JEE MAIN - Chemistry (2025 - 24th January Evening Shift - No. 21)
Consider a complex reaction taking place in three steps with rate constants $\mathrm{k}_1, \mathrm{k}_2$ and $\mathrm{k}_3$ respectively. The overall rate constant $k$ is given by the expression $k=\sqrt{\frac{k_1 k_3}{k_2}}$. If the activation energies of the three steps are 60, 30 and $10 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively, then the overall energy of activation in $\mathrm{kJ} \mathrm{mol}^{-1}$ is _________ . (Nearest integer)
Answer
20
Explanation
$$\begin{aligned} & \mathrm{K}=\sqrt{\frac{\mathrm{K}_1 \mathrm{~K}_3}{\mathrm{~K}_2}} \\ & \mathrm{~A} . \mathrm{e}^{-\mathrm{Ea} / \mathrm{RT}}=\sqrt{\frac{\mathrm{A}_1 \mathrm{e}^{-E \mathrm{Ea}_1 / \mathrm{RT}} \times \mathrm{A}_3 \mathrm{e}^{-E a_3 / \mathrm{RT}}}{\mathrm{~A}_2 \mathrm{e}^{-E \mathrm{Ea}_2 / \mathrm{RT}}}} \end{aligned}$$
By comparinig exponential term
$$\begin{aligned} & \frac{E_a}{R T}=\frac{1}{2} \times\left(\frac{E_{a_1}}{R T}+\frac{E_{a_3}}{R T}-\frac{E_{a_2}}{R T}\right) \\ & E_a=\left(E_{a_1}+E_{a_3}-E_{a_2}\right) / 2 \\ & E_a=(60+10-30) / 2=20 \mathrm{~kJ} \mathrm{~mol}^{-1} \end{aligned}$$
Ans. 20
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