The shared Arrhenius model describes the temperature sensitivity of many thermally activated rates. In chemical kinetics, the rate-like coefficient is the reaction rate constant $k$, the barrier is written $E_a$, and the prefactor is usually written $A$:

$$k=Ae^{-E_a/(RT)}.$$

This connects the kinetic rate law

$$r=k,f(\text{concentrations})$$

to temperature. If reactant concentrations are held fixed, a change in $k$ changes the reaction rate by the same factor.

For two temperatures, the Arrhenius relation gives the useful comparison

$$\boxed{\ln\frac{k_2}{k_1} =-\frac{E_a}{R}\left(\frac1{T_2}-\frac1{T_1}\right)}.$$

Example

A first-order reaction has

$$k_1=1.50\times10^{-3},\mathrm{s^{-1}}$$

at $300,\mathrm K$ and activation energy

$$E_a=50.0,\mathrm{kJ/mol}.$$

At $320,\mathrm K$,

$$\ln\frac{k_2}{k_1} =-\frac{5.00\times10^4}{8.314} \left(\frac1{320}-\frac1{300}\right) \approx1.25,$$

so

$$\frac{k_2}{k_1}\approx e^{1.25}\approx3.49$$

and

$$k_2\approx5.24\times10^{-3},\mathrm{s^{-1}}.$$

At the same concentration, the reaction is therefore about $3.5$ times faster.

An Arrhenius plot of $\ln k$ against $1/T$ can be used to estimate $E_a$ from its slope. In a chemical reaction this measured value is an apparent activation energy for the observed kinetic pathway over that temperature range.

If the dominant reaction mechanism changes with temperature, the kinetic parameters can change as well, producing curvature or different linear regions in an Arrhenius plot. One constant $E_a$ should therefore not be assumed outside the range where the data support it.

The kinetic rate constant $k$ should not be confused with a thermodynamic equilibrium constant $K$. Both can depend on temperature, but $k$ controls how quickly a process proceeds whereas $K$ describes the equilibrium composition.