608
JACKCOHN
Theory of the Radial Distribution Function' by Jack Cohn University of Oklahoma, Department of Physics, Norman, Oklahoma
(Received July $7, 1967)
A method is devised for constructing an integral equation governing the radial distribution function for a general thermodynamic system for sufficiently small number densities. For even smaller number densities, the equation becomes a linear integral equation of the second kind, which is solvable. The symbolic solution is presented and numerical results for argon, using the modified Buckingham potential, are briefly discussed.
I. Introduction I n the present work, a method is devised for the construction of an integral equation governing the radial distribution function for a general thermodynamic system. The method is basically not exact and is expected to apply better at smaller densities. The advantages of the theory are that the various assumptions used have a direct physical significance and that, for sufficiently small number densities, the equation can be converted into a linear integral equation which is solvable by analytical methods. There is a close parallel between the basic scheme of this paper and that used by Debye and Hucke12in their classical theory of electrolytes. Accordingly, the method used here involves an application of a technique previously applied by the author3to the problem of strong electrolytes. Since the context of this paper is somewhat different from that of the other, however, the treatment here will be self-contained. The basic idea of this work is developed as follows. I n the Debye-Huckel theory of strong electrolytes, one is concerned with the average charge density, {, in the neighborhood of any given ion. Replacing the potential of mean force by the average potential, p, about the ion, Debye obtained the well-known inexact relation ecp
sinh kT
v2cp
87rne
Ecp
D
kT
= - sinh -
(3)
Another way of utilizing eq 1 and 2, which is more to our purpose, however, gives an equation on {, instead, as follows. Integrating both sides of eq 2 over a sphere of radius r, we obtain dcp dr
=
-~JC'{(z)47rx2dz 1
(4)
Therefore, we have
where u / 2 is the ionic radius. Next we rewrite eq 1 as
Combining eq 5 and 6 then gives the following equation on i-
-bJCrt-2
dSlCE[(z)47rz2dz - IcT - sinh-' E
2n E
(7)
where e is the magnitude of the ionic charge, n is the average number density of positive or negative ions, k is Boltzmann's constant, and T is the temperature. Since we are dealing with charged particles, there is the additional differential relation between i- and p given by Poisson's equation
I n the present work, we attempt a development similar to the above. We first develop a relation between the average particle number density at any point in the vicinity of any given particle and the average potential energy of a particle at that point. We then establish an integral relation between this average potential energy at a point and the average number density throughout the entire space surrounding the given particle. Between these two relations (for sufficiently small number densities), we are then
where D is the dielectric constant of the solvent. Combining eq 1 and 2 in the usual way then gives a n equation on cp alone as
(1) This work was in part supported by the National Science Foundation. (2) P. Debye and E. Huckel, Phys. Z., 24, 185 (1924). (3) J. Cohn, Phz/s. Fluids, 6,21 (1963).
[ = -2ne
The Journal of Physical Chemistry
THEORY O F THlD RADIAL DISTRIBUTION FUNCTION
609
able to obtain an integral equation for the radial distribution function alone. For even smaller densities, this becomes a linear integral equation, which can be solved analytically. The next section is devoted to the development of the relation betweien the average number density and average potential energy at a point in the vicinity of a particle. Section I11 js then concerned with developing the integral relation between the average number density and average potential and thence the integral equation governing the radial distribution function. In Section IV, the small concentration region is considered, and it is there demonstrated that the equation governing the radial distribution function becomes a solvable linear integral equation. Section V consists of a very brief discussion of numerical results for very dilute argon gas, and section VI consists of (t general summary of the results of the paper. Finally, an A.ppendix is included, where many of the mathematical details of the development are treated.
