Free-energy of activation

For the steady-state case, Z should also give the forward rate of formation or flux of critical nuclei, except that the positive free energy of their formation amounts to a free energy of activation. If one correspondingly modifies the rate Z by the term an approximate value for I results ... 

Within this framework, by considering the physical situation of the electrode double layer, the free energy of activation of an electron transfer reaction can be identified with the reorganization energy of the solvation sheath around the ion. This idea will be carried through in detail for the simple case of the strongly solvated... 

From stochastic molecnlar dynamics calcnlations on the same system, in the viscosity regime covered by the experiment, it appears that intra- and intennolecnlar energy flow occur on comparable time scales, which leads to the conclnsion that cyclohexane isomerization in liquid CS2 is an activated process [99]. Classical molecnlar dynamics calcnlations [104] also reprodnce the observed non-monotonic viscosity dependence of ic. Furthennore, they also yield a solvent contribntion to the free energy of activation for tlie isomerization reaction which in liquid CS, increases by abont 0.4 kJ moC when the solvent density is increased from 1.3 to 1.5 g cm T Tims the molecnlar dynamics calcnlations support the conclnsion that the high-pressure limit of this unimolecular reaction is not attained in liquid solntion at ambient pressure. It has to be remembered, though, that the analysis of the measnred isomerization rates depends critically on the estimated valne of... 

Finally, exchange is a kinetic process and governed by absolute rate theory. Therefore, study of the rate as a fiinction of temperature can provide thennodynamic data on the transition state, according to equation (B2.4.1)). This equation, in which Ids Boltzmaim s constant and h is Planck s constant, relates tlie observed rate to the Gibbs free energy of activation, AG. ... 

As an example, experimental kinetic data on the hydrolysis of amides under basic conditions as well as under acid catalysis were correlated with quantitative data on charge distribution and the resonance effect [13]. Thus, the values on the free energy of activation, AG , for the acid catalyzed hydrolysis of amides could be modeled quite well by Eq. (5)... 

The jump frequency is related in an exponential fashion to the free energy of activation between the ground state and the saddle point ... 

This free energy is referred to as the free energy of activation. The rate of the reaction is then given by... 

The temperature dependence of reaction rates permits evaluation of the enthalpy and entropy components of the free energy of activation. The terms in Eq. (4.4) corresponding to can be expressed as... 

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 identity reactions, the intrinsic barriers are their free energies of activation, which can be determined by tracer studies or less directly by rate-equilibrium correlations. ... 

The dependence of the rate constant on pressure provides another activation parameter of mechanistic utility. From thermodynamics we have (dGldP)T = V, where V is the molar volume (partial molar volume in solutions). We define the free energy of activation by AG = G — SGr. where SGr is the sum of the molar free energies of the reactants. Thus, we obtain... 

A catalyst is a substance that makes available a reaction path with a lower free energy of activation than is available in its absence. (The catalyst does not lower AG the uncatalyzed reaction path remains available.)... 

In Eq. (7-21), AGo is the intrinsic barrier, the free energy of activation of the (hypothetical) member of the reaction series having AG" = 0. It is evident that the Marcus equation predicts a nonlinear free energy relationship, although if a limited... 

The second possible cause of nonlinearity is a change in mechanism. Within a reaction series any change in mechanism must be such as to provide a smaller free energy of activation for the reaction (otherwise the mechanism would not change). If a substituent effect can produce a change in mechanism, the result must therefore be curvature that is concave upward. Figure 7-2 is a per plot for the S l solvolyses... 

Figure 8-3. Plot of free energy of activation for the Menschutkin reaction EtjN + EtI Et4N + 1 against the Kirkwood dielectric constant function. Data are from Table 8-5, where the solvents are identified.

FIGURE 16.1 Enzymes catalyze reactions by lowering the activation energy. Here the free energy of activation for (a) the uncatalyzed reaction, AGu, is larger than that for (b) the enzyme-catalyzed reaction, AG,". 

These large rate accelerations correspond to substantial changes in the free energy of activation for the reaction in question. The urease reaction, for example. 

Based on the reaction scheme shown below, derive an expression for k /k, the ratio of the rate constants for the catalyzed and uncatalyzed reactions, respectively, in terms of the free energies of activation for the catalyzed (AGe ) and the uncatalyzed (AG ) reactions. 

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