CH 368: Unit 2

# REACTION RATES, TRANSITION STATES, AND MECHANISMS

A. Transition State Theory.

1. The Basic Equation.

According to transition state theory, the rates of reactions are given by the following equation:

where k is the rate constant for a given reaction, k is Boltzmann’s constant, T is the absolute temperature,  h is Planck’s constant, and DG* is the free energy difference between the transition state and the reactants, i.e., the activation free energy.

## The value of(kT/h) is 6 x 1012 s-1. According to transition state theory, the fastest reaction, i.e., one which has no activation requirements, has a rate constant of 6 x 1012 s-1. Keep in mind that rates of diffusion in various solvents are typically much less than this, e.g., ca. 10-10 s-1, so that bimolecular reactions in solution which have no activation requirements are diffusion controlled, i.e., they occur at the diffusion rate of the solvent. The maximum rate of transition state theory therefore is attained, insolution, only in intramolecular reactions or quasi-intramolecular reactions (those in which two molecules have initially formed a complex). Enzymic reactions can theoretically also occur at this maximum rate.

The free energy of activation, like the standard free energy change of a reaction, is composed of enthalpic and entropic parts, as shown in the equation below, so that the basic equation of transition state can be re-written in terms of these components.

2. Rate Constants.

The rate constant, k, is independent of the concentrations of any reactants. It is a measure of the intrinsic reaction rate, or the rate of the reaction at unit molar concentrations of all reactants. The overall rate is equal to the rate of consumption of reactants or the rate of formation of products (usually equal). Clearly the rate of product formation in a simple intramolecular reaction is proportional to the concentration of the reactant. The rate constant, however, is independent of the concentration, and this rate constant is what is predicted by transition state theory.

3. Definition of Transition State.

The transition state of a reaction step is considered to be the highest energy point on the minimum (free) energy reaction path (MERP). Reactions are considered to occur via the path from reactants to products which is of lowest energy. The TS is the point of highest energy on this path. As such it represents a saddle point on the free energy surface. On the free energy multidimensional surface, it is an energy minimum with respect to every other variable except the reaction path.

4. Reaction Coordinate.

The reaction coordinate, in a purely qualitative sense, represents the extent to which the coordinates (values of various bond lengths, angles, and dihedral angles) which characterize the reactants have been converted to those of the products as progress is made along the MERP.

5. Reaction Path Diagrams.

Reaction path diagrams, as represented above, are a two dimensional slice of a multidimensional energy surface for the reaction. Conventionally, they are a plot of energy (in the rigorous case, free energy, G) versus the reaction coordinate.

6. Reaction Mechanism.

The mechanism of a reaction is a stepwise description of how reactants are converted to products. Of course, some reactions occur in a single step, as represented in the reaction path diagram above. Such reactions are termed concerted. More generally, reactions are found to occur in a stepwise fashion. They may consist of 2,3 or many more steps. In any stepwise reaction, intermediates (I) must be formed. By definition an intermediate is any species encountered on the reaction path between R and P which  represents a free energy minimum.  As simple illustrations of stepwise mechanisms, we will consider two distinct scenarios for reactions which occur in two steps.

1. First Step Rate-Determining.

Mechanism:

Reaction Path Diagram:

The term “rate-determining step” or “rds” has a specific meaning. An rds is a step of a reaction the rate of which is equal to the rate of the overall reaction (conversion to final product(s)). The importance of a rate-determining step is that if the rate of that step is known, the rate of the overall reaction is known. To put it another way, we do not need to know the rates of any of the other steps of a multi-step reaction if we know the rate of the rds. The conditions for a rate-determining step are, however, quite restrictive. As noted in the scheme above, the intermediate must always go on to product (never back to reactant; so a reversible step can not be rate-determining). To the extent the intermediate reverses to reactants, the rate of final product formation is less than the rate of forming the intermediate. This restriction means that the activation energy for the reverse of the first step must be substantially greater than the activation energy for the second step. Secondly, the intermediate must not build up in concentration, but must go on to product very rapidly. This means that the barrier to the second step must be very small.

