The Arrhenius Equation Calculator computes the rate constant k at any temperature from the pre-exponential factor A, activation energy Ea, and temperature T. Essential for predicting how temperature changes affect reaction rates in chemistry, pharmacy, and materials science.
17.3737
s⁻¹
2.855
-20.1709
0
17.3737
s⁻¹
2.855
-20.1709
0
Temperature doubles and the reaction rate quadruples. Temperature drops by 10°C and the shelf life of your product triples. Behind every such relationship lies the Arrhenius equation — the quantitative heart of chemical kinetics. The calculator for the Arrhenius equation computes k at any temperature given A and Ea, calculates the rate ratio at two temperatures, and projects how any temperature change multiplies or divides the reaction rate.
The rate constant k varies with absolute temperature T (in Kelvin) according to:
k = A × e^(−Ea/RT)
where A is the pre-exponential (frequency) factor (same units as k), Ea is the activation energy (J/mol), R = 8.314 J/(mol·K) is the universal gas constant, and T is temperature in Kelvin (T = °C + 273.15). The exponential term e^(−Ea/RT) is the Boltzmann factor — the fraction of molecular collisions that have sufficient energy to overcome the activation barrier. Use this online calculator with any combination of A, Ea, and T. The activation energy calculator computes Ea from rate constants measured at two different temperatures.
The key insight of the Arrhenius equation is that rate constant varies exponentially with the reciprocal of temperature. Taking the logarithm: ln(k) = ln(A) − Ea/(R × T). A plot of ln(k) vs. 1/T is a straight line with slope = −Ea/R — the Arrhenius plot used to determine activation energy experimentally. The ratio of rate constants at two temperatures T₁ and T₂:
k₂/k₁ = e^[(Ea/R) × (1/T₁ − 1/T₂)]
For Ea = 80 kJ/mol, increasing T from 25°C (298 K) to 35°C (308 K): k₂/k₁ = e^[(80,000/8.314) × (1/298 − 1/308)] = e^2.10 ≈ 8.2×. A 10°C increase almost doubles the rate — more than the popular "rule of thumb" of 2× per 10°C, which applies only at lower Ea values.
The Arrhenius equation is the foundation of accelerated stability testing (AST), which predicts room-temperature product shelf life from elevated-temperature degradation data. By measuring degradation rates at 40°C, 50°C, and 60°C — where reactions are fast enough to observe in weeks — manufacturers extrapolate to 25°C room-temperature shelf life in months to years. ICH guideline Q1A(R2) governs pharmaceutical stability studies; the Arrhenius model is the standard method for establishing accelerated-to-real-time equivalence. A drug degrading at 5% per month at 60°C with Ea = 90 kJ/mol would degrade at: rate₂₅ = rate₆₀ × e^[(90,000/8.314) × (1/298 − 1/333)] ≈ 5% / 73 ≈ 0.07% per month at 25°C — an estimated shelf life of over 10 years. The Q10 temperature coefficient calculator provides the simplified rate-doubling approximation for biologically active compounds.
The Arrhenius equation assumes that Ea and A are temperature-independent — valid for many simple reactions but not all. Deviations occur when: (1) the reaction mechanism changes with temperature; (2) quantum tunneling contributes to the rate (especially for proton and hydrogen atom transfer reactions at low temperatures); (3) diffusion rather than chemical activation controls the rate. Curved Arrhenius plots (non-linear ln(k) vs. 1/T) signal these complications and require modified treatments such as the Eyring equation (transition state theory) or empirical extensions. The rate constant calculator and kinetics calculators provide the complete reaction rate analysis toolkit.
The Arrhenius equation relates the rate constant to temperature:
$$k = A \cdot e^{-E_a / RT}$$
where:
The Boltzmann fraction (e^(−Ea/RT)) represents the fraction of molecules with sufficient energy to overcome the activation barrier:
$$f = e^{-E_a/RT}$$
The logarithmic form is useful for graphical analysis:
$$\ln k = \ln A - \frac{E_a}{RT}$$
A plot of ln(k) versus 1/T (Arrhenius plot) yields a straight line with slope −Ea/R and y-intercept ln(A).
