Temperature Dependence of the Rate of a Reaction: The Arrhenius Equation

The Arrhenius equation. Svante Arrhenius proposed, on the basis of extensive

experimental measurements of how rate constants vary with temperature, that the rate constant $k$ of

almost every reaction obeys the relation

where $E_a$ is the activation energy, $R$ is the universal gas constant ($8.314\ \text{J K}^{-1} \text{mol}^{-1}$), $T$ is the absolute temperature, and $A$ is a constant for a given reaction called

the pre-exponential factor (or frequency factor) -- related, in the language of collision theory

(previous sections), to how often molecules collide and with what fraction of correct orientations, and

having the same units as the rate constant $k$ itself.

The exponential factor $e^{-E_a/RT}$ is exactly the Boltzmann factor from collision theory: the

fraction of molecules, at temperature $T$, whose energy exceeds the threshold $E_a$. Because this

factor rises exponentially, not linearly, with $T$, even a modest increase in temperature produces a

large increase in $k$ -- exactly the strong temperature-sensitivity described qualitatively in the

section on factors affecting rate.

The logarithmic (linear) form. Taking the natural logarithm of both sides of the Arrhenius equation:

or, converting to base-10 logarithms (dividing throughout by $2.303$):

This is the equation of a straight line if $\log k$ is plotted against $1/T$: comparing with

$y = mx + c$, the slope of the line is $-E_a/(2.303R)$ and its $y$-intercept (at $1/T = 0$) is $\log A$.

Measuring the rate constant $k$ of a reaction at several different temperatures and plotting

$\log k$ against $1/T$ therefore gives a straight line whose slope directly yields the activation

energy, $E_a = -2.303\,R \times \text{slope}$, and whose intercept yields the pre-exponential factor …