J. Phys. Chem. 1983,87, 112-118
112
electrodes in 1.0 M KCl solution containing various redox couples examined here. We believe that the electron transfer kinetics characteristic to the semiconductorliquid junction is quite minor in the present study. Further study on photoeffects is
under inve~tigati0n.l~ Registry No. Fe(CN)6s, 1340862-3;Fe, 7439-89-6;Rum(edta), 21687-44-5; IrC162-, 16918-91-5; Fe(CN),4-, 13408-63-4; PB, 14038-43-8;Pt, 7440-06-4; BG, 14433-93-3;PW, 81681-39-2.
Partial Molar Volume from the Hard-Sphere Mixture Model B. Leet Department of Chemistry, University of Kansas, Lawrence, Kansas 66045 and Physlcal Sciences Laboratory, Dlvision of Computer Research & Technology, National Institutes of Health, Bethesda, Maryland 20205 (Received: February 16, 1982: In Find Form: July 26, 1982)
Following Klapper (Biochim.Biophys. Acta, 229, 557-66 (1971)), an expression for the partial molar volume, uz, of the solute in infinitely dilute hard-sphere binary mixtures is derived from a very accurate equation of state for the hard-sphere mixtures (Mansoori et al., J . Chem. Phys., 54,1523-5 (1971)). If the result is written as bZ = (4a/3)(Rz+ a ) 3where Rz is the radius of the solute, a is a slowly varying function of Rz and depends primarily on the two properties of the solvent-the size of the solvent molecule and the packing density of the pure solvent. For a system that models the aqueous solution, a ranges from 0.62 to 0.51 A as the size of the solute varies from half to twice that of the solvent. According to this hard-sphere mixture model, the large volume change that occurs when a hydrophobic solute is transferred from a common nonpolar solvent to water is due to the small size of water molecules. According to the same model, wherein the interior of a globular protein molecule is considered as a pure hard-sphere fluid, the volume change upon protein denaturation is governed as much or more by the difference in the packing density between the protein interior and water as it is by the difference in the sue of an average amino acid residue and a water molecule. An Appendix introduces a new packing density function and gives a precise description of the notion of the cavity around a solute of any shape.
Introduction The partial molar volume (PMV) of a solute in a solution is a thermodynamic quantity and, despite the geometrical nature of the concept of volume, it is not possible to calculate PMV directly from simple geometrical consideration. However, for the simplest system, the hard-sphere mixture, a highly accurate equation of state is now available’ from which an expression for the PMV can be obtained that should be very accurate. This expression is important because it does not explicitly contain temperature or pressure and hence correctly describes the purely geometrical aspect of PMV. Thus, by comparing with experimental data, the equation should yield information on the relative importance of geometrical packing effect on volume. It turns out2 that the geometrical packing effect essentially determines the entire volume behavior as long as the solute is spherical and nonpolar. This fortunate circumstance enables one to examine a number of interesting aspects of volume behavior of such molecules. Derivation The equation of state for a hard-sphere mixture has the following form:'^^,^
where A’ =
td
‘Address all correspondence to the National Institutes of Health address.
Here, P, T , and V are the pressure, temperature, and volume of the system, respectively, k is the Boltzmann constant, m is the number of species in the system, and Ri,Ni,and pi are the radius, total number, and the number density, respectively, of the hard-sphere species i. The parameter h has the value of 0, 1,or 3 depending on how the equation is derived. Scaled particle theory3 gives eq 1 with h = 0. The equation of state can also be derived from the exact solution of the approximate Percus-Yevick equation for the radial distribution function of a hardsphere mixture! When the compressibility relation is used with this solution, eq 1 is obtained with h = 0 while the use of the virial theorem gives the same equation with h = 3. The equation with h = 1represents a weighted average of the above two results, but is also derivable empirically from the computer simulation data.’ It is considered to be very accurate. (1) G. A. Manmri, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr., J. Chem. Phys., 54, 152 (1971). (2) M. H. Klapper, Biochim. Biophys. Acta, 229, 557 (1971). (3)J. L.Lebowitz, E. Helfand, and E. Praestgaard. J. Chem. Phys., 43,774 (1965). (4)J. L. Lebowitz, Phys. Reu. A , 133, 895 (1964).
