J. Phys. Chem. 1984, 88, 4729-4735
4729
Theory of Elementary Bimolecular Reactions in Liquid Solutions. 2. Prediction of Bimolecular Reaction Rates from Fundamental Molecular Parameters A. J. Benesi Department of Chemistry, University of California, Irvine, California 9271 7 (Received: September 2, 1983)
The rates of elementary bimolecular reactions in liquid solutions are shown to depend on the translational and rotational diffusion coefficients of the reactants and on the intrinsic reactivity and size of their reactive spots. For the special case of spherical molecules with axially symmetric reactive spots, a quantitative expression for the reaction rate is found. Thus, it is now possible to calculate bimolecular reaction rates from fundamental molecular properties. Variation of the rotational diffusion coefficients of the reactants from zero to infinity has less than an order of magnitude effect on bimolecular reaction rates, since translational diffusion also causes orientational relaxation. The effects of orientational relaxation on reaction rates are most apparent at high intrinsic reactivities.
In1roduction In liquids, where molecular displacements are small, pairs of solute molecules generally recollide 5-500 times before they escape from each other.’ This provides reactive pairs with many opportunities for reaction. Since reorientation can occur between successive recollisions, rotational diffusion of the molecules affects bimolecular reaction rates. The clustering of collisions in liquids creates a natural division between first collisions and recollisions. The first collisions between molecules in liquids are orientationally random. In the absence of intermolecular forces there is no preferred orientation of reactive spots (corresponding to molecular orbitals) on the molecules with respect to each other, and all relative orientations occur with equal probability. However, after a first collision, the time during which the cluster of recollisions occurs is not long enough to provide complete reorientation of the spots for all the recollisions. Thus, recollisions are not orientationally random with respect to the first collision or to each other. It is possible to calculate the probability of a given relative orientation of spots for two molecules as a function of time, if their relative orientation a t some initial time is known. The time dependence of the relative orientation depends on the rotational diffusion coefficients of the molecules and on a contribution to angular motion from relative translational diffusion. From the time dependence of the relative orientation, the ensemble average number of reactive spot contacts and the reaction rate itself can be calculated. Theory I. Assumptions. Molecules A and B are assumed to be impenetrable spheres of radii rA and rB. It is assumed that there are no attractive or repulsive forces between the A and B molecules. The translational diffusion coefficients, DA and Dg, are assumed to be isotropic. The rotational diffusion coefficients, eA and eg, are also assumed to be isotropic and in units of s-’. eA = I /(27A),where 7 A is the rotational relaxation time of the A molecule, and eB= 1 / ( 2 ~ ~ ) . ’The reactive spots are assumed to occupy solid angles a and @ of the A and B molecules, respectively (see Figure 1). The spots represent particular molecular orbitals of the molecules. In the reactive case, reaction can only occur if the reactive spots make contact. The total collision rate is assumed to be given by the Maxwell collision rate,3 although any total collision rate could be used. I I . Analytical Expression for the Time Dependence of Recollisions. In this section, neither the relative orientational motion of the spots nor their reactivity is considered. First, it is necessary to describe the ensemble-average behavior of a molecular pair as it undergoes a first collision followed by a “recollision cluster”. (1) Benesi, A. J. J. Phys. Chem. 1982, 86,4926-30. (2) Bull, H. B. “An Introduction to Physical Biochemistry”; F. A. Davis: Philadelphia, 1971; 2nd ed, pp 276-302. (3) Levine, Ira N.; “Physical Chemistry”;McGraw-Hill: New York, 1983; 2nd ed.
