Arrhenius Equation: Formula, Derivation, and Activation Energy

What Is the Arrhenius Equation?

Introduced by Swedish chemist Svante Arrhenius in 1889, the Arrhenius equation is a central relationship in chemical kinetics. It describes how the rate constant changes with absolute temperature when a reaction follows Arrhenius behaviour over the temperature range being studied.

Each symbol has a specific meaning:

• \(k\): rate constant. Its units depend on the overall reaction order.
• \(A\): pre-exponential factor, also called the frequency factor. It has the same units as \(k\) and accounts for factors such as collision frequency and molecular orientation.
• \(E_a\): apparent activation energy, usually expressed in J/mol or kJ/mol.
• \(R\): molar gas constant, approximately \(8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}\).
• \(T\): absolute temperature in kelvin (K).
• \(e\): base of the natural logarithm, approximately 2.71828.

Activation Energy and Collision Theory

Activation energy is the energy barrier associated with a reaction pathway. In a simple collision-theory picture, reacting particles need suitable collision conditions and enough energy to reach a configuration from which products can form. The Arrhenius exponential term describes the strong temperature dependence associated with overcoming this barrier.

At a fixed positive activation energy, increasing temperature makes the exponent less negative. The exponential factor therefore increases, and the rate constant generally becomes larger. The factor is a useful model for temperature dependence; it should not be interpreted as an exact count of all successful collisions for every reaction mechanism.

Why does temperature have such a strong effect?

A temperature increase changes the distribution of molecular energies. Although the average kinetic energy rises, the most important change for many activated reactions is the increased fraction of molecules in the high-energy tail. Since the rate constant depends exponentially on \(1/T\), even a moderate temperature increase can produce a substantial change.

Temperature Coefficient of a Reaction

The temperature coefficient describes how much a reaction rate or rate constant changes when the temperature increases by a specified amount, commonly 10 °C. For the rate constant, it can be expressed as:

Many reactions show an approximate \(Q_{10}\) of 2 to 3 over a limited temperature range, but this is a rule of thumb rather than a universal law. The actual value depends on activation energy, the starting temperature, and whether the same reaction mechanism remains dominant.

The Arrhenius equation gives a more specific prediction for a chosen activation energy and pair of temperatures. Use the first calculator below to estimate the ratio of rate constants.

Arrhenius Plot: Determining Activation Energy

Taking the natural logarithm of the Arrhenius equation gives its linear form:

This has the form \(y=mx+c\). Plotting \(\ln k\) on the vertical axis against \(1/T\) on the horizontal axis gives a straight line when the reaction follows the Arrhenius model over the measured range.

• Slope: \(m=-E_a/R\)
• Vertical intercept: \(c=\ln A\)
• Activation energy: \(E_a=-mR\)

Because the horizontal axis is reciprocal temperature, its unit is \(\mathrm{K^{-1}}\). Use consistent units for the rate constant when taking logarithms; changing the units of \(k\) changes the numerical intercept, but not the slope when the same units are used for every measurement.

Effect of Temperature on Molecular Energy Distribution

Increasing temperature broadens and shifts the Maxwell–Boltzmann energy distribution. A greater fraction of molecules occupies the high-energy region, so a larger fraction may be able to cross the energy barrier. This helps explain why the rate constant can rise rapidly as temperature increases.

Two-Point Arrhenius Equation: Derivation

If rate constants are measured at two different temperatures, the activation energy can be estimated without knowing the pre-exponential factor \(A\), assuming the same Arrhenius model applies at both temperatures.

The equivalent base-10 logarithm form is:

How a Catalyst Affects Activation Energy

A catalyst provides an alternative reaction pathway with a different energy barrier. For a pathway with a lower activation energy and otherwise comparable conditions, the Arrhenius equation predicts a larger rate constant at the same temperature.

\[ k=Ae^{-E_a/(RT)} \] \[ E_{a,\mathrm{cat}}

• A catalyst participates in the reaction mechanism and is regenerated overall.
• The amount of catalyst required depends on the reaction and operating conditions.
• A catalyst does not change the equilibrium constant for a given reaction at a fixed temperature.
• A catalyst can accelerate both forward and reverse reactions, helping the system reach equilibrium more quickly.
• The observed rate enhancement depends on the complete reaction mechanism; the pre-exponential factor may also differ between pathways.

Worked Example: Calculate the Rate-Constant Ratio

Suppose a reaction has an activation energy of \(50\ \mathrm{kJ/mol}\). Estimate how much its rate constant changes when the temperature rises from \(300\ \mathrm{K}\) to \(310\ \mathrm{K}\), assuming Arrhenius behaviour and a constant pre-exponential factor.

Given:

• \(E_a=50{,}000\ \mathrm{J/mol}\)
• \(T_1=300\ \mathrm{K}\)
• \(T_2=310\ \mathrm{K}\)
• \(R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}\)

Substitute into the two-point equation:

This gives \(k_2/k_1\approx 1.91\). Under these assumptions, the rate constant is approximately 1.9 times its original value after a 10 K temperature increase.

This is an illustrative calculation, not a universal temperature rule. A reaction with a different activation energy or starting temperature can have a different ratio.

Interactive Arrhenius Equation Calculators

Enter values in the fields below. Use kelvin for temperatures and kJ/mol for the activation energy in Calculator 1. Calculator 2 requires positive rate constants measured at two different absolute temperatures.

Summary of Important Arrhenius Equations

Frequently Asked Questions