This article not subject to U.S. Copyright. Publlshed 1983 by the American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Partial Molar Volume from Hard-Sphere Mixture Model
TABLE I:
TABLE 11: Data Used for Model Solutes u2,0 cmS/mol 2R*,b Ar a in H,O in CC1, in benzene
Data Used for Model Solvents" 01%
H2O
cc1,
benzene
R l , bA
cm3/mol
1.38 2.58 2.51
18.1
[I0
0.366 0.446 0.446
97.1 89.4
Ar N, 0, CH, C,H6 C6H6
From H. Reiss, A d v . Chem. A t or near 25 "C. Phys., 9, 1( 1 9 6 5 ) , except for water which is from R. A. Pierotti, J. Phys. Chem., 69, 2 8 1 ( 1 9 6 5 ) . (I
The PMV of a particular species can be obtained by differentiating eq 1with respect to the number of moles of that species. For an infinitely dilute binary mixture, the procedure leads to
y = C(x3 + 3ax2 + 3apx
+ a&)
(2)
where y =
&/Dl
x = R2/R1
a = (1- 510)/(1
P
= (1- 5?)/(1
+ 251°)
+ 251'
- h5??
Y = (1 - 5?)/51°
c = (1+ 3a + 3aP + .Pr)-' Here O1 and Dz are the partial molecular volume (PMV divided by the Avogadro's number) of the solvent and solute, respectively, and tl0is the average packing density of the pure solvent, which is defined as the fraction of volume actually occupied by the molecules in the system. (SeeAppendix. We omit the bar for notational simplicity.) Thus the partial molar volume of a solute in an infinitely dilute solution, measured in units of the molar volume of the pure solvent, is given by a cubic polynomial in the radius ratio with coefficients that are simple functions of tl0alone. This derivation and result were described earlier by Klapper2i5for the case when h = 0. When h = 0, eq 2 can be simplified to ~2
+
= 4/3~(R2 b)3
+c
3.40 3.70 3.46 3.82 4.42 5.27
32c,d ( 2 8 ) 33c 3 4 ) 3lcJ (29) 37d3e ( 3 7 ) 52d9e 83e(78)
IpO)
43' 5383 ( 4 7 ) 46g(43) 52gh ( 5 0 ) 66: ( 6 8 ) 8 9 (100)
a A t or near 25 "C. The numbers in parentheses are the values calculated from e q 2 with h = 1. F r o m J. 0. Hirshfelder, C. F . Curtiss and R. B. Bird, "Molecular Theory o f Gases and Liquids", Wiley, New York, 1 9 5 4 , as T. selected by Pierotti, J. Phys. Chem., 69, 281 ( 1 9 6 5 ) . Enns, P. F. Scholander, and E. D. Bradstreet, ibid., 69, 389 (1965). E. W. Tiepel and K. E. Gubbins, ibid., 76, 3044 ( 1 9 7 2 ) . e W. L. Masterton,J. Chem. Phys., 2 2 , 1830 ( 1 9 5 4 ) . f J. E. Jolley and J. H. Hildebrand, J. A m . Chem. SOC.,80, 1 0 5 0 ( 1 9 5 8 ) . g As collected by D. D. Eley, Trans. Faraday SOC.,3 5 , 1 4 2 1 ( 1 9 3 9 ) . This reference also gives the aqueous solution values f o r He, H,, 0,, CO, CO,, N,O, and CH,. These were not used because some of these deviate substantially f r o m available more recent measurements. From J. Horiuti, Sci. Pap. Inst. Phys. Chem. Res. ( J p n . ) ,1 7 , 1 2 5 ( 1 9 3 1 ) , as cited by J. Chr. Gjaldbaek and, J. H. Hildebrand, J. A m . Chem. SOC., 7 2 , 1077 ( 1 9 5 0 ) . As reported b y E . B. Smith and J. Walkley, J. Phys. Chem., 66,. 597 ( 1 9 6 2 ) . Probably from Horiuti's data cited above. I Molar volume of pure benzene. The discrepancy between the observed and calculated values occurs because of the different radii used For benzene as a solute and as a solvent.
