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Effect of core nonsphericity on the entropy of simple liquids
CHEMICAL
24 March 1995
PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 235 (1995) 365-369
Effect of core nonsphericity on the entropy of simple liquids Neil Snider Department of Chemistry, Queen's University, Kingston, Ontario, Canada K7L 3N6 Received 16 December 1994
Abstract The lowering of the entropy of simple liquids due to the nonspherical shape of the repulsive molecular cores was estimated quantitatively from a generalized van der Waals equation. For the liquids considered, the entropy lowering relative to an equivalent hard sphere liquid was found to be in the range 0.1R to 0.3R. Such a small decrease can be explained as due to partial compensation of the loss of orientational disorder and the gain in packing disorder which arise from nonsphericity.
1. I n t r o d u c t i o n
Thermodynamic identities, when applied to (1), give for the molar entropy s of the liquid,
It is now well known that the thermodynamic properties of simple liquids are fairly well approximated by an equation of state which treats the effects
s / v ~ _ - 1 - - = ln~ ) - I ~C° R ~-7 "o ~ dr/,
of the repulsive cores accurately while making only the simplest approximations for the effect of the intermolecular attractive forces. The simplest way in which to account for attractive forces is by means of a van der Waals term, which gives for the equation of state
where ~: is Vo/V and v * is a function of T only. The second term in (2) is the entropy relative to what it would be if the system were an ideal gas at the same v and T. This contribution to the molar entropy is hereafter denoted ~s. In a sense, (2) provides a justification for (1): it is physically plausible that 8s is largely determined by the repulsion of the molecular cores [4]. The purpose of the work here reported is to obtain a quantitative estimate of the effect of core nonsphericity on ~s. Eq. (1) contains at least one parameter in addition to a. For hard spheres that parameter is the diameter o-. For nonspherical cores the number of parameters proliferates. Any specific choice of parameters is to some degree arbitrary, but some choices are more plausible than others. This point will be argued further in Section 2.
RT p = ~
[ vo ~ C O~--u- )
a v2,
(1)
where p is pressure, R is the gas constant, T is temperature, v is molar volume, C O is the compressibility factor of a fluid of hard bodies, v 0 is the volume of a single hard body times Avogadro's number, and a is the van der Waals parameter for the attractive forces. Eq. ( 1 ) h a s been employed with success in the calculation of thermodynamic properties of a number of liquids [1-3].
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 0 0 9 - 2 6 1 4 ( 9 5 ) 0 0 1 0 5 - 0
(2)
366
N. Snider / Chemical Physics Letters 235 (1995) 3 6 5 - 3 6 9
2. T h e o r y Attention is here focused on the liquid in equilibrium with a vapor at such a low pressure that it can be considered to be an ideal gas. The vapor pressure Pv is so low that it is negligible compared to the terms on the right of (1), with v understood to be the molar v o l u m e of the liquid. It has been shown [1] that these a s s u m p t i o n s along with (1) and standard t h e r m o d y n a m i c identities give P"r = - l n
=C
r ,c0-1 o+1-~o ~
For classical fluids c o m p o s e d of hard c o n v e x bodies, scaled particle theory gives an equation for C o which is a generalization of the C a r n a h a n - S t a r ling equation for hard spheres [5,6], 1 + a~C+ b~:2 - c~c3 Co = (5) (1 - ~ )3 Eqs. (2) and (5) give for ~s, ~s
-
(c-
1) ln(1 - ~c)
d~, (a+c+2)-l(a-b+3c+3),~
(3) where /z r is the residual chemical potential of the liquid. Eq. (3) implies that the core v o l u m e L,o m a y be derived from experimental values of Pv and t, at a given T. In the calculations here reported, u o was determined in this way. The a r g u m e n t for doing so hinges on the relation tzr = h r - T S s , (4) where h r is the residual molar enthalpy. A more realistic theory of the liquid would very likely introduce corrections of the same sign to hr and to T ~ s . These corrections w o u l d then cancel to some extent in the formula for ~ . Thus one expects (3) to be reasonably accurate for real liquids, even though it is based on a crude model. This expectation has been c o n f i r m e d by calculations which show that ( 3 ) i s an accurate expression for the vapor pressures of the liquefied rare gases other than helium [1].
