In
physical chemistry
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mecha ...
, the Arrhenius equation is a formula for the temperature dependence of
reaction rate
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit ...
s. The equation was proposed by
Svante Arrhenius
Svante August Arrhenius ( , ; 19 February 1859 – 2 October 1927) was a Swedes, Swedish scientist. Originally a physicist, but often referred to as a chemist, Arrhenius was one of the founders of the science of physical chemistry. He received ...
in 1889, based on the work of Dutch chemist
Jacobus Henricus van 't Hoff who had noted in 1884 that the
van 't Hoff equation
The Van 't Hoff equation relates the change in the equilibrium constant, , of a chemical reaction to the change in temperature, ''T'', given the standard enthalpy change, , for the process. It was proposed by Dutch chemist Jacobus Henricus van ' ...
for the temperature dependence of
equilibrium constant
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
s suggests such a formula for the rates of both forward and reverse reactions. This equation has a vast and important application in determining the rate of chemical reactions and for calculation of
energy of activation
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules p ...
. Arrhenius provided a physical justification and interpretation for the formula.
[ Laidler, K. J. (1987) ''Chemical Kinetics'', Third Edition, Harper & Row, p. 42] Currently, it is best seen as an
empirical relationship.
[Kenneth Connors, Chemical Kinetics, 1990, VCH Publishers ] It can be used to model the temperature variation of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions. The
Eyring equation
The Eyring equation (occasionally also known as Eyring–Polanyi equation) is an equation used in chemical kinetics to describe changes in the rate of a chemical reaction against temperature. It was developed almost simultaneously in 1935 by Henr ...
, developed in 1935, also expresses the relationship between rate and energy.
Equation
![KineticConstant](https://upload.wikimedia.org/wikipedia/commons/7/79/KineticConstant.png)
The Arrhenius equation gives the dependence of the
rate constant In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction.
For a reaction between reactants A and B to form product C
the reaction rate is often found to have the f ...
of a chemical reaction on the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
as
where
* is the
rate constant In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction.
For a reaction between reactants A and B to form product C
the reaction rate is often found to have the f ...
(frequency of collisions resulting in a reaction),
* is the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
(in
Kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
or degree
Rankine),
* is the
pre-exponential factor
In chemical kinetics, the pre-exponential factor or A factor is the pre-exponential constant in the Arrhenius equation#Equation, Arrhenius equation (equation shown below), an empirical relationship between temperature and Reaction rate constant, ...
. Arrhenius originally considered A to be a temperature-independent constant for each chemical reaction. However more recent treatments include some temperature dependence - see
Modified Arrhenius equation below.
* is the
activation energy
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules pe ...
for the reaction (in the same units as ''RT''),
* is the
universal gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
.
Alternatively, the equation may be expressed as
where
* is the
activation energy
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules pe ...
for the reaction (in the same units as ''k''
B''T''),
* is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
.
The only difference is the energy units of : the former form uses energy per
mole
Mole (or Molé) may refer to:
Animals
* Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America
* Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
, which is common in chemistry, while the latter form uses energy per
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
directly, which is common in physics.
The different units are accounted for in using either the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, , or the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, , as the multiplier of temperature .
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
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
factor'' or ''attempt frequency'' of the reaction. Most simply, is the number of collisions that result in a reaction per second, is the number of collisions (leading to a reaction or not) per second occurring with the proper orientation to react and
is the probability that any given collision will result in a reaction. It can be seen that either increasing the temperature or decreasing the activation energy (for example through the use of
catalyst
Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
s) will result in an increase in rate of reaction.
Given the small temperature range of kinetic studies, 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
factor; except in the case of "barrierless"
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.
With this equation it can be roughly estimated that the rate of reaction increases by a factor of about 2 or 3 for every 10°C rise in temperature.
The term
denotes the fraction of molecules with energy greater than or equal to
.
Arrhenius plot
Taking the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of Arrhenius equation yields:
Rearranging yields:
This has the same form as an equation for a straight line:
where ''x'' is the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of ''T''.
