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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)
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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

Where Arrhenius Two-Temperature Form fits

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First-Order Integrated Rate LawCSTR Design Equation (First Order)PFR Design Equation (First Order)Damköhler Number (First Order)
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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.Arrhenius Equation: Both use Rate constant at T1 and activation energy.Activation Energy from an Arrhenius Plot: Both use activation energy.CSTR Design Equation (First Order): Both use Rate constant at T1.First-Order Integrated Rate Law: Both use Rate constant at T1.PFR Design Equation (First Order): Both use Rate constant at T1.
1 more relatedDamköhler Number (First Order): Both use Rate constant at T1.
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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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The Arrhenius relations

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

Revised 16 October 2026
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formula.expert. (2026, October 16). Arrhenius Two-Temperature Form Formula & Calculator. https://formula.expert/solvers/arrhenius-two-temperature

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