The Arrhenius relations

activation energyArrhenius equationArrhenius plottemperature dependence of rate constantEa from two temperatures

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

Arrhenius Equation

k = A\,e^{-E_a/RT}

Gives a reaction's rate constant from its activation energy, pre-exponential factor, and absolute temperature — the core law of chemical kinetics.

Arrhenius Two-Temperature Form

\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

Extracts an activation energy from two rate constants measured at two temperatures, eliminating the pre-exponential factor entirely.

Activation Energy from an Arrhenius Plot

E_a = -R \times \text{slope}

Converts the slope of a ln k versus 1/T Arrhenius plot into an activation energy, the standard graphical method for kinetics data.

How they fit together

All three are the same statement: only molecules carrying at least Ea of energy react, and the Boltzmann fraction that do rises exponentially with temperature. That exponential is why a rule of thumb as crude as “rate doubles every 10 °C” works at all — for a typical Ea near 50 kJ/mol it is roughly true around room temperature, and it stops being true for reactions with much larger or smaller barriers.

Choose by what you have. If you know A and Ea, the plain exponential form gives k at any temperature. If you have two rate constants at two temperatures and want Ea, the two-temperature form eliminates A entirely — that is its whole purpose, since A is the hardest of the three to pin down. If you have a series of measurements, plot ln k against 1/T and take Ea = −R × slope, which averages out the scatter that a two-point calculation amplifies. Two traps: T must be absolute, and 1/T being the x-axis means the plot runs backwards, with high temperature on the left. A catalyst lowers Ea and so shifts the whole line; it does not change the equilibrium position, only how fast the system gets there.