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Peak, David

Pearlman, K

Wantuck, P J

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5538 Letters to the Editor leave (1975-76) from the Department of Chemistry, Univer-sity of Minnesota, Minneapolis, Minnesota 55455. tPresent address: Central Research Department, Experimen-tal Station, E. I. duPont DeNemours & Co., Wilmington, DE. 19898. l D. G. Truhlar, F. A. Van-Catledge, and T. H. Dunning, J. Chern. Phys. 57, 4788 (1972L 2D. G. Truhlar and F. A. Van-Catledge, J. Chern. Phys. 59, 3207 (1973). 3For reviews see D. Golden, N. Lane, A. Temkin, and E. Gerjuoy, Rev. Mod. Phys. 43, 642 (1971); D. G. Truhlar, in Modern Theoretical Chemistry, edited by G. A. Segal (Plenum, New York, in pr~ss), Vol. 8. 4For a review see E. Scrocco and J. Tomasi, Top. Curro Chern. 42, 75 (1973). OF. A. Gianturco and J. H. Tait, Chern. Phys. Lett. 12, 589 (1972). 6A. D. McLean and M. Yoshimine, J. Chern. Phys. 46, 38(;2 (1967). 7F . H. M. Faisal and A. C. V. Tench, Comput. Phys. Commun. 2, 261 (1971). 8N. Chandra, Phys. Rev. A 12, 2342 (1975). 9y . Itikawa and K. Takayanagi, J. Phys. Soc. Jpn. 27, 1293 (1969). 10J . A. Pople, D. L. Beveridge, and P. A. Dobosh, J. Chern. Phys. 47, 2026 (1967). Competitive effects on the rate of the diffusion-controlled reaction A + B-lC* David Peak, Karen Pearlman, and Paul J. Wantuck Physics Department, Union College, Schenectady, New York 12308 (Received 27 July 1976) The effects of competition among the reactants in the diffusion-controlled bimolecular reaction A + B - B were recently examined in companion articles in this Jour-nal. 1,2 The treatment presented in these papers was based on an analogy taken from electrostatics, and re-sulted, after substantial analYSiS, in a description of the reaction kinetics which accounts for various cor-relations among the reactants. A primary conclusion of the first of these papers1 (herein referred to as FD) was that the effective rate coefficient for the reaction is enhanced over the value predicted by simpler models which ignore competitive effects. Furthermore, it was demonstrated in FD that the enhancement, to first or-der, varied explicitly as the square root of the B spe-cies (called "sink" in FD) concentration, a result which differed from the concentration dependence elicited elsewhere by other techniques3,4 (see Ref. 14 of FD). As was pointed out in FD (Refs. 4,6,7, and 8), others have attempted to include correlations in their discus-sions of diffusion-controlled reactions. With this in mind, the readers of FD may be interested to note that a treatment of the same problem in terms of a Waite-type formalismS leads to the same square root depen-dence on concentration. Because of its somewhat less detailed nature,6 the Waite-type calculation yields this particular result with a good deal less effort than was presented in FD. We sketch below the details of the calculation, treating at the same time a slightly more general problem than was considered by FD. Consider the reaction A + B - C, occurring in an inert solvent, where the formation of C is assumed to pro-ceed at a steady rate. That is, if we designate by fl the sample-average concentration of species i(i = A, B, or C) then we can write (1) where all quantities on the right hand side of (1) are in-dependent of time. The problem is to determine the form of the rate coefficient k in the circumstance that the reaction is diffusion controlled. Because of the pairwise nature of the reaction, the important spatial dependence in this system can be ac-counted for through the A, B-pair concentration p( r) defined as7 p(r) is the average number of A, B pairs, in the steady state, with A and B separated by a dis-tance r, per unit volume squared. The mutual diffusive flux of A's and B's to within some critical distance R of one another determines the rate of formation of C. Thus, we have dfc/dt= 4rrR2Do(a rP)R , (2) where Do = D A + DB is the sum of the reactant diffusivi-ties in the solvent under consideration. The pair con-centration satisfies the steady state equation 0= Do V2p + (reaction terms) ; (3) we assume, as in FD, that Do is independent of concen-tration. The reaction terms in (3) describe both losses and gains of A, B pairs. A given A, B pair of separa-tion r can be lost due to reactive competition in two ways; the A can react with any other B, the B with any other A. The probability of losing a pair of separation r is then the probability of having such a pair to begin with (which is proportional to p(r)) multiplied by the sum of the probabilities of losing each of the members due to competition. Inasmuch as - kfAfBf~l is the probable rate of loss, on the average, of any molecule of species i (i = A or B) in the sample, the terms in (3) describing competitive losses can be combined as the single ex-The Journal of Chemical Physics, Vol. 65, No. 12, 15 December 1976 Copyright © 1977 American Institute of Physics Downloaded 14 Aug 2011 to 129.123.67.59. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsLetters to the Editor 5539 pression - k(jA + fB)P, In order to attain a steady pro-duction of C's, A's and B's must be replaced as the re-action evolves. Thus, (3) will contain gain terms re-flecting this replacement. If the replacement is as-sumed to occur at random throughout the sample then p(r) is increased at the spatially uniform rate kfAfB(jA + fB)' Defining the dimensionless pair concentration, u(r), by u(r) = 1 - p(r)/fAfB , (4) allows us to write (3) explicitly as 0= v 2u_ ~u ; where rf = k(jA + fJ/D o• (5) A solution to (5) subject to the boundary conditions u(oo) = O-implying that p approaches the sample-aver-age pair concentration for large pair member separa-tions-and Ro( aTU)R = u(R) - I-required by the radiation boundary condition on p at the reaction surface [Ro is the same parameter as in FD, Eq. (4. 15)]-is just When this solution is inserted into the right hand side of (2), and the result equated to the right hand side of (1) we find that k satisfies the nonlinear relation k = 47TDo R2( 1 + (3R) (R + Ro + {3RRor1 (recall that (3 is a fWlction of k). (7) Now, we rewrite (7) in terms of the quantities de-fined in FD. Thus, we introduce O! = k{47TDor1, Y = R2(R +Ror1, T=Ry-1, and €1=(47TY~1)1/2fori=AorB. Equa-tion (7) can be rearranged, and, with these latter defini-tions, be recast as 0= €2(Ro R-1) (O!y -1)3 _(1+2T€2RoR-1)(O!y-1)8+(2+T2€2)(O!y-1)_1, (8) where €2 = €~ + €~. Since the parameters €I are closely related to the volume fractions of the corresponding re-actants, which, in turn, are presumed small, we see from (8) that (O!y-1)-1. A perturbation expansion of (O!y-1) in successive powers of € leads to the result O!= y[l + (€! +€~)1/2+ (€! +€~) G - T) + O(€ )] • (9) As pointed out in FD, the steady state bimolecular rate coefficient predicted by models which accoWlt for diffusion controlling but not for competitive effects is ko = 47TDoY. Consequently, (9) shows that a Waite-type treatment of the reaction A + B - C predicts an enhance-ment of the effective rate coefficient over ko' due to competition, which to first order is (€! + €~)1/2. The result obtained here is easily compared with Eq. (8.7) of FD. In the reaction we have discussed, the species A and B are treated on completely equal footing; (9) is symmetric in A and B. The reaction considered by FD, however, distinguishes between A and B. The B's are taken to be "indestructible"; they are not removed from the sample and are, therefore, not replaced. The analysis involved in obtaining (9) from (5) is all the same in this case except that (3-kfBD~1 and € - €B' Thus, our (9) and Eq. (8.7) of FD agree that the dom-inant enhancement of the effective rate constant for the reaction A+ B - B is just €Bo The two equations do, of course, disagree for higher order corrections, mir-roring the approximations peculiar to the two different approaches. Whether an extremely precise experi-ment could Wliquely discriminate between these two predictions seems to us problematical. * Work supported by a grant from the Research Corporation of the Cottrell Foundation. lB. U. Felderhof and J. M. Deutch, J. Chern. Phys. 64, 4551 (1976). 2J. M. Deutch, B. U. Felderhof, and M. J. Saxton, J. Chern. Phys. 64, 4559 (1976). 3In addition to the references in FD see also D. Peak, H. L. Frisch, and J. W. Corbett, Radiat. Eff. 11, 149 (1971). 4Also, a related calculation is in K. Schroeder, Radiat. Eff. 5, 255 (1970). 5T. R. Waite, Phys. Rev. 107, 463 (1957). 6For a discussion of the limitations of the Waite-type analysis see the review article by R. M. Noyes, Frog. React. Kinet. 1, 129 (1961); also see Ref. 4 of FD. 1We assume the A and B molecules interact through a central potential and that p(r) is spher~cally symmetric. J. Chern. Phys., Vol. 65, No. 12, 15 December 1976 Downloaded 14 Aug 2011 to 129.123.67.59. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

Competitive Effect on the Rate of the Diffusion-Controlled Reaction A+B→C

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Competitive Effect on the Rate of the Diffusion-Controlled Reaction A+B→C

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