The rate constant k tells you the reaction speed at the given temperature. The exponent (−Ea/RT) indicates how restrictive the energy barrier is — more negative values mean a smaller fraction of molecules can react. The Boltzmann fraction shows the proportion of collisions with enough energy: at room temperature with Ea = 50 kJ/mol, only about 2 in 10⁹ molecules have sufficient energy. Higher temperatures dramatically increase this fraction. If your calculated k seems unreasonable, check that Ea is in kJ/mol (not J/mol) and temperature is in Kelvin.
Inputs
Results
With A = 10¹⁰ s⁻¹, Ea = 50 kJ/mol, T = 298.15 K: exponent = −50/(8.314×10⁻³ × 298.15) = −20.17. Boltzmann fraction = e⁻²⁰·¹⁷ = 1.91 × 10⁻⁹. k = 10¹⁰ × 1.91 × 10⁻⁹ = 19.1 s⁻¹.
Inputs
Results
At 373 K (100°C): exponent = −50/(8.314×10⁻³ × 373.15) = −16.12. Boltzmann fraction = 1.08 × 10⁻⁷. k = 10¹⁰ × 1.08 × 10⁻⁷ ≈ 1.08 × 10³ s⁻¹. The rate constant increased ~565-fold with a 75 K temperature increase.
Svante Arrhenius proposed the equation in 1889 based on earlier work by van't Hoff. Arrhenius received the Nobel Prize in Chemistry in 1903, though primarily for his work on electrolytic dissociation theory.
The pre-exponential factor represents the frequency of molecular collisions with the correct orientation for reaction. It has the same units as k and is typically between 10⁸ and 10¹³ s⁻¹ for unimolecular reactions. It is often assumed to be temperature-independent over moderate ranges.
Activation energy (Ea) is the minimum energy that reacting molecules must possess for the reaction to occur. It represents the height of the energy barrier between reactants and products on the potential energy surface. Typical values range from 10 to 200 kJ/mol.
The Arrhenius equation is derived from statistical mechanics and thermodynamics, which require absolute temperature. Using Celsius would give incorrect results because the exponential factor depends on the ratio Ea/RT, where T must be on an absolute scale.
This rule is an approximation. For reactions with Ea ≈ 50-75 kJ/mol near room temperature, a 10°C increase roughly doubles the rate. The exact factor depends on both Ea and the temperature range, and can deviate significantly from 2×.
It works well for most elementary reactions over moderate temperature ranges. Deviations occur for: quantum tunneling reactions, reactions with negative activation energies (some radical reactions), enzyme-catalyzed reactions at high temperatures, and reactions in non-Arrhenius solvents.
The Eyring equation (transition state theory) provides a more detailed molecular interpretation: k = (kB·T/h)·e^(−ΔG‡/RT). It separates activation energy into enthalpic (ΔH‡) and entropic (ΔS‡) contributions, while Arrhenius lumps these into A and Ea.
Catalysts lower the activation energy (Ea) by providing an alternative reaction pathway. The pre-exponential factor may also change if the catalyst alters the reaction mechanism. The net effect is an increase in k, meaning a faster reaction.
At very high temperatures, the Arrhenius equation may break down because the pre-exponential factor can become temperature-dependent. The modified Arrhenius equation k = A·T^n·e^(−Ea/RT) with an additional T^n term is used for combustion and atmospheric chemistry.
An Arrhenius plot graphs ln(k) versus 1/T. If the Arrhenius equation holds, the plot is linear with slope = −Ea/R and y-intercept = ln(A). Non-linearity indicates temperature-dependent Ea, a change in mechanism, or contributions from multiple pathways.
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