Arrhenius Equation Calculator and Derivation

The Arrhenius equation predicts how a reaction’s rate constant changes with absolute temperature, using the activation energy and a pre-exponential factor. Enter those two quantities and the starting and final temperatures to calculate the rate constants and their ratio; the derivation follows from a temperature-dependent activation barrier.

Formula

The Arrhenius equation is k = A exp(−Ea/(RT)). Here k is the reaction rate constant (s⁻¹ for a first-order reaction), A is the pre-exponential factor in the same units as k, Ea is activation energy in J/mol, R = 8.31446261815324 J/(mol·K), and T is absolute temperature in kelvin. In this calculator, Ea is entered in kJ/mol, so the expression converts it to J/mol by multiplying by 1000.

For two temperatures, divide the equations for k₂ and k₁; A cancels:

ln(k₂/k₁) = (Ea/R)(1/T₁ − 1/T₂), or k₂/k₁ = exp[(Ea/R)(1/T₁ − 1/T₂)].

The calculator reports k₂, k₁, and their ratio. The factor 1000 in the ratio expression likewise converts Ea from kJ/mol to J/mol.

Worked Example

Using the default inputs, A = 1.0000 × 10¹² s⁻¹, Ea = 50 kJ/mol, T₁ = 298.15 K, and T₂ = 308.15 K:

• Convert the activation energy: Ea = 50 × 1000 = 50,000 J/mol.
• At the final temperature, k₂ = 1.0000 × 10¹² exp[−50,000/(8.31446261815324 × 308.15)] ≈ 3.362 × 10³ s⁻¹.
• At the initial temperature, k₁ = 1.0000 × 10¹² exp[−50,000/(8.31446261815324 × 298.15)] ≈ 1.682 × 10³ s⁻¹.
• The ratio is k₂/k₁ = exp[(50,000/8.31446261815324)(1/298.15 − 1/308.15)] ≈ 1.998.

Thus the model predicts a rate constant about 1.998 times as large at 308.15 K as at 298.15 K. The calculator rounds its displayed values to three decimal places.

Assumptions

• The reaction follows a single Arrhenius relationship across the temperature interval.
• A and Ea are treated as constant with temperature.
• Temperatures are absolute kelvin, and activation energy is a molar quantity.
• The units of A match the units of the rate constant for the reaction’s rate law.

Derivation from an Activation Barrier

Arrhenius’s empirical relationship can be understood through the fraction of molecular events able to overcome an energy barrier. In a simple activated-process picture, the probability of having the required energy has a Boltzmann factor proportional to exp(−Ea/(RT)). Multiplying this fraction by a factor that represents how often suitably oriented or configured attempts occur gives:

k ∝ exp(−Ea/(RT)).

The proportionality constant is named A, producing k = A exp(−Ea/(RT)). This is a useful physical interpretation, but it is not a complete microscopic derivation for every chemical reaction. Molecular collisions, orientation, solvent effects, and the reaction mechanism can influence the prefactor and even cause deviations from a straight Arrhenius trend.

A practical way to estimate the parameters is to take the natural logarithm:

ln k = ln A − (Ea/R)(1/T).

Plot ln k against 1/T. If the data are linear, the slope is −Ea/R and the vertical intercept is ln A. Therefore Ea = −R × slope, while A = exp(intercept). Use kelvin for T; using Celsius in the reciprocal-temperature plot gives an incorrect slope and activation energy.

Interpreting the Temperature Effect

When Ea is positive, increasing T makes −Ea/(RT) less negative, so the exponential factor and k increase. The effect is not generally a fixed percentage per degree: it depends on both Ea and the temperatures being compared. For a small temperature interval, a useful local approximation is that a larger activation energy produces greater temperature sensitivity.

For example, the worked example has a 10 K increase and nearly doubles k. That does not mean every reaction doubles for a 10 K increase. A low-barrier reaction may change less, while a higher-barrier reaction can change more, all else equal. Also, a rate constant is not itself the observed reaction rate: the rate law and reactant concentrations are needed to calculate that rate.

Units and Common Misconceptions

• Kelvin is essential. Convert Celsius by adding 273.15 before using the equation. A temperature of 25 °C is 298.15 K.
• Match energy units to R. With R in joules per mole-kelvin, Ea must be in joules per mole. This calculator accepts kJ/mol and multiplies by 1000.
• A is not always s⁻¹. Its dimensions depend on the order and form of the rate law. Use units consistent with the particular k being modeled.
• The “rule of doubling” is not universal. A reaction’s temperature response depends on its activation energy and temperature range.
• Activation energy is not necessarily a literal bond-breaking energy. It is a parameter inferred from temperature-dependent kinetics; the mechanism can be more complex than one barrier.

Do not compare reported A values without checking the rate law, units, and model used. An apparent change in A or Ea can arise when the reaction mechanism changes, data span too wide a temperature range, or measurements include transport limitations such as diffusion.

When the Equation Works—and When It Does Not

The equation is commonly used to summarize measured rate constants over a temperature range and to estimate how a modest temperature change affects reaction kinetics. It is most reliable when the same mechanism remains dominant and the reaction conditions are otherwise comparable.

A curved ln k versus 1/T plot is a warning that one constant Ea and A may not describe the data. Possible reasons include a mechanism transition, a change in rate-limiting step, phase or solvent changes, enzyme-like behavior, or experimental limitations. In such cases, fitting one straight line can give misleading parameters; investigate the chemistry and the measurement conditions rather than treating the fitted values as universal constants.

The calculator’s temperatures are restricted to nonnegative kelvin, and its inputs do not establish that a proposed reaction actually follows Arrhenius behavior. For experimental work, follow your lab method, the safety data sheet (SDS), and your supervisor.

Frequently Asked Questions

How is the Arrhenius equation derived?
It is often motivated by the Boltzmann probability that a molecular event has enough energy to cross an activation barrier. Multiplying that probability by an attempt-frequency or collision factor gives k = A exp(−Ea/(RT)); the empirical equation is not a complete mechanism for every reaction.

Why does the rate constant increase with temperature?
For positive activation energy, increasing absolute temperature makes the exponent less negative. A larger fraction of molecular events can then access the activated state, so the predicted rate constant rises.

What is the linear form of the Arrhenius equation?
Taking natural logarithms gives ln k = ln A − Ea/(RT). A plot of ln k against 1/T has slope −Ea/R and intercept ln A when the model is linear over the measured range.

Can I use Celsius in the Arrhenius equation?
No. Use kelvin because the equation contains absolute temperature and its reciprocal. Convert Celsius to kelvin by adding 273.15.

Does the Arrhenius equation calculate reaction rate?
It calculates a rate constant, not the full reaction rate. To obtain a rate, combine k with the rate law and the relevant reactant concentrations.

Related

• Empirical Formula Calculator and Step-by-Step Method
• Heat q in Chemistry: Formula and Calculation
• Arrhenius Equation Graph: How to Plot and Interpret It

in a | Pre-exponential factor A | s^-1 | 1000000000000 | 0
in ea | Activation energy Ea | kJ/mol | 50 | 0
in t1 | Initial temperature T1 | K | 298.15 | 0
in t2 | Final temperature T2 | K | 308.15 | 0
out k2 | Rate constant at final temperature k2 | s^-1 | a * exp(-ea * 1000 / (8.31446261815324 * t2)) | 3
out k1 | Rate constant at initial temperature k1 | s^-1 | a * exp(-ea * 1000 / (8.31446261815324 * t1)) | 3
out ratio | Rate-constant ratio k2/k1 | - | exp(ea * 1000 / 8.31446261815324 * (1 / t1 - 1 / t2)) | 3