## The Arrhenius Equation

#### Learning Objective

• Explain the Arrhenius equation and the meaning of the variables contained within it

#### Key Points

• The equation relates k, the rate constant for a given chemical reaction, with the temperature, T, the activation energy for the reaction, Ea, the pre-exponential factor A, and the universal gas constant, R.
• High temperature and low activation energy favor larger rate constants, and therefore speed up the reaction.
• The equation is a combination of the concepts of activation energy and the Maxwell-Boltzmann distribution.

#### Term

• Exponential DecayWhen a quantity decreases at a rate proportional to its value.

The Arrhenius equation is a simple but remarkably accurate formula for the temperature dependence of the reaction rate constant, and therefore, the rate of a chemical reaction. The equation was first proposed by Svante Arrhenius in 1884. Five years later, in 1889, Dutch chemist J. H. van ‘t Hoff provided physical justification and interpretation for it. The equation combines the concepts of activation energy and the Boltzmann distribution law into one of the most important relationships in physical chemistry:

$k= Ae^{-\frac{E_a}{RT}}$

In this equation, k is the rate constant, T is the absolute temperature, Ea is the activation energy, A is the pre-exponential factor, and R is the universal gas constant.

Take a moment to focus on the meaning of this equation, neglecting the A factor for the time being. First, note that this is another form of the exponential decay law. What is “decaying” here is not the concentration of a reactant as a function of time, but the magnitude of the rate constant as a function of the exponent –Ea /RT.

What is the significance of this quantity? If you recall that RT is the average kinetic energy, it will be apparent that the exponent is just the ratio of the activation energy, Ea, to the average kinetic energy. The larger this ratio, the smaller the rate, which is why it includes the negative sign. This means that high temperatures and low activation energies favor larger rate constants, and therefore these conditions will speed up a reaction. Since these terms occur in an exponent, their effects on the rate are quite substantial.

## Plotting the Arrhenius Equation in Non-Exponential Form

The Arrhenius equation can be written in a non-exponential form, which is often more convenient to use and to interpret graphically. Taking the natural logarithms of both sides and separating the exponential and pre-exponential terms yields: $ln(k)=ln(A)-\frac{E_{a}}{RT}$

Note that this equation is of the form $y=mx+b$, and creating a plot of ln(k) versus 1/T will produce a straight line with the slope –Ea /R. Plot of ln(k) versus 1/T for the decomposition of nitrogen dioxideThe slope of the line is equal to -Ea/R.

This affords a simple way of determining the activation energy from values of k observed at different temperatures. We can plot ln(k) versus 1/T, and simply determine the slope to solve for Ea.

## The Pre-Exponential Factor

Let’s look at the pre-exponential factor A in the Arrhenius equation. Recall that the exponential part of the Arrhenius equation ($e^{\frac{-E_a}{RT}}$) expresses the fraction of reactant molecules that possess enough kinetic energy to react, as governed by the Maxwell-Boltzmann distribution. Depending on the magnitudes of Ea and the temperature, this fraction can range from zero, where no molecules have enough energy to react, to unity, where all molecules have enough energy to react.

If the fraction were unity, the Arrhenius law would reduce to k = A. Therefore, A represents the maximum possible rate constant; it is what the rate constant would be if every collision between any pair of molecules resulted in a chemical reaction. This could only occur if either the activation energy were zero, or if the kinetic energy of all molecules exceeded Ea—both of which are highly unlikely scenarios. While “barrier-less” reactions, which have zero activation energy, have been observed, these are rare, and even in such cases, molecules will most likely need to collide with the right orientation in order to react. In real-life situations, not every collision between molecules will be an effective collision, and the value of $e^{\frac{-E_a}{RT}}$ will be less than one.