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Perturbed hard-sphere equations of state of real fluids. II. Effective hard-sphere diameters and residual properties
Fluid Phase Equilibria, 17 (1984)
Elsevier Science
1- 18 B.V., Amsterdam
Publishers
-
Printed in The Netherlands
PERTURBED HARD-SPHERE EQUATIONS OF STATE OF REAL FLUIDS. 11.EFFECTIVE HARD-SPHERE DIAMETERS AND RESIDUAL PROPERTIES * IV0
NEZBEDA
and KAREL
AIM **
Institute of Chemical Process Fundamentals,
Czechosiovuk
of Sciences,
Academy
165 02 Prague 6 - Suchdol {Czechoslovakia)
(Received
October
25, 1983;
accepted
in revised form December
12, 1983)
ABSTRACT Nezbeda, E. and Aim, K., 1984. Perturbed hard-sphere equations of state of real fluids. If. Effective hard-sphere diameters and residual properties. Fluid Phase Equilibria, 17: l-18. Effective hard-sphere diameters for argon, krypton and xenon have been calculated from the currently most accurate perturbation theories using accurate pair-potential models. Based on the theoretical diameters and on pressure-volume-temperature data for the real fluids, the van der Waals parameter up is examined and two conjectures generalizing the behavior of up are formulated. These conjectures make it possible to evaluate the effective hard-sphere diameters of simple liquids at the triple-point temperature from data for a few low-temperature pressure--volume isotherms. This fact, together with a corresponding-states principle that emerges from results for the theoretical diameters, forms the basis of a simple method which we propose for evaluating temperature-dependent effective hard-sphere diameters of a perturbed hard-sphere equation of state, independently of any pair-potential model and any perturbation theory. The applicability of the method is demonstrated for methane and its extension to nonsimple liquids is discussed. It is also shown that the use of an approximate theory and/or approximate pair-potential model may often result in a qualitatively misleading picture of ap behavior. INTRODUCTION
Most equations
of state for pure fluids can be written in the form
P = P, - a,$
0)
where PO is the pressure of a reference fluid and the parameter ap depends, in general, on both temperature T and density p_ Justification of this form of * Preliminary results of this paper were presented in part at the 5th Int. Conf. on Mixtures of Nonelectrolytes and Intermolecular Interactions (Halle, April, 1983) and in part at the Res. Conf. on the Statistical Mechanics of Liquids (Liblice. Czechoslovakia, June, 1983). ** Author to whom correspondence should be addressed. 0378-3812/84/$03.00
0 1984 Elsevier Science
Publishers
3.V.
2
equation of state (EOS) stems from perturbation theories of liquids, based on expansion of the Helmholtz free energy about that of a reference system after dividing the pair potential into reference and perturbation parts (Boublik et al., 1980). Since the reference part must include a greater portion of the repulsive branch of the potential, in the overwhelming majority of cases the reference pressure P, is the pressure of a fluid of hard spheres (or, generally, of hard nonspherical bodies) of a certain effective diameter d, or
P=J’,,b, d)-a,p2 Modern theories of liquids offer quite accurate expressions for both d and ap given a pair-potential model. Unfortunately, applications of these theories to problems of practical interest are hampered by insufficient knowledge of the intermolecular forces acting in real liquids and by mathematical complexity. Therefore, the usual way of applying eqn. (2) to real liquids consists in choosing an approximate potential model and adjusting its parameters to experimental data for bulk properties through the EOS, thus compensating for inherent deficiencies of the model and for approximations used in evaluating ap_ Consequently, the potential parameters obtained in this way are too much affected by the approximations employed to be interpreted solely in terms of molecular properties. This approach thus imposes severe limits on the use of eqn. (2), especially on its ability to predict the properties of fluids on the basis of the properties of the constituent molecules. In Part I of this series of papers (Aim and Nezbeda, 1983a) we examined a perturbed hard-sphere EOS based on a simplified Barker-Henderson theory of the square-well fluid. We followed the procedure outlined above and, consequently, experienced problems when attempting to interpret the effective potential parameters. At the same time. our study confirmed the extreme sensitivity of the results to d, again underlining the importance of effective diameters in theories of real liquids. Moreover, whereas the parameter up in eqn. (2) is given by complicated integrals involving the reference-system correlation function(s) and the interaction potential, accurate analytical expressions for the hard-sphere pressure P,, are known and the effect of the reference-system potential is built entirely into the effective hard-sphere diameter. Therefore, having an independent source of the effective hardsphere diameter, the ignorance of details of the intermolecular interactions might at least partly be circumvented. All the above facts made us focus our attention on the effective hard-sphere diameter, its dependence on state variables and its effect on the residual term ap, and particularly on the possibility of determining it without reference to intermolecular interactions. We have calculated effective diameters for argon, krypton and xenon using the few available accurate pair-potential models (Maitland et al., 1981)
3
and the currently most accurate perturbation theories. On the basis of the theoretical diameters and experimental pressure-volume-temperature (PVT) data we have tried to find a general pattern of behavior of aP as a function of d. Analysis of the aP results so obtained allows two conjectures to be formulated concerning the behavior of up for real simple liquids. These conjectures make it possible to define an effective hard-sphere diameter d, independent of any perturbation theory and of any pair-potential model. A simple method for evaluating d, for real liquids at the triple-point temperature is proposed. A corresponding-states principle for the diameters emerges which allows determination of temperature-dependent effective hard-sphere diameters for fluids for which no accurate pair potential is known. The applicability of the approach is demonstrated for methane and the possibility of extending the results to more complicated liquids is discussed. THEORY
Given a pair-potential model u(R), eqn. (2) results within the framework of rigorous statistical mechanics from the following two-step scheme. First, the full pair potential u is divided into reference ( uO) and perturbation (ur) parts. Secondly, the properties of the reference fluid are mapped onto the hard-sphere fluid properties by appropriately defining the effective hardsphere diameter d. The crucial point of any perturbation theory is the first step. Proper choice of the reference system guarantees fast convergence of the perturbation expansion and simplicity of the correction terms. Undoubtedly the best reference system for a realistic potential u(R) is that of Weeks, Chandler and Andersen (WCA), which includes the entire repulsive branch of the potential (Chandler and Weeks, 1970; Weeks et al., 1971): UO UO
=24(R)-u(R,)
RGR,
=0
R>,R,
(3)
where R, is the point at which the potential takes the minimum value, (du,‘dR),=. = 0. Also used Frequently is the choice of reference system due to Barker and Henderson (BH) (Barker and Henderson, 1967). In this theory the total potential is divided at the zero-energy point o: u,=u(R)
R
u,=o
Rat7
Concerning the mapping of the reference-fluid properties onto those of the hard-sphere fluid, two commonly used methods are those related to the
4
original WCA and BH theories. The diameter d can also be determined by means of the variational method due to Mansoori and Canfield (1969), but in their approach the hard-sphere diameter is not given exclusively by the reference potential and therefore the method is not considered here. In the WCA theory the reference-fluid radial distribution function g, is approximated first by go = exp(-Puo
+ Pr+,Jghs
= exp(-Pu,)Y,
(5)
where /3 = l/kT, and the subscript hs denotes the fluid of hard spheres of so far undefined diameter d. This diameter is then determined from the requirement that the Helmholtz free energies of the reference and hard-sphere fluids be identical to first order, which leads to the equation /-]exp(-Duo)
- exp( - pu,,)]
Y,dR = 0
(6)
In the BH theory the reference-system radial distribution function is approximated directly by the hard-sphere-fluid distribution function, go = ghs, and the hard-sphere diameter is given by a simple integral:
d=J[1 -
(7)
exp( -bu,)]dR
only on temperature, in contrast to dWCA The diameter d,, is dependent which depends on both T and p_ However, the dependence of dWCA on p is very weak. Since the division of the potential given by eqn. (3) is the most important part of the WCA theory, it is tempting to combine eqn. (3) with the simple expression (7). This method is also considered in this paper and we call it the hybrid BH theory (hBH). Concerning the leading hard-sphere term in eqn. (2), the Carnahan-Starling equation (Carnahan and Starling, 1969) has been used for all the calculations performed: 4&QJ
= 0 +Y +Y2 -Y3)(l
-Y)-3
(8)
where y = 7rpd3/6; for d, we have considered three options: WCA, BH and hBH. Evaluation of dWCA from eqn. (6) requires the hard-sphere radial distribution function. For ghs we have used the parametrized expression given by Labik and Malijevsky (1981), which is based on very extensive Monte Carlo simulation data which are the most accurate to date. Neglecting substances with significant quantum effects, accurate pair potentials are known at present only for argon, krypton and xenon. For the present study we have used Barker-Pompe-type (BP) potentials for Kr and
