What this calculator does
The Arrhenius equation gives the rate constant as the pre-exponential factor multiplied by an exponential term containing the activation energy and temperature. It is the reason reaction rates are so sensitive to temperature.
The sensitivity comes from the exponent. With an activation energy of 75 kJ/mol, warming from 0 °C to 25 °C multiplies the rate constant by about 16, and going from 25 °C to 50 °C multiplies it by about 10. The familiar rule that rates roughly double every 10 °C is a rough summary of this for typical activation energies.
The formula
The rate constant is A times e to the minus activation energy over RT. The exponential term is the fraction of collisions with enough energy to react, which for an activation energy of 75 kJ/mol at room temperature is about 7×10⁻¹⁴. The pre-exponential factor accounts for collision frequency and orientation, and it is what makes the rate constant a usable number despite that tiny fraction.
| Term | Meaning |
|---|---|
| Activation energy (Ea) | The energy barrier a collision must clear to react. |
| Pre-exponential factor (A) | Collision frequency and orientation combined, with the same units as k. |
| Boltzmann factor | e^(−Ea/RT), the fraction of molecules with sufficient energy. |
| Gas constant R | 8.3145 J/(mol·K). Note the joules, since activation energies are often quoted in kJ. |
The inputs explained
| Field | What to enter |
|---|---|
| Pre-exponential factor A (/s) | Pre-exponential factor, in the same units as the rate constant. Typically 10¹² to 10¹⁴ per second for a unimolecular reaction. |
| Activation energy Ea (J/mol) | Activation energy in joules per mole. Multiply a kJ/mol figure by 1,000. |
| Temperature (K) | Temperature in kelvin, not Celsius. Add 273.15 to a Celsius figure. |
When to use it
Predicting a rate at a new temperature
Having characterised a reaction at one temperature, the equation extrapolates to another.
Shelf life estimation
Accelerated stability testing runs products warm and extrapolates back using this relationship.
Understanding catalysis
A catalyst lowers the activation energy, and the exponential shows why a modest reduction transforms the rate.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How sensitive is the rate to temperature?
The same reaction at a range of temperatures.
| Temperature | Rate constant k | Ea / (R·T) | Fraction with sufficient energy e^(−Ea/RT) |
|---|---|---|---|
| 273.15 K | 0.04549719 /s | 33.024 | 4.5497e-15 |
| 298.15 K | 0.72538484 /s | 30.255 | 7.2538e-14 |
| 323.15 K | 7.53494827 /s | 27.914 | 7.5349e-13 |
| 373.15 K | 317.3096597 /s | 24.174 | 3.1731e-11 |
Questions
Why do reaction rates depend so strongly on temperature?
Because temperature appears in an exponent. Raising it slightly increases the fraction of molecules with enough energy to clear the activation barrier, and that fraction changes exponentially. A 10 °C rise typically doubles to triples the rate for ordinary activation energies.
Should activation energy be in J or kJ?
Joules per mole, to match the gas constant of 8.3145 J/(mol·K). Activation energies are usually quoted in kJ/mol, so multiply by 1,000 first. Forgetting this gives an answer wrong by a factor that is easy to miss.
What does the pre-exponential factor represent?
How often molecules collide and how often those collisions have the right orientation to react. It is roughly the rate constant the reaction would have if every collision cleared the energy barrier. For unimolecular reactions it is typically 10¹² to 10¹⁴ per second.
How does a catalyst fit in?
It provides a route with lower activation energy. Because the energy appears in an exponent, a modest reduction produces a large rate increase: dropping Ea by 20 kJ/mol at room temperature multiplies the rate by roughly 3,000. The pre-exponential factor also changes, usually less dramatically.
For finding the activation energy from two measurements, see the activation energy calculator. For rate constants from concentration data, see the integrated rate law calculator.