Editing Reaction rate constant (section)

==Dependence on temperature==
The [[Arrhenius equation]] is an elementary treatment that gives the quantitative basis of the relationship between the [[activation energy]] and the [[reaction rate]] at which a reaction proceeds. The rate constant as a function of thermodynamic temperature is then given by:

<math display="block">k(T) = Ae^{- E_\mathrm{a}/RT}</math>

The reaction rate is given by:

<math display="block">r = Ae^{ - E_\mathrm{a}/RT}[\mathrm{A}]^m[\mathrm{B}]^n,</math>

where ''E''<sub>a</sub> is the [[activation energy]], and ''R'' is the gas constant, and ''m'' and ''n'' are experimentally determined partial orders in [A] and [B], respectively.  Since at [[temperature]] ''T'' the molecules have energies according to a [[Boltzmann distribution]], one can expect the proportion of collisions with energy greater than ''E''<sub>a</sub> to vary with ''e''<sup>{{frac|−''E''<sub>a</sub>|''RT''}}</sup>. The constant of proportionality ''A'' is the [[pre-exponential factor]], or frequency factor (not to be confused here with the reactant A) takes into consideration the frequency at which reactant molecules are colliding and the likelihood that a collision leads to a successful reaction. Here, ''A'' has the same dimensions as an (''m'' + ''n'')-order rate constant (''see'' Units ''below'').

Another popular model that is derived using more sophisticated [[Statistical mechanics|statistical mechanical]] considerations is the [[Eyring equation]] from [[transition state theory]]:

<math display="block">k(T)
= \kappa\frac{k_{\mathrm{B}}T}{h}(c^{\ominus})^{1-M}e^{-\Delta G^{\ddagger}/RT}
= \left(\kappa\frac{k_{\mathrm{B}}T}{h}(c^{\ominus})^{1-M}\right)e^{\Delta S^{\ddagger}/R}
e^{-\Delta H^{\ddagger}/RT},</math>

where Δ''G''<sup>‡</sup> is the free energy of activation, a parameter that incorporates both the enthalpy and [[Entropy of activation|entropy]] change needed to reach the transition state.  The temperature dependence of Δ''G''<sup>‡</sup> is used to compute these parameters, the enthalpy of activation Δ''H''<sup>‡</sup> and the entropy of activation Δ''S''<sup>‡</sup>, based on the defining formula Δ''G''<sup>‡</sup> = Δ''H''<sup>‡</sup> − ''T''Δ''S''<sup>‡</sup>.  In effect, the free energy of activation takes into account both the activation energy and the likelihood of successful collision, while the factor ''k''<sub>B</sub>''T''/''h'' gives the frequency of molecular collision.    

The factor (''c''<sup>⊖</sup>)<sup>1-''M''</sup> ensures the dimensional correctness of the rate constant when the transition state in question is bimolecular or higher.  Here, ''c''<sup>⊖</sup> is the standard concentration, generally chosen based on the unit of concentration used (usually ''c''<sup>⊖</sup> = 1 mol L<sup>−1</sup> = 1 M), and ''M'' is the molecularity of the transition state.  Lastly, κ, usually set to unity, is known as the [[transmission coefficient]], a parameter which essentially serves as a "[[fudge factor]]" for transition state theory.   

The biggest difference between the two theories is that Arrhenius theory attempts to model the reaction (single- or multi-step) as a whole, while transition state theory models the individual elementary steps involved.  Thus, they are not directly comparable, unless the reaction in question involves only a single elementary step.

Finally, in the past, [[collision theory]], in which reactants are viewed as hard spheres with a particular cross-section, provided yet another common way to rationalize and model the temperature dependence of the rate constant, although this approach has gradually fallen into disuse.  The equation for the rate constant is similar in functional form to both the Arrhenius and Eyring equations:

<math display="block">k(T)=PZe^{-\Delta E/RT},</math>

where ''P'' is the steric (or probability) factor and ''Z'' is the collision frequency, and Δ''E'' is energy input required to overcome the activation barrier.  Of note, <math>Z\propto T^{1/2}</math>, making the temperature dependence of ''k'' different from both the Arrhenius and Eyring models.  

=== Comparison of models ===
All three theories model the temperature dependence of ''k'' using an equation of the form

<math display="block">k(T)=CT^\alpha e^{-\Delta E/RT}</math>

for some constant ''C'', where α = 0, {{frac|1|2}}, and 1 give Arrhenius theory, collision theory, and transition state theory, respectively, although the imprecise notion of Δ''E'', the energy needed to overcome the activation barrier, has a slightly different meaning in each theory.  In practice, experimental data does not generally allow a determination to be made as to which is "correct" in terms of best fit.  Hence, all three are conceptual frameworks that make numerous assumptions, both realistic and unrealistic, in their derivations.  As a result, they are capable of providing different insights into a system.<ref>{{Cite book|title=Determination of organic reaction mechanisms|last=Carpenter|first=Barry K.|date=1984|publisher=Wiley|isbn=978-0471893691|location=New York|oclc=9894996}}</ref>