Composite Rate Constants

As presented in Chapters 3-6, a great many systems are characterized by a composite rate constant made up of two or more elementary rate constants or equilibrium constants. Several such situations can be recognized, and we shall take up each in turn. 

The first case is one in which the rate constant is strictly a product or quotient of constants. Two cases illustrate this. One features a rapid prior equilibrium, 

The composite rate constant is k = k2Ka. To explore its temperature profile we write a transition state equation, or Arrhenius equation, for the rate constant k2, and the van t Hoff equation for Ka. In the TST notation, the rate constant for Eq. (7-20) becomes 

Clearly, the temperature profile is linear. The activation parameters are the sums shown in general, a sum of entropies and enthalpies is the result when constants are multiplied. If values of AS% and Aare known independently, from the temperature dependence of Ka for example, one can then calculate AS and AH by difference. 

If AH% is negative, with an absolute value smaller than that of A//f, then the quantity of A// -I- Ais a negative number. In such a case the rate constant for the small reaction will have an apparent negative activation enthalpy (energy). That is, the rate will decrease with increasing temperature. 

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One more case remains to be considered for composite rate constants. This is the one in which the denominator contains a summation. A mechanism and steady-state rate law with this feature are... 

If the rate constant is any sort of a composite, then the pressure profile will not be linear. Very accurate values of both k and P would be needed to resolve the matter. The existence of a composite rate constant is not at all easily distinguished from the effect of... 

Composite rate constants. Consider reaction (7-30). Why is a plot of ln [P ]/[P2] versus l IT linear What is the significance of its slope ... 

The expression for Eq. (8-14) can be interpreted in terms of the activation energy for a composite rate constant as explained in Chapter 7. In those terms Ea is... 

This composite rate constant is predicted to have a different ionic strength dependence for the two schemes. According to the Br0nsted-Debye-Hiickel equation, the composite rate constant for Eq. (9-76) will be independent of ionic strength if Scheme ... 

Acid-base catalysis, 232-238 Brqnsted equation for, 233-236 general, 233, 237 mechanisms for, 237 specific, 232-233, 237 Activated complex (see Transition state) Activation enthalpy, 10, 156-160 for composite rate constants, 161-164 negative, 161 Activation parameters, 10 chemical interpretation of, 168-169 energy of activation, Ea, 10 enthalpy of activation (A// ), 10, 156-160... 

Competition reactions ad eosdem, 106 ad eundem, 105 (See also Reactions, trapping) Competitive inhibitor, 92 Complexation equilibria, 145-148 Composite rate constants, 161-164 Concentration-jump method, 52-55 Concurrent reactions, 58-64 Consecutive reactions, 70, 130 Continuous-flow method, 254—255 Control factor, 85 Crossover experiment, 112... 

If the concentration of the active drug, A, can be monitored, the composite rate constant, k = k + k2 + ky, can easily be determined from the relationship [A] = [A]0e fe where [A]0 is the initial concentration and [A] is the concentration at time t. If the concentrations of A cannot be determined because of assay difficulties, it is still possible to determine k by monitoring one of the degradation products. For example, if the concentrations of B can be assayed as a function of time, and the concentration of B at time infinity, [B], is also determined, the following relationships can be derived ... 

In practice, measurement of the individual rate constants or equilibrium constants for these various chemical steps requires specialized methodologies, such as transient state kinetics (see Johnson, 1992, Copeland, 2000, and Fersht, 1999, for discussion of such methods) and/or a variety of biophysical methods for measuring equilibrium binding (Copeland, 2000). These specialized methods are beyond the scope of the present text. More commonly, the overall rate of reaction progress after ES complex formation is quantified experimentally in terms of a composite rate constant given the symbol km. 

Although fccat is a composite rate constant, representing multiple chemical steps in catalysis, it is dominated by the rate-limiting chemical step, which most often is the formation of the bound transition state complex ES from the encounter complex ES. Thus, to a first approximation, we can consider kCM to be a first-order rate constant for the transition from ES to ES ... 

Also, Higuchi and Senju [7] have proposed that the overall rate constant is composed of three distinct rate constants corresponding to the hydrolysis of the three possible triad configurations, AAA, AAB, and BAB, and have found that the relative reactivity is 1 0.25 0.005. Thus, the overall rate is determined by the relative proportions of these configurations and a relative composite rate constant K can be derived as follows ... 

Table 2. Triad Distributions and Composite Rate Constants (K) for 30...

The composite rate constant k summarized by Equation 7 has a maximum near 302°C. The reaction order r is nearly constant at low temperatures and increases dramatically at higher temperatures. 

Pseudo unimolecular rate constants k for sulfuric acid-catalysed solvolysis of 25c in CD3CN/D20 (adjusted to a constant ratio of 3.8 1) were found to be linearly dependent upon the acid concentration (Fig. 11) and the gradient afforded a composite rate constant of (2.41+0.10) x 10 2lmol 1 s-1 at 308K. From the intercept, ka, the rate constant for uncatalysed solvolysis, was at least three orders smaller and zero within experimental error. A similar linear dependence and near-zero uncatalysed rate constant was demonstrated for other /V-acetoxy-TV-alkoxybenzamides given in Table 3. 

For such dieidic systems we can define a composite rate-constant kp, given by the traditional equation introduced originally for anionic polymerisations... 

On this basis, A is a composite rate constant A = AT.ATjAj... 

Figure 1. Free energy diagram of a three-step process, showing each term in the expression for the composite rate constant /ci,2,s in the forward direction. Reproduced with permission of the authors and the American Chemical Society.

For the more complex situation with other P-hydride transfers, additional terms would be needed to account for those reactions. An alternate approach is to use Eq. 8-50 with ks and ktrjA redefined as composite rate constants for the sum of all P-hydride transfers. 

Setting the right-hand side of Eq. (E) to 0 and using the composite rate constant defined by Eq. (F), one can show (see Problem 11) that... 

Fjg. 37. Composite rate constants for interaction of Argon plasmas with ethylene tetrafluoro-ethylene copolymers obtained from analysis of the F js levels... 

Fig. 39. Three dimensional plot of composite rate constants for surface reaction as a function of power and pressure in an industively coupled RF plasma...

In Equation 11.13, A = k2k3[E]0[S]0/(k2 + k3) / [S]0 + k3Ks/(k2 + k3), which has the form of a Michaelis-Menten equation, B = [E]0[S]0 /(fe + 3) 2/([S]0+ Km(apparent)), and b is a composite rate constant describing the build-up of the acyl enzme intermediate (or, in the general case, the covalently bound enzyme intermediate). The non-linear plot of [Lg ] against time is shown in Fig. 11.10A for a typical substrate of a-chymotrypsin extrapolation of the linear portion gives the intercept shown which allows evaluation of B. 

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