About

The Arrhenius equation quantifies how reaction rate constants depend on temperature. A miscalculated activation energy E_a propagates into orders-of-magnitude errors in predicted rate constants k, leading to failed reactor designs, incorrect shelf-life estimates, or hazardous thermal runaway predictions. This calculator implements both the single-temperature form k = A ⋅ exp(−E_a ÷ RT) and the two-temperature comparison form for extracting E_a from experimental data at two temperatures. It handles unit conversions between J/mol, kJ/mol, cal/mol, and kcal/mol internally. Note: the equation assumes a single dominant reaction pathway and breaks down for diffusion-controlled reactions, quantum tunneling regimes below 200 K, and non-elementary multi-step mechanisms where apparent activation energy is composite.

Formulas

The single-temperature Arrhenius equation expresses the temperature dependence of the rate constant:

k = A ⋅ exp(−E_aR ⋅ T)

where k = rate constant (units depend on reaction order), A = pre-exponential (frequency) factor (same units as k), E_a = activation energy (J/mol), R = universal gas constant = 8.314 J/(mol⋅K), and T = absolute temperature (K).

The two-temperature comparison form eliminates A and relates rate constants at two temperatures:

ln(k₂k₁) = E_aR ⋅ (1T₁ − 1T₂)

This form is used to extract E_a from two experimental measurements without knowing A. When solving for E_a:

E_a = R ⋅ ln(k₂ ÷ k₁)1T₁ − 1T₂

The linearized Arrhenius form for graphical analysis is: ln(k) = ln(A) − E_aR ⋅ 1T, yielding a straight line with slope −E_a÷R on an ln(k) vs 1÷T plot.

Reference Data

Reaction	A (s⁻¹)	E_a (kJ/mol)	Temp. Range (K)	Notes
2 HI → H₂ + I₂	1.0 × 10¹³	186	500-800	Gas-phase bimolecular
CH₃CHO → CH₄ + CO	2.0 × 10¹³	190	700-1000	Thermal decomposition
2 NO₂ → 2 NO + O₂	3.0 × 10¹¹	111	600-900	Second-order gas phase
C₂H₅Br + OH⁻	4.3 × 10¹¹	90	300-400	SN2 in aqueous solution
H₂ + I₂ → 2 HI	1.0 × 10¹⁴	165	500-800	Classic bimolecular
Sucrose hydrolysis (acid)	1.5 × 10¹⁵	108	290-340	First-order in aqueous H⁺
N₂O₅ decomposition	4.0 × 10¹³	103	273-338	First-order gas phase
Cyclopropane → propene	1.6 × 10¹⁵	272	700-900	Unimolecular isomerization
CO + NO₂ → CO₂ + NO	5.0 × 10⁸	132	500-800	Bimolecular gas phase
Enzyme catalyzed (typical)	1.0 × 10¹⁰	40-60	293-313	Protein denaturation limits range
H₂O₂ decomposition (I⁻ cat.)	8.0 × 10⁹	56	293-333	Catalyzed pseudo-first-order
Methane combustion (overall)	1.3 × 10⁸	125	1000-2000	Complex mechanism, apparent E_a
Ester hydrolysis (base)	1.1 × 10¹¹	75	290-350	Second-order in solution
Ozone decomposition	8.0 × 10¹²	18	200-350	Very low barrier
Diels-Alder (butadiene + ethylene)	5.0 × 10⁷	115	400-600	Concerted cycloaddition

Frequently Asked Questions

The Arrhenius equation requires E_a and R to share consistent energy units. If using R = 8.314 J/(mol⋅K), then E_a must be in J/mol. This calculator automatically converts from kJ/mol, cal/mol, or kcal/mol to J/mol internally using the factors: 1 kJ = 1000 J, 1 cal = 4.184 J. A common error is entering kJ/mol values while assuming J/mol, which produces rate constants off by a factor of exp(1000).

Below approximately 200 K, quantum mechanical tunneling allows reactant molecules to pass through the energy barrier rather than over it. The Arrhenius model treats the barrier classically, so it underestimates rate constants in the tunneling regime. Additionally, near absolute zero, quantum effects dominate molecular motion, invalidating the Boltzmann energy distribution assumption underlying the exponential term. For cryogenic kinetics, the Wigner tunneling correction or full quantum transition-state theory is required.

Use the two-temperature comparison mode of this calculator. Measure the rate constant k₁ at temperature T₁ and k₂ at T₂. The equation E_a = R ⋅ ln(k₂÷k₁) ÷ (1÷T₁ − 1÷T₂) gives the activation energy directly without knowing the pre-exponential factor. For higher accuracy, use three or more temperature points and perform linear regression on an Arrhenius plot.

The factor A represents the frequency of molecular collisions with correct orientation, per unit time. In collision theory, A = Z ⋅ p, where Z is the collision frequency and p is the steric factor (0 < p ≤ 1). Typical gas-phase values range from 10¹⁰ to 10¹⁵ s⁻¹. A value outside this range may indicate a non-elementary mechanism or a significant entropy of activation contribution.

Yes. This calculator accepts Kelvin, Celsius, and Fahrenheit. Internally, all temperatures are converted to Kelvin before computation using T_K = T_°C + 273.15 and T_K = (T_°F − 32) × 5÷9 + 273.15. The calculator rejects any input that converts to a non-positive Kelvin value because the Arrhenius equation requires absolute temperature strictly greater than zero.

A common rule of thumb states that reaction rates roughly double for every 10 K increase near room temperature. Quantitatively, the ratio k(T + 10)÷k(T) = exp(E_a ⋅ 10 ÷ (R ⋅ T ⋅ (T + 10))). For E_a = 50 kJ/mol at 300 K, the factor is approximately 1.93. For higher activation energies the effect is more dramatic.