Eyring equation
Encyclopedia
The Eyring equation also known as Eyring–Polanyi equation in chemical kinetics
relates the reaction rate
to temperature
. It was developed almost simultaneously in 1935 by Henry Eyring
, M.G. Evans
and Michael Polanyi
. This equation follows from the transition state theory
(aka, activated-complex theory) and is trivially equivalent to the empirical
Arrhenius equation
which are both readily derived from statistical thermodynamics in the kinetic theory of gases
.
:
where ΔG‡ is the Gibbs energy
of activation, kB is Boltzmann's constant, and h is Planck's constant.
It can be rewritten as:
To find the linear form of the Eyring-Polanyi equation:
where:
A certain chemical reaction is performed at different temperatures and the reaction rate is determined. The plot of
versus 
gives a straight line with slope 
from which the enthalpy
of activation can be derived and with intercept
from which the entropy
of activation is derived.
requires a value of the transmission coefficient
as a prefactor in the Eyring equation above. This value is usually taken to be unity (i.e., the transition state 
always proceeds to products 
and never reverts back to reactants 
and 
). As discussed by Winzor and Jackson in 2006, this assumption invalidates the description of an equilibrium between the transition state and the reactants and therefore the empirical Arrhenius equation
is preferred with a phenomenological interpretation of the prefactor
and activation energy 
. For more details, see discussion in Winzor and Jackson (2006) pages 399-400 in section "Transition-state theory."
To avoid specifying a value of
the ratios of rate constants can be compared to the value of a rate constant at some fixed reference temperature (i.e., 
) which eliminates the 
term in the resulting expression.
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...
relates the reaction rate
Reaction rate
The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
to temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
. It was developed almost simultaneously in 1935 by Henry Eyring
Henry Eyring
Henry Eyring was a Mexican-born American theoretical chemist whose primary contribution was in the study of chemical reaction rates and intermediates....
, M.G. Evans
Evans
Evans may refer to:people*Evans *for a particular person with the surname, see the List of people with surname Evans.place*Evans, California in the United States*Evans, Colorado in the United States*Evans, Georgia in the United States...
and Michael Polanyi
Michael Polanyi
Michael Polanyi, FRS was a Hungarian–British polymath, who made important theoretical contributions to physical chemistry, economics, and the theory of knowledge...
. This equation follows from the transition state theory
Transition state theory
Transition state theory explains the reaction rates of elementary chemical reactions. The theory assumes a special type of chemical equilibrium between reactants and activated transition state complexes....
(aka, activated-complex theory) and is trivially equivalent to the empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
Arrhenius equation
Arrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction 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...
which are both readily derived from statistical thermodynamics in the kinetic theory of gases
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...
.
General form
The general form of the Eyring–Polanyi equation somewhat resembles the Arrhenius equationArrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction 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...
:
where ΔG‡ is the Gibbs energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
of activation, kB is Boltzmann's constant, and h is Planck's constant.
It can be rewritten as:
To find the linear form of the Eyring-Polanyi equation:
where:
= reaction rateReaction rateThe reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
constant
= absolute temperature
= enthalpy of activation
= gas constantGas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
= Boltzmann constant
= Planck's constant
= entropy of activation
A certain chemical reaction is performed at different temperatures and the reaction rate is determined. The plot of
versus gives a straight line with slope from which the enthalpyEnthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
of activation can be derived and with intercept
from which the entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
of activation is derived.
Accuracy
Transition state theoryTransition state theory
Transition state theory explains the reaction rates of elementary chemical reactions. The theory assumes a special type of chemical equilibrium between reactants and activated transition state complexes....
requires a value of the transmission coefficient
as a prefactor in the Eyring equation above. This value is usually taken to be unity (i.e., the transition state always proceeds to products and never reverts back to reactants and ). As discussed by Winzor and Jackson in 2006, this assumption invalidates the description of an equilibrium between the transition state and the reactants and therefore the empirical Arrhenius equationArrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction 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...
is preferred with a phenomenological interpretation of the prefactor
and activation energy . For more details, see discussion in Winzor and Jackson (2006) pages 399-400 in section "Transition-state theory."To avoid specifying a value of
the ratios of rate constants can be compared to the value of a rate constant at some fixed reference temperature (i.e., ) which eliminates the term in the resulting expression.