Arrhenius Equation Calculator

As the temperature is increased, the reaction speeds up. The Arrhenius equation quantifies this speeding up precisely. This equation relates the rate constant of a reaction to the temperature, the size of the energy barrier that molecules must overcome and the frequency with which molecules collide and react in the correct orientation.

If you enter three of the four values this tool will calculate the remaining value. The opposite is also possible. If you give the rate constants at two different temperatures then it calculates the activation energy. This is often what many people actually want to determine.

The Arrhenius equation

Svante Arrhenius formulated this relationship in 1889, and it is commonly expressed per mole as follows:

Each symbol has a specific meaning:

• k is the rate constant. Its units vary with reaction order and range from M/s for zeroth-order reactions to 1/s for first-order reactions to higher-order reaction units.

• A is the pre-exponential factor also known as frequency factor. It has same unit as that of k and gives the frequency with which molecules collide with correct orientation.

• Ea is the activation energy, i.e. the height of the energy barrier between reactants and products, typically given in kJ/mol.

• R is the gas constant and is equal to 8.314 J/mol·K.

• T is the absolute temperature expressed in kelvin. This conversion is often forgotten so calculator tools will convert from celsius or fahrenheit.

The exponential factor gives the proportion of molecules that have enough energy to trigger a reaction when they collide. As this proportion increases dramatically with temperature, even a small change in T can lead to a large change in the rate of reaction.

Example calculation:

Let's consider the decomposition of nitrogen dioxide. At 320 degrees Celsius, the rate constant is 0.5 M/s and the activation energy is 115 kJ/mol. We want to determine the pre-exponential factor.

First we need to convert the units. The temperature is 593.15 K since 320 + 273.15 = 593.15. The energy is calculated by multiplying 115 kJ/mol by 1000 which gives us 115,000 J/mol. To rearrange the formula to solve for A we get:

The result of the exponent calculation is approximately negative 23.32, so A is approximately 6.71 times 10 to the power of nine with units of M/s. If you leave the field for the exponent blank and enter the other three values in a calculator, you will get the same answer without having to consult a logarithm table.

Determination of activation energy from two temperatures.

It is rare that the value of A is known in advance. The activation energy can be determined by measuring the rate constants at two different temperatures. If one writes down the Arrhenius equation for each temperature and divides the two equations, A will completely cancel out.

If you switch the calculator to two-temperature mode and enter both rate constants and corresponding temperatures, it will give you Ea. Conversely, if you leave one of the rate constant or temperature values blank, the calculator will estimate that value based on the already known activation energy. As a rough rule of thumb, at room temperature many reactions double in speed for every 10 degrees Celsius. This corresponds to an activation energy of about 50 kJ/mol.

Linear form and Arrhenius plot

Because exponential curves are difficult to read directly from a graph, chemists take the natural logarithm of both sides of the equation.

This is the linear equation y = mx + c. The vertical axis represents ln (k), the horizontal axis 1/T. The slope is negative and equal to Ea/R, the intercept corresponds to ln(A). By plotting several measurements and fitting an optimal straight line you can determine from the slope of the line the activation energy and from the intercept the pre-exponential factor. If you activate the Arrhenius plot in the calculator you will see this line for your own data.

Per mole or per single molecule?

In chemistry calculations are usually done per mole and the gas constant R is used. In physics one often looks at individual molecules so only a small change is needed. Here R is replaced by the Boltzmann constant, which is 1.380649 multiplied by 10 to the power of minus 23, giving J/K, and the activation energy is given per molecule. In this case electronvolts are a more practical unit. If you select "molecules" in the upper part of the calculator then the Boltzmann constant is used throughout. If a conversion is needed one electronvolt is about 96.5 kJ/mol.

What these numbers mean:

How to use this calculator:

Select the mode in the top section. If you want to calculate a single rate constant value, enter three of four values (A, activation energy, temperature or k), and the fourth will be calculated for you. If you want to use two temperatures, enter two measurements to determine the activation energy, or enter the activation energy and leave one of the values blank to get a prediction.

The activation energy unit should be set to kJ, kcal or eV depending on which is appropriate for your data and the temperature can be entered in any convenient units. Very large numbers like A are best entered using scientific notation (e.g. 1.5e13). The results will appear on the right side of the page and also show ln(k), ln(A) and the absolute temperature used.

These results are based on the assumption that the reaction consists of a single elementary step and that the activation energy is independent of temperature. As actual reaction mechanisms can give curves in an Arrhenius plot, the Ea obtained by fitting should be considered as an effective value within the measured temperature range.