(1-
.
~:) 2 (6)
For hard c o n v e x bodies a, b and ¢ in scaled particle theory are expressible in terms of a single parameter o~ [5]. The extent to which a is greater than one is a measure of the deviation of the shape of the c o n v e x body from that of a sphere. A compilation of formulas pertaining to c o n v e x bodies exists [7]. From these formulas one may calculate the value of ce for practically any interesting c o n v e x body. A n equation of the same form as (5) has also been obtained from fits to simulation data for fused pairs of hard spheres of equal diameters [8]. It has been found for hard body fluids that, at given t, and Vo, nonsphericity increases Co, which implies a decrease in 5s. This decrease in 5 s is expected since nonsphericity implies the possibility of orientational ordering. Calculations based on (6)
Table 1 Experimental temperatures, molar volumes and vapor pressures along with 5s, both calculated and experimental, for selected liquids Liquid CH 4 N2 02 CI 2 C2H 4 C2H 2 CO 2 CS 2 c6n 6
CCI 4
T (K) 111.7 [9] 77.4 [9] 90.2 [9] 239.2 [11] 169.4 [13] 192.6 [14] 216.6 [15] 319.4 [16] 353.3 [9] 350.0 [9]
u (cm 3 mol- 1)
- In( p~ c / R T )
38.0 [9] 34.7 [9] 28.0 [9] 45.4 [10] 49.1 [12] 42.7 [14] 37.3 [15] 62.2 [10] 95.9 [9] 103.9 [9]
5.49 [9] 5.21 [9] 5.58 [9] 6.07 [11] 5.64 [13] 5.68 [14] 4.53 [15] 6.04 [16] 5.71 [9] 5.62 [9]
a Hard sphere, b Convex body. c Fused sphere.
- 5 s/R
calc a
calcb
calc c
3.5 3.3 3.5 3.7 3.6 3.6 3.0 3.8 3.6 3.6
3.4 3.6 3.9 3.7 3.6 3.1 4.0 3.7 3.7
3.4 3.6 4.0
exp. 3.3 [9] 3.5 [9] 3.5 [9] 4.2 [11] 4.0 [13] 4.5 [14] 4.1 [15] 4.1 [16] 4.8 [9] 4.7 [9]
367
N. Snider / Chemical Physics Letters 235 (1995) 365-369
of the amount by which ~s is lowered due to core nonsphericity are reported in Section 3.
Table 2 Fused sphere diameters for the diatomic molecules from this study and from fits to the two-site Lennard-Jones 6-12 potential Molecule o- (nm)
3. Results The calculated values of 8s are listed in Table 1 along with experimental values of ~s and the thermodynamic data from which v 0 were obtained. The only other data required in these calculations were bond lengths, which may be found in standard references. The methane core was modeled as a sphere, i.e. it was assumed that the hydrogens cause negligible distortion of the methane core from spherical shape. This assumption is consistent with the treatment of the other hydrocarbons in Table 1. The cores of the diatomics were modeled both as prolate spherocylinders and as fused hard spheres. Center-tocenter distances in both cases were set equal to bond lengths. The cores for CO2, CS2, C2H 4 and C2H 2 were modeled as prolate spherocylinders with centerto-center distances equal to twice the bond lengths in the case of CO 2 and CS 2 and equal to the c a r b o n carbon bond lengths in the case of C2H 4 and C2H 2. The core of C6H 6 was taken to be the parallel body of a regular hexagon with side equal to the carboncarbon bond length. The CC14 core was taken to be the parallel body of a regular tetrahedron with center-to-vertex distance equal to the carbon-chlorine bond length. The temperatures in Table 1 are the normal boiling temperatures for all of the liquids except CO 2 and C2H2, in which cases they are the triple point temperatures, The lowering of 8s due to nonsphericity is seen to be 0.1R to 0.3R, less than 10% of the total 8s and little more than 10% of the largest discrepancies between the calculated and the experimental values. Only for C12 and CS 2 does accounting for nonsphericity significantly improve agreement with experiment. It is for these two molecules that the deviations of the assumed cores from spherical shape are the greatest, In Table 2 are listed fused sphere diameters derived from (3) for the diatomic molecules. These are contrasted with the o- parameters in the best fits of experimental data to the two-site 6 - 1 2 potential with site-site distance equal to the bond length. For N 2 and 0 2 the fits were to second virial coefficient data
N2 02 Cl2
this work
fit to two-site 6-12 potential
0.308 0.284 0.320
0.320 [18] 0.294 [18] 0.335 [19]
[17]. For C12 the fit was to liquid state properties [18]. The or values derived from (3) are generally smaller, but agreement is within a few percent in all cases.
