedia article on Christianity. Confucianism. Wikipedia article on Confucianism Hinduism. "Hinduism" is a term coined to designate the traditional socio-religious systems of the people of India. This term does not appear in any of the sacred literature of India. Hindus refer to their religion as "Sanatana Dharma" which loosely translated means “The Eternal Path”. "Sanatana" means "eternal", "perpetual" or "sustained". "Dharma" means any method by which one sees reality for what it is, and that by which one is drawn closer to the Absolute Truth and Ultimate Reality — it is the "Philosophia Perenis". In a context of world history, the Hindu emphasis placed upon social divisions as ample means for a productive society led to the highly stratified caste system in which birth and socio-economic position determined semi-permanent placement. There are two world religions which have formed the cultural and ethical basis of the world as we know it. Both have an unbroken history going back thousands of years. Judaism with a 5000 year old tradition is the mother of the western civilization through its offshoot Christianity. Hinduism is the older of the two with a literature going back to the beginning of recorded history. Hindu civilisation originated in the Gangetic and Indus valleys and from there spread out over the entire region of southeast Asia. Its offshoot — Buddhism, shaped and molded the civilizations of Japan, China, Tibet and the rest of Asia. There is evidence to suggest that the Ancient pre-Biblical kingdom of the Mittani in Asia minor was ruled by Kings with Hindu/Sanskrit names. The Hittites were an Indo-European people and according to some sources are said to have originated in the Gangetic Basin of India. Hindu philosophy/theology influenced the ancient Greeks since the time Alexander the Great conquered parts of north India. A remarkable similarity has also been demonstrated between the religion and mythology of the ancient Scandinavian people and that of the people of India. The ancient civilizations such as the Roman, Greek, Egyptian, Sumerian, Babylonian, Mayan, Aztec, and Inca have all passed away. Even the Jewish culture has undergone many radical changes since its inception 5000 years ago – yet the Hindu civilisation continues as a vibrant and living vector, and has remained virtually unchanged for over 6000 years. Today, Hindu communities are to be found in almost every country on earth. Hinduism Islam. Islam Jainism. Jainism Judaism. Judaism Sikhism. Sikhism Shinto. Shinto Taoism. Taoism/Daoism