Consider a complex reaction taking place in three steps with rate constants $k_1, k_2$ and $k_3$ 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}{…

Question

Consider a complex reaction taking place in three steps with rate constants $k_1, k_2$ and $k_3$ 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 $\ldots\ldots$. (Nearest integer)

Aicharya · Accepted Answer

$\mathrm{k}_1, \mathrm{k}_2$ and $\mathrm{k}_3$ are given as rate constants of three steps of a complex reaction. Rate constant (k) of the overall reaction is given as
$k=\sqrt{\frac{k_1 k_3}{k_2}}$
Activation energies of the three steps are given as
$\begin{aligned}
& \mathrm{E}_{\mathrm{a}_1}=60 \mathrm{~kJ} \mathrm{~mol}^{-1}, \mathrm{E}_{\mathrm{a}_2}=30 \mathrm{~kJ} \mathrm{~mol}^{-1}, \\
& \mathrm{E}_{\mathrm{a}_3}=10 \mathrm{~kJ} \mathrm{~mol}^{-1}
\end{aligned}$
From Arrhenius equation, we know that
$\mathrm{k}=\mathrm{Ae}^{-\mathrm{Ea} / \mathrm{RT} T}$
If $E_a$ is the activation energy of the overall reaction, then
$\begin{aligned}
& E_a=\frac{1}{2}\left[E_{a_1}+E_{a_3}-E_{a_2}\right] \\
& =\frac{1}{2}[60+10-30]=20 \mathrm{~kJ} \mathrm{~mol}^{-1}
\end{aligned}$

Consider a complex reaction taking place in three steps with rate constants $k_1, k_2$ and $k_3$…

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