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Corrections to a Hard-Sphere Model of Liquids
by Neil S. Snider Department of Chemistry, Queen’s University, Kingston, Ontario, Canada
(Receiued March $1, 1967)
The configurational partition function Q for a liquid composed of molecules which interact through a pair potential
+(r)
= ’e);(
-(;)
- ah ro
3+x
2?rroa was studied. The leading terms in an expansion of N-I In Q (where N is the number of molecules) in mixed powers of V - l and X were considered. The expansion up to and including linear terms was found to account for much of the observed thermodynamic behavior of liquids composed of small, nonpolar molecules. An overly large value for the configurational C , is given when one neglects terms past linear, but there is some evidence that the quadratic terms in v-l and X account for this discrepancy.
Introduction It has been
that for liquids composed of molecules which interact through a pair potential +(r) given by
the classical configurational partition function Q in the limit v -+ 03 , X -+ 0 is given by
N-I In Q = -In n
+ a([) + an E
(2)
where N is the number of molecules, n is the number density, k is Boltzmann’s constant, T is temperature, [ is (7r/6)nrO3, and u is a function of t characteristic of the hard-sphere fluid. From eq 2 one derives p
=
-nZkT(
bN-’ In Q bn
)
= nkTx(4)
T
U =
bN-’ In Q
kT2(
bT
)
n
= -an
- an2
(3)
(4)
where p is the pressure, u is the configurational internal energy per molecule, and x ( [ ) is 1 - [a’([), where the prime denotes differentiation. Equations 3 and 4 have been used to calculate thermodynamic functions for argon a t its triple pointJ8 and quite good agreement with experiment has been obtained. An expresThe Journal of Physical Chemistry
sion for the v-l term in N-’ In Q has been derived’ and has been employed in the interpretation of properties of fluids a t high temperat~re.’,~An expression for the X term has also been derived.2 I n the present paper we study the expansion of N-‘ In Q up to and including linear terms in v-l and A. We also consider to a lesser extent the quadratic terms in the expansion. We use the linear approximation to interpret properties close to the triple point for liquids composed of small molecules of approximately spherical symmetry. We consider in particular the configurational C,, denoted from now on by a subscript “conf,” the p-T isochores, and the ratio Y given by
Y =
KP - (YT nuK
(5)
where K is the isothermal compressibility and (Y is the coefficient of thermal expansion. We find that the linear approximation reproduces experimental trends for these properties but that it appreciably overestimates Cv,oonf. Finally we give some evidence that this discrepancy in Cv,confis cleared up if one includes the terms quadratic in v-l and X. (1) J. S. Rowlinson, Mol. Phys., 8 , 107 (1964). (2) N. S.Snider, J. Chem. Phys., 45, 378 (1966). (3) H. C. LonguebHiggins and B. Widom, Mol. Phys., 8 , 549 (1964). (4) D. A. McQuarrie and J. L. Katz, J. C h a . Phys., 44, 2393 (1966).
CORRECTIONS TO A HARD-SPHERE MODELOF LIQUIDS
2953
Derivation of the Correction Terms As mentioned above, the v - l and terms in the expansion of N-1 In Q have been derived previously. We denote these here by a superscript (1). They are given by eq 12 of ref 4 and eq 12 and 15 of ref 2. They can
highly favored in the liquid. We also see in both eq 9 and 10 that the correction terms tend to cancel a t low temperatures but that the term in v-' dominates at high temperatures. We now obtain the quadratic terms in v-' and X. Equation 7 of ref 4 can be written
be written as
bN-' In Q
)(
X-Y+2
av-l
In xe-'Z-Y/kTw(xjX, v) dx
dx where y is Euler's constant, ~ ( z [), is the radial distribution function for the hard-sphere fluid expressed in terms of the dimensionless distance x which is equal to r/rOand c(x, E ) is T(X, . E. ) - 1. For the hard-sDhere fluid the virial theorem gives'
m 4 ,E)
=
3[X(O
- 11
(7)
An approximation to ~ ( x ,E ) has been obtained by solving the Percus-Yevick equation for the hardsphere fluid15and numerical values of this approximate expression have been tabulated.6 Numerical values of x ( t ) have been obtained from molecular dynamics cal~ulations,~ and a closed expression for x(E) which agrees well with the molecular dynamics calculations has been obtained both from the Percus-Yevick equation5>*and from scaled particle theory.9 Estimates based on the results of ref 6 have been made2 for the function #(E) which is given by
