J. Phys. Chem. 1983, 87, 1941-1951
cussion of the structure parameters of the Dzd-Na2S04 molecule is in ref 6. Our calculations omit Madelung energies in both sodium chloride and sodium sulfate crystals. On the basis of standard heats of formation, the heat released in the overall reaction 1is -250 kJ/mol. Assuming the Madelung contribution is qualitatively the same for the chloride and the sulfate, our calculated stability gain is in qualitative agreement with the expected value. This suggests that our C1- donor and t-SO, acceptor energy levels are in satisfactory relative positions because the
1941
electron stability gain is expected to dominate. Acknowledgment. We thank Drs. Fred J. Kohl and William L. Fielder of the NASA Lewis Research Center for their encouragement and interest. We also thank Miss Kelly Acker for calculational assistance. Our work was funded by a Select Research Opportunities Grant from the office of Naval Research. Registry No. NaCl, 7647-145; SO2, 7446-09-5; SO3, 7446-11-9; Na2S04,1757-82-6; 02,7782-44-7.
Diffusion-Limited Reactions in One Dimension David C. Torney and Harden M. McConneli' Stauffer Laboratory for phvsical Chemistry, Stanford University, Stanford, Callfornk 94305 (Received: September 10, 1982)
-
The irreversible diffusion-limitedreaction A + A P taking place on a ring is formulated exactly, as a stochastic process. Initially 2N particles are placed at random on a ring of length L. (We assume a dilute system and neglect the length of the particles.) Each diffuses with coefficientD until it collides with another particle which results in the removal of the pair. The expected fraction of the initial number of particles remaining at the dimensionless time { = Dt/L2in an ensemble of such rings is given for all systems containing initially an even number of particles. In the limit of an infinite number of particles put on an infinitely long ring with an initial density Ao,the survival fraction S ( r )is e8r erfc (8r)lI2. = AZDt.) The fluctuations about the mean number are of the order W I 2and the exact rate function is always larger than the Smoluchowski-Noyes rate function with the ratio increasing from 1 at the beginning of the reaction to ~ / at2 its completion. The exact survival fraction S ( r ) is smaller than the prediction of the Smoluchowski-Noyes theory for > 0.
(r
r
Introduction The shortcoming of standard diffusion-limited chemical reaction rate theory, the Smoluchowski-Noyes theory,' is that, although approximations are made, there is no estimate for the order of the error? Confidence in the theory is due to its agreement with experiment.' However, it is conceptually correct only in the limit of pseudo-first-order reaction^.^ With reactants of comparable mobility, finding the rate of a diffusion-limited reaction requires the consideration of a many-body problem with each body acting as a moving sink. Many theoretical treatments assume immobile sinks,4 but we are specifically interested in the reaction A + A P. In this paper the reaction occurring in one dimension is given an exact formulation with an eye to the development of a general method for calculating the rate of diffusion-limited chemical reactions. It is conventional to place the particles initially at random, which we have done. Initially there are 2N type A particles on a ring of length L, and we assume a dilute system with the aggregate length of the particles negligible compared with L. Thereafter each particle diffuses independently with diffusion coefficient D until it collides with another, resulting in the irreversible removal of the pair from the ring so that product molecules do not isolate the remaining reactive particles. (In two or three dimensions the presence of products does not physically block
-
(1) R. M. Noyes, Prog. React. Kinet., 1, 128-60 (1961). (2)D. C. Torney and H. M. McConnell, Proc. R. SOC.London, Ser. A,
the progress of the reaction. Also, the SmoluchowskiNoyes theory cannot be directly applied if unreacted particles do not diffuse freely.) The following method gives the probable number of particles surviving until time t. An ensemble of systems each with 2N particles with the same set of initial coordinates can be described as diffusing away from the initial position in 2N dimensions (2N D). There are 2N - 1 D surfaces of reaction where the coordinates of two particles are identical. In the extreme diffusion limit with all collisions resulting in reaction we can say that there is a vanishing boundary condition at each of the 2N surfaces around the 2N D prism containing the system's initial position. The point is that reaction is accounted for on fixed boundaries in 2N D. Random initial placement of the particles corresponds to uniform initial probability distribution inside each 2N D prism. In the mathematical calculations given in the next section, we begin by giving the survival fraction for two particles S2({). For an ensemble of rings initially containing two particles, &({) is the probability that a ring still contains two particles at the nondimensional time { = Dt/L2. As one can hold one of the particles fixed and consider relative diffusion, there is complete analogy with a previously solved problem; 58 the fraction of the initial heat remaining in an insulated rod which initially had a uniform temperature and its ends maintained at "zero" temperature is equal to the survival fraction S2({)when the parameters of the two problems are made equivalent. The initial probability distribution for the ensemble of
in . . . meas.
(3) E. W. Montroll, J. Chem. Phys., 14, 202-11 (1946). (4)B. U. Felderhof and J. M. Deutch, J. Chem. Phys., 64,4551-8
(1976). 0022-3654/83/2087-1941$01.50/0
(5)(a) H. S.Carslaw and J. C. Jaeger, 'Conduction of Heat in Solids", 2nd ed., Oxford University Press, Oxford, 1959,p 96;(b) ibid., p 97;(c) ibid., p 485;(d) ibid., p 58.
0 1983 American Chemlcal Society
1942
The Journal of Physical Chemisrty, Vol. 87, No. 11, 1983
TABLE I: Survival Fraction Data 1% .c S,(i-) S,(i-) 3.00 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 -0.25 - 0.50 -0.75 - 1.00 -1.25 -1.50 1.75 2.00 - 2.25 - 2.50 - 2.75 -3.00 -3.25 - 3.50 -3.75 -4.00 ~
~
2.169 X 1.225 X 1.577 X 1 O - j 2.423 X lo-’ 0.1126 0.2671 0.4345 0.5744 0.6808 0.7607 0.8205 0.8654 0.8991 0.9243 0.9432 0.9574 0.9681
2.028 X 4.480 X l o w 4 9.322 X l o T 3 5.139 X lo-’ 0.1342 0.2307 0.3192 0.4044 0.4926 0.5798 0.6603 0.7304 0.7891 0.8367 0.8746 0.9042 0.9273 0.9449 0.9584 0.9686
Torney and McConnell
S,(i-)
2.547 x 1.188 x 3.763 X 2.627 X 7.836 X 0.1450 0.2089 0.2730 0.3459 0.4272 0.5120 0.5947 0.6707 0.7372 0.7934 0.8394 0.8762 0.9052 0.9278 0.9452 0.9586 0.9687
S,(T)
10-7
10-4
10.’ 10.’
