J. Phys. Chem. 1982, 86,4926-4930
4926
as described previously.2b A Coherent Radiation CR-12 Ar ion laser was used for 457.9- and 514.5-nm excitation; a CR590-03 dye laser, pumped by the Ar ion laser, was used for 580-nm excitation. Interference filters were used to eliminate laser plasma lines, and solution or glass filters were used in the emission spectrometer to eliminate laser excitation lines; 2b the 580-nm excitation line was filtered out with a Corning 2-60 filter. PL spectra between 750 and 1100 nm were obtained with 457.9-nm excitation by using the McPherson-based spectrometer described previously.zb PL spectra in 5 M OH- solution were acquired by using 457.9-, 514.5-, and 580-nm excitation by 1OX expanding and masking the 2-3-mm diameter laser beam to fill the sample surface; identical intensities (einstein/s; measured with a Tektronix radiometerzb) at the three excitation wavelengths were used without changing the sample geometry. PL spectra resulting from excitation of different crystal strata with 457.9-nm light were obtained in air by the incremental removal of surface material through chemical etching with dilute Br2/H20(1:lOOO v/v). After each etching the PL spectrum was recorded in roughly the same geometry. For probing lateral inhomogeneity, the laser beam was focused to -0.5 mm in diameter; the PL spectra resulting from 457.9-nm excitation at various sites on the crystal surface were then obtained by slight variations in the crystal-detection optics geometry. PEC Experiments. PL properties in a PEC were examined in 1 M OH-/1 M S2-/(0.1 M S) (po1y)sulfide electrolyte by using cells and electrochemical equipment previously reported.2b Typically, 457.9-nm light was used for excitation and was delivered in an expanded, masked beam. PL spectra were recorded while sitting at various potentials without changing the cell geometry. The
open-circuit spectra were run before and after in-circuit spectra to demonstrate reproducibility. Complete i- V curves were also obtained in this geometry. Direct measurements of & required reassembling the cell outside of the spectrometer, as described previously.lsb Photoaction spectra were obtained by using light from a 300-W, tungsten-halogen projector bulb; the lamp’s output was monochromatized by passing the light through a McPherson Model 270,0.35-m monochromator equipped with a grating blazed at 500 nm. Lamp intensity as a function of wavelength (460-800 nm) was measured with a flat-filtered EG&G Model 550-1 radiometer; zb the radiometer’s output was displayed on a Houston Model 2000 x-y recorder and converted to relative einsteins. Photocurrent from a PEC, employing the graded electrode and optically transparent 1 M OH-/l M S2- electrolyte was then measured at -0.30 V vs. SCE (output displayed on the Houston recorder) and corrected for the variation in light intensity to generate the photoaction spectrum. EL Spectra. Uncorrected EL spectra were obtained in 5 M NaOH/O.l M K2S20selectrolyte by pulsing the electrode between 0.00 V (6 s) and a potential cathodic of --LO V (1s), while slowly scanning the emission monochromator (12 nm/min), as described p r e v i o u ~ l y . ~ ~ ~ ~ *
Acknowledgment. This work was generously supported by the Office of Naval Research. A.B.E. gratefully acknowledges support as an Alfred P. Sloan Fellow (1981-1983). We thank Dr. Lee Shiozawa and Professors John Wiley, Ferd Williams, and Giorgio Margaritondo for helpful discussions. Drs. Ngoc Tran and Richard Biagioni are thanked for their assistance with the AES measurements and X-ray powder patterns, respectively.
Theory of Elementary Bimolecular Reactions in Liquid Solutions. 1. Time Spacing of Recollisions between Nonreactive Molecules in Liquid Solutions A. J. Benesl’ Department of Chemistry, University of California, Imine, California 92717 (Received: November 24, 198 1; I n Final Form: August 2, 1982)
In this theory, molecules are assumed to behave as nonreactive spheres diffusing in a continuous liquid medium. The translational diffusion coefficients of the molecules are assumed to have been experimentally determined, and the total collision rate is assumed to be given by the Maxwell collision rate. The rate of first-time collisions between particular molecular pairs is given by the Noyes collision rate which depends on the translational diffusion coefficients, the radii of the molecules, and the Maxwell collision rate. The ratio of the Maxwell collision rate to the Noyes collision rate is a measure of the average number of “recollisions”per first-timecollision of a molecular pair. This phenomenon has been described in the past as the “cage effect”. What has not been known is the average time spacing between recollisions of molecular pairs. In a straightforward mathematical derivation, the time dependence of the recollision rate and the average time spacing between recollisions is found for an ensemble of molecular pairs.
