Reaction rate constant

In chemical kinetics a reaction rate constant or reaction rate coefficient, k, quantifies the rate of a chemical reaction.^[1]

For a reaction between reactants A and B to form product C

a A + b B → c C

the reaction rate is often found to have the form:

$r = k(T)[\mathrm{A}]^m [\mathrm{B}]^{n}$

Here k(T) is the reaction rate constant that depends on temperature. [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 instead moles of A or B per unit area).

The exponents m and n are called partial orders of reaction and are not generally equal to the stoichiometric coefficients a and b. Instead they depend on the reaction mechanism and can be determined experimentally.

Temperature dependence

The Arrhenius equation gives the quantitative basis of the relationship between the activation energy and the reaction rate at which a reaction proceeds. The rate constant is then given by

$k = Ae^\frac{-E_a}{RT}$

and the reaction rate by

$r = Ae^\frac{-E_a}{RT}[\mathrm{A}]^m[\mathrm{B}]^n,$

where E_a is the activation energy, and R is the gas constant. 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_a to vary with e^−E_a⁄_RT. A is the pre-exponential factor, or frequency factor (not to be confused here with the reactant A).

Units

The units of the rate constant depend on the global order of reaction:^[2] If concentration is measured in units of mol·L⁻¹ (sometimes abbreviated as M), then

• For order (m + n), the rate coefficient has units of mol^1−(m+n)·L^(m+n)−1·s⁻¹
• For order zero, the rate coefficient has units of mol·L⁻¹·s⁻¹ (or M·s⁻¹)
• For order one, the rate coefficient has units of s⁻¹
• For order two, the rate coefficient has units of L·mol⁻¹·s⁻¹ (or M⁻¹·s⁻¹)
• And for order three, the rate coefficient has units of L²·mol⁻²·s⁻¹ (or M⁻²·s⁻¹)

Plasma and gases

Calculation of rate constants of the processes of generation and relaxation of electronically and vibrationally excited particles are of significant importance. It is used, for example, in the computer simulation of processes in plasma chemistry or microelectronics. First-principle based models should be used for such calculation. It can be done with the help of computer simulation software.

Rate constant calculations

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. One simple approach to overcome this problem is Divided Saddle Theory.^[3] Other methods as Benett Chandler procedure,^[4][5] Milestoning are also developed for rate constant calculations.

Divided saddle theory

The theory is based on the assumption that the reaction can be described by a reaction coordinate, and that we can apply Boltzmann distribution at least in the reactant state. A new, especially reactive segment of the reactant, called the saddle domain, is introduced, and the rate constant is factored:

$k= k_\mathrm{SD}\cdot \alpha^\mathrm{SD}_\mathrm{RS}$

where αSD
RS is the conversion factor between the reactant state and saddle domain, while k_SD is the rate constant from the saddle domain. The first can be simply calculated from the free energy surface, the latter is easily accessible from short molecular dynamics simulations ^[3]

See also

• Reaction rate
• Equilibrium constant

References

1. http://www.chem.arizona.edu/~salzmanr/480a/480ants/chemkine.html
2. Blauch, David. "Differential Rate Laws". Chemical Kinetics.
3. Daru, János; Stirling, András (2014). "Divided Saddle Theory: A New Idea for Rate Constant Calculation". J. Chem. Theory Comput. 10 (3): 1121–1127. doi:10.1021/ct400970y.
4. Chandler, David (1978). "Statistical mechanics of isomerization dynamics in liquids and the transition state approximation". J. Chem. Phys. 68 (6): 2959. doi:10.1063/1.436049.
5. Bennett, C. H. (1977). Christofferson, R., ed. Algorithms for Chemical Computations, ACS Symposium Series No. 46. Washington, D.C.: American Chemical Society. ISBN 978-0-8412-0371-6.