5
Xe, HFD Aziz (HFD) potentials for Ar and Kr, and Barker-Fisher-Watts (BFW), Parson-Siska-Lee (PSL) and Aziz-Maitland-Smith (AMS) potentials for Ar. All these rather complicated potentials are well described in the literature and we refer readers to the recent monograph by Maitland et al. (1981) for their functional forms and parameters. To compare the qualitative trends of bulk properties evaluated by means of the accurate “true” potentials with those obtained using simple models, two frequently used potential models have been considerd as well, namely, the Mie m--n potential u l?l”
=
f&gqrn
-(o/R)“]
(9)
where K =
[m/(
m -
n)] ( m/n)nnm--n)
and the Kihara spherical-core
potential
By choosing yn = 12 and n = 6 in eqn. (9) the well-known Lennard-Jones (LJ) potential is recovered. Parameters of the Kihara and LJ potentials were taken from the monograph by Reed and Gubbins (1973) and those of the Mie 20-6 potential for methane from the work of Matthews and Smith (1976). RESULTS AND DISCUSSION
By using first the accurate “true” pair potentials available for argon, krypton and xenon we have largely eliminated one of the main factors affecting the results concerning ap, namely, inaccuracies in the intermolecular forces. This enabled us, by using experimental PVT data, to study the behavior of up as it resuhs from eqn. (2) for different theories. The sources of the Pv/T data chosen were the compilations due to Vargaftik (1975) and Rabinovich (1976). In addition, PVT data for argon were also taken from Gosman et al. (1969) and for krypton and xenon those derived from the EOS presented by Juza and Sifner (1976, 1977). The temperature and pressure ranges examined were 85-400 K, O-50 MPa, 120-450 K, O-40 MPa, and 165-450 K, O-40 MPa for argon, krypton and xenon, respectively. All the “true” pair potentials yield practically the same up for a given method used to calculate d. In Figs. l-3 typical examples of the behavior of up over the liquid range of densities ( p a p,) on different isotherms are shown for the three options of evaluating d for argon. For illustrative
6 0.21
0.20
0.16
I
I
10
1
I 20
1
I 30
I
I 40
I
50
P BPa]
Fig. 1. Parameter ap of argon as a function of pressure at 95-200 K, for hard-sphere diameters evaluated from WCA theory using the Barker-Fisher-Watts potential.
purposes and because of physical reasons, pressure has been chosen as an independent variable. It is seen that with the exception of the lowest isotherm the qualitative behavior of aP( dWCA) and a,(d,,,) is identical. Furthermore, for pressures P 5 2P,, these ap’s are also quantitatively nearly the same. These findings confirm the superiority of the reference-system selection over the method of determining effective hard-sphere diameters.
cm--150 --_____-170 ---200
0321
I
I 10
1
I 20
I
I 30
I
I 40
I 50
P [MPa]
Fig. 2. Parameter up of argon as a function of pressure at 95-200 K, for hard-sphere diameters evaluated from BH theory using the Barker-Fisher-Watts potential.
up given by both the WCA and hBH theories is a monotonically increasing function of pressure, and the curves of aP versus P become flat with decreasing temperature. For temperatures close to the triple-point temperature T,, and for the WCA theory, up is constant over the pressure range P 5 2 PC and for larger p!essures it decreases slightly. The hBH theory yields in this temperature region (I, constant over the entire pressure range studied.
0.16L
I
I 10
,
I 20
I
I 30
L +-
1
P[MPa]
Fig. 3. Parameter up of argon as a function of pressure at 95-200 K, for hard-sphere diameters evaluated from hBH theory using the Barker-Fisher-Watts potential.
In contrast to this pattern, the behavior of ap resulting from the BH theory seems more complicated, with no detectable general trends. At Iow temperatures (8a,/i3P), has a sign opposite to that observed for the WCA and hBH theories, and at higher temperatures ap( d,,) exhibits a maximum and different isotherms intersect. Since the properties of ap given by the different theories more or less differ, it is not possible to draw an unambiguous conclusion concerning ap.
9
Nevertheless, taking other facts into account, it is possible to discriminate among the theories. In general, the WCA-type theory should be preferred because it is known (Boublik et al., 1980) that the BH choice of reference system gives rise to much slower convergence of the perturbation expansion compared to the WCA division of the potential. From the practical point of view it is evident that the quite complicated behavior of aP( d,,) must also be difficult for any simple theory to predict or at least to fit analytically. Therefore it seems justified to disregard this theory in further considerations. Concerning the WCA and hBH theories, these give very similar results, with the latter having a great advantage in that d,,, is dependent only on temperature. Even though the density dependence of dWCA is very weak, any application of the WCA theory would require additional information concerning intermolecular forces (de1 Rio, 1981) which may push this theory out of interest for engineers. That is why in the following we focus mainly on the results of the hBH theory. As exemplified by Figs. 1 and 3, a,(d,,,) and a,(d,,,) vary over a narrower range, the lower the temperature. For example, for argon the maximum change in ap over the entire pressure range at 100 K is - 1.2%, while at 90 K it is only 0.2%. Having in mind the approximate nature of the -otherwise quite accurate-theories and being aware of possible inaccuracies in the PI/T data (we mention in passing that most of the recommended data for low-temperature isotherms are extrapolated, rather than directly experimental), we assume that this behavior of ap can be generalized into two conjectures which we claim to hold for real liquids with approximately spherical molecules: >O,forp>p,andT>T,; (1) @a,/aP)r = constant, for p > p, and T = Tt. (11) aP Although these conjectures are simple and quite general, they may prove useful if accepted as working hypotheses. They may serve as a test of precision of PVT data or may provide a filter for various pair-potential models. A consequence of conjecture (II) is that it allows an “absolute” effective hard-sphere diameter d, (the effective size of a molecule at the triple-point temperature) to be defined independent of any potential model and of any perturbation theory. There may be a practical problem in that experimental PVT data at low temperatures are rather scarce compared to other regions, but it must be pointed out that evaluation of molecular parameters by means of macroscopic relations, such as, for example, eqn. (2), at temperatures close to the triple point is the only correct method, as it is just this region in which perturbation theories are justified (Andersen et al., 1976; Henderson, 1979). We have tried to verify conjecture (II) and at the same time to establish a method for evaluating d without resorting to potential models. Our test
10
calculations aimed at finding d such that the standard deviation of up from its mean value on an isotherm be minimal over some pressure range for different temperatures. Indeed, at temperatures close to the triple point a diameter can always be found such that up is virtually constant over the entire pressure range considered. On proceeding to higher temperatures the standard deviations in ap increase. The tendency is the more pronounced, the wider the pressure range covered. However, after passing the critical temperature T, the pressure dependence of ap becomes weaker, and at temperatures of - 3 T, a diameter giving constant up may again be found (see, for example, de1 Rio and Arzola, 1977). Based on a number of test calculations, our procedure for evaluating d, is as follows. First, for several isotherms from the range T, to - [ Tt + (T, T,)/3], values of fictive diameters d, are determined by minimizing the standard deviation in up over the same pressure range on each isotherm, preferably from zero to the critical pressure PC. The resulting “diameters” d, as a function of temperature fall on a nearly straight line, and it is easy to extrapolate them to the triple-point temperature to obtain the value d, = (df)T,. The procedure assumes some reasonable spacing of the data points over the pressure range considered, but is rather insensitive to this factor. Regarding the condition of the same pressure range on each isotherm, the effect of the width of the pressure range involved is also not very large. Going from (0, PC) to (0, 6P,) corresponds to a variation of 0.04-0.15% in the resulting d, , depending upon the species and data source. Moreover, simple trends in d, as a function of the range width can be observed, enabling the d, value to be obtained for the recommended pressure range (0, 0 The evaluated diameters d, are listed and compared with the theoretical values ( dhBH)T,based on different accurate pair-potential models in Table 1. The agreement between the diameters obtained by the two methods is remarkable and lends support to the procedure proposed. Based on conjecture (II) as a check for PVT data precision, we did detect inconsistency between some data sets. For instance, the data of Rabinovich (1976) for neon at 26 K give up as a true constant over the pressure range 0.07-9.5 MPa, whereas ap evaluated from the data due to Vargaftik (1975) over a smaller range, 0.05-2.5 MPa, suffers from significant statistical noise. It can be concluded that the latter data are only fairly precise. Figures 1 and 3 show how simple the function ap is, provided that d has been determined appropriately. Such d is given, for example, by eqns. (3) and (7) if an accurate pair potential is known. However, this is not the usual case, and the question is how, if it is possible at all, to determine the appropriate effective hard-sphere diameter without knowing the accurate potential.
