1934
J. M. Schurr and K. S. Schmitz
Orientation Constraints and Rotational Diffusion in Bimolecular Solution Kinetics. A Simplification J. Michael Schurr* Department of Chemistry, University of Washington, Seattle, Washington 98 195
and Kenneth S. Schmitz Department of Chemlstry, University of Missouri at Kansas City, Kansas City, Missouri 64 I10 (Received January 12, 1976)
A simple analytic expression that summarizes approximately the numerical conclusions of our previously reported investigation of the role of orientation constraints and rotational diffusion in bimolecular solution kinetics has been found. A generalization intended for the case of two species of mobile orientable spheres interacting with one another has been conjectured also. In agreement with Simmons it is found that the unusual data of Ivin et al. can be satisfactorily explained only by postulating (i) an intramolecular step following the bimolecular step, and (ii) a strong dependence of the intramolecular rate constant on the viscosity.
Introduction In a previous paper1 the role of orientation constraints and rotational diffusion in bimolecular solution kinetics was examined for the particular case of mobile orientable spheres, each bearing a single reactive site, interacting with stationary reactive sites on a planar surface. Owing to the complexity of the problem a simple solution in analytic form was not possible, so that considerable recourse to explicit numerical evaluations was necessary to complete the investigation. Consequently, the principal conclusions of that work are contained in the graphs and tables of computational data. However, stimulated by correspondence with Caldin? we have recently discovered a simple analytic expression for the effective bimolecular rate constant that incorporates (approximately) the principal conclusions from those numerical calculations. Moreover, the form of this simple expression permits an immediate, if also partly conjectural, generalization to the case of two mobile orientable spheres of comparable size, each bearing a small reactive surface site, interacting with one another. The relation of the present theory to recently proposed “quasichemical” models3-5 for incorporating the effects of rotational diffusion within the “encounter complex” will be discussed following the theoretical section.
Theory In the model system treated previously the geometrical requirements for reaction were: (i) the center of the mobile orientable sphere of radius R H must lie on the surface of a small target hemisphere of radius RT (RT < R H )centered at a distance R H directly above the reactive site on the planar surface, and (ii) the reactive site vector of the mobile sphere must at the same time lie within the reaction cone defined by the constraint (or half-cone) angle BO. For spheres satisfying both geometrical requirements the (intrinsic) bimolecular rate constant is hao (cm3 molecule-l s-l). For the subsequent generalization to the case of two mobile orientable spheres A and B interacting with each other, it is imagined that precisely the same requirements hold, only then the target hemisphere is centered at a distance R A out from the surface of the larger sphere B on a continuation of the line passing through both its center and reactive surface site. The Journal of Physical Chemistry7Voi. 80, No. 17, 1976
Three important conclusions derived from the original computaJions are the following: (i) (Uh) is essentially a linear function of solvent viscosity 11 with a slope nearly independent of the intrinsic rate constant k aO. (ii) In the diffusion-controlled limit the effect of rotational diffusion is to produce only a modest enhancement of the rate by a factor of 2 (1.65 to 2.23 depending on allowed reaction angle BO) above that calculated by multiplying the Smoluchowski rate k D o (27rR~Din that case) by the fraction !(eo) of all orientations (or solid angle) allowing reaction, and by the ratio RT/RH of the target radius RT to the hydrodynamic radius R H . A physical reasonability requirement led to the approximate relation B&H = RT between the allowed reaction angle BO and the hydrodynamic and target radii. (iii) In the low viscosjty limit 7 0 the biomolecular rate constant becomes just h = haOf(B0) = hao(l - COS B0)/2. These three conclusions may be summarized by the following simple approximate expression for the bimolecular rate constant