ticles in the system, and V is the total potential energy of the system corresponding to the particular distribution in question. I n order for this equation to be valid, it is assumed that as the particles are moved about, each in its own cell, V does not change significantly. This is tantamount to requiring that the pair potential be sufficiently slowly varying. With a hard-core potential (or one that varies rapidly in the neighborhood of the origin), this is clearly true only so long as neighboring cells are not simultaneously occupied. Accordingly, we are assuming then that contributions coming from situations where two particles are simultaneously in neighboring cells may be ignored. This, in turn, implies that our considerations will be made for systems of sufficiently small density. Now V is given by
11. Number Density and Average Potential I n this section, we develop the relation between the average particle number density at any point in the vicinity of any given particle and the average potential energy of a particle at that point. The initial stage of the development is quite general and includes terms related to fluctuations about the average state. Considerations are then specialized to the case of sufficiently small number densities where the fluctuationa can be ignored. Consider, then, a thermodynamic system composed of N particles (identical and structureless) interacting with a pair potential U ( r ) with distance of closest approach u. We are interested in the distribution of particles about any particular particle (hereafter referred to as the central particle). For this purpose divide the space around the central particle into small elements each of volume v. Each such cell is so small that only one particle, at most, can be init. (Wesay that a particle is in a cell when its center of mass is within the cell.:) I t i s to be emphasized here that we are not constructing a so-called cell theory. The term cell is merely used to describe small-volume elements considered in the formulation. From classical statistical mechanics, we can say that the probability of realizing a particular distribution of particles about the central particle (Le., a specification of the numbers of particles in the various cells) is given by
P
== AvN-’(N
- l)!exp(-@V)
(8) where P = l/kT, A is a normalization factor depending on the temperature, N is the total number of par-
(9)
where t is the total number of cells, cpt is the potential energy of a particle (anywhere) in cell i, and Pc = 0 or +1 is an occupation number for cell i; P o refers to the central particle and has the value 1. Combining eq 8 and 9, we have
+
We shall have need of quantities averaged over all distributions about the central particle. Such averages will be denoted with brackets, so that, for example, an average occupation number is denoted by (P,) and is given by
the summations referring to all distributions about the central particle. For further progress, we construct from eq 10 the following identity, which will become our starting point
P
- 1)! exp
= AvN-’(N
t
[
t
-p
p,(cp,) j-1
c (P,)(cp,) 2 j-1
+
- ;Po(Po)
+ A]
(12)
where A
P
= --
c V(r,!J(P, -
2 i#k#O
(Pj))(Pk
- (Pd) (13)
The term A is related to the fluctuation of the given distribution about the average, and (9,)=
c U(rrC,)(Px)
k# j
(13’)
Notice here that (pi) is not the potential of mean
JACK COHN
610 force but rather is really the average potential energy of a particle in cell j calculated as if the particle were not there. I n the following, we shall need eq 12. However, we must first evaluate the sum of the auxiliary quantity
quantities averaged with the function PO. Averages of this type will contain angular brackets together with a superscript zero, viz.
where the summation sign again indicates a sum over all distributions about the central particle.
IIb. Evaluation of ( p , ) O From eq 24 and the defining eq 14, we see that
IIa. Evaluation of C P O For convenience we write
Using eq 20, we then have
where
We shall first be concerned with the evaluation of 2. Defining Et = exp (-p( pi)), we can write t
z = CII$,Bi i- 1 the summation being over all sets {pk;IC = 1, 1
(where pk = 0 or +l), such that
(17)
. ., t ) ,
C P I = N - 1.
i-1
It is convenient to use the method of steepest descents to evaluate 2. We therefore define the following function
where 50-1
=
(nv)-l
-1
(27)
See eq A12. In addition, eq A2 and the fact that (l/A)(bA/b(cp,))is of smaller order than the remaining terms has also been used. (See Appendix 11.)
IIc. Relation between (p,)O and (8,) We are now in a position to develop an approximate relation between (p,) and ( P U ) O . From eq 11 and 12 we have
t
!(E>
=d-1 n (1 + E%)
(18)
where 5 is a complex variable. The term 2 is then given by the expression
If we can ignore the fluctuation (A becomes where the integration is performed in the complex 5 plane around any path surrounding the origin. Using the method of steepest descents, we have then
l dY + - x>U(lz - YI)
= 2?mas, and E@,?) 5 - -urP[ l
+ E~]-~h(uA!,u?) (56)
Written in more convenient form we have dY
(51)
and where G is defined as before. (We could integrate instead from u to r , but then we would have a residual term which is constant. This term could, however, be adjusted to zero to good approximation.) I n obtaining this expression, use has also been made of the very reasonable assumption that lim(yp - yU) = 0. %I+-
The above relation leads to an equation identical in form with that of eq 48, where K(x,y) replaces K(x,y) in the integrand. These equations are equivalent. We are now ready to discuss the solution of eq 48.
IV. Solution for Low Densities The solution of eq 48 (with kernel K or K) poses a The Journal of Physical Chemigtry
+ fo]-’U’(aP),
= w(a?), O(?)= - P [ l
G(y) = f(y)
+ XJmxi(x)K(x,y) 1
dz
(57)
where f(y) = O’(y)
+ XLmxn(z,y)dx
(57’)
If K(z,y) is sufficiently well-behaved (and we expect that it is), the solution to this equation is through first order in X (since X = 5 0 and we are only working to first order in Eo). (The solution here is found by using Volterra or Fredholm’s method.) G(y) = f(y)
+ kJmxX(z,y)0’(x) 1
dx
(58)
THEORY OF THE RADIAL DISTRIBUTION FUNCTION and f ( y ) (5{ ( u y ) ) is then given by f ( y ) = 1 Therefore, we have f(y) = 1
+ f(y) +
+4y).
isrn 1
x&,Y)D’(x) dz
(59)
Writing thifi relation more explicitly we have then as the solution for { at suficiently small number densities the expression
{(r) = 1
- pC1 + t01-’U’(r) +
X ~ m z ~ ( z , ~) [p(l l
+ to)-’U’(uz)l dz
(60)
where
613 Using this data, we evaluated f(r) from eq 60. The resulting curve of { vs. r seems to be qualitatively correct (experimental curves of { vs. r for densities in our range of interest could not be found). From the above values of c ( r ) , we then calculated pV as determined by the general relation
and found the value to be 1.08 atm-cm3/amagat. The experimental value for this product is found in the work of Michels, Wijker, and Wijkere which, for our temperature and number density, yields the value 1.16 atmcm3/amagat. The difference between the two values is about 7%.
