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GATE 2024 CH – Question 21

Chemical Reaction Engineering · Single and multiple reactions in ideal reactors · 1 mark · Multiple choice

Two parallel first-order liquid phase reactions $A \xrightarrow{k_1} B$ and $A \xrightarrow{k_2} C$ are carried out in a well-mixed isothermal batch reactor. The initial concentration of $A$ in the reactor is 1 kmol m$^{-3}$, while that of $B$ and $C$ is zero. After 2 hours, the concentration of $A$ reduces to half its initial value, and the concentration of $B$ is twice that of $C$. The rate constants $k_1$ and $k_2$, in h$^{-1}$, are, respectively

  1. 0.40, 0.20
  2. 0.23, 0.12
  3. 0.50, 0.25
  4. 0.36, 0.18

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Correct answer: (B) 0.23, 0.12

Explanation

The ratio of the products formed is $\frac{C_B}{C_C} = \frac{k_1}{k_2} = 2$. A decays with the total constant, $C_A = C_{A0}e^{-(k_1 + k_2)t}$, so $\frac{1}{2} = e^{-2(k_1 + k_2)}$ and $k_1 + k_2 = \frac{\ln 2}{2} = 0.3466$. With $k_1 = 2k_2$: $k_2 = 0.1155$ and $k_1 = 0.231$.