Entropy of activation

The condition for thermodynamic equilibrium permits us to attach a thermodynamic significance to the constants in the Arrhenius equation, as we have just demonstrated in the case of reversible reaction systems. There is no real objection to extending this in a formal way to all reactions, so that we can rewrite the Arrhenius equation in the following form  

In these equations we have broken down the constant A into a product ve y where v now has the dimensions of the rate constant k and in analogy to the nomenclature for A is referred to also as a frequency factor. S and H are referred to respectively as the entropy and enthalpy of activation, and as we shall see later they are capable of association with more conventional entropy and enthalpy changes. 

Such a formulation of the temperature dependence of the rate constant is more satisfying in that it relates rate constants properly to thermodynamic equilibrium constants in the case of reversible reactions. In the case of reversible reactions we see that the frequency factor v must be the same for the forward and back reactions, since 

If we take the logarithmic partial derivative of k with respect to T [from Eq. (IV.4.1)], we have 

The fact that very careful experimental results in solution show that d In k/d l/T) is not constant with temperature (sec Table IV.l) but does change, indicates that such cancellation, while usual, is not exact. It should be observed that, from Eq. (IV.4.4), a variation of aS, the entropy of activation, with temperature will have no influence on the variations of k with temperature, so that experimentally we can only hope to distinguish the quantities // and the product A = Only some 

Compared to the first version described above, the dissociative electron transfer model has been improved further by a more accurate estimation of the activation entropy, which takes into account that R and X- are formed within a solvent cage from which they diffuse out successively.12 The free energy and entropy of activation are thus obtained from 

ELECTRON TRANSFER, BOND BREAKING, AND BOND FORMATION 

In the preceding sections, interactions between radicals and ions in the product cluster have been neglected. There is little doubt that such interactions exist 

Equation (3.13) indicates that a shallow minimum (DP c Dr) corresponds to a loose cluster (yP yn). Equations (3.11) and (3.12) may be recast as 

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In order to separate the enthalpy and the entropy of activation, the rate is measured as a fiinction of temperature. These data should give a straight line on an Eyrmg plot of log(rate/7) against (1/7) (figure... 

Figure B2.4.2. Eyring plot of log(rate/7) versus (1/7), where Jis absolute temperature, for the cis-trans isomerism of the aldehyde group in fiirfiiral. Rates were obtained from tln-ee different experiments measurements (squares), bandshapes (triangles) and selective inversions (circles). The line is a linear regression to the data. The slope of the line is A H IR, and the intercept at 1/J = 0 is A S IR, where R is the gas constant. A and A are the enthalpy and entropy of activation, according to equation (B2.4.1)...

In principle the use of the entropy of activation as a criterion is straightforward. The electrostatic contribution to this quantity, A5 i, for a reaction between two cations is predicted from simple electrostatic theory to be less than that for a reaction between an ion and a neutral molecule. If the reactions are otherwise similar, the overall entropies of activation can be expected to differ in the same way ... 

An alternative approach is to assume, in the light of the experimental evidence just mentioned, that the reactions of cations and neutral molecules have similar values of (or, equivalently, of log ( /l mol and to try to calculate the difference which would arise from the fact that the observed entropy of activation for a minority free base includes a contribution from the acidic dissociation of the conjugate acid in the medium in question (see (5) above). Consider the two following reaction schemes one (primed symbols) represents nitration via the free base, the other the normal nitration of a non-basic majority species (unprimed symbols) ... 

Hydrolysis of dialkyl sulfites under acidic and alkaline conditions, which is followed by the use of OH2, proceeds by attack at sulfur to give S—O cleavage (72). The rate of hydrolysis is generally faster for cycHc and aryl sulfites than for dialkyl sulfites (73). Activation parameters of hydrolysis are known for some sulfites, and the increased rate for ethylene sulfite results from a reduced entropy of activation which results from a rigid ring stmcture (74). 

It is clear by comparing the uansition state theory with the collision model that the conesponding entropy of activation can be calculated from the value... 

CPR can be used to find continuous paths for complex transitions that might have hundreds of saddle points and need to be described by thousands of path points. Examples of such transitions include the quaternary transition between the R and T states of hemoglobin [57] and the reorganization of the retinoic acid receptor upon substrate entry [58]. Because CPR yields the exact saddle points as part of the path, it can also be used in conjunction with nonnal mode analysis to estimate the vibrational entropy of activation... 

Wide variation in enthalpy and entropy of activation for different reaction systems is possible, as illustrated by the following two reactions. 

Unimolecular reactions that take place by way of cyclic transition states typically have negative entropies of activation because of the loss of rotational degrees of freedom associated with the highly ordered transition state. For example, thermal isomerization of allyl vinyl ether to 4-pentenal has AS = —8eu. ... 

Reductions by NaBKt are characterized by low enthalpies of activation (8-13kcal/mol) and large negative entropies of activation (—28 to —40eu). Aldehydes are substantially more reactive than ketones, as can be seen by comparison of the rate data for benzaldehyde and acetophenone. This relative reactivity is characteristic of nearly all carbonyl addition reactions. The reduced reactivity of ketones is attributed primarily to steric effects. Not only does the additional substituent increase the steric restrictions to approach of the nucleophile, but it also causes larger steric interaction in the tetrahedral product as the hybridization changes from trigonal to tetrahedral. 

The rearrangement of the simplest possible case, 1,5-hexadiene, has been studied using deuterium labeling. The activation enthalpy is 33.5kcal/mol, and the entropy of activation is — 13.8eu. The substantially negative entropy reflects the formation of the cyclic transition state. 

When the expanded expression for Gibb s free energy is substituted for AG. as shown in Eq. 4, it becomes possible to calculate the enthalpy and entropy of activation when rates are available across a range of temperatures. These parameters are calculated in Table 6 using the data supplied by Zavitsas and Ferrero [80,85,90]. 

A number of groups have criticized the ideas of Dauben and Noyce, especially the concept of PDC. Kamernitzsky and Akhrem, " in a thorough survey of the stereochemistry of addition reactions to carbonyl groups, accepted the existence of SAC but not of PDC. They point out that the reactions involve low energies of activation (10-13 kcal/mole) and suggest that differences in stereochemistry involve differences in entropies of activation. The effect favoring the equatorial alcohols is attributed to an electrostatic or polar factor (see also ref. 189) which may be determined by a difference in the electrostatic fields on the upper and lower sides of the carbonyl double bond, connected, for example, with the uncompensated dipole moments of the C—H bonds. The way this polar effect is supposed to influence the attack of the hydride is not made clear. 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

For the isotropic continum model of the reaction of an ion with a neutral molecule [see Eq. (8-25)], obtain an expression for the electrostatic entropy of activation. 

The mechanism of the cycloaddition of phenyl azide to norbornene has been shown to involve a concerted mechanism with a charge imbalance in the transition state (199). In a similar manner the cycloaddition of phenyl azide to enamines apparently proceeds by a concerted mechanism (194, 194a). This is shown by a rather large negative entropy of activation (—36 entropy units for l-(N-morpholino)cyclopentene in benzene solvent at 25°C), indicative of a highly ordered transition state. Varying solvents from those of small dielectric constants to those of large dielectric constants has... 

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