Temperature dependence of k
The Arrhenius equation describes how a rate constant changes with temperature. The activation energy Eₐ and the pre-exponential factor A are assumed constant over the interval studied. Here, k and A are first-order constants, in s⁻¹; in the equations, T is in kelvin and Eₐ in J/mol.
Exponential form
k=Ae−Ea/(RT) Linear form
lnk=lnA−REaT1 The slope of ln k versus 1/T is −Eₐ/R, in K; the intercept is ln A. Inside the logarithm, k and A are divided by their unit (s⁻¹) so that the argument is dimensionless. With Eₐ > 0, k increases with temperature; with Eₐ < 0, it decreases.
Eₐ and A from two temperatures
With k₁ measured at T₁ and k₂ at T₂, subtracting the two linear forms eliminates A:
Ea=1/T1−1/T2Rln(k2/k1) A=k1eEa/(RT1) Eₐ depends only on the ratio k₂/k₁: both constants must belong to the same reaction, with the same definition and unit. Two points always define a line, so they do not show that ln k is linear in 1/T. With three or more temperatures, fit ln k against 1/T in Linear regression and check whether the points curve.
The denominator 1/T₁ − 1/T₂ shrinks when the temperatures are close and amplifies any error in k. With T₁ = 300 K, a 1 % error in k₂ changes Eₐ by:
| T₂ | ΔT/T | Change in Eₐ |
|---|
| 310 K | 3.2 % | ≈ 0.77 kJ/mol |
| 300.3 K | 0.10 % | ≈ 25 kJ/mol |
Measure at well-separated temperatures within the range where the mechanism does not change.
Derived quantities
At a given temperature, or at the mean temperature with two points, Eₐ and A give:
t1/2=kln2 Q10=k(T)k(T+10K)=exp[RT(T+10K)Ea⋅10K] dTdlnk=RT2Ea The half-life ln 2/k is the first-order one, which matches k in s⁻¹; in other orders it depends on the initial concentration, as covered in Integrated rate laws. Q₁₀ compares k at T + 10 K with k at T under the same model and is common in biochemistry and physiology. The rule of thumb that k doubles every 10 K corresponds to Eₐ ≈ 53 kJ/mol near 298 K; with 50 kJ/mol, Q₁₀ ≈ 1.92. d ln k/dT is the relative change of k per kelvin.
What Eₐ and A represent
IUPAC defines Eₐ as an empirical parameter of the temperature dependence of k:
Ea=RT2dTdlnk For an elementary reaction it relates to the barrier to the transition state; for a multistep mechanism it combines those of the steps. In the introductory collision model, A is the frequency of suitably oriented collisions and e^(−Eₐ/RT) the fraction with enough energy. Eₐ values of 40 to 200 kJ/mol are common and, for unimolecular reactions, A lies between 10¹² and 10¹⁶ s⁻¹; values far from these call for checking the data and units.
Apparent negative Eₐ
It means that k decreases as T rises. It can describe a mechanism: with a pre-equilibrium, k_app = K·k₂ and
Ea,app=ΔHeq∘+Ea,2 which is negative if the pre-equilibrium is exothermic enough. It also occurs in radical recombinations stabilized by a third body (2 CH₃ + M → C₂H₆ + M). With a small ΔT, or with constants from different conditions, it can be an artifact of the data. The value of Eₐ alone does not tell these causes apart.
Activation parameters (Eyring)
Transition state theory (Eyring; Evans and Polanyi, 1935) expresses k through the activated complex:
k=κhkBTe−ΔG‡/(RT) With transmission coefficient κ = 1 and a first-order constant:
ΔH‡=Ea−RT ΔS‡=R[lnkBTAh−1] ΔG‡=ΔH‡−TΔS‡ ΔH‡ = Eₐ − RT holds in solution and for unimolecular gas-phase reactions. These are parameters inferred under that assumption, not ordinary thermodynamic quantities: first-order kinetics does not show that the step is unimolecular. If k exceeds k_BT/h (≈ 6.2 × 10¹² s⁻¹ at 298 K), ΔG‡ comes out negative and this interpretation does not apply. ΔG‡ is an activation energy, not the reaction ΔG° computed by Gibbs-Helmholtz.
Limits of the model
Constant Eₐ and A usually hold over narrow temperature ranges. Curvature of ln k versus 1/T can come from a change of mechanism, quantum tunnelling or a temperature dependence of A; attributing it to a cause requires examining the mechanism and the conditions. Extrapolating far from the measured temperatures inherits that uncertainty.
The number of figures shown does not indicate the uncertainty of the measured constants.
Constants and references
- IUPAC · Gold Book: Arrhenius equation, activation energy and transition state theory.
- OpenStax · Chemistry 2e, §12.5: linear form and two-point method.
- Eyring, H. (1938). The theory of absolute reaction rates. Trans. Faraday Soc., 34, 41–48.
- Truhlar, D. G., Garrett, B. C. and Klippenstein, S. J. (1996). Current status of transition-state theory. J. Phys. Chem., 100, 12771–12800.
- Laidler, K. J. (1987). Chemical Kinetics, 3rd edition. Harper & Row.