5212
J . Phys. Chem. 1985, 89, 5212-5217
The rate constant for this reaction, 1.6 X cm3 s-’, is nearly two orders of magnitude greater than that for reaction of the product N, radicals with fluorine atoms. Hence, N, can be made very rapidly for consumption by an added reagent, such as N(4S) atoms in the present case.
Acknowledgment. This work was supported by the US.Air Force Weapons Laboratory, Contract No. F29601-84-C-0094. The authors are grateful to Professor D. H. Stedman of the University of Denver for helpful discussions during the course of the work.
Diffusion-Controlled Reactions of Isotropic Reagents and Molecules wlth Two Active Sites. Effect of Competition of the Active Sites for the Reagent I. A. Pritchin and K. M. Salikhov* Institute of Chemical Kinetics and Combustion, Novosibirsk 630090, USSR (Received: July 23, 1984)
Bimolecular reaction rate constants have been calculated for spheric reagents: isotropic particles and bifunctional molecules. Effective steric factors have been determined for various radii of reagents and various sizes of active sites. The effect of competition of active sites for reagents has been analyzed. The results of numerical calculations have been summarized as plots which can be used for determining the effective steric factors and the active site sizes with the help of experimental data on reaction rate constants for mono- and bifunctional molecules.
Introduction A bimolecular reaction in liquids is featured by the stage of forming a “pair” from the reagents that encounter inside a certain “cage”.1.2 The encountering reagents reside in the “cage” for some time. For uncharged particles this time is within a nanosecond range; for charged ones it can reach thousands of nanoseconds. During these times the reagents inside the cage are liable to reencounters. In time intervals between the reencounters there occur various processes affecting the reaction product generation. The cage effect manifests itself in quite a number of ways. Most prominently it shows up as chemically induced nuclear polarizations and magnetic effects in radical reactions (see, e.g., ref 3-5). Also the cage effect in liquid-phase reactions appears as averaging the anisotropic reactivity of particles by their translational and rotational d i f f ~ s i o n s . ~ - I ~ An interesting cage effect arises in reactions involving particles with several active groups. Such particles can reencounter through different groups. As a result the reaction of one active group depends on whether or not this very reagent has other active groups; Le., the contribution from the reacting groups to the total (1) Franck, J.; Rabinowitch, E. Trans. Faraday Soc. 1934, 30, 120. (2) Noyes, R. M. ’’Progress in Reaction Kinetics”; Pergamon: New York, 1961; Vol. 1, Chapter 5. (3) Bargon, J.; Fischer, H.; Johnsen, U. Z . Nuturfarsch., A 1967, 22A, 1551. (4) Kaptein, R.; Oosterhoff, J. L. Chem. Phys. Lett. 1969, 4, 195. (5) Molin, Yu. N.: Sagdeev, R. 2.; Salikhov, K. M. Rev.Sou. Authors, Chem. Ser. 1979, I , 1 . (6) Solc, K.; Stockmayer, W. H. J . Chem. Phys. 1971,54, 2981. ( 7 ) Solc, K.; Stockmayer, W. H. Int. J . Chem. Kinet. 1973, 5, 733. (8) Doi, M. Chem. Phys. 1975, 11, 115. (9) Salikhov, K. M. Tear. Eksp. Khim. 1977, 13, 732. (10) Samson, R.; Deutch, J. M. J. Chem. Phys. 1978, 68, 285. (11) Zientara, G. P.; Freed, J. M. J . Phys. Chem. 1979, 83, 3333. (12) Levin, P. P., Burlatsky, S. F.; Ovchinnikov, A. A. Teor. Eksp. Khim. 1980, 16, 746. (13) Berdnikov, V. M.; Doktorov, A. B. Tear. Eksp. Khim. 1981, 17, 318. (14) Doktorov, A. B.; Lukzen, N. N. Chem. Phys. Lert. 1981, 79, 498. (IS) Doktorov, A. B.; Yakobson, B. I. Chem. Phys. 1981,60, 223. (16) Shoup, D.; Lipari, G.; Szabo, A. Biophys. J . 1981, 36, 697. (17) Temkin, S. I.; Yakobson, B. I. J. Phys. Chem. 1984, 88, 2679. (18) Temkin, S. I.; Yakobson, B. I. Khim. Fiz. 1984, 3, 1658. (19) Huddleston, R. K.; Mulac, W. A. J . Phys. Chem. 1982, 86, 2279.
