J. Phys.
5052
Chem. 1982, 86, 5052-5067
TABLE IV: Summary of the Thermodynamic Variables for the Vapor-Phase Reaction CH,CRO t HCI CH .CR O . . .H C1 +
exptl i R = CH,)
variable
A E , cal.mol-' A H , cal.mol-'
-4800 i 5 0 0 - 5 4 0 0 k 500 0.14 t 0.05" t 1180 i. 220 -21.7 + 1.8
K,(d03 K j , atm-' A G , cal.mol-' AS,cal,mol-'.K I
selected from usual rangeb
I
cm-'
lla
computed
1
6.22 5.56 3.48 0.46 0.10
600 1000
L't "b
see normal m o d e sets below
eu,mol-'
25 35 100
$,
1'0
-4900 -5600
so
W ,
vibr mode
theor ( R = H)
63 101 180 654 681
4.40 3.46 2.36 0.38 0.34
AS"( vib)c
+ 15.82
+ 10.94
A&"'( rotjc
AS"(trans)c
--2.50 -35.33
-35.33
AS"(total jc
--22.01
-26.89
-2.50
stitution of a H atom for a methyl group in acetone could at most raise the observed frequency only 50 cm-lS2l In addition, the theoretical frequencies refer to harmonic motions, whereas real oscillators are anharmonic. This idealization is not usually of noticeable consequence, but when the frequencies are low enough so that the corresponding oscillators can be thermally excited, the harmonic approximation is less valid in thermodynamic calculations.
Table IV includes an entry of a set of new vibrational frequencies for the CH3CH0.-HC1 complex selected from the ranges given by Thomas.22 The subtle difference between these and the computed frequencies results in a gain of about 5 ewmol-' for the complex, which is sufficiently large to bring the theoretical and experimental A S values into agreement. It is plausible to suggest that a second factor might come into play to account for the difference between the ASo value for CH3CHO-HC1 and the experimental A S value for (CH3)2CO-.HC1. The additional CH3group would be expected to increase the reduced mass of the skeletal framework, which formed the basis of the theoretical frequencies. Then, from Hooke's Law, lower frequencies (and thus larger AS0(vib) values) would be expected for (CH3)2CO--HC1if the force constants for the normal-mode motion could be transferred directly. However, if the motions shown in Figure 4 for the 63- and 101-cm-' modes are separated into opposing components, the reduced mass effect lowers these two values to only about 56 and 97 cm-l, respectively. The corresponding entropy change of 0.29 eusmol-' is insufficient by itself to account for the 5 ew mol-l discrepancy between computed and measured entropies of complex formation. Thus it appears more reasonable to conclude that the low vibrational frequencies due to complex formation are probably overestimated, and thereby do not contribute sufficiently to the computed ASo to bring this value into line with the experimentally based one. Acknowledgment. The Chemistry Department of Youngstown State University most generously supplied the equipment and chemicals used in the experimental work. The support of the Y.S.U. Computer Center is also gratefully acknowledged. Finally, the authors also thank Dr. J. A. Pople for his assistance in this project.
~
(21) L. J. Bellamy, "The Infra-red Spectra of Complex Molecules", Methuen, London, 1954, p 155.
(22) R. K. Thomas, Proc. R SOC.London, Ser. A, 325, 133 (1971).
Nonequllibrium Statistical Thermodynamics and the Effect of Diffusion on Chemical Reaction Rates Joel Kelzer Chemistry Department, Universify of California, Davls, California 95616 (Received: March 2, 1982; In Final Form: July 23, 1982)
The use of nonequilibrium pair correlation functions for calculatingthe effect of diffusion on the rate constants of chemical reactions is described. The customary theory of this effect is based on the Smoluchowski equation which often provides a bad or divergent approximation to the pair correlation function. Instead, recent developments in the theory of nonequilibrium statistical thermodynamics are used as a basis for calculating pair correlation functions. Because of the generality of the theory, it is possible to systematically describe the effect of a finite lifetime for reactive species, nonreactive interactions between reactants, solvent effects, and long-ranged reactive interactions such as occur in fluorescence quenching. The method of calculation is illustrated for several common reaction mechanisms occurring under steady-state conditions and results are compared to the Smoluchowski theory. In three dimensions the Smoluchowski theory is the limiting low-density result of the present theory. In one or two dimensions the present theory-unlike the Smoluchowski theory-is convergent, and a discussion of the effect of diffusion on rapid reactions in membranes is given. When a bimolecular reaction occurs in solution, two effects conspire to determine how rapid the reaction will be. Foremost of these is the intrinsic rate a t which the reaction occurs. The intrinsic rate is determined by the
detailed molecular mechanism of the reaction and involves a knowledge of such things as the distribution of electrons as they rearrangeduring the reaction. The intrinsic rate is a molecular-level quantity, much like a reaction scat-
0022-3654/82/2086-5052$01.25/0@ 1982 American Chemical Society
Effect of Diffusion on Chemical Reaction Rates
tering cross section, and requires a quantum or semiclassical theory for its evaluation.’ The other effect which determines bimolecular reaction rates is statistical and accounts for the probability of the existence of adjacent reactive pairs in the solution. The number of adjacent reactive pairs is determined by several physical factors, including the solvent-reactant interactions, the nonreactive portion of the interaction between reactants, the rate at which reactants approach one another in solution, as well as how the solution was prepared. In the language of statistical mechanics2 this probability factor is the pair distribution function for the reactive molecules. In a previous letter3 a sketch was given of how the theory of nonequilibrium statistical thermodynamics4could be applied to the calculation of molecular pair distribution functions and thence used to calculate chemical rate constants. It is the purpose of this paper to expand on the developments outlined in that note.
The Journal of Physical Chemlstry, Vol. 86, No. 26, 1982 5053
a p B / a t = DV’pB
(1)
with pB(R,t) = 0, = PB, D’= D A + DB is the relative diffusion constant, and R is the radial distance of closest approach of an A molecule and a B molecule. The stationary-state distribution develops as t m and is found to be
-
P B ” ( ~ )= P B ( ~- R / r )
(2)
At steady state the diffusion flux at the distance of closest approach is proportional to the reaction rate, so that 4 ~ R ~ D ’ ( d p ~= ~ ~kobsdpB /dr)~
(3)
Substituting eq 2 into eq 3 gives the Smoluchowski version of the diffusion-controlled rate constant
kObsd = 4TD’R
(4)
I. Review of t h e Smoluchowski Theory The first effort to understand the effect of diffusion on solution-phase reactions appears as an afterthought in Smoluchowski’spaper on the kinetics of the coagulation of colloid^.^ There is an obvious similarity between the coagulation problem and dimerization reactions, although there are also obvious differences. For example, in fluorescence quenching the excited-state donor molecule may have a fluorescence lifetime comparable to the quenching lifetime.6 Nonetheless, Smoluchowski’sidea was simple and modeled the translational diffusion aspect in a clever way. It has come to be the method used by chemists to interpret the rates of rapid reactions in solution.’ Smoluchowski’s basic idea is easy to illustrate for the bimolecular reaction A + B product. Attention is focused on the average spatial distribution of B molecules-which are in great excess-around a central A molecule. While both A and B will be depleted by the reaction as time goes on, the B’s will be depleted more in the immediate neigborhood of an A-at least if A is continuously added to solution so as to keep its concentration fixed. The reason for this is that it is only adjacent to an A molecule that the B molecules are removed by reaction. When an A molecule is added to replace one which reacted, it will be surrounded by a distribution of B molecules which is depleted in the immediate vicinity of the A. Just how depleted the concentration of B will be depends on how rapidly the B molecules can diffuse toward the central A. Thus, under these steady-state conditions an A molecule acts like a sink for the diffusive motion of the B molecules. Smoluchowski made this picture quantitative by using Fick’s law as a model for the relative diffusion of A and B. The average stationary spatial distribution of B around an A was determined by solving the diffusion equation with a sink boundary condition at A and assuming a bulk concentration of B at great distances from A. The basic equation to be solved is, then
With this modification, expression 5 for the spatial distribution of B molecules remains the same. When ko >> 47rD‘R (rapid reaction), these expressions reduce to the Smoluchowski result given by eq 2 and 4 . These results are well-known and some of the strengths and weakness of the theory are described in Noyes’ classic re vie^.^ Since Smoluchowski’s work, the theory has been embellished in a number of ways. For example, in the context of ionic solutions Debye included the effect of molecular interactions by using the full Smoluchowski equation; lo short-ranged potentials of mean force and position-dependentdiffusion “constants” have been used; l1 attempts have been made to include fluorescent lifetime effects; and the theory has been extended to treat interactions among the sinks.13 However, these modifications have done little to alter basic limitations of the theory. One serious limitation is that the same theoretical framework is inapplicable for reactions confined to membranes, such as are commonly encountered in biophysics. Indeed, when one attempts to carry out the Smoluchowski
(1) H. S. Johnston, “Gas Phase Reaction Rate Theory”,Ronald Press, New York, 1961. (2) T. L. Hill, ‘Statistical Mecahnica”;McGraw-Hill,New York, 1956, Chapter 6. (3) J. Keizer, J . Phys. Chem., 85, 940-1 (1981). (4) J. Keizer in “Pattern Formation and Pattern Recognition”, H. Haken, Ed., Springer-Verlag, West Berlin, 1979, pp 266-77. (5) M. V. Smoluchowski, 2. Phys. Chem., 92,129-68 (1917); see also S. Chandresekhar, Rev. Mod. Phys., 15, 1-89 (1943). (6) J. R. Lakowicz and G . Weber, Biochemistry, 12, 4161-70 (1973). (7) R. E. Weston and H. A. Schwarz, ‘Chemical Kinetics”, PrenticeHall, Englewood Cliffs, NJ, 1972, Chapter 6.
