H. 6. Dunford The University of Alberta Edmonton, Alberta, Canada
I
Collision and Transition State Theory Approaches to Acid-Base Catalysis
The collision and transition state theories have been the mainstay of chemical kinetics for the past 40 years and more. An advantage of the former approach is that i t explicitly accounts for the combination of initially separate reactants, an advantage of the latter is that i t enables one to ignore all the steps between initial reactants and the transition state ( I ) in discussing the influence of various factors in the observed rate. In principle, the two approaches are interchangeable as illustrated by the collision and transition state equations for the rate constant of a bimoleeular reaction (compared to the empirical Arrhenius equati~n)'.~ k pze-WRT (collision theory) (1)
-
k
kT h
(transition state)
= -~A"*/"~-"H*'RT
(Arrhenius equation) (3) (The *re-activation energy terms for the three equations, which differ somewhat in their temperature dependence, cannot in practice he distinguished by temperature-dependent studies since the variation of k with temperature is dominated by the activation energy term.) In practice, the transition state theory became popular rapidly, and it is therefore somewhat surprising that i t has not been applied earlier to chemical or enzymatic reactions which are acidor base-catalyzed (3-7). The purpose of the present manuscript is to show that the description of acid-catalyzed chemical reactions in terms of transition state acid dissociation constants is formally equivalent t o the collision theory approach, in which dissociation constants of acid groups on initial reactants are utilized. k
= A~-E.IRT
The Equillbrlum Analogy -Consider, as an analogy to the transition state formulation of a chemical reaction, the equilihrium between enzyme and substrate
* Km
(4) E,,, + S ES,, KO, is an apparent equilihrium constant between total enzyme (which we shall consider here to he represented by two forms of the enzyme E H and E in equilibrium with each other) and total enzyme-substrate complexes. (An equivalent case would involve a proton equilibrium on the suhstrate only and more complicated cases would involve multi-proton equilibria. Although the discussion is formulated in terms of an acid-catalyzed enzymatic reaction, the extension to any chemical reaction, either acid- or hase-catwhich is alyzed is straightforward.) The value of K., pH-dependent, can be expressed in terms of the following equilibria3
HE
+
S
K,
I
(2)
c
1 8
6
4
10
pH Flgure I.(a) Plot of log versus pH (sccndingto eqns. (7H8) w t h K, = lo' W ' . K2 = TO3 M-', PKE = 5. and ~ K E =S 9. (b) Plot of log k- versus pH (according to sqns. (12H13)) w t h k, = lo' M-' s-' k2 lo3 M-' s-', ~ K =E 5 and pK& = 9.
.
-
constants (units, M). For such a cycle only three of the four parameters are independent. Thus if K E is~ chosen as a dependent parameter, its value is KES= K E K ~ I K I
(6)
If Kepp is defined in terms of the independent parameters, KI, KPand KE one obtains
L"
J
- + 1
KF The derivation is given in Appendix I. By substitution of eqn. (6)into eqn. (7), the dependent parameter KEScan he introduced
KO,, =
K*[CH+I + KES [H+]+l
11
(8)
KB and so one can see that eqns. (7) and (8) are of equivalent form. A plot of log Koppversus p H is shown in Figure 1, hased upon eqns. (7) and (8) and utilizing the assumed values of the various parameters listed in the figure legend.
The Collision and Ransltion State Formulations For a chemical reaction, expressed in transition state language
HES
'The conventional symbols in eqns. (1)-(3) are defined in ref. 12). . .
where KI and Kz are pH-independent association constants (units, M-') and KE and KES are acid dissociation
2Equation 2 represents the "thermodynamic" form of absolute reaction rate theory, or transition state theory. It might be expressed more properly but in a more complicated form in terms of partition functions (2). Charges and protons are omitted for convenience.
the pH-dependent rate constant, k,b, can be expressed in terms of the following cycle (again we choose a n example where there is only one proton equilibrium involving one of the r e a ~ t a n t s ) ~ HE
+
- 'i k,
S
HES*
Products
Ij K,.* ii
(10)
E + S % ES* + Products At any given instant in time, the concentrations of species in the transition state will be negligibly small. According t o transition state theory, the magnitudes of k l and kz are proportional to the extent that the reactants associate to form activated complexes, hence products. The rate constants therefore are pseudo-equilibrium constants and the analogy hetween eqns. (4)-(5) and (9)-(10) is readily apparent.5 The value of the dependent parameter KES' is defined by KES* = K ~ h l l k l (11) It is not assumed that an actual proton equilibrium exists between HES' and ESf and indeed it is improbable that such a n equilibrium does exist, hence K E S ~ can be described as a virtual equilibrium constant. This in no way detracts from the validity of eqn. (11). Extending the analogy hetween the equilibrium and rate equations further one can write
L.
