Arrhenius Equation (Activation Energy)
Reference for the Arrhenius equation k = Ae^(-Ea/RT).
Covers activation energy in kJ/mol, pre-exponential factor, and the 10°C temperature-doubling rule.
The Formula
k = A × e^(-Ea / RT)
The Arrhenius equation shows how the rate constant of a reaction changes with temperature. Higher temperature or lower activation energy means faster reactions.
Variables
| Symbol | Meaning |
|---|---|
| k | Rate constant (units depend on reaction order) |
| A | Pre-exponential factor (frequency factor) |
| Ea | Activation energy (J/mol) |
| R | Gas constant (8.314 J/mol⋅K) |
| T | Temperature (Kelvin) |
Example 1
A reaction has Ea = 50,000 J/mol and A = 1 × 10¹⁰ s⁻¹. Find k at 300 K.
k = 10¹⁰ × e^(-50000 / (8.314 × 300))
k = 10¹⁰ × e^(-20.06)
k = 10¹⁰ × 1.96 × 10⁻⁹
k ≈ 19.6 s⁻¹
Example 2
Same reaction at 350 K. How much faster?
k₃₅₀ = 10¹⁰ × e^(-50000 / (8.314 × 350))
k₃₅₀ = 10¹⁰ × e^(-17.19) = 10¹⁰ × 3.41 × 10⁻⁸
k₃₅₀ ≈ 341 s⁻¹
About 17 times faster at 350 K than at 300 K
When to Use It
Use the Arrhenius equation when:
- Predicting how temperature changes affect reaction speed
- Determining the activation energy from experimental data
- Comparing the effectiveness of different catalysts
- Designing industrial processes at optimal temperatures
Key Notes
- A catalyst lowers the activation energy Ea without changing the overall energy difference between reactants and products (ΔG) — it provides an alternative reaction pathway, speeding the reaction without being consumed
- The Arrhenius plot (ln k vs 1/T) is a straight line with slope = −Ea/R — experimentalists use this to determine activation energy by measuring rate constants at two or more temperatures
- The "10°C rule" approximation: many biochemical reactions roughly double in rate for each 10°C increase (Q₁₀ ≈ 2); this comes from typical activation energies of 50–80 kJ/mol in the Arrhenius equation
- T must be in Kelvin, not Celsius — using 300°C instead of 573 K in the exponent would give a completely wrong rate constant