(8) F. C. Collins and G. E. Kimball, J. Colloid Sci., 4,425-39 (1949). (9) R. M. Noyes, Prog. React. Kinet., 1, 129-60 (1961). (10) P. Debye, Trans. Electrochem. SOC.,82, 265-72 (1942). (11) L. Monchick, J. Chem. Phys., 24, 381-5 (1956); S. Northrup and J. T. Hynes, ibid., 71, 871-83, 884-93 (1979). (12) I. Z. Steinberg and E. Katchalski, J . Chem. Phys., 48, 2404-10 (1968). (13) H. L. Frisch and F. C. Collins, J . Chem. Phys., 20, 1797-805 (1952); B. U. Felderhof and J . M. Deutch, ibid., 64, 4551-8 (1976); M. Bixon and R. Zwanzig, ibid., 75, 2334 (1981); T. Kirkpatrick, ibid., 76, 4255 (1982); B. U. Felderhof, J. M. Deutch, and U. M. Titulear, ibid., 76, 4178 (1982).
-
and allows eq 2 to be rewritten as pBas
= jig[l - k o b s d / ( 4 ~ D ’ r ) ]
(5)
It is the sink boundary condition that forces pBsS(r)to vanish at R = r in eq 2. It was recognized by Collins8that for slower reactions the A molecule would act like an imperfect sink and that this could be accounted for by a flux boundary condition at R , namely
4 ~ R ’ D ’ ( a p ~ ( r , t ) / d=r kopB(R,t) )~ where ko is an intrinsic rate constant for the reaction, i.e., the rate constant that one might calculate based on a gas-phase collision model. This modification leads to a new expression for kobad,namely
kobsd= ~ T D ’ R K ~ / ( ~ T+ D ko) ’R
(6)
SO54
The Journal of Physical Chemistry, Vol. 86, No. 26, 7982
Keizer
of view is convenient, it is not necessary and the extension program in two dimensions, it turns out to be impossible of the present theory to include, for example, rotational to satisfy the boundary conditions because of the logarithmic divergence of the solution to Laplace’s e q ~ a t i 0 n . l ~ motions is easily accomplished.20 As eq 7 indicates, the pair distribution function will This has led, among other things, to the mistaken notion generally depend on the time. Nonetheless, as the two that diffusion in two dimensions is too slow to produce a positions in space become well separated, A and B molestationary distribution. Although attempts have been cules act independently. Thus, as Ir - r’l gets large made to remove this two-dimensional di~ergence,’~ they rely on the three-dimensional theory and are ad hoc. PAB(2’( r ,r’,t) = PA‘l’( r ,t) P B qr’,t ) (8) The easiest way out of the difficulties with the Smoluchowski theory has been evident for some times, namely, where the singlet probability densities are PA(l)(r,t)= to keep Smoluchowski’s essential physical idea but to PA(r,t)/NA(t),pB‘’)(r,t) = PB(r,t)/NB(t),with the P’S defmed develop a true calculation of the physical quantities of as the average number densities. Consequently, it is interest. According to Smoluchowski, one should focus natural to define the pair correlation function gAB by2J9 upon a central reactant A and look for the average spatial PAB(r,r’,t) = PA(r,t) PBW,t) gAB(r,r’,t) (9) distribution of B molecules in the neighborhood. This For isotropic fluids interacting through a pair potential, quantity is well-known in the equilibrium theory of fluids where it is called the radial distribution function. Recthe pair correlation function depends on only the relative distance Ir - r’l and is then identical with the radial disognizing this, Waite16 introduced pair distribution functribution function. This implies that the number contions into the theory of diffusion-controlled reactions. In centration of B molecules at a distance F from a central that and more recent work1G1sthe general notion of kinetic A molecule would be given, on the average, by equations satisfied by the pair distribution functions has been discussed. Waite’s work,16 in particular, makes it P ( F , t ) = P&!m(F,t) (10) clear that the Smoluchowskitheory can be interpreted as Consequently, for molecules of spherical symmetry, it is an attempt to devise a kinetic equation for the pair disthe nonequilibrium radial distribution function which one tribution function. should calculate in order to fulfill Smoluchowski’s physical Actually what is needed is a systematic way of calcupicture of diffusion effects on reaction rates. lating the spatial distribution of pairs of molecules in There exists another macroscopic quantity which is systems that are far from equilibrium. In general, that is closely related to pair correlations, the density-density an extremely difficult problem, although a formally exact correlation function. This function is the average of the theory-like that used for simple fluids2-could in prinproduct of deviations in number density from the average ciple be devised. Fortunately a simplier approach to values. Thus, writing 6pi(r,t) = pi(r,t) - Pi(r,t),where pi nonequilibrium problems like this has been developed in is the number density for molecule of type i and pi is its recent years.4 It is based on the theory of nonequilibrium average value, the density-density correlation function is statistical thermodynamics and is described in the foldefined by21 lowing sections. (pi(r,t) +j(r’,t)) (11) 11. Pair Correlation Functions Most of the measurable properties of fluids can be The angular brackets in this equation indicate that the conveniently discussed in terms of the average properties average is taken over an ensemble appropriate to the initial of small clusters of molecules. For example, all the therpreparation of the system. Since attention is being limited modynamic properties of a simple high-temperature fluid to nonquantum fluids, the density-density correlation like argon can be gotten from a knowledge of the equifunction is given in terms of the pair correlation function22 librium pair distribution f u n c t i ~ n p(2)(r,r‘). , ~ ? ~ ~ In general, by the usual relationship the pair distribution function for identical particles is (BPi(r,t) Gpj(r’,t)) = proportional to the probability density of finding a molPi(r,t)6ij6(r - r’) + Pi(r,t) Pj(r’,t)kAB(r,r’,t) - 1) (12) ecule at. r and simultaneously another one at r’. For two solutes in solution we will be concerned here primarily with Indeed, it is the density fluctuations which are the dominant cause of X-ray scattering in fluids,23and it is by using the equilibrium form of eq 12 that the equilibrium radial distribution is obtained from X-ray scattering experiments. Equation 7 is the probability of finding the center of mass The density-density correlation function is an average of an A molecule in r to r d r and a B molecule in r‘ to macroscopic quantity which involves fluctuations in the r’ + dr‘, averaged over their internal degrees of freedom number densities of molecules. Since molecule numbers and the position of all the solvent molecules. In eq 7 pm are extensive variable, their fluctuations can be calculated is the pair distribution function, d r and dr’ represent on the basis of nonequilibrium statistical thermodynamic^.^ infinitesimal volume elements, and NA and N B are the total It is this approach which forms the basis of the following number of A and B molecules in solution at time t. (When calculations. A = B, NB is replaced by NA - 1.) By averaging over the rotational and other internal motions of the molecules, we 111. Spatially Nonlocal Theory of Chemical are effectively replacing molecules of finite extent with Reactions point molecules of spherical symmetry. While this point In dilute gases the spatial dependence of the chemical reactivity of molecules can be satisfactorily accounted for
+
(14) K. Razi Naqvi, Chem. Phys. Lett., 28,28C-3 (1974);C . A. Emeis and P. L. Fehder, J . Am. Chem. SOC., 92, 2246-52 (1970). (15) C. S.Owen, J . Chem. Phys., 62, 3204-7 (1975). (16) T. R. Waite, Phys. Reu., 107, 463-70 (1957). (17) T. R, Waite, J.Chem. Phys., 28,103-6 (1958); T. R. Waite, ibid., 32, 21-3 (1960). (18) G. Wilemski and M. Fixman, J. Chem. Phys., 58,4009-19 (1973). (19) D. McQuarrie, “Statistical Mechanics”, Harper and Row, New York, 1976, Chapter 13.
