4'150
J. Phys. Chem. 1982, 86, 4750-4756
equilibrium conditions. The pulse radiolysis technique can possibly provide information on the absorbance of the ion immediately after their formation. No pulse radiolysis measurements were carried out in HN03, because the possible reactions between e, - and H with HN03 may complicate the interpretation o! the absorbance changes.
of the acid present and on its concentration. Under steady illumination with a 100-W visible light source, the halflifetime of the back-reaction between U02+and Ce(1V) is expected to be of the order of seconds. This system is therefore a potential candidate for photochemical generation of 02,provided a suitable catalyst for water oxidation by Ce(1V) is found, which will be able to compete with the back-reaction. Attempts to use RuOPfor this purpose were not successful.
Conclusions Photochemical electron transfer from Ce(II1) ions to the lowest excited state of the uranyl ion takes place with a quantum yield of about 1. Despite the high negative value of the free energy for the back-reaction, its rate is relatively slow, k E lo7 M-I s-l depending somewhat on the nature
Acknowledgment. We are indebted to Mr. G. Dolan and Y. Ogdan for maintenance and operation of the laser and linear accelerator systems and to Professor G. Czapski for helpful discussions.
Klnetics of Dlffuslon-Medlated Bimolecular Reactions. A New Theoretical Framework K. Rarl Naqvl;
8. Waldonrtrram, and K. J. Mork
Depcrrtnwnt of Fhydcs, Unlvety of Trondhelm (NLHT), N-7066 Dmgwll, Noway (Received: Nowmber 20, 188 1; In FlMl FOmr: July 30, 1882)
We undertake here a study of the applicability of the diffusion equation (DE) to bimolecular reaction kinetics. We point out that the standard procedure, in which one solves the DE subject to the so-called radiation boundary condition (RBC),does not directly give ko = k(O),the initial value of the rate coefficient k ( t ) of a diffusion-mediated bimolecular reaction. To delimit the domain of validity of the DE approach, we derive both the DE and the RBC within a common framework based on the Lorentz model of random flights. We go on to fabricate a modified, but equivalent, form of the RBC which leads, in the t -.,0 limit, to the same expression for ko as that proposed by Noyes; we demonstrate that, with this amendment, the DE approach can be made congruent with the particle-pair standpoint of Noyes.
I, Introduction Let k ( t ) denote the rate coefficient of a diffusion-mediated bimolecular reaction. We wish to point out that
and the radiation boundary condition (RBC)
the conclusion, reached previously by Noyes,' Berg? and US,^ that the formal expression for k ( t ) / k ( O )deduced by Collins and Kimball (CK)' may be equated to that put forward by Noyee was based on an interpretation advanced first by Collins and Kimball and accepted by many subsequent authors-an interpretation that we recant here.
k has been replaced by b.) The solution for c(r,t) satisfying eq 2-4 has been given, for the region r > R, by Carslaw
c(r,O) = co D(dc/dr),.,
= bc(R,t)
(3) (4)
(The notation used above is that of CK, except that their and Jaeger:5
11. Brief Survey of the Works of Collin8 and Kimball and Noyes By dint of a lengthy argument, Collins and Kimball (CK) showed that k ( t ) may be defined as
B
subject to the initial condition (1) (a) Noyen, R. M. J. Chem. Phys. 1964,22,1349. (b) Noyes, R.M. J.Am. Chem. SOC. 1966,78,5486. ( c ) Noyes, R. M. h o g . React. Kinet. 1961, 1, 129. (2) Berg, 0. G.Chem. Phye. 1978,31,47. (3) Razi Naqvi, K.; Mork, K. J.; Waldenstram, S. J.Phya. Chem. 1980, 84, 1315. (4) Collins, F. C.; Kimball, G. E. J. Colloid Sci. 1949,4, 425.
0022-3854/82/2088-4750~0~ .25/0
= b / D + 1/R
Using this solution, CK calculated GM(t) and concluded that it "remains finite even for t = 0. the limiting value being in fact 4.rrR2bco";that is to say,they arriveJat the result k ( t ) / k i = c^(R,t) (6a) (5) Carslaw, H. 5.; Jaeger, J. C. 'Conduction of Heat in Solids", 2nd ed.; Oxford University Press: Oxford, 1959; pp 247-8.
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982 4751
Diffusion-Mediated Bimolecular Reactions
where kistands for the limiting value k ( t - 4 )
ki k(t-0) = 4rR2b
(6b)
Using elementary but highly ingenious reasoning, Noyes was able to show that
k ( t ) = ko[ 1-
1’ 0
dt’h(t?]
