Calculating Activation Energy from the Arrhenius Equation

When rate-constant data are available at only two temperatures, rather than a

full series suitable for a graph, the Arrhenius equation can still be used to extract the activation

energy directly, without needing to plot anything.

Deriving the two-point form. Writing the logarithmic form of the Arrhenius equation once for each

of two temperatures, $T_1$ (where the rate constant is $k_1$) and $T_2$ (where it is $k_2$):

Subtracting the first equation from the second eliminates $\log A$ (since the pre-exponential factor is

assumed unchanged between the two temperatures for the same reaction):

Using the formula. Given the rate constant (or any quantity proportional to it, such as a half-life

or an initial rate) at two different temperatures, this single equation can be solved directly for

$E_a$: compute $\log(k_2/k_1)$ on the left, compute $\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$ on the

right using the two absolute temperatures, and rearrange to isolate $E_a$. This two-point method is

especially convenient when a reaction's rate is reported at its "before" and "after" condition -- for

instance, when it is known that a reaction's rate doubles, or that its half-life falls by a certain …