Derivation

The Arrhenius equation can be written as:

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

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 $T.$

Derive an expression for $E_a$ in terms of $T$ and the associated rate enhancement $(\frac{k_2}{k_1})$ due to the presence of a catalyst, and select an equivalent equation from the list below.

Look closely at how $E_a$ is defined in this problem $(E_{a2}-E_{a1}$ as opposed to $E_{a1}-E_{a2}),$ and be careful with negative signs!

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

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

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

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

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

$E_a=RTln(\frac{k_2}{k_1})$

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