11 TABLE
1
Effective
hard-sphere
Fluid
diameters
dheH and d, at triple-point
hBH theory
(T, WI)
d hBH
temperature
Conjecture Pair
lnrn>
(II)
d, (nm>
potential Argon (83.80)
0.34730 0.34708 0.34610
BFW HFD PSL
0.34628
AMS
Krypton (115.76)
0.36990
BP
0.37028
HFD
Xenon (161.36)
0.40257
BP
7’,
P VT data reference
0.34610 0.34697 0.34625
0.37034
0.40338
Methane
0.38484
a b
C
C
C
d
(90.68) a b ’ d
Gosman et al. (1969). Vargaftik (1975). Rabinovich (1976). Goodwin (1974).
TABLE
2
Parameters of eqn. (ll), expressing as a function of temperature T(K) Pair
R,
(nm)
the theoretical
A
effective
10xB
hard-sphere
diameter ddBH (nm)
c
10’ X sd (nm) a
potential
Argon (dud
W
= 0.376 nrn), for T E (T,,4OO), N = 27
BFW HFD
0.37612 0.37590
0.113997 0.106965
0.424806 0.451875
0.435039 0.437844
0.6 0.6
PSL AMS
0.37600 0.37560
0.112004 0.104832
0.437393 0.470819
0.442716 0.438520
0.5 0.6
Krypton fdudW = 0.404 nm), for T E (T,,450), N = 30 BP 0.40067 0.112287 0.388033
0.444413
1.0
HFD
0.442937
0.4
0.460212
0.8
0.40120
0.112296
Xenon {dud ,,, = 0.432 nm), for T E (T,,450), BP
a s,,=
0.43623
;[d,,B,-d
0.103101
..,..,,,,l’/o)1’2
0.392714 N = 28 0.380650
12
To find a solution to this problem we must begin again with argon, krypton and xenon, for which the correct diameters are available. These diameters can be represented over the temperature ranges examined by the equation
d=R,+A[exp(BFj-11
(11)
where R m has the same meaning as in eqn. (3) ( R m is practically equal to the
Fig. 4. Temperature dependence of effective hard-sphere diameters: -, Barker-Pompetype potential and hBH theory; .-s- ., LennardJones (LJ( V)) potential and hBH theory: - - -, predicted by eqn. (13); - - - - - -, tangent at triple-point temperature (d in 10 X nm).
13
van der Waals diameter dvdW given by Bondi (1964)) and the parameters A, B and C are constants characteristic of the fluid and the pair-potential model. For the diameters calculated from the hBH theory, the constants of eqn. (11) are listed in Table 2 together with the corresponding temperature ranges of validity, standard deviations sd of fit, and the dvdW values. Equation (11) fits the diameters excellently but seems useless for any generalization. However, if In d is plotted versus In T,, where T, is the reduced temperature, T, = T/T,, at temperatures approximately up to (T, + T,)/2 straight lines with identical slopes are obtained for all three substances (see Fig. 4). Even though at higher temperatures In d deviates slightly from the straight line, all three curves differ only by a constant over the entire temperature range considered. This means that the ratio of the effective hard-sphere diameters of two substances (Yand p taken at the same reduced temperature is constant, namely,
Equation (12) expresses a corresponding-states principle for the effective hard-sphere diameters and we claim that it holds for all nonpolar liquids with approximately spherical molecules. A startling finding is that the effective hard-sphere diameters given by the Lennard-Jones potential with parameters adjusted to the second virial coefficient, LJ( V), exhibit the same slope of temperature dependence as those given by the accurate potentials (see Fig. 4). This may explain why the behavior of up based on the LJ( V) potential has been found not very different from that of up obtained for the accurate potentials. For the generalized relationship describing the temperature dependence of effective hard-sphere diameters of simple fluids we have obtained the form ln( d/d,)
= -0.03125
In T, - 0.00540 ln2 c
(13)
which represents dhBH for Ar, Kr and Xe and the BFW, BP, HFD and AMS pair potentials with a maximum deviation of 0.02% at T, < 2 and of 0.05% over the entire temperature range studied. Equation (13), together with the procedure for the evaluation of d, based on conjecture (II), provides a solution to the problem of finding the correct temperature-dependent effective hard-sphere diameters without having the pair-potential model at hand. We have made use of this method to determine the hard-sphere diameter for methane. Based on data of Goodwin (1974), the value of d, obtained for methane is given in Table 1. The temperature dependence of the diameter is plotted in Fig. 4 and the corresponding pattern of the ap = up(P) isotherms is shown in Fig. 5. It is seen that the resulting vdW parameter up exhibits the same simple behavior as those of argon, krypton and xenon based on the “true” pair potentials.
14
Q36-
oak-
.H
I 0
_H
I 10
---
_-'
.-
___---
I
I
I 20 P
I 30
-
_.c--
_/--__A--
,
___---
I 10
__--- _--
I
I 50
IMPaJ
Fig. 5. Parameter up of methane as a function of pressure at 110-240 diameters predicted by eqn. (13): -----, 110 K; .-.-., 160 K; ------, 240 K.
K, for hard-sphere 190 K; ---,
To demonstrate that this result is not so obvious we present in Fig. 6 the aP parameters evaluated for methane by means of the four frequently used potential models. It is evident that none of the potentials is able to give a result comparable to that in Fig. 5. The patterns of a,, based on the Mie 20-6, Kihara and LJ(C) potentials are qualitatively completely different, and only the behavior of a,(~!,(,,) q ualitatively resembles the simple pattern from Figs. 1, 3, and 5. The fact that the approximate potential models are not able to yield at least qualitatively the latter result confirms that up is very sensitive to the actual shape of the pair potential.