Introduction It has been shown that rotational diffusion of reactants in a liquid medium influences the rates of elementary bimolecular reactions,1-3 but the models which have been
developed are difficult to relate to experimentally accessible parameters. It is the purpose of this set of papers to remedy this situation, at least for the case of an elementary bimolecular reaction between idealized spherical molecules in a liquid solution.
(1) Solc, K.; Stockmayer, W. H. J . Chem. Phys. 1971, 54, 2981-90. (2) Solc, K.; Stockmayer, W. H. Int. J . Chem. Kinet. 1973,5, 733-52.
(3) Schmitz, K. S.; Schurr, J. M. J. Phys. Chem. 1972, 76, 534-45.
0022-365418212086-4926$01.25/0
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 25, 1982 4927
Elementary Bimolecular Reactions in Liquid Solutions
In the last decade, several new experimental techniques for the determination of translational and rotational diffusion coefficients (tensors) have been developed. Laser light scattering correlation spectroscopy and laser fluorescence correlation spectroscopy can be used to determine both types of diffusion coefficient^.^^ PulseFourier transform NMR can also be used to find both types of diffusion coefficients.’ In addition to these new techniques, there are also standard techniques for their determination.*v9 It is the accessibility of these descriptors of molecular motion which makes their incorporation into a useful theoretical model so attractive. The dependence of bimolecular solution kinetics on translational diffusion is well understood. In particular, the classical papers by Noyesloand Collins and Kimball’l set the stage for the ideas that follow. These papers first introduced the idea of a local concentration differing significantly from the bulk concentration due to constraints imposed by the translational diffusion coefficients of the molecules. This can be seen in the tendency of two molecules which have collided to remain in each other’s vicinity for a short period of time. The time that they spend together, on the average, is a function of their translational diffusion coefficients. During the time when they are together their proximity makes it probable that they will “recollide” many times. In effect, the fact that the molecules collided in the first place raises their local “concentrations”,promoting recollisions between the two molecules. This phenomenon has been called the “cage effectn.l2 Noyes analyzed this phenomenon mathematically and was able to find expressions for bimolecular reaction rates which took it into account. What was not found, however, was the average spacing in time between the successive recollisions (“encounters”in Noyes’ terminology). This quantity, when obtained, allows the computation of the average amount of reorientation between successive recollisions and thereby allows for the quantitative treatment of the effect of steric requirements and rotational diffusion on bimolecular solution reactions. In this paper, the average spacing in time between subsequent recollisions is found for the case of nonreactive spherical molecules diffusing in a continuous liquid medium. The mathematics in the derivation is straightforward.
Assumptions 1. Molecules A and B are spheres of radius rA and rB, respectively. 2. The liquid solution and the molecules themselves form a continuum; hence, the interior volumes of the A and B molecules are accessible. In this sense, the molecules are “soft” spheres. 3. There are no attractive or repulsive forces between the molecules. 4. A collision is defined when the distance between the centers of the A and B molecules is less than or equal to ?A + rB. (4) Elson, E. L. Biopolymers 1974,13, 1-27. (5)Magde, D.;Elson, E. L.; Webb, W. W. Biopolymers 1974,13,2!3-61. (6) Thomas, J. C.; Fletcher, G. C. Biopolymers 1979, 18, 1333-52. (7)Farrar, T. C.; Becker, E. D. “Pulse and Fourier Transform NMR”; Academic Press: New York, 1971. (8)Weber, G.Ado. Protein Chem. 1953,8,415. (9)Tanford, C. “Physical Chemistry of Macromolecules”, 2nd ed.; Wiley: New York, 1961. (10)Noyes, R.M. J. Chem. Phys. 1954,22,1349-59. (11)Collins, F. C.; Kimball, G. E. J. Colloid Sei. 1949,4, 425-33. (12)Metzler, D.E. “Biochemistry”;Academic Press: New York, 1977; pp 307-10.
5. The translational diffusion coefficients of the molecules, DA and Dg, respectively, are isotropic. 6. The total number of collisions per second between A and B molecules is given by the Maxwell collision rate.I3 Actually, any total collision rate could be used in the development that follows. The Maxwell collision rate is chosen here to be consistent with the spherical approximation.
