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Arrhenius equation

From Wikipedia, the free encyclopedia
Formula for temperature dependence of rates of chemical reactions

Inphysical chemistry, theArrhenius equation is a formula for the temperature dependence ofreaction rates. The equation was proposed bySvante Arrhenius in 1889 on the basis of the work of Dutch chemistJacobus Henricus van 't Hoff, who had noted in 1884 that theVan 't Hoff equation for the temperature dependence ofequilibrium 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 ofenergy of activation. Arrhenius provided a physical justification and interpretation for the formula.[1] Currently, it is best seen as anempirical relationship.[2] It can be used to model the temperature variation ofdiffusion coefficients, population ofcrystal vacancies,creep rates, and many other thermally induced processes and reactions.[citation needed] TheEyring equation, developed in 1935, also expresses the relationship between rate and energy.

The temperature dependence arises because a greater fraction of molecular collisions have sufficient energy to exceed the activation barrier as temperature increases.[3][page needed]

Formulation

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In almost all practical cases,Ea ≫RT andk increases rapidly withT.
Mathematically, at very high temperatures so thatEa ≪RT,k levels off and approachesA as a limit, but this case does not occur under practical conditions.

The Arrhenius equation describes theexponential dependence of therate constant of a chemical reaction on theabsolute temperature ask=Ae−EaRT=Aexp⁡(−EaRT),{\displaystyle k=Ae^{\frac {-E_{\mathrm {a} }}{RT}}=A\exp {\left({\frac {-E_{\mathrm {a} }}{RT}}\right)},}where

Alternatively, the equation may be expressed ask=Ae−EakBT=Aexp⁡(−EakBT),{\displaystyle k=Ae^{\frac {-E_{\mathrm {a} }}{k_{\text{B}}T}}=A\exp {\left({\frac {-E_{\mathrm {a} }}{k_{\text{B}}T}}\right)},}where

The only difference is the unit ofEa: the former form uses energy permole, which is common in chemistry, while the latter form uses energy permolecule directly, which is common in physics.The different units are accounted for in using either thegas constant,R, or theBoltzmann constant,kB, as the multiplier of temperatureT.

The unit of the pre-exponential factorA 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 unitinverse second, s−1, and for that reason it is often called thefrequency factor orattempt 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[6] andexp⁡(−EaRT){\textstyle \exp {\left({\frac {-E_{\mathrm {a} }}{RT}}\right)}} 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 ofcatalysts) 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 factor⁠exp⁡(−EaRT){\displaystyle \textstyle \exp {\left({\frac {-E_{\mathrm {a} }}{RT}}\right)}}⁠; 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 every10 °C rise in temperature, for common values of activation energy and temperature range.[7]

Theexp⁡(−EaRT){\textstyle \exp {\left({\frac {-E_{\mathrm {a} }}{RT}}\right)}} factor denotes the fraction of molecules with energy greater than or equal to⁠Ea{\displaystyle E_{\mathrm {a} }}⁠.[8]

Derivation

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Van't Hoff argued that the temperatureT{\displaystyle T} of a reaction and the standard equilibrium constantke0{\displaystyle k_{\text{e}}^{0}} exhibit the relation:

dln⁡ke0dT=ΔU0RT2{\displaystyle {\frac {d\ln k_{\text{e}}^{0}}{dT}}={\frac {\Delta U^{0}}{RT^{2}}}}1

whereΔU0{\displaystyle \Delta U^{0}} denotes the appositestandardinternal energy change value.

Letkf{\displaystyle k_{\text{f}}} andkb{\displaystyle k_{\text{b}}} respectively denote the forward and backward reaction rates of the reaction of interest, then⁠ke0=kfkb{\displaystyle \textstyle k_{\text{e}}^{0}={\frac {k_{\text{f}}}{k_{\text{b}}}}}⁠,[9] an equation from whichln⁡ke0=ln⁡kf−ln⁡kb{\textstyle \ln k_{\text{e}}^{0}=\ln k_{\text{f}}-\ln k_{\text{b}}} naturally follows.