So, when a reaction has a rate constant that obeys the Arrhenius equation, a plot of ln ''k'' versus ''T''
−1 gives a straight line, whose gradient 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''):
Modified Arrhenius equation
The modified Arrhenius equation makes explicit the temperature dependence of the pre-exponential factor. The modified equation is usually of the form
The original Arrhenius expression above corresponds to ''n'' = 0. Fitted rate constants typically lie in the range . 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 ''T''
1/2 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
The stretched exponential function f_\beta (t) = e^ is obtained by inserting a fractional power law into the exponential function.
In most applications, it is meaningful only for arguments between 0 and +∞. With , the usual exponential function ...
form
where ''β'' is a dimensionless number of order 1. This is typically regarded as a purely empirical correction or ''
fudge factor A fudge factor is an ''ad hoc'' quantity or element introduced into a calculation, formula or model in order to make it fit observations or expectations. Also known as a "Correction Coefficient" which is defined by:
: \kappa_c = \frac
Examples ...
'' to make the model fit the data, 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 Variable-range hopping is a model used to describe carrier transport in a disordered semiconductor or in amorphous solid by hopping in an extended temperature range. It has a characteristic temperature dependence of
:\sigma= \sigma_0e^
where \sigm ...
.
Theoretical interpretation of the equation
Arrhenius's concept of activation energy
Arrhenius argued that for reactants to transform into products, they must first 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
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. 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 calculations for reaction rate constants involve an energy averaging over a
Maxwell–Boltzmann distribution
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and used ...
with
as lower bound and so are often of the type of
incomplete gamma functions, which turn out to be proportional to
.
Collision theory
One approach is the
collision theory
Collision theory is a principle of chemistry used to predict the rates of chemical reactions. It states that when suitable particles of the reactant hit each other with correct orientation, only a certain amount of collisions result in a percept ...
of chemical reactions, developed by
Max Trautz
Max Trautz (19 March 1880 – 19 August 1960) was a German chemist. He was very productive with over 190 scientific publications especially in the field of chemical kinetics. He was the first to investigate the activation energy of molecules b ...
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. The number of binary collisions between two unlike molecules per second per unit volume is found to be
where ''N
A'' is the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
, ''d
AB'' is the average diameter of ''A'' and ''B'', ''T'' is the temperature which is multiplied by the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
''k''
B to convert to energy units, and ''μ
AB'' is the
reduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...
.
The rate constant is then calculated as
so that the collision theory predicts that the pre-exponential factor is equal to the collision number ''z
AB''. However for many reactions this agrees poorly with experiment, so the rate constant is written instead as
. Here ''
'' is an empirical
steric factor, often much less than 1.00, which is interpreted as the fraction of sufficiently energetic collisions in which the two molecules have the correct mutual orientation to react.
[
]
Transition state theory
The Eyring equation
The Eyring equation (occasionally also known as Eyring–Polanyi equation) is an equation used in chemical kinetics to describe changes in the rate of a chemical reaction against temperature. It was developed almost simultaneously in 1935 by Henr ...
, another Arrhenius-like expression, appears in the "transition state theory
In chemistry, transition state theory (TST) explains the reaction rates of elementary chemical reactions. The theory assumes a special type of chemical equilibrium (quasi-equilibrium) between reactants and activated transition state complexes.
T ...
" of chemical reactions, formulated by Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
, Eyring, Polanyi and Evans
Evans may refer to:
People
*Evans (surname)
*List of people with surname Evans
Places United States
*Evans Island, an island of Alaska
*Evans, Colorado
*Evans, Georgia
*Evans County, Georgia
*Evans, New York
*Evans Mills, New York
*Evans City, ...
in the 1930s. The Eyring equation can be written:
where is the Gibbs energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pre ...
of activation, is the entropy of activation In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed ...
, is the enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
of activation, is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and is Planck's constant.
