https://doi.org/10.1351/goldbook.S05962
- In a kinetic analysis of a @C01208@ involving @U06569@ intermediates in low concentration, the rate of change of each such intermediate is set equal to zero, so that the @R05141@ can be expressed as a function of the concentrations of @CT01038@ present in macroscopic amounts. For example, assume that X is an @U06569@ intermediate in the reaction @ST06775@: S05962-1.pngS05962-2.pngConservation of mass requires that: \[[\ce{A}] + [\ce{X}] + [\ce{D}] = [\ce{A}]_{0}\] which, since \([\ce{A}]_{0}\) is constant, implies: \[-\frac{{\rm{d}}[\ce{X}]}{{\rm{d}}t} = \frac{{\rm{d}}[\ce{A}]}{{\rm{d}}t} + \frac{{\rm{d}}[\ce{D}]}{{\rm{d}}t}.\] Since \([\ce{X}]\) is negligibly small, the rate of formation of D is essentially equal to the @R05148@ of A, and the rate of change of \([\ce{X}]\) can be set equal to zero. Applying the steady state approximation (\({\rm{d}}[\ce{X}]/{\rm{d}}t = 0\)) allows the @E02038@ of \([\ce{X}]\) from the kinetic equations, whereupon the @R05156@ is expressed: \[\frac{{\rm{d}}[\ce{D}]}{{\rm{d}}t} = -\frac{{\rm{d}}[\ce{A}]}{{\rm{d}}t} = \frac{k_{1}k_{2}[\ce{A}][\ce{C}]}{k_{-1} + k_{2}[\ce{C}]}\]Note:
The steady-state approximation does not imply that \([\ce{X}]\) is even approximately constant, only that its absolute rate of change is very much smaller than that of \([\ce{A}]\) and \([\ce{D}]\). Since according to the reaction scheme \({\rm{d}}[\ce{X}]/{\rm{d}}t = k_{2}[\ce{X}][\ce{C}]\), the assumption that \([\ce{X}]\) is constant would lead, for the case in which C is in large excess, to the absurd conclusion that formation of the product D will continue at a constant rate even after the reactant A has been consumed. - In a stirred @F02443@ a steady state implies a regime so that all concentrations are independent of time.