rings containing two particles (Figure 1) has a Fourier series representation which plays a central role in the subsequent analysis. The harmonic terms reflect the periodicity of the ring. For two particles (two dimensions) the reaction surfaces (lines) are created by the method of images. For 2N particles (2N D) the reaction surfaces are constructed by using an antisymmetric combination of two-particle probability distributions. The antisymmetry of the initial distribution in 2N D with respect to the interchange of any two particles persists for all time because the solution to the 2N D boundary value problem is given in separated form (a sum of antisymmetric terms decaying exponentially in time). In subsequent calculations,we find the survival fractions for the four- and six-particle problems, S4(f)and s&). We use the probability distribution of the remaining particles at the time of reaction of a pair as a distributed source in a boundary value problem of two fewer dimensions. For example, the rate of reaction of the first two out of four particles is proportional to the total heat flux from a tetrahedron with initial uniform temperature and surfaces held at zero temperature. (The tetrahedron has two opposite edges of length 3’I2L/2 and the other four edges of length L.) The position dependence of the flux on the face of the tetrahedron gives the probability distribution for the separation of the remaining two particles at the time of reaction of the first pair. The flux to the face of the tetrahedron is used as a source for the remaining twoparticle problem, analogous to a distributed heat source in the rod. The flux to the short or long edges is respectively the expected rate of formation of groups of three particles or simultaneously of two pairs, and the flux to the four vertexes is the expected rate at which four particles come together. Because the reaction is assumed to be instantaneous, these fluxes due to combination of more than two particles are negligible. Similarly, for rings with 2M particles we need to consider only the transition rate to rings with two fewer particles. In general, the rings containing 2M out of 2N particles are the source for the probability distribution on rings containing 2M - 2 particles. We use Green’s function for an unbounded region letting the antisymmetry in the probability distribution of the remaining 2M - 2 particles
7.139 X 1.883 X 4.331 X 2.525 X 10” 6 . 8 0 6 x 10‘’ 0.1192 0.1679 0.2202 0.2828 0.3559 0.4365 0.5197 0.6005 0.6748 0.7400 0.7952 0.8405 0.8769 0.9056 0.9280 0.9454 0.9586 0.9688
s,(r) 6.44 x 2.11 x 2.68 X 3.58 x 4.90 X 6.35 X 8.48 X 0.110 0.145 0.189 0.243 0.308 0.380 0.458 0.538 0.601 0.685 0.747 0.800 0.843 0.879 0.907 0.929 0.946 0.959 0.969
10-3
lo-’ lo-’ 10.’
lo-’
10.’ 10.’
ST(r) 9.811 x 3.038 X 4.011 X 5.278 X 6.916 X 9.015 X 0.1167 0.1498 0.1903 0.2386 0.2947 0.3578 0.4263 0.4977 0.5692 0.6379 0.7015 0.7581 0.8069 0.8478 0.8814 0.9083 0.9296 0.9463 0.9592 0.9691
10-3
lo-’ lo-’ 10.’
10.’ 10.’
error, 7% 52 44 50 47 41 42 38 36 31 26 21 16 12 9 6 6 2 1 1 1
create the reaction surfaces in 2M - 2 D. For an ensemble of systems initially containing 2N particles, at a time given by {there is a probability of finding 2N, 2N - 2 , 2N - 4, and so on to zero particles on a ring. These probabilities are used to determine the expected number of particles, 2NS2N({). The solution to this one-dimensional diffusion-limited reaction problem for 2N particles includes the solution to equivalent heat conduction problems in simplexes in 2M - 1 D with N I M > 0. Comparing the formulas for S2({),S,({), and s&f) leads to a hypothesis for Sw(f)which is then proved by induction. In the limit of a large number of particles placed at constant density the solution has a particularly simple form and one can compare the 2N-body solution with the solution for an infinite number of particles. The eightparticle solution is as close to that for an infinite number of particles as is the Smoluchowski-Noyes theory. However, they lie on opposite sides, and for no number of particles is the reaction as slow as predicted by the Smoluchowski-Noyes theory (Table I). To compare reactions on rings initially containing different numbers of particles the initial rate of change of the survival fraction dS,(f)/d{ is made the same in the limit as f goes to zero by adjusting the length of the ring. With 2N particles the length is 2N - 1 times the length L2 of the ring in the ensemble with each initially containing two particles, giving for large N an approximate initial density of L2-l. The method of obtaining a solution is a classical 0118.~9~ The viewpoint is that of the stochastic theory of chemical kineticss although the detailed time-dependent transition rates from a stage containing 2M particles to one with 2M - 2 have not been considered previously. The Smoluchowski-Noyes theory is not stochastic and aims to give only the mean beha~ior.~In our formulation we find the second moment of the probable number of particles surviving until time { about the mean number. In the subsequent formulation for systems initially containing two, (6) J. B. J. Fourier, “Oeuvres de Fourier”, Gauthier-Villars et Fils, Paris, 1889, Tome Premier, p 253. (7) G. Polya, “How to Solve It”, 2nd ed., Princeton University Press, Princeton, NJ, 1957. (8) D. McQuarrie, J. Appl. Prob., 4, 413-78 (1967). (9) J. Keizer, J. Phys. Chem., 85, 940-1 (1981).
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983 1943
Diffusion-Limited Reactions in One Dimension x2
x,=
x,.
!
X,-2L
x,
X,-L
‘Xp
I
I
Figure 1. Two-particle probability distribution at zero time. x axis: x the coordinate of particle one. y axis: x 2 the coordinate of particle two.
four, and six particles, all the details necessary for the solution for a system with 2N particles are developed.
Two Particles An ensemble of rings each with two particles at the coordinates x1 and x2 at time zero which then begin to diffuse with coefficient D can be described as diffusing away from the point (x1,x2) in two dimensions. The twoparticle probability distribution function P2(x1,x2,t) obeys the diffusion equation given in eq 1. The initial proba-
a p 2 / a t = D V ~ P= ~D(a2/axt
+ a2/aX22)p2
(1)
bility distribution which corresponds to random placement of the particles and which creates the reaction boundaries at xl = x2 + nL (with n an integer and L the length of the ring) by the method of images is shown in Figure 1. Because of the axis of translational invariance parallel to x1 = x2 a change of variables reduces this to a one-dimensional problem which has been solved.5a The Fourier series for P2 follows: 8
”
P2(Xl,X2,t) = - C j-le+jDt sin (j7r(xl - x 2 ) / L ) 7rL2j=1,3,5 ... (24
-4
-3
-2
-1
0
1
Figure 2. Survival fractions for systems with initially two, four, eight, and an infinite number of particles. The ring length is chosen so that the initial rate of change of the survival fraction is indistinguishable in the four systems. x axis: log {, { = M/Lz = A t D t . yaxis: S,,(O, 1 3 S*,(j-) 3 0.
0 and change the normalization so that it is +1 or -1 everywhere except on the reaction boundaries where it is 0, giving P2( x ,x2). m
P2(x1,x2)
= (4/7d
C j-l sin (jdxl - x2)/L) j=1,3,5 ...
(4)
If Pm-2(xl,x2 ,...,xm-2) is +1 in the region Rmv2 ( L b x1 b x2 b ... b xm-2 b 0) and antisymmetric upon the interchange of any two variables, then one can construct the function Pm(xl, ...,xm) with analogous properties. PBN(Xl,...,XZN) = 2N
xj
c
(2b)
= 2(j7r/L)2
k=2
Corresponding with Figure 1, if any particle moves an odd number of times around the ring, the probability distribution changes sign because it has passed through an odd number of reaction boundaries. (A sign change upon translation of one particle by a distance (2n + l ) L is found in all systems with an even number of particles.) The normalization of the series in eq 2a is chosen so that the integral of P2(x1,x2,t) over the region R2 (Lb x1 b x2 b 0) gives one at time zero. Also, at time zero, in R2 all values of the separation x1 - x 2 are equally likely. The integral of P2(x1,x2,t)over R2 gives the survival fraction S2(t).
