J. Phys. Chem. 1988, 92, 2016-2018
2016
Pressure of Hard-Sphere Systems Robin J. Speedyt Institut fur Biophysik und Physikalische Biochemie, Universitat Regensburg, 0-8400 Regensburg, West Germany (Received: August 30, 1987)
An equation for the pressure of the hard-sphere fluid or crystal, in terms of the surface to volume ratio So/Voof the available space, p/pkBT = 1 + (u/2D)(So/Vo),is rederived in a way which shows it to be more primitive, and more general, than previous derivations have indicated.
Introduction In ref 1 the pressure of a hard-sphere system was expressed in terms of the surface to volume ratio So/Voof the available space (or spare volume2) as
The available space Vois illustrated in Figure 1. So is the area of the surface which separates the available space from the excluded space V - Vo. u is the diameter of a sphere, D is the dimensionality of the space, p is the pressure, p = N / V is the number of spheres per unit volume and kBT is Boltzmann’s constant times the temperature. Stel13pointed out that B ~ l t z m a n nin , ~his analysis of the lowdensity expansion of the equation of state of the dilute hard-sphere gas: evaluated the first few virial coefficients by way of eq 1. Boltzmann did not have symbols for Soand Vobut he expressed eq 1 using the first few terms in their low-density expansions and no doubt recognized its more general validity. The term “available space”, which is used for Voin Brush’s translation of Ga~theorie,~ is used here in preference to the term “spare volume”, which was coined in ref 2. Reiss and Hammerich5 have recently rederived eq 1 in the language of scaled particle theory. They suggest that it may be possible to establish the necessary existence of the fluid-to-crystal phase transition in hard spheres from a consideration of the constraints which must apply to Soand Vo. For that purpose it is important that the derivation of eq 1 be generalized so as to apply to nonuniform and crystalline states of the system. In this paper eq 1 is rederived in a way which emphasizes that it is both more primitive and more general than the previous derivations indicate. Previous derivations have used the conventional pressure equation
in which B2 is the second virial coefficient and g(u) is the pair correlation function at contact. However, eq 1 is primitive in the sense that it can be derived without using eq 2, without invoking the virial of Clausius on which eq 2 is traditionally based, and without recourse to the distribution function formalism. Thus much of the formal machinery of statistical mechanics, which has evolved to deal with more general systems, can, with advantage, be bypassed. In particular, by avoiding the distribution formalism one avoids the need either to work with the generalization of eq 2 to nonuniform systems or to assume uniformity of the density. The symbolism used here is, with minor changes, that used in ref 2 and that part of the argument which is already given there is stated here with less explanation in the interests of arriving at eq 1 with minimum deviation from the most direct line of reasoning. Some other parts of the argument are already indicated by B ~ l t z m a n n . ~ ‘Permanent address: Chemistry Department, Victoria University of Wellington, New Zealand.
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Derivation The system considered is a set of N Ddimensional classical hard spheres, each of diameter u, whose centers are confined to a D-dimensional space of volume V a t temperature T. As a convenient (but not essential) conceptual device the space is divided into a lattice of V/w very small cells, or sites, each of volume w . There is no loss of rigor involved in “quantizing” space in this way since w cancels out of all measurable quantities and the limit w 0 is implicit. In computer simulation studies space is quantized in a similar way by the finite number of significant figures used to represent positions. A configuration of the system may be specified uniquely by a list of the N sites in which a sphere center is located. The set, and the number, of all possible configurations is denoted by -+
R = R(N,V,u,o) In any one of the k = 1, 2, 3, ..., R configurations of the system the available space ‘/O+ is that volume which is within the space V but not within one diameter of the center of any of the spheres. The available space of the system, Vo,is then defined as the average over configurations of VO,k Vo =
I n
-
c
V0,k
Qk=l
Similarly, SO,k is the surface area of the available space configuration k, and its average is
(3) V0,kin
With those defmitions in place we can now write down an induction formula for the number of configurations in terms of the available space. Consider a system containing M - 1 spheres. An Mth sphere can be added to any one of the Vo,,(M - l)/w available sites of the kth configuration of M - 1 spheres to generate one of the O ( M ) configurations of M spheres. The total number of such configurations is the number of all possible additions to all possible configurations of M - 1 spheres rl(M-1)
O(M) =
2
k= 1
Vo,,(M-l)/wM
= R(M-1) V0(M-l)/wM
(5)
The second equality uses eq 3, and we divide by M because the Mth sphere is not distinguishable. Equation 5 , together with the M = 0 and M = 1 terms R(0) = 1 and Q(1) = n(0) Vo(0)/w (1) Speedy, R. J. J . Chem. SOC.,Faraday Trans. 2 1980, 76, 693. (2) Speedy, R. J. J . Chem. SOC.,Faraday Trans. 2 1977, 73, 714. (3) Stell, G. Chapter 6 in “The Wonderful World of Stwhastics: A tribute to Elliot W. Montroll”, Vol. 111 of Studies in Statistical Mechanics; Shlesinger, M. F., Weiss, G. W., Eds.; North-Holland: Amsterdam, 1985. (4) Boltzmann, L. Lectures on Gas Theory (translated by S . Brush); University of California Press: Berkeley, CA, 1964. ( 5 ) Reiss, H.; Hammerich, A. D. J . Phys. Chem. 1986, 90, 6 2 5 2 .
