Arrhenius Equation: Temperature Dependence of Reaction Rates

The Arrhenius equation is a cornerstone of chemical kinetics. It quantitatively describes how the rate constant of a chemical reaction varies with temperature and activation energy. Proposed by Svante Arrhenius in 1889, it provides a deep connection between molecular energy distributions and observable reaction rates, and is essential for predicting reaction behaviour in industrial processes, atmospheric chemistry, and biological systems.

where:

• \(k\) = rate constant (units depend on reaction order)
• \(A\) = pre‑exponential factor (frequency factor) – same units as \(k\); represents the frequency of collisions with proper orientation
• \(E_a\) = activation energy (J/mol or kJ/mol)
• \(R\) = universal gas constant = 8.314 J·mol⁻¹·K⁻¹
• \(T\) = absolute temperature (K)

Activation Energy and Collision Theory

According to collision theory, a reaction proceeds only when reactant molecules collide with sufficient kinetic energy and proper orientation. The threshold energy is the minimum energy required for an effective collision. Activation energy (\(E_a\)) is the extra energy that reactant molecules must acquire to reach this threshold.

The fraction of molecules with kinetic energy equal to or greater than \(E_a\) is given by the Boltzmann factor \(e^{-E_a/(RT)}\). Only this fraction can undergo a successful reaction.

Fig. 1: Maxwell‑Boltzmann distribution at temperatures T₁ and T₂ (T₂ > T₁). The shaded area represents molecules with energy ≥ Ea. At higher temperature, the fraction exceeding the activation energy increases, leading to a faster rate.

Temperature Coefficient

For many reactions, the rate constant roughly doubles for every 10 °C rise in temperature. This empirical observation is expressed as the temperature coefficient:

Typical values range from 2 to 3, though variations exist. The Arrhenius equation provides the theoretical basis for this behaviour.

Graphical Representation: The Arrhenius Plot

Taking natural logarithms on both sides of the Arrhenius equation yields a linear form:

When \(\ln k\) is plotted against \(1/T\) (Kelvin⁻¹), a straight line is obtained:

• Slope \(m = -\dfrac{E_a}{R}\)
• Intercept \(= \ln A\)

Thus, the activation energy can be determined from the slope: \(E_a = -m \times R\).

Fig. 2: Arrhenius plot – linear relationship between ln k and 1/T. The slope gives \(-E_a/R\), and the intercept gives ln A. This plot is widely used to determine activation energies experimentally.

Effect of Temperature on Molecular Energy Distribution

As temperature increases, the kinetic energy distribution shifts to higher energies. The area under the tail of the curve beyond \(E_a\) grows significantly, explaining the exponential increase in the rate constant.

Fig. 3: As temperature rises from T₁ to T₂, the distribution shifts rightward, and the proportion of molecules with energy ≥ Ea (shaded area) increases dramatically. This directly explains the temperature sensitivity of reaction rates.

Two‑Point Form of the Arrhenius Equation

When rate constants \(k_1\) and \(k_2\) are known at temperatures \(T_1\) and \(T_2\), the activation energy can be calculated without needing \(A\). The derivation proceeds as follows:

These relationships allow prediction of rate constants at new temperatures if \(E_a\) is known, or determination of \(E_a\) from two experimental rate measurements.

Effect of a Catalyst on Activation Energy

A catalyst provides an alternative reaction pathway with a lower activation energy, thereby increasing the rate constant without being consumed. Key points:

• The catalyst forms temporary bonds with reactants, creating an intermediate complex that decomposes to products and regenerates the catalyst.
• Only a small amount of catalyst is needed to accelerate a large quantity of reactants.
• A catalyst does not change the thermodynamic equilibrium constant (\(K\)); it only speeds up the attainment of equilibrium.
• Catalysts cannot make a non‑spontaneous reaction occur; they only accelerate spontaneous reactions.

Fig. 4: Schematic showing the influence of temperature or catalyst on the reaction progress and energy barrier. The lower activation energy (catalyzed path) leads to a much faster rate.

Example: Decomposition of Nitrogen Dioxide

The decomposition of nitrogen dioxide, \(2\text{NO}_2 \rightarrow 2\text{NO} + \text{O}_2\), follows Arrhenius behaviour. Experimentally, the activation energy for this reaction is approximately 114 kJ/mol. The rate constant increases exponentially with temperature, consistent with the Arrhenius equation.

Interactive Arrhenius Calculators

Summary of Key Equations