rate constant becomes nonadditive: the active groups compete for the reagent. That competition has been found out experimentally in studying the reactions of solvated electrons with molecules involving two functional acceptor groups.19 The measured reaction rate constant KI2for bifunctional molecules has been comparedI9 with the constants K 1 and K2 for analogous monofunctional molecules ( K , # K2 if the acceptor groups are different). The nonadditive contribution from two acceptor groups in one molecule is characterized by the parameter If the functional groups of the same molecules did not compete, we should have 4 = 1. For a number of bifunctional molecules reacting with solvated electrons 4 < 119 which proves the nonadditive contribution from active groups to the reaction rate constant, In the present paper the effect of competition of two active sites for a reagent has been analyzed theoretically. It has been shown that comparative studies on reactions of mono- and bifunctional molecules allow independently effective steric factors. The Model: Calculation Technique Assume the reacting particles A and B to be spheres with radii RA and Rg. Let the B particles have isotropic reactivity and move with a diffusion coefficient Dg. For example, solvated electrons may be such parti~1es.I~ An A particle reacts only by a portion of its surface, namely, by two axially symmetric active sites with an angular size 0, (Figure 1). The A particles move translationally with a diffusion coefficient DA and rotate with a rotational diffusion coefficient DR. The relative location of the reagents is set by the distance r between their centers and the polar angle B between the direction r and the axis of symmetry for A molecules. Taking into account translational and rotational particle diffusions, the stationary density distribution C(r,0) for B particles relative to a particular A molecule is6 ec = 0
1 = DV2 + DR sin-’ 0
0022-3654185 12089-5212%01.50/0 , 0 1985 American Chemical Society I
,
ae
where D = DA + Dg is the coefficient of relative diffusion of A and B particles.
The Journal of Physical Chemistry, Vol. 89, No. 24, 1985 5213
Diffusion-Controlled Reactions of Isotropic Reagents
Appendix 1). In the present paper we employ the approximate method (6-8).16 Calculations of the Rate Constant The general solution of eq 2 can be presented as m
C(r,O) = a
+ /=0 Cah(r)Pr(cos 0,)
(9)
where P ( x ) is Legendre’s polynomial, f ( r ) is expressed via the modified spherical Bessel functions of the third kind:20
fi(4 = (a/2t/)’/zK/+l,2(t/)
(10)
Here tl = r(l(1 + ~ ) D ~ / D ) I / ~
‘P Figure 1. Geometric sizes of active sites and relative location of reagents.
The condition for infinity gives a =
The chemical reaction is modelled as a flow of B particles to active sites of A particles when the reagents draw together at the RB. In this model we have the following distance R = R, boundary conditions
+
D(dC/dr),,R = KC(R,O) at
o 6 e 6 eo s 712,
o(ac/dr),=R = 0 at Bo C-C,
- eo 6
os7
(3)
< 0 < .rr - 0,
where integration is carried out through the active surface So. The rate constants for reactions of anisotropic reagents with isotropic or anisotropic particles have been calculated several time^.'^^'^ The reaction with one active site has received much study. In the limiting case of a very small active site, the solution has been found in analytic form.I4 However, in the general case the problem (2, 3) has not been solved yet, and hence one has or search for approximate either to use numerical ways of solving the problem. In the limiting case of complete m, one often employs the approximation diffusion control K proposed by Dois (see, e.g., ref 13, 15, 17, 18):