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It should be emphasized that eq 1 was not in any ordinary sense derived, but rather was inferred directly from the approximate conclusions (i-iii) above, which are sufficient to uniquely specify the effective bimolecular rate constant in this case. It should be remarked that the quantity 2 appearing in the denominator of eq 1 applies accurately (to within 4% or less) only to constraint angles 00 in the range from 0.3 to 0.15 radians (or 17 to 9’). However, it is valid to better than 20% accuracy over the range of BO values from 1to 0.04 radians (or 57 to 3O), as can be seen from the data in Table I of ref 1. The expression (1)reduces to h = ~ ~ ( B o ) ( R T / R Hin) ~the DO diffusion-controlled limit, defined by h a o ( R ~ / 2 R>>~ DO, ) in accord with (ii) above, and in the 7 0 limit gives h = kaof(Bo) in accord with (iii) above. Moreover, the quantity
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1 1 l/i = l + ha0f(Oo) ~ ~ ( O O ) ( R T / R H )
kDO
(2)
1935
Bimolecular Solution Kinetics is linear in 9 with a slope independent of kaO, in accord with (i) above. For the generalization to a reaction between two mobile orientable spheres of comparable size obeying the reaction requirements described above it is assumed that the substitutions R H = R A and kDo = ~ ~ R A ( DDAB ) (where R A is the radius of the smaller sphere) should be employed in eq 1and 2. Then, in the diffusion controlled-limit, there results
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&diiY
~ ~ ( O O ) ( R T I R A ) ~ T R+AD( D B )A
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in which the factor D A D B evidently accounts for the simultaneous translational diffusion of both species, and in which the additional factor of 2 has been incorporated to account for both the simultaneous rotational diffusion of both species and the departure from planar geometry, as proposed in eq 38 of our previous paper. The angle 00 here represents the allowed half-cone angle for reaction of the smaller sphere A with the localized surface site on the larger sphere B, when its center has diffused to the target hemisphere of radius RT which is centered at a distance R A directly above the reactive site. Converting to units of M-l s-l and utilizing directly the relations 00 = R T / R A and f ( 6 0 ) = (1- cos 00)/2 there results finally
wherein the Stokes-Einstein relation D = kTf6aqR has been employed, and where k , = (1- cos 00)k,~N~/2000. Thus, it is possible to estimate the reaction half-cone angle 00 directly from the observed slope of a (kM)-l vs. 9 plot, provided the molecular radii are known. Such estimates have already been presented for proton-transfer data by Burfoot and Caldin. Unfortunately, the discovery of an error by one of us (J.M.S.) in a communication to Caldin necessitates an upward revision of the calculated range of 00 values from 34-46” to 42-55’. It should also be noted that another interpretation of the same data involving a three step mechanism is in better overall agreement with the combined viscosity dependence and thermal activation data. The picture of the reaction process employed here is valid so long as the linear dimensions of the reactive surface sites are appreciably smaller than the radius R A of the smaller sphere. For surface sites of larger extent, the approximate relation 00 = R T / R H is no longer valid, and one must deal separately with f ( 0 0 ) and RTIRH in eq 1-3. For broad reactive surface regions of linear extent greater than the radius A of the smaller sphere the substitutions RT R A R B and f ( 0 o ) fA(oO)fB(OO), where fA(0O) and f B ( 8 0 ) are the fractions of reactive surface (or solid angle) of spheres A and B, respectively, might be employed. However, in such a case it should be born in mind that one has strayed rather far from the small reactive site picture for which the model was originally intended, and upon which these formulae are based, so that errors of factor of 2 or more are to be expected.
