Arrhenius equation

The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the rate constant, and therefore rate, of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation for it. Nowadays it is best seen as an empirical relationship. It can be used to model the temperature-variance of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions.

The general rule of thumb, without solving the equation, is that for every 10°C increase in temperature the rate of reaction doubles. As with any rule of thumb, it does not always work.

Overview
In short, the Arrhenius equation gives "the dependence of the rate constant k of chemical reactions on the temperature T (in Kelvin) and activation energy Ea", as shown below:


 * $$k = A e^{{-E_a}/{RT}}$$

where A is the pre-exponential factor or simply the prefactor and R is the gas constant. The units of the pre-exponential factor are identical to those of the rate constant and will vary depending on the order of the reaction. If the reaction is first order it has the units s-1, and for that reason it is often called the frequency factor or attempt frequency of the reaction. When the activation energy is given in molecular units, instead of molar units, e.g. joules per molecule instead of joules per mole, the Boltzmann constant is used instead of the gas constant. 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.

Given the small temperature range in which kinetic studies are carried, it is reasonable to approximate the activation energy as being independent of the temperature. Similarly, under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor is negligible compared to the temperature dependence of the $$\scriptstyle \exp(-E_a/RT)$$ factor; except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.

Some authors define a modified Arrhenius equation, that makes explicit the temperature dependence of the pre-exponential factor. If one allows arbitrary temperature dependence of the prefactor, the Arrhenius description becomes overcomplete, and the inverse problem (i.e. determining the prefactor and activation energy from experimental data) becomes singular. The modified equation is usually of the form


 * $$k = A (T/T_0)^n e^{{-E_a}/{RT}}$$

where T0 is a reference temperature and allows n to be a unitless power. Clearly the original Arrhenius expression above corresponds to n = 0. Fitted rate constants typically lie in the range -1<n<1. Theoretical analyses yield various predictions for n. It has been pointed out that "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted T½ dependence of the pre-exponential factor is observed experimentally." However, if additional evidence is available, from theory and/or from experiment (such as density dependence), there is no obstacle to incisive tests of the Arrhenius law.

Another common modification is the stretched exponential form


 * $$k = A \exp (-(\frac{-E_a}{RT})^{\beta})$$

where &beta; is a unitless number of order 1. This is typically regarded as an experimental fit, but can have theoretical meaning, for example showing the presence of a range of activation energies or in special cases like the Mott variable range hopping.

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 Ea 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):


 * $$\ E_a \equiv -R \left( \frac{\partial \ln k}{\partial ~(1/T)} \right)_P $$

Kinetic theories interpretation of Arrhenius equation
Arrhenius argued that in order for reactants to be transformed into products, they first needed to acquire a minimum amount of energy, called the activation energy Ea. At an absolute temperature T, the fraction of molecules that have a kinetic energy greater than Ea can be calculated from the Maxwell-Boltzmann distribution of statistical mechanics, and turns out to be proportional to $$\ e^{\frac{-E_a}{RT}}$$. The concept of activation energy explains the exponential nature of the relationship, and in one way or another, it is present in all kinetic theories:

Collision theory
One example comes from the "collision theory" of chemical reactions, developed by Max Trautz and William Lewis in the years 1916-18. In this theory, molecules are supposed to react if they collide with a relative kinetic energy along their line-of-centers that exceeds Ea. This leads to an expression very similar to the Arrhenius equation.

Transition state theory
Another Arrhenius-like expression appears in the "transition state theory" of chemical reactions, formulated by Wigner, Eyring, Polanyi and Evans in the 1930s. This takes various forms, but one of the most common is


 * $$\ k = \frac{k_BT}{h}e^{-\frac{\Delta G^\Dagger}{RT}}$$

where ΔG‡ is the Gibbs free energy of activation, kB 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, which is multiplied by the absolute temperature, as well as an enthalpy term. Both of them 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 reaction, and can be calculated using formulas from statistical mechanics involving the partition functions of the reactants and of the activated complex.

Notes and references

 * Laidler, K. J. (1997) Chemical Kinetics,Third Edition, Benjamin-Cummings
 * Laidler, K. J. (1993) The World of Physical Chemistry, Oxford University Press

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