Rival Rate Equations for Steady States Sidney I. Miller Illinois Institute of Technology, Chicago, IL 60616
In examining the literature on the catalyzed exchange of o e ~ t i d hvdroeen e . .. in conformationallvmohile macromolecules. i iound a puzAing rate expression-for a standard sequence (eqn. ( I ) ) . If thr steady t a t r condition applies to R,
i.e., d[B]ldt
-- 0 and [B] -0,
to one product (P) by competing first- and second-order steps:
then A is an activated haloaromatic, N is a nucleophile and k3 and ha are pseudounimolecular constants that have absorbed the concentration of catalyst and possibly an equilibrium constant (8).Again, the expressions for k,, are different:
This is the usual chemist's formulation of k,, ( 1 , Z ) . According to one of the simpler models of hydrogen exchange in a macromolecule, a closed (folded) form (A) undergoes a conformational change to become open (unfolded) (B). The latter then exchanges3H or 2H for lH to yield C. If the reaction is acid- or hase-catalyzed at constant pH and if the steady state restriction applies to B, workers often give the appropriate rate constant for exchange as ( 3 , 4 ) hex= k ~ k z l ( k k-I ~+
+ kz)
(5)
and set he, = k,,. Admittedly k l must be relatively small to he consistent with a steady state in B hut the fact is that k, fk,. Provided that one makes the substitution [A] = [A],, - [B] - [C] in eqn. (2), one can obtain
+k d d[Clldt -- klkd[AIo - [C])l(k~+ k-I + kz) IBI
k~([Alo- [Cl)l(kl+k - I
(6) (7)
Both versions of d[C]/dt, i.e., eqns. (4) and (I), are valid and independent of the observation that reapplication of the steady state assumptions, [Ale - [C] [A] and kl