J
KE
+1
or, by combination of eqns. (11) and (12)
h , [ g hob.
=
+ 11
[H+l+l
(13)
KF (See Appendix 1). Equation (12) represents the collision theory approach and eqn. (13) that of the transition state theory. Since eqns. (12)-(13) are analogous to eqns. (7)-(8) they also are represented in Figure 1,simply by interchanging analogous parameters. Thus KI, Kp, and KES from eqns. (7)-(8) are replaced by kl, kz, and K&, respectively, in eqns. (12)-(13) while the definition of K E remains unchanged. 'One should not draw an analogy between the transition state and an enzyme-substrate intermediate. Steady-state enzyme kinetics does not yield information as to the number of intermediates which might be formed during the course of the reaction (8) so that the measurements of individual rate constants and acid dissociation constants becomes an almost impossible task. On the other hand, with relaxation techniques (9) or by certain chemical tricks combined with stopped flow studies (lo), it is possible to a t l d v a sin& steo of reactions. Each sin. in a multi-steo . seauence . ole rt.en is -~ an elementan, transition state -....-~. ~ - reaction ~ ~ with ~ its own ~ which is the concern of this paper. 5 A rate constant is formulated in terms of a ratio of partition functions times an activation energy term according to transition state theory. The ratio of partition functions is identical to an equilibrium constant expressed in terms of partition functions, excent that one demee of vibrational motion is replaced by translati;nsl motion through the saddle point at the mz&num of the potential energy harrier. Caretul attention has been paid to the delinitron of the actwated romplex when s light atom or ion such aa H' rs being transferred (111. An analogous situation, in which the low concentration of a less domrnant species parallels that uf its dominant parent containing the same total number of protons, which involves group (mieroscopie) and molecular (macroscopic) ionization constants, is given in ref. (7). The computed concentrations of species in the transition state using the equilibrium analogy are in fact proportional to the numher of molecules which pass through the transition state. A~~~
~~
4
6
8
10
pH Figwe 2. (upper)Plot of log kohl versus pH as in Figure 1. (lower)Plot of log concentration versus pH for hspecles E, HE. HES*,and ES'. Total enzyme M, [S] = lo-' M. me poshive inflection concenbatlon ([El + [HE])B in the loo - k*.-.versus oH curve can be acwunted for in eimer of two wavs. According to coill~mtheory the reanwe specles changes hom E to K wnh lnness ng acidity at pH 9 Transhon state meory. on tne other hand, am ou t . ~ me porhve m f l ~ t m to addcon of a proton to ES' to form HES' React b ~ p d e according s to the two theories are shown by the heavier lines. The absolute concentrations of HES* and ESt are not correct but their bend in mncsntratians versus pH is proportional to hnumber of molecules which p s s through the hansition state. (seeAppendix 11).
pH Dependence of Concentrallons of Actlvaled Complexes
I t is instructive t o examine how the concentrations of the various species HE, E, HES', and ESf change as a function of pH. The concentration of each activated complex, deoendent uoon a rate constant and the concentrations of its parent species, the initial reactants, varies in an unusual way with p H as shown in Figure 2. Instead of the concentrations of HESf and ES' being symmetric about the value of p K ~ s ' , as the concentrations of H E and E are about ~ K Ethe , concentration of HESf parallels that of its parent HE, and the concentration of ESf parallels that of E in such a way that all of the relationships in eqns. @-(lo) are ~ b e v e dDerivations .~ are given in Appendix 11. The relative roles of thb independent concentration variables, those of the initial reactants (dominant species), and dependent concentration variables, those of the activated complexes (recessive species), are thus made clear. It is also dear from this diaeram that collision theorv and transition .-.-state theory explanations of acid catalysis are completely equivalent. According to collision theory, which is not concerned with the transition state, the positive inflection in Fieure 2 corresnonds to the ooint where a proton is added is lost f r o m * ~depending ~, upon whether to% (or a one discusses the effect of increasing or decreasing acidity). Transition state theory which neglects the collisional processes on the other hand accounts for the same inflection by addition of a proton to ESf to form HESf. ~
~~
-
More Complex Reactions Both eqns. (12)-(13) can be written in the semi-empirical form Volums 52, Number 9. September 1975 / 579
