Arrhenius Equation Calculator
Calculate the rate constant k of a chemical reaction at a given temperature using the Arrhenius equation. Used in reactor design, catalyst evaluation, and reaction kinetics studies.
About this calculator
The Arrhenius equation describes how the rate constant k of a chemical reaction depends on temperature: k = A × exp(−Eₐ / (R × T)), where A is the pre-exponential (frequency) factor (s⁻¹ for first-order reactions), Eₐ is the activation energy (J/mol), R is the universal gas constant (8.314 J/mol·K), and T is the absolute temperature (K). The exponential term exp(−Eₐ/RT) represents the fraction of molecular collisions that have sufficient energy to overcome the activation energy barrier. A higher activation energy makes k more sensitive to temperature changes — a classic rule of thumb is that reaction rates roughly double for every 10 °C rise in temperature for typical Eₐ values around 50–60 kJ/mol. Plotting ln(k) vs. 1/T gives a straight line (Arrhenius plot) with slope −Eₐ/R, which is used experimentally to determine Eₐ.
How to use
A first-order reaction has a pre-exponential factor A = 1.0 × 10¹³ s⁻¹ and an activation energy Eₐ = 75,000 J/mol. Find k at T = 500 K. Apply k = A × exp(−Eₐ / (R × T)): exponent = −75,000 / (8.314 × 500) = −75,000 / 4,157 = −18.04. So k = 1.0 × 10¹³ × exp(−18.04) = 1.0 × 10¹³ × 1.455 × 10⁻⁸ ≈ 1.46 × 10⁵ s⁻¹. Increasing T to 550 K gives k ≈ 9.5 × 10⁵ s⁻¹ — about a 6.5-fold increase for a 50 K rise, illustrating the strong temperature sensitivity.
Frequently asked questions
What is the pre-exponential factor A in the Arrhenius equation and how is it determined?
The pre-exponential factor A (also called the frequency factor or attempt frequency) represents the maximum possible rate of reaction if every molecular collision led to a product — essentially the collision frequency corrected for geometric orientation. Its units match those of the rate constant (s⁻¹ for first-order, m³/mol·s for second-order, etc.). A is determined experimentally by measuring k at multiple temperatures, constructing an Arrhenius plot of ln(k) vs. 1/T, and extrapolating the y-intercept to give ln(A). Collision theory and transition state theory both provide frameworks for estimating A from molecular properties.
How does activation energy affect how sensitive a reaction rate is to temperature?
Activation energy Eₐ controls the steepness of the Arrhenius exponential. A reaction with high Eₐ has a rate constant that changes dramatically with even small temperature shifts, because the fraction of molecules exceeding the energy barrier is very small and grows rapidly with T. Conversely, a low Eₐ reaction is less temperature-sensitive. In industrial reactor design, highly temperature-sensitive reactions require precise temperature control to avoid runaway or poor selectivity. Catalysts work by providing an alternative reaction pathway with a lower Eₐ, increasing k at the same temperature without changing the thermodynamic equilibrium.
Why must temperature be in Kelvin when using the Arrhenius equation?
The Arrhenius equation contains the ratio Eₐ/(RT), where R is the gas constant in J/mol·K and T must be in absolute temperature units (Kelvin) to make this ratio physically meaningful. Using Celsius or Fahrenheit would give incorrect and non-physical results because those scales have arbitrary zero points not anchored to the absence of thermal energy. At 0 K (absolute zero), molecular motion ceases and the reaction rate is theoretically zero, which is consistent with the Arrhenius equation as exp(−Eₐ/0) → 0. Always convert measured temperatures to Kelvin by adding 273.15 before applying the equation.