JEE Main 2024 — 31 January, Shift 2 — Question 63

$$\text{Given equilibrium:} \quad \mathrm{A}_{(g)} \rightleftharpoons \mathrm{B}_{(g)} + \frac{1}{2}\mathrm{C}_{(g)}$$ $$\text{Let initial moles of A = 1}$$ $$\text{At equilibrium:}$$ \text{Moles of A} &= 1 - \alpha \\ \text{Moles of B} &= \alpha \\ \text{Moles of C} &= \frac{\alpha}{2} \\ \text{Total moles} &= 1 - \alpha + \alpha + \frac{\alpha}{2} = 1 + \frac{\alpha}{2} \end{aligned}$$

\text{Let total pressure be } P

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\text{Mole fractions:}

$$\begin{aligned} \chi_A &= \frac{1 - \alpha}{1 + \frac{\alpha}{2}} = \frac{2(1 - \alpha)}{2 + \alpha} \\ \chi_B &= \frac{\alpha}{1 + \frac{\alpha}{2}} = \frac{2\alpha}{2 + \alpha} \\ \chi_C &= \frac{\alpha/2}{1 + \frac{\alpha}{2}} = \frac{\alpha}{2 + \alpha} \end{aligned}$$

\text{Partial pressures:}

$$\begin{aligned} P_A &= \chi_A \cdot P = \frac{2(1 - \alpha)}{2 + \alpha} P \\ P_B &= \frac{2\alpha}{2 + \alpha} P \\ P_C &= \frac{\alpha}{2 + \alpha} P \end{aligned}$$

\text{Equilibrium constant expression:} \quad K_P = \frac{P_B \cdot P_C^{1/2}}{P_A}

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K_P = \frac{\left( \frac{2\alpha P}{2 + \alpha} \right) \cdot \left( \frac{\alpha P}{2 + \alpha} \right)^{1/2}}{\frac{2(1 - \alpha) P}{2 + \alpha}}

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= \frac{2\alpha \sqrt{\alpha P}}{(2 + \alpha)^{3/2}} \cdot \frac{2 + \alpha}{2(1 - \alpha) P}

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= \frac{\alpha^{3/2} \cdot P^{1/2}}{(1 - \alpha)(2 + \alpha)^{1/2}}

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\boxed{K_P = \frac{\alpha^{3/2} P^{1/2}}{(2 + \alpha)^{1/2}(1 - \alpha)}}

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