4. Discussion Two striking results of this study are evident. The first is that introduction of a plausible amount of core nonsphericity lowers 8s by only OAR to 0.3R. The other is that for liquids composed of diatomic molecules agreement with experiment is good whereas for liquids composed of polyatomic molecules agreement is, in some cases, not good. The first of these results is understandable in terms of what is known about the effect of core nonsphericity on the pair correlation function. Computer simulations [19] have shown that the peaks and the troughs in the spherically averaged pair correlation function are diminished by the introduction of nonsphericity. This is an indication that the packing disorder is increased. This effect partially offsets the orientational ordering so that the net ordering, and hence the net decrease in 8s, is not large. The second of these results is more difficult to explain. Corrections to (2) and (3) tend to cancel. Correction to (2) due to softening of the intermolecular repulsion is positive and that due to the intermolecular attraction is negative. In (3) the signs of the corrections may be reversed. As seen from (4), this depends on the relative magnitudes of the corrections to h r and to T S s . Thus the method here employed underestimates both o- and T ~ s if the correction for softening of the repulsion is dominant. Prior investigations [1,20] indicate that the method does slightly underestimate both o- and ~s for the
N. Snider / Chemical Physics Letters 235 (1995) 365-369
368
120•
~
- -
•
fC~a
•
0.a0
,
o4o .
v 000
-o4o200.00
•
'
° •
I
300.00
'
"
I
400.00
'
I
500.00
'
I
600.00
(u,/~'r)2 Fig. 1. Plot of difference between calculated and experimental entropy versus square of residual internal energy divided by ~RT. Data points are for the liquids in Table 1 along with the liqueficd rare gases Ne to Xe. The least squares fitted line has no significance except as a reference line.
liquefied rare gases Ar, Kr and Xe. Data for the diatomics in Tables 1 and 2 indicate no clear dominance of one or the other of these features of the interaction potential, The effect of the intermolecular attraction, at least on 8s, apparently dominates in the liquids of polyatomic molecules here investigated. A semiquantitative argument based on perturbation theory [21] lends partial support to this assertion. The leading correction term to ~s from the attractive forces is proportional to ( a / v o R T ) 2. A rough estimate of a / v o is given by a ur v0 ~ ' where u r is the residual internal energy. As seen from Fig. 1, there is some correlation between (Ur//~RT) 2 and the discrepancies between calculated and experimental ~s. It is not a close correlation, but is probably all that may be expected from such a gross oversimplification of the theory. Another possible source of discrepancy is the assumption of convex bodies. It has been argued in the case of C C I 4 that concave portions of the repulsive cores give rise to interlocking of C1 atoms on adjacent molecules, an effect which manifests itself
in features of the pair correlation function for liquid CC14 [22]. This interlocking may account for an appreciable part of the discrepancy between the calculated and experimental 8s for C C I 4. That a similar effect on the correlation function for CS 2 was not found [22] is in accord with the results listed in Table 1. The calculations here presented indicate that for fused hard spheres of equal diameter the ratio of the center-to-center distance to the diameter must exceed 0.5 before ~s is lowered by as much as 0.1R relative to the 8s of an equivalent convex body fluid. A value of 0.46 for this ratio was chosen for CS 2 in Ref. [22]. Neither the results of Ref. [22] nor the results of this study suggest any appreciable effect of interlocking in liquid benzene. Thus one must look elsewhere for an explanation of the large discrepancy between calculated and experimental ~s for this liquid. 5. C o n c l u s i o n The work here reported is based on an equation of state of the van der Waals form. The equation, as here modified, implies via (2) a ~s which arises from packing and from orientational ordering of hard, nonspherical bodies. It is lower than the ~s of a comparable hard sphere fluid. The major finding of this study is that this lowering is only by 0.1R to 0.3R for hard bodies with shapes and sizes which are realistic for a number of molecules. Many experimental ~s for simple liquids with nonsperical repulsive cores are lower than those for comparable hard sphere fluids. In some cases the discrepancies can be accounted for in large part by nonsphericity, in other cases only in small part. It was argued herein that the success of an equation of the van der Waals form is due to cancellation of terms due to opposing effects. No quantitative estimates o f these terms were made in this study. Such estimates would be useful by way of explaining the successes and the failures of the calculations here reported.
