# Arrhenius equation

The rate law, rate = k[X]m[Y]n, shows how the reaction rate depends on the concentration of reactants but how about other parameters such as temperature. The dependence of the reaction rate with temperature comes from the variation of the rate constant, k, with temperature.

The rate constant, k, usually increases exponentially with temperature.

In 1888, Swedish chemist Arrhenius suggested that molecules must possess a certain minimum amount of energy in order to react. This energy is the kinetic energies of the colliding molecules. Upon collision, the kinetic energy of the molecules can be used to stretch, bend, and break bonds, leading to chemical reactions.

If molecules are moving slowly with little kinetic energy, they will bounce off one another without changing. In order to react, colliding molecules must have a total kinetic energy equal to or greater than some minimum value. The minimum energy required to initiate a chemical reaction is called the activation energy, Ea. The value Ea varies from reaction to reaction. Only faster moving particles will collide with sufficient kinetic energy to overcome the activation barrier.

If activation energy is high, only a few molecules have sufficient kinetic energy to overcome the barrier and the reaction is slower. As temperature increases, more of the molecules will have sufficient kinetic energy to overcome the barrier, therefore, the reaction rate increases.

Arrhenius demonstrated an empirical relationship, called the Arrhenius equation. It is used to describe the temperature dependence of the rate constant, k.

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

where Ea is the activation energy,

R is the universal gas constant,
T is the temperature (in Kelvin), and
A is the proportionality constant called the frequency factor, or pre-exponential factor.

Taking the natural logarithm of the Arrhenius equation gives:

$ln(k)= -\left(\frac{E_a}{R}\right)\left(\frac{1}{T}\right)+ln(A)$
y = m x + b

If k is determine from several temperatures, a plot of ln(k) vs. 1/T(K) can be made.. The data should give a straight line. Fitting the data points to a linear equation, y = mx + b, can be used to determine the slope, m, and y-intercept, b. Then the values of Ea can be determine from Ea = -Rm and A from a = eb. A is related to the frequency of collision. It has units of the rate constant, k, and is related to the probability that the collisions are favourably oriented for the reaction. (i.e. correct spatial orientation, steric factor, with respect to each other is required). The more constrained the transition state is the smaller the steric factor.