(20) U. Steiger and J. Keizer, J. Chem. Phys., 77, 777-88 (1982). (21) L. D. Landau and E. M. Lifshitz, “Statistical Physics”, 2nd ed., Pergamon Press, Elmsford, NY, 1969, p 360. (22) J. P. Boon and S.Yip, ’Molecular Hydrodynamics”,McGraw-Hill, New York, 1980, p 24. (23) B. J. Berne and R. Percora, ‘Dynamic Light Scattering”,Wiley, New York, 1976.
Effect of Diffusion on Chemical Reaction Rates
The Journal of Physical Chemistry, Vol. 86, No. 26, 1982 5055
in terms of collision cross sections.’ This is due to the fact that in dilute gases the time between collisions is much larger than the duration of a collision. This means that the rate of a reaction factorizes into the number of collisions per second times the fraction of collisions which cause reaction. This simple separation no longer is possible in condensed phases, because solvent interactions keep reactants in close proximity for times which can exceed the duration of a collision. This sort of caging effect24is quite general in liquids or solids and means that the spatial dependence of chemical reactivity becomes important. An especially striking example of this is the spatial dependence of energy transfer through dipolar mechanisms.I2 The transition rate for energy transfer has been calculated by F o r ~ t e to r ~have ~ the form (1/ d ( r o / r ) 6
(13)
where ro depends on the angular orientation of the dipoles and 7 is the lifetime of fluorescence. Since ro can be the order of 25 A, this effect cannot be subsumed in a rapidtime-scale collision term in solution. To describe these spatially nonlocal reactivity effects in solution we introduce here a spatially nonlocal kinetic description of reaction^.^ In this description it is assumed that the intrinsic rate of reaction can be accounted for by a spatially dependent reactivity function, ko(r,r’). This function gives the spatially dependent bimolecular reaction rate per unit volume according to the following definition V+(r,t) = lkO(r,r’)pAB(r,r’,t)dr’
(14)
-
where, for illustration, the bimolecular reaction is taken to be A B products. The definition in eq 14 splits the reaction rate up into two terms: ko(r,r’),the intrinsic reactivity; and pAB(r,r’,t),the joint statistical distribution of A, B pairs. The exact meaning of ko(r,r’) can be seen by substituting eq 9 into eq 14. This gives
+
V+W = JkO(r,r’) pB(r’,t) gAB(r,r’,t) p A ( r , t ) dr’
(15)
Since ~ B ~ AisBthe average density of B a t r’ given that there is an A a t r, it is clear that ko(r,r?dr’ is the reaction rate constant for A, B pairs with A at r and B at r’. Thus, the reactivity function, ko(r,r‘),is the rate-constant density for these pairs. To obtain the reactivity function it is necessary to calculate the rate a t which a molecular pair would react if A were located a t r and B at r’. For the Forster mechanism of fluorescence quenching, this is given by expression 13. For collisional-type mechanisms, the calculation of ko would involve an average over momenta and internal degrees of freedom and so would be dependent on the temperature, T. For example, for atomic recombination one might expect the reactivity function to depend on the overlap of electron density and so to have a form like2s k o ( r ) = kO(T)e-‘iro (16) where ro is the order of a few angstroms. The simplest model one might choose for the reactivity function would have a hard-sphere form (24) See ref 9, pp 148-9; for a recent simulation of caging effects, see J. T. Hynes, R. Kapral, and G. M. Torrie, J. Chem. Phys., 72, 117 (1980) or M. Schell, R. Kapral, and R. Cukier, ibid., 75, 5879 (1981). (25) T. Forster, Ann. Phys., 2, 55-75 (1948). (26) N. J. Turro, ‘Modern Molecular Photochemistry”, W. A. Benjamin, New York, 1978, pp 305-9. (27) D. Nesbitt and J. T. Hynes, submitted to J . Chem. Phys.,; for a treatment at the level of the Boltzmann equation, see M. Schell and R. Kapral, ibid., 75, 915 (1981). (28) M. Medina-Noyola and J. Keizer, Physica A Amsterdam, 107, 437-63 (1981).
ko(r) = k06(r - R)/(4?rr2)
(17)
which involves reaction only between pairs a t a distance of closest approach R. As is shown in later sections this reactivity function will give rise to results that agree with the Smoluchowski approach at low d e n ~ i t i e s . Conse~ quently, we shall call the reactivity function in eq 17 the “Smoluchowski reactivity”. As has been noted previously, our considerations here are based on an average over internal degrees of freedom. In other words, it is assumed that the internal degrees of freedom relax more rapidly than the spatial distributions and so they appear only as average parameters which affect the dynamics of the spatial distribution functions. This is the simplest assumption that can be made but is clearly not necessary. If there are slow rotational motions or excitation of other internal degrees of freedom,n then their explicit dependence in the reactivity function as well as the time dependences of the densities of these variables must also be considered. Results for the coupling of rotational and translational diffusion using the ideas have already been obtaineda20 With the neglect of internal degrees of freedom, the only other motion which contributes to the change in density of A and B is translational diffusion. Combining this effect with the rate expression for the bimolecular reaction in eq 15 gives the nonlocal kinetic equations dpi(r,t)/dt = uiJko(r,r’) gAB(r,r’,t)pB(r’,t) dr’ pA(r,t)+ D?V2zi (18) where vi is the signed stoichiometric coefficient of i = A or B and zi = exp(pi - p?)/kBT) is the activity of i = A or B evaluated a t the average local densities. For ease of writing the back-reaction has been left out of eq 18, although it will be included in several examples in the following ~ections.~ The form for the diffusion term is based on the theory of nonequilibrium statistical thermodynamics and is appropriate even in dense solutions.2s The transport coefficient is not the usual density-based diffusion constant Di but is related to it byz8
D? = Di(dpi/dZi)T (19) In dilute solutions Zi = pi, and one recovers the usual Fick’s law diffusion term. The form of the transport equation 18 is reminiscent of the Enskog transport equation29which was devised to describe the effects of binary collisions on space and momentum relaxation in dense fluids. Actually eq 18 is somewhat more complicated than the Enskog equation in that it involves the actual two-body correlation function rather than the local equilibrium form of gAB. Thus, eq 18 expresses something more like the intrinsic coupling of the one-body and two-body dynamics that appears in the first of the BBGKY hierarchy of kinetic equations.30 Finally, it should be emphasized that eq 18 involves the average values of the densities of A and B. Just which “average” will be discussed in detail in the following sections. While eq 18 is the basis of our treatment of diffusion effects, we will be primarily concerned with the analysis of uniform systems in which p A and pB are independent of position. In that case eq 18 reduces to dpi/dt = ~ik’~~~(t)pApB (20) (29) P. Resibois and M. de Lenner, “Classical Kinetic Theory of Fluids”, Wiley-Interscience, New York, 1977, Chapter VI. (30) G. E. Uhlenbeck and G. W. Ford, “Lectures in Statistical Mechanics”, American Mathematical Society, Providence, RI, 1963, Chapter VII.