(7)
He defined ko as “the rate constant applicable for an equilibrium molecular distribution”,lcand h(t)dt as the “probability two molecules separating from a nonreactive encounter at time zero will react with each other between t and t dt”.lb Throughout this paper, Ro will stand for the distance between “two molecules separating from a nonreactive encounter” (see ref 3); Noyes himself equated Ro to R , a step taken later also by Monchicksa and When expounding the DE approach, NoyeslCdefined 47rR2b as “the rate constant that would describe the reaction if equilibrium distributions maintained so that c(R,t) equalled con and went on to identify 47rR2b with k,. (NoyeslCused c, for our c(R,t), c, for co, and k for I t must be added here that, in order to be useful, eq 7 must be supplemented with expressions for ko and h(t); likewise, the parameter b entering the RBC must be linked to physically significant quantities. On physical grounds, Noyesla deduced the following definition of ko:
+
ko = &,
(84
where a is the reaction probability per encounter and 12, the rate constant for the formation of encounter pairs; he argued further that k, may be written as
k, = 7rR2uo
(ab)
where uo is the average speed of the diffusing particle. Equating ko = a7rR2uowith 47rR2b,3one accomplishes the task-left unfinished by CK-of fiiding an expression for b: b = aUo/4
(9)
in agreement with Monchick; who had already derived the above expression for b from different considerations.
111. Validity of the Standard DE Approach The interpretation of ki = 47rR2b in eq 6b as an initial value rests evidently on the assumption that the initial condition in eq 3 holds also for r = R and reads c(r, 0) = co for r 1 R . But if this were so, one would be forced to conclude that, at t = 0, ac/& = 0 for r 1 R and admit-in consequence of the definition of k ( t ) in eq 1-that k(t=O) = 0, which contradicts the former conclusion. This inconsistency arises from disregarding the fact, stated clearly by Carslaw and Jaeger? that the solution for c(r,t) in eq 5 may be used for all times t provided that r > R , but it can be employed to extract c(R,t) = limFRc(r,t) only for t > 0. A detailed exposition of these restrictions can be found in the sections entitled “Initial and Boundary Conditions” and “The Mathematical Interpretation of Initial and Boundary Conditions” in the treatise of Carslaw and Jaeger; according to these authors, the initial condition in eq 3 holds only for r > R , and consequently the RBC, Le., eq 4, applies only for t > 0. (See also eq 441 in the review article by Chandrasekhar.8) (6) (a) Monchick, L. J. Chem. Phys. 1956.24, 381. (b) Zbid. 1975,62,
1907. (7) Razi Naqvi, K.; Waldenstram,S.;Mork, K. J. J. Chem. Phys. 1979, 71, 73.
Hence, without further assumptions, the standard DE approach cannot be used to calculate the initial value of the rate coefficient. The usual strategy is to invoke the continuity of k ( t ) and equate hi to lim,.++k(t); this gives ki = 47rR2b. However, eq 6 shows that k ( t ) = 47rR2bE(R,t)
(10)
and since
c^(r,O) = 1
for r
>R
(11)
it follows that a given initial value, ko, can be realized only by imposing a discontinuity on c^(r,O)at r = R in such a manner that c^(R,O)= ko/(47rR2b)
(12)
Thus, before any progress can be made, one must determine the parameter b, which amounts to deriving, in a consistent manner, the DE and the boundary condition at the reaction surface. As we have shown elsewhere, this can be done within the framework of a random-walk model in which the free paths are governed by an exponential distribution; a summary of our work is given below. A. Derivation of the DE and the RBC. In a comparatively unknown paper,l0 CK used a random-walk model to derive the DE and an expression for the total flux into the sink at r = R . By extending their arguments, we constructed expressions for @*(r,t+T), the total outward (+) and inward (-) flux through a fictitious sphere of radius r at time t + T , where T , the mean free time, is the reciprocal of the scattering frequency v ( T = l / ~ )the ; ~fre~ quency satisfies the relation uo = vl
(13)
where 1 denotes the scattering mean free path, and uo is the average speed of the diffusing particle (cf. eq 8b). Since the diffusion equation follows on combining Fick‘s law
J(r,t) = -D[&(r,t)/dr]
(14)
which is an approximate relation, with the exact continuity equation
it is sufficient for our purpose to derive only Fick’s law and the boundary condition at the reaction surface. Starting from the expressions for @*and defining the (radial) current densities J* by the relation
the net outward current entering the continuity equation can in principle be constructed as
As demonstrated in ref 9a, this procedure leads to the following expression for the current J ( 1 < r assumed):
(8) Chandrasekhar, S. Reu. Mod. Phys. 1943, 15, 1. (9) (a) Razi Naqvi, K.; Waldenstram, S.; Mork, K. J. Ark. Fys. Semin. Trondheim 1981,9,1-60. (b) Ibid. 1981,11,1-59. At the time of writing these articles, we did not know that the model of random walk used by us is often called the Lorentz model. (IO) Collins, F. C.; Kimball, G.E. Znd. Eng. Chem. 1949, 41, 2551.
