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)

<p>To determine the overall energy of activation for the given complex reaction with rate constants $k_1$, $k_2$, and $k_3$, we start with the expression for the overall rate constant $k$:</p> <p>$ k = \sqrt{\frac{k_1 k_3}{k_2}} $</p> <p>The rate constant can also be expressed in terms of the Arrhenius equation:</p> <p>$ A \cdot e^{-E_a / RT} = \sqrt{\frac{A_1 e^{-E_{a_1} / RT} \cdot A_3 e^{-E_{a_3} / RT}}{A_2 e^{-E_{a_2} / RT}}} $</p> <p>By comparing the exponential terms from both sides, we have:</p> <p>$ \frac{E_a}{RT} = \frac{1}{2} \left( \frac{E_{a_1}}{RT} + \frac{E_{a_3}}{RT} - \frac{E_{a_2}}{RT} \right) $</p> <p>This simplifies to:</p> <p>$ E_a = \frac{E_{a_1} + E_{a_3} - E_{a_2}}{2} $</p> <p>Substituting the given activation energies—$E_{a_1} = 60 \, \text{kJ mol}^{-1}$, $E_{a_2} = 30 \, \text{kJ mol}^{-1}$, and $E_{a_3} = 10 \, \text{kJ mol}^{-1}$:</p> <p>$ E_a = \frac{60 + 10 - 30}{2} = \frac{40}{2} = 20 \, \text{kJ mol}^{-1} $</p> <p>Therefore, the overall energy of activation is 20 kJ/mol.</p>