Table of
Concentrations in M Units at a Given Instant in Time
where only the 'parameter A differs in interpretation accordine t o the two kinetic theories. For more complex reactions &e would obtain
[H+l %h"
=
1
+ -[HCl Kn
+
[H+l2
+
,.,
B
+
[Hfl'
+
]
(15) ,,,
K ~ Z E
where k is the rate constant a t high pH. Each term involving [H+] in the numerator corresponds to a positive inflection in the log hob. - p H profile (increase of one unit in slope) and each term involving [H+] in the denominator to a negative inflection.? It is suggested that an appropriate equation to fit an experimental log h o b - p H profile of the form of eqn. (15) which contains the minimum number of adjustahle parameters can always be written by inspection. The equation can then be interpreted according to either collision theory or transition state theory. A more detailed discussion of more complex pH-dependent reactions is given elsewhere (7). Slgnnicance of a given log k& - pH Profile The larger the difference between the positions of the positive and negative inflections in Figure 2, the larger the extent of acid catalysis. (There are certain rules t o he followed to ensure that the observed catalysis can be attributed to a proton added to the enzyme and not the substrate (12).) However, heyond establishing that acid catalysis is occurring, the type of profile shown in Figures 1and 2 tells nothing about the mechanisms of uncatalyzed and catalyzed reactions. The distinction of identical or different mechanisms can in principle be made for a series of reactions on the basis of whether a Brqnsted relationship is obeyed, ((13), also ref. (I), pp. 170-182) provided some of the rate constants are not approaching the diffusion-controlled limit (14). Acknowledgment The author is indebted to Profs. Colin Hubbard, W. P. Jencks, and K. J. Laidler for helpful discussions. Appendix I The conservation equations relating eqns. (4)-(5) are
The equilihrium constant corresponding to eqn. (4) is
and similarly the equilihrium constants for eqn. (5) are defined as
(6). Substitution of the conservation relations from (16) into eqn. (17) yields
Multiplication and division by [El and [ESJyields
By use of eqn. (6),eqns. (8) and (7) are readily interconverted (see text). The derivations of the relationshipa among eqns. (9)-(13) are identid if one assumes that k l and k z are association equilibrium constants, identical in form to K1 and K2. Appendix I1 It is not the purpose of this paper to carry out a quantitative ealculation of We concentrations of HESf and ESf as a function of pH. It is the trend in concentrations of HES' and ESt relative to each other and ta their parent species HE and E, all as a function of pH, which is of interest here. Therefore hy use of eqns. (7)-(8) the following tahle of values was computed on the basis that [Eltot = 10-5 M, [SJ = 0.1 M,K, = 107 M - I , K? = 1n3 M - l , K~ = M,and KES=l O W M. Because of the analogy between eqns. (7)(8) and (12)-(13), [HESI has been replaced by [HES'] and [ES] by [ES*] in the tahle. Thus the absolute concentrations of HESf and ES! in the tahle are not correct, hut the trend in concentrations as a function of pH is proportional to the number of molecules which pass through the transition state. The results are plotted in Figure 2. The reader can verify from this table and its derivation that all of the equilihrium constants in eqn. (18) are obeyed, that the concentration of HES does indeed parallel that of HE, and ES that of E for a chemical equilihrium, and therefore by analogy that the concentration of HESf parallels that of HE and ESf that of E in a chemical reaction. Literature Cited (1) Jeneks, W . P.. ''Cstalwi8 in Chemiatry and Enmolmgy," MeGrsw-Hill.New York. 1W9.p~.602605. (2) Glasstono. S., Laidler, K. J., and Ey.ing, H.. "The Theory 01 Re* Ploeuuwa," MeGraw-Hill. New York. 1941. pp 13-16. (3) Kurr,J. L..J Amen Chem. Soe., 85,987 (1963). (4) Kurz. J. L., Accta. Chem. Re$., 5. l(1972). (5) Hammett, L. P., "Phyaieal Organic Chemistry? 2nd Ed.,McGraw-HiU, New YorL. 1910. no lR&ld2 -~~ - ~ -
Combination of the four equilihrium constants in (18) yields eqn.
16) Critchlow.J. E., and Dunford, H. B., J. Theor. Bid,37.337 (1972). (7) Dunford. H.B., Critchlow,J. E., Mwire. R. J.. and Roman, R., J. T k o r B i d , (8. 293 11974). (8) PeUer, L., and Alberty, R A,, J Amen Chem. Soc, 81,5901 (1959). (9) Hammo8. G. G., Aeefl. Ckm. Res, 1.321 (1968). (10) Dunford,H.B..Phyalal. V&€fole, 12.13(1974). (11) Bell, R. P., Tmrrp Pornday Soc., 66,2770 (1970). (12) Duoford. H. B.. J. Theor B i d , 46.467 (1974). (13) Bell, R.P., "The Proton in Chemistry? Cornell University Pnaa. Itham 1959. pp
."-
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