If for a hypothetical reaction, $E_a=0$ at $273 \mathrm{~K}$, then find the ratio of the rate constants at $383 \mathrm{~K}$ and $273 \mathrm{~K}$.

Relation between activation energy $\left(E_a\right)$ and rate constant (k), $\Rightarrow \quad k=A e^{-\frac{E_a}{R T}}$ Here, A = frequency factor or pre-exponential factor. T = temperature (in K) R = gas constant e = mathematical quantity Now, at $273 \mathrm{~K}, E_a=0$ $$ \begin{aligned} k_1 & =A e^{-\frac{0}{R \times 273}} \\ k_1 & =A \cdot 1 \\ k_1 & =A...(i) \end{aligned} $$ Similarly at, 383 K, $k_2=A e^{-\frac{E_a}{383}}$...(ii) Ratio of Eqs. (i) and (ii) is $$ k_2=k_1 e^{-\frac{E_a}{383}} $$ If, $\Delta E_a=0$ $$ k_2=k_1 $$ Hence, ratio of rate constant at $383 \mathrm{~K}$ is 1 .