The Arrhenius Equation: How Temperature Changes Reaction Rates

A reaction mixture can look unchanged while its microscopic traffic has transformed. Raise the temperature and molecules move through a wider range of energies; more encounters may reach configurations that can become products. The Arrhenius equation turns that temperature sensitivity into a compact exponential model.

Its compactness invites shortcuts. Temperature entered in degrees Celsius, activation energy left in kilojoules beside a joule-based gas constant, or a rate coefficient mistaken for the whole reaction rate can produce a plausible-looking but meaningless number. Arrhenius is most useful when its units, fitted range, and empirical status stay visible.

A chemist warms a molecular system while paired particles cross a large activation-energy hill and rearrange on the far side — Activation energy is an empirical measure of temperature sensitivity. The hill is a useful metaphor, but a fitted value is not automatically one literal microscopic barrier.

Simple Arrhenius modelk = A exp(−E_a / RT)

Temperature changes the exponential factor; the original simple form treats A and E_a as constant over the fitted range.

Every symbol carries a condition and a unit

krate coefficient at Tunit depends on the rate law

Apre-exponential factorsame unit as k

E_aArrhenius activation energyJ mol⁻¹

Rmolar gas constant8.314462618… J mol⁻¹ K⁻¹

Tthermodynamic temperatureK, never °C in the formula

The IUPAC Gold Book defines the Arrhenius equation as a relationship between a rate coefficient and thermodynamic temperature. It separately calls activation energy an empirical parameter. That wording matters: the equation summarizes how a measured k changes; it does not by itself identify a molecular mechanism.

The exponent must be dimensionless:

activation energyJ mol⁻¹

R × T(J mol⁻¹ K⁻¹)(K)

exponentdimensionless

Using R = 8.31446261815324 J mol⁻¹ K⁻¹ from NIST’s 2022 CODATA adjustment means an activation energy reported in kilojoules per mole must first be multiplied by 1000. A temperature such as 25 °C must become 298.15 K. A 10 °C interval equals 10 K, but an absolute temperature of 10 °C is not 10 K; the BIPM SI Brochure distinguishes those two ideas.

A rate coefficient is not the complete reaction rate

For a concentration-based rate law such as rate = k[A]^m[B]ⁿ, Arrhenius models k. Concentrations and reaction orders remain in the rate law. If the total order is m + n, the unit of k changes so the final rate retains concentration per time.

zero orderk, A: M s⁻¹

first orderk, A: s⁻¹

second orderk, A: M⁻¹ s⁻¹

There is no universal unit for k or A. Pressure-based, surface, number-density, and other kinetic conventions use other units. IUPAC’s definition of reaction order and the OpenStax rate-law treatment supply the link between exponents and coefficient units.

Compare two temperatures without knowing A

For the same fitted model at two temperatures, divide one Arrhenius equation by the other. The pre-exponential factor cancels:

rate-coefficient ratioln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂)

predict k₂k₂ = k₁ exp[(E_a/R)(1/T₁ − 1/T₂)]

infer E_aE_a = R ln(k₂/k₁) / (1/T₁ − 1/T₂)

The logarithm is legitimate because k₂/k₁ is dimensionless when the coefficients use the same unit. A useful sign check follows immediately: for positive E_a, if T₂ > T₁, then 1/T₁ − 1/T₂ is positive and k₂/k₁ should exceed one.

Identical molecular systems sit along one rail, with sparse short paths in a cool vessel and dense energetic motion in a warmer vessel — The model compares compatible rate coefficients while temperature changes. Concentration, medium, pressure, phase, and kinetic definition must not silently change between the two points.

Worked example 1: predict k after a 10 K increase

Suppose a first-order reaction has E_a = 50.0 kJ mol⁻¹ and k₁ = 2.50 × 10⁻³ s⁻¹ at T₁ = 298.15 K. Predict the coefficient at T₂ = 308.15 K.