Theory Noyes has shown that the correct expression for translationally diffusion-limited reactions isl8
where DA, DB = translational diffusion coefficients of A and B molecules, respectively (cm2/s),rA, r B = radii of A and B molecules, respectively (centimeters), nB = concentration of B molecules (molecules/cm3),and
where cFe1= (8kgT/~p)1/2, KB = Boltzmann’s constant (erg/K), T = absolute temperature (K), p = mAmB/(mA + n g ) (grams), and mA, mB = masses of A and B molecules, respectively (grams). Equation 1is a modified form of the Smoluchowski equation14which contains a correction term to account for concentration gradients which are created by the reaction itself. Equation 2 is the expression for the Maxwell collision rate.13 Equation 1 is useful for the following reason: It is equivalent to the “first-time” encounter rate between A particles and B particles. Thus, a “Noyes collision” corresponds to the first time a particular A molecule collides with a particular B molecule. This is true because any first-time collision is by definition “successful” in being a first-time collision. Hence, a particular B molecule which collides with a particular A molecule “disappears” in terms of that particular pair ever being a first-time collision again. Since “reaction” occurs with every collision, the boundary conditions used by Noyes pertain.18 The ratio ZA /ZA gives the total number of collisions per first-time &%ision. It also gives the average number of recollisions per first-time collision by the relation12
or (4)
where N = average number of recollisions per first-time collision. Although eq 4 gives the average number of recollisions per first-time collision, it contains no information about the spacing in time between them. This information is obtained below. Consider a statistical ensemble of molecular pairs which share the common feature that they all suffered first-time collisions with their partners at t = 0. For convenience, let us lock all the centers of all the A molecules at r = 0 (13)Hirschfelder, J. D.;Curtis, C. F.; Bird, R. B. ‘Molecular Theory of Gases and Liquids”; Wiley: New York, 1954. (14)Smoluchowski, M. Z.Phys. Chem. 1918,92,129-68. (15)Carslaw, H.S.;Jaeger, J. C. ’Conduction of Heat in Solids”,2nd ed.; Clarendon Press: Oxford, 1959;p 257. (16)Noyes, R. M., personal communication. (17)P6lya, G. Math. Ann. 1921,84, 149. (18)Noyes, R. M. B o g . React. Kinet. 1961,1, 129-60.
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The Journal of Physical Chemistry, Vol. 86, No. 25, 1982
in a polar coordinate system, and let the translational diffusion of the B molecules relative to their particular partners be described by the relative diffusion coefficient D = DA+ DB Although all of the A molecules are centered at r = 0, the recollisions of each pair are considered separately, and the sum of the recollisions for all the separate pairs constitutes the recollisions for the entire ensemble. No new first-time collisions are considered. A collision sphere of radius a = rA rB is also centered a t the origin. A t t = 0, the probability of finding the molecular centers of the B molecules in the collision sphere is 1, by definition. At times later than zero, the fraction of the B molecules whose centers are in the collision sphere is given byI5
since the recollision rate at any time t is due only to molecules in the collision sphere at that time, the inverse of k is the average time spacing between recollisions which we seek. Since @ ( t )is not directly measurable, an alternative approach for finding k must be found. The integrated form of eq 7b is
+
F(r,t) dr
(5)
where
F(r,t) = ilerf 2
(z)(s) - erf
-