W t ) = ~ L d ~ 1 ~ z 1Pd2 (x~21 , ~ 2= ,t) (8/7r2)
f ...f2ee-”jDt( 3 )
j=1,3,5
The survival fraction for two particles is plotted as a function of ( = D t / L 2 in Figure 2. Initial Probability Distribution for 2N Particles. One can construct the initial probability distribution for 2N particles from that for two particles. Take eq 2a at t =
( - l ) k P 2 ( x 1 , x k ) p2N-2(3c2,...,xk-1,xk+l,..,x2N)
(5)
This assertion is proved in Appendix A. Apart from the normalization, the function Pm is the correct initial probability distribution for 2N particles. One can assign a value of +1, -1, or 0 to all points in the 2N D hypercube ( L b xl,x2, ...,xm b 0) by interchanging the variables to generate all (2N)! permutations and by periodicity to all points in space. The value 0 occurs on the 2N - 1 D surfaces of reaction where the coordinate of two particles is the same or different by an integral multiple of L. Pm can be seen by induction to be the sum of (2N-l)!!terms Each term is one of the each a product of N P;s.l0 possible ways to group 2N objects by twos and contains each variable only once. Therefore, if the time dependence is restored to the Pis, the resulting PZNconstructed according to eq 5 will be the solution to the 2N D diffusion equation with the initial probability distribution described above. P m (the time-dependent Pm) is normalized so that at time equal zero, when it is proportional to Pm, the (10) The !! notation has the following definition: (!m)!/(2”.).
(2N - l)!! =
IS44
Torney and McConnell
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983
volume integral of the probability in Rm ( L 1 x1 2 x 2 L ... 1 xZhr 2 0) is 1. This is accomplished by the coefficient n(N) which multiplies the summation.
ditional distribution of particles around other particles, this is implicit in the probability distributions used to obtain the expected fraction of particles remaining as a function of time. Previously it has been necessary to use Q(N) = ( ~ N ) ! ( ~ / T L ~ ) ~ ( 6 ) a closure hypothesis in a hierarchy of equations for the distribution of pairs, triplets, and so on to get the SmoThe examples of four and six particles show the form luchowski-Noyes solution to this prob1em.l’ of functions generated by eq 5. It is useful to introduce It is convenient to follow the course of the reaction in the following notation for the spatial part of eq 2a, now the region & ( L 2 xl 2 x2 2 ... 2 xzN 2 0) and in &with including a time development obeying eq 1. 2M out of 2N particles remaining. One can construct a n segment ( L 2 X2M 2 0) of the 2M D prism in which the system diffuses from 2M regions, each equivalent to RN because of the periodicity of the probability distribution. (7) &(xl,x2) = sin ( j r ( x l - x 2 ) / L ) Therefore, the transition rate from a system containing 2M particles to one containing 2M - 2 is proportional to the Then by eq 5 and 6 the four-particle probability P4 is rate of loss of probability from Rm, and the two can be P4(xl,X2,x3,X4,t)= made equivalent by careful attention to normalization. In the hypercube (L2 xl, x 2 , ...,xzN 2 0) there are (f“) equivalent 2N - 1 D surfaces of reaction. The normalization of PzNhas been chosen so that initially the absolute &(x1,x3) P $ ( x 2 , x 4 ) + &(x1,x4) P $ ( x Z , X S ) ) value of the probability within RzN or any of the 2 N ! equivalent regions obtained by a permutation of the var= ( 3 8 4 / ( ~ L ~ ) ~ I ’ O ’ k ) - 1 e - A j k D t ~ k ( x l , x 2 , x 3 , (8a) ~4) iables is 1 (eq 6 ) . The integral of the absolute value of the j , k = 1,3,5 ... flux density (from one side of the surface) over the surface X j k = 2G2 + k 2 ) ( r / L ) 2 (8b) of reaction ( L 2 x1 = x 2 2 0) by infinite time is one-half the integral of the absolute value of the initial probability The prime on the sum in eq 8a indicates that terms with density in the hypercube,J2(2iV)!, divided by the number j equal k are omitted. This is not a consequence of the of surfaces of reaction, (2 ). This division results in (2N method of construction which multiplies two sums over - 2)!. Because there is a sign change for each permutation all odd values of the indexes, but it is found by inspection of the remaining 2N - 2 variables, half of this total flux of eq 8a that summands with j equal k are zero. In Apis positive and half is negative. The flux to each region pendix B it is proved that in the sum over N indexes in in a hypercube of two fewer dimensions corresponding to PzN(xl,...,xzN) the net contribution from summands with one permutation of the 2N - 2 variables will be fl by at least two indexes equal is zero. It simplifies subsequent infinite time. Therefore,the flux to the surface of reaction operations to explicitly drop these summands. Equation x1 = x 2 is correctly normalized to be the source for the 8a defines plqk(xl,xz,x3,x4)in terms of which P6 is next stage of reaction with 2N - 2 particles. Moreover, because of the antisymmetry in Puv the 2N - 3 D surfaces P ~ ( X ~ , X Z , X ~ , X ~=, X( 6~!,/XL~6 ,)x~ ) ( 4 / 7 ~ ) ~C ’ ~ k ~ ) ~ 1 e ~ X j k ~ D t { ~ ( x l , x 2 ) ~ ~ -1 ( x z , x of 3 , reaction x ~ , x 6 )are established by images from the periodic flux J,k,1=1,3,5 ... to the surface x1 = x 2 from Pm when one integrates the flux with Green’s function for an infinite region. One can P~(x1,x3)P~1(xZ,x4,x5,x6) + P~(x1,x4)P~1(x2,x3,X5,~~) repeat the procedure of finding the flux to one surface of &(x1,x5) P i 1 ( x 2 , x 3 , x 4 i x 6 ) + d(X1,xB) P i 1 ( x 2 , x 3 , x 4 , X 6 ) / reaction and making it the source for a system with two = ( 4 6 0 8 0 / ( ~ LX~ ) ~ ) fewer particles to find the probability distribution on rings m containing 2M out of 2N particles, NPm, C ’ ( j k l ) - 1 e A j k l D t ~ k ~ ( x l , ~ 2 , ~(9a) 3 , ~ 4 , ~ 5 , ~To 6 )calculate the transition rate T(x3,...,xZN,t)from a j , k , l = 1 , 3 , 5 ... system with 2N particles to one with 2N - 2, one f i s t finds the flux to the reaction surface x1 = x 2 by taking the inward X j k / = 2G2 k 2 12)(?r/L)2 (9b) normal derivative. Because probability is lost from RZhr Again in the six-particle probability distribution the prime only on part of the surface x1 = x 2 , L 2 x1 = x 2 0, one on the sum means that all summands with two or more then integrates to find the correctly normalized transition indexes equal are omitted, and in all subsequent expresrate T(XS,...,X2N,t)to rings containing 2N - 2 particles. sions for Puv and functions derived from it the prime will T(X3,.*.,X2N,t) = have the same meaning. In P2N the eigenvalue X is m
f
+ +
21/2&Ldx1{2-lizD(d/dx1 - d/dX2)PZN(xl,...,X2N,t)lx2=x,) Equations 8a and 9a describe the four- and six-particle probability distributions until the first pair reacts. Then one uses the distribution of the remaining particles as a source in two fewer dimensions. Transitions and Distributed Sources The initial random distribution of 2N particles is modified by the combination of diffusion and reaction. Whenever two particles react, it is essential to use the exact distribution of the remaining particles as a distributed source in the boundary value problem in two fewer dimensions. Then the rings containing 2N, 2N - 2 , and so on through two particles incorporate the “history” of all preceding events. Although we do not analyze the con-
(11) To calculate the probability distribution on the ring with 2N - 2 particles 2N-2PzN, we integrate the source over an infinite 2N - 2 D domain with Green’s function for an infinite domain and integrate on time from zero up to time t . The single integral sign represents all of the spatial integrations. 2N-2P2N(x 3 , . . ,x 2N,t) =
L ‘ d t ‘J+mdx3‘...dxm’ G ( x ~ - x ~ / , . . . , x ~ ~ ’?~ Xx ~ N ’ , ~ - ~ -m
T(X3’r*.*,X2N’,t? (12) (11) T.R. Waite, Phys. Reu., 107, 463-70 (1957).
