Chapter 4: Stoichiometry
Rate Law in terms of Partial Pressures
An example of a typical rate law found in the catalystic engineering literature for the gas phase catalyzed reaction.
\( A + B \xrightarrow{\text{Cat}} C \)
If the reaction follows an Eley-Rideal mechanism (Ch 10) it will take the form
\(-r'_A = \frac{kP_A P_B}{1 + K_A P_A + K_C P_C}\)
where k is the specific reaction rate \(k = 0.1 \frac{(\text{mol})}{\text{kg} \cdot \text{cat} \cdot \text{s} \cdot (\text{atm})^2}\) and KA and KC are the equilibrium adsorption constant of KA = 2 atm-1 and KC = 0.5 atm-1.
(a) Express as a function of conversion for an entering partial pressure of 8 atm Using the ideal gas law we know
\(y = \frac{P}{P_0}\)
\(P_{A0} = C_{A0} R T_0\)
\(P_A = C_A RT = \frac{C_{A0}(1-X)}{(1+\epsilon X)} \left( \frac{P}{P_0} \frac{T_0}{T} \right) RT\)
\(P_A = C_{A0} R T_0 \frac{(1-X)}{(1+\epsilon X)} y\)
| \(P_A = P_{A0} \frac{(1-X)}{(1+\epsilon X)} y\) |
\(P_B = C_B RT = C_{A0} \frac{(\Theta_B - X)}{(1+\epsilon X)} \frac{P}{P_0} \frac{T_0}{T} RT\)
| \(P_B = P_{A0} \frac{(\Theta_B - X)}{(1+\epsilon X)} y\) |
similarly
| \(P_C = P_{A0} \frac{(\Theta_C + X)}{(1+\epsilon X)} y\) |
| \(-r'_A = \frac{k P_{A0}^2 (1-X)(\Theta_B - X)}{(1+\epsilon X)^2 + K_A P_{A0} (1-X)(1+\epsilon X) + K_C P_{A0} X (1+\epsilon X)}\) |
For an equal molar feed
\(\Theta_B = 1, \; \Theta_C = 0 \; \text{and} \; \epsilon = y_{A0} \delta = \frac{1}{2}(1-1-1) = -\frac{1}{2}\)
\(-r'_A = \frac{\frac{10 \; \text{mol}}{\text{kg cat} \cdot \text{s} \cdot \text{atm}^2} (8 \; \text{atm})^2 (1-X)^2}{(1-0.5X)^2 + (0.5)(8)(1-X)(1-0.5X) + (0.25)(8)(X)(1-0.5X)}\)
\(-r'_A = \frac{640 (1-X)^2}{(1-0.5X)^2 + 4(1-X)(1-0.5X) + 2X(1-0.5X)}\)
We now can solve for the catalyst weight to achieve a specific conversion by combining -r`A in terms of X with the mole balance
\(\frac{\text{d}X}{\text{d}W} = \frac{-r'_A}{F_{A0}}\)