Question

A hydrogenation reaction is carried out at $500$ $K$. If the same reaction is carried out in presence of a catalyst at the same rate, the temperature required is $400$ $K$. Calculate the activation energy of the reaction is the catalyst lowers the activation barrier by $20$ $kJ$ ${mol}^{-1}$.

Answer 1

Arrhenius equation may be given as,
$k=A{e}^{-E_a/RT}$
Let $k_{500}$ and $k_{400}$ be the rate constants at temperature $500$ $K$ and $400$ $K$ (in presence of catalyst) respectively. $E_{500}$ and $E_{400}$ be the activation energies at temperatures $500$ $K$ and $400$ $K$ respectively.
$k_{500}=A{e}^{-E_{500}/R\times 500}$ ...(i)
$k_{400}=A{e}^{-E_{400}/R\times 400}$ ...(ii)
Given, $k_{500}=k_{400}$ (same rates in presence and absence of a catalyst).
On comparing equation (i) with equation (ii),
$\frac{E_{500}}{R\times 500}=\frac{E_{400}}{R\times 400}$
or $\frac{E_{500}}{5}=\frac{E_{400}}{4}$
or $E_{500}=\frac{E_{400}}{4}\times 5$ ...(iii)
Given, $E_{500}=E_{400}+20$
Substituting in equation (iii),
$E_{400}+20=E_{400}\times 1.25$
or $E_{400}=\frac{20}{0.25}=80$ $kJ$ ${mol}^{-1}$
So, $E_{500}=80+20=100$ $kJ$ ${mol}^{-1}$