GATE 2025 CH – Question 59

Chemical Reaction Engineering · Heterogeneous catalysis, diffusion effects and catalyst deactivation · 2 marks · Numerical answer

The catalytic gas phase reaction $A \to products$ is carried out in an isothermal batch reactor of 10 L volume using 0.1 kg of a solid catalyst. The reaction is first-order with $(-r_A) = k'a(t)C_A$ where $k' = 1\ \frac{\text{L}}{(\text{kg catalyst})(\text{h})}$ and $C_A$ is the concentration of $A$ in mol/L.
The catalyst activity $a(t)$ undergoes first-order decay with rate constant $k_d = 0.01$ per hour and $a(0) = 1$.
The reactant conversion after 1 day of operation is ____ (rounded off to 2 decimal places).

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Correct answer: 0.18 to 0.20

Explanation

The mole balance for the batch reactor gives $V\frac{dC_A}{dt} = -k'a(t)WC_A$, so $\frac{dC_A}{dt} = -\frac{k'W}{V}aC_A = -0.01\,aC_A$ per hour. With $a = e^{-0.01t}$: $\ln\frac{C_{A0}}{C_A} = 0.01\int_0^{24}e^{-0.01t}dt = 1 - e^{-0.24} = 0.2134$. So $\frac{C_A}{C_{A0}} = e^{-0.2134} = 0.8079$ and the conversion is $1 - 0.8079 = 0.19$.