Degree of rate control

Campbell (29) suggested that the kinetic importance of a particular step in a reaction scheme can be ascertained by computing the effect on the overall rate of increasing the forward and reverse rate constants for that step while maintaining the same value of the equilibrium constant for the step. According to Campbell, the degree of rate control for step i, Arc, is equal to... 

We now compute Campbell s degree of rate control in terms of the sensitivities for the forward and reverse rate constants. We compute the change in the overall rate, Sr, resulting from a change in kt by Ski and a change in k, by SkilKifiq to maintain the same equilibrium constant ... 

Therefore, Campbell s degree of rate control, XRCl, provides an excellent measure of the sensitivity of the overall reaction rate to the kinetic parameters for each step. The value of A"rc>( approaches zero as step i becomes quasi-equilibrated, and the value of 3lrC, becomes small as the preceding steps that produce the reaction intermediates for step i become irreversible. 

Table XI shows the steps with the highest degree of rate control for representative points at low and high temperatures. The most sensitive steps involve hydride transfer of surface species with isobutane and oligomerization/ P -scission steps involving predominantly adsorbed C and C species.

Steps with the Highest Degree of Rate Control (Xrqi)... 

The concept of a rate-determining step (RDS) is common and useful in heterogeneous catalysis. It has been used as an a priori assumption in the development of reduced rate expressions from experimental data [4]. Knowledge of the RDS can provide insights into how to improve a catalyst. The definition of the RDS has been discussed in the past two decades [51, 52]. Considering the definition proposed by Campbell [51], the degree of rate control by an elementary step is... 

Figure 8.8 displays the three reactions in the WGS model with the largest normalized sensitivity coefficients. These were computed using Equation (8.35), except that f. is the preexponential factor of reaction j, This is similar to the degree of rate control defined in Equation (8.34) except that the overall conversion is used rather than the reaction rate. Important insights from this plot are that the sensitivity (kinetic relevance) of a reaction depends on reaction conditions, for example, temperature, and that there is not always a single rate-determining step rather, multiple reactions can be simultaneously kinetically important. 

Reuter K, Scheffler M. First-principles kinetic Monte Carlo simulations for heterogeneous catalysis Application to the CO oxidation at RuO2(110). Phys Rev B 2006 73 045433. Stegelmann C, Andreasen A, Campbell CT. Degree of rate control How much the energies of intermediates and transition states control rates. J Am Chem Soc 2009 131 8077. 

This derivative corresponds to a change in the overall rate resulting from a change in k, by Ski a change in Sk-i by dfe,/K, gq maintaining the same equihbrium constant. The degree of rate control is then given by ... 

Figure 1.15 Illustration of the generalized degree of rate control Xrc and degree of catalyst control Xcc (dashed PES) for CH4 steam reforming. Xrc(CH4-TS) = 0.8, Xrc(C) = -0.26, Xcc(C) = 0.11.

Equivalently, one can change the free energy of an intermediate n and calculate its degree of rate control. This is schematically shown for the free energy of C in Figure 1.15 and one can define the thermodynamic degree of rate control, Ztrc, , for an intermediate n as... 

Now that we are able to predict a reaction rate for a single catalyst, only a few lines of extra code are needed to calculate the degree of rate control and the degree of catalyst control. However, we need to solve the microkinetic model repeatedly for different parameters and it is more convenient if we first write a function, solve ode, that takes the rate eonstants as input and returns the solution of the model. This function may look like this. 

The degree of rate control, Zrc, is then calculated by repeatedly calling solve ode with rate constants that are systematically changed for each step. 

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