Acknowledgement This work was supported by a grant from the Natural Sciences and Engineering Research Council
N. Snider / Chemical Physics Letters 235 (1995) 365-369
of Canada. Ms. Daphne Ripley assisted with the calculations and with the literature search. References [1] H.C. Longuet-Higgins and B. Widom, Mol. Phys. 8 (1964) 549. [2] M. Rigby, J. Phys. Chem. 76 (1972) 2014. [3] P. Svejda and F. Kohler, Ber. Bunsenges. Physik. Chem. 87 (1983) 672. [4] S.J. Yosim and B.B. Owens, J. Chem. Phys. 39 (1963) 2222. [5] N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 635. [6] T. Boublik, J. Chem. Phys. 51 (1975) 4084. [7] T. Boublik and I. Nezbeda, Collection Czech. Chem. Commun. 51 (1985) 2301. [8] D.J. Tildesley and W.B. Streett, Mol. Phys. 41 (1980) 85. [9] J.S. Rowlinson, Liquids and liquid mixtures, 2nd Ed. (Butterworth, London, 1969). [10] E.W. Washburn ed., International critical tables, Vol. 3 (McGraw-Hill, New York, 1928).
369
[11] W.F. Giauque and J.M. Powell, J. Am. Chem. Soc. 61 (1939) 1970. [12] O. Maas and C.H. Wright, J. Am. Chem. Soc. 43 (1921) 1098. [13] C.J. Egan and J.D. Kemp, J. Am. Chem. Soc. 59 (1937) 1264. [14] D. Ambrose, Trans. Faraday Soc. 52 (1956) 772. [15] W. Duscheck, R. Kleinrahm and W. Wagner, J. Chem. Thermodynamics 22 (1990) 841. [16] G. Waddington, J.C. Smith, K.D. Williamson and D.W. Scott, J. Phys. Chem. 66 (1972) 1074. [17] A. Das Gupta, S.J. Sandier and W.A. Steele, J. Chem. Phys 62 (1975) 1769. [18] K. Singer, A. Taylor and J.V.L. Singer, Mol. Phys. 33 (1977) 1757. [19] S.S. Wang, C.G. Gray, P.A. Egelstaff and K.E. Gubbins, Chem. Phys. Letters 21 (1973) 123. [20] H.H. Chen, C.C. Lim and R.A. Aziz, J. Chem. Thermodynamics 7 (1975) 191. [21] R.W. Zwanzig, J. Chem. Phys. 22 (1954) 1420. [22] L.J. Lowden and D. Chandler, J. Chem. Phys. 61 (1975) 5228.
24 March 1995
PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 235 (1995) 365-369
Effect of core nonsphericity on the entropy of simple liquids Neil Snider Department of Chemistry, Queen's University, Kingston, Ontario, Canada K7L 3N6 Received 16 December 1994
Abstract The lowering of the entropy of simple liquids due to the nonspherical shape of the repulsive molecular cores was estimated quantitatively from a generalized van der Waals equation. For the liquids considered, the entropy lowering relative to an equivalent hard sphere liquid was found to be in the range 0.1R to 0.3R. Such a small decrease can be explained as due to partial compensation of the loss of orientational disorder and the gain in packing disorder which arise from nonsphericity.