Diff usion-Limited Reactions in One Dimension
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983 1945
Green's function for 2N - 2 D is G(x3-~3/,...,~~hr~ZN',t-t 9 = ( 4 ~ D (-t t?)-""' X (x, - ~ ~ ' ) ~ ] / [ 4 D t?]) ( t (13) exp{-[(x3-
+ ... +
When one takes the derivatives inside the integral of eq 11after expressing PUv as a s u m according to eq 5 and uses eq 2a for Pzwith its index j,
function. We illustrate these general formulas with the solution of the four- and six-particle problems.
Four and Six Particles For four particles we begin with the function Pq(x1,x2,x3,x4,5) (eq 8a). Performing the operations given in detail in the last section, we calculate the probability distribution on rings with two out of four particles remaining, 2P4. 2P4(x3,3e4,E)
=
To calculate the probability of finding zero particles OS4([), one must perform the operations in eq 11 to find
Every term in the two summations in eq 14 has an x1 dependence of the form sin (jl?r(xl - xk)/L) cos (j,?r(xl - xl)/L)
the rate of production of rings with zero particles at time t'and then integrate on t'from 0 to t. The net result is equivalent to multiplying by kL2/a and integrating on 5' from 0 to 5 (5 is defined in eq 18).
(15)
5
OS4([) = (384/?r4) 'j-2(k-2[1- (j2+ j,k=1,3,5... The integral over x1 from 0 to L of eq 15 is 0 because jl k2)-1[1 - e-CiZ+kz)E]J (21) + j, is an even number and because jl cannot be equal to j , because these summands have been omitted from Puy One way to calculate the probability of finding two par(Appendix B). Therefore ticles on a ring ?S4(5) is to integrate 2P4over the region & T(x3,...,x ~ , t =) 2?rjlDP2N-2(~3,...,x2N,t) (16) (L 2 x 3 2 x4 2 0). P2N-z is the sum of terms each of which is a product of N - 1sines whose argument is proportional to the difference of two variables, each variable occurring only once in a term. Green's function (eq 13) is also of separated variable I2 = J L d x 3 ~ ' ' d x 4 pa(x3,x4) = L2/ka form so that the spatial integrations in eq 12 for each term are a product of N - 1 identical integrals of the form However, because the spatial part of 2P4is p i ( x 3 , x 4 ) , the integral in eq 22a has been done implicitly in solving the two-particle problem (eq 22b). When one compares eq 2a Xl')/L) exP(-[(xk' - x J 2 + (xII - XJ2I/[4D(t - t311 and 3, the value of the integral Iz is L2/k?r. Therefore
= sin (jn?r(xk- xJ/L) exp{-2(jn?r/L)9(t - t?)
(17)
Therefore, the spatial part of uy-zP2N is identical with that of PUv+We express this using the notation introduced in eq 6,7b, 8a, and 9a. Also in the final integral in eq 12 we change to the dimensionless variable 5.
5 = 2(a/L)'Dt
' (jd3...jN)-1&3p(x3 ,...,xm) x
j19...j.,.= 1,3,5...
+ + ... + j N z ) [ ) J
e ~ p { - ( j ~j 3~z
C ' Gk)-2[e-kz[- e-v2+k2)E] (23) j,k=1,3,5 ...
Using the fact that a ring has probability 1of having either four, two, or zero particles and that OS,(=) equals 1
4s4(5)= 1 - 2s4(5)- OS,([) = (384/*4)
= (L2/*)s2(N) x
2N-2PzN(x3,...,x2&
5
(18)
0)
2S4(t)= (384/?r4)
E
d r e-jl'r (19)
In summary, taking the flux derivatives and integral to find the source (eq 11)and then integrating the source with Green's function over space and time to generate the correct probability distribution for two fewer particles (eq 12) is equivalent to the following operations: (1)multiply by L2/?r, (2) replace p ! ! N with jlp?!-f, (3) integrate over the dimensionless variable from 0 to 5 the product of PUv expressed as a function of r and an exponential,of the sum of the squares of the indexes retained in dM-3 multiplying 5 - 5'. Because of the spatial part of 2N-pPuyis d -if, one can perform the same set of operations to generateg4Pm from 2N-2P2N and so on until all of the N Probability dmtributions which completely describe the time evolution of the reaction of the ensemble of rings each initially containing 2N particles have been generated. The transition from 2Puyto OPuy does not involve integrating over Green's
5
' j - 2 0 ' 2 + k2)-le-0"+k2)t(24)
j,k=1,3,5...
Finally, the survival fraction S4(5)for the ensemble with four particles per ring initially is the probability to be on a ring with four particles 4S4(5)plus half the probability on rings with two particles '/Z2S4(E).
(25) The values of the s u m s required in the derivation of S4(5) and &(5) apart from the values of Riemann's { function given elsewhere12*are given in Table 11. The same procedure is applied to the ensemble of rings each initially with six particles beginning with (eq 9a). Performing the operations given in detail in the last section we calculate the proba(12) (a) I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products", 2nd ed.,Academic Press, New York, 1980,pp 7,1074; (b) ibid., p 649. The following integral was used in the derivation of eq 48: J~=
[erfc zle-~z'
1948
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983
Torney and McConnell m
TABLE 11: Useful Summations"
S,([) =
n4/192 = z ' ( j k ) - z = 2 g'j-z(jz + ha)-* j , k=
1,3,5..
.
j,k=i,3,5
...
S4([)=
I ,3,5...
n 6 / 7 6 8 0 = ? , ' ( j k l ) - z = 6 t ' j - * ( j z + h 2 ) - ' ( j Z+ h Z +
...
j,k,l=1,3,5
j,k,l=i,3,5
(8/r2)f2e-J2[
(3)
m
n 6 / 3 4 8 0= t ' ( j 2 h ) - ' J,k'
...
j=1,3,5
...
C {(24/r2)f2- (192/~~)j-~)e-j'C (36)
j=1,3,5...
lZ)-l
s6(.$) =
" The prime means that terms with a n y of the indexes equal are o m i t t e d ,
5
j=1,3,5...
bility distributions on rings containing four and two out of six particles, 4P6and "6.