1
0
b = aRl
+ 510)
If the water molecule is assumed to be a hard sphere of radius R1 = 1.38 A, the liquid water is modeled by a hard-sphere fluid with D1 = 18 cm3/mol = 30 A3 and 510 = 0.366. (See Table I.) The value of b is then 0.51 A and that of c is 0.74 A3 = 0.44cm3/mol. If c is ignored since it is quite small compared to the PMV of most solutes in aqueous solution (see Table 11), eq 3 has the same form as eq 4 and the border thickness, a, defined by that equation becomes equal to b and independent of solute. Since eq 1 is the most accurate with h = 1, we expect the same with eq 2. We shall henceforth use h = 1. However, the difference between the volumes with h = 1 and h = 0 is small. The relative PMV, y, is plotted as a function of x for the two cases in Figure 1. When x is between 0.5 and 2, the maximum difference in volume between the two cases is less than 3% for all values of solvent packing density between 0.3 and 0.6. ( 5 ) M. H. Klapper, Prog. Bioorg. Chem., 2 , 55 (1973).
44f ( 4 2 ) 53g,h(49) 458(42) 52g*h ( 5 2 ) 66h (70)
(3)
where c = y3rb3(1
113
2
3
R,/R, Figure 1. The ratio of the partial molar volume of the solute to that of the solvent, at infinite dilution, as a function of the ratio of the radius of the solute to that of the solvent, according to eq 2. Each doublet of curves is for a particular value of the solvent packing density, from 0.3 to 0.6 in steps of 0.05, some of which are indicated to the right of the curves. The two curves within each doublet are for h = 0 (lower curve when x > 1) and h = 1 cases.
The solute-solvent border thickness, a (see Appendix), is defined by the relation 02
+
= 4/3~(R2
Measured in units of R1, it is given by a / R 1 = (Y/{10)1/3 - x
(4) (5)
This function is plotted in Figure 2. The border thickness is generally a function of the solute radius as well as of the properties of the solvent. Between x = 0.5 and 2, the variation with the solute radius amounts to about 10-20% over the solvent packing density range of 0.3-0.6, the variation being larger the smaller the packing density. For a large solute, this much variation in a translates into
Lee
The Journal of Physical Chemistty, Vol. 87, No. 1, 1983
114
1
1
I
I
06
!x\ ld
0.4
0.2
0
Y 0
2
1 R2
3
1R,
Flgure 2. The ratio of the solute-solvent border thickness to the solvent radius as a function of the solute-solvent radius ratio, according to eq 5. The solvent packing density ranges from 0.3 to 0.6 in steps of 0.05 as in Figure 1.
relatively small variation in R2 mines the volume.
+ a which actually deter-
Comparison with Experimental Results Real molecules are, of course, not hard spheres and eq 2 is not expected to describe real situation exactly. Rather it represents that portion of the partial molar volume that is due strictly to the geometrical packing effect alone. However, since the structure of a dense simple fluid seems to be governed mainly by the repulsive interaction between molecules6 and since volume is a structural quantity, it is not unreasonable to expect the geometrical packing effect to be the dominant factor in determining the partial molar volume of a solute in nonpolar system. Whether the hard-sphere result is applicable even to aqueous solutions is clearly debatable. However, geometrical packing effects must surely exist even in this case. When the solute is nonpolar, so that strong interactions such as hydrogen bonding do not exist between the solute and solvent, packing effect may again become dominant. In any case, it should be of interest to see how much the aqueous system deviates from what is expected from purely geometrical packing consideration alone. Klapper2q5has already compared the experimental data against theory using eq 2 with h = 0. His result shows that the experimental values of the partial molar volume of nonpolar solutes in aqueous solution as well as in CC4, in fact, fit the hard-sphere result very well. Since the difference between h = 0 and h = 1 cases is small, it is unnecessary to present this comparison for h = 1case here. However, the quality of the agreement can be seen from the sample given in Table 11. Other supporting evidence for this agreement exist in the literature. For a model of aqueous solution (Table I), the value of a is 0.55 A for a solute of the size of the water molecule, is 0.53 for a solute of the size of the methane (Table II), and generally ranges from 0.62 A for x = 0.5 to 0.51 f\ for x = 2. GlueckauP first proposed form (4) for the so-called “intrinsic volume” of ions with a = 0.55 A. The experimental partial molar volumes of ions8 are generally consistent with the hypothesis that their “intrinsic volume” is given by (4) with a in the range of 0.3-0.6 A. This comparison is, however, compiicated because of the (6)D. Chandler, Annu. Rev. Phys. Chem., 29,441 (1978). (7)E. Glueckauf, Trans. Faraday Soc., 61,914 (1965). (8)F. J. Millero in “Water and Aqueous Solutions”, R. A. Horne, Ed., Wiley, New York, 1972,pp 546-51.
uncertainty in estimating the effect of ionic charge and because a number of other semiempirical relations seem to give equally good fit to the experimental data. On the other hand, Edward and Farrellg independently proposed the same form for the partial molar volume of small approximately spherical organic molecules in water. By a least-squares fit of the experimental data, they obtain a = 0.53 A. All this correspondence between theory and experiment depends, of course, on the values chosen for the hardsphere radius of each molecule in the system. The force of the agreements cited above must, therefore, be reduced to the extent that the proper choice for the value of the hard-sphere radius is not altogether unequivocal. It should be mentioned, however, that the values used by the authors cited above are those obtained from experimental data that are independent of the partial molar volume. It seems, therefore, that the hard sphere mixture model does provide a useful mode of approximately describing the general PMV behavior of spherical nonpolar molecules. (See also the comments made at the end of next section.) Origin of t h e Solvent-Dependent Volume Change It is well-known1° that, when a small nonpolar organic molecule is transferred from a nonpolar solvent to water, a large reduction in its partial molar volume is observed. Along with the changes in the solubility, enthalpy, entropy, heat capacity, and other thermodynamic quantities, this volume change is considered to be a manifestation of the hydrophobic effect, on which a large body of literature exists.” In this section, we consider the physical origin of the volume effect, assuming that the volume behavior can indeed be described by the hard-sphere packing effect alone. In the hard-sphere mixture model, the effect of solvent is characterized by just two parameters, the size of the solvent molecule and the packing density of the pure solvent. So that the different contributions made by these two properties could be investigated, it is convenient to transform eq 2 into the following form u = (C/[lo)(l
+ 3at + 3a@t2+ a@7t3)
(6)
where u =
&/UZW
=
1/[2
t = R1/R2 = l/x This function is plotted in Figure 3 for the two different values of given in Table I. The points labeled a represent the experimental values for the PMV of methane in CC14and benzene, assuming that the molecules are hard spheres. The point labeled b similarly represents the experimental PMV of methane in water. The radius of the methane molecule was assumed to be 1.91 A. Other data used are given in Tables I and 11. The point labeled c represents the PMV of methane in a hypothetical solvent which is made of molecules of the same size as water but has the packing density of the organic solvents. We shall call this hypothetical solvent “high-density water” for convenience. When a molecule of methane is transferred from the organic solvents to water, its volume is reduced by an amount given by the difference in the vertical height between a and b. This difference can be broken into two (9)J. T.Edward and P. G . Farrell, Can. J. Chem., 53, 2965 (1975). (10)W.Kauzmann, Adu. Protein Chem., 14, 1 (1959). (11)See, for example, F. Franks, Water Compr. Treatise, 4,l (1975).
The Journal of Physical Chemistty, Vol. 87, No. 1, 1983
Partial Molar Volume from Hard-Sphere Mixture Model
115
TABLE 111: Volume Change u p o n Transfer of a Solute from Nonpolar Solvents to Water“ transfer from benzene
transfer from CCl,
AVIb 4 5 4 5 6 8
Ar N2 0 2
CH, ‘ZH6 C6H6
A
vZc
Avtd
- 18 -20 -18 - 20 -25 -3 1
-13 -15 -14 -15 -19 -23
AVobsde -12 -20 -14 -15 -14
AvIb
4 5 4 5 6 8
A
vzc
-16 -18 -17 -19 -23 -29
A Vtd
A Vobsde
-12 -13 -12 -14 -17 -21
-11 -20 -15 -15 -14 -6
“ In c m s / m o l , using the d a t a o f Tables I and 11.
Calculated volume of transfer f r o m the “high-density water” t o ordinary water. See the text. Calculated volume of transfer f r o m the nonpolar solvent t o the “high-density water”, Calculated total volume of transfer from the nonpolar solvent to water. e Experimental volume of transfer.
-
3.5
1
P
.;a 2.5
0
0.25
0.5
0.75
1
1.25
1.5
R, 1R2 Flgure 3. The parthl molar volume of the solute, in units of its physical volume, as a function of the radius of the solvent, in units of the solute radius, according to eq 8. The top curve is for a model water solution, the middle for the model organic solvents solution, and the bottom for a model protein interior. The experimental partial molar volumes of methane in CCI,, benzene, and water are represented by the points labeled a (right), a (left), and b, respectively. Other points are explained in the text.
components-a reduction that accompanies the transfer from the organic solvents to the “high-density water”, given by the vertical distance between a and c, and an increase that accompanies the transfer from the “high-density water” to ordinary water, given by the distance between b and c. The latter is the effect of the packing density difference. Water is a more coarsely packed solvent than the organic solvents and its packing efficiency around a given solute is correspondingly less. Other things being equal, this effect alone will make the PMV of a solute larger in water than in other solvents. This effect is, however, overwhelmed by the large and opposite solvent size effect (the vertical distance between a and c). The size of a water molecule is essentially that of an oxygen atom and is considerably smaller than that of any other commonly used organic solvent molecules. The packing efficiency around a given solute is larger, the smaller the solvent molecules. This effect is much larger than the packing density effect and results in the large decrease in the volume of the solute upon transfer from the nonaqueous solution to water. Similar observations can be made with all small, approximately spherical, nonpolar molecules for which relevant data are available. Table I11 lists the data assembled from the literature cited and calculated by using eq 2 with h = 1. The total volume changes upon transfer, calculated from the model, generally agrees well with the experimental value with the exception of the case of benzene as solute.12 In all cases, it is seen that the negative volume
change originates from size difference of the solvent molecule. To the extent that the geometrical packing effect may be considered as the controlling factor in determining the PMV of a nonpolar solute, we conclude that the large hydrophobic volume effect arises from the small size of the water molecule compared to that of other organic solvent molecules. This is the same conclusion reached earlier by Assarsson and Eirich.I3 This conclusion is not necessarily in conflict with the “iceberg” model of hydrophobic hydration.’l Even in hard-sphere mixtures, there will be more solvent immediately around a solute than in the bulk as indicated by the pronounced peak in the radial distribution function and the packing density function. The molecules in this shell could have transient hydrogen bond structures akin to those seen in clatherate crystals. These hydrogen bonds are obviously important in determining most thermodynamic properties of the solute. However, except for its role in determining the packing density of the bulk solvent, its effect on volume could well be insignificant. The situation is also undoubtedly aided by the fact that PMV is determined by a relation between certain integrals over the density variation around the solute and is insensitive to the detailed structure variation as long as it leaves certain averages unchanged. (See Appendix.)