#(E)
= JmC(x,
dx
0;
u = -an
+ 3v-'kT[x([)
- 11 - Xun#([)
where w(x, X, v) is g(x, A, v)eCx-Y'kTand g is the radial distribution function. Of course, w also depends on density and temperature, but for the sake of brevity we do not designate this dependence. For large v and small X the right side of eq 11 can be written
bN-' In Q
x - ~ + 2e -rZ-'/kT
B V -1
[w(l, 0, m ) Xwx(1, 0, m )
(10)
The physical significance of the correction terms in eq 9 is clear. The term in v - l is an upward correction which accounts for the fact that pairs of molecules can now experience an interaction potential which is positive and finite. The term in X is a downward correction which accounts for the fact that the pair potential is particularly deep a t pair distances which are
X
+ v-lwY(l, 0, + + - 1)wz(1, 0, m ) l dx (X
(12)
where w is (bw/bzJ-')Z= I , X = O , ~ m= and wx and wX are defined similarly. Performing the integration over z in eq 12 and discarding terms Past linear in V-' and X, we obtain
bN-' In Q = -12t bv-1
K
y
+ In-
w(1, 0, m )
8 2
g - ln2 ,&)[3w(lj
v-'{(y2
-
wd11 0,
,3J)l -
(8)
For the purposes of this paper little more need be said about x and $ other than that they are positive, nionotonically increasing functions of [. Combining ecl 2, 6, 7, and 8, we obtain, after the appropriate partial differentiations, the relations
(11)
0,
( + In IcT) w,O, 0, y
-
From eq 13 we obtain terms in the expansion of N-' In &. From eq 15 of ref 6 we have
v-l,
m>
-
+
m)}
v - ~ , and
+
v-l
X in
where again one should keep in mind that c depends also on density and temperature. For large v and small X eq 14 becomes approximately (5) M.S. Wertheim, Phys. Rea. Letters, 10, 321 (1963) (6) G. J. Throop and R. J. Bearman, J. Chem. Phys., 42, 2408 (1965). (7) B. J. Alder and T. E. Wainwright, ibid., 33, 1439 (1960). (8) E. Thiele, ibid., 39, 474 (1963). (9) H.Reiss, Advan. C h m . Phys., 9, 1 (1965).
Volume 71,Number 9 Augual 1967
NEILS. SNIDER
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Connection with Experiment X[c,(x, 0,
a)
- 2c(x, 0,
a ) In
x]
+
where c, and cA are defined similarly to w, and wx. From eq 15 we obtain terms in X, X2, and v-lX in the expansion of N-' In Q. Equations 13 and 15 provide us with all of the quadratic terms in our expansion. We denote these terms by superscript (2) and find on integration of eq 13 and 15 that they are given by
g l ' n [ c A ( x0, ,
a)
- 2c(x, 0,
a ) In
dx x ] - (16) X
where we have assumed that (bN-' In Q)/(dv-l bX) is equal to (3N-l In Q)/(bXbv-'). We shall later be concerned with the quadratic terms in the expansion of u. These are found on differentiation of eq 16 to be given by ~ ( 2 )=
(
y
Gv-2tkT
[(
7 2
a2 -- ln2
:>
12v-IXtkT
The physical significance of this contribution is that at higher temperatures more pairs of molecules are sufficiently close together that they have appreciable positive potential energy. Linear terms in X contribute nothing to Cv,oonf,and, as nearly as can be estimated, quadratic terms contribute negligibly. Such results are in accord with the findings of ref 2 which indicate that attractive forces do not give rise to appreciable correlation between molecules at liquid densities. For 5 values typical of liquids x(f) is about 10, and a reasonable value of v is 12. Inserting these numbers into eq 18, we get that CV,,,,f/Nk is about 2. For the aforementioned liquids Cv,conf/Nk is about 1.14 I n eq 3 we see that (bp/dT), is a constant for fixed density. Therefore, this equation gives p-T isochores which are linear. For actual liquids the isochores are very nearly linear, but precise measurements indicate that (bp/bT), decreases slightly with te1nperat~re.l~ From eq 9 we obtain
6
+ In i T ) T - + 2 In -(3w + w,) + w, kT --
We have noted above that eq 3 and 4 have been used successfully to calculate thermodynamic properties for liquid argon. Since argon, krypton, xenon, N2, 02, CO, CH,, and, to a lesser extent, neon very nearly obey the principle of corresponding states as liquids,13 eq 3 is a satisfactory equation of state for all of these liquids. It has, however, been pointed out that eq 3 and 4 have shortcomings, the most notable one being that eq 3 implies Cv,oonf is equal to zero. From eq 10 we obtain a finite contribution to Cv,oonf
I( y
E
1
-
i?