((4O/r2)f2- ( 1 9 2 0 / ~ ~ )+j -(~1 5 3 6 0 / ~ ~ ) j ~ ) e - j ~ [
(37) These last three equations were used (in a programmable hand calculator) to generate the date in Table I plotted in Figure 2. The time axis is S:
t = Dt / L 2 2
(38) As discussed above to make the initial rate of change of the survival fraction the same, L4 is 3 times and L6 is 5 times L2, the length of the ring initially containing two particles.
2N Particles In this section we give the probability for 2M particles to remain out of an initial number 2N, YSZh,([). The probability distribution over M is characterized by the following two functions which are related to the mean and mean square number of pairs remaining: N s2N(.$)
= (1/N)
M=l
flS2N(f)
(39)
N v2N(.$) 0'2
+ k2)-1(e-12t- e-Ci2+k2+W))(29)
Comparing eq 8a and 24 gives the value of the integral needed to compute 4S6([).
= (N(N - I))-'
= m
4S6([) = (46080/n6)
5
( L 4 / a 2 ) k [ l ( k+2 12)]-1 (30)
' (jl)-2(k2+ 12)-1(e-(k2+12)t -
j,k,l=1,3,5 ...
e-G2+k2+l')E
) (31)
The probability that six particles remain, 6s&),is now determined. 6s6([)=
1 - 4s6- 's6 - ' 8 6 - O 6 s = (46080/r6) x
" .
2 ... (jkl)-2e-j2t(33)
(7680/r6)
j,k,l=l,3,5
The comparison of the survival fractions for two, four, and six particles, eq 3, 25, and 33, leads directly to a hypothesis for the form of the survival fraction for 2N particles s,(.$). S,N(.$)
= E(N)
' (j1~2,**JN)-2e-j1''(34)
jlj2,...j N = 1,3,5...
E ( N ) = ( 2 - ~1 ) ! ! 2 3 ~ ~ - 2 ~ (35) In the next section eq 34 is proved to be the solution by induction. In eq 25 and 33, one can do all but the final summation leading to the following formulas (see Table 11).
M ( M - 1)2Ms2N([) (40)
S,([) is the survival fraction. In Appendix C it is proved that for all M and N ( M I N) 2 M S 2 N ( f ) is "!32N(.$)
".
M=2
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983 1947
Diffusion-Limited Reactions in One Dimension
The first of the two terms of eq 43b exactly cancels the term with M = 2 in eq 42. If one uses a relation derived in Appendix C, eq C10, it is clear that another integration by parts on the remaining term in eq 43b will cancel the M = 3 term and so on until only one term will remain. It is the function given in eq 34. This completes the proof of the hypothesis of the form of S&). Combination of eq 40 and 41 and the same reduction based on integration by parts lead to the following expression for V,(f') ( E ( N ) from eq 35):
v,([)
2
= E(N)
...
Gj2...jhr)-2e-0'1z+~zz)~
(44)
j,...jN=1,3,5
In the next section it will be shown that the formulas for S&[) and V&[) have a simple form in the limit that both N and L go to infinity at a constant ratio A,, the initial particle density. To make the initial rate of change of the survival fraction the same for S&[) and S 2 ( [ ) the , length of the ring containing 2N particles must be 2N 1times L2,the length of the ring with two particles. The density for large N is approximately L2-l. Therefore { = Dt/L22 -AO2Dt
(38')
Infinite Number of Particles We find first-order differential equations for S,(O and V&{) in the limit as the initial number of particles, 2N, and the ring length, L, go to infinity with a constant ratio of the initial particle density, A,,. In this section the symbol = is used whenever terms are dropped which are of order N-' with respect to the retained terms. Other approximations will be stated. Expressed as a function of { (eq 38) Sw is
S,,M = E N
5
' ~ j ~ . . . j N ) -e2x p [ - ~ ~ r / ~ ) 2 1 { / 2
j,. ..jN= 1,3,5...
(34') and its derivative follows. d ~ w ( t ) / d { =-{E(N)/2)[r/N12
X
m
The s u m over jl is performed independently of the values of the other N - 1indexes and N - 1 terms are subtracted in which jl equals one of the other indexes. The sum over the last N - 1 indexes in eq 45 with no two equal but independent of j , is the reciprocal of the normalization (eq 35). coefficient, (Z(N-
".
convenient to use a formula equivalent to the sum in eq 46c which contains a sum of the first integral of the complementary error function.5b Neglecting terms of order e-N dS/d{ = -8N-'(N/(8~{)l/~- NSJ = -(8/7~{)'/~ 85' (464 The solution to the inhomogeneous first-order differential equation which has the value 1 when { equals 0 is
+
S({) = e8ferfc (8{)lI2 (47) The function S({) has been tabulatedek Values obtained by linear interpolation are given alongside those for the S,({), N = 1,2,3,4,in Table I and are plotted in Figure 2 (-D). Tie same procedure gives V( {), Vw as N goes to infinitye12b,13
V ( { ) = (e8rerfc (8{)1/2)2= S(02 (48) The following formula characterizes the fluctuation of the number of pairs about the mean number:
c(l)= {N(N- 1)vW + NSN) - (Ns(0)2J1/2(49) Using eq 41 and 48 s(l))J1/2
(50) Because terms of order N-' were dropped in the derivation of S({) and V({), the { dependence for C({) in eq 50 is incomplete. However, the fluctuations are of an order not greater than W / 2and are negligible as N goes to infinity. Comparison with the Smoluchowski-Noyes Theory. In terms of the diffusion coefficient for one type A particle the Smoluchowski-Noyes rate function kT(t),which can be found by consideration of the heat flux from a semiinfinite rod with uniform initial temperature and zero temperature maintained at the e n d , l ~ is ~as~follows: =w " 1
-
kT(t) = ( 8 D / ~ t ) ' / ~ (51) The rate function predicts the survival fraction of A in the equation dA /dt = (8D/?rt)1/2A2 (52) The integral of eq 52 gives the Smoluchowski-Noyes theory survival fraction, ST( {) ST({) = (1 (32{/~)'/']-' (53)
+
The ratio (dS/d{):(dST/d{) is 1as {goes to 0. The values of ST({)are always too large, and the discrepancy increases as the reaction proceeds. When S({) equals ST({)is 9% too large. Considering the approximate nature of the Smoluchowski-Noyes theory the agreement with the exact solution is very good. Another way to compare the theoretical and exact rates of reaction is to calculate an exact rate function k&) from the survival fraction s ( t ) and to take the ratio kE(t)/kT(t). First we give the exact rate equation. dA/dt = A,(dS/dt) = -kE(t)(A,S(t))2 (54) Equation 54 defines kE(t). Using eq 46d and 51 and expressing the ratio R as a function of { (eq 38')
R(f) = kE({)/kT({)
In the limit as N goes to infinity, Sw is not different from S2N-2.If we call the survival fraction in the limit as N goes to infinity S({) dS/d{
&
-8N-'(
5 ... e-Ul*/h')'f/z - N q
(46~)
j1=1,3,5
As N goes to infinity, the sum diverges. In this case it is
= (s({))-2[1( 8 d 1 / 2 S ( t ) l (55) R({)is plotted as a function of the progress of the reaction, 1- S({), in Figure 3. The ratio increases from 1 initially to a12 by the completion of the reaction, so the exact rate (13) M. Abramowitz and I. S t e p , Eds., "Handbook of Mathematical Functions", US.Government Printing Office, Washington, DC, 1964, p 302. The following integral was used in the derivation of eq 48:
Jz = L ' d x eaz2(1- .z*)-l
The Journal of Physical Chemlstty, Vol. 87, No. 11, 1983
1048
Torney and McConneil 2N
1.6,
P2N =
c (-l)k = + 1
(AI)
k=2
Second, to show that P, changes sign if any two variables are interchanged, we consider two cases. In the first case the exchanged variables do not include x 1 and in the second case one of the exchanged variables is xl. We consider both cases for completeness; although the form of eq 5 appears to distinguish x1 from the remaining variables, in fact all the variables in PW are equivalent. Case I. Exchange x 1 and x q in eq 5. There is no loss of generality to assume q > 1.