-
Here u is the reaction zone and T the mean residence time of particles in the zone (with the volume v) wherein they may react. In calculations of T the particle reactions are neglected. The data obtained by eq 5 coincide fairly well with numerical calculations by eq 2 and 3.1° There is one more approximate way of calculating K.I6 Here the flow of particles to the active surface is set to be &independent. As a result, instead of eq 2 and 3 one solves eq 2 with simpler boundary conditions D(ac/dr),,R = DQ at D(ac/ar),,,
o 6 o s e, 6 7/2, 7 - eo 6 e 6 n = o at eo < e < 7 - eo (6)
C-Co
c,
Substituting eq 9 into eq 6, we obtain the coefficients a/. a/
=
QU + !&‘(R)[
s_r””p,(x) dx
+ J1P,(x) xo dx]
(13)
Hence ai = (Q/fiW)(f‘/-i(xo)
- P/+I(xo))
(14)
with an even 1 and
atr-m
Here K is the rate constant of B ”sedimentation” on active sites of A particles. The required rate constant of bimolecular reactions is
(11)
a/ = 0
with an odd 1, where xo = cos Bo, P-,(x0) = l,A(R) = (dfi(r)/ &),=R. Thus, the solution (9) involves only even Legendre’s polynomials because the problem under study is symmetrical relative to the reflection at the plane perpendicular to the A molecule axis of symmetry. Using eq 7, we find from eq 9 and 14 m
Q
= 2KCo( 1 - xO)
/ [ 2D( 1 - xO) - K C B z / ] /=0
(1 5)
4 = [(P/-I(X,) - P/+l(Xo))2fi(R)1/Lj;(R)(~ + 1/2)1 At last, using eq 8 we obtain the bimolecular reaction rate constant for isotropic B particles and anisotropic A ones having two active sites m
K12 = 8rKR2D(1 - ~ 0 ) ~ / [ 2 D (-l xO)
- KC&] /=0
(16)
To discuss the competition of active groups we shall need the constant K, for reagents with one active site which has been determined and analyzed numerically by Shoup et al.:16 m
K1 = 8*KRzD(1 - X0)’/[40(1 - XO) - K C B J /=0
(17)
The Effective Steric Factor Let us discuss the limiting cases of diffusion and kinetic controls. Setting the inherent rate constant K (see (3)) to be sufficiently small, we have from eq 16 the rate constant in the kinetic limit K12kin = 4xRZ(1 - xo)K = Kso
(18)
In the other limiting case of diffusion control eq 16 affords
atr-m
m
Kl2diff = -87R2D( 1 - XO)’/ x B z /
(19)
where Q is a &independent parameter. The solution C(r,e) of eq 2 and 6 depends on the parameter Q determined from the exact boundary condition (3) integrated through the active surface So,
In the general case eq 16 can be written as
Further, with Q determined, eq 4 yields the rate constant required
(20) = KlZkin-l + KIZdif 0.15 it is possible to consider only the terms up to 1 = 12 accurate to 1%. For small sites the series in eq 24 and 25 converge slowly and numerical calculations off for small 0, are hampered (see also ref 10). We calculated numerically only the case of 0, 5 0.15 (Figures 2 and 3). For comparison Figures 2 and 3 depict also the geometric steric factors f120, flo. The translational and rotational diffusions of the reagents are seen from Figures 2 and 3 to average the reactivity anisotropy. Curve 1
ANGLE
FA (rad1
Figure 4. Plots of 6 vs. the angular size of an active site. Curves 1 and 2 pertain to kinetic-controlled reactions. Curves 3-7 conform to the limiting case of diffusion control for various Y: (3) 0, (4) 0.4, (5) 0.8, (6) 1.4, (7) 10.
