Physical chemistry

Activation energy, Arrhenius equation

Why reactions speed up when heated, and how the height of an energy barrier controls the rate.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: only energetic collisions count

Molecules must collide to react, but most collisions just bounce: bonds can only break if the colliding molecules bring enough energy, the activation energy $E_a$ . Temperature does not change how many molecules there are; it changes how many of them are energetic enough.

Raise the temperature, add the catalyst or increase Ea and compare how quickly C appears. Energies are scaled (Ea ÷ 8) so that reactions are visible in seconds.

SchoolSchool level: temperature and rate

As a rough rule, many reactions near room temperature go two to three times faster for every 10 °C rise. The rule is only approximate: the real factor depends on $E_a$ and on the temperature itself. A catalyst speeds a reaction by offering a path with a lower $E_a$ ; it is not used up.

UndergraduateUndergraduate: the Arrhenius equation

k = A\,e^{-E_a/RT}\qquad\Longleftrightarrow\qquad \ln k = \ln A - \frac{E_a}{R}\cdot\frac{1}{T}

The factor $e^{-E_a/RT}$ is the fraction of collisions with energy above $E_a$ (Boltzmann factor); $A$ collects the collision frequency and the orientation requirement. A plot of $\ln k$ against $1/T$ is a straight line of slope $-E_a/R$ , which is how activation energies are measured.

Example: Rate doubles for 10 K

The rate constant doubles between 300 K and 310 K. Estimate $E_a$ .

Solution

$E_a = R\ln 2 \big/ \left(\tfrac1{300} - \tfrac1{310}\right) = 8.314 \times 0.693 / 1.075\times10^{-4} \approx 5.4\times10^{4}$ J/mol, about 54 kJ/mol.