From two rate constants to activation energy
The Arrhenius equation defined by IUPAC is
Here T is absolute temperature, R is the molar gas constant, and A is the pre-exponential factor. Assuming A and apparent activation energy are constant over the interval, subtracting the logarithmic equations for two measurements cancels A:
Use Kelvin in the reciprocal terms. The header buttons convert Celsius to Kelvin with the exact relation T(K) = t(°C) + 273.15, as specified in the NIST SI guide. The example's temperatures are 26.85 °C and 76.85 °C. Putting those Celsius numbers directly into the formula would describe different absolute temperatures and give the wrong energy.
For the illustrative pair above, the rate ratio is 10 and
The calculator uses R = 8.31446261815324 J mol⁻¹ K⁻¹, derived from the exact Avogadro and Boltzmann constants in the NIST fundamental constants listing. Divide J/mol by 1000 for kJ/mol. Thermochemical kcal/mol uses 4184 J per kcal; it is distinct from other historical calorie definitions.
Swapping both complete temperature/rate pairs leaves the answer unchanged. Swapping only the rates reverses the sign. Mixing a rate constant in minutes with one in seconds changes the ratio and invalidates the comparison; use matching units first.
Match both axes when using an Arrhenius slope
OpenStax's Arrhenius linearization writes the relationship as
Therefore a slope m from ln(k) against 1/T gives Eₐ = −mR. A graph using common logarithms or 1000/T has a different numerical slope:
| Vertical axis | Horizontal axis | Activation energy in J/mol |
|---|---|---|
| ln(k) | 1/T | −mR |
| log10(k) | 1/T | −mR ln(10) |
| ln(k) | 1000/T | −1000mR |
| log10(k) | 1000/T | −1000mR ln(10) |
For example, m = −2.5 from log10(k) versus 1000/T gives
Reading that same number as a natural-log slope against 1/T would omit both conversion factors. Select the axes exactly as they appear on the graph. A slope alone gives energy but does not determine the intercept or absolute rate constants.
The calculator's paired-data diagram uses ln(k/k1) versus 1/T. This shifts the vertical origin without changing the slope and makes the logarithm dimensionless. Slope-only diagrams also use natural-log/reciprocal-Kelvin coordinates, with an explicitly arbitrary vertical offset. Their extra result reports the equivalent natural-log slope in Kelvin.
Read zero, negative and sensitive results
Equal positive rate constants at different temperatures give zero apparent activation energy for that pair. Equal temperatures cannot determine a slope, even if the rate constants also match; the calculator asks for distinct temperatures.
A negative result means the fitted rate constant decreases as temperature rises for the entered pair. It does not, by itself, establish a negative elementary reaction barrier. Check the pairing, units and whether one Arrhenius relationship describes the measurements. IUPAC defines activation energy as an empirical temperature-dependence parameter; this two-point estimate is not a direct measurement of a transition-state energy difference.
When temperatures are very close, small measurement changes can strongly affect the answer. This tool flags pairs less than 1 K apart as a sensitivity prompt, rather than assigning an uncertainty that the supplied data cannot establish. Two points always define a line, so they cannot reveal curvature or a changing reaction mechanism. Additional measurements and an appropriate regression are needed to assess that behavior.
Rates accept scientific notation from 1e-100 to 1e100, with up to 15 significant decimal digits. Temperatures and plot slopes accept up to six decimal places. The six-significant-digit result is a display choice, not a claim about experimental precision. The model does not calculate vacancy-formation energies or supply reaction-specific material data.
Read the formula definitions, supported input ranges, and limitations before using this estimate.