Substituting the expression forln⁡ke0{\textstyle \ln k_{\text{e}}^{0}} in eq.(1), we obtain⁠dln⁡kfdT−dln⁡kbdT=ΔU0RT2{\displaystyle \textstyle {\frac {d\ln k_{\text{f}}}{dT}}-{\frac {d\ln k_{\text{b}}}{dT}}={\frac {\Delta U^{0}}{RT^{2}}}}⁠.

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

dln⁡kfdT=constant+EfRT2{\displaystyle {\frac {d\ln k_{\text{f}}}{dT}}={\text{constant}}+{\frac {E_{\text{f}}}{RT^{2}}}}2

and

dln⁡kbdT=constant+EbRT2{\displaystyle {\frac {d\ln k_{\text{b}}}{dT}}={\text{constant}}+{\frac {E_{\text{b}}}{RT^{2}}}}3

whereEf{\displaystyle E_{\text{f}}} andEb{\displaystyle E_{\text{b}}} are the activation energies associated with the forward and backward reactions respectively, with⁠ΔU0=Ef−Eb{\displaystyle \Delta U^{0}=E_{\text{f}}-E_{\text{b}}}⁠.

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

dln⁡kfdT=EfRT2{\displaystyle {\frac {d\ln k_{\text{f}}}{dT}}={\frac {E_{\text{f}}}{RT^{2}}}}

and

dln⁡kbdT=EbRT2{\displaystyle {\frac {d\ln k_{\text{b}}}{dT}}={\frac {E_{\text{b}}}{RT^{2}}}}

Integrating these equations and taking the exponential yields the results⁠kf=Afe−Ef/RT{\displaystyle \textstyle k_{\text{f}}=A_{\text{f}}e^{-E_{\text{f}}/RT}}⁠ and⁠kb=Abe−Eb/RT{\displaystyle \textstyle k_{\text{b}}=A_{\text{b}}e^{-E_{\text{b}}/RT}}⁠, where eachpre-exponential factorAf{\displaystyle A_{\text{f}}} orAb{\displaystyle A_{\text{b}}} is mathematically the exponential of the constant of integration for the respectiveindefinite integral in question.[10]

Arrhenius plot

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Main article:Arrhenius plot
Arrhenius linear plot:lnk against1/T.

Taking thenatural logarithm of Arrhenius equation yields:ln⁡k=ln⁡A−EaR1T.{\displaystyle \ln k=\ln A-{\frac {E_{\text{a}}}{R}}{\frac {1}{T}}.}

Rearranging yields:ln⁡k=−EaR(1T)+ln⁡A.{\displaystyle \ln k={\frac {-E_{\text{a}}}{R}}\left({\frac {1}{T}}\right)+\ln A.}

This has the same form as an equation for a straight line:y=ax+b,{\displaystyle y=ax+b,}where⁠x{\displaystyle x}⁠ is thereciprocal of⁠T{\displaystyle T}⁠.

So, when a reaction has a rate constant obeying the Arrhenius equation, a plot of⁠ln⁡k{\displaystyle \ln {k}}⁠ versus⁠T−1{\displaystyle T^{-1}}⁠ gives a straight line, whose slope and intercept can be used to determine⁠Ea{\displaystyle E_{\mathrm {a} }}⁠ and⁠A{\displaystyle A}⁠ respectively. This procedure is common in experimental chemical kinetics. The activation energy is simply obtained by multiplying by (⁠−R{\displaystyle -R}⁠) the slope of the straight line drawn from a plot of⁠ln⁡k{\displaystyle \ln {k}}⁠ versus⁠1T{\displaystyle \textstyle {1 \over T}}⁠:Ea≡−R[∂⁡ln⁡k∂⁡(1/T)]P.{\displaystyle E_{\text{a}}\equiv -R\left[{\frac {\mathop {\partial } \ln k}{\mathop {\partial } (1/T)}}\right]_{P}.}

Modified Arrhenius equation

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The modified Arrhenius equation[11] makes explicit the temperature dependence of the pre-exponential factor. The modified equation is usually of the formk=ATne−EaRT.{\displaystyle k=AT^{n}e^{\frac {-E_{\text{a}}}{RT}}.}

