The Arrhenius equation can be written as:

$k=A{e}^{-\frac{E_0}{RT}}$

Consider a reaction with some activation energy that we will call $E_{a1}.$ Suppose a catalyst is added to the reaction that binds to the transition state and lowers the activation

energy to a new $E_a$ that we can call $E_{a2}.$ We can define the difference in catalyzed and uncatalyzed activation energies as $E_a=E_{a2}-E_{a1}.$

The rate enhancement due to the presence of a catalyst can be defined as $\frac{k_2}{k_1}$ where $k_1$ is the rate constant of the reaction without the catalyst, and $k_2$ is the rate constant of the

process with the catalyst, both at some constant temperature $T.$ That is:

$k_1=A{e}^{-\frac{E_{a1}}{RT}}$

$k_2=A{e}^{-\frac{E_{q2}}{RT}}$

Derive an expression for the rate enhancement $\frac{k_2}{k_1}$ in terms of $T$ and $E_a,$ and select an equivalent equation from the list below.

$\frac{k_2}{k_1}=e^{\frac{-E_a}{RT}}$

$\frac{k_2}{k_1}=e^{\frac{E_a}{RT}}$

$\frac{k_2}{k_1}=\frac{E_a}{RT}$

$\frac{k_2}{k_1}=ln(\frac{E_a}{RT})$

$\frac{k_2}{k_1}=ln(\frac{-E_a}{RT})$

$\frac{k_2}{k_1}=-\frac{E_a}{RT}$

$\frac{k_2}{k_1}=ln(e^{\frac{-E_a}{RT}})$