/
1 2
/ -0 2
loo
0 4
0.6 1
-
0 8
10
S(5)
Figure 3. Ratio R(0 of the exact rate function to the Smoiuchowski-Noyes rate function as a function of the progress of the reaction. x axis: the progress of the reaction, 1 - S ( 0 . yaxis: R(0,r / 2 Z R(0 3 1.0.
function can be considerably larger than the theoretical rate function. For small { S ( { ) 1 - (32{/.~.)'/' u([) (564
R(c)
- + -+ 1
+
((128/.~.)'/*- (B.R.)'/~){'/~ + a({) 1
1.37{1/2 +
(56b)
CT([)
Conclusion All of the features required for the formulation of a diffusion-limited chemical reaction have been employed P on a ring. in the formulation of the reaction A + A For a more complicated reaction occurring in two or three dimensions (instead of one) the boundary value problem which can be constructed is also more complicated. Having an exact result gives support to the Smoluchowski-Noyes theory, which comes close to the solution to our problem-the theory is not bad in a demanding new setting. Other approximate treatments of irreversible chemical reactions and irreversible thermodynamics may similarly benefit from a favorable comparison with an established result. The result for an infinite number of particles (eq 47)is so simple that it may suggest a new approach to the problem. Qualitatively, the same reaction on a two-dimensional surface should always be closer to completion than predicted by the Smoluchowski-Noyes theory. Finally, much more information about this system can be extracted from the formulation which contains much detail of the many-body dynamics.
-
Acknowledgment. From conception through completion this formulation was inspired by the interest and encouragement of Dr. Harry Frisch. Also, we thank Dr. Joseph Keller for his lucid examination of the problem at critical stages and Dr. Harold Levine for assistance with analytical methods. This work has been supported by National Science Foundation Grants 77-23586 and 8021993. D.C.T. has been supported by the National Institutes of Health Training Grant CMB GM 07276. Appendix A Construction of P2Nfrom P2 In this inductive proof we assume the two properties-+l in &-2 and a sign change if any two variables are interchanged-for P2N-2 and then show that they are true for Pm constructed according to eq 5. Because they are true for P2every Pw has the two properties. First, PW is + 1 in Ruv ( L 3 x1 3 x 2 3 ... 3 xw 3 0) because both PZ(X1,Xk) and P2N-2(x2,...,xk-l,xk+11...xw) are and therefore
p2N(x 1
,*a*
vXi-1 ,xq, *
9.
2N
-
9xq-l,x 1,- * ,XZN) =
' (-1 )kp2( x 1 ,xk)
k=2 P2N-2(X2,...,X1-1,Xq,...,Xk-l,Xk+l,...,~q-l,~1,,..,~2N) +
(-l)%l,xq) PzN-2(x2,...,x1-1,x1+1,..., X*-l,XbXq+l,...,XW) + (-1)qP2(Xl,xl) P2N-2(~2,...,~1-1,~q,~1+l,...,~q-l,~q+l,...,~2N~ (A21 The prime on the summation sign in eq A 2 means that terms with k = 1 and k = q are omitted. One must rearrange the terms of eq A 2 by interchanging variables to make them the same form as the terms in eq 5. Interchanging x 1 and x7 in the terms in the summation of eq A 2 and the resulting sign change make those terms the negative of those in eq 5. Similarly, the last two terms of eq A2 are the negative of those in eq 5 after q - 1 - 1 interchanges. The result is a sign change for PZN Case II. Exchange x 1 and x l in eq 5. PUv(Xl
,..a,
..
X&1,X1,. xm)
=
P~+ N (-l)'Pz(xi,xJ P~N-~(x~,...,xI-~,xI+~,...,~ (A3) P$& = 1-1
(-l)kP2(X1,Xk) P2N-2(~2,...,Xk-l,Xk+l,...,~1-1,~1,...,~ZN) +
k=2
2N
(-l)kP2(X1,Xk) PZN-2(XZ,...,~1-l,Xl,...,~k-l,~k+lt...,~2N) k=l+l
(A41 Moving x1 to the left inside P2N-2 in eq A4 results in a sign change of (-1)l-l for the first summation and of (-1)l in the second. Then Pm-zis decomposed into P2and Pm4 using eq 5. k-1
1-1
PiN
=[
(-l)kP2(X1,Xk)(-1)"'(
(-1)@2(Xl,xj) x
k=2 j=2 P2N-4(~Z,,..,Xj-l,Xj+l,...,Xk-l,Xk+l,...,X1-l,~i+l,...,X2N)-
c I-1
(-1)jP2(x1,xj) x
j=k+l
.
9 x 2 ~121 ) (A51 The signs in front of the summation signs inside the curly brace in eq A 5 follow from eq 5 because the terms in P2N-2 p2~-4(X2,** 9x1-1 ,xi+l,*.txk-1txk+l,* J j - 1 J j + 1,. *
The Journal of Physical Chemlstty, Vol. 87, No. 11, 1983 1949
Diffusion-Limited Reactions in One Dimension
must alternate in sign. The next step is to change the order of summation, doing the sum on 12 first.
summands with two or more indexes equal can be omitted, then in the sum over N indexes in Pw(xl,...,xm) the summands with any two or more indexes having the same value can be omitted. It follows that these summands can also be omitted from P2N(x1,...,xw,t). We prove the conclusion in & (L 3 x1 B x2 B ... 2 xw),and, since PZNonly changes sign in other regions of 2N D space, the conclusion is true for all values of the variables. Using the notation introduced in eq 7b, Sa, and 9a, eq 5 reads OJ
P2N(X1,-..,XZN)
= (4/a)N
...
