THE ACTIVATED COMPLEX IN CHEMISORPTION AND CATALYSIS BY HENRY EYRING, CHARLES B. COLBURN AND BRUNO J. ZWOLINSKI Received 61h February, 1950 The basic assumptions underlying the theory of absolute reaction rates have been analyzed in terms of recent studies. A general treatment of chemi- sorption and catalysis is given in terms of the activated complex theory of chemical kinetics. A mechanism is proposed for the hydrogenation of ethylene which appears to be in agreement with the available data. Through the efforts of investigators such as Rideal, Taylor, Schwab, Langmuir, Hinshelwood and others, the broad outlines of the field of catalysis have been defined and the importance of parameters like extent of surface, nature and condition of surface, coverage, active sites, geometry put in their proper perspectives.However, one is still not in the desir- able position whereby for any one case one may predict and calculate the extent of a surface reaction from first principles. In this regard it may be worth while to Ye-examine the assumptions underlying the theory of absolute rates of reaction and show its applicability in the interpreta- tion of heterogeneous reactions. In formulating the rate of any chemical reaction, consider reaction complexes passing from one region of configuration space to another. Along any one reaction path there will always be a region of phase space corresponding to a maximum free energy separating the initial and final states. The number of reaction complexes passing through this region is given by - dC/dt = XniCnpnt, .(1) n, i where C = total number of complexes in initial state at any time t, X = probability that the reaction complex crosses the barrier in any v = frequency with which the complexes cross the energy barrier. one attempt, The subscripts n and i refer to the quantum numbers associated respectively with the degree of freedom along the reaction co-ordinate and with the remaining internal degrees of freedom of the reaction com- plex. The above equation applies exactly to any rate process under any set of conditions. Assuming a normal distribution of energy among the internal degrees of freedom of the complex or the equivalent idea of an equilibrium existing between the reactants and the activated state at the top of an energy barrier, the well-known expression obtained for the specific rate constant €or a reaction of any order is : Here, X is the average transmission coefficient, F* the partition function of the activated complex, Fg the partition functions for the reactants and c0 the activation energy of the reaction at the absolute zero of tem- perature.Although the strict validity of the equation rests on the establishment of an equilibrium, the interpretation of many varied rate 3940 THE ACTIVATED COMPLEX IN CHEMISORPTION phenomena in terms of the above relation lends support to its applicability to most irreversible processes. The concept for which one could wish more definite proof before proceeding to a study of the mechanics of heterogeneous processes is the equilibrium postulate. For adiabatic reactions involving a rearrangement of matter as em- bodied in the idea of a reaction complex surmounting an energy barrier, it appears to be sufficient to ascribe X, the transmission coefficient, a value of unity.Hirschfelder 1 g and Wigner have considered more closely the shape of the potential energy surfaces in the region of the activated state to find that only in rather exceptional cases can one expect the % to deviate from unity. Born and Weisskopf formulated a mechanism of surface reactions based on the idea of tunnelling through the potential barrier which would be reflected in values of X less than unity. A t the present time there is little evidence to indicate that this may be an important step in catalysis. The Equilibrium Postulate.