Application of Kinetic Approximations to the A 2 B -+ C Reaction System Gregory I. Gellene Texas Tech University, Lubbock, TX 79409-1061
Two of the most often discussed procedures for simplifying the kinetic analysis of chemical reaction systems are the steady-state (ss) and the preequilibrinm (pel assumptions ( I ) . The domains of applicability of these kinetic assumptions a s applied to the title reaction system are determined by a comparison of the exact solution of the rate equations to the corresponding approximate solutions. The Simple Steady State The steady-state approximation is often introduced ( 1 , 2 ) by considering the consecutive first-order reaction system,
Using eq 8 to eliminate [AISsfrom eq 10 gives upon rearrangement
Substitution of eq 11 into eq 9 allows [ C F to be determined by direct integration. This result allows [BY to be determined from eq 11,which in turn allows [AIS3obe determined from eq 8. The final results are
for which the corresponding coupled differential equations
can be analytically solved by elementary techniques to yield the exact solutions.
The steady-state assumption for the reaction intermediate (i.e., d[Bl""/dt = 0) is usually applied to the special case of [Blo = 0; however, this condition is not required. Because the general approach for applying kinetic approximations is not widely treated and will be used throughout, it will be outlined here. If the steady-state equations
where k,, = klkz/(kl + kz). These approximate results can be compared to the exact solutions (eqs 5-7) to identify the domain of applicability of the steady-state assumption as kz>> k, (i.e., kzf k1 = kz and h., = kl) and t >> Ilkz (i.e., e-ks' = 0 and [Clg = [Blo + [Clo). Under these conditions the familiar steady-state solutions for this reaction system are recovered. Although this approach cannot be generalized to explore the conditions under which the steady-state approximation for an arbitrary kinetic system is valid (31, the analysis does reveal the two critical minimal requirements: a n intermediate that is destroyed faster than i t is produced and sufficient time for the steady state to be established. If the reaction scheme is extended to include the reverse of the first step of reaction 1,that is,
the validity of the preequilibrium assumption can be similarly explored by this pedagogically satisfying approach of comparing exact and approximate sdutions of the coupled rate equations. Although a numerical analysis of this kinetic system is available (41, a n analytical approach appears not to have been explicitly treated previously ( 5 ) and is thus considered here. Although the solution to the more general case, which includes the reverse of the second step, that is,
(where the superscript denotes the kinetic approximation invoked) are combined with the mass-balance relationship
(which must hold for any valid kinetic approximation), a steady-state solution can be obtained for arbitrary initial concentrations. 196
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is available (61,the rate equations for k_2=0 case are easier to solve and better suited for treating the preequilibrium assumption. With apologies to those readers familiar with the techniques of solving elementary differential equations, some intermediate steps will be explicitly indicated
to clarify the presentation for those with a less sophisticated mathematical background.
Approximate Solutions Preequilibrium
The Exact Solution The coupled rate equations for the title reaction system are
The equations describing the application of the preequilibrium approximation (29) k,[AIP' = k.,LBIPe
combined with the mass-balance relationship (eq 10 with the superscript "ss" replaced by "pen), can be simultaneously solved by the largely algebraic approach that led to eqs 12-14. When this is done the results are which can be solved exactly by the following approach (7). Solving eq 16 for [A1 gives
which upon substitution into eq 15 followed by differentiation and rearrangement yields
This homogeneous second-order differential equation with constant coefficients is readily solved by substitution of the assumed solution, [BIZ e", into eq 19, which generates the characteristic equation,
having the solutions
where k,= klkz/(k-, + k J and the familiar preequilibrium results are recovered for [Clg' = [Clo. Anticipating that the domain of applicability of eqs 3133 will require the equilibrium reactions to be fast (in a sense to be determined) relative to the rate of C formation, we rewrite eq 21 a s
If kz > I4k.r + k2) Because the condition of k z ll(k-, + kzl
(i.e., e-lk~'+ k''' = 0 )is sufficient for the steady-state approximation to be valid. Under these conditions [CI; where kss = k l k z / ( k ~+, k l + k z ) . With the expectation that the validity of the steady-state approximation will require k l ll(k-l + kz)). Concluding Remarks A comparison of the exact solution of the kinetic system
Proceeding a s before yields the results to first order in k ~ / ( k . ~ k+z ) .
to the approximate solutions obtained under the assumptions of preequilibrium and steady-state indicate that the preequilibrium assumption is valid for k z > lI(k-, + k , ) , whereas the steady-state assumption is valid for k 1 > l i ( k - ~ +k z ) .
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Table 2. Relative Magnitudes of ki, kl, k2 Required for the Validity of the These conditions are not mutually Pre-equilibrium (pe) and Steady-State (ss) Kinetic Approximations exclusive. For example, both kinetic assumptions a r e simultaneously valid under the conditions k l V(k.l + k l ) as can be seen readily by noting that eq 42 rek, k, > 141111 k) ss 1 duces to eq 29 under these condikl l 4 l c t + k,) pe 2 tions. Less obvious, perhaps, is that t>> l/(h., + k t ) ss and pe 2 k~
