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(13) PERROTT I S D KISSEY:J . .4m. Ceram. Sac. 6. 417 (1923). (11) P I C H L E R“Syiithesis : of Hydrocarbons from Carbon Xonoxide and Hydrogen,” Bureau of Mines, Special Report. (15) SCHLXTER A K D FCLL.AM Ind. Eng. Chem.. Anal. Ed. 18, 653 (1946). (16) SELIGMAX ASU ASDERSOS: T o lie published. (17) STORCH et a i . : L-.S . Bur. Mines Tech. Paper No.709 (1948). (18) TEICHSER: Compt. rend. 227, 178 (1948). (19) VISSER A S D D E LASGE:De Ingenieur 68, 24 (1946). (20) WEISER:Inorganic Colloid Chemistry. Vol. I I . T h e Hydrous Oxides and Hydroxides, p. 22. J o h n U’iley and Sons. Inc., Sen. York (1938).
KINETIC I,.lWS I N CATALYZED SYSTEMS. I
REACTIOSS ISVOLVISG Ii. J. IAIDLER
ASD
A S I S G L E SUBSTRATE
I R E S E 11. SOCQUET’
Deparlinenl sf Cheniistr?, T h e Catholic Cniaersily qf d m e r i c a , W a s h i n g t o n , D . C Receiijed J u n e 10, 1949
Theoretical kinetic laws for catalyzed reactions have previously been developed in an ad hoc manner, each of the various resulting expressions being applicable only to a limited set of experimental data. Surface catalysis, for example, is usually treated by equilibrium theory, using Langmuir isotherms; acid-base catalysis by either steady-state or equilibrium theory according to circumstances; and enzyme catalysis by it special type of treatment in which allowance is made for the amount of enzyme (but not of substrate) combined in the intermediate complex. The treatments lead to quite different types of kinetic laws in the various cases: thus the usual theory of acid-base catalysis will allow a nonlinear dependence of the rate on the catalyst concentration but a linear dependence on the substrate concentration, while the Michaelis-Menten treatment of enzyme reactions leads to a nonlinear dependence on substrate concentration and a linear dependence on catalyst concentration. In the present paper there will be developed a more consistent set of kinetic laws which \vi11 be applicable to many types of surface, homogeneous, and enzyme catalysis involving a single substrate. The treatment will emphasize the very close resemblance between these systems and will reveal new types of relationships; some of these have been exemplified by existing experimental data, while the conditions leading to others have not yet, been achieved. The work t o be described was undertaken with the primary object of seeing what light can be thrown by the study of heterogeneous and acid-base catalysis on the more elusive problem of the kinet,ic mechanism of enzyme action. I’rpsent address: Sister Irene \ h i e , S.S.A.,. h n a Maria College, 3Iarlboro. Massachuset ts.
520
K. J. LAIDLER .4SD I R E Y E Y. SOCQUET THE MECHANISM O F CATALYSIS
A catalyst diminishes the free energy of activation of a process by introducing an alternative mechanism, in most cases by forming an intermediate complex with one or more of the substrate molecules, this complex subsequently undergoing reaction. I n the simplest type of catalyzed reaction, in which only one substrate is involved, the substrate and catalyst form a complex which subsequently reacts, either unimolecularly or bimolecularly, in a single stage; the overall reaction thus consists of two elementary processes,-the formation of the complex and its reaction to give the final products. Similarly, when two substrates are involved these may form with the catalyst a complex which subsequently reacts in a single step. The case of the single substrate is treated in the present paper, and that of the double one in the following paper; more complicated mechanisms, involving chain reactions and free radicals, will not be treated, as they cannot be reduced to a single scheme of elementary processes. A kinetic mechanism to account for the mechanism of reactions involving a single substrate was put forward by E(. F. Herzfeld (7), but a slightly more general mechanism, which appears to apply to most types of nonchain catalysis with only one substrate, is as follows:
c+
ki
S---'X+Y
(1)
k-1
Here C represents the catalyst and S the substrate; X is the intermediate complex and Y some substance formed in addition t o it. W is a molecule which reacts with the complex to give the product or products P with elimination of the molecule Z, which may sometimes (if W is identical with Y) be C itself. It is assumed that the reaction products are a t such a low free-energy level that the reverse reaction P Z 4 X W can be neglected. The treatment to be given will apply only to initial rates of reaction; otherwise very clumsy kinetic expressions would be obtained, and the experimental data could only with difficulty be analyzed with respect to them. In surface catalysis X is an adsorption complex and Y and W are nonexistent; k-l and k 2 are in this case first-order rate constants, while kl is a second-order constant. The species Y and Z are also probably nonexistent in most cases of enzyme catalysis involving only one substrate. Y and Z do, however, play roles in acidic and basic catalysis in solution; thus, if C is an acid, reaction 1 involves the transfer of a proton to S, so that Y is the base conjugate to C. Similarly, in basic catalysis, Y is theacid conjugate to the baseC. I n acid catalysis W is a basic or amphoteric substance which accepts a proton from X and is frequently a solvent molecule; it may also, however, be a basic solute. In basic catalysis W transfers a proton to X and may be the solvent or another acidic substance present in the system.