-Consider a unimolecular decomposi- tion of a molecule on a surface.For a surface to be effective, it is im- perative that a much closer and more specific interaction takes place between the molecule and the surface atoms than can be ascribed to dispersion forces. Regarding this intermediate as the reaction complex assume that reaction consists in the complex passing as a result of molec- ular collisions or vibrations into a set of levels corresponding to the activated state or the final state of the products. The following set of linear equations with constant coefficients is obtained : for the n possible energy levels of the system. The vi5’s are the specific rates of transition from the level i to the level j , which, in principle at least, can be calculated. Solution of the above set of linear differential equations can be carried out readily and takes the following form n C4 = C AkB5, ebkt, .(4) k = l where k f k are the arbitrary constants in the general solution determined by the initial values of the C,’s ; B,, and b, are functions of the transition constants vij, which arise from solution of the secular or characteristic determinant. Each Ci is now a completely determined function of time and an overall rate of reaction can be calculated under any set of equi- librium conditions. To compare the actual rate with the equilibrium rate as determined by a Maxwell-Boltzmann distribution of reacting complexes, a function r is defined whose deviation from unity with the extent of reaction pro- vides a measure of the validity of the equilibrium postulate in chemical rate theory.The actual rate for the forward reaction is given by where k and I refer to the levels of the initial and final states, respectively. Assuming a Maxwell-Boltzmann distribution function, the equilibrium rate for the forward process is then given by the following : €1 E2 v, =- i=l Hirschfelder and Wiper, J . Chem. Physics, 1939, 7, 616. Hulburt and Hirschfelder, ibid., 1943, 1 1 , 276. Born and Weisskopf, Physik. Chem., 1931, 12, 206.EYRING, COLBURN AND Z'CVOLINSKI 41 where C, is the total concentration of complexes in the levels of the initial state a t any time t. Hence, the ratio of the actual rate to the equilibrium rate is 5 i virci i = l j = k + l r = k 1 { h : 1'4 n * . (7) 20-i &i 2 c (Vi,e-s)}; k 2 e-m s = l j = k + l i =1 Calculations of this nature * have been carried out for an extremely simple model of reacting system (with n equals 4) based on chosen values of proper magnitudes for the transition constants.The results are shown graphically in Fig. I. Though the calculations were made on an over- simplified model under rather extreme conditions of non-equilibrium, only an error of 20 yo occurs in the specific rate constant as determined by the theory of absolute rates of reaction for I yo of the material reacted. Essentially similar results were obtained by Kramer who employed classical diffusion theory. Hulburt and Hirschfelder in a recent study regarded the reaction complexes in configuration space as a compressible fluid and employed hydrodynamic theory t o show qualitatively the correctness of the equilibrium postulate.In view of the above studies, a complete rate theory is available which, except for minor refinements, can be applied with confidence to all rate processes. General Treatment of Chemisorption and Catalysis .-The application of the activated complex or transition state theory to surface reactions was carried out by Kimball,' Temkin, 8 Eyring 9~ l1 and more recently by Laidler.ll9 12* 13 A slightly more generalized approach will be developed in the following paragraphs. Zwolinski and Eyring, J . Amer. Chem. SOC., 1947, 69, 2702. 5 Kramer, Physicu, 1940, 7, 284. Hulburt and Hirschfelder, ibid., 1949, 17, 964. Kimball, ibid., 1935, 6, 447. * Tempkin, Actu Physicochim., 1938, 8, 141. Eyring, J.Chem. Physics, 1935, 3, 107. lo Glasstone, Laidler and Eyring, The Theory of Rate Processes (McGraw-Hill l1 Laidler, Glasstone and Eyring, J . Chem. Physics, 1940, 8, 659, 667. l2Laidler, J . Physic. Chem., 1948, 53,712. Book Co., Inc., New York, 1941). Schuler and Laidler, J. Chem. Physacs, 1949, 17, 1212. B"42 THE ACTIVATED COMPLEX IN CHEMISORPTION The following assumptions are made : (I) In every surface process, the activated complex consists of the reactants and the catalyst. In certain specific cases, where-except for the catalyst atoms-the activated state contains the same number and kind of atoms of the reactants as the homogeneous reaction, the differ- ence in energies of activation bet ween the homogeneous and heterogeneous reactions is ascribed to the heat of adsorption of the homogeneous activ- ated complex to the surface of the catalyst.