0022-3654/84/2088-4729$01.50/0
This phenomenon depends on the radial part of the relative translational diffusion coefficient, D = DA Dg. The total collision rate is3
+
Ztot = ~a’creinAn~ (1) where a = rA + rB,crel= (8kgT/Tp)’/’,kBis Boltzmann’s constant, T i s the absolute temperature, p = mAmB/(mA mB),mA and mg are masses of the A and B molecules, and n A and ne are the number concentrations of A and B molecules, respectively. The first-time or Noyes collision rate is1v4
+
where D = DA + Dg. The average number of recollisions per first collision is’ (3)
Any valid expression for the total collision rate could be used in eq 2 and 3 . With eq 1 and 2 N = acrei/(4D) (4) With eq 1 chosen as the total collision rate, the average frequency of recollisions in a cluster is1 k = 3 c r e ~ / ( 4 a c r e l 160) (5)
+
The average time between recollisions is thus 4acrel+ 160 t,, = l / k = 3cr2 If 4 is defined as the probability that the molecules in a pair will not escape before making another collision, then (1 - 4) is the probability that the molecules will escape before making another collision. If we start with a s~perensemble~ of Nsupens molecular pairs that undergo first-time collisions a t t = 0, the fraction of the superensemble which will make just one recollision is (1 - 4)4. The fraction which will make just two recollisions is (1 - 4)&. The fraction which will make just j recollisions is (1 - +)@. It follows that N, the average number of recollisions per first-time collision, is given by m
A
v,
(1 - $)CjI#J= --N j=o
1- 4
(7)
Thus
N N+l 1-4=- 1 N+ 1
4=-
(4) Noyes, R. M. Prog. React. Kinet. 1961, I , 129-60. (5) TF superensemble will be separated into different ensembles in the next section.
0 1984 American Chemical Society
4730 The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
Benesi
I/
a
b
Figure 1. Cross sections through the spherical A and B molecules, showing that the spot sizes are determined by the angles and 0 between (Y
the centers of the spots and the edge of the spots.
6ot
7 =i C
d
e
f
1 I
I
I
I
I
I
I
I
I
I
,
Oo
5
IO
15
20
25
30
35
40
45
50
I
.
55
(
m
t (set/ lo-")
Figure 2. The time dependence of recollisions for a superensemble of molecular AB pairs which suffered first-time collisions at t = 0: ( 0 ) calculated from eq 11; (0) calculated as in ref 1 by assuming interpenetrability of the molecules. N,,(?) is the total number of recollisions is the number of molecular made by the superensemble at time t , Nsupns pairs in the superensemble. rA = 5 A, rB = 2 A, DA = 4.90 x 104 cmZ/s, DB = 1.23 X cm2/s, N = 58.71, k = 6.07 X lo1' s-'.
If there was no loss of molecular pairs from the superensemble due to escape, one would expect the number of recollisions made by time t to be (N,,,,kt). But, due to the escape of molecules, there is a time-dependent decrease in the number of molecular pairs making recollisions. With the fact that the ensemble-average first recollision occurs at a time corresponding to kt = 1, that the second occurs when kt = 2, etc, it is found that
where NsUw is the number of molecular pairs in the superensemble at t = 0, and N,,(t) is the total number of recollisions which have occurred for the superensemble by time t . In eq 10, k is the fundamental frequency of recollisions, and $kT is the fraction of pairs which have not escaped (e.g., are still ~ns the making recollisions) by time 7. N r ~ ( t ) / N s urepresents average number of recollisions made by a molecular pair by time t after a first collision. The solution to eq 10 is
Figure 3. (a) A molecular pair with reactive spots in contact. (b) Relative translational diffusion, without rotational diffusion, causes loss of reactive spot contact. (c) Rotational diffusion, without translational diffusion, causes loss of reactive spot contact. (d) Translational diffusion and rotational diffusion occur simultaneously. (e) An ensemble of seven molecular pairs which start with reactive spot contact can be represented in terms of seven intermolecular vectors for each of the two molecules. (f) The combination of relative translational diffusion and rotational diffusion can be represented in terms of rotational diffusion of the intermolecular vectors relative to the reactive spots of the separated molecules.
angular diffusion at a constant intermolecular distance is treated. The intermolecular distance between centers is chosen as a = rA rB,the radius of the collision sphere, since the relative orientation of spots at this distance determines whether the spots make contact in a collision. The phenomenon is shown to depend on the rotational diffusion coefficients of the A and B molecules and on the angular component of the relative translational diffusion coefficient in spherical polar coordinates. Consider an ensemble of first collisions in which nonreactive spots of the A and B molecular pairs make contact! The situation is shown schematically in Figure 3. It is apparent that the vector which connects the centers of a molecular pair (hereafter called the intermolecular vector) must intersect both spots for spot contact to occur. For this ensemble, the initial probability is 1 that this will be the case. And, since there is no preferred relative orientation of the pairs, the intermolecular vectors for a large enough ensemble will be distributed evenly over the surface of each of the spots at d = 0 (as in Figure 3e). Thus, for the A molecules at t = 0
+
J a p ( 8 ~ , o ) Sin 8~ d8A = 1 Nrec(t!/Nsupens is plotted against time in Figure 2. The corresponding plot according tQ the error function method used in ref 1 is plotted for comparison. Note that no assumptions of interpenetrability are required to obtain eq 11, since eq 1 and 2, which lead to the expressions for k and 9, apply for "hard" spheres. III. Orientational Diffusion of Molecules with Nonreactive Spots. This section introduces the ensemble-average relative orientational motion of a molecular pair. The particular ensemble of pairs treated is the population of AB pairs which undergo first collisions in which nonreactive spots make contact. Relative radial translational diffusion as manifested in the recollision clusters of section I1 is not treated. Neither is reactivity. Only relative
(12)
where LY is the solid angle defining the spot of the A molecule, 0, is the angle between a given intermolecular vector and the vector through the center of A's spot, and p(OA,o) sin OA de, is the probability of finding an intermolecular vector at angle BA at t = 0, (p(OA,O) = constant = 1/(1 - cos a ) , 0 IBA Ia; p(Op,O) = 0, a < 0 I T). An analogous expression describes the initial distribution of intermolecular vectors for the B molecules in the ensemble. (6) This is one of the ensembles in the superensemble of all first-time collisions which occur at t = 0.
Bimolecular Reactions in Liquid Solutions
The Journal of Phydcal Chemistry, Vol. 88, No. 20, 1984 4731
It is apparent from Figure 3 that there are two types of orientational diffusion which contribute to spreading of intermolecular vectors relative to the center of the spot of the A molecule. One type arises from the translational diffusion of the B molecule relative to the A molecule (compare Figure 3, a and b), which is characterized by the relative translational diffusion coefficient D = DA + DE. After transforming to spherical polar coordinates, we define an analogue of a rotational diffusion coefficient, e t , = D / $ . Because it is angular diffusion at the surface of the collision sphere that determines the intermolecular vector distribution for collisions, r = a = rA + rB,and
et,= D / a 2
(13)
where D = DA + DB and a = rA + rB. It is assumed that the rotational diffusion of the intermolecular vectors relative to the center of the A molecule’s spot can be correctly described in terms of a total rotational diffusion coefficient which is the sum of 8, and the “true” rotational diffusion coefficient of the A molecule, which also contributes to the spreading of intermolecular vectors (compare Figure 3, a and c). Thus 8Atot
= 8tr + e
A
(14)
where BAmtis the total rotational diffusion coefficient which applies to diffusion of the intermolecular vectors relative to the center of the spot of the A molecule. Similarly, for the B molecule it is assumed that %tot
= e t , + OB
(15)
The choice of the centers of the respective spots as 8 = 0 is a convenient one, because the initial conditions are always symmetrically distributed around 0 = 0, and there is no dependence on the longitudinal angle 4. There are two diffusion equations to be solved, one for the A molecule and one for the B molecule. They both have the same form
1 5 h
I0l \I
t = 5.0 x
____ 0
40
20
sac
/:=-+
-
_ _ - __ _ _ _ - ______ ----- _ - _ - _
60
EO
100
120
Thus a single phenomenon, the relative random motion of two molecules with respect to each other, has been separated into three different parts. The relative radial translational diffusion was separated from the relative angular motion to obtain the results of section 11. Here it is proposed that the treatment of relative angular diffusion is simplified by focussing on the rotational diffusion of the intermolecular vector relative to the spots of the separate molecules. The rotational diffusion of this vector is described in two separate differential equations, one for each molecule. These equations are “coupled” through e,, the angular component of the relative translational diffusion coefficient in spherical polar coordinates, which contributes to the rotational diffusion of the intermolecular vector for both molecules. The general solution to eq 16 is known to be the Fourier-Legendre series7q8
where 8 , = 8Am or 8Btot, PAWS 8) is the lth Legendre polynomial, 1 = 0, 1, 2, ..., and cI is the coefficient of the lth Legendre polynomial. The coefficients cI can be evaluated by using the or(7) Poole, C. P.; Farach, H. A. “Relaxation in Magnetic Resonance”; Academic Press: New York, 1971; p 226. (8) Butkov, E. “Mathematical Physics”; Addison-Wesley: Reading, 1968; pp 350-2.
160
I80
Figure 4. p ( 8 ) , the probability of finding an intermolecular vector at angle 8, plotted as a function of 8 for an ensemble of intermolecular vectors which start in contact with the A molecules’ spots (see Figure 3e). = 4.973 x io9 s-~,eA= 1.472 x io9 s-I, e,, = 3.501 x io9 s-I, (Y = 13.00°. p ( 8 ) was calculated numerically by using eq 17 and 18.
thogonality property of the Legendre polynomials, along with the initial conditions p(8,O). Thuss 21+ 1 cI = ,-Jrp(8,0) P,(cos 8) sin 8 d8 (18) From the initial conditions given in eq 12, p(8A,t) was calculated numerically for the A molecule of Figure 2 with eq 17 and 18.9 The results are shown in Figure 4. Note that p(OAt) does not vanish as t m, but instead approaches By time t, the fraction of intermolecular vectors which intersect the spots of the A molecules is
-
PAspt(t) = J‘P(~A,~) 0 sin^ OA
where
140
6 (degrees)
(19)
An analogous expression holds for PBspt(t). I K The Number of Repeat Spot Contacts per First-Time Spot Contactfor Nonreactive Spots. In this section, the collision cluster phenomenon of section I1 is combined with the orientational relaxation phenomenon of section I11 to give the ensemble-average number of first-spot contacts and repeat spot contacts for a collision cluster. The number of first-spot contacts per cluster is always less than or equal to one, since the cluster is the record of collisions for a single pair of molecules. Reactivity is not considered. The product PAspt(t)PBspt(t) gives the probability that the intermolecular vectors contact both spots by time t . That the intermolecular vectors touch both spots does not imply that the spots are in physical contact, however. To calculate the actual number of spot contacts, one must know the fraction of pairs undergoing recollisions by time t . This fraction is given by @‘, the “escape” factor in eq 10. Combining terms and integrating with the fundamental recollision frequency gives the average number of spot contacts made in recollisions by time t for a molecular pair:
where Nspt ( t ) is the number of spot contacts which have occurred in recollisions by time t for an ensemble, and Ne,, is the initial number of molecular pairs in an ensemble. Nspt(t)/Nens has been (9) Each c, value from I = 0 to I = 100 was evaluated according to eq 18. One degree increments in 8 were used for all numerical integrals in this paper. Truncating the infinite series of eq 17 at I = 100 provides sufficient accuracy for the purpose of this work.
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
4732
Benesi For the ensemble which was originally considered in section
-
111, we can calculate the average number of repeat spot contacts m in eq 20. Let per first spot contact by letting t
Since all spot contacts are equivalent, regardless of the ensemble from which they originate, it follows that the ratio of repeat-spot contacts to first spot contacts must be Q for the entire superensemble of first collisions. Thus, one can divide the total number of spot contacts made by the superensemble into first spot contacts and repeat spot contacts. Letting t + m in eq 22 and multiplying by the appropriate factors, we obtain
*
zs
10.80 c
Y
c 0 0
0.60
-
z 0.40
1
(1
Nfirst
--
Nsupens
0.00 o 0
'IO
20 2
0
~
-Figure 5. Nspt(t)/Nens, calculated numerically from eq 20, = 2.299 x 1010s-l and eBtot = 2.649 x 1O'O s-l, and 8, same as in Figure 4.
calculated numerically for the same molecular parameters used in Figures 2 and 4.1° The results are plotted in Figure 5. Up to this point we have been considering an ensemble of pairs which had first collisions (at t = 0) in which the intermolecular vectors intersected the spots of both molecules. The methods developed in eq 12-20 are also applicable to other ensembles (for example, the ensemble of molecular pairs which have first-time collisions in which the intermolecular vectors intersect neither spot). The superensemble of all first-time collisions is composed of four ensembles: the ensemble of pairs with intermolecular vectors which intersect both spots, the ensemble of pairs with intermolecular vectors which intersect A molecules' spots but not B molecules' spots, the ensemble of pairs with intermolecular vectors which intersect B molecules' spots but not A molecules' spots, and the ensemble of pairs with intermolecular vectors which intersect neither molecule's spot. For the superensemble as a whole and in the absence of reaction, the fraction of unescaped pairs with spot contact is a constant, since each intermolecular vector which rotationally diffuses out of a spot is replaced by one diffusing in. This fraction is given by the product of the fractions of the molecular surfaces which are occupied by the spots, (1 - cos a)(1 - cos @)/4(see section I). When combined with the time dependence of molecular escape (eq 1 l), one obtains the time dependence of the average number of recollisions with spot contact for a molecular pair chosen at random from the superensemble: Nspot(t) -- (1 -cos Nsupens
a ) ( 1 -cos p ) [ p - 11 4 In 9
(21)
where Nspt(t)is the number of spot contacts which have occurred in recollisions by time t for the superensemble, and Nsupens is the initial number of molecular pairs in the superensemble at t = 0. Adding spot contacts from first collisions gives, after rearrangement Ntot spot(t)
Nsupens
- (1 - cos a)(l - cos p) 4
(bkt+ In 4 - 1) In 9
(22)
This represents the average number of spot contacts, including those from first collisions, as a function of time for a molecular pair chosen a t random from the superensemble of first collisions. (10) For short times T , PASPI and P B change ~ ~rapidly, ~ so small time increments must be used. At longer times, PASPIand PBspor change slowly, so larger time increments can be used.
+
Q( 1 - COS a)(1
Nrepcat Nsupens
t (see/ IO-")
-
a)(l - C O S p) 4(1 Q ) (1
-COS
-
4(1
- COS p)
+ Q)
(1
-A) -&)
(24) (25)
where Nfirst/Nsupens is the average number of first spot contacts made by a molecular pair chosen a t random from the superensemble, Nfi,,/NS,,,, is the average number of repeat spot contacts made by a molecular pair chosen at random from the superensemble of first collisions, and Note that when a = /3 = T , Nfirst/Nsupens attains its maximum possible value of 1, because (1 + Q ) = (1 - 1 /In 4) in this case. Also note that a superensemble of first collisions is not created instantaneously at t = 0. It is created continuously at a rate given by eq 2. Despite this, the results obtained so far apply to the average molecular pair. With this in mind, one can immediately write the rates of nonreactive spot contacts: Nfiirst
zfirst
= -ZNoyes Nsupens
where ZfirSt is the rate of first spot contacts for nonreactive spots (contacts cm-3 s-l) Nrepeat
Zrepeat
= -ZNoyes Nsupens
where ZrePtis the rate of repeat spot contacts, nonreactive spots (contacts cm-3 s-l). V. Bimolecular Reaction Rates. In this section, the effect of reactivity on the number of spot contacts which can occur in a collision cluster is found. This result leads directly to an expression for the bimolecular reaction rate. In considering reactive spots, it will be assumed that the probability of reaction is K if the reactive spots make contact (0 5 K 5 1). The reactivity of a spot (hereafter called the intrinsic reactivity) is assumed to be uniform throughout the spot, with zero reactivity elsewhere on the molecule's surface. It may be possible to calculate K if the potential energy surface connecting reactants and products is known. It will be seen that K can be found from an experimental measurement of the bimolecular reaction rate if other molecular parameters (diffusion coefficients and spot sizes) are known. One would expect that K values for similar types of reactive spot combinations would be comparable. The ability to separate diffusion effects on the reaction rate from quantum mechanical effects (manifested in K ) may find practical application in the study of bimolecular solution reactions. Before dealing with spot reactivity, one must again consider Q, the average number of repeat spot contacts per first spot contact for nonreactive spots. Since first spot contacts are equivalent to repeat spot contacts in that the molecules move with the same translational and rotational diffusion coefficients, it follows that the probability that the molecules will not escape before making another contact, &on, is the same for all spot contacts.
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4133
Bimolecular Reactions in Liquid Solutions
TABLE I 1 Effect of Angular Diffusion on Bimolecular Reaction Ratesa r, = rB = 1.0 x cm DA = Dg = 2.45 X cd/s D,, = 4.90 X cm2/s e,, = 1.23 x 1 0 1 1 S-I CY = 0 = 69.00’ N = 22.86 k = 8.03 X 10l2s-]
TABLE I: Effect of Angular Diffusion on Bimolecular Reaction Ratesa rA = 5.0 X cm; rg = 2.0 X lo-* cm DA = 4.902 X lod cm2/s; DB = 1.226 X cm2/s D,, = DA DB = 1.716 X 10” cm2/s e,, = Dtr/(rA rg)2= 3.5 X lo9 s-’ CY = 13.00’; 0 = 33.00’ N = 58.71 k = 6.07 X 10” s-’
+
+
4 1 + Qreac,). = eBtot = 1.23 x 10” s-l, 0 = 9.87, Nfirst/Nsu,ns = 0.23, NrepcatlNsupens = 2.28 0 0.23 0.83 0.21 4.5 0.13 8.9 0.023 9.8 0.0025
Case 1: 1.3 X 0 = 3.71, Nfi,,,t/Nsupens Nrewl/Naupns = 4.9 X lo-’ 0 1.3
1 0.50 0.10 0.01 0.001
1 0.50 0.10 0.01
X
1.1 x 4.5 x 6.0 x 6.2 x
0.65 2.4 3.5 3.7
10-2
10-3 10-4
0.001
10-5
NfirstlNsupens
Oreact
K
8Atot
Case 2: eAtot = eBtat = 3.06 X lo1], Q = 6.59, Nfirst/Nsupens = 0.33,
Case 2: eAtot = 4.973 x io9, ea,,, = 2.649 x 1010. Q = 1.57,
= 2.18 0 0.33 0.77 0.29 3.6 0.15 6.1 0.024 6.5 0.0025 Case 3: eA = OB = m, 0 = 2.40, Nfirst/Nsupens = 0.74, NrepeatlNsupens = 1.77 1 0 0.74 0.50 0.55 0.57 0.10 1.7 0.20 0.01 2.3 0.025 0.001 2.4 0.0025 NrepeatlNsupens
1 0.50 0.10 0.01 0.001
..
0.50
0.44 1.2 1.5 1.6
0.10 0.01
0.001 Case 3:
1.7 X lo-’ 5.4 x 10-3 6.1 x 10-4 6.2 x 10-5
eBtot
= = m, 0 = 6.12 NfirstlNsupena = 5.9 X lo-’, NrepeatlNsupens = 3.6 X lo-’ OAtot
5.9 x 3.0 X 6.2 x 6.2 x 6.2 x
0 0.030 0.055 0.061 0.061
1
0.50 0.10 0.01 0.001
x lo-’, 10-2
10-3
10-4 10-5
+
“The reaction rate, in units of ZNoyes,is given by ~ ( 1 QreBCtiVe)Nfirst/NSUpcns (see eq 32). Case 1 gives the results obtained when e A = eg = 0; case 2 for e A = kBT/8*vrA3,e g = kgT/8*vrg3; and case 3 for 8, = eB = m.
The fraction of molecular pairs which make just one repeat spot The fraction of pairs which will make contact is (1 - &,n)q5mn. just two repeat contacts is (1 - &,n)4m:. The fraction which will ,&.:, Thus make just j repeat spot contacts is (1 - &J
’The reaction rate, in units of ZNoycs,is given by ~ ( 1+ O,,c,ivc)Nr~st/Ns~pcns (see eq 32). Case 1 gives the results obtained when e A = eg = 0; case 2 for e A = kBT/8rvrA3,eg = keT/8rvrg3; and case 3 for e A = OB = a. ZNoyes for several values of f3,,0B, and K are shown in Tables I and 11. The data in Table I were calculated from the same rA, rB, DA, DE, a, and /3 values that were used in Figures 2, 4, and 5. Reduced to the level of fundamental parameters, one obtains Zreaction
=
52
9con = ~+n If the intrinsic reactivity of the spots is incorporated, the probability that the molecules in a pair will not escape and not react before making another reactive spot contact is (1 - K)&,~. This implies that the probability that the pair will escape or react is [ l - (1 - K)$con]. By analogy with eq 7 and 29, the average number of reactive spot repeat contacts, areact, is thus m
(33)
j-0
+ I[]: Q,,,, is the average number of repeat spot contacts per first spot contact and takes into account both reaction and diffusion effects. The reaction rate, in uits of molecules cm-3 s-l, can be written immediately as Zreaction
+
=~ (+ 1
Nfirst QreacJ-ZNoyes Nsupens
(32)
The product (1 nreBCt)NfrSt/NSUpCns gives the average number of reactive spot contacts per cluster. Reaction rates in units of
The only fundamental parameter which requires computer evaluation is 52. Effects of angular diffusion on bimolecular reaction rates are manifested entirely in s2. Quantum mechanical effects on the reaction rate are manifested entirely in K . Effects of radial translational diffusion are manifested entirely in D. Discussion The average number of recollisions made by a molecular pair is plotted as a function of time in Figure 2 for A and B molecules with radii of 5 and 2 A, respectively. The translational diffusion coefficients were calculated from the Stokes-Einstein equation,2
4734 The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
Benesi
and N a n d k were calculated from eq 4 and 5. Two sets of data reaction rate on the rotational diffusion coefficients essentially are presented. The solid circles were calculated from eq 11 for vanishes. hard spheres, and the open circles by assuming interpenetrability It is difficult to put this theory in the perspective of previous as in ref 1. theoretical or experimental results. Experimental data on biThere is a considerable difference in the time dependence of molecular reaction rates in solutions have not been tabulated as recollisions as calculated by the two methods, although they apfunctions of reactant translational diffusion coefficients, rotational m. The analytical solution (eq 11) proach similar values as t diffusion coefficients, reactive spot sizes, and intrinsic reactivities m and predicts that more recollisions approaches - l / h 4 as t of reactive spots. Even if such data have been tabulated, one would occur at earlier times. The interpenetrability method assumes have to be careful about effects of intermolecular forces, anisothat the centers of colliding B molecules are distributed evenly tropic diffusion, nonspherical molecular shape, and intramolecular within the collision sphere a t t = 0 and that there is no concensteric effects in comparing experiment with theory. This theory tration gradient of B molecules within the collision sphere at t is good for making quantitative predictions only when the latter = 0. This leads to lower recollision rates at early times, because effects are insignificant. The theory does make the interesting molecules escape more slowly with a lower concentration gradient. prediction that rotational diffusion is relatively unimportant in its effect on solution reaction rates. Figure 4 shows p ( 8 ) for three different times (eq 12 and 17). Previous theoretical treatments take a different approach than For an axially symmetric distribution of intermolecular vectors, the one presented here. In previous treatments, a steady-state .forp(8) sin 8 d8 is constant, but . f o f f p ( 8 )d8 is not. In Figure 4, approach is The focus of attention is on the ensemble p ( 8 ) is shown for the A molecule. e A t 0 t was calculated from eq of unreacted A molecules and on the angular dependencies of the 13 and 14 and by assuming that the rotational diffusion coefficient steady-state concentration of B molecules and the radial conof the A molecule is given by F)A = kT/8?rqrA3.’ centration gradient of B molecules surrounding them. The difThe spot size is a very important parameter. A spot surface fusive flux of B molecules into the reactive spots of unreacted A area of about 4 A2 was chosen for both the A and B molecules, molecules is equal to the overall reaction rate, so a steady state since this value corresponds to what is expected for a “typical” is obtained. The steady-state concentration and radial concenlocalized molecular orbital. This results in values of CY = 13O and tration gradient of B molecules are asymmetrically distributed /3 = 33’ for the A and B molecules of Figures 2, 4, and 5. over the ensemble-average unreacted A molecule. At the reactive The results plotted in Figure 5 give the average number of spot the local concentration of B molecules is lower than the rest repeat spot contacts as a function of time for pairs which start of the molecular surface because reaction has preferentially with spot contact (as in section 111). In this case, Nspt!)