The Eyring equation

The equation above is identieal to Equation (6.16) describing the frequency of a molecular event. 

The resemblance of the large bracket to the sinh function should be noted. 

Assuming that the net flow in the forward direction is directly related to the rate of change of strain, we have 

The rate of strain equation (Equation (10.29)) defines an activated viscosity. 

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This guarded assessment does not mean that the effort involved in deriving the Eyring equation was wasted-far from it. Several points might be noted ... 

The effect of temperature on diffusivities in zeolite ciystals can be expressed in terms of the Eyring equation (see Ruthven, gen. refs.). 

The free enthalpy of activation, aG, of the ring inversion at 253 K is calculated from the logarithmic form of the Eyring equation ... 

The values of the apparent rate constants kj for each temperature and the activation enthalpies calculated using the Eyring equation (ref. 21) are summarized in Table 10. However, these values of activation enthalpies are only approximative ones because of the applied simplification and the great range of experimental errors. Activation entropies were not calculated in the lack of absolute rate constants. Presuming the likely first order with respect to 3-bromoflavanones, as well, approximative activation entropies would be between -24 and -30 e.u. for la -> Ih reaction, between -40 and - 45 e.u. for the Ih la reaction and between -33 and -38 e.u. for the elimination step. These activation parameters are in accordance with the mechanisms proposed above. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

According to the Eyring equation [In P= — AAH RT + AAS R, where P -A (major diol)/ /c(minor diol)], an asymmetric reaction proceeding through a pair of diastereomeric transition states shows a linear temperature effect on the enantioselectivity. However, if the asymmetric... 

The Arrhenius theory (above) was wholly empirical in terms of it derivation. A more rigorous, but related, form of the theory is that of Eyring (also called the theory of absolute reaction rates). The Eyring equation is... 

It is extremely common, but wrong, to see the Eyring equation written in a similar form, with the Arrhenius ordinate of ln(fc) (as y ) and the final logarithmic term written as In(fcBr -t-fi). Although such an equation might be readily achieved from the brief derivation in Justification Box 8.6, it is seen straightaway to be nonsensical, for the following three reasons ... 

We will employ the form of the Eyring equation written as Equation (8.56). 

Experimental values.b Calculated from the Eyring equation. 

It is important to remember that the Marcus model refers to a weakly electronically coupled model, as embodied in the term outer-sphere ET. Thus it must be assumed that the electronic overlap between the two reactants is so small that no quantum-chemical effects ensue, yet that there must be enough overlap for the transmission coefficient k of the Eyring equation to be equal to 1 (the reaction must be adiabatic). Usually, this minimum overlap requirement is put at a fairly low level, around 0.1 kcal mol-1, which causes no problems for most reactions involving at least one organic species. 

Tinker D., Lesher C.E., Baxter G.M., Uchida T., and Wang Y. (2004) High-pressure vis-cometry of polymerized silicate melts and limitations of the Eyring equation. Am. Mineral. 89, 1701-1708. 

The analysis of the curvature of the experimental parabola led to very reasonable determinations of the intrinsic barrier. The measured values are relatively large, ca. 10-13 kcal moP, i.e. larger than usually found in stepwise dissociative processes but still not as large as found with other dissociative-type acceptors, such as halides. On the other hand, if the intrinsic barriers are calculated by the Eyring equation (equation 4) the values are larger by a few kcal mol (using the collision frequency factor Z). This is because the heterogeneous ET is actually non-adiabatic (which means that the actual pre-exponential factor is smaller). This is a very important aspect, which will be covered below. 

From the derivative of these plots, the potential dependence of a was obtained and the for each disulfide was estimated, using the approach described in Section 2. The double-layer uncorrected E° values and corresponding standard rate constants were optimized by reproducing the experimental curves by digital simulation. The data are reported in Table 11. By using the Eyring equation (4) with the pertinent pre-exponential... 

The values of the standard rate constants vary considerably along the series of disulfides, the logk°i,et going from —0.80 for the nitro to —4.36 for the methoxy derivatives. Again, as observed for the other class of disulfides, most of them are unusually low for substrates undergoing a stepwise mechanism. By using the Eyring equation (4), the values of the intrinsic barriers AG, reported in Table 12 were obtained. 

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