Arrhenius Equation Calculator
Use the Arrhenius equation to find a reaction rate constant (k) from its activation energy and temperature, solve for the activation energy from two rate constants at two temperatures, or predict how the rate constant shifts when the temperature changes. Switch between solving for k, Ea, or k2, and toggle between the per-mole form (using the gas constant R) and the per-molecule form (using the Boltzmann constant kB). The show-your-work panel walks through every step, and the chart plots ln(k) against 1/T so you can see the Arrhenius slope at a glance.
Formula
Worked example
For the decomposition of NO2 at 320 C (593 K) with Ea = 111 kJ/mol and A = 6.7e9 M^-1 s^-1: exponent = -111000 / (8.314 * 593) = -22.53; k = 6.7e9 * exp(-22.53) = 6.7e9 * 1.605e-10 = 1.08 s^-1. Using the two-temperature form with k1 = 0.0045 s^-1 at 298 K to find k2 at 350 K: k2 = 0.0045 * exp(-75000/8.314 * (1/350 - 1/298)) = 0.0045 * exp(4.00) = 0.0045 * 54.6 = 0.246 s^-1.
What the Arrhenius equation describes
The Arrhenius equation, published by Svante Arrhenius in 1889, quantifies how the rate constant of a chemical reaction depends on temperature. In its standard form k = A * exp(-Ea / (R * T)), the rate constant k increases exponentially with temperature because more molecules gain enough energy to overcome the activation barrier Ea. The pre-exponential factor A (also called the frequency factor or collision frequency) captures how often molecules collide in the right orientation, R is the universal gas constant (8.314 J/(mol*K)), and T is the absolute temperature in Kelvin. The equation explains why even a small temperature rise dramatically speeds up many reactions, a principle exploited in industrial catalysis, pharmaceutical stability studies and food preservation.
The linearised form and Arrhenius plots
Taking the natural logarithm of both sides gives ln(k) = ln(A) - (Ea/R) * (1/T). This is a straight-line equation (y = mx + b) when ln(k) is plotted against 1/T: the slope equals -Ea/R (always negative for an endothermic barrier) and the y-intercept equals ln(A). Chemists use this linearised form to determine Ea and A from experimental data - two or more rate constants measured at different temperatures give two points on the Arrhenius line, and the slope yields Ea without knowing A at all. The two-temperature form ln(k2/k1) = (Ea/R)(1/T1 - 1/T2) is derived directly from subtracting two linearised equations, and it is the most common way to predict how a rate constant shifts with temperature.
Per-mole vs. per-molecule form
Chemists almost always work with molar quantities, so R = 8.314 J/(mol*K) and Ea is in J/mol or kJ/mol. Physicists and computational chemists sometimes work per molecule: the gas constant R is replaced by the Boltzmann constant kB = 1.381 * 10^-23 J/K (which equals R / Avogadro's number), and Ea is expressed in joules per molecule rather than per mole. The two forms give identical k values; the only difference is the unit convention for the energy barrier and the constant used to scale it. The "Constant basis" selector in this calculator handles the conversion automatically.
Activation energy and reaction speed
The activation energy Ea is the most chemically informative parameter: it sets how steeply k responds to temperature. Low Ea (below about 40 kJ/mol) means the rate barely changes with temperature; these reactions are typically diffusion-limited or involve radical intermediates. Moderate Ea (40-80 kJ/mol) covers most solution-phase reactions and enzyme-catalysed processes. High Ea (above 100 kJ/mol) signals a thermally demanding reaction that needs elevated temperatures or a catalyst to proceed at a practical rate. Catalysts work by providing an alternative pathway with a lower Ea, not by changing the thermodynamics of the overall reaction.
Typical activation energies for common reactions
| Reaction type / example | Ea (kJ/mol) | Speed category |
|---|---|---|
| Radical chain reactions (H + H2) | 5-25 | Extremely fast |
| Enzyme-catalysed reactions | 25-60 | Very fast |
| Acid-base neutralisation | 10-40 | Very fast |
| SN2 substitution reactions | 50-100 | Moderate |
| NO2 decomposition | 111 | Moderate-slow |
| Unimolecular gas-phase decomposition | 100-160 | Slow |
| Protein unfolding | 200-400 | Very slow |
| Solid-state diffusion | 100-250 | Slow-very slow |
Approximate Ea values. Actual values vary with solvent, catalyst and experimental conditions.
Frequently asked questions
What units does the rate constant k have?
The units of k depend on the overall reaction order. For a first-order reaction k has units of s^-1 (reciprocal seconds). For a second-order reaction it is M^-1 s^-1 (L/(mol*s)). The pre-exponential factor A must have the same units as k, so the exponent (dimensionless) scales correctly. This calculator does not impose units - enter A in the units appropriate to your reaction order and k will come out in the same units.
Does the Arrhenius equation work for all reactions?
It works well for most elementary reactions over a modest temperature range. It can break down at very high temperatures (where the Boltzmann distribution changes shape), at very low temperatures (where quantum tunnelling through the barrier becomes important), and for complex multi-step reactions where the apparent activation energy changes with temperature. The modified Arrhenius equation k = A * T^n * exp(-Ea/RT) adds a temperature power-law prefactor to improve fit over wider ranges.
How do I find activation energy from experimental data?
Measure the rate constant at two or more temperatures, then apply the two-temperature form: ln(k2/k1) = (Ea/R)(1/T1 - 1/T2). With two data points that gives Ea directly. With many measurements, plot ln(k) against 1/T on a graph - the slope of the best-fit line equals -Ea/R, so Ea = -slope * R. This calculator does the algebra for you when you select "Activation energy Ea" mode.
Why does a catalyst not appear in the Arrhenius equation?
A catalyst changes the reaction pathway, which changes the activation energy Ea and the pre-exponential factor A. You would simply run the Arrhenius equation with the new, lower Ea that applies to the catalysed pathway. The equation itself is the same - the catalyst shifts the parameters, not the mathematical form.
What is the pre-exponential factor A?
A (also called the frequency factor or collision frequency) represents how often molecules collide with the correct orientation and sufficient proximity for a reaction to occur, independent of whether they have enough energy to cross the barrier. In collision theory A = Z * P, where Z is the collision frequency and P is a steric factor (0 to 1) accounting for orientation. Values of A for unimolecular reactions are typically 10^10 to 10^15 s^-1.
How does temperature affect the rate constant?
The effect is exponential and depends strongly on Ea. A rough rule of thumb: for Ea near 50 kJ/mol, each 10 K rise near room temperature roughly doubles the rate constant. At higher Ea the factor per 10 K is larger; at very low Ea the increase is smaller. You can compute the exact factor using the "Rate at new temperature" mode in this calculator.