5058
Keizer
The Journal of Physical Chemistry, Vol. 86, No. 26, 1982
where the time-dependent rate “constant” is defined as
to 6pA and 6pB. The purely randop Gaussian contribution
kobed(t) = l k o ( r , r ’ ) gAB(r,r’,t)dr’
to the time derivative is written fi(r,t) and its correlation function is28331932
(21)
Clearly kobsd(t)will be independent of r in a uniform solution as long as no external fields are present. Because kohd(t) depends on time, it will generally depend on the initial conditions of the experiment and, consequently, it is difficult to give a comprehensive analysis of its form. Thus, we specialize our attention in the following to conditions of steady state. In order to achieve a true steady state, one must add molecules of A and B to the solution at a rate, e.g., viK, to compensate for their loss (or gain) by reaction. Thus dpi/dt = vikobsd(t)pdB- V&
(22)
After a transient period a steady state will be achieved in which kobsdpAsspBss= K
(23)
At steady state the rate constant, kobsd,becomes independent of time and is given by kobsd l k O ( r , r ’ )gm(r,r’) dr
(24)
with gm the steady-state pair correlation function. Clearly koM will be the experimentally measured rate constant in a steady-state experiment. To calculate the observed steady-state rate constant, it is necessary to obtain an expression for the steady-state pair correlation function.
IV. Steady-State Correlation Functions The theory of nonequilibrium statistical thermodynami c ~ ~is used ~ ~to ~calculate , ~ ~ two-point - ~ ~correlation functions at steady state. In doing so, one relies upon eq 12 to make the connection between the density correlation function and the radial distribution function. To obtain the density correlation function it is necessary to examine the time dependence of local fluctuations in the density around their average values. According to the nonlocal kinetic desorption in the previous section, near steady state the average value of the local densities will satisfy equations of the form
(fi(r,t) fjW,t)) = (
v
1 1
~
k
~ - 26i,D/’~:sV,2)6(r ~ ~ ~ p ~- r’) ~ 6 (~t - pt? (27) ~ ~
again, corresponding to the single elementary chemical reaction and diffusion. When other elementary processes change the density of A and B (i.e., if the back-reaction is not negligible) or if other variables are changing (i.e., the density of products), then these are to be added to eq 25-27. To illustrate how these equations are solved for the density correlation function, the simplest reaction scheme A + A products is treated in detail. For this dimerization reaction p A is the only density, UA = -2, and, to begin with, it will be assumed that the solution is ideal. Thus, zA = pA. Equations 25-27 can then be written
-
dPA(r,t)/dt = -2kobsdpA2+ DV2-pA(r,t) + 2K (28)
a 6pA(r,t)/ a t
+
= -4kobsd~AB88~A(r,t)Dv26pA(r,t) + f(r,t)
(29) (f(r,t) f(r’,t?) = (4kobsdpAsS2 - 2DpAssv:)6(r
-
r’) 6(t - t ? (30)
where D is the diffusion constant of A. To solve eq 29 it is necessary to have the steady-state value, pAss. This is given by the solution to eq 28 as t a. Since we have assumed a uniform system on the average this gives
-
= (K/kobsd)lP
(31)
To calculate the density correlation function, one must solve eq 29 for 6pA(r,t) and calculate the average of its product at two different spatial points with the help of eq 30. This is most easily done by first Fourier transforming3*34eq 29 and 30, a legitimate step since a uniform system becomes infinite in the thermodynamic limit. The time-space Fourier Jransform is denoted by 2 and the space transform by f , where 1 +2(k,w) = 2a B(k,t)eiutdt =
xm
+
3pi(r,t)/et = vikobsdpA(r,t)PB(r,t) D/’V2Zi(r,t) - viK (25) This equation is in the canonical form4 corresponding to the occurrence of the elementary chemical reaction A + B product and the elementary process of diffusion by the solute i. Thus, the Rrinciples of mechanistic nonequilibrium statistical t h e r m ~ d y n a m i c s ~can l - ~be ~ used to examine density fluctuations. According to that theory the average densities in eq 25 must be understood as the averages which develop in an ensemble in which the initial densities were precisely known to be pt(r). The averages are thus called conditional (on the initial values). Furthermore, close by the steady state deviations from the conditional average, 6pi(r,t) = pi(r,t) - pi(r,t), satisfy the kinetic equations
-
a6pi(r,t)/at = vikobsdpAgs6pB(r,t) + vikobsdpBs86pA(r,t)+ DyV26zi(r,t)+ fi(r,t) (26) where 6zi(r,t) is the variation of the activity with respect (31)J. Keizer, J. Chem. Phys., 63,390-403,5037-43 (1975). (32)J. Keizer, J. Chem. Phys., 64, 1679-87 (1976). (33)J. Keizer, J. Chem. Phys., 64,4466-74 (1976). (34)R. F. Fox, Phys. Rep., 48,179-283 (1978),especially section 1.7.
Applying the space-time transform to eq 29 once and twice to eq 30 gives i ~ 6 b A= -(4kobsdpss+ Dk2)6bA + f
E
(f(k,w)f W , 4 ) = 2pss(2kobsdpss+ Dk2)6(k E
y ( k ) 6(k
H(k)GbA + f
(33)
+ k’)6(o’ + w ) / ( 2 ~ ) ~
+ k’) 6(w + w ’ ) / ( 2 ~ ) ~
(34)
where k = Ikl. Solving eq 33 for 8PA and employing eq 34 then gives (6bA(k,w) abA(k’,w’) ) =
2pss(2kobsdpss+ Dk2)6(k + k’) 6(w + w’) (35) (zn)4(iw 4kobsdpss + ~ k 2 ) ( - i ~4kob5dpss + DP)
+
+
The time transform is most easily carried out first. Equations 29 and 30 describe a scalar stationary, Gaussian, Markov process so that the time correlation function is certainly a single e x p ~ n e n t i a l . ~ ~ Using * ~ ~the , ~ ~inverse (35)J. Keizer, J. Chem. Phys., 65,4431-44 (1976). (36)Reference 21,pp 37C-84.