VI. Summary or
(UX
- Z ) U ( ~ U-X 21) dZ +
J u ~ G ( u z , Z ) ( Z- az)U(lux - 21) dZ}
(62)
and where X == 2nna3. The evaluation of { from this equation is not difficult to obtain, but involves very lengthy calculations for any realistic type of intermolecular potential. I n the following section, we briefly report on a numerical result found by means of the preceding formulation.
V. Numerical Result I n this section, we report on the single numerical result that has so far been calculated using the preceding theory. We consider argon gas a t a temperature of 50” and number density 0.865 X loz1atoms/cm3. This density is sufficiently small that the approximation (to2 r. Adding up the contributions given in eq A21 and A23 and changing variables in an obvious manner then yield the relation cp(r) - U ( r ) = 2rnJ
- r2 + 2cr) dc +
dxL+'U(dx2
Figure 2. Contribution to rp'(P) when X
xp(x)F(x,r) dx
+
C
(A21) where
where u2 L 3 E
- x 2 + r2 2r
F(x,r) =
(A219
The first term in the above comes from regions where x 2 ut r - 5 2 U. The second term comes from regions where r - x 6 u, r > x.
and
H(z,r)
J-*U ( d x 2 - r2 + 2cr) dc
=A
r+2
r-x
U ( d x 2 - r2
+ 2cr) dc
(A25)
(A26)
Volume 76,Number 6 February 1968
RALPHKLEINAND MILTOND. SCHEER
616 where V ( r ) is the intermolecular potential, specified only to the extent of allowing a distance of closest approach, 5. Before going any further, however, an important point is to be noted. The average potential, cp, which is of interest here, must go to zero as r goes to infinity, since this requirement was used in the derivation of eq 39. However, if we consider the cp as developed above, we find that for large r it has the form p(r) = f(r)
where f(r)
+
0 as r
-t m
.
+ constant
L427)
The above constant term
arises in connection with contributions from particles in two regions: those for which x is given by lx - T I 6 u and those for which x 2 r 5. As r becomes very large, this environmental contribution approaches a constant value. I n order then to have a cp(r) which goes to zero as r + , we must subtract this constant term from p(r) and use instead a cp’ = cp - constant. In the text (eq 40), we have subtracted the constant term from the right-hand side of the equation and have called cp’ just cp, This is the reason for replacing U(r) (on the left-hand side of the equation) by U’(r) = V ( r ) - constant.
+
Mechanism of O(3P) Addition to Condensed Films. 11. Propene, 1-Butene, and Their Mixtures’ by Ralph Klein and Milton D. Scheer National Bureau of Standards, Washington, D . C.
(Received July 87, 1967)
The kinetics of O(3P) addition to condensed films of propene and l-butene has been investigated at 90°K. The reaction rate was found to be independent of olefin concentration for both propene and l-butene using propane as an inert diluent. Films containing mixtures of propene and l-butene yielded mostly C3 products at high olefin concentration and approached a limiting value of about 1.6 for the C3/C4 product ratio at infinite dilution in propane. When C3Ds films were used, the ratio of carbonyl to epoxide was measurably less than that obtained with CBHB. The results are interpreted in terms of a model which depicts the olefin film to be a nearly perfect sink for O(3P) atoms. The primary process is assumed to be the formation of a triplet adduct which undergoes either ring closure to form the epoxide or an intramolecular migration of a hydrogen atom to produce a carbonyl compound. The latter exhibits the expected isotope effect when D is substituted for H in the olefin reactant. The behavior of propene-l-butene mixtures can be accounted for qualitatively by assuming that propene not only reacts more rapidly but diffuses more easily than does the l-butene.
The addition of ground-state oxygen atoms to olefins occurs with a sufficiently low activation energy that the reaction may be conveniently studied below100°K.2~3 The results of such studies with propene and the 2butenes are given in some detail.4 Several general observations were made on the characteristics of the 0 atom-condensed olefin reaction. By analogy to the H atom-olefin reaction, the mechanism of which has been established from rather extensive experimental data, it was tacitly assumed that the addition reaction occurred on the surface of the deposit. It will be The Journal of Physical Chemistry
shown that with 0 atoms, however, diffusion into the bulk of the condensed layers occurs. The general features of the 0 atom addition to olefins have been established in a series of publications by Cvetanovic and co-workers.5 In the gas phase the (1) Supported by the U.S. Public Health Service. (2) (a) A. N. Ponomarev, Izv. Akad. Nauk SSR, Otd. Khim. Naulo, 1307 (1962); (b) A. N. Ponomarev, Kinetiloa i Kataliz, 6, 859 (1965). (3) V. M. Orlov and A. N. Ponomarev, ibid., 7, 419 (1966). (4) A. N. Hughes, M. D. Soheer, and R. Klein, J. Phye. Chem., 70, 798 (1966).