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Discussion Although eq 1 is a concise summary of our numerical conclusions for the case of mobile orientable spheres interacting with stationary sites on a plane, eq 3 and 4,intended to apply to the case of two species of mobile orientable spheres interacting with each other, contain a substantial element of conjecture. Solc and Stockmayer have proposed a general quasichemical model, based on their numerical solutions of
the diffusion-reaction problem for orientable spheres of nonuniform surface reactivity interacting with uniformly reactive spheres, that accounts for rotational diffusion by means of transitions within the encounter complex. Here, too, the generalization applicable to the case when both species possess nonuniform surface reactivities is based partly on conjecture. The primary advantage of our formulation is that it permits directly (and simply) an estimate of the constraint angle 00 from the observed slope of kM-l with respect to 7, even for the circumstance when the spheres are of dissimilar size. Calculations of the pertinent parameters in the Solc and Stockmayer model are simply unavailable for a range of ratios RA/RB when simultaneously the linear dimensions of the corresponding reactive surface areas are constrained to be equal, that is when CARA E B R Bwhere , E A and CBare the respective half-cone angles defining the reactive surface patches. Since such a constraint should be approximately valid whenever the molecules are large enough to justify the use of Stokes law for rotation, the lack of such data is a serious, though temporary, drawback to use of that model in some instances. The peculiar “homogenizing” of the surface reactivity of the smaller sphere in the limit of large radius ratio RB/RA, as reported by solc and Stockmayer, results from use of the constraint CA = CB,rather than the one suggested above. Indeed, if the linear dimensions (i.e., C B R Bof ) the site on the larger sphere B remain small compared to RA, then increasing the ratio RBIRAsufficiently should lead to a situation equivalent to that of mobile orientable spheres (A) interacting with localized sites (B) on a “plane”, for which there certainly is no “homogenization” of the reactivity of the smaller sphere. In the limit of large radius ratio RB/RA>> 1our eq 1above should be quite a satisfactory approximation. The similar, though simpler, quasichemical model proposed by Ivin et al.4 and by Simmons5is formally contained within the theory of Solc and Stockmayer,3 though Simmons5 seems unaware of that work, which, as noted above, not only provides a rational foundation for the quasichemical model, but also permits the numerical evaluation of model parameters from the molecular radii and constraint angles in, at least, some instances. However, because fewer parameters are contained in the model of Ivin et al. and of Simmons than in that of Solc and Stockmayer, it is not clear just how these are (rigorously) related to the molecular variables. Ivin et al. were in part motivated to formulate their quasichemical model by the observation7 of a negative temperature coefficient for the reaction between 2,4-dinitrophenol and tri-n-butylamine in chlorobenzene, despite a viscosity dependence indicating a substantial degree of diffusion control. Moreover, the observation of such a large degree of diffusion control (in the sense defined in ref 1)for a reaction proceeding with a rate constant kf = 4.0 X lo8 M-l s-l, more than a factor of 20 smaller than the Smoluchowski limit of M-l s-l, certainly suggests the presence of a significant angular constraint, or small target effect.6With an appropriate selection of parameters, R A = R B ,00 = 34”,k , = 2.5 X 109 M-1 S-1, and hao = 5 X 1 O l o M-l s-l, eq 4 above can account for the observed viscosity dependence of kf. However, to account for a negative temperature coefficient using eq 4 it is necessary to postulate the existence of one or more intermediates following the bimolecular (combined diffusion-reaction) step, thus
The quantity & M in eq 4 may be written in terms of the molar The Journal of Physical Chemistry, Vol. BO, No. 17, 1976
J. M. Schurr and K. S.Schmitz
1936
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rate constants k, and k~ as