~
Effect of Diffusion on Chemical Reaction Rates
The Journal of Physical Chemlstty, Vol. 86, No. 26, 7982 5057
Fourier time transform gives
exp[-(4kobsdpss+ Dk2)lt - t l ] (36) Actually we are interested only in the correlations at equal times, since this correspondsto the steady-state correlation function. Thus
Equation 37 is an example of a general result, called the generalized fluctuation dissipation theorem,4~~~ that is known to describe the statistics of molecular systems at steady state in the nonequilibrium thermodynamic theory. To exhibit the general relationship, the left-hand side of eq 37 is written as a(k,k’), the relaxation rate in eq 33 is written as H(k,k’) = H ( k ) 6(k + k’), and the strength of the random term in eq 34 is written as y(k,k’) = y ( k ) 6(k k ’ ) / ( 2 ~ ) ~It. is simple to see that a(k,k’) solves the following equation
+
There are two differences between this expression and the expression for the radial distribution function obtained by using the Smoluchowski theory. For identical molecules, the Smoluchowski result in eq 5 is modified by a factor of 2 and becomesg gAASmol(lr - r’l) = 1 -
kobsd
27rDlr - r’l
(43)
The Smoluchowskiexpression is missing the effect of the lifetime in solution of the A molecule, since the term e-[’ does not appear; it also involves D ’ = 2 0 , rather than simply the diffusion constant D. Thus, the Smoluchowski theory takes into account the identity of the molecules in a way which differs from the present calculation by a factor of 2. This difference does not appear in our calculation when nonidentical molecules A and B are treated. We infer from this that the usual Smoluchowski theory does not correctly account for the indistinguishability of identical molecules. Another defect of the Smoluchowski radial distribution function can be seen by examining eq 41. Indeed, if the A-A term of that equation is integrated over the volume of the container for both r and r’, one obtains
1 l m [ H ( k , k ” )a(k”,k’) + a(k,k”) ZP(k”,k’)] dk” = -dk,k’) (38) where T means to interchange the two sets of variables on which H depends. The function u is the Fourier transform of the density-density correlation f u n ~ t i o n .For ~ ~this ~~~ example, eq 38 leads to
-dk)
a(k,k’) = -6(k + k ’ ) / ( 2 ~ ) ~ 2H(k)
(39)
which, using the definitions of H(k) and y ( k ) in eq 33 and 34, is seen to be equivalent to eq 37. Equation 38 is a quite general result of the nonequilibrium statistical thermodynamic theory and holds even when u is a matrix of density correlations, ( 6&(k)6Jj(k’)). We make use of this in applications which follow. Having obtained the Fourier transform of the densitydensity correlation function, we simply invert it using standard integrals.% The desired result is then obtained: kobsde-[lr-r’l
(6pA(r) 6pA(r’)) = pss6(r - r’) - pss2
2 4 r - r’l0 (40)
where $. = (4k0b”pss/D)1/2and has the units of (length)-l. The meaning of this length can be seen by recalling that (4kobsdpss)-l is the lifetime of a small perturbation in the concentration of A near steady state. If one recalls also the expression (6D7)’I2 for the root mean square (rms) distance which an A molecule moves in time 7,the length 1/[ is seen to measure the expected distance that an A molecule moves before it is consumed by the reaction. To find the stationary-state pair correlation function, gAA(r,r’),one uses the steady-state version of eq 12, i.e. (pi(r) 6pj(rf)) = Gijp?G(r
- rf) + pypjSs(gij(r,r’)- 1) (41)
Comparing this to eq 40 gives kObsde-Elr - r’l gAA(r,rf) = 1 - 2alr - r’lD
This is an analogue of the compressibility equation in the theory of simple fluidslgand expresses the number fluctuations of A in terms of its radial distribution function. If the Smoluchowskiresult in eq 43 is substituted into eq 44, the right-hand side is infinite, implying that the number fluctuations have an infinite variance. It is known that this anamolous situation occurs at steady states away from equilibrium only very close to points of i n ~ t a b i l i t y . ~ t ~ ~ - ~ ~ Since the present steady state is dynamically stable, the form of the Smoluchowski radial distribution function is clearly incorrect. Notice that the radial distribution function in eq 42 does not lead to a divergence but instead gives ( (6NA)2)/NA= 1/2, which agrees with the value based on the calculation of reaction fluctuations using standard appro ache^.^^^^^ To complete the calculation of the rate constant, kobsd, for this reaction requires that the form of the reactivity function ko(r,r’)be specified. We will illustrate how to do this with two simple choices. First, the Smoluchowski form of ko in eq 17 is used. When the expression in eq 24 is used, the observed rate constant for this reactivity function is given by
Changing variables of integration to r - r’ and using spherical coordinates then gives kobsd = kO[1 - kObsde-[R / (2rRD)I (45) Recalling that $, = (4k0bsdpss/D)’/2
(46)
one sees that eq 45 is a transcendental equation for the observed rate constant. It can be written in a more familiar way as (42)
(37) For a review see ref 22. (38) 1. S.Gradshteyn and I. M. Ryzhik, Table of Integrals,Series, and Products”, Academic Press, New York, 1965.
kobsd = 2aRe~RDk0/(27rRe~RD + ko)
(47)
(39) J. Keizer, J. Chem. Phys., 67, 1473-6 (1977). (40) D. McQuarrie and J. Keizer in “TheoreticalChemistry: Advances and Perspectives”,D. Henderson, Ed.,Academic Press, 1981, pp 184-210.
5058
Keizer
The Journal of Physical Chemistry, Vol. 86, No. 26, 1982
Equation 47 is reminiscent of the Smoluchowski result obtained from the flux boundary condition (cf. eq 6), except for the appearance of the term etR. Although eq 45 or 47 must be solved for kobsdwith the effect of this term included (see sections IX and X), its size can be estimated by reverting to the Smoluchowski theory. In that theory, the value of kobsdfor an intrinsically rapid reaction (ko >> 2rRD) is 23rRD. Because of the definition of 5 in eq 46, this will approximate the largest value of etR, Le. exp(fR) i= e ~ p ( 8 x R ~ p ~ ~ ) ' / ~ (48)
- -
This factor is of no importance in extremely dilute solutions, since as pss 0, 4 0 and the correction is unity. A t higher densities, its value can be significant. For example, take the spherical close-pack density, 2lI2/R3,as an estimate of the maximum density of A, pssmax. Then eq 48 estimates that exp(ER) if ps8/pssmax= 2.8 X 1.1. Such a density corresponds to about M, and, as will be seen in the following, it is a t about that concentration that these correlation-length effects begin to become important. Indeed, at that density ER = 0.1, so that the correlation length 1/5 is comparable to molecular dimensions. Another choice of reactivity function is the exponential one in eq 16. Using this form in eq 24 along with the calculated radial distribution in eq 42 gives kobsd
= (8.rrro3k0)(1 - kobnd/[2.rrDrO(1 + r4)2]) (49)
Equation 49 is similar in form to eq 45 with 8rro3k0replacing ko,ro replacing R, and (1 r&)2replacing exp(R6). = 1.2 when For this model reactivity function, (1+ rot = 0.1. Consequently, significant modifications of the usual Smoluchowski result can be expected for this sort of reactivity function, too. Notice, however, for both reactivity functions that the correction in the present theory always appears as a multiplicative factor with the radius R or the falloff length rP Thus, the corrections could be artificially absorbed into R or ro by an appropriate choice of those parameters. Notice, however, too that the correction factors are density dependent in a way that should be capable of experimental d e t e ~ t i o n . ~
