Arrhenius equation

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In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius 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 for the temperature dependence of equilibrium constants 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. Arrhenius provided a physical justification and interpretation for the formula. [1] [2] [3] [4] Currently, it is best seen as an empirical relationship. [5] :188 It can be used to model the temperature variation of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally induced processes and reactions. The Eyring equation, developed in 1935, also expresses the relationship between rate and energy.

Contents

Equation

In almost all practical cases,
E
a
[?]
R
T
{\displaystyle E_{a}\gg RT}
and k increases rapidly with T. NO2 Arrhenius k against T.svg
In almost all practical cases, and k increases rapidly with T.
Mathematically, at very high temperatures so that
E
a
[?]
R
T
{\displaystyle E_{a}\ll RT}
, k levels off and approaches A as a limit, but this case does not occur under practical conditions. KineticConstant.png
Mathematically, at very high temperatures so that , k levels off and approaches A as a limit, but this case does not occur under practical conditions.

The Arrhenius equation gives the dependence of the rate constant of a chemical reaction on the absolute temperature as

where

Alternatively, the equation may be expressed as

where

The only difference is the unit of Ea: the former form uses energy per mole, which is common in chemistry, while the latter form uses energy per molecule directly, which is common in physics. The different units are accounted for in using either the gas constant, R, or the Boltzmann constant, kB, as the multiplier of temperature T.

The unit of the pre-exponential factor A 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 unit s −1, and for that reason it is often called the frequency factor or attempt frequency of the reaction. Most simply, k is the number of collisions that result in a reaction per second, A is the number of collisions (leading to a reaction or not) per second occurring with the proper orientation to react [7] 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 catalysts) 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-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 to 3 for every 10 °C rise in temperature, for common values of activation energy and temperature range. [8]

The factor denotes the fraction of molecules with energy greater than or equal to . [9]

Derivation

Van't Hoff argued that the temperature of a reaction and the standard equilibrium constant exhibit the relation:

 

 

 

 

(1)

where denotes the apposite standard internal energy change value.

Let and respectively denote the forward and backward reaction rates of the reaction of interest, then , [10] an equation from which naturally follows.

Substituting the expression for in eq.( 1 ), we obtain .

The preceding equation can be broken down into the following two equations:

 

 

 

 

( 2 )

and

 

 

 

 

( 3 )

where and are the activation energies associated with the forward and backward reactions respectively, with .

Experimental findings suggest that the constants in eq.( 2 ) and eq.( 3 ) can be treated as being equal to zero, so that

 

 

 

 

()

and

 

 

 

 

()

Integrating these equations and taking the exponential yields the results and , where each pre-exponential factor or is mathematically the exponential of the constant of integration for the respective indefinite integral in question. [11]

Arrhenius plot

Arrhenius linear plot: ln k against 1/T. Arrhenius plot with break in y-axis to show intercept.svg
Arrhenius linear plot: ln k against 1/T.

Taking the natural logarithm of Arrhenius equation yields:

Rearranging yields:

This has the same form as an equation for a straight line:

where x is the reciprocal 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 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):

Modified Arrhenius equation

The modified Arrhenius equation [12] 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 −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 T1/2 dependence of the pre-exponential factor is observed experimentally". [5] :190 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[ citation needed ]

where β is a dimensionless number of order 1. This is typically regarded as a purely empirical correction or fudge factor 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.

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 Ea. At an absolute temperature T, the fraction of molecules that have a kinetic energy greater than Ea can be calculated from statistical mechanics. 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 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 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. The number of binary collisions between two unlike molecules per second per unit volume is found to be [13]

where NA is the Avogadro constant, dAB is the average diameter of A and B, T is the temperature which is multiplied by the Boltzmann constant kB to convert to energy, and μAB is the reduced mass.