corresponding to Y = 0 describes the averaging only due to translational diffusion of the reagents. The deviations of curves with Y # 0 from curve 1 reflect the contribution from rotational relaxation to averaging the reactivity anisotropy. Figure 2 shows the rotation makes an appreciable contribution to the process at Y 5 0.4 only which, according to eq 27, corresponds to RB k O.1RA. The rotational motion is especially important when the radii of anisotropic particles become smaller than those of isotropic ones. If the rotational contribution tof12 is negligible ( Y 5 0.4; RB 5 O.lRA), it is possible to derive an approximate formula forf,, which is in good agreement with numerical calculations for Y = 0 and Bo k 0.15 (curve 1, Figure 2) and gives a precise asymptotic dependence for small Bo (see Appendix 2). With the results from Appendix 2, we obtain the following expression for the effective steric factor (16) for small Bo values f12
= 00/[16/W
- (00/2) In (1/00)1
(28)
Expression 28 is the doubled value off, obtained by Doktorov and Lukzen.14 It means that in the range of small angles the contribution from active sites to the total rate constant becomes additive. The above formula makes it possible to extend curve 1 in Figure 2 to the range of small 0,. The Competition of Active Sites As stated above, the competition of active sites for a reagent can be analyzed with the parameter @ (1). Here two cases are possible: 0, < a / 2 and Oo > r / 2 . If 0, > r/2, active sites of different group overlap when a monofunctional reagent is r e p l a d by a bifunctional one. Figure 4 (curves 3-7) shows plots of @dip vs. the active site size and the parameter Y for the case of full diffusion control. For small 0, two active sites react practically independently and hence @ = 1. As Bo increases from 0 to */2, t#J reduces to t#Jdin(*/2). If the reagent rotation is negligible, RB5 O.lRA, &,i&~/2) z 0.7. Due to rotational diffusion &iff reduces and 0.7 > @Pdiff(~/2) > 0.5. When the ranges of the active group effects overlap, &iff
The Journal of Physical Chemistry, Vol. 89, No. 24, 1985
Diffusion-Controlled Reactions of Isotropic Reagents
10
0.9
08 0.7 0.6 Parameter @
0.5
Figure 5. Plots of the effective steric factors for monofunctional (curves 1-3 forfi) and bifunctional (curves 4-6 forfi2) reagents vs. the parameter 6 in the limiting case of diffusion control. Every curve corresponds to a definite Y: (1) 0.8, (2) 0.4, (3) 0, (4) 0.8, (5) 0.4, (6) 0.
decreases to 0.5. The limiting value of 4 = 0.5 conforms to the case offl2 = l,f, = 1; i.e., the molecules with both one and two active sites show isotropic reactivity. In the kinetic limit &in = 1 until Bo < r / 2 . When the ranges behavior is determined of the active group effects overlap, the by the reactivity of the overlapping ranges. If the overlapping active sites result in doubling the inherent rate constant K (see eq 3), &in = 1 for any Bo, 0 < Bo < r (Figure 4, curve 1). If K of a bifunctional particle in the overlapping regions is the same as that on the active surface of a monofunctional particle, then +kin
= 1/(1
5215
- cos 80) for r / 2 < Bo r / 2 (cf. Figure 4, curve l), we obtain K ~ ( K l 2 - l- (2KI)-') =f12-' - (2f,)-'
10 15 20 2 5 A N G L E 8. I r a d l
30
Figure 6. Plots of KD(KIC1- (2KJ') vs. the angular size of an active site for various Y: (1) 0, (2) 0.4, (3) 0.8, (4) 1.4.
(29)
Figure 4 shows that in the diffusion limit 4 is lower than that in the kinetic limit, &iff < $kin. In the general case &iff < 4 < and 4 runs from 1 (no competition for a reagent) to 0.5 (maximum competition). Note that the competition effect depends on the relative location of active sites on a particle surface. The competition is minimum if the active sites are at opposite poles. Therefore, the effective steric f a ~ t o r f and , ~ the competition parameter 4 for reagents with some other relative location of its active sites are lower than those calculated in the present paper.
Kdiff = K d e f f
05
(31) Here the left-hand side includes values measured experimentally
ANGLE
0, (rad)
Figure 7. The ratio Kl/KD against size of an active site calculated for various K R / D (1) m, (2) 10, (3) 5, (4) 3, (5) 2, (6) 1.4, (7) 1, (8) 0.714, (9) 0.5, (10) 0.333, (11) 0.2, (12) 0.1. The curves are for Y = 0 and thus applicable only to systems with negligible rotation.
and the right-hand side is a theoretically calculated function depending on the sizes of molecules and active sites. Figure 6 gives the plot of \k = f12-' - (2A)-l. With experimental values of K , , K 1 2 ,K,, and Figure 6, it is possible to obtain Bo and then K by using Figure 7. Figure 7 shows K,/KD against Bo calculated for various parameters KRID. However, one should remember that \k depends weakly on 80 for Bo 5 r/3 and hence the accuracy of 80 is here not high. As stated above, the competition effect has been found out e~perimentallyl~ in the reaction of solvated electrons with monoand bifunctional reagents. For two different pairs of reagents it = 0.86 f 0.08 and c # J ~=~ 0.65 ~ ~ ~f has been obtained that 0.04 Huddleston and Mulaclg assume the reactions to be fully diffusion controlled. The reagents studiedTgare stretched spheroids with acceptor groups at poles. If a spheroid is interpreted in terms of the model of reacting spheres with the anisotropic sphere radius set to equal the major semiaxis, the active site size can be determined from Figure 4. The value of 4 = 0.86 corresponds to Bo = 45O, and 4 = 0.65 to Bo = 100' which means that the bifunctional molecule has isotropic reactivity. However, when applied to spheroid molecules, the theory developed for spheric molecules overestimates the space angle of an active site on a spheroid surface. Indeed, the competition between two active sites at poles of a sphere is lower than that on a spheroid since in the former case the reagents have to cover longer distances between opposite poles for reencounters. To obtain a more correct quantitative interpretation of experimental data,Ig it is necessary to calculate the model of anisotropic spheroids.
