H. F. Gilbert' Department of Biochemistry Brandeis University Waltham. Massachusens 02154
I
The "Rule of Thumb" for Deriving
I
I n the course of discussing the kinetic behavior observed for chemical reactions, i t frequently becomes desirable or necessary to derive the rate law for the proposed mechanism in terms of individual kinetic constants. For reaction mechanisms involving intermediates, the rigorous derivation of the kinetic expression is often complicated and may not lead to a direct sdution.2 A simplification of the rate law is often obtained when the steady state approximation can be applied. This approximation assumes that the absolute rate of change in concentration of an intermediate is zero: i.e. the intermediate decomnoses a t the same rate that it is formed. This approximation generallv holds when the intermediates are unstable and exist only-at low concentrations during the course of a chemical reaction." Even when the steady state approximation may be applied, derivation of the rate expression for reaction mechanisms in\,olving more than one intermediate is laborious. A simple method ior deriving srpads state rate equations for firs-order and pseudo-first-oider reactions is For "linear" mechanisms (mechanisms where reactants and products are connected through a linear sequence of intermediates), the method allows derivation of the observed first-order rate constant in terms of individual kinetic constants by inspection. Using a few additional rules, the steady state rate equation for complex mechanisms involving branched and cyclic paths may be easily derived. The Method Consider a "linear" reaction mechanism for the first-order conversion of A to Z which involves n steps proceeding
'Publication No. 1139 from the Graduate Department of Biochemistry Brandeis University. Supported by a postdoctoral fellowship (PF-1111)from the American Cancer Society. Frost, A. A., and Pearson, R. G., "Kinetics and Mechanism; a Study of Homogeneous Reactions," 2nd Ed., Wiley, New York, 1961. 'The asstrmptim 181 a steady sfnte ~pprorimationdoes nut implicitlv aiiume thnt thecon~entrationof stead) state intermediates is small. However, xhendenlmg wrh tunstable inlrrmrdlate~it i r dwn valid to assume that the concentrations of all steady state intermediates are small compared to the total concentration of starting material and product. The "rule of thumb" makes these assumptions for analysis of systems which approach equilihrium. The steady state approximation is not valid for all kinetic systems. Therefore,application of this approximation must he made with caution. See also: Jeneks, W. P., "Catalysis in Chemistry and Enzymology," McGrawHill, New York, 1969, p. 590 and Admur, K., and Hammes, G. G., "Chemical Kinetics: Principles and Selected Topics," McGraw-Hill, New York, 1966, p. 14.
492 1 Journal of Chemical Education
Figure 1. The ''rule of ihumb." Thumb placements are shown for a linear steady state system for Ihe canversion of A to E through three intermediates. The term in the denominator of klxdand k,, corresponding to each thumb placement is shown at the right.
through n - 1intermediates
Ak - IA B...[SXI ...+*. k-2
k-"
z
(1)
The experimentally observed rate constant for the approach of a system to equilihrium is2 bobs = kfWd+ krsv
(2) where kf,a is the first-order rate constant for the reaction in the forward direction and k,, is the first-order rate constant for the reaction in the reverse direction. The values of k k d and k,, in terms of individual kinetic constants may he evaluated by the "rule of thumb." 1) The numerator for krwd is simply the product of the n forward rate constants,k l k 2 . . . kn. 2) The denominator for kf,d is the sum of n terms. Each of then terms consists of the product of n - 1 rate constants. 3) Write the mechanism. Place your left thumb over one of then stem. Write the oroduct of the forward rate constants for steos to the right dynur thmnhtimrs thr product of the reverse rate conttnnts t'or all steps u, the lefr ut' your rhumb. Kepent this procedure for the remaining steps. The denominator is then the sum of all the terms generated using the "rule of thumb." ~
~~
The expression fork, is simply the product of all reverse rate constants divided by the same denominator as k r d . The ratio of kf,d to k , is the equilihrium constant for the reaction. The sum of k~ and k,, is the ohserved rate constant for approach to equilihrium from either direction2
Examples
Consider the mechanism k1
k*
kg
k4
k-2
*-a
k-4
A+B&CeD&E k
(3)
The numerator of kr,d is klk2k3k4. The denominator of kr,d will contain the sum of four terms, each term cnnsistingof the product of three rate constants. Thumb placements and the terms resulting from each thumh placement are shown in Figure 1.The denominator for kr,d is then the sum of all the terms shown in Figure 1.The overall expression for kr,d is then
and fork.,,
I R X N Coordinate
The resulting steady state expression can be simplified if certain steps are fast relative to other steps. Consider the ahove system (eqn. (3)), in which the reverse reaction may be ignored (k-4 = O), and the conversion of C to D occurs though the transition state of highest energy (the third step is rate determining) (Fig. 2). With the system so defined, the transition states for conversion of A to B, B to C, and D to E, must he of lower energy. I t can then he shown that ks
