Activation Energy Calculator
Enter two rate constants measured at two temperatures and this calculator returns the activation energy Eₐ using the two-point Arrhenius equation, with the pre-exponential factor A shown alongside for sanity checks.
Activation energy Eₐ (kJ/mol)
52.95
- Activation energy (J/mol)
- 52,948.86
- Pre-exponential factor A (same units as k)
- 1,889,032,006.52
- ln(k₂ / k₁)
- 0.69
- 1/T₁ − 1/T₂ (1/K)
- 0
- T₁ (K)
- 298.15
- T₂ (K)
- 308.15
Two-point Arrhenius: Eₐ = R · ln(k₂/k₁) / (1/T₁ − 1/T₂), with R = 8.314462618 J/(mol·K). Here ln(k₂/k₁) = 0.6931 and 1/T₁ − 1/T₂ = 1.088e-4 K⁻¹, giving Eₐ = 52949 J/mol = 52.95 kJ/mol. The pre-exponential factor A = k₁ · exp(Eₐ/(R·T₁)) = 1.889e+9 (same units as your k values).
How to use this calculator
Type the two temperatures in degrees Celsius and the two measured rate constants k₁ and k₂ in any matching units (s⁻¹, M⁻¹·s⁻¹, etc.). The activation energy is independent of the rate-constant units provided both values use the same units. The result updates instantly in kJ/mol; the breakdown also shows Eₐ in J/mol, the pre-exponential factor A, and the intermediate ln(k₂/k₁) and 1/T₁ − 1/T₂ terms so you can audit the maths.
How the calculation works
The Arrhenius equation, k = A · exp(−Eₐ / (R·T)), describes how a reaction rate constant varies with absolute temperature. Taking the logarithm at two temperatures and subtracting cancels the unknown pre-exponential factor A, leaving Eₐ = R · ln(k₂/k₁) / (1/T₁ − 1/T₂), where R = 8.314462618 J/(mol·K) is the universal gas constant (CODATA 2018) and temperatures are in kelvin. Once Eₐ is known, A is recovered from A = k₁ · exp(Eₐ / (R·T₁)). This is the standard textbook treatment (Atkins & de Paula §17C, Brown/LeMay §14.5) and assumes the rate-determining step is the same at both temperatures.
Worked example
A reaction has k₁ = 1.0 at 25 °C and k₂ = 2.0 at 35 °C — a doubling of rate per 10 K rise, the classic biological Q₁₀ rule. T₁ = 298.15 K, T₂ = 308.15 K. Then 1/T₁ − 1/T₂ = 1/298.15 − 1/308.15 = 1.0884 × 10⁻⁴ K⁻¹ and ln(2) = 0.69315, so Eₐ = 8.3145 × 0.69315 / (1.0884 × 10⁻⁴) ≈ 52 949 J/mol ≈ 52.95 kJ/mol. This matches the textbook rule of thumb that reactions doubling per 10 K near room temperature have Eₐ ≈ 50 kJ/mol — characteristic of enzyme-catalysed steps in metabolism.
Frequently asked questions
What is activation energy?
Activation energy Eₐ is the minimum kinetic energy that colliding molecules must carry to react. It corresponds to the height of the energy barrier separating reactants from products on the reaction coordinate. Higher Eₐ means fewer collisions cross the barrier at a given temperature, so the rate constant k is smaller and the reaction is slower. Typical values are 50–250 kJ/mol for chemical reactions; enzyme-catalysed biological reactions often sit near 50 kJ/mol, while bond-breaking gas-phase reactions can exceed 300 kJ/mol.
Which formula does this calculator use?
The two-point Arrhenius equation Eₐ = R · ln(k₂/k₁) / (1/T₁ − 1/T₂), with R = 8.314462618 J/(mol·K) and temperatures in kelvin (the calculator converts your °C inputs internally). It also reports the pre-exponential factor A = k₁ · exp(Eₐ/(R·T₁)), which carries the same units as your rate constants and represents the collision frequency multiplied by the steric factor.
Do the rate constants need to be in any particular units?
No — Eₐ is independent of the units of k, provided both k₁ and k₂ use the same units. The calculator only ever sees the dimensionless ratio k₂/k₁. So if k₁ = 1.2 × 10⁻⁴ s⁻¹ and k₂ = 8.7 × 10⁻³ s⁻¹, just enter those numbers. The reported pre-exponential factor A will then carry the same units as your k inputs.
Should I use Celsius or Kelvin for the temperatures?
Enter temperatures in degrees Celsius — the calculator adds 273.15 internally to convert to kelvin before applying Arrhenius. The result is identical to entering kelvin directly, but °C is what most laboratory thermometers and notebooks record. Never use Fahrenheit; the Arrhenius equation requires an absolute thermodynamic temperature scale.
When does the two-point Arrhenius method break down?
Whenever the mechanism changes with temperature. If a different rate-determining step takes over between T₁ and T₂ — for example a competing pathway, an enzyme denaturing, a phase change, or diffusion control replacing kinetic control — the apparent Eₐ is a weighted average and is not the true barrier of either step. For reliable values, measure k at several temperatures, plot ln k versus 1/T (an "Arrhenius plot"), and use the slope only over the linear region.
What is the pre-exponential factor A used for?
A combines the collision frequency between reactant molecules and a steric (orientation) factor — the fraction of collisions with the right geometry for reaction. For simple bimolecular gas-phase reactions, A is roughly 10⁹ to 10¹¹ M⁻¹·s⁻¹; values much smaller indicate strong steric requirements, while values much larger usually mean the simple Arrhenius model is breaking down. Reporting A alongside Eₐ lets reviewers sanity-check both numbers against transition-state theory.