CONCLUSIONS
Effective size of molecules is an important molecular property frequently used in liquid-state physics and chemistry_ Unlike the methods based on its physical interpretation (see, for example, Bienkowski and Chao, 1975; de1 Rio and de Longi, 1976), the present method determines temperature-depen-
Fig. 6. Parameter ap of methane as a function of pressure at 110-240 K, for hard-sphere diameters evaluated from the hBH theory using the Mie 20-6, Kihara, Lennard-Jones LJ(C) (with parameters adjusted to combined data for the second virial coefficient and viscosity) and Lennard-Jones LJ( V) (with parameters adjusted to the second virial coefficient) poten100 K; - . - . -, 160 K; - - - - - -, 190 K; - - -, 240 K. tials: -,
dent hard-core diameters which can be employed directly in perturbed hard-sphere equations of state. Our procedure makes use of (i) two conjectures that resulted from a detailed analysis of the behavior of the residual term up in dependence on d, and (ii) a corresponding-states principle that emerged from theoretical calculations of d for accurate pair-potential models. The method is valid for liquids made up of spherical or nearly spherical molecules and as input information requires only PI/T data on a few (at least two) isotherms from the lower third of the temperature range of existence of a liquid. The procedure has been demonstrated for methane and the evaluated effective diameter gave a simple pattern of behavior of aP, analogous to those obtained for argon, krypton and xenon from the theory. Such a result cannot be attained by any of the commonly used pair-potential models for methane. The idea of analyzing the properties of the up parameter by means of PC!” data for real liquids is not new (see Hlavaty, 1974; Ree, 1980). However, the present results are in sharp contrast with those reported by Hlavaty (1974),
14
due to the fact that the latter author employed only an approximate theory for the P,, term in eqn. (2) in connection with the Lennard-Jones potential. As we have demonstrated, the application of either an insufficiently accurate theory or of an approximate pair-potential model (or both) may not lead to a qualitatively correct picture of aP behavior. The present results can be extended to compounds of nonspherical molecules by replacing the Carnahan-Starling equation (eqn. (8)) by an EOS for hard nonspherical bodies. The thickness of the hard-body outer layer then takes on the role of the hard-sphere diameter, and parameter(s) of nonsphericity may be estimated from molecular structure or treated as additional adjustable parameter(s). In accordance with conjecture (II), preliminary calculations for nonpolar liquids show that effective hard bodies can be found such that the corresponding aP parameters remain constant along low-temperature isotherms. Details of these calculations, together with recommended values of effective hard-sphere diameters and hard-body parameters for a series of real liquids, are the subject of a separate communication (Aim and Nezbeda, 1984). We believe that by evaluating a priori the effective hard-sphere diameter we have managed to some extent to bridge the gap between statistical-mechanical expressions for liquid properties and inadequate knowledge of intermolecular interactions: a perturbed hard-sphere EOS employing the hardsphere diameters evaluated by the proposed method reflects our contemporary understanding of liquids. To get a complete theoretical EOS for real “intermolecular-interaction-independent” theory for aP remains liquids, an to be developed. This is part of a current project conducted in our laboratory and the results will be reported in due course. LIST OF SYMBOLS
Y
vdW parameter in pressure-explicit effective hard-sphere diameter radial distribution function Boltzmann’s constant pressure center-to-center distance temperature pair potential packing fraction ( y = rpd3/6)
P
= l/(/z)
aP
d 8
k
P
R T U
&
P (I
EOS
characteristic energy (depth of potential number density zero-energy point (u( 0) = 0)
well)
17
Subscripts
a C
f hs m r t VdW 0
“absolute” property critical point fictive property hard-sphere property minimum of a potential reduced quantity, such as T, = T/T, triple point van der Waals property of a reference system
Superscripts (a), (p)
related
to species CYand /3
REFERENCES Aim, K. and Nezbeda, I., 1983a. Perturbed hard sphere equations of state of real liquids. I. Examination of a simple equation of the second order. Fluid Phase Equilibria, 12: 235-251. Aim, K. and Nezbeda, I., 1984. Effective hard cores of normal real fluids. submitted to Fluid Phase Equilibria. Andersen, H.C., Chandler, D. and Weeks, J.D., 1976. Roles of repulsive and attractive forces in liquids: the equilibrium theory of classical fluids. Adv. Chem. Phys., 14: 105-156. Barker, J.A. and Henderson, D., 1967. Perturbation theory and equation of state for fluids. II. A successful theory of liquids. J. Chem. Phys.. 47: 4714-4721. Bienkowski, P.R. and Chao, K.-C., 1975. Hard-cores of molecules of simple fluids. J. Chem. Phys., 62: 615-619. Bondi, A., 1964. Van der Waals volumes and radii. J. Phys. Chem., 68: 441-451. Boublik, T., Nezbeda, I. and Hlavatjr, K., 1980. Statistical Thermodynamics of Simple Liquids and Their Mixtures. Elsevier, Amsterdam. Carnahan, N.F. and Starling, K.E., 1969. Equation of state for nonattracting rigid spheres. J. Chem. Phys., 51: 635-636. Chandler, D. and Weeks, J.D., 1970. Equilibrium structure of simple liquids. Phys. Rev. Lett., 25: 149-152. Del Rio, F., 1981. Effective diameters and corresponding states of fluids. Mol. Phys., 42: 217-230. Del Rio, F. and Arzola, C., 1977. On the equation of state of argon. J. Phys. Chem., 81: 862-865. Del Rio, F. and de Longi, D.A., 1976. The mean distance of closest approach in the theory of liquids. Phys. Lett., 56A: 463~-464. Goodwin, R.D., 1974. The thermophysical properties of methane, from 90 to 500 K at pressures to 700 bar. Natl. Bur. Stand. (U.S.), Tech. Note, 653. Gosman, A.L., McCarty, R.D. and Hust, J.G., 1969. Thermodynamic properties of argon from the triple point to 300 K at pressures to 1000 atmospheres. Natl. Bur. Stand. (U.S.), Rep. NSRDS-NBS 27.
1x
Henderson, D., 1979. Practical calculations of the equation of state of fluids and fluid mixtures using perturbation theory and related theories. In: K.-C. Chao and R.L. Robinson, Jr. (Editors), Equations of State in Engineering and Research. Adv. Chem. Ser., 182, Am. Chem. Sot., Washington, DC.. Hlavaty, K., 1974. Semirational method for the construction of equations of state for liquids and gases. Collect. Czech. Chem. Commun., 39: 2927-2934. Juza, J. and Sifner, O., 1976. Modified equation of state and formulation of thermodynamic properties of krypton in a canonical form in the range from 120 to 423 K and 0 to 300 MPa. Acta Tech. CSAV, 21: l-32. J&a, J. and Sifner, O., 1977. Modified equation of state and formulation of thermodynamic properties of xenon in the range from 161.36 to 800 K and 0 to 350 MPa. Acta Tech. CSAV, 22: l-32. Labik, S. and Malijevsky, A., 1981. Monte Carlo simulations of the radial distribution function of fluid of hard spheres. Mol. Phys., 42: 739-744. Maitland, G.C., Rigby, M., Smith, E.B. and Wakeham, W.A., 1981. Intermolecular Forces: Their Origin and Determination, Clarendon, Oxford. Mansoori, G.A. and Canfield, F.B., 1969. Variational approach to the equilibrium properties of simple liquids. I. J. Chem. Phys., 51: 4958-4972. Matthews, G.P. and Smith, E.B., 1976. An intermolecular potential energy function for methane. Mol. Phys., 32: 1719-1729. Rabinovich, V.A. (Editor), 1976. Teplofizicheskiie Svoistva Neona, Argona, Kriptona i Ksenona. Izdatel’stvo Standartov, Moscow (in Russian). Ree, F.H., 1980. Reexamination of van der Waals model for soft-core molecules. Physica, 104A: 197-209. Reed, T.M. and Gubbins, K.E., 1973. Applied Statistical Mechanics. McGraw-Hill, New York. Vargaftik, N.B., 1975. Tables on the Thermophysical Properties of Liquids and Gases. 2nd edn. Wiley, New York. Weeks, J.D., Chandler, D. and Andersen, H.C., 1971. The role of repulsive forces in determining the equilibrium structure of simple liquids. J. Chem. Phys., 54: 5237-5247.
Elsevier Science
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PERTURBED HARD-SPHERE EQUATIONS OF STATE OF REAL FLUIDS. 11.EFFECTIVE HARD-SPHERE DIAMETERS AND RESIDUAL PROPERTIES * IV0
NEZBEDA
and KAREL
AIM **
Institute of Chemical Process Fundamentals,
Czechosiovuk
of Sciences,
Academy
165 02 Prague 6 - Suchdol {Czechoslovakia)
(Received
October
25, 1983;
accepted
in revised form December
12, 1983)
ABSTRACT Nezbeda, E. and Aim, K., 1984. Perturbed hard-sphere equations of state of real fluids. If. Effective hard-sphere diameters and residual properties. Fluid Phase Equilibria, 17: l-18. Effective hard-sphere diameters for argon, krypton and xenon have been calculated from the currently most accurate perturbation theories using accurate pair-potential models. Based on the theoretical diameters and on pressure-volume-temperature data for the real fluids, the van der Waals parameter up is examined and two conjectures generalizing the behavior of up are formulated. These conjectures make it possible to evaluate the effective hard-sphere diameters of simple liquids at the triple-point temperature from data for a few low-temperature pressure--volume isotherms. This fact, together with a corresponding-states principle that emerges from results for the theoretical diameters, forms the basis of a simple method which we propose for evaluating temperature-dependent effective hard-sphere diameters of a perturbed hard-sphere equation of state, independently of any pair-potential model and any perturbation theory. The applicability of the method is demonstrated for methane and its extension to nonsimple liquids is discussed. It is also shown that the use of an approximate theory and/or approximate pair-potential model may often result in a qualitatively misleading picture of ap behavior. INTRODUCTION
Most equations
of state for pure fluids can be written in the form
P = P, - a,$
0)
where PO is the pressure of a reference fluid and the parameter ap depends, in general, on both temperature T and density p_ Justification of this form of * Preliminary results of this paper were presented in part at the 5th Int. Conf. on Mixtures of Nonelectrolytes and Intermolecular Interactions (Halle, April, 1983) and in part at the Res. Conf. on the Statistical Mechanics of Liquids (Liblice. Czechoslovakia, June, 1983). ** Author to whom correspondence should be addressed. 0378-3812/84/$03.00
0 1984 Elsevier Science
Publishers
3.V.
2
equation of state (EOS) stems from perturbation theories of liquids, based on expansion of the Helmholtz free energy about that of a reference system after dividing the pair potential into reference and perturbation parts (Boublik et al., 1980). Since the reference part must include a greater portion of the repulsive branch of the potential, in the overwhelming majority of cases the reference pressure P, is the pressure of a fluid of hard spheres (or, generally, of hard nonspherical bodies) of a certain effective diameter d, or
P=J’,,b, d)-a,p2 Modern theories of liquids offer quite accurate expressions for both d and ap given a pair-potential model. Unfortunately, applications of these theories to problems of practical interest are hampered by insufficient knowledge of the intermolecular forces acting in real liquids and by mathematical complexity. Therefore, the usual way of applying eqn. (2) to real liquids consists in choosing an approximate potential model and adjusting its parameters to experimental data for bulk properties through the EOS, thus compensating for inherent deficiencies of the model and for approximations used in evaluating ap_ Consequently, the potential parameters obtained in this way are too much affected by the approximations employed to be interpreted solely in terms of molecular properties. This approach thus imposes severe limits on the use of eqn. (2), especially on its ability to predict the properties of fluids on the basis of the properties of the constituent molecules. In Part I of this series of papers (Aim and Nezbeda, 1983a) we examined a perturbed hard-sphere EOS based on a simplified Barker-Henderson theory of the square-well fluid. We followed the procedure outlined above and, consequently, experienced problems when attempting to interpret the effective potential parameters. At the same time. our study confirmed the extreme sensitivity of the results to d, again underlining the importance of effective diameters in theories of real liquids. Moreover, whereas the parameter up in eqn. (2) is given by complicated integrals involving the reference-system correlation function(s) and the interaction potential, accurate analytical expressions for the hard-sphere pressure P,, are known and the effect of the reference-system potential is built entirely into the effective hard-sphere diameter. Therefore, having an independent source of the effective hardsphere diameter, the ignorance of details of the intermolecular interactions might at least partly be circumvented. All the above facts made us focus our attention on the effective hard-sphere diameter, its dependence on state variables and its effect on the residual term ap, and particularly on the possibility of determining it without reference to intermolecular interactions. We have calculated effective diameters for argon, krypton and xenon using the few available accurate pair-potential models (Maitland et al., 1981)
3
and the currently most accurate perturbation theories. On the basis of the theoretical diameters and experimental pressure-volume-temperature (PVT) data we have tried to find a general pattern of behavior of aP as a function of d. Analysis of the aP results so obtained allows two conjectures to be formulated concerning the behavior of up for real simple liquids. These conjectures make it possible to define an effective hard-sphere diameter d, independent of any perturbation theory and of any pair-potential model. A simple method for evaluating d, for real liquids at the triple-point temperature is proposed. A corresponding-states principle for the diameters emerges which allows determination of temperature-dependent effective hard-sphere diameters for fluids for which no accurate pair potential is known. The applicability of the approach is demonstrated for methane and the possibility of extending the results to more complicated liquids is discussed. THEORY
Given a pair-potential model u(R), eqn. (2) results within the framework of rigorous statistical mechanics from the following two-step scheme. First, the full pair potential u is divided into reference ( uO) and perturbation (ur) parts. Secondly, the properties of the reference fluid are mapped onto the hard-sphere fluid properties by appropriately defining the effective hardsphere diameter d. The crucial point of any perturbation theory is the first step. Proper choice of the reference system guarantees fast convergence of the perturbation expansion and simplicity of the correction terms. Undoubtedly the best reference system for a realistic potential u(R) is that of Weeks, Chandler and Andersen (WCA), which includes the entire repulsive branch of the potential (Chandler and Weeks, 1970; Weeks et al., 1971): UO UO
=24(R)-u(R,)
RGR,
=0
R>,R,
(3)
where R, is the point at which the potential takes the minimum value, (du,‘dR),=. = 0. Also used Frequently is the choice of reference system due to Barker and Henderson (BH) (Barker and Henderson, 1967). In this theory the total potential is divided at the zero-energy point o: u,=u(R)
R
u,=o
Rat7
Concerning the mapping of the reference-fluid properties onto those of the hard-sphere fluid, two commonly used methods are those related to the
4
original WCA and BH theories. The diameter d can also be determined by means of the variational method due to Mansoori and Canfield (1969), but in their approach the hard-sphere diameter is not given exclusively by the reference potential and therefore the method is not considered here. In the WCA theory the reference-fluid radial distribution function g, is approximated first by go = exp(-Puo
+ Pr+,Jghs
= exp(-Pu,)Y,
(5)
where /3 = l/kT, and the subscript hs denotes the fluid of hard spheres of so far undefined diameter d. This diameter is then determined from the requirement that the Helmholtz free energies of the reference and hard-sphere fluids be identical to first order, which leads to the equation /-]exp(-Duo)
- exp( - pu,,)]
Y,dR = 0
(6)
In the BH theory the reference-system radial distribution function is approximated directly by the hard-sphere-fluid distribution function, go = ghs, and the hard-sphere diameter is given by a simple integral:
d=J[1 -
(7)
exp( -bu,)]dR
only on temperature, in contrast to dWCA The diameter d,, is dependent which depends on both T and p_ However, the dependence of dWCA on p is very weak. Since the division of the potential given by eqn. (3) is the most important part of the WCA theory, it is tempting to combine eqn. (3) with the simple expression (7). This method is also considered in this paper and we call it the hybrid BH theory (hBH). Concerning the leading hard-sphere term in eqn. (2), the Carnahan-Starling equation (Carnahan and Starling, 1969) has been used for all the calculations performed: 4&QJ
= 0 +Y +Y2 -Y3)(l
-Y)-3
(8)
where y = 7rpd3/6; for d, we have considered three options: WCA, BH and hBH. Evaluation of dWCA from eqn. (6) requires the hard-sphere radial distribution function. For ghs we have used the parametrized expression given by Labik and Malijevsky (1981), which is based on very extensive Monte Carlo simulation data which are the most accurate to date. Neglecting substances with significant quantum effects, accurate pair potentials are known at present only for argon, krypton and xenon. For the present study we have used Barker-Pompe-type (BP) potentials for Kr and
5
Xe, HFD Aziz (HFD) potentials for Ar and Kr, and Barker-Fisher-Watts (BFW), Parson-Siska-Lee (PSL) and Aziz-Maitland-Smith (AMS) potentials for Ar. All these rather complicated potentials are well described in the literature and we refer readers to the recent monograph by Maitland et al. (1981) for their functional forms and parameters. To compare the qualitative trends of bulk properties evaluated by means of the accurate “true” potentials with those obtained using simple models, two frequently used potential models have been considerd as well, namely, the Mie m--n potential u l?l”
=
f&gqrn
-(o/R)“]
(9)
where K =
[m/(
m -
n)] ( m/n)nnm--n)
and the Kihara spherical-core
potential
By choosing yn = 12 and n = 6 in eqn. (9) the well-known Lennard-Jones (LJ) potential is recovered. Parameters of the Kihara and LJ potentials were taken from the monograph by Reed and Gubbins (1973) and those of the Mie 20-6 potential for methane from the work of Matthews and Smith (1976). RESULTS AND DISCUSSION
By using first the accurate “true” pair potentials available for argon, krypton and xenon we have largely eliminated one of the main factors affecting the results concerning ap, namely, inaccuracies in the intermolecular forces. This enabled us, by using experimental PVT data, to study the behavior of up as it resuhs from eqn. (2) for different theories. The sources of the Pv/T data chosen were the compilations due to Vargaftik (1975) and Rabinovich (1976). In addition, PVT data for argon were also taken from Gosman et al. (1969) and for krypton and xenon those derived from the EOS presented by Juza and Sifner (1976, 1977). The temperature and pressure ranges examined were 85-400 K, O-50 MPa, 120-450 K, O-40 MPa, and 165-450 K, O-40 MPa for argon, krypton and xenon, respectively. All the “true” pair potentials yield practically the same up for a given method used to calculate d. In Figs. l-3 typical examples of the behavior of up over the liquid range of densities ( p a p,) on different isotherms are shown for the three options of evaluating d for argon. For illustrative
6 0.21
0.20
0.16
I
I
10
1
I 20
1
I 30
I
I 40
I
50
P BPa]
Fig. 1. Parameter ap of argon as a function of pressure at 95-200 K, for hard-sphere diameters evaluated from WCA theory using the Barker-Fisher-Watts potential.
purposes and because of physical reasons, pressure has been chosen as an independent variable. It is seen that with the exception of the lowest isotherm the qualitative behavior of aP( dWCA) and a,(d,,,) is identical. Furthermore, for pressures P 5 2P,, these ap’s are also quantitatively nearly the same. These findings confirm the superiority of the reference-system selection over the method of determining effective hard-sphere diameters.
cm--150 --_____-170 ---200
0321
I
I 10
1
I 20
I
I 30
I
I 40
I 50
P [MPa]
Fig. 2. Parameter up of argon as a function of pressure at 95-200 K, for hard-sphere diameters evaluated from BH theory using the Barker-Fisher-Watts potential.
up given by both the WCA and hBH theories is a monotonically increasing function of pressure, and the curves of aP versus P become flat with decreasing temperature. For temperatures close to the triple-point temperature T,, and for the WCA theory, up is constant over the pressure range P 5 2 PC and for larger p!essures it decreases slightly. The hBH theory yields in this temperature region (I, constant over the entire pressure range studied.
0.16L
I
I 10
,
I 20
I
I 30
L +-
1
P[MPa]
Fig. 3. Parameter up of argon as a function of pressure at 95-200 K, for hard-sphere diameters evaluated from hBH theory using the Barker-Fisher-Watts potential.
In contrast to this pattern, the behavior of ap resulting from the BH theory seems more complicated, with no detectable general trends. At Iow temperatures (8a,/i3P), has a sign opposite to that observed for the WCA and hBH theories, and at higher temperatures ap( d,,) exhibits a maximum and different isotherms intersect. Since the properties of ap given by the different theories more or less differ, it is not possible to draw an unambiguous conclusion concerning ap.
9
Nevertheless, taking other facts into account, it is possible to discriminate among the theories. In general, the WCA-type theory should be preferred because it is known (Boublik et al., 1980) that the BH choice of reference system gives rise to much slower convergence of the perturbation expansion compared to the WCA division of the potential. From the practical point of view it is evident that the quite complicated behavior of aP( d,,) must also be difficult for any simple theory to predict or at least to fit analytically. Therefore it seems justified to disregard this theory in further considerations. Concerning the WCA and hBH theories, these give very similar results, with the latter having a great advantage in that d,,, is dependent only on temperature. Even though the density dependence of dWCA is very weak, any application of the WCA theory would require additional information concerning intermolecular forces (de1 Rio, 1981) which may push this theory out of interest for engineers. That is why in the following we focus mainly on the results of the hBH theory. As exemplified by Figs. 1 and 3, a,(d,,,) and a,(d,,,) vary over a narrower range, the lower the temperature. For example, for argon the maximum change in ap over the entire pressure range at 100 K is - 1.2%, while at 90 K it is only 0.2%. Having in mind the approximate nature of the -otherwise quite accurate-theories and being aware of possible inaccuracies in the PI/T data (we mention in passing that most of the recommended data for low-temperature isotherms are extrapolated, rather than directly experimental), we assume that this behavior of ap can be generalized into two conjectures which we claim to hold for real liquids with approximately spherical molecules: >O,forp>p,andT>T,; (1) @a,/aP)r = constant, for p > p, and T = Tt. (11) aP Although these conjectures are simple and quite general, they may prove useful if accepted as working hypotheses. They may serve as a test of precision of PVT data or may provide a filter for various pair-potential models. A consequence of conjecture (II) is that it allows an “absolute” effective hard-sphere diameter d, (the effective size of a molecule at the triple-point temperature) to be defined independent of any potential model and of any perturbation theory. There may be a practical problem in that experimental PVT data at low temperatures are rather scarce compared to other regions, but it must be pointed out that evaluation of molecular parameters by means of macroscopic relations, such as, for example, eqn. (2), at temperatures close to the triple point is the only correct method, as it is just this region in which perturbation theories are justified (Andersen et al., 1976; Henderson, 1979). We have tried to verify conjecture (II) and at the same time to establish a method for evaluating d without resorting to potential models. Our test
10
calculations aimed at finding d such that the standard deviation of up from its mean value on an isotherm be minimal over some pressure range for different temperatures. Indeed, at temperatures close to the triple point a diameter can always be found such that up is virtually constant over the entire pressure range considered. On proceeding to higher temperatures the standard deviations in ap increase. The tendency is the more pronounced, the wider the pressure range covered. However, after passing the critical temperature T, the pressure dependence of ap becomes weaker, and at temperatures of - 3 T, a diameter giving constant up may again be found (see, for example, de1 Rio and Arzola, 1977). Based on a number of test calculations, our procedure for evaluating d, is as follows. First, for several isotherms from the range T, to - [ Tt + (T, T,)/3], values of fictive diameters d, are determined by minimizing the standard deviation in up over the same pressure range on each isotherm, preferably from zero to the critical pressure PC. The resulting “diameters” d, as a function of temperature fall on a nearly straight line, and it is easy to extrapolate them to the triple-point temperature to obtain the value d, = (df)T,. The procedure assumes some reasonable spacing of the data points over the pressure range considered, but is rather insensitive to this factor. Regarding the condition of the same pressure range on each isotherm, the effect of the width of the pressure range involved is also not very large. Going from (0, PC) to (0, 6P,) corresponds to a variation of 0.04-0.15% in the resulting d, , depending upon the species and data source. Moreover, simple trends in d, as a function of the range width can be observed, enabling the d, value to be obtained for the recommended pressure range (0, 0 The evaluated diameters d, are listed and compared with the theoretical values ( dhBH)T,based on different accurate pair-potential models in Table 1. The agreement between the diameters obtained by the two methods is remarkable and lends support to the procedure proposed. Based on conjecture (II) as a check for PVT data precision, we did detect inconsistency between some data sets. For instance, the data of Rabinovich (1976) for neon at 26 K give up as a true constant over the pressure range 0.07-9.5 MPa, whereas ap evaluated from the data due to Vargaftik (1975) over a smaller range, 0.05-2.5 MPa, suffers from significant statistical noise. It can be concluded that the latter data are only fairly precise. Figures 1 and 3 show how simple the function ap is, provided that d has been determined appropriately. Such d is given, for example, by eqns. (3) and (7) if an accurate pair potential is known. However, this is not the usual case, and the question is how, if it is possible at all, to determine the appropriate effective hard-sphere diameter without knowing the accurate potential.
