) is a proportionality constant which quantifies the rate and direction of a chemical reaction by relating it with the concentration of reactants.
is the reaction rate constant that depends on temperature, and [A] and [B] are the molar concentrations of substances A and B in moles per unit volume of solution, assuming the reaction is taking place throughout the volume of the solution.
(For a reaction taking place at a boundary, one would use moles of A or B per unit area instead.)
The exponents m and n are called partial orders of reaction and are not generally equal to the stoichiometric coefficients a and b.
There are few examples of elementary steps that are termolecular or higher order, due to the low probability of three or more molecules colliding in their reactive conformations and in the right orientation relative to each other to reach a particular transition state.
Most involve the recombination of two atoms or small radicals or molecules in the presence of an inert third body which carries off excess energy, such as O + O2 + N2 → O3 + N2.
[3][4][5] In cases where a termolecular step might plausibly be proposed, one of the reactants is generally present in high concentration (e.g., as a solvent or diluent gas).
Transition state theory gives a relationship between the rate constant
, a quantity that can be regarded as the free energy change needed to reach the transition state.
) changes that need to be achieved for the reaction to take place:[7][8] The result from transition state theory is
As useful rules of thumb, a first-order reaction with a rate constant of 10−4 s−1 will have a half-life (t1/2) of approximately 2 hours.
For a one-step process taking place at room temperature, the corresponding Gibbs free energy of activation (ΔG‡) is approximately 23 kcal/mol.
where Ea is the activation energy, and R is the gas constant, and m and n are experimentally determined partial orders in [A] and [B], respectively.
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.
Another popular model that is derived using more sophisticated statistical mechanical considerations is the Eyring equation from transition state theory:
where ΔG‡ is the free energy of activation, a parameter that incorporates both the enthalpy and entropy change needed to reach the transition state.
The factor (c⊖)1-M ensures the dimensional correctness of the rate constant when the transition state in question is bimolecular or higher.
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.
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.
where P is the steric (or probability) factor and Z is the collision frequency, and ΔE is energy input required to overcome the activation barrier.
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.
[10] If concentration is measured in units of mol·L−1 (sometimes abbreviated as M), then Calculation of rate constants of the processes of generation and relaxation of electronically and vibrationally excited particles are of significant importance.
Rate constant can be calculated for elementary reactions by molecular dynamics simulations.
One possible approach is to calculate the mean residence time of the molecule in the reactant state.
Although this is feasible for small systems with short residence times, this approach is not widely applicable as reactions are often rare events on molecular scale.
[11] Such other methods as the Bennett Chandler procedure,[12][13] and Milestoning[14] have also been developed for rate constant calculations.
A new, especially reactive segment of the reactant, called the saddle domain, is introduced, and the rate constant is factored:
The first can be simply calculated from the free energy surface, the latter is easily accessible from short molecular dynamics simulations [11]