Implications on the Volume Change upon Protein Denaturation The occurrence of the large hydrophobic volume change discussed in the previous section led to the expectation that a large volume reduction would occur upon denaturation of globular protein molecules because these have many hydrophobic amino acid side chains that are in the interior, removed from the solvent, but which presumably become exposed to solvent water upon denaturation. The volume changes upon protein denaturation have since been measured and found to be negative but very small in magnitude.14J5 Klapper2 found a way to resolve this paradox using eq 2 with h = 0. Here we consider a different presentation of the elements of his argument. A basic assumption in expecting a large volume decrease from the observed hydrophobic volume change of small molecules is that the protein interior is like a nonpolar solvent. However, the protein interior and a nonpolar (12) The poor fit in the case of benzene is partly attributable to the fact that the radius of benzene is poorly estimated, e.g., the radius estimated from the surface tension and compressibility data via the scaled particle theory given in Table I is substantially different from that estimated from the viscosity data given in Table 11. Another reason, not unrelated to the difficulty of assigning an effective radius to the molecule, is probably that it has the ring structure with its polarizable *-electron system and significantly nonspherical shape. (13) P. Assarsson and F. R. Eirich, J. Phys. Chem., 7 2 , 2710 (1968). (14) J. F. Brandts, R. J. Oliveira, and C. Westort, Biochemistry, 9,
1038 (1970). (15) A. Zipp and W. Kauzmann, Biochemistry, 12, 4217 (1973).
116
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
solvent are different in one important aspect-the packing density of the protein interior is near 0.74, the packing density of close-packed spheres,16whereas that of common organic liquids is about 0.45 (Table I). Thus the protein interior is like a crysta12J7and denaturation process is akin to that of dissolving a solid into water. The effect on volume due to the packing density change can be estimated, again in the hard sphere model, by eq 6. The lowest curve in Figure 3 is the plot of this equation for a hypothetical solvent which has a packing density of 0.72.17 At this density, the solvent would no longer be a fluid but a solid, either glass or crystalline. Considering the protein interior as a pure solvent of identical spheres, the molar volume of an average amino acid residue in the protein would be given by the point labeled d. The point labeled e represents the volume of an average amino acid residue exposed in the aqueous environment, again considering the residue and water as hard spheres. The point labeled f is that in the “high-density water” similar to point c. The size of the average amino acid residue is assumed to be 2.9 A in radius calculated from the average van der Waals volume of 102.5 A3 given by Liquori and Sadun.17 The vertical distance between points d and e gives the volume change upon denaturation expected from the hard-sphere model. In this case the packing density effect (distance between e and f) is large and dominates over the solvent size effect (distance between d and f), resulting in an expectation of an increase in volume upon denaturation. The value of this increase, according to this model, is 14 cm3/mol of residue or, using Liquori and Sadun’sI7 value for the average molecular weight of an amino acid residue, 0.13 cm3/g. This is a huge increase. This model is obviously very crude. However, it serves to indicate the direction and a rough estimate of the magnitude of the packing density effect. As with Klap~ e r , we ~ Jconclude that this effect is large and makes the hydrophobic volume change observed in the small molecule system inapplicable to the protein system.
