Arrhenius Two-Temperature Form

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 Ea = 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.