where Nenwmble = the number of B molecules (number of pairs) in the ensemble. P ( t ) decreases with t. P ( t ) gives no trajectory information about the molecular centers of the B molecules. Thus, we cannot tell whether the same molecular centers present in the collision sphere a t time t , are present at time t2. Despite this limitation, it is reasonable to assume that the recollision rate for the ensemble, @ ( t )is, proportional to the number of B centers in the collision sphere at that time. Alternatively, one could say that the number of B centers in the collision sphere at time t is proportional t,o the recollision rate at time t. Thus kP(t)Nensembie
0
(8)
where
According to eq 4, NR(T)/Nensembie approaches N , the average number of recollisions per first-time collision, for long T . This relationship provides the necessary criterion for verifying k . If nB, the number of B molecules per cm3 is constant, the probability of collision for a single A molecule with all B molecules is
Zrecoll
F(r,t) is the solution to the diffusion equation for a spherical volume source in an infinite three-dimensional medium, with a finite uniform distribution of probability within the sphere at t = 0, and a zero probability everywhere outside the sphere at t = 0. As required, P(0) = 1. The number of B molecules with centers in the collision sphere at time t is
@ ( t )=
J ' p ( t )dt
(9)
From eq 1 and 2, it follows directly that the total recollision rate for a single A molecule with all B molecules is
+ DB
@ ( t )= kNB(t)
=k
pcou = (47r/3)a3?2B
a = rA + rB
D = DA
N R ( T ) /"semble
(74 (7b)
k represents the recollision rate at t per molecular pair colliding a t t. A recollision is identical with a fiist-time collision in that it requires that the center of the recolliding B molecule be in the collision sphere during the recollision. Since the same relative diffusion coefficient applies to the B molecule whether it is making a first-time collision or a recollision, it must be true that the average time spacing between recollisions is a constant.16 Thus, the decrease in recollision rate for the ensemble with time does not arise from increasing average times between recollisions for individual molecular pairs. Rather, it arises from escape of pairs of molecules from each other.16-18 If the constant of proportionality k could be found, it would be possible to give a numberical value for the recollision rate of the ensemble at any time t . Furthermore,
= ZA,~ ZA
(10)
The total recollision rate per unit of collision probability is thus k = Zrecoli/Pcoll (11) Substituting from eq 1, 2, and 9 gives
k = 3 ~ , , 1 * / ( 4 ~ ~+, , l16D)
(12)
k has units of s-l. In general, 16D is much less than 4acrel, SO k E 3cr,i/4a. This choice for k is subject to verification in eq 8. Due to the nonelementary expression for P ( t ) ,eq 8 must be evaluated numerically. Results of the integration are shown for several molecular size combinations in Figures 1 and 2. In all cases (ignoring 2-10% errors due to nu/Nensembie approaches N for long merical artifacts), NR(T) T . This verifies the choice of k made in eq 11 and 12. The average time between recollisions, t,,, is thus 1 4acr,l + 16D t,, = - = (13) k 3crel2 Values of k and t,, for some different molecular size combinations are shown in Table I. It is interesting to note that the ensemble-averagetimes of occurrence of later recollisions are significantly longer than Nt, (compare the data in Figures 1 and 2 to the data in Table I). This observation is consistent with the view that a small fraction of the ensemble of molecular pairs make many more recollisions than N prior to escape, just as a small fraction make no recollisions (or a very few). If Arecou = Creitav (14) is defined as the mean path between recollisions, and Asoi,
= 1/ [.rra2(ntot)l
(15)
is defined as the mean free pathlg (n,,, is the total mo(19)Kittel, C.; Kromer, H. 'Thermal Physics", 2nd ed.; w. H. Freeman: San Francisco, 1980; p 395.
