Arrhenius equation

The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of a chemical reaction rate. It was first proposed by the Dutch chemist J. H. van't Hoff in 1884; five years later, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation for it.

Arrhenius argued that in order for reactants to be transformed into products, they first needed to aquire enough energy to form an "activated complex". This minimum energy is called the "activation energy" E_a for the reaction. In thermal equilibrium at an absolute temperature T, the fraction of molecules that have a kinetic energy greater than E_a can be calculated from the Maxwell-Boltzmann distribution of statistical mechanics, and turns out to be proportional to $e^{\frac {-E_{a}}{RT}}$, where E_a is measured in molar units (e.g. joules per mole) and R is the gas constant. This leads to the Arrhenius formula for the reaction rate constant k:

$k=Ae^{\frac {-E_{a}}{RT}}$.

(If the energy is given in molecular units (e.g. joules per particle or per molecule), then R (the gas constant) is replaced by Boltzmann's constant $k_{B}$, which is just R divided by Avogadro's number) The A factor or the frequency factor A is a constant specific to a particular reaction.

It can be seen that either increasing the temperature or decreasing the activation energy (for example through the use of catalysts) will result in an increase in rate of reaction.

While remarkably accurate in a wide range of circumstances, the Arrhenius equation is not exact, and various other expressions are sometimes found to be more useful in particular situations. One example comes from the "collision theory" of chemical reations, developed by Max Trautz and William Lewis in the years 1916-18. In this theory, molecules react if they collide with a relative kinetic energy along their line-of-centers that exceeeds E_a This leads to an expression very similar to the Arrhenius equation, with the difference that the preexponential factor "A" is not constant but instead is proportional to the square root of temperature. This reflects the fact that the overall rate of all collisions, reactive or not, is proportional to the average molecular speed which in turn is proportional to T^1/2. In practice, the square root temperature dependence of the preexponential factor is usually very slow compared to the exponential dependence associated with E_a.

Another Arrhenius-like expression appears in the Transition State Theory of chemical reactions, formulated by Wigner, Eyring *[1], Polanyi and Evans in the 1930's. This takes various forms, but one of the most common is:

$k={\frac {k_{B}T}{h}}e^{-{\frac {\Delta G^{\ddagger }}{RT}}}$

where ΔG^‡ is the Gibbs free energy of activation, $k_{B}$ is Boltzmann's constant, and h is Planck's constant.

At first sight this looks like an exponential multiplied by a factor that is linear in temperature. However, one must remember that free energy is itself a temperature dependent quantity. The free energy of activation includes an entropy term as well as an enthalpy term, both of which depend on temperature, and when all of the details are worked out one ends up with an expression that again takes the form of an Arrhenius exponential multiplied by a slowly varying function of T. The precise form of the temperature dependence depends upon the reation, and can be calculated using formulas from statistical mechanics (it involves the partition functions of the reactants and of the activated complex)

Taking the natural logarithm of the Arrhenius equation yields

$\ln(k)={\frac {-E_{a}}{R}}{\frac {1}{T}}+\ln(A)$.

So, when a reaction has a rate constant which obeys the Arrhenius equation, a plot of ln(k) versus T^-1 gives a straight line, whose slope and intercept can be used to determine E_a and A. This procedure has become so common in experimental chemical kinetics that practitioners have taken to using it to define the activation energy for a reaction. That is the activation energy is defined to be (-R) times the slope of a plot of ln(k) vs. (1/T):

$E_{a}\equiv -R\left({\frac {\partial ~lnk}{\partial ~(1/T)}}\right)_{P}$

This results in an E_a that is in principle a function of T (since the Arrhenius equation is not exact) but in practice the T dependence is very weak.

References

Laidler, K. J. (1997) Chemical Kinetics,Third Edition, Benjamin-Cummings

Laidler, K. J. (1993) The World of Physical Chemistry, Oxford University Press

See also

• Kinetics