Chapter 10: Catalysis and Catalytic Reactors

Strictly Speaking

The surface reaction is rate limiting. We need to further discuss setting (r_AD/k_A)~0 to arrive at C_C.S=K_IP_IC_v. We are going to do this by first using the total number of sites, C_T [i.e. C_T=C_v+C_A.S+C_B.S], and then use the fractional surface coverage [1=f_v+f_A.S+f_B.S].

\( C + S \leftrightarrow C \bullet S \)

\( C \bullet S \leftrightarrow B \bullet S + P \)

\( B \bullet S \leftrightarrow B + S \)

For steady state operation we have:

\( -r'_C = r_{AD} = r_S = r_D \)

where:

\( r_{AD} = k_A P_C \left[ C_{\vee} - \frac{C_{C \bullet S}}{K_C P_C} \right], \quad k_A = \left( \frac{1}{\text{s} \cdot \text{atm}} \right), \quad k_A P_C = \left( \frac{1}{\text{s}} \right) \)

\( r_S = k_S \left[ C_{C \bullet S} - \frac{C_{B \bullet S} P_P}{K_S} \right], \quad k_S = \left( \frac{1}{\text{s}} \right) \)

\( r_D = k_D \left[ C_{B \bullet S} - \frac{C_{\vee} P_B}{K_D} \right], \quad k_D = \left( \frac{1}{\text{s}} \right) \)

NOTE: Strictly speaking, We really cannot compare the magnitudes of k_A, k_S, and k_D directly, because k_A has different units than k_S and k_D. Consequently, we must compare the product (k_AP_C) with k_S and k_D to determine which reaction step may be limiting. If the surface reaction is limiting, we say k_AP_C and k_D are very large with respect to k_S:

\( r_S = \frac{k \left[ P_C - \frac{P_B P_P}{K_P} \right]}{1 + K_C P_C + K_B P_B} \)

Strictly speaking, We only take ratios of quantities that have the same units.

\( r_{AD} = k_A \left[ P_A C_{\vee} - \frac{C_{A \bullet S}}{K_A} \right] \)

If \( f_{\vee} \) and \( f_{A \bullet S} \) are the fraction of free sites and the fraction of covered sites, respectively, and \( y_A \) is the gas phase mole fraction of species A, then:

\( f_{A \bullet S} = \frac{C_{A \bullet S}}{C_t} \)

\( f_{\vee} = \frac{C_{\vee}}{C_t} \)

\( P_A = y_A \, P \)

and

\( r_{AD} = k_A P C_t \left[ y_A f_{\vee} - \frac{f_{A \bullet S}}{P K_A} \right] \)

For surface reaction control:

\( \frac{r_{AD}}{k_A P C_T} \approx 0 = y_A f_{\vee} - \frac{f_{A \bullet S}}{P K_A} \)

\( f_{A \bullet S} = y_A f_{\vee} P_A K_A = f_{\vee} P_A K_A \)

Multiply both sides by C_t:

\( C_{A \bullet S} = C_{\vee} K_A P_A \)

which is identical to the expression derived in the text, assuming that:

\( \frac{r_{AD}}{k_A} \approx 0 \)

Developing the Rate Law

Now let's consider developing the rate law on the basis of fractional suface coverage, i.e. (r_AD/k_A)~0 and (k_ADB/k_B)~0.

\( \text{A} + \text{S} \leftrightarrow \text{A} \bullet \text{S} \)

\( \text{B} + \text{S} \leftrightarrow \text{B} \bullet \text{S} \)

\( \text{A} \bullet \text{S} + \text{B} \bullet \text{S} \leftrightarrow \text{C} \bullet \text{S} + \text{S} \)

\( \text{C} \bullet \text{S} \leftrightarrow \text{C} + \text{S} \)

Assume that the surface reaction is limiting, then:

\( -r'_A = r_S = \frac{k_S \left[ P_A P_B - \frac{P_C}{K_P} \right]}{\left(1 + K_A P_A + K_B P_B + K_C P_C \right)^2} \)

Derive Derive this equation

Table 10-4 Irreversible Surface-Reaction-Limited Rate Laws.

NOTE: This is the same rate law we would get by comparing the k's directly. Return to Chapter