Rate equilibrium

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

From the intercept at AG° = 0 we find AGo = 31.9 kcal mol , and the slope is 0.77. As we have seen, if Eq. (5-69) is applicable, the slope should be 0.5 when AG = 0. In this example either the data cover too small a range to allow a valid estimate of the slope to be made or the equation does not apply to this system. Such a simple equation is not expected to be universally applicable. Recall that it was derived for an elementary reaction, so multistep reactions, even if showing simple rate-equilibrium behavior, introduce complications in the interpretation. The simple interpretation of Eq. (5-69) also requires that AGo be constant within the reaction series, but this condition may not be met. Later pages describe another possible reason for the failure of Eq. (5-69). 

Figure 5-17. Rate-equilibrium relationship according to Eq. (5-69) for a hypothetical system with AG = 20.

Figure 5-18. Rate-equilibrium plot for the isomerization of substituted 5-aminotriazoles at 423 K. ...

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. ... 

There are several equations other than the Marcus equation that describe rate-equilibrium relationships. Murdoch writes all of these equations in the general form... 

A prediction of AE /AEq to within 0.1 kcal/mol may produce a AG /AGq accurate to maybe 0.2 kcal/mol. This corresponds to a factor of 1.4 error (at T = 300 K) in the rate/equilibrium constant, which is poor compared to what is routinely obtained by experimental techniques. Calculating AG /AGq to within 1 kcal/mol is still only possible for fairly small systems. This corresponds to predicting the absolute rate constant, or the equilibrium distribution, to within a factor of... 

The first procedure is to use the rate equilibrium equations for electrons and ions at the quasi-steady state for a new guess of the particle densities. In the final result, quasi-steady balances of the positive ions and negative ions integrated over... 

A limit to mass transfer is attained if two phases come to equilibrium and the net transfer of material comes to a halt. For a process in practice, which must have a reasonable production rate, equilibrium must be avoided, as the rate of mass at any point is proportional to the compelling or driving force, which is the departure from equilibrium at that point. In order to evaluate driving forces, a knowledge of equilibria between phase is therefore fundamentally important. Several kinds of equilibria are important in mass transfer. 

More advanced scale was proposed by Kamlet and Taft [52], This phenomenological approach is very universal as may be successfully applied to the positions and intensities of maximal absorption in IR, NMR (nuclear magnetic resonance), ESR (electron spin resonance), and UV-VS absorption and fluorescence spectra, and to many other physical or chemical parameters (reaction rates, equilibrium constant, etc.). The scale is quite simple and may be presented as ... 

Table 2 gives rate and equilibrium constants for the deprotonation of and nucleophilic addition of water to X-[6+]. These data are plotted as logarithmic rate-equilibrium correlations in Fig. 5, which shows (a) correlations of log ftp for deprotonation of X-[6+] and log Hoh for addition of water to X-[6+] with logXaik and log KR, respectively (b) correlations of log(/cH)soiv for specific-acid-catalyzed cleavage of X-[6]-OH (the microscopic reverse of nucleophilic addition of water to X-[6+]) and log( H)aik for protonation of X-[7] (the microscopic reverse of deprotonation of X-[6+]) with log Xafc and log XR, respectively. 

Fig. 5 Logarithmic plots of rate-equilibrium data for the formation and reaction of ring-substituted 1-phenylethyl carbocations X-[6+] in 50/50 (v/v) trifluoroethanol/water at 25°C (data from Table 2). Correlation of first-order rate constants hoh for the addition of water to X-[6+] (Y) and second-order rate constants ( h)so1v for the microscopic reverse specific-acid-catalyzed cleavage of X-[6]-OH to form X-[6+] ( ) with the equilibrium constants KR for nucleophilic addition of water to X-[6+]. Correlation of first-order rate constants kp for deprotonation of X-[6+] ( ) and second-order rate constants ( hW for the microscopic reverse protonation of X-[7] by hydronium ion ( ) with the equilibrium constants Xaik for deprotonation of X-[6+]. The points at which equal rate constants are observed for reaction in the forward and reverse directions (log ATeq = 0) are indicated by arrows.

We propose that potential surfaces for gas-phase S 2 reactions may be correlated using Marcus rate-equilibrium... 

These results suggest that the Marcus equations can be applied quite successfully to gas phase displacement reactions, as suggested by Professor Brauman. We are currently generating more cross reactions and intend to test other rate-equilibrium relationships using our data. 

Many possible permutations of rate, equilibrium, and contacting pattern can be imagined however, only some of these are important in the sense that they are widely used on the technical scale. 

Figure 1.3. Rate-equilibrium correlation for hydration of carbocations triarylmethyl and 9-aryl-9-fluorenyl ( and , respectively, slope =—0.60) diarylmethyl (A, —0.54) aryltropylium (O, —0.68) 9-xanthylium and cyclic phenyldialkoxycarbocations ( and A, respectively, slope = —0.63).

When do rates and equilibria correlate in organic chemistry, and why do rate-equilibrium relationships break down ... 

Normal rate-equilibrium relationships are expected for methyl SN2 processes but the Bronsted parameter a is not a measure of TS charge development in these cases (Pross, 1984). 

The existence of rate-equilibrium relationships in the SN2 reactions of simple alkyl derivatives is well established (Arnett and Reich, 1980 Lewis and Kukes, 1979 Lewis et al., 1980 Bordwell and Hughes, 1982). A plot of the rate constants for a family of nucleophiles against the pKa for the nucleophiles generates linear Brensted plots whose slopes lie in the range 0.3 to 0.5. A typical example, taken from Bordwell s work (Bordwell and Hughes, 1982) is the reaction of a family of aryl thiolates with -butyl... 

An attempt to establish a rate-equilibrium relationship shows that a plot of E X -I- CH3X) as a function of the methyl cation affinity (MCA) of X-yields the linear correlation (44), where 7P(CH3) represents the ionization... 

---