F. del Rio and C. Arzola
862
On the Equation of State of Argon Fernando del Rjo“ Universidad Aut6noma Metropolitana, Apdo. 55-534, Iztapalapa 13, Mexico
and Carlos Arrola Universidad Naclonal Mayor de San Marcos, Lima, PerC (Received March 29, 1976; Revised Manuscript Recelved February 8, 1977) Publication costs asslsted by Universidad Aut6nom Metropolitana
The properties of the square-well model of a liquid, calculated using molecular dynamics by Alder et al., are used to analyze the equation of state of argon. From experimental PVT data at high temperatures one may then obtain values of the equation-of-stateparameters. These values are found in a self-consistent way, and show a reasonable temperature dependence.
I. Introduction The subject of the equation of state (ES) of simple fluids has continued to draw the attention of chemical physicists. Since publication of the results of Longuet-Higgins and Widom on the ES of argon,’ there have been several papers on the subject.2 The main reasons for this progress are (1) the theoretical basis furnished by the perturbation theory which has shown the adequacy of a hard-sphere system as the zeroth order contribution, and (2) the availability of closed forms for the pressure of a hardsphere system, e.g., those obtained from the scaled particle and Percus-Yevick t h e ~ r i e s or , ~ the Pads4 and Carnahan-Starling5 approximations to the computer simulated results. The mixture of these ingredients leads to an equation of state of the form
p = Po(& T ; 0 ) - 4 P , T)PZ (1) where Po is the pressure of a system of hard-spheres of diameter u at temperature T and density p, and a takes into account the effect of the attractive interaction among the molecules. The best choice for the function Po seems to be nowadays the Carnahan-Starling formula5 = P k ~+ ( ~+ y * - y 3 ) / ( 1 - y ) 3 (2) where k is the Boltzmann constant, y = bp, and b = ao3/6 is the volume of a hard sphere. An important point learned from the perturbation theory is that, in order to represent the properties of realistically soft molecules, b or u has to be taken as a temperature-dependent effective parameter. The precise form of b ( T ) depends on the pair interaction potential and the type of perturbation theory selected; e.g., for the Lennard-Jones 12-6 potential, the theories of Weeks-Chandler-Andersen,6 Barker-Henderson,’ and Mansoori-Canfield8lead to different functions of b(7‘). In some cases, in order to have a rapidly convergent perturbation expansion, b has been taken as a function of density and temperature. In contrast, the best choice for the attractive term a(p, T ) of the equation of state is not as well defined. Taking Zwanzig’s high-temperature expansiongas a basis, one can write
(3) where c is an energy parameter characteristic of the intermolecular interaction. From a thermodynamic point of view, one would like a closed-form expression for a(p, T ) which would depend on thermodynamic parameters The Journal of Physical Chemistry, Vol. 8 I , No. 9 I 1917
characteristic of the molecular species. A more ambitious program would contemplate the possibility of finding those parameters which could be physically interpreted as representing some kind of average of a molecular characteristic. With this aim in mind, the series given by eq 3 presents two main problems: its lack of uniqueness and the complicated expressions for higher-order terms. The lack of uniqueness of the series, which depends on the particular theory selected, stems from the separation of the intermolecular potential into a “repulsive” reference part and an “attractive” perturbations, and also from the related choice of the definition of the effective size b(T). Nevertheless, it is important to point out that Zwanzig’s high-temperature expansion, defined by eq 3, has been shown to provide a convergent representation of the thermodynamic data of Ar and Xe, if the terms a,(p) are directly fitted to the experimental results (Clippe and Evrard).lo The second difficulty, the rather complicated expression for the second and higher-order terms an(p),makes it necessary to introduce one of several available approximations. The simplest choice at hand, which corresponds to the classical van der Waals approximation, is to assume a constant a. and to neglect all higher-order terms. This simple choice has been widely studied in combination with different forms of P ( p ) in the equation of state.” Carnahan and Starling 15 have also considered the empirical Redlich-Kwong recipe13 and have shown that it improves the equation over the van der Waals case when used with a realistic Po(p),as their own eq 2. More recently, Kreglewski et al.14 used the Barker-Henderson” macroscopic compressibility approximation for a l ( p ) , and Renon and Ponce16have used the Percus-Yevick approximation for the evaluation of a. and al obtaining excellent results for the thermodynamic properties of methane at low and moderate densities. The purpose of this paper is to show that the PVT data of a simple substance may be used to obtain information about the effective size of b(T) and the attractive energy term a(p, T ) at high densities and temperatures. The procedure followed stems from, and is compatible with, the insight gained by the application of perturbation theory to a square-well fluid in the molecular dynamics study of Alder et a1.l’ This procedure could be regarded as an attempt to complement equivalent information obtained from low-density data, i.e., from virial coefficients. In section I, we introduce the procedure used to reduce the experimental PVT data. The results for b(T) and a(p, T ) are given in section 111. These resulta are discussed in
Equation of State of Argon
863
the last section, where the critical parameters are evaluated independently of any fitting at the critical point.