/Nens consumed A molecules with high local B concentrations at their 1.571 as t m (eq 20), so Q = 1.571, N~rst/Nsupns - 2.421 reactive spots. X loe2, and Nrept/Nsun8 = 3.803 X Nfirst/Nsupens and The reaction rate per A molecule is expressed as the product Nrrt/Nsuw represent tge average numbers of first spot contacts of a specific rate constant and the local steady-state concentration an repeat spot contacts made for an AB molecular pair picked of B molecules a t the reactive neither of which is a t random from the superensemble of pairs which made first measurable. However, the product of the two is simply the excollisions. Q represents the average number of repeat contacts perimental reaction rate per A molecule. The diffusive flux of per first contact for nonreactive spots. B molecules into the reactive spots is given by the product of the Because e A m t and OBmtboth receive contributions from relative relative translational diffusion coefficient (D = D A DB)and the translational diffusion (eq 14 and 15), and because relative radial radial concentration gradient of B molecules at the reactive spot. translational diffusion determines the values of N and k (eq 4 and Multiplying the diffusive flux by the spot surface area again gives can take for a given 5), the minimum values that e A t o t and eBtot the reaction rate per A molecule. set of N and k values is @Atot = e B t o t = et, = D/a2 (eq 13). In It is important to point out that the local concentration of B the present case, e,, = 3.501 X lo9 s-l, which gives Q = 3.707. This yields values of Nf,,/Nsum= 1.322 X 10” and Nrept/Nsupens molecules cannot be zero at the reactive spot of the ensembleaverage unreacted A molecule, since this would give a zero reaction = 4.902 x lo-’. When e A t o t = e ~ t =~ m, t Q = 6.122 x lo-’, rate. Both translational and rotational diffusion bring B molecules and Nrepat/Nsupns = 3.591 X Nfirst/Nsupns = 5.865 X to the reactive spot surface, thereby reducing the local steady-state Thus, in the absence of reaction, variation of the “true” rotational concentration gradient and increasing the local steady-state diffusion coefficients, @A and eB, from zero to infinity causes a concentration. Due to angular concentration gradients in the 4.44-fold increase in the number of first spot contacts, and a vicinity of the reactive spot, rotational diffusion and angular 13.65-fold decrease in the number of repeat spot contacts. Larger translational diffusion cause a net angular flux of B molecules effects are precluded because angular translational diffusion also into the reactive spot region. This flux reduces the concentration contributes to Q. of B molecules over the nonreactive part of the molecular surface, In Table I, data are presented for the reaction between a large especially in the region adjacent to the reactive spot. This also molecule and a smaller molecule. All molecular parameters are increases the radial concentration gradient over the nonreactive the same as for Figures 2, 4, and 5. surface, again especially near the reactive spot. Higher rotational In Table 11, data are resented for molecules which have equal diffusion coefficients tend to cause more “smearingn of radial radii of 1.0 A. The 4-1, reactive spot assumption results in CY concentration gradients into the nonreactive regions by promoting = /3 = 69.0° in this case. Other molecular parameters are listed. higher angular flux. In both tables, the values of K ( 1 + Qreact)NfirSt/NSUpenS give the The steady-state diffusion equation is expressed in spherical reaction rate in units of ZNoyes. polar coordinates, and its solution is given by a series of Legendre The effects of orientational relaxation are clearly seen in both polynomials. l 3 The coefficients of the Legendre polynomials are are each allowed to take three different tables, where and eBmt evaluated by imposing boundary conditions on the angular devalues. The highest reaction rates occur when @Atot = e B t 0 t = m. pendence of the radial concentration gradient. These The variation in reaction rate in going from e A t o t = eBtot = e,, = m) depends on to eAtot = OBtot = m (0, = e B = 0 to e A = a < 8 I T (nonreactive part) =0 K as well as the size and translational diffusion coefficients of the reacting molecules. The variation is most pronounced when K = 1. For example, the ratio of the case 3 to case 1 reaction rates is 4.4 in Table I and 3.2 in Table 11, when K = 1. When K = 1 = constant 0 I8 ICY (reactive part) the variation in reaction rate with eAtot and OBtot is due entirely the average number of first spot to variation in Nfirst/Nsupcns, contacts per collision cluster. When K < 1, the dependence of the (11) Solc, K.;Stockmayer, W. H. J . Chem. Phys. 1971, 54, 2981-90. reaction rate on the rotational diffusion coefficients of the mol(12) Solc, K.;Stockmayer, W. H. Int. J . Chem. Kinet. 1973, 5 , 733-52. ecules is less pronounced. When K
