The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of a chemical reaction rate, more correctly, of a rate coefficient, as this coefficient includes all magnitudes that affect reaction rate except for concentration.^[1] The equation 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. Nowadays it is best seen as an empirical relationship ^[2]. In chemical kinetics a reaction rate constant quantifies the speed of a chemical reaction. ... Jacobus Henricus van t Hoff (August 30, 1852 - March 1, 1911) was a Dutch physical and organic chemist and the winner of the inaugural Nobel Prize in Chemistry. ... Svante August Arrhenius (February 19, 1859 – October 2, 1927) was a Swedish chemist and one of the founders of the science of physical chemistry. ...

Overview

In short, the Arrhenius equation is an expression that shows the dependence of the rate constant k of chemical reactions on the temperature T and activation energy E_a, as shown below:.^[3] In chemical kinetics a reaction rate constant quantifies the speed of a chemical reaction. ... A chemical reaction occurs when vapours of hydrogen chloride in a beaker and ammonia in a test tube meet to form a cloud of a new substance, ammonium chloride A chemical reaction is a process that results in the interconversion of chemical substances [1]. The substance or substances initially involved... Absolute zero is the lowest temperature that can be obtained in any macroscopic system. ... The sparks generated by striking steel against a flint provide the activation energy to initiate combustion in this Bunsen burner. ...

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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 mol, 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. In chemical kinetics, the frequency factor or A factor is the pre-exponential constant in the Arrhenius equation, which indicates how many collisions between reactants have the correct orientation to lead to the products. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... The sparks generated by striking steel against a flint provide the activation energy to initiate combustion in this Bunsen burner. ... It has been suggested that this article or section be merged into Catalysis. ...

Given the small temperature range in which kinetic studies are carried, it is reasonable to approximate the activation energy as being independent of 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 factor; except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable. This article or section does not cite its references or sources. ...

Some authors define a modified Arrhenius equation,^[4] 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. 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 T^½ dependence of the pre-exponential factor is observed experimentally"^[2] ... but 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.

Taking the natural logarithm of the Arrhenius equation yields: The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...

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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):

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 E_a. 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 . 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: The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

Collision theory

Main article: 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 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 pre-exponential factor is usually very slow compared to the exponential dependence associated with E_a, to the point that some think it can not be experimentally proven. Collision theory is a theory, proposed by Max Trautz and William Lewis in 1916 that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. ...

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 1930's. This takes various forms, but one of the most common is Henry Eyring (February 20, 1901 - December 26, 1981) was a Mexican-American theoretical chemist whose primary contribution was in the study of chemical reaction rates and intermediates. ...

where ΔG^‡ is the Gibbs free energy of activation, k_B is Boltzmann's constant, and h is Planck's constant. In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

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 reaction, and can be calculated using formulas from statistical mechanics involving the partition functions of the reactants and of the activated complex. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

Notes and references