4752
The Journal of Physical Chemistty, Vol. 86, No. 24, 1982
where D(l),D(2),etc., can be arranged as a series in ascending powers of llr. Designating as the nth-order approximation for J that in which the coefficients D(k)are calculated to order uOln,only the two lowest-order approximations ( n = 1, 2) turn out to be compatible with Fick‘s law, and even this compatibility obtains only at large values o f t , i.e., when t >> T . These conclusions follow by a careful inspection of the corresponding expressions for J*; calculating to order u0l2,which in fact is consistent with the assumption t >> r , one finds
Razi Naqvi et al.
In this approximation, where the currents are evaluated only to order uol ( E D ) ,the expressions for J* in eq 19 reduce to
Noting that J- can be written as
and using Fick’s law, eq 20, one can also deduce the boundary condition in eq 24 from the relation
J ( R , t ) = -(I
-
K)J-(R,t)
(26)
which follows on combining eq 17 and 21. The upshot of the preceding discussion is that the standard DE approach will provide a faithful description only if from which one arrives at Fick’s law, with or without taking into account the second-order term proportional to (Z/r)2, by forming J = J+ - J-:
J(r,t) = - ~ [ a c ( r , t ) / a r l D E U&/3 (20) If terms of order uOl3are retained in eq 19, the coefficient D(l)in eq 18 becomes r dependent, and the terms involving a2C/ar2and d3c/dr3 on the right-hand side of the latter equation will contribute. The boundary condition at r = R needed to solve the DE can be formulated in a physically plausible way in terms of the basic quantities J+ and J-: J+(R,t)= K J J R , ~ ) (0 5 K I 1) (21) where K = 0 ( K = 1) corresponds to a perfect absorber (perfect reflector). The relation resulting from eq 21 and 19 can be stated in the form
where A
E
21/3
w
E
+
(1- ~ ) / ( 1 K )
(22b)
or equivalently as
(a) l / R
> 7
I t is worth noting that the results and conclusions presented in this section can be reached also by starting from the Boltzmann equation for the Lorentz model of a random walk12and applying the lowest-order version, referred to as the P1approximation, of the so-called spherical harmonics method for solving this equation.” Marshak,12who pioneered the use of the method in neutron transport theory, indeed alluded to the P1approximation as “the diffusion approximation” and in fact was the first to recommend, as an alternative to the boundary condition c(R,t) = 0 (Le., Smoluchowski’s boundary condition), the RBC, which he called “the improved boundary Condition”. It should finally be mentioned here that the restrictions spelled out in eq 27 concur with those set forth by Morse and Feshbach, who derive the DE as well as the RBC (for a perfectly absorbing plane boundary) from the transport equation13aand state,13b“The diffusion equation is approximately obeyed by diffusing particles ... as long as the distances involved are large compared to the mean free path of the particles and as the times involved are large compared to the mean free times between collisions”. B. Alternative Form of the Boundary Condition. Temporal Validity of the DE Results. The fact that terms involving u0l2 (and higher powers of 1) are specifically discarded in the approximation considered allows one to replace eq 25 by the succinct form
J*(r,t) = ( ~ 0 / 4 ) c ( r ~ A , t )
with (cf. eq 20) D = u,A/2 = u&/3 bEw~0/2 (23b) Since 0 IK 5 1, it follows that 0 5 w I 1; further, K = 0 corresponds to w = 1 (perfect sink) and K = 1 to w = 0 (perfect reflector). It is now seen that the standard RBC emerges by stepping down to the lowest-order approximation consistent with Fick’s law, i.e., by neglecting the term containing the factor ( l / R ) 2in eq 23a: l1 D(ac/&),,R = bc(R,t) b E ~ ~ 0 / 2 (24) (11) In another context, a boundary condition similar to that in eq 24 was derived by Monchick and Reies: Monchick, L.; Reiss, H. J. Chem. Phys. 1954,22, 831.
