KEITHJ. LAIDLER
3 18
Since, very closely, ZZM (r) = 221M(r) and
(r) =
~ Z I M ( ~ ) ,and SZM = (~)‘/%sIM and S3M = ( ~ ) % I M ,
the values wx2 and tx for f l M will satisfy condition I for f2M and f3M; providing that the model does not change in shape or hydration. In this event the abscissa ( M ) of Fig. 2 relating R, to M for five and eight hours centrifugation can be multiplied by a constant factor f by choosing appropriate values of wx and k. For example, the range shown in Fig. 2 may be extended by f = 3 (3,000 to 9,000) by using w = 3.61 X lo3 radians per second and t = 7.21 hours (curve 1) and t = 11.54 hours (curve 2) and by f = 4 (4,000 to 12,OQO) by using w = 3.13 X los and t = 7.94 hours (curve 1) and t = 12.70 hours (curve 2). In a similar manner the calculations may be made to apply to any new combination of the parameters V , u, p , 17, etc., as well as M. It should be noted that a change in shape can be treated in a like manner to a change in u. In this respect it should also be noted that a change in u alone may be compensated for largely by a change in viscosity (therefore temperature) and that new values of w and t need not be sought (see eq. 13).
VOl. 57
When an accurate independent measurement of D (Dm) is available, the calculations and a measurement of Rs may be used to obtain an anhydrous molecular weight not subject to assumptions of shape or hydration. From this value a shapehydration factor DO/D, may be obtained in the usual way. Dm is used in eq. (2) along with M = 1,000, M = 2,000 and M = 3,000 to obtain values of s (sl, $2, 83). The ratios s/Dm so obtained will give, for each M, the correct values of zs, and Zb. Equation (15) is now employed using these values of s, the value of w used in calculations and experiments, and the experimental value of t, to obtain values of ~ ( 7 1 , ,TZ, 73). Through the assumed values of M , each T value corresponds to a value of R, which can be obtained by interpolation in Table V. The paired values of Rs and M have been calculated for the experimental condition used and the assumption that D = Dm. The value of M corresponding to the measured value of R, may now be determined. The extension of calculations given at the start of this section likewise applies to the situation where a value Dm is available.
KINETICS O F SURFACE REACTIONS IN THE CASE OF INTERACTIONS BETWEEN ADSORBED MOLECULES BY KEITHJ. LAIDLER Department of Chemistry, The Catholic University of America, Washington, D . C . Received July 1& lg.58
The influence of interactions between adsorbed molecules on the absolute rates of surface reactions, with special reference to the arahydrogen conversion, is considered. It is shown that these interactions enter in two ways: they affect the number of Xual surface sites and also the partition function for the surface sites. The rate equations are developed for the Bonhoeffer-Farkas mechanism for the parahydrogen conversion on tungsten and the two effects are shown to cancel exactly for a fully covered surface. In calculating absolute rates i t is therefore correct to omit interaction terms.
Introduction Recent work on ad~orptionl-~ has indicated that there may frequently be strong repulsive interactions between atoms or molecules adsorbed on neighboring surface sites. It is therefore important to consider the effect of these interactions on the kinetics of surface reactions. Most treatments of surface reactions4-* neglect these effects: for example, in the calculations of the absolute rates of surface reactions interactions have usually not been considered, without any particular justification. In spite of this omission calculations of absolute rates have for the most part been remarkably successful and an explanation for this success is given in the present paper. (1) J. K.Roberts, Proc. Roy. Soc. (London), 8158,445 (1935). (2) E. K. Rideal and B. M. W. Trapnell, J . chim. phys., 47, 126 (1950); Faraday Soc. Diecussion, 8 , 114 (1950); B. M. W.Trapnell, Proc. Roy. Soc. (London), 8206,39 (1951). (3) J. Weber and K. J. Laidler, J . Chem. Phys., 18, 1418 (1950); 19, 1089 (1951). (4) K. J. Laidler, S. Glasstone and H. Eyring, ibid., 8, 659, 667 (1940). (5) S. Glasstone, K.J. Laidler and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill Book Co., Inc., New York, N. Y., 1941. (6) K. E. Shuler and K. J. Laidler, J . Chem. Phys., 17, 1212 (1949). (7) K.J. Laidler, TAIBJOURNAL,67, 320 (1953). ( 8 ) M C.Markham, M. C. Wall and K. J. Laidler, ibid., 67, 321 (1958).
In two recent papersg-10absolute rate calculations have been made in which explicit account has been taken of the repulsive interactions. These calculations have been made for the BonhoefferFarkas mechanism11 for the parahydrogen conversion on tungsten. The equation employed for calculating the rates involves the application of collision theory and employs the expression derived by Peierls12for the number of bare neighboring sites on a surface. Employing an interaction energy of 4.4 kcal. derived from Trapnell’s isotherm Eleyio calculates that the number of bare dual sites is about of the number that would be present if there were no repulsive forces. His absolute rates moreover are about of the observed ones and on this basis Eley rejects the Bonhoeffer-Farkas mechanism. Laidler’ on the other hand has neglected the repulsive forces, obtains good agreement and concludes that the Bonhoeffer-Farkas mechanism is acceptable. At first sight, neglect of the repulsive forces of this magnitude would not appear to be correct. (9) B. M. W. Trapnell, Proc. Roy. Soc. (London), A806,39 (1951). (10) A. Couper and D. D. Eley, ibid., A811,536 (1952). (11) K. F. Bonhoeffer and A. Farkas, Z . physik. Chem., Bl8, 231 (1931). (12) R. Peierls, Proc. Camb. Phil. Soc., 32, 471 (1936).
. 4
Mar., 1953 KINETICS OF SURFACE REACTIONS IN INTERACTIONS BETWEEN ADSORBED MOLECULES319 Further consideration, however, reveals a serious objection to the rate equation employed by Trapne11 and by Eley. The only way in which the repulsive forces appear in this equation is to diminish, by IOw6or so, the concentration of bare dual sites. It is clear, however, that they should also enter in another way: by making the dual sites intrinsically improbable they should increase the rate of reaction per dual site. Expressed differently, the rate equation contained in its denominator the partition function for the dual site and this will be diminished by the repulsive interactions. There will clearly be some compensation between these two effects. On intuitive grounds the writer has previously assumed the compensation to be complete and has therefore neglected the interactions. The formal justification of this is now given. The proof will be given for the special case of the parahydrogen conversion 011 a well-covered surface, the rate of which is assumed to be controlled by the rate of adsorption of hydrogen. this^ case of the well-covered surface is the one of chief interest; when the surface is sparsely covered the interactiohs are of course unimportant owing to the small number of neighboring adsorbed species. The statistical treatment of the adsorption is equivalent to that of Peierls12 and Roberts.13 Consider a central site Sc surrounded by four neighboring sites S1,Sz,S3 and Sd, The isotherm for dissociative adsorption without interaction is e / ( i - e) = ~ p ~ z (1) Suppose that So is bare: then the probability P that SI is covered is given by P/(1
- P)
=
K
