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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.