+
V. Extensions of the Simple Theory The calculation of the radial distribution function in the preceding section neglects the effect of the products, the back-reaction, and intermolecular interactions. Any or all of these can give rise to important effects, and here the calculation of gAA(r)is extended to include these processes. First, the back-reaction-and an explicit mechanism for the generation of A-is examined. In this calculation the generation of A is assumed to be photochemical from the dimer A2. Thus A2 -!L 2A 2A-
ko" ku
A2
aspA/at = -4kobsdp~Ss6p~ + 2(k* aapA,/at = 4kobsdp~"6p~2(k*
b(r-r')6(t-f')
(-2, (55)
v(r,r') 6(t - t 3 where
R = kobsdpAas2+ kUpAF No term for the photochemical reaction 50 appears in eq 55 since that process is nonthermodynamic (the light is externally supplied). Clearly by including the product A2 in the reaction scheme, the calculation has been seriously complicated since 6pA is now explicitly coupled to 6pA, by eq 54 and 55. Thus, the equation of the fluctuation-dissipation theorem (eq 38) becomes a matrix integral equation.36 Indeed, the relaxation matrix H(k,k') in eq 38 is defined by d@(k,t)/dt = JH(k,k')
Gp(k',t)
dk
+ f(k,t)
H(k,k')= -4kobdp~" - 2 D ~ k ' 2(k* t k,)
i
4k o b d p ~ S S
-2(k*
-+
1
i(k + k') K,) - 2 D ~ ~ k '
Similarly Fourier transforming with respect to r and r' in eq 55 gives r ( k , k ' )= 4 8 t 2 D ~ p ~ ' ~ -2E k' 5(k t k ' ) / ( 2 7 ~ )(58) ~ (-28
~ ( k6(k ) + k')/(2~)~
Using these expressions, one can obtain the equation which is solved by the covariance matrix, u, of the fluctuations and @A,. Substituting into eq 38,one obtains H ( k ) u(k,k') u(k,k') HT(k') = -r(k) 6(k + k ' ) / ( 2 ~ ) ~
+
(59)
And at steady state
where (53)
Two partial differential equations are required to describe
the fluctuations near steady state. These are4,31-33v39
(57)
H ( k ) 6(k + k')
dpA/dt = 2k*pA2- 2kobsdp~2 4- 2 k 9 = ~ -2dp~,/dt ~ (52) (kobSd/k*+ kU)pABB2
(56)
where 6fi is the spatial Fourier transform of the column vector of fluctuations (6pA,6pA2). If one takes the spatial Fourier transform of eq 54, it is found that for the present reaction
Equation 59 implies that u must have the form o(k,k') = u ( k ) 6(k + k')/(27~)~
PA? =
(54)
tii(r,t) l T ( r ' , t ' ) ) = 4 8 - 2 D ~ p ~ ' ' ( 7 -~2' 8
3
The reverse reaction has a unimolecular rate constant k , which, excluding solvent, is unaffected by nearest neighbors. The average equations in a uniformly illuminated container will be
+ k,)6p~,+ DA,V26pA, + ?A,
?A
which are gotten by linearizing eq 52 around steady state after including the effect of diffusion. According to the principles of mechanistic nonequilibrium thermodynami c ~the , ~ random terms will have the correlation matrix
(50) (51)
+ k,)6pA, + DAV26pA +
H(k) u ( k )
+ u(k) P ( k ) = - r ( k )
(60) (61)
This is a linear matrix equation for u. Because Y(k) is a symmetric matrix, eq 61 can also be thought of as three
The Journal of Physical Chemistry, Vol. 86, No. 26, 1982 5059
Effect of Diffusion on Chemical Reaction Rates
independent linear e q u a t i ~ n s ~for v ~ the l components all&) (6$A(k) aFA(k)), UlZ(k) = 021(k) E (6$A(k) 6pA2(k)), 022(k) (b$A,(k)6$A,(k)). For purposes of calculation this is the easiest way to regard eq 61. The three equations then may be solved for ull, u12,and u22by using by Cramer's rule. Since only the spatial correlations of the density of A with itself affect the rate constant, only the result for ull(k) is given explicitly. It is
VI. Interaction among Reactants In the Smoluchowski theory of diffusion-controlled reactions, the effect of interaction between reactants is taken into account by including a mobility term in the diffusion equation.lOJ1 In the present theory these effects are included already in the elementary processes of diffusion and reaction as described, e.g., by eq 25. To illustrate this we again treat the chemical reaction A + A products but now remove the restriction of dilute solution. This means that the average equation (eq 28) must be written as28,32
-
dyA(r,t)/dt= -2k*0bsdZA2+ DoV2ZA + 2K
where, the activity zA = exp[(p - po)/kB7'l having been introduced, the bimolecular rate constant klobsd is now activity based. The relationship between the density- and activity-based rate constants is
where
ti2= [(~DA~/PA")(DA/PAF)]K/(DADA 52'
(68)
= [(4/pAss) + (l/pA,"")]R/(DA
+ DA2)
k,obsd
When the inverse Fourier transform is applied to eq 62, the steady-state density of A-density of A correlation function is found to be kobsdpAss2k*
+ k*)
e-€zlr-r'l
= (5'
+ k*/DA)'/'
with yAthe activity coefficient. In concentrated solutions klobadis the more fundamental transport c ~ e f f i c i e n t . ~ ~ ~ ~ ~ If we apply the principles of the mechanistic fluctuation theory3 the conditional fluctuation, 6pA, satisfies
-
(65)
Equation 66 can be used with an appropriate reactivity function to calculate kobsd. For example, using the Smoluchowski reactivity in eq 17 would yield = kogu(R)
(67)
The presence of the diffusion constant of A2, the unimolecular rate constant k,, and the illumination rate in eq 66 and 67 make it clear that all these factors play a part in determining the rate of dimerization reaction in solution. How large a part depends on the magnitude of the various transport coefficients. The previous calculation has been extended42to include interactions among the species A and A2. The details, however, are lengthy and will not be reported here. Instead the effects of nonideality are illustrated with a simpler model in the following section. (41) J. Keizer, J. Chem. Phys., 69,2609-20 (1978). (42) J. Keizer, unpublished.
+ D0V26zA+ 7
(70)
(4k*obadZ~s82 - 2D0Z~""o~)6(r - r') 6(t - t ? (71)
Thus, in the limit of low illumination intensity (k* 0) and negligible product, the result of the previous calculation in eq 40 is recovered. The radial distribution function gAA(r)is obtained by combining eq 64 with eq 41. This gives
kobsd
(69)
(7(r,t)?(r',t?) =
+
where eq 52 was used and D' = DA + DAz. Notice that, if DA = k, = 0, that is, if both diffusion of the product and its Lack-reactionare negligible, eq 64 reduces to eq 40 with 5 replaced by 52
kobsdiyA2
dsPA(r,t)/dt = -4k*obsd-ZA 6zA
(6PA(r)h ( r ' ) ) = pAss6(r- r') 27rDlr- r'l(k,
=
(63)
The most significant difference between eq 28-30 and 68-71 is that the latter involve fluctuations in the activity, 6zk For nonideal solutions these fluctuations are related in a spatially nonlocal way to fluctuations in the densit y . ' * ~ ~In~ a uniform fluid at stationary state this can be written as
6zA(r,t)= zAaslE(r,r') 6pA(r',t)dr'
(72)
where (73) and 6 denotes the functional d e r i ~ a t i v e . ~For ~ an ideal solution3gE(r,r') = 6(r - r')/pAss,%Aas = pAss, and = 1. Thus, for an ideal solution eq 68-71 reduce to the previous eq 28-30. To make headway with the nonlocal term, recall that attention has been restricted to radially symmetric fluids, so that E(r,r') = E(lr - r'l), Thus, substituting eq 72 into eq 70, Fourier transforming the resulting expression with respect to space, and using the convolution theorem gives
+
+
d66A(k,t)/dt = -[4k*obsdZ~ss2k(k) D 0 Z ~ s s k ( k ) k 2 ] 6 $7~ (74) (3(k,t) ?(k',t?) =
Proceeding, then, as in Section V, one can use the fluctuation-dissipation theorem (cf. eq 59-61) to obtain
~~~
(43) A. A. Frost and R. G. Pearson, "Kinetics and Mechanism", 2nd ed., Wiley, New York, 1965. (44) J. Keizer and M. Medina-Noyola, Physica A (Amsterdam),115, 301-38 (1982). (45) V. Volterra, "Theory of Functionals", Dover Publications, New York, 1958.