The rate constant is then calculated as , so that the collision theory predicts that the pre-exponential factor is equal to the collision number zAB. 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. [13]

Transition state theory

The Eyring equation, another Arrhenius-like expression, appears in the "transition state theory" of chemical reactions, formulated by Eugene Wigner, Henry Eyring, Michael Polanyi and M. G. Evans in the 1930s. The Eyring equation can be written:

where is the Gibbs energy of activation, is the entropy of activation, is the enthalpy of activation, is the Boltzmann constant, and is Planck's constant. [14]

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 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. [15]

Another situation where the explanation of the Arrhenius equation parameters falls short is in heterogeneous catalysis, 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. [16]

There are deviations from the Arrhenius law during the glass transition in all classes of glass-forming matter. [17] 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 of the material.

See also

Related Research Articles

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In the Arrhenius model of reaction rates, activation energy is the minimum amount of energy that must be available to reactants for a chemical reaction to occur. The activation energy (Ea) of a reaction is measured in kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol). Activation energy can be thought of as the magnitude of the potential barrier (sometimes called the energy barrier) separating minima of the potential energy surface pertaining to the initial and final thermodynamic state. For a chemical reaction to proceed at a reasonable rate, the temperature of the system should be high enough such that there exists an appreciable number of molecules with translational energy equal to or greater than the activation energy. The term "activation energy" was introduced in 1889 by the Swedish scientist Svante Arrhenius.

<span class="mw-page-title-main">Reaction rate</span> Speed at which a chemical reaction takes place

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 time. Reaction rates can vary dramatically. For example, the oxidative rusting of iron under Earth's atmosphere is a slow reaction that can take many years, but the combustion of cellulose in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds. A reaction's rate can be determined by measuring the changes in concentration over time.

Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is different from chemical thermodynamics, which deals with the direction in which a reaction occurs but in itself tells nothing about its rate. Chemical kinetics includes investigations of how experimental conditions influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition states, as well as the construction of mathematical models that also can describe the characteristics of a chemical reaction.

In physical organic chemistry, a kinetic isotope effect (KIE) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes. Formally, it is the ratio of rate constants for the reactions involving the light (kL) and the heavy (kH) isotopically substituted reactants (isotopologues):

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 towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

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<span class="mw-page-title-main">Collision theory</span> Chemistry principle

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 the correct orientation, only a certain amount of collisions result in a perceptible or notable change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known as activation energy, at the moment of impact to break the pre-existing bonds and form all new bonds. This results in the products of the reaction. The activation energy is often predicted using the Transition state theory. Increasing the concentration of the reactant brings about more collisions and hence more successful collisions. Increasing the temperature increases the average kinetic energy of the molecules in a solution, increasing the number of collisions that have enough energy. Collision theory was proposed independently by Max Trautz in 1916 and William Lewis in 1918.

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In theoretical chemistry, Marcus theory is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron transfer reactions – the rate at which an electron can move or jump from one chemical species (called the electron donor) to another (called the electron acceptor). It was originally formulated to address outer sphere electron transfer reactions, in which the two chemical species only change in their charge with an electron jumping (e.g. the oxidation of an ion like Fe2+/Fe3+), but do not undergo large structural changes. It was extended to include inner sphere electron transfer contributions, in which a change of distances or geometry in the solvation or coordination shells of the two chemical species is taken into account (the Fe-O distances in Fe(H2O)2+ and Fe(H2O)3+ are different).

The Eyring 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 Henry Eyring, Meredith Gwynne Evans and Michael Polanyi. The equation follows from the transition state theory, also known as activated-complex theory. If one assumes a constant enthalpy of activation and constant entropy of activation, the Eyring equation is similar to the empirical Arrhenius equation, despite the Arrhenius equation being empirical and the Eyring equation based on statistical mechanical justification.

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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.

In chemical kinetics, an Arrhenius plot displays the logarithm of a reaction rate constant, (, ordinate axis) plotted against reciprocal of the temperature (, abscissa). Arrhenius plots are often used to analyze the effect of temperature on the rates of chemical reactions. For a single rate-limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy and the pre-exponential factor can both be determined.