61,j2**.jN)-'
j,...j9 1 , 3 , 5
2N
k=2
(-1) ' d ( X
1Jk)
M$(x2,.* * yxk-1 ,xk+ I,.. .,x UV) (B1)
We do not omit any terms in the summation over the indexes j2through jNalthough by hypothesis the net contribution from summands with any two of the indexes equal is zero. To prove the conclusion for PW, we show ~ 2 N - 4 ( ~ 2 , ~ . . , ~ 1 - 1 , ~ I + l , ~ ~ ~ , ~ k - l , ~ k + l , ~ ~ ~ , ~ j - l ~ ~ j + l below ~ ~ ~ ~ ~ ~that 2 N ~ from eq B1 the sum 2 of the summands with j, equal j2,j, equal j3,and so on through j, equal jNis zero 2N for every value of j,. (-1YP2(xI,xk) x k=j+l The function p!!-j(x2,...,xk-l,xk+l,...,xuv) has (2N - 3)!! P2~-4(x~,...,~1-1,x1+l,...~~j-l,xj+l,..~,~~-l,~~+l,...,~2~)I2l (A6) terms, each a product of N - 1 4 ' s . In the sum 2 each pair of variables x l and x, occur with the index jl in the form In the first curly brace in eq A6 we change variables calling p!'&cl,x,) (2N - 5)!!ti,mes. By rearranging eq B1 we show x 2 x j and x 1 x i . With this change the first brace becomes that the expression &(xl,xk) p!!(xl,x,) occurs net one time {I1 = p2(~2',~1') P2~-4(x3,...,xj-1,xj+l,...,xk-1,~k+l,...,x2~) + with either a positive or negative sign determined by j-1 summing over the remaining N - 2 indexes. We apply eq (-l)kP2(x21,xk) x 5 again to eq B1. k=3 P2N(x1,*.*,x2N) = p2(xl,x2) P2N-4(~I',~3,~..,~k-lt~k+l,...t~j-l,~j+lt~..,~I-l,~l+l,.~.,~2N) -
c
c
2N
1-1
c
k=j+l
(-1)kP2(xi,xk) x
k=4
+
p2N-4(~~,x3,...,xj-l,~j+l,...,~k-l,~~+l,..~,~l-l,~l+l,~~.,x2N) 2N
E
(-l)kP2(xi,xk) x
k=l+l p2N-4(xI',x3,...,xj-l,~j+l,...,~1-l,~1+l,...,~k-l,~k+l,...,~2N) (A7)
Then x ( is transposed so that it occurs between xITland xI+lin the sums where all three occur in Pw-4;after this rearrangement the first two sums are multiplied by (-1)l-l and the third by (-1)l. If we multiply by the factor (-1)l occurring in eq A6, we find (-1)'{I1 = P2~-2(x2'ix3,...,xj-l,xj+li...,x1-l,xI',...,xw) = -p2N-2(~Z,x3,...,xj-l,~j+l,.~~,~1-l,xl,.~.,xZN) (AS) The same two steps with the second curly brace in eq A6 give (-1)I-'{j2 = -Pw-2(~2,...,x ~ - ~ , x ~ + ~ , . . . , z Z N ) (A9) Substituting eq AS and A9 into eq A6 gives directly that an interchange of the variables x1 and X I in Pw results in a sign change. As explained in the text we can now assign a definite value of +1, -1, or 0 to every point in space by permuting the variables from the order in which they occur in Rw and using what we know about the periodicity of Pw Because in each of the (2N- l)!!terms in Pw each variable occurs only once and it is in a P2,the periodicity of Pw with respect to any variable is the same as P2. Increasing any variable by (2n 1)L with n an integer results in a sign change for Pw. Also, because of P2's periodicity, PZhr is 0 if two variables differ by an integral multiple of L.
+
Appendix B Omission of Summands in PZN with Two Indexes. If in the sum over N - 1 indexes in P2N-2(xl,...,x2N-2) the
(-l)kP2(x3,xk) P 2 N - 4 ( ~ 4 , . . . , ~ k - 1 , X k + ~ , . . . , X Z N + ) 2N
k-1
(-l)kP2(x1,xk)(
(-l)mf1P2(x2,x,)
k=3 m=3 p2N-4(x3,...,~m-1,xm+l,...,ltk-1,xk+l,...,~2N)+ 2N
c
(-1)mP2(x2,x,) x
m=k+l P2N-4(~3,...,~k-l,~k+l,.~.,~~-l,~m+l,...,~2N)} (B2)
One can see from eq B2, that in Ruv the sign associated with the term p!!(xl,xz)p!!(x3,xk) is (-1)'. Also, the sign for d$(x1,~3) p!$(x#k) is (-l)k-l, and the sign for p!!(xl,xk) p!!(x2,x3) is (-l)k. In these three terms one evaluates the sum over the remaining N - 2 indexes by noting that in Rw the Pw4 multiplying each term is +l. We use the trigonometric identity
- d(x1,xl) & ( x k , x m ) + d(xl,xm)d(xk,xi) = sin (Yi - Y k ) sin (YI - Ym) sin (Y1 - yl) sin ( Y k - y,) + sin (yl - y,) sin ( Y k - yl) = 0 (B3) Here eq 7 has been used and Y k = jlmk/L. By grouping the terms, which are pairs of pairs, in threes one can eliminate all the terms containing xl, x2, and x 3 from 2. It will be shown below how to group the terms so that there are none remaining. Because all the variables in PZNare equivalent, we expect that all the terms grouped according to eq B3 with k < 1 < m will have the correct signs for cancellation; this is demonstrated below. To consider a second special case with the terms including the four variables xl, x2,x l , and x, with 1 < m and I , m # 3, one must exchange x 1 with x 3 in the first sum of eq B2 and multiply by -1 because of the antisymmetry of PW-> Then one exchanges the variable x 3 1 - 4 times (resulting in a sign change of (-1)l4 in P2N-4to move it all the way to the left with the rest of the variables arranged
d(x1,xk) d t ( x l , x m )
Torney and McConnell
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983
1950
sequentially so that the value of P,, is then + l . Then, in Rm the sign of the term p!!(xl,x2) p!!(xl,x,) is (-l)l+"'+l, the sign of the term p!!(xl,xl)p!!(x2,x,) is (-l)l+"',and the sign of the term p!!(xl,x,) p!!j(x2,xl) is (-l)",+l. Again from eq B3 the sum of these terms is zero. In the general case with k < 1 < m and k, I , m # 2 one interchanges x 2 and X I in the two sums inside the curly brace in eq B2 and multiplies by -1 with the result
pressed). If for all M IN Im has the value given in eq C1, then it does also for M = N + 1 . To prove the conIm =
2N
clusion we start with the general formula for "PW obtained by following the procedure following eq 19 ( R ( N ) given in eq 6 )
k=3
(-1)kp2(xl,xk){~ = 2N
-C
k-1
(-l)kP2(x1,Xk)(
k=3
m=3
' (-l)m+1P2(Xl,xm)
m
p2N-4(~3,...,~m-1,~m+1,...,~l-l,~2,...,~k-l,~k+l,~~~,~2N) + 2N
C ' (-1lmPz(Xl,xm)X
m=k+l p 2 N - 4 ( ~ 3 , . . . , ~ k - l , ~ k + l , . ~ ~ , ~ l - l , ~ 2 , . . . , ~ ~ - l , ~ m( B + l4, )~ ~ ~ , ~ W ) )
The prime means that the term in the summation with m equal 1 is omitted; since that P2 also contains x 2 , we are concerned neither with it nor about the first sum in eq B2 in this case. After transposing x 2 all the way to the left in PW4 making its value +1 as above, we find the following signs for the three terms. The sign of p ! ! ( x l , x k ) p!!(xl,xm) is ( - ~ ) ~ + l + ~ the + l , sign of p!!(xl,xl) p!!(xk,x,) is (-l)k+l+m, and the sign of p!!(x,,x,) p ! ! ( x k , q ) is (-l)k+l+m+l. Using eq B3 the sum of these terms is zero, and the above special cases are consistent with the general case. All that remains to show that the sum 2 is zero is to demonstrate that one can group the pairs of pairs by threes according to eq B3 and have none remaining. One can arrange the pairs of pairs of the form
(B5)
plf(x1,xk) plf(xl,xm)
in a matrix with k going from 2 to 2N in 2N - 1 rows and the p+(xl,x,) arranged from left to right in ascending order with the terms for the lowest value of 1 arranged in ascending order with respect to m, then the terms with the next lowest value of 1, and so on in 1 / 2 ( 2 N- 2)(2N - 3) columns. As an example the matrix of the terms for the six particle problem is constructed in eq B6. We use the (12){(34) -(13){(24) (14){(23) - ( 1 5 ) (23) (16) ( 2 3 )
i
-(35) -(25) -(25) -(24) -(24)
+(36) +(26) +(26) +(26) +(25)
+(45) +(45) +(35) +(34) +(34)
-(46) -(46) -(36) -(36) -(35)
+(56)} +(56)} +(56) +(46) +(45)
i
From eq C2 wSW+z is obtained from the expression for 2MPuv+2 by integrating over the region Rw ( L 2 x1 2 x 2 2 ...xu 2 0) with the value of the integral given by eq C1 for M < N . 2MS2N+2
f
= Q ~ ( N + ~ ) [ L ~ / X I ~ + ~{ j N + l 2 O ' N + l 2 + j l . . . j ~ + l =1,3,5
+
...