5060
The Journal of Physical Chemistry, Vol. 86, No. 26, 1982
where D is the density-based diffusion constant of A (see eq 19), E.’ = 4k,oMzA”/D,and de is the Fourier transform of the so-called “local equilibrium” structure factor,28i.e., ale(k) = E(k)-’. The local equilibrium structure factor is identical in its functional form with the equilibrium result, except that the average values corresponding to the steady-state condition must be substituted for equilibrium value^.^^,*^ Thus, eq 76 gives corrections to the local equilibrium result. Indeed, applying the inverse Fourier transform to eq 76 and using eq 41 gives k*obsd(ZAss / P A S S ) e-[* W’l gAA(lr - r’l) = g,le(lr - r’l) 2.rrDlr - r’I
Since for an ideal solution gMle 1,it is easily checked that eq 77 reduces to the ideal solution radial distribution function given by eq 42. The result in eq 77 can be compared with the radial distribution function based on the Smoluchowski theory. That result was first obtained by Debyelo using the Debye-Huckel potential. In general, one should use the local equilibrium potential of mean force as the potential in the Smoluchowski theory,ll i.e. uAAle(r)= -kBT In gAAle(r)
(78)
Using this potential and following Debye one then obtains7 gAASmOl(r)= gMle(r)[1 - kobsdLm(xzgAA1e(x))-l dx/(4i~D)](79) While both eq 77 and 79 give rise to corrections to the local equilibrium radial distribution function, the two corrections are qualitatively quite different. The Smoluchowski result in eq 79 is comparable to the first two terms in eq 77 but is missing the convolution integral correction. That correction involves the propagation of pair correlations which exist a t equilibrium through the nonequilibrium chemical processes. The magnitude of the various terms in eq 77 can be estimated given an explicit form for gAAle(r).An approximate form, for which analytical results can be obtained, is gAAle(r)= 1 + e - K ‘ / ( 4 ~ ~ r )
(80)
Such a form is suggested by the Debye-Huckel theory of electrolyte solutions or the Ginzberg-Landau theory of first-order phase transitions.* The correlation length at equilibrium is then K - ~ . Substituting the expression for gAAlegiven in eq 80 into eq 77 gives ---Yr
gAA(r)= 1 + L--49kr
Keizer
&-’ = r = 10 A, this occurs at a concentration of about 0.1 M. Thus, the present theory predicts corrections to the radial distribution function due to intermolecular forces and reaction beyond those given by the Smoluchowski theory. VII. Lifetime Effects Some molecules which undergo bimolecular reaction in solution can participate in unimolecular decomposition. Consequently, these reactants have an intrinsic lifetime. Fluorescence quenching is typical of such a process.6 In the Smoluchowski theory this effect amounts to a limited lifetime for the sink and can be treated only by adding ad hoc procedures.12 In the present theory the existence of a competing unimolecular reaction path simply means that fluctuations are generated dynamically by another elementary reaction. Consider, for example, a simplified model of fluorescence quenching: A
how
+B A
k“
products
products
(82)
(83)
A should be thought of as an excited state which is quenched (either by reaction or by energy transfer) to B and which can fluoresce to its ground state by reaction 83. For simplicity it is assumed that the back-reactions are negligible and the concentration of the quencher B is assumed much larger than A so that it can be taken as constant. The average equations for this model are then where jig is taken as fixed and K is the rate of production of A from its ground state by applied radiation. Although this scheme neglects the average change of B with time and the fate of B after it reacts, both of these effects can be included by adding the appropriate elementary processes to the kinetic equations. To gauge the effect of the unimolecular lifetime of A on the bimolecular rate constant, we must obtain the nonequilibrium radial distribution function for an A-B pair. Thus, even though the concentration of B is fixed on the average, its fluctuations around pABB must be examined. To obtain the radial distribution function, we follow the procedure outlined in earlier examples. First the partial differential equations solved by the conditional fluctuations a t steady state ( 6 p A , 6 p B ) are obtained by using the principles of mechanistic nonequilibrium thermodynamics; these equations are then Fourier transformed with respect to space and the relaxation matrix H(k,k’)is identified icf. eq 56) as well as y(k,k’), the correlation matrix of the f s ; when H a n d y are known, the fluctuation dissipation theorem (cf. eq 59-61) is solved for 0. This gives
(81)
The term from the convolution integral in eq 77 corresponds to the second factor in the square brackets. This term can be significant under a variety of conditions. For example, when r = E.-’ = K - ~ ,the term will be large when rpABB/(K(I) is the order of unity, in other words, when the density is such that the number of molecules in the parallelepiped of volume rK-l&-’ is the order of 1. For K-’ = (46) See ref 21, pp 362-5.
where K 1 = k, + kobsdpBss, K 2 = kobsdpAsS, D’ = DA + D,, a = ( K $ B + K&J/(DADB), P = (Ki + K z ) / D ’ , Y = 2k&/(&&), A1 = (cY/2)(1 + [1 - 2A/0(]liZ), 1 2 = (Lu/2)(1 - [l - 2 y / ( ~ ] ’ / ~ )The . inverse Fourier transform of eq 85 can be done by using standard integrals. Equation 41 then shows that the radial distribution function is3
Effect of Diffusion on Chemical Reaction Rates
The Journal of Physical Chemistry, Vol. 86, No. 26, 1982 5081
only because
In eq 86 there are three correlation lengths, p1I2, X1-lI2, and The unimolecular lifetime, 7 = l / k , , appears in an important way in these correlation lengths and disappears only when k , = 0; i.e., when the lifetime is infinite. In case the lifetime of A is infinite, eq 86 reduces to the simplified expression
with a = kobsd(pgmDB+ pAsSDA)/(DADB), @ = kobad(pAs+ pBBS)/D!Since eq 87 refers only to the bimolecular reaction 82, it is the result which should be compared to the radial distribution function given by the Smoluchowski theory in eq 5, namely
gABSmol(r) = 1 - kobsd/(4~D'~)
(88)
The difference between these equations is the squarebracketed factor in eq 87 which depends on the correlation lengths a-lI2and p1j2. In the dilute-solution limit a and P go to zero and the term in square brackets reduces to unity. Thus, the usual Smoluchowski theory is the dilute-solution limit of the present theory. At higher densities-even when the lifetime is infinite-eq 87 involves a multiplicative correction to the Smoluchowski theory. For rapid reactions the effect of this correction becomes important above a total concentration of A and B of about M, as we have seen already for the dimerization reaction in section IV. Lifetime effects do not necessarily dissappear in dilute solution. For example, if we assume that pASS 1, and for correlation lengths, CY-'/^ = (D7)'12, which are comparable to R. As discussed in section IV, short correlation lengths occur when the unimolecular lifetime is short and the viscosity is high. The limiting effect under these conditions is a factor of increase in the observed rate constant over the Smoluchowski result. As Figure 2 shows, more than a 20% increase over the Smoluchowski theory is predicted even when the correlation length is 5 times the reaction radius R. Thus, this effect should be observable. Our second example is for the bimolecular scheme, A B products. The radial distribution function for this reaction is given by eq 87. To calculate the bimolecular rate constant, one uses a truncated form of the exponential reactivity function in eq 16, namely
+
D ' r , = 1 0 .k2
-
r ko(r) = ( 1 / ~exp(-r/ro) ~) r IRo ko(r) = 0
> Ro
~
O '
D'r, :10i2 k'/kD= 364
336
-8
-'6
-15
-14
-3
-1
-'I
loQ,o[el
Flgure 3. Graph of kad/D'vs. log (molar concentration of 6)for the reaction model described by eq 108. At low concentrationsthe results converge to the value given by the Smoluchowski theory in eq 6 with R fo([Ro/ro) 11' l)/[Ro/fo) 11, k o 3 (47rf:/70)([(Ro/fo) 11' 1) exp(-Ro/ro), and k, 47rRD'. Values of the parameters which were fixed in the calculation were R o = 6 A, r o = 3 A, D' = 2.8 X cm2 s-', DA = 2 X lo5 cm2 s-'. Three curves are given corresponding to different values of the parameter r0 (see eq 107). The largest deviations from the Smoluchowski theory are seen when k o is large, Le., for rapid reactions.
