Arrhenius Equation
The Arrhenius equation k = Ae^(-Ea/RT) describes how reaction rate constants change with temperature.
Learn with examples.
The Formula
The Arrhenius equation describes how the rate constant of a chemical reaction changes with temperature. Higher temperatures lead to exponentially faster reactions because more molecules have enough energy to overcome the activation energy barrier.
Swedish chemist Svante Arrhenius proposed this equation in 1889. He observed that even a modest temperature increase can dramatically speed up a reaction. A common rule of thumb is that reaction rates roughly double for every 10°C increase in temperature.
The pre-exponential factor A (also called the frequency factor) represents how often molecules collide with the correct orientation. The exponential term gives the fraction of molecules that have sufficient energy to react. At higher temperatures, this fraction increases, leading to faster reactions.
Variables
| Symbol | Meaning |
|---|---|
| k | Rate constant (units depend on reaction order) |
| A | Pre-exponential (frequency) factor |
| Ea | Activation energy (joules/mol or kJ/mol) |
| R | Gas constant (8.314 J/(mol·K)) |
| T | Absolute temperature (kelvin, K) |
Example 1
A reaction has Ea = 75 kJ/mol and A = 4.0 × 10¹³ s⁻¹. What is k at 300 K?
Ea/RT = 75,000 / (8.314 × 300) = 75,000 / 2,494 = 30.07
k = 4.0 × 10¹³ × e⁻³⁰·⁰⁷
e⁻³⁰·⁰⁷ = 8.6 × 10⁻¹⁴
k = 4.0 × 10¹³ × 8.6 × 10⁻¹⁴ = 3.44 s⁻¹
Example 2
A reaction is 5 times faster at 350 K than at 300 K. What is the activation energy?
Two-temperature form: ln(k₂/k₁) = (Ea/R)(1/T₁ − 1/T₂)
ln(5) = (Ea/8.314)(1/300 − 1/350) = (Ea/8.314)(0.000476)
1.609 = Ea × 5.73 × 10⁻⁵
Ea ≈ 28,100 J/mol = 28.1 kJ/mol
When to Use It
Use the Arrhenius equation to predict how temperature affects reaction speed.
- Determining activation energies from experimental kinetics data
- Predicting shelf life of food and pharmaceuticals
- Optimizing industrial reaction temperatures
- Understanding enzyme kinetics and biological reaction rates