Exemplification. The addition of hydrogen chloride to an alkene such as isobutene is a typical example of a two-step reaction in which the first step is rate-determining. Note that the overall reaction requires breaking two bonds (the C=C pi bond and the H-Cl bond) and forming two bonds (the C-H bond and the C-Cl bond). However, these four primary bond changes do not occur simultaneously in a concerted reaction; this pathway is of much higher energy (we will see why later), and is not the minimum energy reaction path. In the minimum energy path, two bonds are broken in the first step and only one is formed. Therefore, the intermediate carbocation/chloride ion pair is at a much higher energy than the starting materials. Consequently, the activation free energy for this step is rather large (it has to be at least as large as the energy difference between the intermediate and the reactants). The intermediate, once formed, has a very low barrier to going on to the product, since no bonds are broken in this second step, while one new bond is formed. The intermediate essentially always goes on to product because the barrier toward the reverse of step 1, in which the chloride ion would remove a proton from the intermediate carbocation to yield back the reactant alkene and hydrogen chloride, is substantially larger than the very small barrier to step two. Therefore, both conditions for a rate-determining step are fulfilled, viz., the intermediate always goes on to product, and it doesn’t build up in significant amounts.

b. Second Step Rate-Determining.

### Reaction Path Diagram

In this mechanistic scenario, the intermediate is formed in a reversible equilibrium, and then slowly goes on to the products. The effective barrier for the reaction is the energy difference between the second TS and the original reactants. Neither the energy of the 1st TS not the energy of the intermediate is of immediate consequence in this scenario. The intermediate may be formed in detectable amounts or it may be formed in minute amounts. Either way, the energy of the intermediate is irrelevant to the rate of the reaction . This principle is called the Curtin-Hammett principle. If the intermediate is of relatively low energy, it will be formed more extensively, but the activation energy for promoting it to the second TS will then be correspondingly higher.

B. Transition State Models.

1. Resonance Theoretical Approach.

a. A Polar Transition State. For relatively simple reaction systems, transition states and their energies (therefore activation energies) can be calculated. For qualitative purposes, resonance theory provides a very convenient approach to transition state modeling. The basic principles involved are as follows:

q      Transition states are generally intermediate between reactant and product in their geometry and, for the most part, in electron distribution. Consequently, the TS should be reasonably represented as a resonance hybride of reactant and product-like structures (at the unique geometry of the TS).

q      In some cases, since the TS is a unique entity, and since resonance theory indicates that a given species is best represented as a resonance hybrid of all reasonably good resonance structures, one or more non-reactant and non-product-like structures may be a further improvement.

q      Since the real TS is intermediate between the canonical structures, some of the bonds will be partial, because a given bond is made in some structures and broken in others. These partial bonds can be represented as dashed lines in a dashed line/partial charge structure. In the same way, charges on some atoms will be intermediate between unit charge and zero charge, and these can be represented as partial charges using the d formalism.

q      The DL/PC model can then be examined and characterized by means of the salient characteristics of the model. Subsequently, the Hammond Principle can be applied to refine the TS character(s).

Consider, as an example, the addition of HCl to ethene.

The structures of ethene and HCl, as reactants, are oriented in the proper manner to qualitatively represent the anticipated TS, i.e., protonation of one of the carbons of ethene. Then use the electron flow formalism to generate the product structure, i.e., the ethyl carbocation and chloride anion, still in the same orientation.

Following this, summarize the model as a single DL/PC structure, using dashed lines for partial bonds and Greek delta’s for partial charges.

q      Since the C-C bond is double in the reactant structure and single in the product structure, it must be intermediate between single an d double in the TS. This is represented as a full line for the sigma bond and a dashed line for the partial pi bond. Simliarly, the new C-H bond is made in the product but unformed in the reactant, so it is also intermediate between no bond and a single bond. Again, a dashed line represents this partial bond, as it does also in the case of the H-Cl bond, which is full (bond order of 1.0) in the reactant and broken (bond order of 0) in the product.

q      In a similar way, unit positive charge appears on the passive carbon (the one not bonding to the proton) in the product, but there is no charge on that carbon in the reactant. Consequently, this is represented as a partial positive charge on the passive carbon. When we apply the Hammond Principle, we will be able to be more specific about how much charge is present there. Correspondingly, there isi partial negative charge on the chlorine.

q      Finally, the formal characterization is presented, i.e., carbocation character at the passive carbon.

