Question

CO and H\(_2\) participate in a catalytic reaction. The partial pressures (in atm) of the reacting species CO and H\(_2\) in the feed stream are \(p_{CO}\) and \(p_{H_2}\), respectively. While CO undergoes molecular adsorption, H\(_2\) adsorbs via dissociative adsorption (as H atoms). The equilibrium constants (in atm\(^{-1}\)) corresponding to adsorption of CO and H\(_2\) to the catalyst sites are \(K_{CO}\) and \(K_{H_2}\), respectively. Total molar concentration of active sites per unit mass of the catalyst is \(C_t\) (in mol\(\cdot\)(g cat)\(^{-1}\)). Both the adsorption steps are at equilibrium. Which one of the following expressions is the CORRECT ratio of the concentration of catalyst sites occupied by CO to that by hydrogen atoms?

• A. \(\displaystyle \frac{K_{CO}\,p_{CO}}{\sqrt{K_{H_2}\,p_{H_2}}}\)
• B. \(\displaystyle \frac{K_{CO}}{\sqrt{K_{H_2}}}\)
• C. \(\displaystyle \frac{p_{CO}}{\sqrt{p_{H_2}}}\)
• D. \(\displaystyle \frac{K_{CO}\,p_{CO}}{K_{H_2}\,p_{H_2}}\)

Hint:

Molecular adsorption: \(K=\dfrac{\theta_A}{p_A(1-\theta)}\approx\dfrac{[A^*]}{p_A[S]}\).
Dissociative adsorption: \(K=\dfrac{[A^*]^2}{p_A[S]^2}\Rightarrow [A^*]\propto \sqrt{p_A}\).
Ratios often cancel the unknown free-site concentration \([S]\).

Answer 1

The Correct Option is A

Step 1: Write equilibrium relations using free site concentration \([S]\). \[ \text{Molecular adsorption: } CO(g)+S \rightleftharpoons CO^*, \quad K_{CO}=\frac{[CO^*]}{p_{CO}[S]} \;\Rightarrow\; [CO^*]=K_{CO}\,p_{CO}\,[S]. \] \[ \text{Dissociative adsorption: } H_2(g)+2S \rightleftharpoons 2H^*, \quad K_{H_2}=\frac{[H^*]^2}{p_{H_2}[S]^2} \;\Rightarrow\; [H^*]=\sqrt{K_{H_2}\,p_{H_2}}\,[S]. \] Step 2: Take the required ratio of site concentrations: \[ \frac{[CO^*]}{[H^*]}=\frac{K_{CO}\,p_{CO}\,[S]}{\sqrt{K_{H_2}\,p_{H_2}}\,[S]} =\boxed{\frac{K_{CO}\,p_{CO}}{\sqrt{K_{H_2}\,p_{H_2}}}}. \] (The free-site concentration cancels; \(C_t\) is not needed for this ratio.)