The Journal of Physical Chemistry, Vol. 86, No. 25, 1982 4929
Elementary Bimolecular Reactions in Liquid Solutions
TABLE Ia 'BI A
'A,
D A ,cm"s DB,cmlls N k , s-l taw s creb cm/s hrecol, cm Asoh, cm hi-ecollhsok 102.04 4 . 6 4 X l o 9 2.15 X lo-'' 2.50 X lo' 5.38 X 1.23 X 6.22 X 10'" 8.65 X lo3 20 20 1.23 X 1.11 X lo-'' 3.61 X l o 3 2.45 X loF6 108.02 1.31 X 10'' 7.61 X lo-'] 5.30 X l o 3 4.03 X 20 10 1.23 X 145.80 4.26 X 10'' 2.36 X lo-" 1 . 4 3 X 10' 3.37 X lo-' 1.59 X lo-'' 2.12 X l o 3 4.90 X 20 5 1.23 X 1.23 X lo-' 227.24 1.90 X 10" 5.27 X lo-" 5.59 X lo4 2.95 X lo-' 2.06 X lo-'' 1.43 X l o 3 20 2 1 . 2 3 X 7.07 X l o 3 2.70 X l o - ' 2.49 X lo-'' 1.08 X l o 3 2.45 X 72.14 2.61 X 10" 3.82 X 10'" 10 10 2.45 X 76.53 7.40 X 10'' 1.35 X lo-" 1.50 X l o 4 2.03 X lo-' 4.42 X lo-'' 4.59 X l o 2 4.90 X 10 5 2.45 X 5.61 X l o 4 1.62 X lo-' 6.91 X 10"' 2.34 X lo2 1.23 X l o - ' 113.72 3.48 X 10" 2.88 X 10 2 2.45 X 51.02 1.47 X 10" 6.80 X lo-'* 2.00 X lo" 1.36 X 9.95 X lo-'" 1.37 X l o 2 4.90 X 5 5 4.90 X 58.71 6.07 X 10" 1.65 X lo-" 5.77 X lo4 9.52 X lo-'' 2.03 X 1.23 X 4.69 X 10' 5 2 4.90 X 7.90 X lo4 5.50 X lo-' 6.22 X 32.24 1.44 X 10" 6.96 X 1.23 X lo-' 8.84 2 2 1.23 X 19.75 1.41 X 10'' 7.11 X lo-'' 7.90 X l o 4 5.62 X lo-' 6.22 X 2 2 2.00 X lo'' 2.00 X 9.04 2.45 X 10.' 22.86 8.03 X 1OI2 1.24 X lo-'' 2.24 X l o 5 2.79 X l o - * 2.49 X 19-8 1.12 1 1 2.45 X a Values of N , k , and ta, for molecular pairs of different sizes (calculated from eq 4, 12, and 13, respectively). D A and D B were calculated from the Stokes-Einstein equation for 25 "C, q = 0.89 CP(water solution). c1, was calculated (see eq 2 ) for 25 "C and by assuming the density of the spherical molecules to be 1.00 g/cm3. h r e c 0 n / h , ~were calculated from eq 14 and 15, respectively. ntot, the total molecular concentration including n ~ng, , and n,bent, is assumed to be 3.20 X lo2' molecules/cm3,which approximates aqueous solutions at 25 "C. A
I
1 1
IO
T Isel
20 I
10'
Flgure 1. NR(T)/Ncalculated numerlcally from eq 8 for two spherical m~lecules,rA= rB= 20 b;, D, = DB = 1.23 X 10" cm2/s. D, and D, were calculated by using the Stokes-Einstein equation, assuming that the density of the molecules was 1.00g/cm3, the temperature was 25 OC,and the solvent was water (9 = 0.89 cP). ,c,
was calculated for the same temperature and molecular density. crd = 2.50 X lo3cm/s. Note the discrepancy between the limit calculated numerically and the average number of recollisions per first-time collision calculated from eq 4. The last value of NR(7)/Nmcalculated numerically was lll.95, but an approximate value of k was used (k = 3cd4a). When cwrected, the last value was 110.85, which still exceeds the value from eq 4 by 8.6%. The value of N from eq 4 is 102.04. The dsaepancy is probably due to errors caused by fitting an equation for a parabola (the numerical integration was carried out by Simpson's method)to the early, rapidly changing part of P(t). Also, it is probable that there are significant errors in the library error function for low t values (see eq 5). I n any case, the same behavior is observed for all molecular size combinations and diffusion coefficients. I n all cases the numerically calculated limit exceeds N from eq 4 by 2-10%.
lecular concentration in molecules/cm3 including solvent molecules), the mean number of collisions with other molecules per recollision of a particular pair is given by
Flgure 2. NR(~)/Nens6mbl. calculated numerically from eq 8 for (A) f A = r B = 2 A, D, = = 1.23 X cm2/s; (B) f A = r p = 2 A, DA = DB = 2.00 x OIcm2/s; (c)r p = 5 A, f B = 2 A, D, = 4.90 x 10" cm2/s, D, = 1.23 X lo-' cm 1s. Again, an approximate value of k (3cr,,/4a) was used to obtain the results plotted. Even when corrected to account for the true k value, the apparent numerical limits exceeded the values of N from eq 4 by 2-6%.