1. Match energy units50.0 kJ mol⁻¹ = 50,000 J mol⁻¹

2. Form the exponent(50,000 / 8.314462618)(1/298.15 − 1/308.15) = 0.654543981

3. Find the multiplierk₂/k₁ = e^0.654543981 = 1.924264816

4. Predict k₂k₂ = (2.50 × 10⁻³)(1.924264816) = 4.81066 × 10⁻³ s⁻¹

Substituting the unrounded k₂ into the activation-energy rearrangement returns 50,000 J mol⁻¹. The example also disproves a common shortcut: this particular 10 K increase multiplies k by about 1.92, not exactly two.

Worked example 2: infer Eₐ and A from two measurements

For an overall second-order reaction, suppose k₁ = 0.150 M⁻¹ s⁻¹ at 300.0 K and k₂ = 1.20 M⁻¹ s⁻¹ at 340.0 K. The ratio is 8.00.

Activation energyE_a = 8.314462618 ln(8.00) / (1/300.0 − 1/340.0) = 44,088.1 J mol⁻¹ = 44.09 kJ mol⁻¹

Pre-exponential factorA = k₁ exp(E_a/RT₁) = 7.11797 × 10⁶ M⁻¹ s⁻¹

Midpoint predictionk(320.0 K) = A exp[−E_a/(R·320.0)] = 0.45275 M⁻¹ s⁻¹

Forward substitution at 300.0 K and 340.0 K recovers 0.150 and 1.20 M⁻¹ s⁻¹. The units of A match the second-order coefficient; reporting it in s⁻¹ would silently change the rate law.

Why “10 °C doubles the rate” is not a law

OpenStax describes approximate doubling for many homogeneous reactions as a useful observation, not a universal identity. The multiplier also depends on activation energy and the starting temperature. From 25 °C to 35 °C:

E_a	temperature interval	predicted k₂/k₁
40 kJ mol⁻¹	298.15 → 308.15 K	1.688
50 kJ mol⁻¹	298.15 → 308.15 K	1.924
80 kJ mol⁻¹	298.15 → 308.15 K	2.850

Exact doubling over that specific interval would require E_a ≈ 52.949 kJ mol⁻¹. Change the starting temperature and the required value changes too. Moreover, Arrhenius predicts a change in k; the observed rate can also move because concentrations, phases, or other conditions changed.

The straight-line plot is a test, not a guarantee

Taking logarithms gives a linear form. Strictly, a logarithm needs a dimensionless argument, so choose a fixed reference coefficient k° with the same unit as k:

ln(k/k°) = −(E_a/R)(1/T) + ln(A/k°)

slopem = −E_a/R

interceptb = ln(A/k°)

recover E_aE_a = −mR

Textbooks often write ln k as shorthand after fixing a unit. Changing from s⁻¹ to min⁻¹ multiplies every numerical coefficient by 60, shifting the intercept by ln 60; the slope and inferred E_a stay unchanged. If the horizontal axis uses 1000 K/T instead of 1/T, the slope also changes by a factor of 1000, so axis scaling must be stated.

Hands stretch a straight thread through central data points while markers at both extremes curve away and a folded layer reveals another path — A straight middle region can support one fitted slope. Departures at the ends show that the simple model is inadequate there, not which alternative mechanism is correct.

Where the simple model stops earning trust

Temperature range

A fit belongs to the measured interval. NIST kinetics records publish expressions with explicit ranges rather than promising unlimited extrapolation.

Constant parameters

The simple form assumes one A and one E_a. IUPAC also defines modified forms when temperature dependence needs more structure.

Compatible conditions

Phase, solvent, pressure regime, catalyst state, mixing, diffusion, and the rate-law definition must remain comparable.

Mechanism restraint

Curvature proves that one constant slope is inadequate. It does not by itself prove which step, pathway, or transport process changed.

Narrow temperature ranges create a second problem: the small spread in 1/T makes the fitted slope sensitive to scatter, while A is extrapolated to the distant intercept at 1/T = 0. Fitted A and E_a can therefore be strongly correlated. Report the range and uncertainty when they are available, and do not treat a precise-looking intercept as independent molecular evidence.

The NIST Chemical Kinetics Database illustrates the right reporting habit: an Arrhenius expression appears with reaction order, units, and a fitted temperature interval. A usable calculation carries those conditions forward. It can interpolate compatible data, compare sensitivities, and expose a model’s assumptions. It cannot certify a mechanism, replace the rate law, or make an extrapolation safe merely because the exponential returns a number.