11. Evaluation of the Equation of State Parameters We thus consider the problem of the evaluation of b ( T ) and a(p, T). The most common criterium, already used by several authors, is to propose a particular form for a(p, T), usually density independent, and to fit the equation to the experimental data by a least-squares techniques." In this work we will use a different approach. We will first look a t the general behavior of a(p, T ) as predicted by the perturbation theory, and in particular choose a system of particles interacting with a hard-sphere plus square-well potential as a model to gain insight into the behavior of the terms an. To do so we will refer to the molecular dynamics study of these terms by Alder et aI.I7 The first term in the expansion of a(p, T ) for a perturbation potential 4(r) can be written as
-
/4
51 E
1
'L- -, 1.5
- - - - - -- - -bo,=? 2.5 3.0
2.0 Pr
Figure 1. Mean-field parameter a as a function of density for different values of the effective molecular volume b. The value b = 9.60 cm3/mol makes the mean-field parameter approximately constant over a wide density interval.
where =
b0=15.7
P) ~ ( r )
l/2.rd3r go@,
(5)
and go(r, p ) is the radial distribution function of the reference system. At high liquid densities the change of go(r, p ) with density amounts to a redistribution of particles a few diameters away from the central one. This behavior tends to make the integral in eq 5 approximately constant with density. This tendency will be particularly noticeable if 4(r) is a square-well potential, since then any change in the position of a neighboring particle within the well will leave 6 invariant. The constancy of 6 with changes in p for the square well potential, over limited density ranges, is apparent from the calculations of the first perturbative term in the Helmholtz free-energy e~pansion.'~Further, the constancy of a. is implied by that of 6. The density dependence of the second-order term to the pressure, al, is known from the molecular dynamics square well data of Alder et al.17 If one writes p 2 = alp2as in ref 17, one finds that p z is at most 3% of the first-order term p 1 = aop2in the whole density range up t o y N 0.38, just below the fluid-solid transition where p 2 rises steeply. As a consequence al is small and will decrease with p2. al and all higher order terms will be neglected in this work. Keeping this information in mind, the following conjecture may be tested: from the experimental PVT data of a simple substance and an equation of state such as eq 1and 2, it is possible to obtain values of b ( T ) and a(p, T ) which agree with the following requirements: (I) a(p, T ) is constant along an isotherm within a density range between the critical and the melting densities; (11) the effective size of b ( T ) is a monotonic decreasing function of T; and (111) a(p, T ) is a linear function of 1 / T along a high-density isochore. In order to ensure a reasonable hope for the conjecture to be true, one has to examine the high-density, hightemperature region of a simple substance. In this case, the terms n, n 1 2, can be assumed negligible and the leading term a0 constant. To test the conjecture one can determine b and a(p, T ) self-consistently at several constant temperatures by imposing requirement (I), and then testing if (11) and (111) are satisfied. Hence, we propose the following procedure for the evaluation of b(Q and a b , T ) at any constant temperature: from eq 1 we write
4 P , T)= CPOtP, T;b ) -PI P-2
(6)
with Po given by eq 2. Then, an experimental P, p, T point gives a value of a(p, T j once b is chosen. To satisfy (I) one has to fiid a value of b which makes a(p, T j constant along a high-density portion of an isotherm. The PVT data of argon at T = 4.47Tc were chosen18and the value of b was determined by iteration. This procedure was effective since an overestimation of the value of b leads to a predicted hard-sphere pressure Powhich is increasingly larger than the experimental pressure. This behavior is shown in Figure 1for b = 15.70 cm3/mol. On the other hand, an underestimation of b, as b = 7.50 cm3/mol in Figure 1, leads to values of a(p, T ) that decrease with density and may even become negative. Hence, the iteration to produce the desired value of b is rapidly convergent. One immediately notices the high sensitivity of the approach on the precise value of b. This sensitivity is physically obvious: at liquid densities, the molecules are quite close to each other, so that a small increase in size produces an enormous increment in pressure. Frisch et a1.l' noticed the same effect in one of the earlier applications to the ES of argon. This general behavior was pointed to many years ago by Alder and Dymond" and has been noticed by other authors.21 For T = 4.47Tc, the desired value of b is 9.60 cm3/mol, which gives a = 1.375 X lo6 cal cm6/mo12.An uncertainty in these values arises from two independent facts. First, because of the scattering of the experimental data themselves. This is noticeable in Figure 2 where a(p, T ) is lotted on a blown-up scale for values of b close to 9.60 cmB/mol. Secondly, from Figure 2, one is left with two options to refine requirement (I): one can look for a constant behavior of a(p, T ) over a wide density interval, as for b = 9.66, but with a definite increase at still higher densities, or instead, look for a vanishing slope of a(p, T ) at the highest possible density, This occurs at p = 2 . 8 for ~ ~b = 9.55. In general, the a vs. p curves for different values of b diverge increasingly when higher and higher densities are considered. Thus we chose the second criterium in order to obtain a more precige value of the parameters. The fact that a(p, T ) shows a definite increase after p = 2.8pc, also apparent from Figure 2, may be attributed to the high-density behavior of the second-order term a,. This increase appears at a value of the density that, for the selected value of b = 9.55, corresponds to y = 0.36, which is close to the value found from the square well data." We must stress the fact that the whole The Journal of Physical Chemistry, Vol. 81, No. 9 , 1977
884
F. del Rio and C. