Chemistry

Arrhenius Equation

Estimate a reaction rate constant from temperature and activation energy.

k = A e^{- E_{a} / (R T)}

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Source & review

Difficulty

Intermediate

Read time

4 min

Prerequisites

Algebra, logarithms, and molar quantities

Review statusNeeds review

Source

OpenStax, Chemistry 2e, §12.5

Adapted from Teng-Jui Lin’s CHEM 155 equation sheet. The linked textbook reference was used to check the equation and its physical context. Pending editorial review.

Adapted from Teng-Jui Lin, Equation Sheets, under CC BY 4.0.

FormuLab adapted notation and chemical symbols, added variable meanings, units, applicability conditions, and an original worked example.

Definition

Understanding the core concept

The Arrhenius equation models how a rate constant depends on temperature through an activation-energy barrier. The pre-exponential factor has the same units as the rate constant, which depend on the reaction order.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$k$	Reaction rate constant; s⁻¹ for this first-order example	$s^{- 1}$
$A$	Pre-exponential factor, with the same units as k	$s^{- 1}$
$E_{a}$	Molar activation energy	$J mo l^{- 1}$
$R$	Molar gas constant	$J mo l^{- 1} K^{- 1}$
$T$	Absolute temperature	$K$

Real-World Applications

Where this formula is used in practice

Temperature dependence

Estimate a rate constant at a stated temperature.

Activation barriers

Compare the effects of activation energy on a kinetic model.

Worked Example

Step-by-step calculation with real numbers

Problem

For a first-order Arrhenius model, take A = 1.00 × 10¹² s⁻¹ and E_a = 75.0 kJ/mol. Estimate k at 500 K.

Given

A = 1.00 \times 1 0^{12} s^{- 1}

E_{a} = 75000 J mo l^{- 1}

T = 500 K

R = 8.314462618 J mo l^{- 1} K^{- 1}

Solution

k = A e^{- E_{a} / (R T)}

k = (1.00 \times 1 0^{12} s^{- 1}) exp [- \frac{75000}{( 8.314462618 ) ( 500 )}]

k \approx 1.462 \times 1 0^{4} s^{- 1}

Final Answer

k \approx 1.462 \times 1 0^{4} s^{- 1}

Formula Guide

Info

DifficultyIntermediate

Read Time4 min

PrerequisitesAlgebra, logarithms, and molar quantities

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, Chemistry 2e, §12.5

Adapted from Teng-Jui Lin’s CHEM 155 equation sheet. The linked textbook reference was used to check the equation and its physical context. Pending editorial review.

Adapted from Teng-Jui Lin, Equation Sheets, under CC BY 4.0.

FormuLab adapted notation and chemical symbols, added variable meanings, units, applicability conditions, and an original worked example.

Assumptions

• Use an Arrhenius regime in which A and activation energy are approximately constant over the interval of interest.
• Use kelvin temperature and consistent molar energy units; the example’s k and A units assume first-order kinetics.

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Arrhenius Equation

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