0 1988 American Chemical Society
Pressure of Hard-Sphere Systems I
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 2017
A
1
\ \
Ql
= Q(N,V,a+du,o)
= Q(l + (a In
Q/du)N,V
da)
(1 1)
The derivative ND (a In Q / ~ u ) , ,= --(p/pkBT U
- 1)
(12)
is evaluated in the Appendix. We now turn to the calculation of Q2. Q2 is the number of configurations of N (indistinguishable) spheres of diameter u in the space V, with the constraint that at least one unspecified pair of spheres are in contact. The logic used here is similar to that used by Boltzmann to calculate the probability of a pair of spheres being in contact (Gastheorie; Book 11, Section 51). In the argument leading to eq 5 the Nth sphere added to a system of N - 1 spheres was allowed to go anywhere in the available space VO,k(N-l). Here we construct the set of Q3 configurations in which the Nth sphere is constrained to contact another sphere. That is, the Nth sphere can be placed only in those So,k(N-l) du/w cells which lie within d o of the surface of the available space. Boltzmann4 called Sodo the favorable space in this context. The number of configurations which result from adding the Nth sphere to the favorable space is Q(N-I)
Q3
=
E
So,k(N-l) du/w
k= I
Figure 1. The available space Vo,kof a particular configuration, k, of hard disks is defined as the space which is not within one diameter u of a disk center. The disks are represented by the small circles of diameter u and the larger arcs, of radius u drawn about each disk center, trace out the surface SO,kof the available space. A disk which is added to the system must have its center in the available space.
(where Vo(0)= V), constitutes the induction formula for Q(N), which yields
Contact with thermodynamics may be made through the canonical partition function Z
Z
Z(N,V,T,u) = (X-Dw)NQ(N)
(7)
= Q(N-1) So(N-1) du/w
where the last two steps incorporate eq 4 and 5. Q3 differs from 0,by a combinatorial factor because the last sphere is, at this stage, still distinct. To get the correct combinatorial factor it is helpful to first consider all the spheres to be labeled and distinct, when there are ( N - l)!Q3 configurations in which the Nth sphere contacts another (unspecified) sphere and (N - l)!Q3/(N - 1) in which it contacts a specified sphere. There are N ( N - 1)/2 ways of choosing a pair so there are ( N ( N 1)/2)(N - l)!Q3/(N - 1) configurations in which an unspecified pair is constrained to contact. Dividing by N! removes the labels to give Q2 Q2
=
'/zQ3
~ s ~ ( N - 1du)
= Q -
2 The term A%, where X = (2rmkBT/h2)-'/*, is the partition function for a particle of mass m confined to a cell of volume w. Note that on combining eq 6 and 7 to get eq 8 w cancels out, so (for a classical system) the limit w 0 can be taken implicitly. Expressions for all the thermodynamic properties in terms of the available space follow2from eq 8 but need not be stated here. For the present purposes we require only the pressure
Vo(N-1)
Combining eq 10-14 gives
-
P
E kBT(a
In
2/8V)N,T,u
=
kBT(a
In
n/aV)N,T,u
(9)
We now wish to relate the pressure to the chance of finding a pair of spheres in contact, and to relate that chance to the ratio So/V,. The set of all configurations of a given system Q = Q(N,V,u,w) can be written as the sum of two sets
Q = R,
+ R2
(10)
where Q l collects all those configurations in which no pair of spheres are in contact and Q2 is the set of configurations in which at least one pair of spheres are in contact. A pair of spheres are said to be in contact when their centers are separated by u to u du. The set of configurations, QI, in which no pair of spheres are closer than u du is the same as the set of all configurations of a system of N spheres of diameter u + du in the space V, so
+
+
which is the same as eq 1 for large N a n d is the result that we set out to derive.
