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Arrhenius Two-Temperature Form

ln⁡k2k1=EaR(1T1−1T2)\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

Worked example: Rate doubles 300 K → 310 K: Ea = 53.597 kJ/mol — press Try an example to run it live, then adjust anything.

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Constant used — built into this formula, no need to enter
R=8.31446261815324 J/(mol⋅K)R = 8.31446261815324\ \text{J/(mol}{\cdot}\text{K)}Molar gas constant · exact
See also
van 't Hoff Equation (K at Two Temperatures) — The identical algebra applied to the equilibrium constant instead of the rate constant — how far a reaction goes, rather than how fast.
Here the solver did the work — could you?

Two temperatures →

Grade 12Grade 12 Chemistry

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Arrhenius Two-Temperature Form explained

T1T2k1k2Ea

Write the Arrhenius equation twice, once at each temperature, and divide: the pre-exponential factor A cancels and only the ratio of rate constants survives. That is a gift to the experimentalist, because A is hard to measure but a rate ratio needs only two runs on the same apparatus. Since only k₂/k₁ appears, the units of the rate constants are irrelevant as long as both are the same — half-lives work just as well, inverted.

The classic result: a reaction whose rate exactly doubles between 300 K and 310 K has EaE_a = R ln 2 / (1/300 − 1/310) = 8.314 × 0.693 × 9300 = 53.6 kJ/mol. That is where the "rates double every 10 degrees" rule of thumb comes from — it is only true for activation energies near 50 kJ/mol, and it fails badly for very fast or very slow reactions. The two traps are using Celsius instead of kelvin (this calculator converts for you, but a hand calculation will be wildly wrong) and reversing the reciprocal difference, which flips the sign and hands you a negative activation energy.

Arrhenius Two-Temperature Form formula

ln⁡k2k1=EaR(1T1−1T2)\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)
Where
  • k1k_1= Rate constant at T1 (Hz)
  • k2k_2= Rate constant at T2 (Hz)
  • EaE_a= Activation energy (kJ/mol)
  • T1T_1= First absolute temperature (°C)
  • T2T_2= Second absolute temperature (°C)

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Part of this formula set

The Arrhenius relations

3 formulasThe exponential rate-constant law, its two-temperature form and the activation energy read off an Arrhenius plot's slope.