Arrhenius Equation Calculator
Find how fast a reaction goes at a given temperature. Enter the pre-exponential factor A, the activation energy Eₐ and the temperature, and the Arrhenius equation k = A·exp(−Eₐ/RT) returns the rate constant.
Rate constant k (same units as A)
0.7254
- Exponent −Eₐ/(R·T)
- -30.2547
- Exponential factor exp(−Eₐ/RT)
- 7.2538e-14
- Temperature (K)
- 298.15
- Activation energy (J/mol)
- 75000
Arrhenius equation: k = A·exp(−Eₐ/RT), with R = 8.314462618 J/(mol·K). The rate constant shares the units of A; raising the temperature or lowering the activation energy increases k.
How to use this calculator
Enter three values. First, the pre-exponential factor A (also called the frequency factor), which sets the overall scale of the rate constant and carries the same units as k — for a first-order reaction that is per second (s⁻¹), while a second-order reaction uses L·mol⁻¹·s⁻¹. Typical first-order gas-phase values are around 10¹³ s⁻¹. Second, the activation energy Eₐ in kilojoules per mole (kJ/mol); this is the energy barrier the reactants must clear, and for most reactions it lies somewhere between about 40 and 150 kJ/mol. Third, the temperature in degrees Celsius, which the calculator converts to kelvin (K = °C + 273.15) before using it. The result is the rate constant k in the same units as A, along with the exponent −Eₐ/(R·T), the exponential factor exp(−Eₐ/RT), the temperature in kelvin, and the activation energy expressed in J/mol. If you know k at two temperatures and want the activation energy instead, use the Activation Energy calculator, which inverts the same equation.
How the calculation works
The Arrhenius equation describes how a reaction’s rate constant depends on temperature: k = A·exp(−Eₐ/RT). Here A is the pre-exponential factor, Eₐ is the activation energy, R is the molar gas constant (8.314462618 J·mol⁻¹·K⁻¹), and T is the absolute temperature in kelvin. The exponential term exp(−Eₐ/RT) is the fraction of molecular collisions that carry enough energy to overcome the activation barrier; A represents how often collisions occur with the right orientation. Because the activation energy sits in a negative exponent divided by T, two things follow. Raising the temperature makes −Eₐ/(RT) less negative, so the exponential grows and the reaction speeds up — this is why a modest temperature rise can produce a large increase in rate. And a larger activation energy makes the exponent more negative at any given temperature, so high-barrier reactions are both slower and more sensitive to temperature changes. When Eₐ is zero the exponential equals 1 and k simply equals A, meaning the rate no longer depends on temperature. The equation is the foundation of chemical kinetics and appears throughout chemistry, biology and materials science, from enzyme activity to the shelf life of food and the ageing of batteries.
Worked example
Take the thermal decomposition of dinitrogen pentoxide, 2 N₂O₅ → 4 NO₂ + O₂, a classic first-order reaction with Arrhenius parameters A = 4.94×10¹³ s⁻¹ and Eₐ = 103.4 kJ/mol. At 25 °C the absolute temperature is T = 298.15 K. The exponent is −Eₐ/(R·T) = −103400 / (8.314462618 × 298.15) = −41.71, so the exponential factor is exp(−41.71) = 7.68×10⁻¹⁹. Multiplying by A gives k = 4.94×10¹³ × 7.68×10⁻¹⁹ = 3.79×10⁻⁵ s⁻¹, which matches the experimentally measured value of about 3.8×10⁻⁵ s⁻¹. Warm the reaction to 35 °C (308.15 K) and the exponent rises to −40.36, so the exponential factor increases roughly four-fold and k climbs to about 1.5×10⁻⁴ s⁻¹ — a ten-degree rise nearly quadruples the rate, a direct consequence of the equation’s exponential temperature dependence.
Frequently asked questions
What is the Arrhenius equation?
The Arrhenius equation, k = A·exp(−Eₐ/RT), gives the rate constant k of a chemical reaction as a function of temperature. A is the pre-exponential (frequency) factor, Eₐ is the activation energy, R is the molar gas constant (8.314462618 J·mol⁻¹·K⁻¹), and T is the absolute temperature in kelvin. Named after Svante Arrhenius, it captures the experimental fact that reaction rates rise sharply with temperature, because the exponential term is the fraction of collisions energetic enough to react.
What is the pre-exponential factor A?
A, also called the frequency or pre-exponential factor, represents how frequently reactant molecules collide with the correct orientation to react. It sets the maximum possible value of the rate constant — the value k would take if every collision had enough energy — and carries the same units as k, which depend on the reaction order (s⁻¹ for first order, L·mol⁻¹·s⁻¹ for second order). Typical first-order gas-phase reactions have A around 10¹³ s⁻¹.
Why does temperature have such a large effect on reaction rate?
Temperature appears inside the exponential term exp(−Eₐ/RT). A small rise in T makes the exponent −Eₐ/(RT) less negative, and because the relationship is exponential rather than linear, the rate constant can change dramatically. A common rule of thumb is that near room temperature the rate roughly doubles for every 10 °C increase, which corresponds to an activation energy of about 50 kJ/mol.
What units should I use for activation energy?
Enter the activation energy in kilojoules per mole (kJ/mol); the calculator multiplies by 1000 to get J/mol so it is consistent with the gas constant R in J·mol⁻¹·K⁻¹. Many textbooks quote Eₐ in kJ/mol, and most reactions fall between roughly 40 and 150 kJ/mol. If a source gives Eₐ in kcal/mol, multiply by 4.184 to convert to kJ/mol before entering it.
How is this different from the Activation Energy calculator?
This calculator uses the Arrhenius equation in its forward direction: you supply A, Eₐ and T, and it returns the rate constant k. The Activation Energy calculator works backwards from the two-point form, Eₐ = R·ln(k₂/k₁)/(1/T₁ − 1/T₂): you supply two rate constants measured at two temperatures, and it returns the activation energy. They are two views of the same underlying equation.
What happens when the activation energy is zero?
If Eₐ = 0, the exponent −Eₐ/(RT) is zero and exp(0) = 1, so k = A regardless of temperature. Physically this means a barrierless reaction whose rate does not depend on temperature. Real reactions almost always have a positive activation energy, so in practice k is always smaller than A and grows as the temperature rises.