In the same way, radical addition, abstraction, or other transition states can be modeled using resonance theory. For example, the abstraction of hydrogen atoms from an alkane by bromine atoms:

c. Inclusion of Non-R non-P Structures.

Resonance theory indicates that the best representation of a structure is a combination of all canonical forms of reasonable energy. Although essentially any TS should have R and P character, the TS may have additional characters not present in either the reactant or product, though these additional structures, which we will term X structures. We will consider two important examples of this type.

i. Hydroboration. The addition of borane to an alkene has been shown to be influenced, in its regiochemistry, by polar effects which uniquely exist in the transition state. Besides the reactant and product-like structures, an additional polar structure can be written which recognizes the Lewis acidity or electrophilicity of trivalent boron. As a consequence, there is carbocation character at the carbon to which the hydrogen becomes attached.

ii. E2 Eliminations. Although E2 eliminations are typically discussed in terms of predominant alkene (P-like) character in the TS, there is another significant character in that TS which, in some cases, becomes the dominant character. As a result of the carbanion character at the beta carbon, certain eliminations prefer to yield the less thermodynamically stable alkene.

2. The Hammond Principle. The treatment of transition states as resonance hybrids of reactant-like and product-like structures (along with any other “good” structures) implicitly takes into account the circumstance that virtually any transition state has some reactant-like and some product-like character. The treatment does not, in itself, indicate how much of each one or even whether the TS more closely resembles one or the other or is very centrally located between the two. The Hammond Principle is extremely useful in refining transition state models in the sense of knowing whether the TS strongly resembles reactants or products or is rather centrally located between them. In its most succinct form, the Hammond Principle states that “The transition state of an elementary  reaction step more closely resembles the reaction partner (i.e., reactant or product of that step) which is of higher energy.” Consequently the TS of a reaction step which is endothermic more closely resembles the product of the step, and a the TS of a reaction step which is exothermic more closely resembles the reactant.  As a corollary of the Principle, reaction steps which are highly endothermic, have TS’s which closely resemble the product of that step, and reaction steps which are highly exothermic have TS’s which closely resemble the reactant in that step. Reaction steps which are neither strongly endothermic nor exothermic, but roughly thermoneutral are considered to have centrally located transition states.

As an example of a highly endothermic step we can consider the protonation of an alkene by hydrogen chloride. In this reaction step, two bonds are broken (the H-Cl bond and the C-C pi bond), and only one new bond is formed (the C-H bond). Consequently, the formation of the carbocation intermediate in this step is highly endothermic, and the TS for this step closely resembles the product of the step, which is a carbocation. We can then refine our characterization of the TS model to say that the TS has “extensive carbocation character at the passive carbon.”  Experimental evidence indicates that the TS has indeed developed between 80-90% of the unit positive charge present in the full-fledged carbocation intermediate. The special importance of this refinement in our characterization lies in the implication that it has for selectivity. If there were only a small fractional positive charge developed on the passive carbon in the TS, the distinction between  TS’s which have primary, secondary, or tertiary carbocation character would not be quantitatively very great, so that selectivities (e.g., regioselectivity) and relative rates of addition of HCl to various alkenes would be quite small. Selectivity is, of course, a matter of prime importance in the utilization of organic reactions.

As an example of a highly exothermic reaction, we can consider the second step in the reaction of HCl with an alkene, i.e., the reaction of the intermediate carbocation with chloride ion. In this step, no bonds are broken, and one is formed, so that the reaction is highly exothermic. In this case, the Hammond Principle indicates that the TS resembles the reactant of the step, which is (again) the carbocation. Interestingly, the TS’s of both reaction steps strongly resemble the carbocation intermediate.

As examples of reactions which are neither highly exothermic nor highly endothermic, we can consider either the hydroboration of an alkene or the elimination of HCl in an E2 reaction, both of which have been previously discussed. Note that in both cases, the mechanism is a concerted one. In both cases, also, two bonds are broken and two are formed, so that neither reaction is highly exo- or endothermic.  The reaction, since it goes to completion, must be exergonic (and it is also exothermic), but it is not a highly exothermic process. Consequently, the TS does not strongly resemble either the reactants or products, but is centrally located.