48
uniform probability distribution consistent with a continuum model. Whether a spherical volume source or an infinitesimal point source is used, it can be shown that for long times P(t) varies as t"I2 (see eq 5); Attempts to determine average times of recollisions by using the form JmtP(t) dt t =
therefore result in divergent values. This is why previous models have put too much emphasis on the few recollisions which occur at long times, and yield intractable results. The use of diffusion out of a small spherical source yields information that concentrates on recollisions that occur a t short times (see Figures 1 and 2 , for example). FurDiscussion thermore, taking the average time between recollisions as Numerical integrations in which the inifiitesimal point the inverse of k (see eq 7) ensures that only the population of unescaped pairs contributes to the average time. source diffusion equation15is used rather than the spherical volume source result in values of N R ( T ) / Nwhich ~ ~ ~ ~ ~ Of ~ ~course, the real time spacings between successive recollisions are random. Each recollision "starts the clock" approach about 2.75N for long T. This result is not over again, since the molecular centers are again in the unexpected, since an infinitesimal point source located at collision sphere. Thus, although the time spacings are the center of the collision sphere at t = 0 does not give the
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The Journal of Physical Chemistry, Vol. 86, No. 25, 1982
random, there is a finite average time spacing between successive recollisions until the pair escapes from each other forever. In infinite three-dimensional space, a pair of molecules can escape from each other forever. This was first shown by Pdya,” who demonstrated that there is a finite escape probability in an infinite three-dimensional lattice. The concept of escape forever is also intrinsic to the treatment by Noyes, in which the quantity /3 is defined as the probability that two molecules separating from an encounter will undergo at least one more encounter.1° That /3 is less than 1 implies that molecules can escape forever. Thus, the finite limit to the number of recollisions for an ensemble, as well as the decreasing recollision rate, are explained by the concept that pairs from the ensemble ultimately escape from each other. The times necessary for escape of the ensemble of pairs are consistent with previous result@ and are on the order of 10-’o-10-6 s depending on molecular size. Since first-time collisions and recollisions are equivalent in that the molecular centers are in the collision sphere and move with the same diffusion coefficients, it follows that the probability that the molecules will not escape, /3, is the same for all collisions of A and B molecules. Furthermore, since we know that the ensemble average number of recollisions is N (eq 4 and 8), it follows that N = p p2 p3 ... = /3/(1 - p).l0 Thus, the probability of escape from any collision is 1 - p = 1 / ( N + 1). As will be shown in the next paper of this pair, both the probability of escape and the average time between recollisions are of fundamental importance to bimolecular solution kinetics. The probability of escape determines how many recollisions can occur; the time between recollisions determines how much orientational relaxation can occur. In Table I, there are several relations worth mentioning. First, it is apparent that t, can be considerably longer than the average time between collisions in general (which includes collisions with solvent molecules and with A and B molecules other than the pair of interest). This is directly reflected in the ratio of It is also apparent that this ratio decreases as the sum of the A and B radii approaches the diameter of the solvent molecules. Thus, large ponderous molecular pairs such as globular proteins collide with many smaller molecules before recolliding with one another, on the average. Similarly, a large ponderous enzyme molecule collides many times with other molecules before recolliding with its substrate, on the average. Small molecular pairs recollide with each other almost as much
+ + +
Benesi
as with other molecules. Note that, when rA = rB = 1 A, rA + rB I dHnO, Xrecol/Xsolv = 1.12, a recollision is almost certain to occur before a collision with another molecule. This may, in fact, be the origin of the so-called solvent cage phenomenon. Note that no intermolecular forces or special solvent properties are required to obtain this result, although properties of the solutes and solvent are reflected in the values of DA and D g . Thus, the continuum model is not in conflict with the concept of a solvent cage effect for molecules smaller than solvent molecules. Also interesting is the fact that Xremu/(rA+ rB)was between 1.345 and 1.405 for all molecular pairs. Thus, it appears that molecules in a pair “take a walk” of approximately 1.37 times the sum of their radii (path length, not intermolecular distance) before recolliding, on the average. Apparently if they “take a longer walk” than this they usually escape from each other forever! In conclusion, the most important observation that may be made is that the source of the decay in the recollision rate of an ensemble of molecular pairs is the variation in the numbers of recollisions made by different molecular pairs. The source of the decay is not the variation in the times between recollisions. If variation in the times between recollisions was the source of the decay, one would be forced to believe that the average time spacing between the first-time collision and the first recollision would be shorter than the average time spacings between later recollisions. Since all collisions are equivalent in that the molecular centers are in the collision sphere and move with the same diffusion coefficients, the average time spacings between all recollisions must be equal. Thus, the source of the decay must be variation in the numbers of recollisions made by different molecular pairs in the ensemble. This view is consistent with the short average path lengths and short average times between recollisions predicted by eq 13 and 14, respectively. Implicit in these short lengths and times is the prediction that molecular pairs which take longer “walks”usually lose each other forever! Thus, it seems that molecules do not have to be separated very far or for very long to get lost. Conversely, molecules which have not escaped from each other must remain close to each other, with a short average time spacing between recollisions.
Acknowledgment. I would particularly like to thank R. M. Noyes at the University of Oregon for his help. I would also like to thank Gerald Young at Eastern Oregon State College for providing me with the computer time and real time necessary to complete this project.