+
+
KINETIC LAWS I K CATALYZED SYSTEMS.
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THE INTERUEDIlTE COMPLEX
The role of the intermediate complex in catalysis has been much emphasized in surface catalysis particularly by I. Langmuir (9) and C. K . Hinshelwood (8), in acid-base catalysis by J. S . Bransted (3), N . Bjerrum ( l ) ,and H . von Euler ( 5 ) , and in enzyme catalysis by L. Michaelis and R I . L. Menten (10). Two extreme possibilities for the stability of the complex have been considered, and the kinetic laws obtained depend in an important manner upon what is assumed in this connection. In the first place the complex may be one which decomposes to regenerate the catalyst and the substrate a t a rate which is much greater than the rate Tvith which it undergoes reaction 2 to give the final products; under these circumstances the complex, known as an Arrhenius complex, may exist at comparatively high concentrations during the course of reaction. The rate of reaction can then be obtained by calculating the concentration of X from the equilibrium
I
L(d) k.,
> k, > k ,
FIG.2 FIG.1 FIG.1. Schematic free-energy profiles for catalyzed systems, showing six possible cases. FIG. 2. Relationship between rate and substrate concentration for a catalyzed reaction (equation 8) (except in acid-base catalysis, for which the relationship is always linear). A similar variation with catalyst concentration is found (equation 12).
1 above and multiplying this by Icp[W], this being the procedure, for example, with surface reactions. The second possibility is that the intermediate complex is a much less stable species (a van't Hoff complex), the rate of its reaction 2 not being small compared with the reverse rate of reaction 1. It is then not permissible t o calculate [XI by making use of equilibrium 1, since the rate of reaction 2 cannot be ignored. Since the concentration of X must in this case be low, the steady-state treatment may be employed. The distinction between the two cases may be shoivn by means of standard free-energy diagrams. Six cases which are of special interest are illustrated in figure 1. Curve (a) corresponds t o kl > k-1 > k,, in which case X lies a t a low freeenergy level with respect to both reactants and products and is therefore a stable ilrrhenius intermediate; its concentration can therefore be calculated accurately using equilibrium theory. This case is probably found in most surface reactions
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K. J . LAIDLER AND IRENE M . SOCQUET
and in many enzyme-catalyzed reactions. The complex is also an Arrhenius one in cases (c) and (d), and equilibrium theory is therefore applicable. In cases (b), (e), and (f), on the other hand, the steady-state treatment must be applied. The kinetic expressions for the cases of equilibrium treatment and steadystate treatment will now be derived, and will later be applied t o the various types of catalysis. TH9 EQUILIBRIUM TREATMENT
In the case that the rate of Teaction 2 can be neglected in calculating the concentration of X, the concentrations of X, Y, C, and S are related by
where K is the equilibrium constant. If [Ch and [SI0 are the initial concentrations of C and S before some of them have been used in complex formation, it follows that [CI = [CIO - [XI [SI = [SI0 - [XI
(2)
Therefcre
This is a quadratic in [XI, and can be solved for [XI. An expression for the rate, equal to k2[W][X],can then be written down. However, it is more convenient to consider two special cases. Case 1: If the substrate is initially present in much higher concentrations than the catalyst, i.e., if [SI0 >> [C],, it follows that [SI0 - [XI is very close to [SI,, since [XI cannot exceed [C],; equation 4 therefore approximates to