( 2 ) The surface reaction is considered as a homogeneous reaction between the gaseous reactants and the atoms, molecules or ions of the catalyst. In this analysis, we are neglecting volume phenomena, such as solution of reactants, and the effect of reverse reactions. Consider a reacting system consisting of two different gaseous species and a catalyst. The following mechanism can be written based on a model of the two different gaseous reactants interacting w-ith the surface. The kind and number of atoms in the activated complex will determine the kinetics of the overall process, so as to agree with experimental data. Thus one obtains nCi -k mCj + Sk ( c n i c m j s k ) * -% ci+ 1 + cj- 1 + sk, where the C i , C j = gaseous concentration of reactants, Ci+ Ct-r = gaseous concentrations of products, 1 .(8) Sk = kth surface site. J The subscripts n and m in the activated complex are used to denote any fragmentation or dissociation of the original reactants. As is well known, the rate-determining step in catalysis can be one of a number of different processes, e.g. diffusion to the surface, diffusion on the surface (excluding solubility in catalyst), adsorption of reactants, surface dissociation, combination of radicals on the surface and desorption of products. Assuming no alteration in surface characteristics with re- action, it is possible to write, in the most general form, fcr the initial velocity of reaction, the following equation without assuming any one specific activated complex for the mechanism : v = X(Cn,C,,S,)+v .(9) Expressing the concentration of activated complex in terms of the react- ants, one has the result Ff - By proper choice of an activated complex, i.e. specifying the values of n and m the rate equation for any kind of a surface process is obtained. If i t is assumed that all the products and reactants establish equilibrium concentrations on the surface, the effect of pressure and poisons can readily be accounted for by modification of the above equation with the ap- propriate adsorption isotherms. Mechanism of Ethylene Hydrogenation.-To illustrate more fully the application of the above theory the procedure appropriate to the determination of the mechanism of the hydrogenation of ethylene may here be outlined.This reaction was chosen because a successful mechanism for the catalytic hydrogenation has not been proposed, even though the reaction has been studied extensively by a number of workers.EYRING, COLBURN AND ZWOLINSKI 43 (Beeck l4* 16* l6 and co-workers investigated this reaction on numerous catalysts such as chromium, iron, cobalt, nickel, rhodium, palladium, platinum, tantalum and tungsten ; Twigg l7 on nickel ; Taylor and co- workers la, l9 on various oxides ; A. Farkas,,OS 21 L. Farkas 20e 21 and Rideal 22 on nickel ; Toyama 23s 24 on nickel at temperature of oo C and in the temperature range 99-165" ; Pease 25 on copper ; Wheeler and Pease 26 have studied the relative rates of hydrogenation by deuterium and hydrogen on copper.Many other workers have also contributed much to the understanding of the mechanism of hydrogenation. work on the nature of adsorbed layers is fundamental to the knowledge of all surface processes.) From these studies the following facts will have to be explained by or be in agreement with any proposed mechanism : (I) H, and C2H, are rapidly adsorbed by metallic surfaces even a t liquid air temperatures, however, in the case of the hydrogen about 20 yo of the H, can be removed by pumping at room temperature. (2) AHads for H, varies from 30 to 18 kcal. per mole on Ni; AHade for C,H, varies from 60 to 23 kcal. per mole on Ni. The larger value is the heat of adsorption on a clean metal surface and the smaller value is the heat of adsorption on a nearly complete monolayer.