The original Arrhenius expression above corresponds to⁠n=0{\displaystyle n=0}⁠. Fitted rate constants typically lie in the range⁠−1<n<1{\displaystyle -1<n<1}⁠. Theoretical analyses yield various predictions for⁠n{\displaystyle 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 predicted⁠T1/2{\displaystyle T^{1/2}}⁠ dependence of the pre-exponential factor is observed experimentally".[12] 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 thestretched exponential form[citation needed]k=Aexp⁡[−(EaRT)β],{\displaystyle k=A\exp \left[-\left({\frac {E_{a}}{RT}}\right)^{\beta }\right],}where⁠β{\displaystyle \beta }⁠ is a dimensionless number of order 1. This is typically regarded as a purely empirical correction orfudge 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 Mottvariable range hopping.

Theoretical interpretation

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Arrhenius's concept of activation energy

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Arrhenius argued that for reactants to transform into products, they must first acquire a minimum amount of energy, called theactivation energy⁠Ea{\displaystyle E_{\mathrm {a} }}⁠. At an absolute temperature⁠T{\displaystyle T}⁠, the fraction of molecules that have a kinetic energy greater than⁠Ea{\displaystyle E_{\mathrm {a} }}⁠ can be calculated fromstatistical mechanics. This fraction increases with temperature because molecular energies follow a Maxwell–Boltzmann distribution, which broadens as⁠T{\displaystyle T}⁠ rises and increases the proportion of molecules with kinetic energies equal to or greater than the activation energy.[3][page needed]The concept ofactivation energy explains the exponential nature of the relationship, and in one way or another, it is present in all kinetic theories.

From a physical perspective, activation energy represents an energy barrier that must be overcome for reactant molecules to reach a transition state. Increasing temperature raises the fraction of molecules with sufficient kinetic energy to overcome this barrier, which explains the strong temperature dependence of reaction rates described by the Arrhenius equation.

The calculations for reaction rate constants involve an energy averaging over aMaxwell–Boltzmann distribution with⁠Ea{\displaystyle E_{\mathrm {a} }}⁠ as lower bound and so are often of the type ofincomplete gamma functions, which turn out to be proportional to⁠exp⁡−EaRT{\displaystyle \textstyle \exp {\frac {-E_{\mathrm {a} }}{RT}}}⁠.

Collision theory

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Main article:Collision theory

One approach is thecollision theory of chemical reactions, developed byMax Trautz andWilliam Lewis in the years 1916–18. In this theory, molecules are supposed to react if they collide with a relativekinetic energy along their line of centers that exceeds⁠Ea{\displaystyle E_{\mathrm {a} }}⁠. The number of binary collisions between two unlike molecules per second per unit volume is found to be[13]zAB=NANBdAB28πkBTμAB,{\displaystyle z_{AB}=N_{A}N_{B}d_{AB}^{2}{\sqrt {\frac {8\pi k_{\mathrm {B} }T}{\mu _{AB}}}},}where⁠NA{\displaystyle N_{A}}⁠ and⁠NB{\displaystyle N_{B}}⁠ are the number densities of⁠A{\displaystyle A}⁠ and⁠B{\displaystyle B}⁠,⁠dAB{\displaystyle d_{AB}}⁠ is the average diameter of⁠A{\displaystyle A}⁠ and⁠B{\displaystyle B}⁠,⁠T{\displaystyle T}⁠ is the temperature which is multiplied by theBoltzmann constant⁠kB{\displaystyle k_{\mathrm {B} }}⁠ to convert to energy, and⁠μAB{\displaystyle \mu _{AB}}⁠is thereduced mass of⁠A{\displaystyle A}⁠ and⁠B{\displaystyle B}⁠.