1. I n t r o d u c t i o n
Thermodynamic identities, when applied to (1), give for the molar entropy s of the liquid,
It is now well known that the thermodynamic properties of simple liquids are fairly well approximated by an equation of state which treats the effects
s / v ~ _ - 1 - - = ln~ ) - I ~C° R ~-7 "o ~ dr/,
of the repulsive cores accurately while making only the simplest approximations for the effect of the intermolecular attractive forces. The simplest way in which to account for attractive forces is by means of a van der Waals term, which gives for the equation of state
where ~: is Vo/V and v * is a function of T only. The second term in (2) is the entropy relative to what it would be if the system were an ideal gas at the same v and T. This contribution to the molar entropy is hereafter denoted ~s. In a sense, (2) provides a justification for (1): it is physically plausible that 8s is largely determined by the repulsion of the molecular cores [4]. The purpose of the work here reported is to obtain a quantitative estimate of the effect of core nonsphericity on ~s. Eq. (1) contains at least one parameter in addition to a. For hard spheres that parameter is the diameter o-. For nonspherical cores the number of parameters proliferates. Any specific choice of parameters is to some degree arbitrary, but some choices are more plausible than others. This point will be argued further in Section 2.
RT p = ~
[ vo ~ C O~--u- )
a v2,
(1)
where p is pressure, R is the gas constant, T is temperature, v is molar volume, C O is the compressibility factor of a fluid of hard bodies, v 0 is the volume of a single hard body times Avogadro's number, and a is the van der Waals parameter for the attractive forces. Eq. ( 1 ) h a s been employed with success in the calculation of thermodynamic properties of a number of liquids [1-3].
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 0 0 9 - 2 6 1 4 ( 9 5 ) 0 0 1 0 5 - 0
(2)
366
N. Snider / Chemical Physics Letters 235 (1995) 3 6 5 - 3 6 9
2. T h e o r y Attention is here focused on the liquid in equilibrium with a vapor at such a low pressure that it can be considered to be an ideal gas. The vapor pressure Pv is so low that it is negligible compared to the terms on the right of (1), with v understood to be the molar v o l u m e of the liquid. It has been shown [1] that these a s s u m p t i o n s along with (1) and standard t h e r m o d y n a m i c identities give P"r = - l n
=C
r ,c0-1 o+1-~o ~
For classical fluids c o m p o s e d of hard c o n v e x bodies, scaled particle theory gives an equation for C o which is a generalization of the C a r n a h a n - S t a r ling equation for hard spheres [5,6], 1 + a~C+ b~:2 - c~c3 Co = (5) (1 - ~ )3 Eqs. (2) and (5) give for ~s, ~s
-
(c-
1) ln(1 - ~c)
d~, (a+c+2)-l(a-b+3c+3),~
(3) where /z r is the residual chemical potential of the liquid. Eq. (3) implies that the core v o l u m e L,o m a y be derived from experimental values of Pv and t, at a given T. In the calculations here reported, u o was determined in this way. The a r g u m e n t for doing so hinges on the relation tzr = h r - T S s , (4) where h r is the residual molar enthalpy. A more realistic theory of the liquid would very likely introduce corrections of the same sign to hr and to T ~ s . These corrections w o u l d then cancel to some extent in the formula for ~ . Thus one expects (3) to be reasonably accurate for real liquids, even though it is based on a crude model. This expectation has been c o n f i r m e d by calculations which show that ( 3 ) i s an accurate expression for the vapor pressures of the liquefied rare gases other than helium [1].
(1-
.
