How much faster is a reaction whose activation energy is "52.3 kJ/mol" compared to one at "62.3 kJ/mol", both at 37^@ "C"?

Mar 3, 2016

Recall the Arrhenius equation, which is:

$\setminus m a t h b f \left(k = A {e}^{- {E}_{a} \text{/RT}}\right)$

where:

• $A$ is the pre-exponential factor.
• ${E}_{a}$ is the activation energy in $\text{kJ/mol}$.
• $R$ is the universal gas constant. We will be using $8.314472 \times {10}^{- 3}$ $\text{kJ/mol"cdot"K}$.
• $T$ is the temperature in $\text{K}$.

Now suppose we had two activation energies ${E}_{a 1}$ and ${E}_{a 2}$ and respective rate constants ${k}_{1}$ and ${k}_{2}$, but for the same reaction at the same temperature.

Then, the only things that would change are $k$ and ${E}_{a}$:

${k}_{2} = A {e}^{- {E}_{a 2} \text{/RT}}$
${k}_{1} = A {e}^{- {E}_{a 1} \text{/RT}}$

Now, let's take the ratio of these to determine the "factor" by which the reaction rate is changed.

$\left({k}_{2} = A {e}^{- {E}_{a 2} \text{/RT"))/(k_1 = Ae^(-E_(a1)"/RT}}\right)$

$\textcolor{g r e e n}{\frac{{k}_{2}}{{k}_{1}}} = \left({e}^{- {E}_{a 2} \text{/RT"))/(e^(-E_(a1)"/RT}}\right)$

Using the properties of exponents, we get:

$= {e}^{- {E}_{a 2} \text{/RT" + E_(a1)"/RT}}$

$= {e}^{- \left({E}_{a 2} - {E}_{a 1}\right) \text{/RT}}$

= color(green)(e^((E_(a1) - E_(a2))"/RT")

Because we are solving for the ratio of the rates of reaction, we have to also relate $k$ back to the rate law of the reaction to get:

r_2(t) = k_2["reactant"]^"order"

r_1(t) = k_1["reactant"]^"order"

Since we are looking at ${k}_{2} / {k}_{1}$, we don't really care what the reaction order is; it'll cancel out. Comparing these reactions we get:

color(green)((r_2(t))/(r_1(t)) = k_2/k_1 = e^((E_(a1) - E_(a2))"/RT"

We know that ${E}_{a 1} = \text{62.3 kJ/mol}$ and ${E}_{a 2} = \text{52.3 kJ/mol}$ at $T = \text{310.15 K}$. Therefore:

color(blue)((r_2(t))/(r_1(t))) = e^(-(52.3 - 62.3 "kJ/mol")"/"(8.314472xx10^(-3) "kJ/mol"cdot"K"cdot"310.15 K")

$\approx \textcolor{b l u e}{48.32}$

That means the new, catalyzed reaction is about 48 times as fast as the regular reaction.