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Chemical

Arrhenius rate constant

How a reaction rate constant changes with temperature.

Ref: CHE-05
Rev: v1
Field: Chemical
Result: k [1/s]

k = k_{0} e^{- \frac{E_{a}}{R_{u} T}}

k rate constant, k₀ pre-exponential factor (same units as k), E_a activation energy in J/mol, R_u = 8.314 J/(mol·K) and T absolute temperature in kelvin.

Reactions speed up sharply with temperature because more molecules have enough energy to react. A high activation energy makes the rate especially sensitive to temperature.

Calculator

Solve for

kRate constantComputed

1/s

k₀Pre-exponential factor

1/s

1,0001e12

E_aActivation energy

kJ/mol

20150

TTemperature

250500

Exponent E_a/(R_uT)20.62
Rate at T + 10 K ÷ rate at T1.773 ×

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Rate constant (1/s) current value 0.0111 1/s

E_aActivation energyKey driver

4.1e-1610,400

k₀Pre-exponential factor

1.11e-61,110

TTemperature

2.91e-65.39

A rough rule: the reaction rate rises by a factor of about 1.5 to 3 for each 10 K, depending on the activation energy.
A plot of ln k against 1/T is a straight line whose slope is −E_a/R, used to find E_a from experiments.
Storing energetic materials cool is effective precisely because of this exponential temperature dependence.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (1/s)	Local step	Change in result
E_a Activation energy	20 to 150 kJ/mol	4.101e-16	10,350	10,350	+10%	−87.3 %
k₀ Pre-exponential factor	1,000 to 1e12 1/s	1.11e-6	1,110	1,110	+10%	+10 %
T Temperature	250 to 500 K	2.906e-6	5.39	5.39	+10%	+552 %

Sweep of Temperature

Temperature (K)	Rate constant (1/s)
250	2.906e-6
281.3	7.181e-5
312.5	9.345e-4
343.8	0.007627
375	0.04387
406.3	0.1928
437.5	0.6857
468.8	2.059
500	5.39

Sweep one variable

Vary

Rate constant (1/s)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Rate constant

Result · 1/s

What the equation gives you. Everything else feeds it.

k₀

Pre-exponential factor

1/s · typical 1,000 to 1e12

Frequency factor. Directly proportional.

E_a

Activation energy

kJ/mol · typical 20 to 150

Energy barrier for the reaction. Strong: higher barrier lowers k exponentially.

Temperature

K · typical 250 to 500

Absolute temperature. Strong: k rises roughly 1.5–3× for every 10 K at typical activation energies.

Worked example with the values above

Exponent: E_a/(R_uT) = 60,000 / (8.314 × 350) = 20.62.
e^−20.62 = 1.11×10⁻⁹.
k = 1×10⁷ × 1.11×10⁻⁹ = 0.0111 s⁻¹.

Assumptions and limits

Single-step reaction with constant E_a.
Absolute temperature in kelvin.
Range where Arrhenius behaviour holds.

Common mistakes

Using °C.
Mixing kJ/mol with J/mol.
Extrapolating widely beyond measured temperatures.

Statistics and uncertainty

The exponent magnifies errors: a 10 kJ/mol error in E_a at 350 K changes k by a factor of about 31. Temperatures must be known to within a degree or so to be useful. Treat the result as an order-of-magnitude estimate unless fitted to your own data.

Where it is used

Ageing and shelf-life prediction of propellants and batteries.
Thermal runaway estimates.
Accelerated life testing.

Maths you need

Exponentials: e^(−x)
Absolute temperature: kelvin

Related formulas

Practise this formula