whence K[CIo[SIo [Y]
= Z> K[Z], so that equation 24 applies. At high catalyst concentrations, on the other hand, [XI >> [SI,since most of the substrate is used up in forming the complex; it then follows from equation 26 that K[Z] >> K'[W], so that equation 25 applies. The conclusions arrived at above may be summarized as follows, the remarks applying t o the equilibrium case and to acid catalysis only: (1) If the acceptor W in reaction 2 is the solvent, there is specific hydrogen-ion catalysis, the rate becoming independent of the hydrogen-ion concentration at high acid concentrations. ( 2 ) If W is a solute species there is general acid catalysis, the rate varying linearly with the acid concentrations at all concentrations. The attainment of a limiting rate is therefore evidence against general acid catalysis in any particular reaction. Whether the catalyst C1 is solvent or solute is seen to have no bearing on the type of catalysis. The idea that the complex X is in equilibrium with the reactants is due to H. von Euler (G), who proposed it as a universal explanation of acid-base catalysis. Evidence for the mechanism has been obtained (6) for the acid-catalyzed hydrolysis of acetamide, the apparent catalytic constant decreasing with increasing hydrogen-ion concentration which finally has no influence on the rate. Since the attainment of a limiting rate is not obtained xhen the steady-state treatment is applied (see below), this result appears to indicate unequivocally that, the equilibrium treatment is applicable; it also proves that the catalysis is specific, i.e., that the mater is the acceptor W.In no other case, however, has good evidence been obtained for the attainment of a limiting rate, and the steady-state treatment is probably more generally applicable. THE STEADY-ST.%TE TREATMEST
The alternative hypothesis that the second step is not slow \vas proposed by
J. N. BrBnsted (3) and the rate equations may be formulated as follows: The general steady-state expression for the rate of a catalyzed reaction is equation 16, and since by hypothesis the rate of formation of the complex is slow in comparison with the rate of its reaction the term k,[C]o kl[S]o may he neglected; the rate equation then becomes
+
Two special cases of this are important
528
K. J . LAIDLER AXD IREPjE M. SOCQUET
Case 1: If kz[W]>> kJY] the rate becomes:
(28) The rate is thus completely controlled by the first step, and there is general acid catalysis. Case 2: If k-JY] >> kz[W] the rate is given by: d[Sl - kikz [Clo[SldWl (29) dt k-i [Y] However, all of the acidic and basic species (other than the substrate) present in solution are a t equilibrium, and in particular (cf. equation 19)
TABLE 2 Szcmmary of kinetic behavior in acid and base catalysis DEPENDENCE OF RAIE ON CATALYST CONCENTRATION ASSUMING
TYPE OF
Equilibrium theory
S A T U R E OF \$'
CAIALYSIS
High
LOW
I
Steadystate theory (All concentrations)
,_____I
Acid
1
Solvent
hld Base
1
Solute Solvent
Base
1
Solute
~
1
[HaO-] Z[acid] [OH-I Xlbase]
S o depend-
ence Z[acid] No dependence z[base]
since [C]Ois in this case approximately equal to [C]. The rate equation therefore becomes:
In acid catalysis Z may be either the lyonium ion (if W is the solvent) or a solute acid (if W is a solute); there is therefore either specific or general acid catalysis according as W is the solvent or a solute. I n the steady-state case there is linear dependence of the rate on catalyst concentrations a t all concentrations, in contradistinction to the equilibrium case for which there is nonlinear dependence if W is the solvent. The variation of the rate with catalyst concentration is shown for the various cases in table 2, in which Z [acid] and Z [base] indicate general acidic and basic catalysis. Whether equilibrium or steady-state theory is the valid one, the results can be summarized as follows: (1) If the mechanism is prototropic, Le., if W is a solute, there is general acid catalysis and the rate is always linear in the catalyst concentrations. (k') If the mechanism is protolytic, there may be either general or specific
