## rate-controlling step

https://doi.org/10.1351/goldbook.R05139
A rate-controlling (rate-determining or rate-limiting) step in a reaction occurring by a composite reaction sequence is an elementary reaction the rate constant for which exerts a strong effect — stronger than that of any other rate constant — on the overall rate. It is recommended that the expressions rate-controlling, rate-determining and rate-limiting be regarded as synonymous, but some special meanings sometimes given to the last two expressions are considered under a separate heading. A rate-controlling step can be formally defined on the basis of a control function (or control factor) CF, identified for an elementary reaction having a rate constant ki by: $\text{CF}=\frac{\partial (\ln \nu)}{\partial \ln k_{i}}$ where v is the overall rate of reaction. In performing the partial differentiation all equilibrium constants Kj and all rate constants except ki are held constant. The elementary reaction having the largest control factor exerts the strongest influence on the rate v, and a step having a CF much larger than any other step may be said to be rate-controlling. A rate-controlling step defined in the way recommended here has the advantage that it is directly related to the interpretation of kinetic isotope effects. As formulated this implies that all rate constants are of the same dimensionality. Consider however the reaction of A and B to give an intermediate C, which then reacts further with D to give products:
 R05139-1.png (1) R05139-2.png (2)
Assuming that C reaches a steady state, then the observed rate is given by: $\nu = \frac{k_{1}\,k_{2}\,\left[\text{A}\right]\left[\text{B}\right]\left[\text{D}\right]}{k_{-1}+k_{2}\left[\text{D}\right]}$ Considering k2[D] a pseudo-first order rate constant, then k2[D] ≫ k-1, and the observed rate v = k 1 A B and kobs = k1. Step (1) is said to be the rate-controlling step. If k2[D] ≪ k-1, then the observed rate: $\nu = \frac{k_{1}\ k_{2}}{k_{-1}}\left[\text{A}\right]\left[\text{B}\right]\left[\text{D}\right]=K\ k_{2}\left[\text{A}\right]\left[\text{B}\right]\left[\text{D}\right]$ where K is the equilibrium constant for the pre-equilibrium (1) and is equal to k1/k-1, and kobs = K.k2. Step (2) is said to be the rate-controlling step.
See also: Gibbs energy diagram, microscopic diffusion control, mixing control, rate-determining step
Source:
PAC, 1994, 66, 1077. 'Glossary of terms used in physical organic chemistry (IUPAC Recommendations 1994)' on page 1156 (https://doi.org/10.1351/pac199466051077)