# Calculate the new activation energy at 37^@ "C" if the catalyzed reaction is 200 times as fast, and the uncatalyzed activation energy was "100 kJ/mol"?

## The equation is $\ln \left({k}_{\text{cat")/(k_"uncat")) = -1/(RT) xx [E_(a,"cat") - E_(a,"uncat}}\right]$

Jun 12, 2017

$\text{86.34 kJ/mol}$.

Remember to convert $R$ to $\text{kJ/mol"cdot"K}$ and $T$ to $\text{K}$. The units must work out so that both sides of the $\ln$ form of the Arrhenius equation have no units.

I would start from the original form of the Arrhenius equation, just to verify first that the given equation is correct.

You know that you are at the same temperature twice, but you have two different rate constants and two different activation energies.

So:

${k}_{u n c a t} = A {e}^{- {E}_{a , u n c a t} \text{/} R T}$

${k}_{c a t} = A {e}^{- {E}_{a , c a t} \text{/} R T}$

Assuming the pre-exponential factor is the same and only the mechanism changes,

$\frac{{k}_{c a t}}{{k}_{u n c a t}} = \left(\cancel{A} {e}^{- {E}_{a , c a t} \text{/"RT))/(cancel(A)e^(-E_(a,uncat)"/} R T}\right)$

$= {e}^{- {E}_{a , c a t} \text{/"RT + E_(a,uncat)"/} R T}$

Take the $\ln$ of both sides:

$\boldsymbol{\ln \left(\frac{{k}_{c a t}}{{k}_{u n c a t}}\right)} = - {E}_{a , c a t} / \left(R T\right) + {E}_{a , u n c a t} / \left(R T\right)$

$= \boldsymbol{- \frac{1}{R T} \times \left[{E}_{a , c a t} - {E}_{a , u n c a t}\right]}$

Okay, your equation is correct. Now, if you know that the reaction rate increased by a factor of $200$, recall that the rate law for a general reaction

$A \to B$

in two scenarios is

${r}_{u n c a t} \left(t\right) = {k}_{u n c a t} \left[A\right]$,

${r}_{c a t} \left(t\right) = {k}_{c a t} \left[A\right]$,

where $\frac{{r}_{c a t}}{{r}_{u n c a t}} = 200$.

Therefore, with the same reactants and products, divide these rates to get:

$\frac{{r}_{c a t}}{{r}_{u n c a t}} = \frac{{k}_{c a t}}{{k}_{u n c a t}}$

So, the ratio of the rates is the ratio of the rate constants at the same temperature.

That means you can plug in the ratio of the two rates into the Arrhenius equation:

=> ln(200) = -1/(8.314472 cancel"J""/mol"cdotcancel"K" xx "1 kJ"/(1000 cancel"J") xx (37 + 273.15 cancel"K"))

$\times \left[{E}_{a , c a t} - \text{100 kJ"/"mol}\right]$

Simplify the expression by evaluating some of it:

ln(200) = -"0.3878 mol/kJ" xx (E_(a,cat) - "100 kJ"/"mol")

Solve for the catalyzed activation energy:

color(blue)(E_(a,cat)) = ln(200)/(-"0.3878 mol/kJ") + "100 kJ/mol"

$=$ $\textcolor{b l u e}{\text{86.34 kJ/mol}}$