Discussion In deriving eq 1 no assumptions were made concerning the translational or rotational invariance of the singlet or pair distribution functions, because the distribution function formalism was bypassed. The result needs to be qualified only on the grounds that the system must be large enough for thermodynamic arguments to apply and for those parts of the surface SO,kwhich lie on the surface of the space V to be negligible in comparison to those parts of SO,kwhich lie on the surface of the exclusion spheres. With those qualifications the result is applicable to the hard-sphere fluid or crystal, to systems in which the fluid and crystal coexist, and to spaces of arbitrary (but connected) shape in which surface-induced nonuniformities may affect the properties. The relation between eq 15 and the conventional pressure equation, eq 2, is easy to understand. The radial distribution function, g(u), at the contact distance, is defined such that
J . Phys. Chem. 1988, 92, 2018-2022
2018
4?r$pg(u) du is the probability that a reference sphere has another sphere center within a distance u to u du. (In a nonuniform system g(u) is averaged over all positions of the reference sphere, weighted by the density at each position.) But by the argument leading to eq 13, the probability that a given sphere (the Nth sphere in that argument) contacts another is just So(N-1) du/ Vo(N-1) so we have
+
4ru2pg(u) d o
So(N-1) do/ V&N-1)
(16)
Substituting eq 16 into eq 15 and noting that the second virial coefficient B2 = 2 / 3 r u 3yields eq 2.
Acknowledgment. Support of the Deutsche Fortschungsgemeinschaft and the hospitality of Professor H.-D. Ludemann are gratefully acknowledged. Appendix The purpose of this Appendix is to evaluate the derivative (13 In Q/au)N,vwhich appears in eq 11. The variable y defined by
represents the difference between the Helmholtz free energies of the hard-sphere system and an ideal gas, on a per particle basis. Since y is an intensive variable it depends only on the density
variable x = NuD/V, regardless of whether N, u, or V is the independent variable, sobne can write
(aY/dU)N,v= (dy/dx)(ax/d4v,v with dY/dX = (aY/av)N,u(av/wN,o The required partial derivatives are (ay/wN,v=
kB T 7 (a In Q / W N , V
(ax/aU),, = ND+/V (av/ax),, = - V L / N ~ D ( a Y / ~ v ) N , ,=
kBT kBT 7 (8 In Q / w h , u - 7
= p / N - k,T/V The last step incorporates eq 9. Substituting (A3)-(A7) into (A2) yields ND (a In Q / a ~ ) , , y = --(p/pkBT
- 1)
U
which is eq 12 of the text.
Reaction Entropies and Acid-Base Behavlor of Transltlon-Metal Complexes in Recast Naflon Fllms Michael H. Schmidt and Nathan S. Lewis* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: March 10, 1987; In Final Form: August 27, 1987)
The effect of Donnan potentials on the apparent formal potentials of polymer-bound metal ions has been investigated for recast Nafion films and for spin-cast Nafion films on graphite electrodes. For the probe ions R u ( N H ~ ) ~ ~and + / R~(bpy),~+/~+, ~+ the behavior generally correlates with Donnan potentials expected for ideally permselective domains, and plots of the apparent Eo' vs ionic strength ( p ) have slopes of 59 mV/(decade increase in p ) for 1:l monovalent electrolytes. The observed reaction entropies, ASobsd,have been determined as a function of solution ionic strength. The measured AS,, values are found to consist of the intrinsic reaction entropy for the reaction and an entropy of transfer term. The latter term arises from the permselectivity properties of the Nafion films. Extraction of the intrinsic reaction entropy, ASr,, yields a revised interpretation of the reaction entropy behavior of Ru complexes in Nafion films. The pH vs El,1 properties of Nafion-bound Ru(NH3)5(H20)3+/2+, Ru(NH3),(isna)'+/*+ (isna = isonicotinic acid), and R U ( N H ~ ) ~ ( ~ Z(pz )~= + /pyrazine) ~+ have been investigated, and apparent pK, values have been determined for these complexes. Apparent pK, values in Nafion are generally higher than those in aqueous media by 1.5-2 pK units.
Although a great deal is now known regarding the charge transport behavior of polymer-coated e l e ~ t r o d e s , relatively '~ little
information is available concerning the thermodynamic and kinetic properties of metal ions which are bound in polymer layers. Such
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(3) (a) Facci, J. S.; Schmehl, R. H.; Murray, R. W. J. Am. Chem. SOC. 1982,104,4959. (b) Daum, P.;Murray, R. W. J. Phys. Chem. 1981,85,389. (c) Daum, P.; Lenhard, J. R.; Rolion, D.; Murray, R. W. J. Am. Chem. Soc. ls%O,lO2,4649.(d) Ikeda, T.;Leidner, C. R.; Murray, R. W. J. Electroanal. Chem. 1982,138,343.(e) Leidner, C.R.; Murray, R. W. J. Am. Chem. Soc. 1984,106,1606. (0 Pickup, P. G.; Kutner, W.; Leidner, C. R.; Murray, R. W. J. Am. Chem. SOC.1984,106,1991. (9) Denisevich, P.;Abruna, H. D.; Leidner, C. R.; Meyer, T. J.; Murray, R. W. Inorg. Chem. 1982,21, 2153. (h) Kuo, K. N.; Murray, R. W. J. Electroanal. Chem. 1982,131, 37. (i) Murray, R.W. Annu. Rev. Mater. Sci. 1984,14,145. ij)Murray, R. W. Ace. Chem. Res. 1980,13, 135.
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