Free energy of activation for

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... 

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. 

The formation of the complex is expected to decrease the free energy of activation for the homolysis of the peroxide bond, and the decomposition of TBHP would occur at a lower temperature. It was further observed that at a higher concentration of mineral acid, the decomposition of TBHP occurs via an ionic pathway, as reported by Turner [27]. 

In the context of Scheme 11-1 we are also interested to know whether the variation of K observed with 18-, 21-, and 24-membered crown ethers is due to changes in the complexation rate (k ), the decomplexation rate (k- ), or both. Krane and Skjetne (1980) carried out dynamic 13C NMR studies of complexes of the 4-toluenediazo-nium ion with 18-crown-6, 21-crown-7, and 24-crown-8 in dichlorofluoromethane. They determined the decomplexation rate (k- ) and the free energy of activation for decomplexation (AG i). From the values of k i obtained by Krane and Skjetne and the equilibrium constants K of Nakazumi et al. (1983), k can be calculated. The results show that the complexation rate (kx) does not change much with the size of the macrocycle, that it is most likely diffusion-controlled, and that the large equilibrium constant K of 21-crown-7 is due to the decomplexation rate constant k i being lower than those for the 18- and 24-membered crown ethers. Izatt et al. (1991) published a comprehensive review of K, k, and k data for crown ethers and related hosts with metal cations, ammonium ions, diazonium ions, and related guest compounds. 

FIGURE 1-10 Effect of a change in the applied potential on the free energies of activation for reduction and oxidation. 

These techniques are known as linear free energy relations, LFER. Imagine that one has determined the rate constants, or the Gibbs free energies of activation, for a series of reactions. The reactions are all the same, save for (for example) a different substituent on each reactant. The substituent is not a direct participant in the reaction. In an LFER, the values of log k or AG are correlated with some characteristic of the substituent as manifested in another reaction series. If the correlation is successful, then the two series of reactions have a common denominator. This technique has proved to be a powerful one for systematizing reactivity. We shall see a number of such correlations. 

The free energies of activation for the one reaction series are directly proportional to the standard free energy changes for another. This form is emphasized by Eq. (10-3), and is what gives rise to the designation of this approach as an LFER. 

One derivation, devised by Ratner and Levine,31 elegant in its simplicity, is based on thermodynamics and TST. They assume that the transition states for Eqs. (10-67)-(10-69)-— AA / BB +, and AB —have components with individual and transferable free energies that is, they assume that the partners are activated independently. We represent the components as G(A f), G(A ), G(B t), and G(B ). Then the free energies of activation for the EE andET reactions are... 

FIGURE 6.2 (a) Free energy profile for a reaction with an intermediate. AG and AG are the free energy of activation for the first and second stages respectively, (b) Free energy profile for a reaction with an intermediate in which the first peak is higher than the second. 

The iminophosphoranes (181) with imide bromides gave the amidino-phosphonium bromides (183) which, it appears from their temperature-variable n.m.r. spectra, are interesting fluxional molecules. The free energy of activation for the interconversion of (183a) and (183b) (R R = Pr R3 = Ph) is 17.2 0.9kcalmol-i. 

H/D exchange of H and Hg protons of sulfone 86 and estimated the difference in the free energies of activation for 79a and 79b to be < 1.2 kcal mol , based on the kjk value of 3 0.5. In the base-catalyzed H/D exchange of 87, kjk = 1.6, where k and k are the rate constants of H/D exchange of H, and H, respectively. Based on the small kjk value. Brown and colleagues suggested that if the carbanion is pyramidal, the steric stabilities of 79a and 79b are almost identical. Meanwhile, based on their C-NMR study Chassaing and Marquet proposed that the hybridization of the carbon atom of the sulfonyl carbanion, PhSOjCHj , would be between sp and sp . 

According to Marcus theory [69-71], in the absence of work terms, the Gibbs free energy of activation for an ET reaction is given by ... 

Alternatively one can make use of No Barrier Theory (NBT), which allows calculation of the free energy of activation for such reactions with no need for an empirical intrinsic barrier. This approach treats a real chemical reaction as a result of several simple processes for each of which the energy would be a quadratic function of a suitable reaction coordinate. This allows interpolation of the reaction hypersurface a search for the lowest saddle point gives the free energy of activation. This method has been applied to enolate formation, ketene hydration, carbonyl hydration, decarboxylation, and the addition of water to carbocations. ... 

Our results show that the calculated potential energies for the TS obtained from the combined procedure are around 4 kcal/mol lower than the corresponding ones calculated with the CD method. In contrast, the calculated free energies of activation for all four paths differ by not more than 1.6 kcal/mol. These results show that the inclusion of the fluctuation of the MM environment dramatically improves the results of the calculations of enzymatic catalysis, even if the calculated PES is not highly accurate. In addition, the calculation of free energies for multiple paths using the QM/MM-FE method can serve as an alternative to more expensive sampling methods such as QM/MM-MFEP and QM/MM-MD. 

Figure 6.7 A free-energy diagram for the SnI reaction of tert-butyl chloride with water. The free energy of activation for the first step, AC (l), is much larger than AG (2) or AG (3). TS(1) represents transition state (1),...

The order of free energy of activation for dehydration of alcohols is3°>2°>l°> methyl ... 

Figure 7.8 Free-energy diagram for the hydrogenation of an alkene in the presenceof a catalyst and the hypothetical reaction in the absence of a catalyst. The free energy of activation [AG (i)] is very much larger than the largest free energy of activation for the catalyzed reaction [AG (2,].

The free energies of activation for these two reactions of carbocations are not very different from one another. 

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