+
+
+
+
+
(107)
Substituting eq 87 and 107 into eq 24, which defines koM, yields
kobsd
C Y [ R ~ ( C+Yl/ro) ' / ~ 4- l]e-Roaliz + 2(a + l/ro)z
E{ (CY
- 2/3)[R0(/3'/2+ l/ro) + l ] e - R ~ l / z (108) CY - p)(/3'/2 + l/ro)2
and explicit expressions for CY and /3 are given following eq 87. Since CY and /3 depend on kobsd,to solve this equation for kobsdrequires an iterative procedure. We have done this on a hand calculator, iterating kobad/D'startingwith an initial estimate of the Smoluchowski theory value for this quantity. Representative results are given in Figure 3, for the special case that D'/DA = 1.4 and pAss > / DR ) ' / ~ (110) kobsd = kO In the rapid intrinsic reactivity limit ( Y >> 1) eq 109 becomes
x2e-X = 87rR3psE
(111)
Because the derivation of this expression was restricted to dilute solution, the right-hand side is much less than 1. A graph of the left-hand side reveals a maximum value of 0.54 a t x = 2, so that two solutions to eq 111exist: one with x > 2 and with x < 2. Only the solution with x < 2 is physically meaningful as x = 2 corresponds to a correlation length of one-half the molecular diameter R. Representative values of f-' and kobsd/Das a function of pas are given in Table I1 for R = 4 A and R = 8 A. The increase in size of kobsdover the dilute-solution value is given by the factor ex and should be experimentally observable a t about M.
X. Calculation of Rate Constants in Membranes The radial distribution functions obtained for two dimensions in section VI11 can be used to calculate bimolecular rate constants in membranes. The calculations are similar to those described in the previous section. For radially symmetric distributions in two dimensions, the defining equation for kobd becomes KObsd
= 27rJcrk0(r) gij(2)(r) dr
(112)
with the superscript 2 referring to two dimensions. Analytical conclusions of kohd in two dimensions require integrals over the McDonald function of order zero. For simplicity we restrict ourselves to the two-dimensional Smoluchowski reactivity ko(r) = k06(r - R)/(27rR) (113) which gives = kOgij(2)(R)
,
(114)
,
O L -2
1
-3
1 -4
I
I
-5
-6
1
-7
I -8
I
-9
'OQI O D A
Flgure 5. Unimolecular lifetime effect on kObsdlko in two dimensions as a function of the diffusion constant of A, as described by eq 116. Parameter values were chosen to be k , = 2 X lo8 s-', R = 4 A, and ko = 2 X cm2 s-'. The units of D A are cm2 s-'. The two inflection points appear near values of 0,for which k o = D, and R = (D,A/k,)"*. As the infledon points approach one another (i.e., when k o and K , are smaller), the minimum in the curve becomes less prominent.
In general, eq 114 must be solved iteratively to obtain a value for kobsd. There is one case in which eq 114 can be explicitly solved for koW. That will occur when the correlation length does not depend upon kO". An example of that is given by the fluorescence quenching model in eq 82 and 83 when the unimolecular lifetime is much shorter than the bimolecular lifetimes, Le., k , >> kobsdpAw,kobsdpBss.Such a condition, of course, is favored by dilute solution. If it is further assumed that DA = DB, the two-dimensional radial distribution function in eq 100 reduces to
with a = k,/DA. Thus, combining eq 114 and 115 yields
As a function of the diffusion constant, the asymptotic forms of eq 116 are easy to recognize. When DA is large, a is small, and D A / [ K o ( ~ c Y 'is/ ~large ) ] (Ko(x) -1n x / 2 , see eq 96). Thus, for DA large, eq 116 implies that kobsd = ko, a result similar to that in three dimensions. On the other hand, when DA is small, a is large, and K o ( R ~ 1 /is2 ) exponentially small (see eq 97). Thus, when DA is small, one finds again that kobsd= k". This result also holds in three dimensions (if a is independent of kohd)and is easy to understand physically. When DA is small, the rms distance traveled during the unimolecular lifetime, (DA/ k,)lI2 = a-lI2,is so small that reaction occurs only with neighbors which preexist when the reactant A is created by the external input. That is, the reactant A does not live long enough to build up correlations within the membrane or fluid. An example of the dependence of kobsdon DA in the regime intermediate between the high- and low-viscosity regimes is given in Figure 5. The dip in the ratio of koM/ko is enhanced when ko is larger, i.e., intrinsically fast reactions. This result, incidentally, is compatible with recent experiments on the quenching of the fluorescence of parinaric acid by bimolecular reaction in phospholipid bi1aye1-s.~~ In that system the unimolecular lifetime is about 5 X s and the intrinsic reactivity is very high.
-
The Journal of Physical Chemistry, Voi. 86, No. 26, 1982 5065
Effect of Diffusion on Chemical Reaction Rates
TABLE 111: Correlation Lengths E-' for the Diffusion-Controlled Membrane Reaction Described by Eq 120 as a Function of the Density p 1og (F'iR)
(PiPC,)"
0.087 0.490 0.795 1.07 1.34 1.59
-1.60 -2.03 -2.46 -2.90 -3.33 -3.77
log
log
(E-lIR) 2.07 2.55 3.03 3.26 3.72 3.95
" p C P is the close-packing density
2.0r
log (PiPC,)"
-4.64 -5.50 -6.31 -6.81 -7.68 -8.11
2/(3"'R2). 0'
Indeed, when the diffusion constant is increased by a factor of over lo4, kobsd increases only by a factor of 3. The present calculation suggests that no change (or perhaps a slight decrease) might be caused by diffusion and that the increase in kobedis probably a result of other effects, e.g., the extreme change in the rigidity of the fluid on rotational diffusion. As our second and final two-dimensional example, we consider the bimolecular reaction A + B product. This is the infinite unimolecular lifetime case of the scheme in eq 82 and 83. The corresponding radial distribution function can be gotten from eq 100. If it is assumed that DA = DB and pAss + pBSs is set equal to p , eq 115 is again obtained, except now
-
= kobsdp/DA
(117)
Consequently, kobsd satisfies eq 116, which becomes an implicit equation since a depends on kobd as given by eq 117. If one writes Y = k 0 / ( 4 r D A )2 , = 47rR2pY, and x = R a 1 f 2 the , equation for kobd becomes
kobsd = h D ~ k ' & - ~ ( X ) / [ k '+ ~ ~ D A K { ' ( X )(118) ] or 2 = x(xKo(x)Y + x )
(119)
The low-density behavior of eq 118 and 119 is easy to identify since then 2 = 0. Under this condition the physical ( x < 1) solution of eq 119 must be x = 0, since the factor xKo(x)Y + x is nonnegative and bounded. 0, K0-'(x) 0 (cf. eq 96); thus, eq 118 However, as x implies that koW = 0. At higher densities, the magnitude of kobd depends on the magnitude of both Y and 2. For example, when Y cyclohexane.
Introduction
Earlier studies have established the coordination chemistry of benzene, toluene, cyclohexane, cyclohexene, and 1,3- and 1,4-cyclohexadiene on the atomically flat Pt(lll), stepped Pt 6(111)X(111),as well as the low and high Miller index planes of nickel These studies were (1) Friend, C. M.; Muetbrties, E. L. J.Am. Chem. SOC.1981,103,773. (2) Tsai, M.-C.; Muetterties, E. L. J.Am. Chem. SOC.1982,104 2534.
0022-365418212086-5067$0 1.2510
conducted under ultrahigh vacuum conditions with thermal desorption spectroscopy, isotopic labeling, chemical displacement reactions, low-energy electron diffraction, and Auger electron spectroscopy as primary diagnostic techniques. Electronic and stereochemical features that facilitate carbon-hydrogen bond breaking and making were (3) Tsai, M A ; Friend, C. M.; Muetterties, E. L. J. Am. Chem. SOC. 1982,104, 2539.
0 1982 American Chemical Society
