Arrhenius Equation

Links the rate constant to temperature: k = Ae^(−Ea/RT), or the two-point form ln(k2/k1) = (Ea/R)(1/T1 − 1/T2).

k = Ae^{-E_a/RT} \quad ; \quad \ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

k = Ae^(−Ea/RT) ; ln(k2/k1) = (Ea/R)(1/T1 − 1/T2)

What it means

The Arrhenius equation explains why heating speeds reactions up so dramatically: only molecules with energy above Ea can react, and that fraction grows exponentially with temperature. The pre-exponential factor A captures collision frequency and orientation, while the exponential term captures the energy requirement — for typical activation energies a 10 K rise roughly doubles the rate. The two-point form is the practical version: measure k at two temperatures and solve for Ea, or predict k at a new temperature from a known Ea. When to use it: finding activation energy from rate constants at two temperatures, predicting how much faster a reaction runs when heated, or explaining why catalysts (which lower Ea) have such large effects.

Variables

Symbol	Name	Unit	Meaning
$k$	Rate constant	varies with reaction order	The rate constant at temperature T.
$A$	Frequency factor	same as k	The pre-exponential (frequency) factor: collision rate times the fraction of properly oriented collisions.
$Ea$	Activation energy	J/mol	The energy barrier the reactants must clear to react.
$R$	Gas constant	J/(mol·K)	The gas constant, 8.314 J/(mol·K) here.
$T$	Temperature	K	Absolute temperature — kelvin required.

Assumptions

The activation energy Ea is constant over the temperature range studied.
Temperatures are absolute (kelvin) in the two-point form.
Ea must be in joules per mole when paired with R = 8.314 J/(mol·K).
The mechanism does not change between the two temperatures.

Worked example

Problem

A reaction has Ea = 50.0 kJ/mol. By what factor does its rate constant increase from 300 K to 310 K?

Given:

Ea = 50.0 kJ/mol = 50,000 J/mol
k1 at T1 = 300 K, k2 at T2 = 310 K
R = 8.314 J/(mol·K)

Write the two-point form: ln(k2/k1) = (Ea/R)(1/T1 − 1/T2).
Convert Ea to joules: 50.0 kJ/mol = 50,000 J/mol.
Compute 1/T1 − 1/T2: 1/300 − 1/310 = (310 − 300)/(300 × 310) = 10/93,000 = 1.0753 × 10⁻⁴ K⁻¹.
Compute Ea/R: 50,000 / 8.314 = 6013.95 K.
Multiply: ln(k2/k1) = 6013.95 × 1.0753 × 10⁻⁴ = 0.64666.
Exponentiate: k2/k1 = e^0.64666 = 1.9092, which rounds to 1.91.

k2 ≈ 1.91 k1 (the rate nearly doubles for a 10 K rise)

Raising the temperature by just 10 K almost doubles the rate constant — the exponential sensitivity that makes temperature the most powerful kinetic lever.