The rate constant is then calculated as⁠k=zABexp⁡−EaRT{\displaystyle \textstyle k=z_{AB}\exp {\frac {-E_{\mathrm {a} }}{RT}}}⁠, so that the collision theory predicts that the pre-exponential factor is equal to the collision number⁠zAB{\displaystyle z_{AB}}⁠. However for many reactions this agrees poorly with experiment, so the rate constant is written instead as⁠k=ρzABexp⁡−EaRT{\displaystyle \textstyle k=\rho z_{AB}\exp {\frac {-E_{\mathrm {a} }}{RT}}}⁠. Here⁠ρ{\displaystyle \rho }⁠ is an empiricalsteric 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

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TheEyring equation, another Arrhenius-like expression, appears in the "transition state theory" of chemical reactions, formulated byEugene Wigner,Henry Eyring,Michael Polanyi andM. G. Evans in the 1930s. The Eyring equation can be written:k=kBThe−ΔG‡RT=kBTheΔS‡Re−ΔH‡RT,{\displaystyle k={\frac {k_{\mathrm {B} }T}{h}}e^{-{\frac {\Delta G^{\ddagger }}{RT}}}={\frac {k_{\mathrm {B} }T}{h}}e^{\frac {\Delta S^{\ddagger }}{R}}e^{-{\frac {\Delta H^{\ddagger }}{RT}}},}whereΔG‡{\displaystyle \Delta G^{\ddagger }} is theGibbs energy of activation,ΔS‡{\displaystyle \Delta S^{\ddagger }} is theentropy of activation,ΔH‡{\displaystyle \Delta H^{\ddagger }} is theenthalpy of activation,kB{\displaystyle k_{\mathrm {B} }} is theBoltzmann constant, andh{\displaystyle h} is thePlanck constant.[14]

At first sight this looks like an exponential multiplied by a factor that islinear in temperature. However, free energy is itself a temperature-dependent quantity. The free energy of activationΔG‡=ΔH‡−TΔS‡{\displaystyle \Delta G^{\ddagger }=\Delta H^{\ddagger }-T\Delta S^{\ddagger }} 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 of⁠T{\displaystyle T}⁠. The precise form of the temperature dependence depends upon the reaction, and can be calculated using formulas fromstatistical mechanics involving thepartition functions of the reactants and of the activated complex.

Limitations of the idea of Arrhenius activation energy

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Both the Arrhenius activation energy and the rate constant⁠k{\displaystyle 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 molecules⁠A{\displaystyle A}⁠ and⁠B{\displaystyle 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 molecule⁠AB{\displaystyle AB}⁠. Macroscopic measurements of⁠E{\displaystyle E}⁠ and⁠k{\displaystyle 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][page needed]

Another situation where the explanation of the Arrhenius equation parameters falls short is inheterogeneous catalysis, especially for reactions that showLangmuir-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 theglass 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 toviscous flow of the material.

See also

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References

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  1. ^Arrhenius (1889a);Arrhenius (1889b);Laidler (1984);Laidler (1987), p. 42.
  2. ^Connors (1990), p. 188.
  3. ^abAtkins & de Paula (2014).
  4. ^IUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Arrhenius equation".doi:10.1351/goldbook.A00446
  5. ^Arrhenius (1889a);Arrhenius (1889b);Laidler (1987), p. 42.
  6. ^Silberberg (2006), p. 696;Silberberg (2013), p. 522.
  7. ^Avery (1974), pp. 47–58: "However, the rate of reaction varies greatly with temperature, since for a typical process the rate doubles or trebles for a rise in temperature of10 °C."
  8. ^"6.2.3.3: The Arrhenius Law – Activation Energies".Chemistry LibreTexts. 2013-10-02.
  9. ^"15.2: The Equilibrium Constant (K)".Chemistry LibreTexts. 2016-03-11. Retrieved2023-06-27.
  10. ^"Arrhenius Equation"(PDF).University of Pennsylvania. Retrieved27 June 2023.
  11. ^IUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "modified Arrhenius equation".doi:10.1351/goldbook.M03963
  12. ^Connors (1990), p. 190.
  13. ^abLaidler & Meiser (1982), pp. 376–378.
  14. ^Laidler & Meiser (1982), pp. 378–383.
  15. ^Levine (2005).
  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.

Cited works

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General references and further reading

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  • Pauling, L. C. (1988).General Chemistry. Dover Publications.
  • Laidler, K. J. (1993).The World of Physical Chemistry. Oxford University Press.

External links

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Nucleophilic substitutions
Electrophilic substitutions
Elimination reactions
Addition reactions
Unimolecular reactions
Electron/Proton transfer reactions
Medium effects
Related topics
Chemical kinetics
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