~:) 2 (6)
For hard c o n v e x bodies a, b and ¢ in scaled particle theory are expressible in terms of a single parameter o~ [5]. The extent to which a is greater than one is a measure of the deviation of the shape of the c o n v e x body from that of a sphere. A compilation of formulas pertaining to c o n v e x bodies exists [7]. From these formulas one may calculate the value of ce for practically any interesting c o n v e x body. A n equation of the same form as (5) has also been obtained from fits to simulation data for fused pairs of hard spheres of equal diameters [8]. It has been found for hard body fluids that, at given t, and Vo, nonsphericity increases Co, which implies a decrease in 5s. This decrease in 5 s is expected since nonsphericity implies the possibility of orientational ordering. Calculations based on (6)
Table 1 Experimental temperatures, molar volumes and vapor pressures along with 5s, both calculated and experimental, for selected liquids Liquid CH 4 N2 02 CI 2 C2H 4 C2H 2 CO 2 CS 2 c6n 6
CCI 4
T (K) 111.7 [9] 77.4 [9] 90.2 [9] 239.2 [11] 169.4 [13] 192.6 [14] 216.6 [15] 319.4 [16] 353.3 [9] 350.0 [9]
u (cm 3 mol- 1)
- In( p~ c / R T )
38.0 [9] 34.7 [9] 28.0 [9] 45.4 [10] 49.1 [12] 42.7 [14] 37.3 [15] 62.2 [10] 95.9 [9] 103.9 [9]
5.49 [9] 5.21 [9] 5.58 [9] 6.07 [11] 5.64 [13] 5.68 [14] 4.53 [15] 6.04 [16] 5.71 [9] 5.62 [9]
a Hard sphere, b Convex body. c Fused sphere.
- 5 s/R
calc a
calcb
calc c
3.5 3.3 3.5 3.7 3.6 3.6 3.0 3.8 3.6 3.6
3.4 3.6 3.9 3.7 3.6 3.1 4.0 3.7 3.7
3.4 3.6 4.0
exp. 3.3 [9] 3.5 [9] 3.5 [9] 4.2 [11] 4.0 [13] 4.5 [14] 4.1 [15] 4.1 [16] 4.8 [9] 4.7 [9]
367
N. Snider / Chemical Physics Letters 235 (1995) 365-369
of the amount by which ~s is lowered due to core nonsphericity are reported in Section 3.
Table 2 Fused sphere diameters for the diatomic molecules from this study and from fits to the two-site Lennard-Jones 6-12 potential Molecule o- (nm)
3. Results The calculated values of 8s are listed in Table 1 along with experimental values of ~s and the thermodynamic data from which v 0 were obtained. The only other data required in these calculations were bond lengths, which may be found in standard references. The methane core was modeled as a sphere, i.e. it was assumed that the hydrogens cause negligible distortion of the methane core from spherical shape. This assumption is consistent with the treatment of the other hydrocarbons in Table 1. The cores of the diatomics were modeled both as prolate spherocylinders and as fused hard spheres. Center-tocenter distances in both cases were set equal to bond lengths. The cores for CO2, CS2, C2H 4 and C2H 2 were modeled as prolate spherocylinders with centerto-center distances equal to twice the bond lengths in the case of CO 2 and CS 2 and equal to the c a r b o n carbon bond lengths in the case of C2H 4 and C2H 2. The core of C6H 6 was taken to be the parallel body of a regular hexagon with side equal to the carboncarbon bond length. The CC14 core was taken to be the parallel body of a regular tetrahedron with center-to-vertex distance equal to the carbon-chlorine bond length. The temperatures in Table 1 are the normal boiling temperatures for all of the liquids except CO 2 and C2H2, in which cases they are the triple point temperatures, The lowering of 8s due to nonsphericity is seen to be 0.1R to 0.3R, less than 10% of the total 8s and little more than 10% of the largest discrepancies between the calculated and the experimental values. Only for C12 and CS 2 does accounting for nonsphericity significantly improve agreement with experiment. It is for these two molecules that the deviations of the assumed cores from spherical shape are the greatest, In Table 2 are listed fused sphere diameters derived from (3) for the diatomic molecules. These are contrasted with the o- parameters in the best fits of experimental data to the two-site 6 - 1 2 potential with site-site distance equal to the bond length. For N 2 and 0 2 the fits were to second virial coefficient data
N2 02 Cl2
this work
fit to two-site 6-12 potential
0.308 0.284 0.320
0.320 [18] 0.294 [18] 0.335 [19]
[17]. For C12 the fit was to liquid state properties [18]. The or values derived from (3) are generally smaller, but agreement is within a few percent in all cases.
