James P. Birk University of Pennsylvonia Philadelphia, 19104
Mechanistic Implications and Ambiguities of Rate Laws
T h e primary mechanistic information provided by a rate law is the composition of the activated complexes (1-3). Although the information can be summarized in this manner, or in terms of a net activation process (4), i t is frequently more satisfying to interpret this information in terms of a mechanism for the reaction. The form of the rate law will suggest the pattern of a mechanism consistent with the data, but the question arises as to whether there might be other mechanisms equally consistent with the rate law. Newton (3) has described a method, using electrical analogs to reaction rates, for the determination of all the mechanistic patterns consistent with a given rate law. It will be assumed throughout this discussion of mechanistic ambiguities that the rate laws will be written in terms of the actual, rather than stoichiometric or total concentrations of species, so we need not be concerned about ambiguities arising from the appearance of terms in a rate law due to a rapidly established equilibrium, with equilibrium quotient sufficientlylarge that the predominant form of a species changes with concentration. If a rate law contains more than a single term, this will be interpreted in terms of more than one aotivated complex. The compositions of the activated complex are determined from the limiting forms of the rate law, which are obtained by successively assuming that each term in a sum of terms is the predominant one. I n the electrical analogs to reaction rates (3), resistors oorrespond to activated complexes, junctions between resistors to steady-state intermediates, and terminals to reactants and products. For a given number of activated complexes and intermediates, the number of indistinguishable mechanisms is given by the number of ways in which the corresponding number of resistors and junctions can be connected between the two terminals. The electrical analogs to various reaction patterns are given in Figure 1. I n the pattern designation, the first number indicates the number of activated complexes, while the second number indicates the number of steady-state intermediates. For any analog which contains a junction (i.e., an intermediate), the mirror image provides another indistinguishable mechanism. The R, in Figure 1 are rate laws of the form k,[A]",[Bl6' . . . where A, B, . . . are species (reactants, catalytic species, and sometimes products) present initially and a', br, . . . are positive or negative integers (or fractions in certain special cases). These R, will correspond to the limiting forms of the experimental rate law and can be derived in the case of consecutive steps by assuming that the step in question is ratedetermining and that any preceding steps rapidly reach
an equilibrium state lying far towards reactants. A further extension of the electrical analogs is the calculation of rate laws by assuming the resistances to be analogs of the reciprocals of the corresponding rate terms Rt. Hence, the Rr can be combined to give the overall rate law by following the rules for combining the reciprocals of the resistances. The following procedure should be followed in applying the electrical analogs to the determination of reaction mechanisms: (a) Write any mechanism which is consistent with the form of the rate law. (b) Determine the reaction pattern and consult the figure to decide whether there are indistinguishable mechanisms. ( c ) Using the electrical analogs as a guide to alternate sequences of the formation of activated complexes, construct all other mechanisms consistent with the rate law. (d) Write the Ri,using the limiting form technique, such that the R, involve only concentrations of stable species and not intermediates. The R, then correspond to the rates of formation of the various aotivated complexes. The R, can be used, with the last column in the figure, to derive the form of the rate law for each mechanism. It should be noted that reactions will occur which, according to the electrical analogs, should have more than one indistinguishable mechanism, but for which the alternate mechanisms are not readily conceived.
Na of Pattern lndist. Mech.
3 -2
2
Electrical Analog
Rate Law
(R;'+R;I+R;I)'
Electrical analogs to reaction rater. Resirtor5 correspond to activated complexes, m a l l circles (iunctions) to steadystole intarmediates and large circle, Lterminahl to reactants and produch
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Although it should always be possible to write these alternate mechanisms, occasionally they will make little chemical sense and, hence, their conception may require a good deal of imagination. The electrical analogs become especially useful in relation to reactions in aqueous solution having rate laws with a complex [H+] dependence. Because of the ubiquitous nature of the hydrogen ion, the appropriate number of indistinguishable mechanisms can always be constructed quite readily in these cases. A series of successively more complex rate laws will now he examined and the'application of electrical analogs will be demonstrated.
complex, one Fe(II1) must be consumed after the ratedetermining step, e.g.
Since the rate law provides no information about processes occurring after the rate-determining step, the above mechanism cannot be distinguished without further data from others such as eqns. (9) and (10) or eqns. (11) and (12) rapid
Rate Laws Involving O n e Activated Complex ( O n e Term Rate Laws) Rote lows Consistent with One-Step
The composition of the activated complex is given by {IrC161rBra5-)*. The rate equation is consistent with a one-step mechanism in which the overall reaction (eqn. (1))is an elementary reaction. I t is not possible to rule out other mechanisms such as
where kt = k Q . However, there is no evidence in this case to indicate that the mechanism does consist of two steps and, in general, a mechanism should be written which is of the least complexity necessary to accommodate the available data. The application of the electrical analog is trivial in this c a s e t h e one-step mechanism corresponds to pattern 1-0, for which there is only one type of mechanism.
ZFe(I1)
(10)
When the stoichiometry of the' reaction is diierent from the composition of the activated complex, there must necessarily be a sequence of steps involving one or more intermediates which occur after the rate-determining step. At constant [H+],the reaction U(IV) ZFe(II1) = U(V1) ZFe(I1) (5)
+
+
k[U(IV)I[Fe(III)l
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Rote Lows Implying 0 Rote-Determining Step Preceded by One or More Ropid Steps
A single term rate law containing a simple inverse dependence on the concentration of some substance indicates that this substance is produced from some or all of the reactants in a rapid step prior to the ratedetermining step. This rapid step approximates an equilibrium with very small equilibrium quotient (Q
