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How can an Arrhenius plot be used to calculate activation energy?

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A graph showing a potential energy curve for a chemical reaction, with the y-axis labeled potential energy and the x-axis labeled progress of reaction. The curve rises from the initial state (reactants) to a peak labeled transition state (activated complex), indicating activation energy, then drops to the final state (products). — chemical kinetics In the Arrhenius equation, the activation energy (E) represents the minimum amount of energy required to transform reactants into products in a chemical reaction. On a potential energy curve, the value of the activation energy is equivalent to the difference in potential energy between particles in an intermediate configuration (known as the activated complex, or transition state) and particles of reactants in their initial state. The activation energy thus can be visualized as a barrier that must be overcome by reactants before products can be formed.

To calculate the activation energy ( $E$ ) using an Arrhenius plot, you utilize the Arrhenius equation, which relates the reaction-rate constant ( $k$ ) to the absolute temperature ( $T$ ):

$k = A \exp (- E / R T)$

In this equation, $A$ is the pre-exponential factor and $R$ is the molar gas constant.123

To create the plot, you take the natural logarithm of the equation, which yields a linear relationship:

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

By plotting the natural logarithm of the rate constant ( $\ln k$ ) on the y-axis against the reciprocal of the absolute temperature ( $1 / T$ ) on the x-axis, you obtain a straight line. According to this linear form ( $y = m x + c$ ), the slope ( $m$ ) of the line is equal to $- E / R$ . Therefore, you can calculate the activation energy by multiplying the negative slope of the line by the gas constant:

$E = - slope \times R$

This graphical method allows chemists to determine the kinetic parameters of a reaction by measuring rate constants at various temperatures.23