( 3 ) The activation energy of the hydrogenation is 10.7 kcal. on Ni, Rh, Co, Fe, Pd, Pt, W, Ta. (4) The value for the average heat of adsorption of ethylene on nickel is 41-5 kcal. (5) The reaction rate may be expressed as V Q : P ~ ~ P & ~ at low tem- peratures. (6) The reaction rate may be expressed as z ~ a ~ ~ ~ P & ~ ~ at high tem- peratures. (7) The overall heat of reaction is 32 kcal. per mole. (8) The hydrogenation appears to be molecular in hydrogen. These observations which must be explained are apparently satisfied only by the following mechanism. Both ethylene and hydrogen are adsorbed on the surface but ethylene being much more tightly held tends to displace hydrogen molecules. The activated complex is an ethane just desorbed from its two positions on the surface.Suppose that for an adsorbed ethylene there are b neighbouring positions which a hydrogen molecule can occupy and still react with the ethylene to form ethane. We let a represent the number of positions per square centimetre accessible to both hydrogen or ethylene and we take u1 and u, as the fraction of these positions covered by ethylene and hydrogen respectively ; k' is the specific rate of reaction of an adsorbed ethylene with an adsorbed hydrogen mole- cule. We shall not take explicit account of the co-operative forces which tend to make adsorbed molecules bunch on the surface. Then the velocity of reaction v in molecules per square centimeter is v = au1u,bk'. - ( 1 4 l4 Beeck, Smith and Wheeler, Proc.Roy. Soc. A , 1940, 177, 62. 1S Beeck, Rev. Mod. Physics, 1945, 17, 61. 16 Beeck, ibid., 1948, 20, 127. l7 Twigg, Trans. Faraday Soc., 1935, 35, 934. l9 Woodman and Taylor, ibid., 1940, 62, 1393. 2O Farkas, Farkas and Rideal, Proc. Roy. Soc. A , 1934. 146, 630. 21 Farkas and Farkas, J . Amer. Chem. Soc., 1938, 60, 22. 22 Rideal, Chem. and Ind., 1943, 62, 335. *3Toyama, Rev. Phys. Chem., Japan, 1937, 1 1 , 152. 24 Toyama, ibid., 1938, 12, 115. 25 Pease, J. Amer. Chem. Soc., 1923, 45, 1196, 2235. 26 Wheeler and Pease, ibid., 1936, 58, 1665. 27 Roberts, Proc. Roy. Soc. A , 1935, 152, 445. Woodman, Taylor and Turkevich, J . Amer. Chem. Soc., 1940, 62, 1397.44 THE ACTIVATED COMPLEX IN CHEMISORPTION To calculate the U'S we define a partition function f, for ethylene such that p1 = - kT In f,, where p1 is the chemical potential of ethylene.Here V is the total gaseous volume which contains n, molecule of C2H4 so that V/n, = kT/P,, where p , is the partial pressure of ethylene. The other quantities in (13) have their usual meaning. f a is defined analogously for hydrogen gas. fs represents the partition function for the surface atoms before they combine with hydrogen or ethylene ; hl and faz repre- sent the partition functions for the compound ethylene-surface and hydro- ge. -surface respectively. Then f S l f 1 =l = fSflf2 + f S l f 2 + f s 2 f i ' and f d l f 2 =2 = f S f l f 2 + f S l f Z + . f , 2 f l ' Thus substituting in (12) gives or Here f38S is the combined partition function for activated ethane with all the surface atoms involved in reaction and f * = f3s8h-2. Since under all experimental conditions v is proportional to the pressure of hydrogen we must suppose the quantity K p = - may always be neglected in comparison with I and with 1% = K,p,.Further, if we write fl = C / p , f s z - f 8 f 2 f S f 1 At low pressures of ethylene ZI ot 9, and at high pressures v Q I @ , in agree- ment with experiment. By setting -(p1(1 + K , p , ) - 2 ) = o we find that for K,pl = I the value of v is independent of pl. Substituting p1 = I / K , gives d dPl Substituting for C and K , gives We now substitute for these quantities. We assume b = 2 corre- sponding to two neighbouring hydrogen molecules being able to com- bine with an adsorbed ethylene. Then ZJ = 170/~ x 1015 x z x 5-6 x rolZ I 8.rr2(8a3ABC*)'~2(kT)512 - 3 e RT h v f 6h3 -- (16) I - e kT r"r i = l I I h v fi h v ( z .r r ~ n ~ ~ k T ) ~ / 2 kT 8dIH2kT -~ h3 P 2 2h2 1 - e-=&i=lI - e - ~ C ~ H 4EYRING, COLBURN AND ZWOLINSKI 45 Here E , is the energy required to pass from gaseous hydrogen and absorbed ethylene to gaseous ethane. This is the heat cf adsorption of ethylene minus the heat of reaction of gaseous hydrogen with gaseous ethylene t o make gaseous ethane and agrees with the experimental activation energy E,, = 10,700 cal. as nearly as we can estimate it. This point is discussed further below. The internal degrees of freedom of