+ In lcT) - T - - wx ]
:anlrn(cA
- 2c In x - T bT x
Anything said about the terms on the right of eq 17 is necessarily based on tenuous assumptions. However, a little is known about the behavior of w and c from the solution of the Percus-Yevick equation for hard spheres6J and from a growing body of numerical ~ork,6~1&'2On the basis of these results it appears approximately true that the coefficients of v-lx and x 2 in eq 17 are close to zero and that the coefficient of v - ~ decreases with temperature* More be said concerning these points in the next section. The Journal of Physical Chemistry
The term in v-I is seen to decrease with temperature in accordance with experimental results. Now let us consider the ratio Y given by eq 5. The following is an easily obtained t,hermodynamic identity
Thus Y is 1 if u is -an.
Experimentally Y is found to
(10) A. A. Broyles, J. Chem. Phys., 35,493 (1961). (11) A. A. Khan and A- A. Broyles, ibid., 43, 43 (1965). (12) G. J. Throop and R. J. Bearman, Physica, 3 2 , 1298 (1966). (13) E. A. Guggenheim, J. Chem. Phys., 13, 253 (1945). (14) J. s. ~ ~ ~ ~ l ~and i i ~ ~i ~~~~~ i i ~d t~~~~ ~ ~, t ~ t ~ ~ , ~" and Co. Ltd., London, 1959.
~
CORRECTIONS TO A HARD-SPHERE MODEL OF LIQUIDS
be close to 1 for a number of l i q ~ i d s . ~ Equation J~ 10 gives Y as
2955
Table I: Values of the Ratio Y for Liquids of Moderately Large and Symmetric Molecules T,
un - 3v-’kT‘x’(‘) + an - :3v-’lcT[x(E) - 11
+ ‘“(E)1
+ XU~$(()
(21)
The right side of eq 21 is 1 if terms in Y-l and X are negligible, greater than 1 if the terms in dominate the terms in v - l , and less than 1 if the terms in v-l dominate. It has been pointed out in ref 2 that for small molecules the experimental value of Y is somewhat less than 1. For nloderately large molecules of approximate spherical symmetry Y turns out to be somewhat greater than 1 as is seen in Table I. I n the last column of this table u was assumed to be given by
where A H , is the heat of vaporization. Equation 22 is valid provided the liquid is in equilibrium with an ideal vapor of negligible density. The right side of eq 21 is greater than 1 if the repulsive part of the pair potential is very steep. From second virial coefficient measurements one concludes that the repulsive potential is steeper for moderately large molecules than it is for smaller ones.15 This result also follows from the assumption that the repulsive interaction is more properly a function of the distances from the atoms which comprise the molecule than of the distance from the center of mass of the molecule.16 Thus eq 21 is in qualitative accord with other relevant experimental and theoretical findings. All of the aforementioned results show that eq 9 and 10 account for a number of the significant trends observed in experiments on liquids. The one major discrepancy which has been noted, and this a quantitative one, is the overly large C,,conf estimated from eq 10. There is some indication that this discrepancy is accounted for by the quadratic terms in our expansion of N-’ In Q. Arguments concerning the quadratic terms in this expansion are necessarily shaky because so little is known about the distribution functions which appear on the right side of eq 16. Two features of c(x) are known, however, from the solution of the Percus-Tevick equation for the hard-sphere systeni5*6s8 and from solutions of the Percus-Yevick equation and the hypernetted chain equation for systems wherein the pair potential has some more realistic form.1°-12 First, one notes that the major difference in c(x) is that the maxima are
Molecule
OK
Benzene Cyclohexane CCla
298 298 298
OK-’
io-dn, atm-1
.von-l,
AH^,
cc‘mole
cal/mole
Y
1,240“ 0.953“ 59.40b 809Ob 1.12 1.214’ 1.113d 108.7fia 7895* 1 . 1 7 1.226‘ 1.096d 97.08” 7830’ 1 . 1 2
‘N. A. Lange, “Handbook of Chemistry,” 6th ed, Handbook American Petroleum Publishers, Inc., Sandusky, Ohio, 1956. Institute, “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Carnegie Press, Pittsburgh, Pa., 1961. I. Prigogine, “illoleciilar Theory of So1utions,” Interscience Publishers, Inc., Xew York, S . Y., 1957. “International Critical Tables,” Tol. 111, McGraw-Hill Rook CO., Inc., New York, N. Y., 1926-1928. e Reference 14. J. H. Hildebrand arid R. L. Scott, “Regular Solations,” Prentice N. J,, 1962, Hall, Inc., Englewood
