J . Phys. Chem. 1989,93, 5745-5750 subsequent chemistry contributes to the disilane yield. Silylene Insertion and Subsequent Reactions. After the initial dissociation of silane, the silylene radical rapidly insertsa into the Si-H bond of another silane molecule. Three possible processes are open to the excited disilane produced in this reaction: (1) dissociation back to the reactants (reaction 3), (2) unimolecular dissociation to the silylsilylene radical and hydrogen (reaction 4), and (3) collisional stabilization to disilane (process 5 ) . The first of these is equivalent to no reaction but its occurrence contributes to the establishment of an equilibrium between reactants and products, the second results in production of higher silanes, and the third gives the desired stable disilane. As the disilane product accumulates, the disilane produced on previous pulses may also acquire sufficient vibrational energy to dissociate (reaction 3). Because the activation energy is lower for disilane dissociationa by reaction 3, it will probably occur more rapidly than silane dissociation by reaction 4. Figure 1 shows that the disilane production does tend to plateau after many laser pulses. The actual number of pulses necessary to reach that plateau depends on the cell used, the pressure, and the pulse energy. The experiments with the short cell (Figures 3, 5, and 7) are well beyond the turnover to the plateau. A comparison of the energy efficiencies (Figures 6 and 7) demonstrates that this is so. The short-cell experiments (Figure 7) with the large number of pulses is inefficient. Clearly, the product distribution is near a steady state (on the plateau). Under those steady-state conditions we can get some measure of the relative importance of the reaction paths of Si2H6* decomposition as a function of pressure. Process l and thermal excitation of disilane produce the species Si2H,*. The main source of disilane is collisional stabilization of that species by process (40) Jasinski, J. M.; Chu,J. 0. J . Chem. Phys. 1988, 88, 1678.
5745
5. Because the rate of collisional stabilization is proportional to the total pressure, we expect the reaction selectivity to increase linearly with pressure. That is what we see in Figure 5. The high selectivity a t the highest pressures indicates that collisional stabilization is much faster than dissociation by reaction 4. The disilane yields under steady-state conditions suggest that the rate of collisional stailization (process 5 ) is about 3% of the rate of dissociation to silane and silylene (reaction 3). The high selectivities at high pressure indicate that reaction 4 and subsequent reactions are much slower than either reaction 3 or process 5. The high disilane selectivity a t a pulse energy of 1 . I 8 J and a pressure of 100 Torr (Figure 4) deserves some comment. Note that the energy efficiency is very low for that experiment (Figure 6) and that the pulse energy was higher than most of the other experiments. It appears that the high pulse energy not only dissociates silane but also converts the disilane and higher silanes back to monosilane. The result is a high net selectivity and a low energy efficiency. Conclusions W e have demonstrated the highly selective synthesis of disilane from silane by an infrared-laser-induced chemistry technique. Earlier experiments with infrared irradiation gave high yields of solid products and higher silanes. The higher pressures in our experiments created an environment where absorption of laser energy is sufficient for silane dissociation, and collisional energy transfer is sufficient to stabilize the fragile disilane product. The collisional stabilization terminates the chemical reaction sequence to solid products. Reaction termination by expansion cooling prevents further degradation of the reaction products.
Acknowledgment. W e acknowledge fruitful discussions with Dr. John Clark. Registry No. SiH4, 7803-62-5; H,SiSiH,, 1590-87-0; H,, 1333-74-0.
Theory of Elementary Bimolecular Reactions in Liquid Solutions. 3. Essential Results and Comparison with Experiment Alan J. Benesi Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: October 12, 1988; In Final Form: February 23, 1989)
A clear picture of the phenomena that govern elementary bimolecular reaction rates in liquids is presented. Translational diffusion of solute molecules determines the rate of creation of encounter pairs and, in combination with the total collision rate, the average number and frequency of collisions in a nonreactive encounter. Rotational diffusion determines-within limits set by the reactive spot sizes, the reactivity, and the average number of nonreactive collisions-the number of reactive spot contacts in an encounter. Higher rotational diffusion coefficients yield more reactive spot contacts per encounter and therefore greater bimolecular reaction rate constants. Calculated and experimental rate constants agree within a factor of 2 for those reactions which have been examined. Graphical and tabular data are presented which allow the estimation of maximum rate constants for elementary bimolecular reactions between small molecules with limited reactive sites.
Introduction Rates of elementary bimolecular reactions in liquids are determined by both translational and rotational diffusion. Due to the cage of enclosing solvent molecules, pairs of solute molecules collide with each other many times prior to escape from one another.' For reactants, any of the collisions in such an encounter can result in reaction. Maximum reaction rates occur when reaction is guaranteed prior to escape. In this case, translational diffusion determines the encounter rate between surviving reac-
tants, and thus the reaction rate. However, when reaction is not guaranteed prior to escape, the dynamics of the encounter must be considered. If a functional group, orbital, or an atom within a reactant is responsible for reactivity, it must make contact with the reactive spot on the other molecule. Then rotational diffusion determines the rate of reorientation and thereby affects the average In previous number of reactive spot contacts in an papers, the theoretical expression for the rate constant was found (2)
( I ) Franck, J.; Rabinowitch, E. Trans. Faraday SOC.1934, 30, 120.
0022-3654/89/2093-5745$01.50/0
(3)
Benesi, A. J . J. Phys. Chem. 1982, 86, 4926. Benesi, A. J. J. Phys. Chem. 1984, 88, 4729.
0 1989 American Chemical Society
5746
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
Benesi contacts per encounter, and kNoys is the bimolecular rate constant for the formation of encounter In units of cm3 molecule-' s-', kNoyer is given by2.'x5 kNoyes
=
4a[DA
+ DBl
[rA
/(I +
+
r4T[DA
+
[rA
+ r~I/ktotI) (2)
where DA and DB are the translational diffusion coefficients of the reactants, and rA and rBare their radii. k,, is the bimolecular rate constant for the total Maxwell collision rate (cm3 molecule-' s-'):7
=
ktot
Figure 1. Cross sections of the spherical reactants, molecules A and B, with uniformly reactive spots characterized by polar angles 01 and 0 (0 6 a , 5 180'). The fractions of the surfaces covered by the spots, 4A and de, are respectively (1 - cos 01)/2 and ( 1 - cos p ) / 2 .
for the simplest case of spherical reactants with reactive spots, isotropic translational and rotational diffusion, and no intermolecular force^.^,^ The parameters needed to calculate the rate constant are the reactant radii, reactive spot sizes, translational and rotational diffusion coefficients and/or solvent viscosity, reactant masses, temperature, and the probability of reaction if the reactive spots make contact. The goals here are to make this theory accessible, to discuss its predictions, and to compare theoretically calculated rate constants with experimental rate constants. In part 1 the essence of the theory of bimolecular reactions is restated in clear terms. I n part 2, the effects of fundamental parameters on the rate constant are discussed, graphical results are presented, and the theory is compared with experiment.
Part 1 The reactants are assumed to be spheres. Reactive spots on the reactants must make contact in order for reaction to be possible (Figure The corresponding parameters are the reactant radii and spot sizes, as well as ti, the probability of reaction if the spots make contact (hereafter called the reactivity). In order to simulate experimental rate constants, reactant radii were calculated from atomic volume increments given by E d ~ a r d .The ~ same approach may be used to approximate the size of the reactive spot. The reactivity ti is treated here as an empirical constant (0 Iti II ) . It depends on the relative energies of the transition state and the collisions. Translational and rotational diffusion coefficients were calculated from Stokes-Einstein equations which correct the frictional coefficient to account for ellipsoidal molecular shape^.^ Accurate values for the solvent viscosity are necessary to simulate experimental data. By definition, elementary bimolecular reactions require physical contact of the reactants. The total collision rate between the reactants is therefore of great importance. Due to the cage effect,' one must divide the total collision rate into two parts-the collision rate due to first collisions between reactant pairs and the collision rate due to recollisions between reactant pair^.^,^ The first collision and the subsequent recollisions of a pair constitute a single encounter. The first collisions are orientationally random, but they define the starting relative orientation of the reactive spots. Because there is limited time for reorientation to occur before the reactants escape from one another, the rate of reorientation and hence the rotational diffusion coefficients of the reactants become important From the fundamental parameters, the following expression for the bimolecular rate constant may be derived? .233
kcalc = KcreactivekYoyes
(1)
where k,,', is the calculated bimolecular rate constant, K is the reactivity, crreactiveis the ensemble average number of reactive spot (4) Edward, J. T. J . Chem. Educ. 1970, 47, 261
T(rA
+ rB)2creI
(3)
+
where creI= ( 8 k b T / ~ [ m A m B / [ m?nB]])"2 A is the mean relative speed of the reactants, and m A and mB are the reactant masses. It is not widely recognized that eq 2 is the correct rate constant for fast reactions which are "diffusion-limited'' by translational diffusion, Le., those in which reaction is guaranteed prior to escape. Most authors incorrectly use only the numerator of eq 2 (Le., the Smoluchowski equation6). The Smoluchowski equation incorrectly yields rate constants that exceed k,,, if the translational diffusion coefficients are very large such as would occur in gases under standard conditions. Equation 2 applies to gases also. The translational dynamics of the encounter are determined by eq 2 and 3 and the constants therein. The ensemble average number of collisions per encounter in the absence of reactivity is Ne,, = kto,/kNoyes. The average frequency of the collisions in the encounter is vent = 3 ( c r , J 2 / ( 4 [ r A rS]c,,, 16[DA D e ] ) , and the average duration of a nonreactive encounter is T,,,, =
+
+
+
Nenc/~enc.~'~
T o find the bimolecular reaction rate constant, it is necessary to determine the ensemble average number of reactive spot contacts per encounter, ureactive. ureactive depends on many variables--?\i,,,, T,,,, rA, rB, dA, $B, K-and on the rotational diffusion of the reaciants during the encounter. Rotational diffusion and translational diffusion can bring reactive spots which were not initially oriented for contact into correct orientation before the end of an encounter. To calculate ureactiver one must first solve rotational diffusion equations for both reactants in order to calculate R,the ensemble average number of repeat spot contacts per first spot contact during an encounter for nonreactive spots.3 The calculation of R consumes most of the computer time in the calculation of theoretical rate constants. The range of R is limited to
-l/h
t
2
n2
- $A4B/ln
(4)
where dA and dB are the respective fractions of the molecular surfaces occupied by the spots and = (Ne,, - I)/Nenc.Because (A',,,, - 1) = = -1 /In t, inequality 4 may be written in approximate form as (Ne,,,- 1 ) 2 R IdA+B(Nenc- 1). If spot sizes are given as solid angle fractions (Figure I ) , $A = (1 - cos a ) / 2 and $B = (1 - cos fi)/2.3 The limit R (Ne,, - 1) is obtained when the rotational diffusion coefficients for both reactants approach 0. I n this case, each recollision of the encounter has the same orientation as the preceding collision. The limit 'Cl $A$B(Nenc 1) is obtained when the rotational diffusion coefficients approach m . In this case, each recollision of the encounter has a completely random orientation with respect to the preceding collision. The parameter R depends entirely on the orientational dynamics which occur during the encounter. The effect of rotational diffusion on the rate constant is manifested entirely in the parameter 0. Reorientation of the molecules' spots relative to each other, as shown in ref 3 , can be brought about by translations as well as rotations. This gives rise to a contribution of the translational diffusion coefficients to the effective relative rotational However, rotational diffusion coefficients used to calculate diffusion is dominant in the liquid state. Maximum reaction rates
-
-
( 5 ) Noyes, R . M . f r o g . React. Kinet. 1961, I , 129. (6) Smoluchowski, M . Z . Z . Phys. Chem. 1918, 92, 129.
(7) Levine, 1. N. Physical Chemistry, 2nd ed.; McGraw-Hill: New York, 1983.
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5747
Elementary Bimolecular Reactions in Liquid Solutions 0.80
0.60
c
I b
."
." -
. I
0.40
-
"
tl
I
0.20
0.0
0.0
0.2
0.6
0.4
0.8
0.2
0.6
0.4
1 .O
Q A
Figure 2. Theoretical bimolecular rate constants k,,, in units of kNoyes as a function of the fractional reactive spot size of molecule A (molecule B has fixed spot size) for the following free-radical reactions:* (i) 2 CH,' C2H6; rA = rB = 1.89 A, T = 298 K, 9 = 0.90 CP, m A = mB = 2.49 X g, 0 ( p = 90°, K = l), (6 = 56", K = 1). (ii) 2t-Bu' t-Bu-t-Bu; = rB = 2.67 A, T = 298 K,n = 0.90 CP, m A = mB = 9.47 X g, 0 ( p = 60°, K = I ) , ( p = 40°, K = 1).
-
-
+
I .o
0.8
@ Figure 3. Theoretical bimolecular rate constants kc,,, in units of kNoyss as a function of the fractional reactive spot size of molecule A for the reactions of (i) N,N-tetramethylthiourea and (ii) N,N-tetraisopropylthiourea with molecular i ~ d i n e .Molecular ~ iodine is assumed to be uniformly reactive, i.e., p = 1800 and +B = 1. (i) r A = 3.17 A, rB= 2.50 A , T = 2 9 8 K , ~ = 0 . 4 3 ~ P , m A = 2 . 1 9 X l O - ~ ~ g , m g =X10-22g, 4.21 ( K = l), 0 ( K = 0.1). (ii) Same conditions as in (i) except r A = 4.01 A and mA = 4.07 X g, 0 ( K = l ) , ( K = 0.1).
+
are obtained with small Q values and fast rotational diffusion. After Q is calculated, ureactive,the number of reactive spot contacts per encounter, is obtained from the equation
I
-
Combining eq 1 and eq 5 yields a simple expression for the bimolecular reaction rate constant: kcalc
=
Kd)Ad)B(l
-
5)kNayes/(1
-k
Kn)
(6)
Again, replacing ( 1 - l / l n E ) with Ne,, in eq 6 yields a good approximation. When Q = 0 the reaction rate is proportional to K . This value 1. In liquids, for Q is only approached in gases, for which Ne,, 500 1 Ne,, L 1 . 5 , * 9 j so Q 0 only if d)Ad)B 0. The maximum value for Q is obtained if the spots cover the surfaces of both molecules, in which case Q = Ne,, - 1. When Q >> 0, reaction may be almost certain even when K < 1 , because the spots make contact many times and in each contact the probability of reaction is K . A correction to ureactive is necessary a t high reactant concentrations, because in this case a given reactant A molecule may be undergoing encounters with more than one B molecule. In this case ureacliveis reduced, because there is a greater chance for reaction prior to escape. During the "overlap" of two encounters, the probability of reaction is exactly doubled; during the overlap of three, the probability is tripled; etc. This analysis leads to the following approximation:
-
koverlap
= pKd)Ad)B(l - l / l n
--
E)kNayes/(l
+ pKQ)
(7)
where p is the overlap correction for the probability of reaction, with p II . It is beyond the scope of this paper to give a detailed analysis of this phenomenon. However, no overlap correction is necessary if the average duration of an encounter is much less than the average time between encounters. Stated quantitatively in terms of fundamental parameters, this criterion is nX([4r(rA
+ rBI3/31 + [ 1 6 * ( r ~+ ~ B ) ~ (+D DB)/3crell) A