(27)
(28)
Inserting from the above the expression for J- into eq 26 and using Fick’s law, one gets D(&/ar),,R = U C ( R + A , ~ ) u E (1 - ~ ) ~ , 3 / 4 (29) It must be underlined here that the transformation of eq 24 into eq 29 is based on the assumption that, for AIR 0 ) (31) In the last step, use has been made of the definitions of D , b, and A in eq 20,24, and 25. It must be recognized that, by construction, "eq 31" can never be made to hold exactly so long as w # 0, A # 0, and R is finite and that it cannot at all be realized for small values of t if w is not very much smaller than unity, Le., unless the boundary a t r = R acts as an almost perfect reflector. The reason for this is, of course, that the solution to the DE with the RBC will not faithfully portray the steep concentration profile close to a strongly absorbing boundary at short times; only at sufficiently long times will the concentration vary smoothly enough to justify a truncated expansion in terms of A/R. Indeed, we are now able to quantify the restriction t >> T by defining "large times" as those for which "condition 31" is reasonably fulfilled. Writing c ( R + A , t ) / [ ( l + o)c(R,t)]= 1 - 6 (32)
+
-
-
one can calculate E with the aid of eq 5;9a for the limiting cases t a and t 0, one finds
W
€0
= t(t-0) = -
l+w Since E decreases with increasing values of t , one must require (at least) that AIR is small enough to make t, negligible compared to unity; for w = 1 (perfect sink)when q,attains its maximum value 'I2-it follows that AIR < 1/10 gives t, < 0.04, which sets a reasonable upper bound on the value of A/R. (Further details can be found in ref 9a.)
IV. Reexamination of the Equivalence of the Concentration-Gradient Approach and the Particle-Pair Standpoint of Noyes. Expressions for k ( t )and h ( t ) The expression for k ( t ) in eq 1 follows, by invoking Fick's law, from the more fundamental definition k(t) = -4rR'J(R,t)/~o (34) which can be recast, on account of eq 26, into the form
k ( t ) = ( 1 - ~)47rR'J_(R,t)/~o (t > 0 )
(35)
A. Equilibrium State. Under equilibrium conditions, i.e., when the surface at r = R is a perfect reflector and c(r,t) = constant = co, the currents J* defined in eq 16 become identicalgaand define the equilibrium current J+(r) = J J r )
J,(r)
(36)
from which the rate constant for the formation of encounter pairs, k,, can be deduced as
k , = 4?rR2J,,(R)/co (37) Equation 19 shows that, within the approximation necessary to obtain Fick's law k , = 7rR2uo[1- '/(Z/R)'] (38) which reduces, in the approximation defining the standard DE approach, to
k, = aR2uo the expression for k , given by Noyes (cf.eq 8b).
(39)
One may visualize the instant t = 0 as the very moment at which the surface at r = R starts to absorb and the equilibrium is spontaneously broken; assuming that eq 35 applies at t = 0 when J-(R,O) is replaced by the equilibrium current J,(R) inferred from eq 36, one gets, with the aid of eq 37 ko I k(t=O) = ( 1 - ~ ) k ~ (40) which agrees with Noyes' definition, eq 8a, provided that "the transmission coefficient" ( 1 - K ) is identified with a, the reaction probability per encounter: CYE1-K (41) It is seen that the parameter w, defined in eq 22b, is related to CY as follows: CY = 2 w / ( l + w ) (424 or w =
a / ( 2 - a)
(42b)
B. Rate Coefficient, k ( t ) . Recalling the definition of the parameter b entering the RBC, eq 24, and using eq 42, it follows that the limiting value ki defined in eq 6b can be expressed as ki E k ( t - 4 ) = 47rR'b = ( 1 w)a7rR2uo= ( 1 o)ko (43)
+
+
where ko = a?rR2his the initial value of the rate coefficient proposed by Noyes; the DE result for k ( t ) in eq 10 can thus be written in the form k ( t ) / k o = (1 + o)c^(R,t) (44) In order to discuss the equivalence of the two approaches, one must therefore demand first of all that the right-hand side of eq 44 become unity at t = 0; as emphasized at the end of the introductory part of section 111, this can be achieved only by introducing a discontinuity in c^(r,O)( = 1 for r > R ) at r = R (cf. eq 12): c^(R,O)= 1 / ( 1 w ) (45)
+
The rate coefficient defined by setting k(t=O) = It, and using eq 44 for all times larger than zero will have a discontinuity at t = 0; however, as stressed in section IIIB, eq 44 should not be used at all for small values of t. Replacing the RBC in eq 24 by the equivalent condition in eq 29, where the parameter a can be expressed as a = au0/4 according to eq 41, one arrives at the formula k ( t ) a~R'uoc^(R+A,t) i.e.
k ( t ) / k o = c^(R+A,t) A I 21/3 (46) If the two requirements for the validity of the standard DE approach are met-Le., AIR