11 TABLE
1
Effective
hard-sphere
Fluid
diameters
dheH and d, at triple-point
hBH theory
(T, WI)
d hBH
temperature
Conjecture Pair
lnrn>
(II)
d, (nm>
potential Argon (83.80)
0.34730 0.34708 0.34610
BFW HFD PSL
0.34628
AMS
Krypton (115.76)
0.36990
BP
0.37028
HFD
Xenon (161.36)
0.40257
BP
7’,
P VT data reference
0.34610 0.34697 0.34625
0.37034
0.40338
Methane
0.38484
a b
C
C
C
d
(90.68) a b ’ d
Gosman et al. (1969). Vargaftik (1975). Rabinovich (1976). Goodwin (1974).
TABLE
2
Parameters of eqn. (ll), expressing as a function of temperature T(K) Pair
R,
(nm)
the theoretical
A
effective
10xB
hard-sphere
diameter ddBH (nm)
c
10’ X sd (nm) a
potential
Argon (dud
W
= 0.376 nrn), for T E (T,,4OO), N = 27
BFW HFD
0.37612 0.37590
0.113997 0.106965
0.424806 0.451875
0.435039 0.437844
0.6 0.6
PSL AMS
0.37600 0.37560
0.112004 0.104832
0.437393 0.470819
0.442716 0.438520
0.5 0.6
Krypton fdudW = 0.404 nm), for T E (T,,450), N = 30 BP 0.40067 0.112287 0.388033
0.444413
1.0
HFD
0.442937
0.4
0.460212
0.8
0.40120
0.112296
Xenon {dud ,,, = 0.432 nm), for T E (T,,450), BP
a s,,=
0.43623
;[d,,B,-d
0.103101
..,..,,,,l’/o)1’2
0.392714 N = 28 0.380650
12
To find a solution to this problem we must begin again with argon, krypton and xenon, for which the correct diameters are available. These diameters can be represented over the temperature ranges examined by the equation
d=R,+A[exp(BFj-11
(11)
where R m has the same meaning as in eqn. (3) ( R m is practically equal to the
Fig. 4. Temperature dependence of effective hard-sphere diameters: -, Barker-Pompetype potential and hBH theory; .-s- ., LennardJones (LJ( V)) potential and hBH theory: - - -, predicted by eqn. (13); - - - - - -, tangent at triple-point temperature (d in 10 X nm).
13
van der Waals diameter dvdW given by Bondi (1964)) and the parameters A, B and C are constants characteristic of the fluid and the pair-potential model. For the diameters calculated from the hBH theory, the constants of eqn. (11) are listed in Table 2 together with the corresponding temperature ranges of validity, standard deviations sd of fit, and the dvdW values. Equation (11) fits the diameters excellently but seems useless for any generalization. However, if In d is plotted versus In T,, where T, is the reduced temperature, T, = T/T,, at temperatures approximately up to (T, + T,)/2 straight lines with identical slopes are obtained for all three substances (see Fig. 4). Even though at higher temperatures In d deviates slightly from the straight line, all three curves differ only by a constant over the entire temperature range considered. This means that the ratio of the effective hard-sphere diameters of two substances (Yand p taken at the same reduced temperature is constant, namely,
Equation (12) expresses a corresponding-states principle for the effective hard-sphere diameters and we claim that it holds for all nonpolar liquids with approximately spherical molecules. A startling finding is that the effective hard-sphere diameters given by the Lennard-Jones potential with parameters adjusted to the second virial coefficient, LJ( V), exhibit the same slope of temperature dependence as those given by the accurate potentials (see Fig. 4). This may explain why the behavior of up based on the LJ( V) potential has been found not very different from that of up obtained for the accurate potentials. For the generalized relationship describing the temperature dependence of effective hard-sphere diameters of simple fluids we have obtained the form ln( d/d,)
= -0.03125
In T, - 0.00540 ln2 c
(13)
which represents dhBH for Ar, Kr and Xe and the BFW, BP, HFD and AMS pair potentials with a maximum deviation of 0.02% at T, < 2 and of 0.05% over the entire temperature range studied. Equation (13), together with the procedure for the evaluation of d, based on conjecture (II), provides a solution to the problem of finding the correct temperature-dependent effective hard-sphere diameters without having the pair-potential model at hand. We have made use of this method to determine the hard-sphere diameter for methane. Based on data of Goodwin (1974), the value of d, obtained for methane is given in Table 1. The temperature dependence of the diameter is plotted in Fig. 4 and the corresponding pattern of the ap = up(P) isotherms is shown in Fig. 5. It is seen that the resulting vdW parameter up exhibits the same simple behavior as those of argon, krypton and xenon based on the “true” pair potentials.
14
Q36-
oak-
.H
I 0
_H
I 10
---
_-'
.-
___---
I
I
I 20 P
I 30
-
_.c--
_/--__A--
,
___---
I 10
__--- _--
I
I 50
IMPaJ
Fig. 5. Parameter up of methane as a function of pressure at 110-240 diameters predicted by eqn. (13): -----, 110 K; .-.-., 160 K; ------, 240 K.
K, for hard-sphere 190 K; ---,
To demonstrate that this result is not so obvious we present in Fig. 6 the aP parameters evaluated for methane by means of the four frequently used potential models. It is evident that none of the potentials is able to give a result comparable to that in Fig. 5. The patterns of a,, based on the Mie 20-6, Kihara and LJ(C) potentials are qualitatively completely different, and only the behavior of a,(~!,(,,) q ualitatively resembles the simple pattern from Figs. 1, 3, and 5. The fact that the approximate potential models are not able to yield at least qualitatively the latter result confirms that up is very sensitive to the actual shape of the pair potential.
