Activation energy is the amount of energy that needs to be supplied in order for a chemical reaction to proceed. The example problem below demonstrates how to determine the activation energy of a reaction from reaction rate constants at different temperatures.

## Activation Energy Problem

A second-order reaction was observed. The reaction rate constant at three degrees Celsius was found to be 8.9 x 10^{-3} L/mol and 7.1 x 10^{-2} L/mol at 35 degrees Celsius. What is the activation energy of this reaction?

### Solution

The activation energy can be determined using the equation:

ln(k_{2}/k_{1}) = E_{a}/R x (1/T_{1} - 1/T_{2})

where

E_{a} = the activation energy of the reaction in J/mol

R = the ideal gas constant = 8.3145 J/K·mol

T_{1} and T_{2} = absolute temperatures (in Kelvin)

k_{1} and k_{2} = the reaction rate constants at T_{1} and T_{2}

**Step 1:** Convert temperatures from degrees Celsius to Kelvin

T = degrees Celsius + 273.15

T_{1} = 3 + 273.15

T_{1} = 276.15 K

T_{2} = 35 + 273.15

T_{2} = 308.15 Kelvin

**Step 2** - Find E_{a}

ln(k_{2}/k_{1}) = E_{a}/R x (1/T_{1} - 1/T_{2})

ln(7.1 x 10^{-2}/8.9 x 10^{-3}) = E_{a}/8.3145 J/K·mol x (1/276.15 K - 1/308.15 K)

ln(7.98) = E_{a}/8.3145 J/K·mol x 3.76 x 10^{-4} K^{-1}

2.077 = E_{a}(4.52 x 10^{-5} mol/J)

E_{a} = 4.59 x 10^{4} J/mol

or in kJ/mol, (divide by 1000)

E_{a} = 45.9 kJ/mol

**Answer: **The activation energy for this reaction is 4.59 x 10^{4} J/mol or 45.9 kJ/mol.

## How to Use a Graph to Find Activation Energy

Another way to calculate the activation energy of a reaction is to graph ln k (the rate constant) versus 1/T (the inverse of the temperature in Kelvin). The plot will form a straight line expressed by the equation:

m = - E_{a}/R

where m is the slope of the line, Ea is the activation energy, and R is the ideal gas constant of 8.314 J/mol-K. If you took temperature measurements in Celsius or Fahrenheit, remember to convert them to Kelvin before calculating 1/T and plotting the graph.

If you were to make a plot of the energy of the reaction versus the reaction coordinate, the difference between the energy of the reactants and the products would be ΔH, while the excess energy (the part of the curve above that of the products) would be the activation energy.

Keep in mind, while most reaction rates increase with temperature, there are some cases where the rate of reaction decreases with temperature. These reactions have negative activation energy. So, while you should expect activation energy to be a positive number, be aware that it's possible for it to be negative as well.

## Who Discovered Activation Energy?

Swedish scientist Svante Arrhenius proposed the term "activation energy" in 1880 to define the minimum energy needed for a set of chemical reactants to interact and form products. In a diagram, activation energy is graphed as the height of an energy barrier between two minimum points of potential energy. The minimum points are the energies of the stable reactants and products.

Even exothermic reactions, such as burning a candle, require energy input. In the case of combustion, a lit match or extreme heat starts the reaction. From there, the heat evolved from the reaction supplies the energy to make it self-sustaining.