In thermodynamics, enthalpy–entropy compensation is a specific example of the compensation effect. The compensation effect refers to the behavior of a series of closely related chemical reactions, which exhibit a linear relationship between one of the following kinetic or thermodynamic parameters for describing the reactions:

  1. Between the logarithm of the pre-exponential factors and the activation energies where the series of closely related reactions are indicated by the index i, Ai are the preexponential factors, Ea,i are the activation energies, R is the gas constant, and α, β are constants.
  2. Between enthalpies and entropies of activation where H
    i
    are the enthalpies of activation and S
    i
    are the entropies of activation.
  3. Between the enthalpy and entropy changes of a series of similar reactions where Hi are the enthalpy changes and Si are the entropy changes.

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In chemical kinetics, the pre-exponential factor or A factor is the pre-exponential constant in the Arrhenius equation, an empirical relationship between temperature and rate coefficient. It is usually designated by A when determined from experiment, while Z is usually left for collision frequency. The pre-exponential factor can be thought of as a measure of the frequency of properly oriented collisions. It is typically determined experimentally by measuring the rate constant at a particular temperature and fitting the data to the Arrhenius equation. The pre-exponential factor is generally not exactly constant, but rather depends on the specific reaction being studied and the temperature at which the reaction is occurring.

In chemistry, NMR line broadening techniques can be used to determine the rate constant and the Gibbs free energy of exchange reactions of two different chemical compounds. If the two species are in equilibrium and exchange to each other, peaks of both species get broadened in the spectrum. This observation of broadened peaks can be used to obtain kinetic and thermodynamic information of the exchange reaction.

In chemical kinetics, the Aquilanti–Mundim deformed Arrhenius model is a generalization of the standard Arrhenius law.

References

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  2. 1 2 Arrhenius, S. A. (1889). "Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren". Z. Phys. Chem. 4: 226–48. doi:10.1515/zpch-1889-0416. S2CID   100032801.
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  4. 1 2 Laidler, K. J. (1987) Chemical Kinetics, Third Edition, Harper & Row, p. 42
  5. 1 2 Kenneth Connors, Chemical Kinetics, 1990, VCH Publishers Chemical Kinetics: The Study of Reaction Rates in Solution at Google Books
  6. IUPAC Goldbook definition of Arrhenius equation.
  7. Silberberg, Martin S. (2006). Chemistry (fourth ed.). NY: McGraw-Hill. p.  696. ISBN   0-07-111658-3.
  8. Avery, H. E. (1974). "4. Dependence of Rate on Temperature". Basic Reaction Kinetics and Mechanisms. Springer. pp. 47–58. Retrieved 18 December 2023. However, the rate of reaction varies greatly with temperature, since for a typical process the rate doubles or trebles for a rise in temperature of 10 °C.
  9. "6.2.3.3: The Arrhenius Law – Activation Energies". Chemistry LibreTexts. 2013-10-02.
  10. "15.2: The Equilibrium Constant (K)". Chemistry LibreTexts. 2016-03-11. Retrieved 2023-06-27.
  11. "Arrhenius Equation" (PDF). University of Pennsylvania. Retrieved 27 June 2023.
  12. IUPAC Goldbook definition of modified Arrhenius equation.
  13. 1 2 Laidler, Keith J.; Meiser, John H. (1982). Physical Chemistry (1st ed.). Benjamin/Cummings. pp. 376–78. ISBN   0-8053-5682-7.
  14. Laidler, Keith J.; Meiser, John H. (1982). Physical Chemistry (1st ed.). Benjamin/Cummings. pp. 378–83. ISBN   0-8053-5682-7.
  15. Levine, R.D. (2005) Molecular Reaction Dynamics, Cambridge University Press
  16. Slot, Thierry K.; Riley, Nathan; Shiju, N. Raveendran; Medlin, J. Will; Rothenberg, Gadi (2020). "An experimental approach for controlling confinement effects at catalyst interfaces". Chemical Science. 11 (40): 11024–11029. doi: 10.1039/D0SC04118A . ISSN   2041-6520. PMC   8162257 . PMID   34123192.
  17. Bauer, Th.; Lunkenheimer, P.; Loidl, A. (2013). "Cooperativity and the Freezing of Molecular Motion at the Glass Transition". Physical Review Letters. 111 (22): 225702. arXiv: 1306.4630 . Bibcode:2013PhRvL.111v5702B. doi:10.1103/PhysRevLett.111.225702. PMID   24329455. S2CID   13720989.

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