3. The Method of Competing Transition States.  Chemistry, like much else, is a competitive world. Competition determines the outcome of chemical reactions, i.e., whether a given reaction is successful in generating a desired product in good yield, or this product is obtained as a mixture or in very low yield. In general, the outcome of any competition is determined by therelative activation free energies of each competing mechanism. If the reactants are the same for each of the competing processes, the situation simplifies further: the results of the competition are determined by the relative free energies of the transition states of each mechanism. It is transition states which actually compete!! At room temperature, for example, for each 1.37 kcal/mol of difference in the free energies of the TS’s, a ten-fold rate ratio results.  This would provide an approximately 91:9 ratio of the products. If one desires higher selectivity, e.g., 99:1, a 2(1.37) = 2.74 kcal/mol energy difference would be required.

A. Regioselectivity (or Regiospecificity).  The reaction of an unsymmetrically substituted alkene, such as isobutene, with an unsymmetrical reagent such as hydrogen chloride could potentially generate either or both of two products, i.e., tert-butyl chloride or isobutyl chloride. These are products which results from combining the two unsymmetrical reactants in the two different orientations. Orientational selectivity such as this is often referred to as regioselectivity. In reality, one of these products (tert-butyl chloride) is obtained with a very high level of selectivity, i.e., essentially none of the other product (isobutyl chloride) is obtained. Nevertheless, the results of the reaction of isobutene with hydrogen chloride are determined by the outcome of a competition (it turns out not a very close competition) between two distinct reaction mechanisms and, in particular, two distinct transition states. To understand the strong preference for formation of tert-butyl chloride over isobutyl chloride we need to create TS models for both TS’s and compare their refined and extended characters. To do this, we have only to utilize our previously derived TS model for the addition of HCl to ethene and convert the  hydrogens of the ethene TS to the desired substitutents, as shown in the scheme below. The difference between the two TS’s is easily evident. One has tertiary carbocation character, and the other has primary carbocation character. Since tertiary carbocation character is more favorable than primary carbocation character, the TS on the left is favored, and the preferred product is tert-butyl chloride. The high selectivity is rationalized by two considerations: (1) a full-fledged tertiary carbocation in solution is more stable than a primary one by at least 30 kcal/mol, a very large preference (2) the TS’s both have extensive carbocation character (via the Hammond Principle), so that a very large fraction of the total energy difference between a tertiary carbocation intermediate and a primary carbocation intermediate is incorporated into the TS. Consequently, the TS on the left is powerfully favored, and the selectivity is extremely high.

B. Relative Rates. When comparing the relative rates of reaction of different molecules, e.g., the relative rates of addition of HCl to a series of different alkenes,  differences in the free energies of activation of the competing reactions could arise from either transition state energy differences or differences in energy of the different reactants. In the case of HCl addition to alkenes, the energy differences between simple alkenes are quite small compared to TS energy differences, which are based upon differential carbocation character. Consequently, the Method of Competing TS’s is still a very appropriate approach to use in discussing the relative rates of addition to alkenes.

Consider the following series of simple alkenes: ethene, propene, isobutene, and trans-2-butene. Again using the basic TS model for the rate-determing step of these reactions, we can perform the requisite substitution of alkyl groups for the four hydrogens of ethene or for the four unspecified bonds of a general alkene and set up a TS for each of these alkenes.

In the case of unsymmetrically substituted alkenes (propene and isobutene), we will ignore the disfavored regioisomeric TS. We can then characterize these TS’s and rank them according to their expected relative energies. We can also apply the Hammond Principle to gain insights into the expected magnitudes of the relative rate differences. We note that isobutene has thebest TS, since it has extensive tertiary carbocation character, while propene has extensive secondary carbocation character, and ethene has extensive primary carbocation character. Trans-2-butene is especially interesting because, although it is a disubstituted alkene, like isobutene, its TS has only secondary carbocation character, like that of propene. This is because the carbocation character is generated on only one carbon atom of the original alkene double bond, specifically the passive carbon, so that the second methyl group of 2-butene is not attached to a carbocation center and does not have a major effect upon the rate. Consequently, the rate of addition to this alkene is similar to that of propene.

The relative rate ratios will be large (i.e., isobutene reacts very much faster than propene, e.g.), because there is extensive carbocation character, and the energy difference between tertiary carbocations and secondary carbocations is very large. In contrast, relative rate ratios for hydroboration are rather small, because the TS has only a modest amount of carbocation character.