4. Discussion Two striking results of this study are evident. The first is that introduction of a plausible amount of core nonsphericity lowers 8s by only OAR to 0.3R. The other is that for liquids composed of diatomic molecules agreement with experiment is good whereas for liquids composed of polyatomic molecules agreement is, in some cases, not good. The first of these results is understandable in terms of what is known about the effect of core nonsphericity on the pair correlation function. Computer simulations [19] have shown that the peaks and the troughs in the spherically averaged pair correlation function are diminished by the introduction of nonsphericity. This is an indication that the packing disorder is increased. This effect partially offsets the orientational ordering so that the net ordering, and hence the net decrease in 8s, is not large. The second of these results is more difficult to explain. Corrections to (2) and (3) tend to cancel. Correction to (2) due to softening of the intermolecular repulsion is positive and that due to the intermolecular attraction is negative. In (3) the signs of the corrections may be reversed. As seen from (4), this depends on the relative magnitudes of the corrections to h r and to T S s . Thus the method here employed underestimates both o- and T ~ s if the correction for softening of the repulsion is dominant. Prior investigations [1,20] indicate that the method does slightly underestimate both o- and ~s for the
N. Snider / Chemical Physics Letters 235 (1995) 365-369
368
120•
~
- -
•
fC~a
•
0.a0
,
o4o .
v 000
-o4o200.00
•
'
° •
I
300.00
'
"
I
400.00
'
I
500.00
'
I
600.00
(u,/~'r)2 Fig. 1. Plot of difference between calculated and experimental entropy versus square of residual internal energy divided by ~RT. Data points are for the liquids in Table 1 along with the liqueficd rare gases Ne to Xe. The least squares fitted line has no significance except as a reference line.
liquefied rare gases Ar, Kr and Xe. Data for the diatomics in Tables 1 and 2 indicate no clear dominance of one or the other of these features of the interaction potential, The effect of the intermolecular attraction, at least on 8s, apparently dominates in the liquids of polyatomic molecules here investigated. A semiquantitative argument based on perturbation theory [21] lends partial support to this assertion. The leading correction term to ~s from the attractive forces is proportional to ( a / v o R T ) 2. A rough estimate of a / v o is given by a ur v0 ~ ' where u r is the residual internal energy. As seen from Fig. 1, there is some correlation between (Ur//~RT) 2 and the discrepancies between calculated and experimental ~s. It is not a close correlation, but is probably all that may be expected from such a gross oversimplification of the theory. Another possible source of discrepancy is the assumption of convex bodies. It has been argued in the case of C C I 4 that concave portions of the repulsive cores give rise to interlocking of C1 atoms on adjacent molecules, an effect which manifests itself
in features of the pair correlation function for liquid CC14 [22]. This interlocking may account for an appreciable part of the discrepancy between the calculated and experimental 8s for C C I 4. That a similar effect on the correlation function for CS 2 was not found [22] is in accord with the results listed in Table 1. The calculations here presented indicate that for fused hard spheres of equal diameter the ratio of the center-to-center distance to the diameter must exceed 0.5 before ~s is lowered by as much as 0.1R relative to the 8s of an equivalent convex body fluid. A value of 0.46 for this ratio was chosen for CS 2 in Ref. [22]. Neither the results of Ref. [22] nor the results of this study suggest any appreciable effect of interlocking in liquid benzene. Thus one must look elsewhere for an explanation of the large discrepancy between calculated and experimental ~s for this liquid. 5. C o n c l u s i o n The work here reported is based on an equation of state of the van der Waals form. The equation, as here modified, implies via (2) a ~s which arises from packing and from orientational ordering of hard, nonspherical bodies. It is lower than the ~s of a comparable hard sphere fluid. The major finding of this study is that this lowering is only by 0.1R to 0.3R for hard bodies with shapes and sizes which are realistic for a number of molecules. Many experimental ~s for simple liquids with nonsperical repulsive cores are lower than those for comparable hard sphere fluids. In some cases the discrepancies can be accounted for in large part by nonsphericity, in other cases only in small part. It was argued herein that the success of an equation of the van der Waals form is due to cancellation of terms due to opposing effects. No quantitative estimates o f these terms were made in this study. Such estimates would be useful by way of explaining the successes and the failures of the calculations here reported.
Acknowledgement This work was supported by a grant from the Natural Sciences and Engineering Research Council
N. Snider / Chemical Physics Letters 235 (1995) 365-369
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