Variables

Symbol	Quantity	Unit
k	Rate constant
A	Frequency factor
Ea	Activation energy	J/mol
R	Gas constant	J/(mol·K)
T	Absolute temperature	K

What it means

The Arrhenius equation is a foundational formula in chemical kinetics that relates the rate constant k to temperature T and activation energy Ea. It is written as k = A exp(−Ea/(R T)), where A is the pre‑exponential factor (frequency factor), R is the universal gas constant, and Ea is the activation energy. This equation arises from the collision theory and transition state theory. It shows that even a small increase in temperature can greatly increase the reaction rate because of the exponential term. The activation energy is the minimum energy that reactant molecules must have to undergo reaction. In practice, the Arrhenius equation is used to extrapolate rate constants to different temperatures, to determine Ea from experimental data (Arrhenius plot), and to design reactors with temperature control. It is also applied in materials science, biology (enzyme kinetics), and environmental chemistry to model degradation rates. The equation highlights the trade‑off between reaction speed and energy input, which is crucial for process economics and safety.

Worked example

Arrhenius Equation – Two Examples

Real‑World

Scenario: A reaction has A = 1×10¹⁰ s⁻¹, Eₐ = 50 kJ/mol, T = 298 K. Find rate constant k.

Parameter	Value
A	1×10¹⁰ s⁻¹
Eₐ	50,000 J/mol
R	8.314 J/mol·K
T	298 K

1Eₐ/(RT) = 50000/(8.314×298) = 20.18

2k = 1e10 × e⁻²⁰·¹⁸ ≈ 16.2 s⁻¹

Result k ≈ 16.2 s⁻¹ ✓ Reasonable

Scenario: Eₐ = 75 kJ/mol, A = 5×10¹² s⁻¹, T = 310 K. Determine k.

Parameter	Value
A	5×10¹² s⁻¹
Eₐ	75,000 J/mol
T	310 K

1Eₐ/(RT) = 75000/(8.314×310) = 29.11

2k = 5e12 × e⁻²⁹·¹¹ ≈ 3.11 s⁻¹

Result k ≈ 3.11 s⁻¹ ✓ Higher Eₐ → slower

Key insight: k increases with temperature and decreases with Eₐ – Arrhenius describes this dependency.

Common mistakes

Activation energy Ea: Must be in J/mol (not kJ/mol) when using R=8.314 J/(mol·K) – convert kJ to J.
Temperature T: Must be in Kelvin – never Celsius or Fahrenheit.
Frequency factor A: Has the same units as k; check consistency.
Exponential term: The exponent −Ea/(RT) is dimensionless; ensure R and Ea use consistent units.
Assumption of constant Ea: This equation assumes Ea is independent of T; for some reactions it varies.

Applications

The Arrhenius equation, k = A·e^(−Ea/RT), is the cornerstone of temperature‑dependent reaction kinetics. It relates the rate constant k to the activation energy Ea, the absolute temperature T, and the pre‑exponential factor A, which accounts for collision frequency and orientation. This equation is vital for predicting how reaction rates change with temperature, enabling engineers to design reactors that operate safely and economically. It is used to determine optimal operating temperatures, to assess catalyst performance, and to calculate the shelf‑life of pharmaceuticals and food products. In materials science, it models diffusion and creep processes. By understanding the Arrhenius relationship, engineers can extrapolate rates from lab data to industrial conditions and can diagnose issues like catalyst deactivation or unwanted side reactions that may become significant at elevated temperatures.

Reactor design and temperature optimisation
Kinetic parameter estimation from experimental data
Stability and shelf‑life prediction for thermally sensitive products
Catalyst activity and deactivation analysis
Diffusion and solid‑state kinetics modelling

Arrhenius Equation

Arrhenius Equation Calculator k = A · e−Ea/(R·T)