K I S E T I C LAWS IN CATALYZED SYSTEMS. I1
529
catalysis and, in the latter case only, the rate may sometimes (if equilibrium theory is applicable) become independent of the catalyst concentration. The fact that the kinetic behavior depends upon the nature of JT7 has long been recognized in the distinction betn-een prototropic reactions and protolytic reactions. According to the present treatment protolytic reactions may be said t o be those in vhich IT7 is the solvent, and prototropic reactions those in which it is a solute. The above analysis has shown that protolytic reactions may show either general or specific catalysis according to the relative magnitudes of kzpV] and k-,yU], while a prototropic reaction always exhibits general catalysis whatever the relative magnitudes. It is difficult on the experimental side to distinguish betvieen the two types of reaction however, since failure to detect general catalysis, which might be taken as evidence for a protolytic mechanism, does not necessarily imply that acids other than the hydrogen ion have absolutely no catalytic action. SCMM.4RY
The kinetic equations for catalyzed reactions involving a single substrate are developed in a generalized form, the object being to demonstrate analogies betTT-een the behavior of surface-catalyzed, enzyme-catalyzed, and acid-basecatalyzed reactions. Both the equilibrium and the steady-state treatments are used, and it is shown that, vhere the former is applicable, the rate becomes limiting a t high substrate concent'rations for both surface-catalyzed and enzymecat,alyzed reactions; v i t h acid-base catalysis, on the other hand, the rate is always linear in the substrafe concentration. .A nonlinear dependence of the rate on the catalyst concentration is found to be possible for all types of catalysis; examples are quoted, and the necessary conditions defined. It is shown that in acid-base catalysis the general type of kinetic behavior does not depend upon the nature of the catalyst n-hich initially interacts with the substrate, except as t o whether it is acidic or basic, but depends in a critical manner on the nature of the species which reacts with the intermediate complex to form the final products. REFERESCES (1) BJERRTJI,S . :Z. physik. Chem. 108, 82 (1924). (2) BRIGGS,0 . E., ASD H ~ L D ~ JS.EB .. S . : Biochem. J.19,338 (1925). (3) B R ~ S S T E J. D ,S: Chem. Revs. 6, 271 (1925) (4) CHAKCE. B . : J. Biol. Chem. 131, 553 (1943); Science 109,211 (1949). ( 5 ) EL-LER.H . vox: Svensk. Vet. .Iliad. Forh. No. 4 (1899); Z.phvsik. Chem. 28,619 (1899); 32,381 m o o ) ; 3 6 , 6 4 1 (mol) (6) EULER.H . v a s , .ASD OLLSDER, A : Z . physik. Chem. 131, 107 (1927). (7) HERZFELD, IC. F . : Z . phrsik. Chem. 98, 161 (1921) (8) HIMHELWOOD. C. 5 . :Kinetics of Chemical C h a n w in Gaseoils Systems. Oxford University Press. London (1926. 1929. 1933, 1911). (9) T . & K G M ~ I R . I. : J. Am, Chem. SOC.40, 1351 (1918) (10) ~ I I C N A E LT,,, I S ,I S D MENTES.31. I,.:Biochein Z.49,373 (1913. Ind. E n g . Chem. 41,378 (1949). (11) SEYER,W .F.. A S D YIP, C. W.: I. lf...AND LAIDLER. K . J . :Arch. Biochrm. 26, 171 (1950). (12) SOCQCET. (13) STERS,K . G . : J. Biol. Chem. 114,473 (1935). (14)S T R A I TF. , R . : Fhzrrnoloqin 9, 147 (1941,