CONCLUSIONS
Effective size of molecules is an important molecular property frequently used in liquid-state physics and chemistry_ Unlike the methods based on its physical interpretation (see, for example, Bienkowski and Chao, 1975; de1 Rio and de Longi, 1976), the present method determines temperature-depen-
Fig. 6. Parameter ap of methane as a function of pressure at 110-240 K, for hard-sphere diameters evaluated from the hBH theory using the Mie 20-6, Kihara, Lennard-Jones LJ(C) (with parameters adjusted to combined data for the second virial coefficient and viscosity) and Lennard-Jones LJ( V) (with parameters adjusted to the second virial coefficient) poten100 K; - . - . -, 160 K; - - - - - -, 190 K; - - -, 240 K. tials: -,
dent hard-core diameters which can be employed directly in perturbed hard-sphere equations of state. Our procedure makes use of (i) two conjectures that resulted from a detailed analysis of the behavior of the residual term up in dependence on d, and (ii) a corresponding-states principle that emerged from theoretical calculations of d for accurate pair-potential models. The method is valid for liquids made up of spherical or nearly spherical molecules and as input information requires only PI/T data on a few (at least two) isotherms from the lower third of the temperature range of existence of a liquid. The procedure has been demonstrated for methane and the evaluated effective diameter gave a simple pattern of behavior of aP, analogous to those obtained for argon, krypton and xenon from the theory. Such a result cannot be attained by any of the commonly used pair-potential models for methane. The idea of analyzing the properties of the up parameter by means of PC!” data for real liquids is not new (see Hlavaty, 1974; Ree, 1980). However, the present results are in sharp contrast with those reported by Hlavaty (1974),
14
due to the fact that the latter author employed only an approximate theory for the P,, term in eqn. (2) in connection with the Lennard-Jones potential. As we have demonstrated, the application of either an insufficiently accurate theory or of an approximate pair-potential model (or both) may not lead to a qualitatively correct picture of aP behavior. The present results can be extended to compounds of nonspherical molecules by replacing the Carnahan-Starling equation (eqn. (8)) by an EOS for hard nonspherical bodies. The thickness of the hard-body outer layer then takes on the role of the hard-sphere diameter, and parameter(s) of nonsphericity may be estimated from molecular structure or treated as additional adjustable parameter(s). In accordance with conjecture (II), preliminary calculations for nonpolar liquids show that effective hard bodies can be found such that the corresponding aP parameters remain constant along low-temperature isotherms. Details of these calculations, together with recommended values of effective hard-sphere diameters and hard-body parameters for a series of real liquids, are the subject of a separate communication (Aim and Nezbeda, 1984). We believe that by evaluating a priori the effective hard-sphere diameter we have managed to some extent to bridge the gap between statistical-mechanical expressions for liquid properties and inadequate knowledge of intermolecular interactions: a perturbed hard-sphere EOS employing the hardsphere diameters evaluated by the proposed method reflects our contemporary understanding of liquids. To get a complete theoretical EOS for real “intermolecular-interaction-independent” theory for aP remains liquids, an to be developed. This is part of a current project conducted in our laboratory and the results will be reported in due course. LIST OF SYMBOLS
Y
vdW parameter in pressure-explicit effective hard-sphere diameter radial distribution function Boltzmann’s constant pressure center-to-center distance temperature pair potential packing fraction ( y = rpd3/6)
P
= l/(/z)
aP
d 8
k
P
R T U
&
P (I
EOS
characteristic energy (depth of potential number density zero-energy point (u( 0) = 0)
well)
17
Subscripts
a C
f hs m r t VdW 0
“absolute” property critical point fictive property hard-sphere property minimum of a potential reduced quantity, such as T, = T/T, triple point van der Waals property of a reference system
Superscripts (a), (p)
related
to species CYand /3
REFERENCES Aim, K. and Nezbeda, I., 1983a. Perturbed hard sphere equations of state of real liquids. I. Examination of a simple equation of the second order. Fluid Phase Equilibria, 12: 235-251. Aim, K. and Nezbeda, I., 1984. Effective hard cores of normal real fluids. submitted to Fluid Phase Equilibria. Andersen, H.C., Chandler, D. and Weeks, J.D., 1976. Roles of repulsive and attractive forces in liquids: the equilibrium theory of classical fluids. Adv. Chem. Phys., 14: 105-156. Barker, J.A. and Henderson, D., 1967. Perturbation theory and equation of state for fluids. II. A successful theory of liquids. J. Chem. Phys.. 47: 4714-4721. Bienkowski, P.R. and Chao, K.-C., 1975. Hard-cores of molecules of simple fluids. J. Chem. Phys., 62: 615-619. Bondi, A., 1964. Van der Waals volumes and radii. J. Phys. Chem., 68: 441-451. Boublik, T., Nezbeda, I. and Hlavatjr, K., 1980. Statistical Thermodynamics of Simple Liquids and Their Mixtures. Elsevier, Amsterdam. Carnahan, N.F. and Starling, K.E., 1969. Equation of state for nonattracting rigid spheres. J. Chem. Phys., 51: 635-636. Chandler, D. and Weeks, J.D., 1970. Equilibrium structure of simple liquids. Phys. Rev. Lett., 25: 149-152. Del Rio, F., 1981. Effective diameters and corresponding states of fluids. Mol. Phys., 42: 217-230. Del Rio, F. and Arzola, C., 1977. On the equation of state of argon. J. Phys. Chem., 81: 862-865. Del Rio, F. and de Longi, D.A., 1976. The mean distance of closest approach in the theory of liquids. Phys. Lett., 56A: 463~-464. Goodwin, R.D., 1974. The thermophysical properties of methane, from 90 to 500 K at pressures to 700 bar. Natl. Bur. Stand. (U.S.), Tech. Note, 653. Gosman, A.L., McCarty, R.D. and Hust, J.G., 1969. Thermodynamic properties of argon from the triple point to 300 K at pressures to 1000 atmospheres. Natl. Bur. Stand. (U.S.), Rep. NSRDS-NBS 27.
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Henderson, D., 1979. Practical calculations of the equation of state of fluids and fluid mixtures using perturbation theory and related theories. In: K.-C. Chao and R.L. Robinson, Jr. (Editors), Equations of State in Engineering and Research. Adv. Chem. Ser., 182, Am. Chem. Sot., Washington, DC.. Hlavaty, K., 1974. Semirational method for the construction of equations of state for liquids and gases. Collect. Czech. Chem. Commun., 39: 2927-2934. Juza, J. and Sifner, O., 1976. Modified equation of state and formulation of thermodynamic properties of krypton in a canonical form in the range from 120 to 423 K and 0 to 300 MPa. Acta Tech. CSAV, 21: l-32. J&a, J. and Sifner, O., 1977. Modified equation of state and formulation of thermodynamic properties of xenon in the range from 161.36 to 800 K and 0 to 350 MPa. Acta Tech. CSAV, 22: l-32. Labik, S. and Malijevsky, A., 1981. Monte Carlo simulations of the radial distribution function of fluid of hard spheres. Mol. Phys., 42: 739-744. Maitland, G.C., Rigby, M., Smith, E.B. and Wakeham, W.A., 1981. Intermolecular Forces: Their Origin and Determination, Clarendon, Oxford. Mansoori, G.A. and Canfield, F.B., 1969. Variational approach to the equilibrium properties of simple liquids. I. J. Chem. Phys., 51: 4958-4972. Matthews, G.P. and Smith, E.B., 1976. An intermolecular potential energy function for methane. Mol. Phys., 32: 1719-1729. Rabinovich, V.A. (Editor), 1976. Teplofizicheskiie Svoistva Neona, Argona, Kriptona i Ksenona. Izdatel’stvo Standartov, Moscow (in Russian). Ree, F.H., 1980. Reexamination of van der Waals model for soft-core molecules. Physica, 104A: 197-209. Reed, T.M. and Gubbins, K.E., 1973. Applied Statistical Mechanics. McGraw-Hill, New York. Vargaftik, N.B., 1975. Tables on the Thermophysical Properties of Liquids and Gases. 2nd edn. Wiley, New York. Weeks, J.D., Chandler, D. and Andersen, H.C., 1971. The role of repulsive forces in determining the equilibrium structure of simple liquids. J. Chem. Phys., 54: 5237-5247.