At first sight this looks like an exponential multiplied by a factor that is ''linear'' in temperature. However, free energy is itself a temperature dependent quantity. The free energy of activation is the difference of an enthalpy term and an entropy term multiplied by the absolute temperature. The pre-exponential factor depends primarily on the entropy of activation. The overall expression again takes the form of an Arrhenius exponential (of enthalpy rather than energy) 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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
involving the partition functions of the reactants and of the activated complex.
Limitations of the idea of Arrhenius activation energy
Both the Arrhenius activation energy and the rate constant ''k'' are experimentally determined, and represent macroscopic reaction-specific parameters that are not simply related to threshold energies and the success of individual collisions at the molecular level. Consider a particular collision (an elementary reaction) between molecules A and B. The collision angle, the relative translational energy, the internal (particularly vibrational) energy will all determine the chance that the collision will produce a product molecule AB. Macroscopic measurements of E and ''k'' are the result of many individual collisions with differing collision parameters. To probe reaction rates at molecular level, experiments are conducted under near-collisional conditions and this subject is often called molecular reaction dynamics.
Another situation where the explanation of the Arrhenius equation parameters fall short is in heterogeneous catalysis
In chemistry, heterogeneous catalysis is catalysis where the phase of catalysts differs from that of the reactants or products. The process contrasts with homogeneous catalysis where the reactants, products and catalyst exist in the same phase. Ph ...
, especially for reactions that show Langmuir-Hinshelwood kinetics. Clearly, molecules on surfaces do not "collide" directly, and a simple molecular cross-section does not apply here. Instead, the pre-exponential factor reflects the travel across the surface towards the active site.
There are deviations from the Arrhenius law during the glass transition
The glass–liquid transition, or glass transition, is the gradual and reversible transition in amorphous materials (or in amorphous regions within semicrystalline materials) from a hard and relatively brittle "glassy" state into a viscous or rubb ...
in all classes of glass-forming matter. The Arrhenius law predicts that the motion of the structural units (atoms, molecules, ions, etc.) should slow down at a slower rate through the glass transition than is experimentally observed. In other words, the structural units slow down at a faster rate than is predicted by the Arrhenius law. This observation is made reasonable assuming that the units must overcome an energy barrier by means of a thermal activation energy. The thermal energy must be high enough to allow for translational motion of the units which leads to viscous flow
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
of the material.
See also
* Accelerated aging
Accelerated aging is testing that uses aggravated conditions of heat, humidity, oxygen, sunlight, vibration, etc. to speed up the normal aging processes of items. It is used to help determine the long-term effects of expected levels of stress wit ...
* Eyring equation
The Eyring equation (occasionally also known as Eyring–Polanyi equation) is an equation used in chemical kinetics to describe changes in the rate of a chemical reaction against temperature. It was developed almost simultaneously in 1935 by Henr ...
* Q10 (temperature coefficient)
The ''Q''10 temperature coefficient is a measure of temperature sensitivity based on the chemical reactions.
The ''Q''10 is calculated as:
: Q_=\left( \frac \right )^
where;
: ''R'' is the rate
: ''T'' is the temperature in Celsius degrees o ...
* Van 't Hoff equation
The Van 't Hoff equation relates the change in the equilibrium constant, , of a chemical reaction to the change in temperature, ''T'', given the standard enthalpy change, , for the process. It was proposed by Dutch chemist Jacobus Henricus van ' ...
* Clausius–Clapeyron relation
* Gibbs–Helmholtz equation
The Gibbs–Helmholtz equation is a thermodynamic equation used for calculating changes in the Gibbs free energy of a system as a function of temperature. It was originally presented in an 1882 paper entitled " Die Thermodynamik chemischer Vorgang ...
* Cherry blossom front
The refers to the advance of the cherry blossoms across Japan. The Japan Meteorological Agency records the opening and full bloom of the blossoms from Kyūshū in late March to Hokkaidō in the middle of May. The advancing front is also the subjec ...
predicted using the Arrhenius equation
References
Bibliography
*
*
*
External links
Carbon Dioxide solubility in Polyethylene
– Using Arrhenius equation for calculating species solubility in polymers
{{Reaction mechanisms
Chemical kinetics
Equations
Statistical mechanics