The Arrhenius Equation (College Board AP® Chemistry): Study Guide

The Arrhenius Equation

• The effect of temperature on reaction rate can be explained in terms of kinetic theory

  • Increasing temperature greatly increases the fraction of molecules with very high velocity and high kinetic energies

  • These molecules are more likely to have energy equal to or greater than the activation energy when they collide, so a greater proportion of collisions reach the transition state and react

  • Hence, the higher the temperature, the larger the fraction of molecules that can provide the activation energy required for reaction

• The collision model gives the rate constant for an elementary reaction as

k = p × Z  ×  f            (equation 1)

• Where:

  • Z, the collision frequency

  • f, the fraction of collisions with energy equal to or greater than the activation energy

  • p, the steric factor (the fraction of collisions with a suitable orientation)

• The steric factor, p, does not depend on temperature

• The collision frequency, Z, is relatively insensitive to temperature

  • For example, when the temperature increases from 500 to 600 K, Z increases by less than 10%

• On the other hand, the relationship between the fraction of collisions with energy greater than the activation energy and temperature has been shown as

f = e^-Ea/RT                   (equation 2)

• Where:

  • e is a mathematical constant with an approximate value of 2.718

  • E_a is the activation energy, in J/mol

  • R is the gas constant, 8.314 J mol^-1 K^-1

  • T is the absolute temperature in K

• A combination of equations 1 and 2 gives the Arrhenius equation, which shows how the rate constant is dependent on temperature

k = Ae^-Ea/RT

• Where A is a constant and called the frequency factor

• This factor is related to the frequency of collisions,* Z* and the steric factor, p

• In general, as the value of E_a increases, k decreases

  • This is because the fraction of molecules that possess the required energy is smaller

• On taking the natural log of both sides of the Arrhenius equation, the equation becomes:

ln k = ln A - E_a/RT

• A plot of ln k against 1/T gives a straight line with a negative slope:

The Arrhenius plot

• slope = -E_a/R

• y-intercept = ln A

• A larger activation energy gives a steeper line