Temperature Dependence

The Arrhenius equation in its most common form is k = A · exp(−Ea / RT), where R = 8.314 J mol⁻¹ K⁻¹ is the universal gas constant and T is temperature in kelvin. The exponential term e^(−Ea/RT) equals the Boltzmann fraction — the proportion of molecules with kinetic energy ≥ Ea at temperature T. Because this fraction grows rapidly with T, so does k. Note that A has the same units as k (s⁻¹ for first-order, M⁻¹ s⁻¹ for second-order, etc.).

Taking the natural logarithm of both sides gives ln k = ln A − (Ea / R)(1/T). This is the equation of a straight line with y = ln k, x = 1/T, slope = −Ea/R, and y-intercept = ln A. Experimentally, rate constants are measured at several temperatures, ln k is plotted against 1/T, and the slope of the best-fit line is used to calculate Ea = −slope × R. Using log base 10 instead: log k = log A − (Ea / 2.303R)(1/T), so the slope equals −Ea / (2.303R).

When only two temperatures are available, subtracting the Arrhenius equation at T₁ from that at T₂ eliminates A entirely: ln(k₂/k₁) = −(Ea/R)(1/T₂ − 1/T₁). Rearranging: Ea = −R · ln(k₂/k₁) / (1/T₂ − 1/T₁). This is the most common exam formula. Remember: temperatures must always be in kelvin, not Celsius.

Ea is the height of the energy hill (transition state) that reactants must climb. A high Ea (e.g. >100 kJ mol⁻¹) means the reaction is very temperature-sensitive — small increases in T cause large jumps in k. A low Ea (e.g. <40 kJ mol⁻¹) means the rate is less sensitive to T. The pre-exponential factor A is related to the collision frequency Z and steric factor p (from collision theory): A = p·Z. A catalyst lowers Ea without changing A, increasing k and therefore the reaction rate at constant T.

The Maxwell–Boltzmann speed distribution shows that at higher temperature the distribution shifts to higher energies and broadens. The fraction of molecules with energy ≥ Ea, represented by the area under the curve beyond Ea, grows exponentially — not linearly — with T. This is why a 10 °C rise can double the rate: even though the average energy increases modestly, the high-energy tail of the distribution grows dramatically.

• k — Rate constant (varies (e.g. s⁻¹ or M⁻¹ s⁻¹)): Proportionality constant in the rate law; depends strongly on temperature.
• A — Pre-exponential (frequency) factor (same as k): Reflects collision frequency and steric requirements; approximately constant over moderate temperature ranges.
• E_a — Activation energy (J mol⁻¹ (or kJ mol⁻¹)): Minimum energy that colliding molecules must have for a successful reaction; determines temperature sensitivity.
• R — Universal gas constant (J mol⁻¹ K⁻¹): R = 8.314 J mol⁻¹ K⁻¹; connects macroscopic energy units to molecular temperature.
• T — Absolute temperature (K (kelvin)): Must be expressed in kelvin; T(K) = T(°C) + 273.15.

Problem. The first-order decomposition of dinitrogen pentoxide, \ce{2N2O5(g) -> 4NO2(g) + O2(g)}, has k = 3.46 × 10⁻⁵ s⁻¹ at 25 °C and k = 4.87 × 10⁻³ s⁻¹ at 65 °C. (a) Calculate the activation energy Ea. (b) Predict k at 45 °C.

Answer. Ea ≈ 103.5 kJ mol⁻¹; k at 45 °C ≈ 4.31 × 10⁻⁴ s⁻¹

• Activation Energy and Transition State Theory — Ea in the Arrhenius equation is directly defined as the transition-state energy barrier; understanding potential energy profiles deepens interpretation of the equation.
• Collision Theory — The pre-exponential factor A in the Arrhenius equation derives from collision frequency and steric factors in collision theory.
• Rate Laws and the Rate Constant k — The rate constant k whose temperature dependence is described by the Arrhenius equation is defined through the rate law for a reaction.
• Maxwell–Boltzmann Energy Distribution — The exponential term in the Arrhenius equation is the Boltzmann fraction from the Maxwell–Boltzmann distribution — the fraction of molecules exceeding Ea.
• Catalysis — Catalysts increase reaction rates by lowering Ea in the Arrhenius equation, which exponentially increases k at constant temperature.