GATE 2022 CH – Question 64
An elementary irreversible gas-phase reaction, $A \to B + C$, is carried out at fixed temperature and pressure in two separate ideal reactors: (i) a 10 m$^3$ plug flow reactor (PFR), (ii) a 10 m$^3$ continuous-stirred tank reactor (CSTR). If pure $A$ is fed at 5 m$^3$ h$^{-1}$ to the PFR operating at 400 K, the conversion is 80%. If a mixture of 50 mol% of $A$ and 50 mol% of an inert is fed at 5 m$^3$ h$^{-1}$ to the CSTR operating at 425 K, the conversion is 80%. The universal gas constant $R = 8.314$ J mol$^{-1}$ K$^{-1}$. Assuming the Arrhenius rate law, the estimated activation energy is __________ kJ mol$^{-1}$ (rounded off to one decimal place).
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Correct answer: 47.26 to 47.74
Explanation
The reaction is elementary and unimolecular, so it is first order. With pure A the volume expansion factor is $\varepsilon = 1$, and for the 50% mixture it is $\varepsilon = 0.5$. PFR: $\tau = \frac{10}{5} = 2$ h and $k_1\tau = (1 + \varepsilon)\ln\frac{1}{1 - X} - \varepsilon X = 2\ln 5 - 0.8 = 2.419$, so $k_1 = 1.209$ h⁻¹ at 400 K. CSTR: $\tau = 2$ h and $k_2\tau = \frac{X(1 + \varepsilon X)}{1 - X} = \frac{0.8 \times 1.4}{0.2} = 5.6$, so $k_2 = 2.8$ h⁻¹ at 425 K. Then $E_a = \frac{R\ln(k_2/k_1)}{\frac{1}{400} - \frac{1}{425}} = \frac{8.314 \times 0.8394}{1.4706 \times 10^{-4}} = 47.5 \times 10^3$ J/mol.