The steady-state and partial-equilibrium approximations

Although numerical integrations of equation (14) usually can be performed accurately now, some systems remain that are too complex or too stiff for the computational methods to succeed. To analyze such systems, as well as to understand better the behaviors of various systems amenable to accurate numerical integrations, rational approximations producing simplifications are wanted. Two important approximations of this type are the steady-state and partial-equilibrium approximations. 

A general criterion for applicability of the steady-state approximation to species i may be stated by writing equation (6) in the form — 

The classical example of a complex straight-chain reaction for which the results of the steady-state approximation agree with experimental measurements is the hydrogen-bromine reaction H2 T Br2 2HBr [5]. The inferred mechanism is 

The last relation in equation (34) shows that the fairly complicated reaction-rate expression [constant c J j -h constant which is 

The partial-equilibrium approximation differs from the steady-state approximation in that it refers to a particular reaction instead of to a particular species. The mechanism must include the forward and backward steps of any reaction that maintains partial equilibrium, and the approximation for a reaction k is then expressed by setting = 0 in equation (11). It is not always proper to conclude from this that when equations (6), (10), and (11) are employed in equation (14), the terms may be set equal to zero for each k that maintains partial equilibrium partial equilibria occur when the forward and backward rates are both large, and a small fractional difference of these two large quantities may contribute significantly to dcjdt. The criterion for validity of the approximation is that be small compared with the forward or backward rate. 

A general criterion for applicability of the steady-state approximation to species i may be stated by writing equation (6) in the form (h, = , + — o),, where + is the sum of the positive terms on the right-hand side of the equation and (u, denotes the sum of the magnitudes of the negative terms. The approximation is = ch,, which—according to equation (14)— will be good if dci/dt +. It should be realized that the approximation 

To apply a partial-equilibrium approximation for a reaction k = I, identify a speeies, say = 1, for which v — v, j 0, and solve equation (11) with co = 0 for Cj as a function of the other concentrations C . Also write equation (14) for i 1 in the form 

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