Acknowledgment. We are thankful to A. B. Doktorov and B. I. Yakobson, who called our attention to ref 8 and also to the possible equivalence of approximate methods (5) and (6). We also thank Professor A. Szabo of Princeton University; in reviewing the manuscript, he pointed out that the structure of the solution of the problem studied reflects its symmetry properties.
5216 The Journal of Physical Chemistry, Vol. 89, No. 24, 1985
Appendix 1 Let us demonstrate that approximations 5 and 6 in fact coincide. Write eq 5 in another form. Let Soand A be the active surface area and width, respectively. In this case, the reaction zone volume is v = SOA (Al-1)
Pritchin and Salikhov
Q = (C$,/DR4)/S
(Al-12)
d 9 S d\k’G(R,\k’lR,\k) *n
yo
Having substituted it into eq 8, we obtain eq A1-6. So the two approximations discussed are indeed equivalent. Appendix 2 If Y 0 and the role of rotational diffusion in averaging the reactivity anisotropy is negligible, thenfi(R)/f(R) = - l / ( l + 1) and eq 24 for the steric factor, using eq 13, is (A2-1) f i 2 = (1 - x o ) / 2 -+
The residence time of a nonreacting particle in the reaction zone is expressed via its conditional probability P(7li;t) of passing from the point r f to the point r during the time t2I
where m
C(1+ 1/2)C//(I
Z =
(Al-2)
/=0
where G is the Green function of the problem (2) which is the solution of the equation LG = 4 ( 7 - 7’) (Al-3)
m
G-Oatr-m
( R 4 A / S 0 ) 1 d q s d W G(R,\k’lR,\k)
Zl = /=0 ? ( L ; P / ( x ) dx)2 = 2 5 S ” d 0 , sin
*n
d\E’ G(R, \k‘lR,\k)
(A2-3)
Here the first right-hand term can be summed up. Let us consider
*n
= ( S O 2 / R 4 ) / 1 d\kJ
U/T
m
/= 1
(Al-5)
with the integration carried out over the space angles of the active surface. As a result, eq 5 takes the form Kdirr=
(A2-2)
P/(x)P!(y)
z = /=0 zc/- 1 / 2 E C / / ( I + 1) - (1 - XO)*
(Al-4)
If the reaction zone is sufficiently thin, the mean-value theorem affords *a
+ xddxJ-xndy -1
This sum can be written in the form
with the boundary conditions (aG/a?),=R = 0
T
C, = ( x j P , ( x ) dx)2
+ 1)
/=o
el ~ ‘ * d O ,sin 02PI(cos Bl)Pl(cos 6,)
(A2-4)
0
Substituting the integral representation of P,(cos P/(COS e) =
*o
(Al-6)
( 2 / r ) x 6 d p cos [ ( I
+ 1/2)p]/(2(cos
cp -
cos e))’/* (A2-5)
Let us show now that approximation 6 leads to a similar result. With the Green formulaZ2we can solve eq 2
into eq A2-4 and using the well-known sum of the series23
C(?) = - D ~ [ C ( P ) a C / a i i S , - GdC(P)/a&] dS’
c c o s [(I + y 2 ) c p 1 ~ , (e)~ ~=s
(Al-7)
-
/=0
integrating through both the sphere and an infinitely far surface, and differentiating with respect to the outer normal to the surfaces, restricting a domain out of the sphere with radius R. Note that
G
aG/ans aC/ans
Y4ar
-1/,a$
-
1 /r2 for r
-
-
for r
1/(2(cos
(Al-8)
m
at 0
< p < 0 < ?r
(A2-6)
=OatO