j N 2 ) , . , 0 ' ~ + ~ 2 ,,. jN+2-M)2)-le-Ci,v+12+. ..+j~+z-d)E x
We can get an expression for 2N+2P2N+2 from the relation 2N+2~ 2N+2
=1(2NSW+z+ 2N-2s2N+2
+ 2N-4s2N+2 + + 'S2N+Z + OS2N+2] 1..
((24) The expression in braces in eq C4 simplifies with integration of 0Suy+2by parts. ' S Z N +=~Q ( N + 1 ) [ L 2 / ~ ] N x+1
(B6)
notation (Im) = p!!(xl,x,). Returning to the general matrix of terms of the form given in expression B5 which for PUvis (2N - 1 ) X 1 / 2 ( 2 N - 2)(2N- 3 ) , one can remove all the terms which contain the variable x 2 in groups of three according to eq B3 with a total sum zero. What remains is a (2N - 2 ) X 1 / 2 ( 2 N3)(2N - 4 ) matrix of terms of the remaining variables. Repeating this process will yield a final 3 X 1 matrix containing three terms with x1 and the last three variables. There are no remaining terms so the sum 2 is zero. Therefore, the conclusion follows that in PUv one can omit all summands with two or more equal indexes. This ensures that the spatial part (pw) of a reaction at a stage with 2M particles remaining out of 2N is the same regardless of the value of N . Appendix C
General Form for the Survival Fraction. The following conclusion is preliminary to the development of the general formula for the survival fraction 2MSzN(the [ is sup-
The first of the two terms obtained by integration by parts exactly cancels 2Suv+z in eq C 4 . Repeating this process N - 1 times gives 2N+2s
ZN+2
= m
J. Phys. Chem. 1983, 87, 1951-1954
((2N)!( 2 /*)2N)-' m
C
jp..jp1,3,5...
To derive eq C10 we begin with an equality symmetric in j , and j 2 .
=
' jN2(jN2+ jN-12)...(jN2+ jN-12 + ... + j12)-I
m
2, = f/2
(C7)
As usual the prime on the summation means that terms with equal indexes are omitted. We cancel the two constant terms in eq C6 and compare the formula for 2N+2S2N+2 with that for P2N+2.
P
~ = +S ~ (~+ N 1)
5
jl...j~t1=1.3,5...
1951
c ' jl..jN=1,3,5... (0'12
G1-2
+ j2-2) x + ... + jiVZ))-'
+ j 2 2 ) ...0"2
m
f/2
C
' Lj12j220'12
...
jl...jp1,3,5
=
+ j 2 2 + j 3 2 ) ( j 1 2 + ... + j N 2 ) ) - I
G N + N...jl)-l j x
((3)
~ ~ l ( ~ ~ , . . , , ~ ~ ~ + ~ ) e - ~ N t i ' + . . . + j i ' ) ~
Because
is the relation between the two formulas
luv+2 = [L2/7r]N+1LjN+l(jN+12 + jN2)...cjN+12+ ... + j12) I-' O'NjN-l.. .jl)(Cg) This proves the conclusion. Because the value of I2 (eq 22b) is consistent with eq C1, by induction all the IUvare given by eq C1. And for all M and N ( M 5 N)2MS2Nis given by eq C3. We can transform eq C3 with the following formula:
Rate Constants and Products of the Reactions of e&, 02-,and H with Ozone in Aqueous Solutions K. Sehested,' J. Holcman, and Edwln J. Hartt Accelerator Depaffment, Rls 0 National Laboratory, DK 4000 RosklMe, Denmark (Recelved: October 11, 1982)
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Ozonide ions, identified by their optical absorption band at 430 nm, form in electron-irradiatedaqueous ozone solutions by the reactions eaq-+ O3 03-and 02-+ O3 03-+ 02.Both reactions are quantitative and the rate constant for the hydrated-electron reaction with ozone was determined in competition with oxygen to be We,,- + 0,) = (3.60 f 0.2) X 1O'O dm3mol-' s-'. The rate constant for the peroxy radical reaction with ozone was measured from the pseudo-first-orderbuildup of the ozonide ion to be k ( 0 2 - + 0,) = (1.52 f 0.05) X lo9 dm3mol-' s-l. The rate constant of hydrogen-atom reaction with ozone was measured in acid solution, pH 2, by competition with oxygen to be k(H + Os) = (3.65 f 0.40) X 1O'O dm3 mo1-ls-l. The initial product is an OH radical.
Introduction The free-radical-induceddecomposition of O3in aqueous solutions has recently been reported.lr2 In these pulse radiolysis studies, rate constants for the reaction of O3with e, - and OH radicals were measured.' In studies on the Oh--induced decomposition of 03:it was concluded that the free radicals, OH, 0-, 02-, and 03-, participated in the chain reactions involved. Except for the observation that 03-was a transient species in the alkaline decomposition of 03,no attempt was made to identify products in these studies. In the present paper we have redetermined k(e,; + 0,)and also report on the identification of the product 0; from the reactions of e, with O3 In addition, we have 'Present address: 2115 Hart Road, Port Angeles, WA 98362. 0022-365418312087-195 1$01.50/0
measured k(H + 0,)and the rate constant for electron transfer from 02-to 03. As expected, the OH and 03radicals are the products of these two reactions.
Experimental Section Pulse Radiolysis. The 10-MeV HRC linear accelerator at Riser was used as radiation source with a pulse width of 0.2-0.5 ps. The optical detection system has been described previ~usly.~It consists of a 150-W Varian highpressure xenon lamp and a Perkin-Elmer double quartz prism monochromator with a 1P28 photomultiplier. The (1)Bahnemann, D.; Hart,E. J. J. Phys. Chem. 1982,86, 252. (2) Fomi, L.; Bahnemann, D.; Hart,E. J. J . Phys.Chem. 1982,86,255.
(3) Sehested,K.; Corfitzen, H.; Christensen, H.C.; Hart, E. J. J.Phys. Chem. 1975, 79, 310.
0 1983 American Chemical Society
