Arrhenius Equation Calculator - Reaction Rate Constant

What Is the Arrhenius Equation?

The Arrhenius equation describes how the rate of a chemical reaction changes with temperature. The Swedish scientist Svante Arrhenius proposed it in 1889. Chemists use it to predict how much faster a reaction runs when heated, or how much slower it runs when cooled.

Arrhenius Equation Formula

$k = A \times e^{-E_a/RT}$

Where:

• $k$ is the rate constant, a number that measures how fast the reaction proceeds
• $A$ is the pre-exponential factor, also called the frequency factor. It reflects how often molecules collide in a way that could react
• $E_a$ is the activation energy, the minimum energy molecules need to react, usually given in kilojoules per mole (kJ/mol)
• $R$ is the molar gas constant, 8.31446261815324 J/(mol·K). Since the 2019 revision of the SI this value is exact, because it is the product of two defined constants
• $T$ is the absolute temperature in kelvin

Chemists often use a logarithmic version of the equation to find activation energy from lab data:

$\ln(k) = \ln(A) - \frac{E_a}{R} \times \frac{1}{T}$

Plotting ln(k) against 1/T gives a straight line. The slope of that line equals $-E_a/R$. This is called an Arrhenius plot.

How to Calculate the Reaction Rate Constant

1. Find the activation energy in kJ/mol, from a reference table or from lab measurements taken at several temperatures.
2. Convert the temperature to kelvin. Add 273.15 to a Celsius reading.
3. Find the pre-exponential factor. It is often written in scientific notation, such as 1.0 × 10¹³ s⁻¹.
4. Convert the activation energy to joules per mole by multiplying by 1000, so the units match the gas constant R.
5. Divide the activation energy in J/mol by R times T.
6. Raise e to the negative of that result, then multiply by A.

This calculator performs steps 4 through 6 automatically. Enter the activation energy, the temperature, and the pre-exponential factor, and it returns the rate constant k.

All three inputs must be greater than zero, so the calculator shows no answer for a zero or negative entry. The answer appears in scientific notation with five significant figures. With the starting values of 50 kJ/mol, 298 K and 1.0 × 10¹³ s⁻¹, the result is 1.7217 × 10⁴ s⁻¹.

Worked example

Take a reaction with an activation energy of 75 kJ/mol, running at 350 K, with a pre-exponential factor of 5.0 × 10¹² s⁻¹.

First convert the activation energy to joules per mole: 75 kJ/mol equals 75,000 J/mol.

$k = 5.0 \times 10^{12} \times e^{-75{,}000/(8.31446261815324 \times 350)}$

R times T is 2910.06 J/mol, so the exponent works out to -25.7726.

$k = 5.0 \times 10^{12} \times e^{-25.7726}$

$k = 5.0 \times 10^{12} \times 6.4133 \times 10^{-12}$

$k \approx 32.07 \text{ s}^{-1}$

The rate constant is about 32.07 s⁻¹ at 350 K. The calculator displays it as 3.2066 × 10¹.

Rounding the exponent too early changes the answer. The exponential is steep, so cutting -25.7726 down to -25.77 already shifts k by about 0.3%.

Why Temperature Changes Reaction Rate

Molecules must collide with enough energy to react. At any given temperature, only a fraction of molecules have that much energy. The exponential term $e^{-E_a/RT}$ in the equation stands for that fraction.

Raising the temperature gives more molecules enough energy to react, so the fraction grows and the reaction speeds up. Because the relationship is exponential rather than straight-line, even a small rise in temperature can noticeably raise the rate, especially for reactions with a high activation energy. This same effect explains why cooling food, as a refrigerator does, slows the reactions that cause spoilage.

Where the Arrhenius Equation Is Used

• Chemical engineering. Engineers use it to choose operating temperatures for industrial reactors.
• Pharmaceuticals. Drug makers store samples at high temperatures for a short time, measure how fast they break down, then use the equation to estimate shelf life at normal storage temperature.
• Food science. It explains why warm food spoils faster than refrigerated food, and it guides pasteurization time and temperature.
• Materials science. It helps predict how fast plastics, coatings, and electronic parts degrade at their working temperature.
• Earth science. Soil scientists use it to model how warming changes the rate at which microbes release carbon dioxide from soil.

History

Svante Arrhenius (1859–1927) proposed the equation in 1889 while studying how reaction rates depend on temperature. He received the 1903 Nobel Prize in Chemistry, mainly for his theory of electrolytic dissociation. In the 1930s, Henry Eyring and Michael Polanyi developed transition state theory, which gave the exponential term in Arrhenius's equation a firmer theoretical basis.

Limits of the Arrhenius Equation

The equation fits many simple reactions well over a moderate temperature range, but not every case:

• Enzyme-catalyzed reactions slow down at high temperature because the enzyme denatures, so the rate does not keep rising the way the equation predicts.
• Reactions limited by how fast molecules diffuse toward each other show weaker temperature dependence than the equation predicts.
• Reactions involving light atoms, especially hydrogen, can run faster than predicted at low temperature because of quantum tunneling.

A plot of ln(k) against 1/T that curves, rather than forming a straight line, is a sign the reaction does not follow the Arrhenius equation over that temperature range.

Frequently asked questions

What units does the Arrhenius equation use?

Activation energy is entered in kJ/mol and converted to J/mol inside the calculator, temperature is in kelvin, and the molar gas constant R equals 8.31446261815324 J/(mol·K). The pre-exponential factor A and the rate constant k always share the same units. The calculator labels the answer s⁻¹, which is correct for a first-order reaction. For another reaction order, k carries whatever units A was given in.

Why must temperature be in kelvin, not Celsius?

The equation comes from the Boltzmann distribution, which measures energy relative to absolute zero. Kelvin starts at absolute zero, so it gives a meaningful ratio between energy and temperature. Using Celsius would give incorrect results.

What does the pre-exponential factor represent?

It represents the theoretical maximum rate constant, the value k would reach if every molecular collision carried enough energy to react. It accounts for how often molecules collide and whether they are lined up correctly when they do.

How is activation energy found from experimental data?

By measuring the rate constant at several temperatures, plotting ln(k) against 1/T, and reading the slope of the resulting line. Activation energy equals the slope multiplied by −R.

Is the rate constant the same as the reaction rate?

No. The rate constant k depends only on the temperature and on the reaction itself. The reaction rate also depends on how concentrated the reactants are. For a first-order reaction the rate equals k times the reactant concentration. Between two reactions of the same order at the same concentrations, the larger k is the faster one.

Can the Arrhenius equation predict the rate at any temperature?

Only within the range where the reaction mechanism stays the same. Extrapolating far outside the temperatures used to measure Ea and A can give inaccurate results, since the mechanism or the rate-limiting step may change.