Appendix Packing density is defined as the fraction of volume that is actually occupied by the molecules in the system.ls It is widely used in the study of volume relationship in solids, liquids, and random heaps of hard spheres,21p22 and protein structure^^^^^^ as well as in the statistical mechanical theories of l i q ~ i d s . ~It~will , ~ ~be shown here that this concept can be generalized and defined as a function of the position in the system. The usual packing density then becomes a local or system-wideaverage of this packing density function. The motivation for such a generalization is the desire to precisely define the notion of “border” between the solute and the solvent. For a dilute solution of simple solute, one intuitively divides the volume of the system into (16)F. M. Richards, J . Mol. B i d , 82, l(1974). (17)A. M. Liquori and C. Sadun, Int. J. Biol. Macromol., 3,56 (1981). (18)For this definition to be meaningful, molecules must be supposed to have a sharp impenetrable boundary. Usually this boundary is established by considering a molecule as a system of interlocking spheres which represent atoms (or sometimes groups of atoms) in the molecule. The radii of these spheres are chosen to be equal to the van der Waals radii. See ref 19. (19)A. Bondi, J.Phys. Chem., 68,441 (1964). (20)A.Bondi, ‘Physical Properties of Molecular Crystals, Liquids, and Glasses”, Wiley, New York, 1968. (21)S.Debbas and H. Rumpf, Chem. Eng. Sci., 21, 583 (1966). (22)J. L. Finney, Proc. R. SOC.London, Ser. A, 319,479 (1970). (23)F. M. Richards, Annu. Rev. Biophys. Bioeng., 6 , 151 (1977). (24)H. Reiss, Adu. Chem. Phys., 9,1 (1965). (25)J. P. Hansen and I. R. McDonald, ‘Theory of Simple Liquids”, Academic Press, New York, 1976.
Lee
three regions-the region physically occupied by the solute, the region assignable to the solvent, and an “empty” border region between the solute and s ~ l v e n t . The ~ ~ ~question ~ is whether this intuitive notion can be made more precise. It will be shown here that this is indeed possible with the help of the packing density function. The distinguishing characteristic of above intuitive notion is that it focuses attention on the solute and considers the solvent essentially as a continuum. The packing density function appears especially suited for use in an analysis along this line of approach. We begin by defining this function and exploring several of its properties before considering the notion of the border between solvent and solute.
Definition We consider a general system made of any number of any kinds of molecules. The molecules may or may not be spherical in shape and the system may be a solid or fluid. However, we assume that the molecules are rigid and have sharp, impenetrable boundary. For a given configuration X of such a system, define &(r,X)as follows: Ei(r,x) = 1 if the position r is inside the molecular species i =0
otherwise
(AI)
Now define the packing density function, &(r),as the average of &(r,X)over all configurations, i.e.
where P(X)is the probability that the system is in the configuration X. Obviously, &(r)gives the probability that the position r will be “covered by”, Le., inside, a molecule of species i . Let Et be defined similarly except that it gives the probability that a given position is covered by any molecule in the system. Since a given position r cannot be covered by more than one molecule at a time
&(r,X)= C[i(r,X) i
(-43)
1
where the summations are over all species in the system. A three-dimensional plot of the function &(r)for crystals and glasses will look like the crystallographer’s electron density map except that the Gaussian profile of the electron density near the center of each atom is replaced by a plateau of unit height. The average of &(r)over the whole system, Et, is the packing density defined and used by Bondi20and others. In an isotropic fluid, with a space-fixed coordinate system, &(r)is a constant and given by where pi and uiw are the number density and the physical (van der Waals) volume of the species i. The packing density of the fluid is given by ft(r)= Ev With respect to the coordinate system fixed a t a particular molecule (the origin molecule), &(r)will be a function of the position vector r even in an isotropic fluid. The value of E(r) for the origin molecule will be unity inside (26)J. T. Edward, J . Chem. Ed., 47,261 (1970).
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Partial Molar Volume from Hard-Sphere Mixture Model
5
0’50 0
2
4
6
r = Radial Distance
8
10
(fi)
the molecule and zero everywhere else. The value of 4(r) for a nonorigin species will be zero inside the origin molecule, rise outside the boundary, and generally oscillate around the average value f , which it approaches as Irl becomes large. An example is shown in Figure 4 for a spherical molecular system. Relation t o the Solvent-Solute Correlation Function When Solvent Molecules Are Spherical Choose a coordinate system fixed a t a particular molecule. Call this the origin molecule and denote it by subscript 2. Consider a nonorigin molecular species i which is a sphere of radius Ri.The value of fi(r,X)will be unity when and only when the spherical region of radius Ri centered at r contains a center of a molecule of species i. Since there cannot be more than one such center within this spherical region, the probability that there will be such a center is given by P i s IrT-r(
