Arrhenius Equation Calculator: Activation Energy & Temperature Dependence

simulator intermediate ~10 min
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k = 7.4 × 10⁻² s⁻¹ — t½ = 9.4 s at 298 K

With Ea = 75 kJ/mol and A = 10¹² s⁻¹ at 298 K, the rate constant is 7.4 × 10⁻² s⁻¹, giving a first-order half-life of 9.4 seconds. Raising temperature by 10°C approximately doubles the rate.

Formula

k = A × exp(−Ea / RT)
ln(k) = ln(A) − Ea/(RT) — Arrhenius plot
k₂/k₁ = exp[(Ea/R)(1/T₁ − 1/T₂)]

The Exponential Barrier

The Arrhenius equation reveals why temperature is the master variable of chemical kinetics. The exponential factor exp(−Ea/RT) represents the fraction of molecular collisions with sufficient energy to overcome the activation barrier. Even a modest activation energy of 75 kJ/mol means that at room temperature, only about 1 in 10¹³ collisions succeeds — yet reactions still proceed because molecules collide trillions of times per second.

The Arrhenius Plot

Taking the natural logarithm of k = A × exp(−Ea/RT) gives ln(k) = ln(A) − Ea/(RT), a straight line when plotted against 1/T. The slope gives −Ea/R and the intercept gives ln(A). This linearization, proposed by Arrhenius in 1889, remains the standard method for extracting activation energies from experimental rate data. Deviations from linearity reveal non-Arrhenius behavior such as quantum tunneling or complex mechanisms.

Temperature Sensitivity

The common rule of thumb that reaction rates double per 10°C increase is only approximately correct, and only near room temperature for moderate activation energies. This simulation lets you explore the actual rate ratio for any Ea and T combination. High activation energies produce extreme temperature sensitivity — enzyme-catalyzed reactions with low Ea are deliberately temperature-insensitive, while explosives have high Ea for safety.

Energy Diagram

The visualization shows a potential energy surface with reactants, transition state, and products. The activation energy appears as the height of the barrier, and the Boltzmann distribution of molecular energies is overlaid to show what fraction exceeds Ea. As you increase temperature, the distribution broadens and more molecules clear the barrier — the visual origin of the Arrhenius exponential.

FAQ

What is the Arrhenius equation?

The Arrhenius equation k = A × exp(−Ea/RT) describes how the rate constant k depends on temperature T. Ea is the activation energy (minimum energy needed for reaction), A is the pre-exponential factor (related to collision frequency), and R is the gas constant. It was proposed by Svante Arrhenius in 1889.

Why does a 10°C increase roughly double the rate?

For typical activation energies (50–100 kJ/mol) near room temperature, the exponential factor exp(−Ea/RT) approximately doubles when T increases by 10°C. This is a rule of thumb — the actual factor depends on Ea and the specific temperature.

How do you determine activation energy experimentally?

Plot ln(k) vs 1/T (an Arrhenius plot). The slope equals −Ea/R. By measuring rate constants at several temperatures and fitting a straight line, you extract Ea from the slope and A from the intercept.

What is the pre-exponential factor?

The pre-exponential factor A represents the frequency of collisions with correct orientation, typically 10⁸–10¹⁶ s⁻¹ for unimolecular reactions. It includes the collision frequency and a steric factor. A is roughly temperature-independent compared to the exponential term.

Sources

Embed

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