ethane plus its rotational degrees have an entropy of 18.46 cal./mole deg. at 300° K. The internal vibrations of ethylene at 300' have the value of 0.57 cal./mole deg.If we use these values and the known values for hydrogen the only things left undetermined in (16) are the two vibrations associated with the motion of ethane tangent to the surface and these same two vibra- tions for ethylene plus its motion normal to the surface. Thus we obtain for the velocity of reaction h v 2 (1 - e-=) 'LI = 1.4 x 10-~p, mole/cm.2 sec. . * (17) I hvi e-kr i = 1 1 - T? This is to be compared with ZI = 1-2 x I O - ~ ~ ~ found by Beeck for un- oriented nickel where the surface roughness was found to have the value 170 which we used in our calculation, Thus, if we assume the vibration frequencies in (17) cancel agreement is too good since many things includ- ing the experimental values may be slightly different. Quite apart from the numerical agreement it does appear that this must be the reaction mechanism.L FIG. 2. If we had an exact value for the heat of adsorption of ethylene our postulated mechanism would enable us to calculate a value for E , to compare with the experimental value E , = 10,700 cal./mole. The situ- ation is as follows. The homogeneous hydrogenation of ethylene requires an activation energy of 43 kcal. together with a AH of reaction of 32-5 kcal. and occurs readily on the surfaces of metals of Group VII such as Rh, Fe, Ni, Pt, Pd as also W, Ta and Cr, with a constant energy of activ- ation at 10.7 kcal. Whereas the AE* remains constant, the temperature- independent factor of the rate constant shows variation by a factor of 10,000 as we progress from W to the much more active Rh.This has been shown by Beeck and co-workers 143 159 who studied this reaction on evaporated films of the above metals. The average heat of adsorption46 THE ACTIVATED COMPLEX IN CHEMISORPTION of ethylene is 41-5 kcal. whereas that of hydrogen is 24 kcal. on prepared Ni evaporated films; thus, the more strongly adsorbed ethylene pre- ferentially covers the surface of the metal catalyst in the place of the hydrogen. The 10.7 kcal. of energy represents the amount of work necessary to form the gaseous ethane from the adsorbed ethylene and the gaseous hydrogen for the condition where the rate of reaction was measured independent of the ethylene pressure. The temperature co- efficient for the reaction will take on different values depending on the magnitudes of the pressure of the two reactants.This is shown in Fig. 2, where the potential energy changes are shown along the reaction co- ordinate. Such specific considerations of the energies of the process are limited to the specially prepared catalytic surfaces as employed by Roberts and by Beeck. One has only to observe the variations in the heats of ad- sorption of ethylene, as summarized in Table I, to realize the importance TABLE I.-HEATS OF ADSORPTION OF ETHYLENE Metal Au Fe . Ni . Cr,O, . CU Pt . ZnO . ZnO-Cr,O, (reduced) ZnO-Cr,O, (oxidized) Pt-Black . Ni . Ni . Heat (kcal./mole) t 8.8- 6.9 16 - 8 10.8 16 9 25 -19 20 -17 9 15 60 -20 I 2 I1 -I0 Reference Beebe-Schwab * 3 3 .I I D I , I , , a , I 9 , Maxted, Moon Toyama 23 Beeck l4 *Schwab, Handbuch der Katnlyse (Springer, 1943). -f Variation of heats of adsorption with extent of surface coverage. Maxted and Moon, Trans. Faraday SOC., 1936, 32, 1375. of catalyst preparation, as is well known. Changes that have been meas- ured in the heat of adsorption indicate the heterogeneity of the catalysts, whose activity is conditioned by the sintering, crystallization and im- purities, e.g. hydrogen and oxygen resulting from the mode of preparation of catalysts from their salts. Such catalysts, however, as pointed out by H. S. Taylor, are of prime importance in industrial application and will remain a constant challenge to the chemist. The authors wish to acknowledge a grant from the United States Naval Research Office which made this investigation possible. Special thanks are due to Dr. R. B. Parlin for reading the paper in manuscript and €or helpful discussions during the course of the study. Department of Chemistry, University of Utah, Salt Lake City, Utah.