’
higher and the minima deeper for the hard-sphere potential than for more realistic potentials.6s10 Second, one notes that for realistic potentials the first maximum in c decreases with increasing temperature while the other maxima and minima are insensitive to temperature changes. For the hard-sphere fluid c is of course independent of temperature as is w. We now assunie that the attractive portion of the potential causes changes in the niaxima and minima of c other than the first and that the softening of the repulsive portion causes the decrease in the first maximum. Then wxis zero, and the v-’X terms in eq 17 are zero. Combining our assumption with the abovementioned observations about c and w,we may also conclude the following: w, is negative, cx is opposite in sign to c, and the temperature derivatives of W , and cx are zero. An estimate based on the solution of the Percus-Yevick equation for hard spheres5 shows that at high densities we have (‘23)
We may assume that the same is true of the integral of cx since the positive and negative portions of c will cancel to a great extent. Thus in first approximation we assume that the terms in X 2 in eq 17 are also zero. The terms in v - ~cannot be neglected so readily. If we assume that order of differentiation of w with respect to T and 2 is immaterial b2W ____
bTbx
b2W - ___
bxbT
(24)
(15) A. E. Sherwood and J. RI. Pmusnitz, J . Chem. Phys., 41, 429 (1964). (16) T. Kihara, Reo. Mod. Phvs., 25, 831 (1953).
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NEILS. SNIDER
2956
and if we take the expressions for w and w,to be given by the solution of the Percus-Yevick equation for hard spheres,* then eq 17 reduces t o 1 - 2 i - 2f2 In (1 -
1'
kT
(25)
The first term in brackets on the right of eq 25 is negative, and the second is negative a t high density and low T . If we assume eq 25 to be a valid approximation to u ( ~ then ) , the contribution to Cv,oonf from the quadratic terms is given by
so that an expression for Cv,conf derived on the assumption that the molecules are spherical is appreciably in error when applied to molecules which are even slightly nonspherical.
Conclusions I n summaryj the following results were obtained from this investigation. (1) The configurational partition function for a liquid composed of niolecules which interact through a pair potential of the form
was obtained as an expansion in mixed powers of v - l and X up to and including quadratic terms. Good estimates of the leading term and of the coeffcients of the linear terms can be obtained. Thus the quadratic terms in our expression appear to give an appreciable negative contribution to Cv,conf (2) The major features of the thermodynamic behavior of liquids composed of simple molecules are at low temperature and high density. given by the leading term in the aforementioned exOne can also verify that the expression for the quadpansion. A number of secondary features are acratic term in u as given by eq 25 does not imply a counted for by the linear terms. An exception is the substantial change in our foregoing conclusions about configurational C, which comes out considerably larger the isochores and about Y , conclusions which were made than typical experimental values if terms past linear in on the assumption that the quadratic terms in the exthe expansion are neglected. pansion of N-' In Q are negligible. However, the equations derived here do not explain the experimentally (3) A crude estimate of the quadratic terms in the expansion indicates that they give an appreciable negaobserved values of C,,,,,f/Nk for benzene and carbon tetrachloride, values which range between 2 and 3.14 tive contribution t o the configurational C,. Any one of a number of limitations in our treatment Acknowledgment. This work was supported by a might account for this discrepancy. For example, it grant from the National R,esearch Council of Canada. is possible that Cv,oonfis sensitive t o molecular shape
The J O U Tof ~Phyaieal Chemistry
