10.3.1 Is the Adsorption of Cumene Rate-Limiting?

To answer this question we shall assume that the adsorption of cumene is indeed rate-limiting, derive the corresponding rate law, and then check to see if it is consistent with experimental observation. By postulating that this (or any other) step is rate-limiting, we are assuming that the reaction rate constant of this step (in this case k_A) is small with respect to the specific rates of the other steps (in this case k_S and k_D).¹⁰ The rate of adsorption is

10-25

Because we can measure neither C_v or C_{C · S}, we must replace these variables in the rate law with measurable quantities for the equation to be meaningful.

For steady-state operation we have

10-30

For adsorption-limited reactions, k_A is very small and k_S and k_D are very large. Consequently, the ratios r_S/k_S and r_D/k_D are very small (approximately zero), whereas the ratio r_AD/k_A is relatively large.

The surface reaction rate law is

10-31

Again, for adsorption-limited reactions, the surface-specific reaction rate k_S is large by comparison, and we can set

10-32

and solve Equation (10-31) for C_{C · S}:

10-33

To be able to express C_{C · S} solely in terms of the partial pressures of the species present, we must evaluate C_{B · S}. The rate of desorption of benzene is

10-29

However, for adsorption-limited reactions, k_D is large by comparison, and we can set

10-34

and then solve Equation (10-29) for C_{B · S}:

10-35

After combining Equations (10-33) and (10-35), we have

10-36

Replacing C_{C · S} in the rate equation by Equation (10-36) and then factoring C_v, we obtain

10-37

We observe that at equilibrium that r_AD = 0 and Equation (10-37) rearranges to

We also know from thermodynamics (Appendix C) that for the reaction

also at equilibrium we have the following relationship for partial pressure equilibrium constant K_P:

Consequently, the following relationship must hold

10-38

The equilibrium constant can be determined from thermodynamic data and is related to the change in the Gibbs free energy, ΔG°, by the equation (see Appendix C)

10-39

where R is the ideal gas constant and T is the absolute temperature.

The concentration of vacant sites, C_v, can now be eliminated from Equation (10-37) by utilizing the site balance to give the total concentration of sites, C_t, which is assumed constant¹¹:

Because cumene and benzene are adsorbed on the surface, the concentration of occupied sites is (C_{C · S} + C_{B · S}), and the total concentration of sites is

10-40

Substituting Equations (10-35) and (10-36) into Equation (10-40), we have

Solving for C_υ, we have

10-41

Combining Equations (10-41) and (10-37), we find that the rate law for the catalytic decomposition of cumene, assuming that the adsorption of cumene is the rate-limiting step, is

10-42

We now wish to sketch a plot of the initial rate of reaction as a function of the partial pressure of cumene, P_C0. Initially, no products are present; consequently, P_P = P_B = 0. The initial rate is given by

10-43

If the cumene decomposition is adsorption rate limited, then the initial rate will be linear with the initial partial pressure of cumene, as shown in Figure 10-15.

Figure 10-15. Adsorption-limited reaction.

Before checking to see if Figure 10-15 is consistent with experimental observation, we shall derive the corresponding rate laws for the other possible rate-limiting steps and then develop initial rate plots for the case when the surface reaction is rate-limiting and then for the case when the desorption of benzene is rate-limiting.