4 a ( R ~4- R B ) ( D A D ~ ) N ~ / 1 0 0 0mons, though it apparently is capable of rationalizing these data, can only do so with the aid of two rather unusual assumptions regarding the elementary rate constants k,, k,' for "rotational" transitions within the encounter complex. These rate constants are assumed to be (a) strictly proportional to 7, and a t the same time, (b) to possess activation energies that are appreciably different from that of the viscosity itself. If we are allowed to make here the single assumption that k c in reaction 5 is inversely proportional to the viscosity, that is, k c 0~ 7-l, then the reaction scheme 5 and its corresponding eq (7) 8 also suffice to qualitatively explain the experimental data. Upon rewriting eq 8 in the form where kd kdo(l - cos 00)/2 is the dissociation rate in the limit of small viscosity 7 0. The usual steady-state analysis for the complex (AB) leads to an expression for the overall effective forward rate constant it is apparent that kf-l will be linear in 7 with an intercept ka-l. Furthermore, writing eq 8 in the form
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which is analogous to eq 58 and 60 of ref 1. When (a) the degree of diffusion-control of the overall reaction is small, that is when ka > k c , corresponding to a preequilibration of the bimolecular process, then
which may exhibit a negative temperature coefficient provided the bimolecular step is sufficiently exothermic. Although the formulation of eq 5-9 can provide under appropriate conditions for (i) a linear dependence of kf-l on 7, (ii) a substantial degree of diffusion control with an effective rate constant still well-below the Smoluchowski value, and (iii) a negative temperature coefficient, it cannot account for all of these features under the same set of conditions. The difficulty, as noted by Ivin et al. in their original report, is that the observed degree of diffusion control is so large that the temperature coefficient ought to be essentially that of the reciprocal viscosity 7-l. Indeed, from their Figure 2 it is apparent that kf in chlorobenzene at 22 "C has a value only about 15% of the (7 0) limiting value k,(l kd/kC)-l, which implies that the forward reaction is already proceeding with 85% of its theoretical maximum rate constant WkD at that viscosity. Using their value (9075 kJ mol-l deg-l) for the viscosity activation energy, it is readily calculated that 7 is increased by a factor of 1.72 when the temperature is lowered from 22 to -15.6 "C. Evidently, the theoretical maximum rate constant WkD is correspondingly decreased by a factor of (1.72)-l = 0.58 over the same temperature range. Thus, kf ought to decrease substantially as the temperature is lowered from 22 to -15.6 "C, regardless of the degree of diffusion control at the lower temperature, in contrast to the experimental data. Thus, this simple model with k c independent of 7 fails to account for the experimental facts. I t should be pointed out that the model discussed by Sim-
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The Journal of Physical Chemistry, Vol. 80, No. 17, 1976
(11) it is clear that a negative temperature coefficient results whenever the following three conditions hold: (i) the bimolecular step (with equilibrium constant ka/kd) is sufficiently exothermic; (ii) kd >> k c corresponding to a preequilibration of the bimolecular step, and (iii) WkD Z kc[k,/(kd + k c ) ] ,corresponding to a degree of diffusion-control of the overall reaction somewhat below the maximum value of 1.0 (achieved when kc[ka/(kd+ k c ) ] a). The data of Ivin et al. for the proton-transfer reaction between 2,4-dinitrophenol and tri-n-butylamine evidently require an intermediate unimolecular reaction step with a rate constant proportional to the reciprocal viscosity. A possible cause of such unusual behavior is the volume change expected for such a dual ionization reaction. The resulting solvent flow requirement may be the source of the 7-l dependence. If this interpretation is valid, then in view of eq 10, one can obtain only a lower limit for the constraint angle 8 0 from the plot of (observed) kf-l vs. 7. Thus, in this case 34" would be just a lower limit to the actual value of 00. A less likely, but still possible, explanation of the unusual temperature dependence of k f would be a negative temperature coefficient for the constraint angle 8 0 itself.
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References and Notes (1) (2) (3) (4) (5) (6) (7)
(8)
K. S. Schmitz and J. M. Schurr, J. Phys. Chem., 76, 534 (1972). E. F. Caldin, personal communications. K. Solc and W.Stockmayer, Int. J. Chem. Kinet., 5, 733 (1973). K. J. Ivin, J. J. McGarvey, E. L. Simmons, and R. Small, J. Chern. SOC.,Faraday Trans. 1, 69, 1016 (1973). E. L. Simmons, 2.Phys. Chem. (Frankfurtarn Main), 96, 47 (1975). C. G. Burfoot and E. F. Caidin, J. Chern. SOC., Faraday Trans. 1, 72, 963 (1976). K. J. ivin, J. J. McGarvey, E. L. Simmons, and R. Small, Trans. Faraday Soc., 67, 104 (1971). J. M. Schurr, Biophys. J., I O , 700 (1970).
