Decomposition of ethane, $\frac{d\left[\mathrm{C}_2 \mathrm{H}_6\right]}{d t}=k\left[\mathrm{C}_2…
Decomposition of ethane, $\frac{d\left[\mathrm{C}_2 \mathrm{H}_6\right]}{d t}=k\left[\mathrm{C}_2 \mathrm{H}_6\right]$ proceeds through a complex mechanism, which includes 5 steps. The overall rate constant $(k)$ can be expressed as $k=\frac{k_1 k_2 k_3}{k_2 k_5}$. where $k_1, k_2, k_3, k_4, k_5$ are the rate constant of the 5 steps.
If the activation energies of each of the steps respectively are $E_1=1 E, E_2=2 E$,
$E_3=3 E, E_4=4 E, E_5=5 E$, where $E=20 \mathrm{~kJ} / \mathrm{mol}$. Then find the overall activation energy of the decomposition.
$6.67 \mathrm{~kJ} / \mathrm{mol}$
$3.33 \mathrm{~kJ} / \mathrm{mol}$
$20 \mathrm{~kJ} / \mathrm{mol}$
$10 \mathrm{~kJ} / \mathrm{mol}$
Solution
Given that, $k=\frac{k_1 k_2 k_3}{k_2 k_5}$
According to Arrhenius equation, $k=A e^{-\frac{E_d}{R T}}$
$
\begin{aligned}
\text { Overall } E_a & =\frac{\left(E_1+E_2+E_3\right)}{\left(E_2+E_5\right)} \\
& =-[(20+40+60)-(40+100)] \\
\Rightarrow \quad & =20 \mathrm{~kJ} / \mathrm{mol}
\end{aligned}
$
Hence, the correct option is (3)