A rate-controlling (rate-determining or rate-limiting) step in a reaction occurring by a @C01211@ @ST06775@ is an @E02035@ the @O04322@ for which exerts a strong effect — stronger than that of any other @O04322@ — 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 @E02035@ having a @O04322@ \(k_{i}\) by: \[\text{CF}=\frac{\partial (\ln \nu)}{\partial \ln k_{i}}\] where \(\nu\) is the overall @R05156@. In performing the partial differentiation all equilibrium constants \(K_{j}\) and all rate constants except \(k_{i}\) are held constant. The @E02035@ having the largest control factor exerts the strongest influence on the rate \(\nu\), 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 @K03405@. 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 @S05962@, 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 \(k_{2}\left[\text{D}\right]\) a pseudo-first order @O04322@, then \(k_{2}\left[\text{D}\right]\gg k_{-1}\), and the observed rate \(\nu = k_{1}\ \left[\text{A}\right]\left[\text{B}\right]\) and \(k_{\text{obs}}=k_{1}\). Step (1) is said to be the rate-controlling step. If \(k_{2}\left[\text{D}\right]\ll k_{-1}\), 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 @E02177@ for the pre-equilibrium (1) and is equal to \(\frac{k_{1}}{k_{-1}}\), and \(k_{\text{obs}}=K\ k_{2}\). Step (2) is said to be the rate-controlling step.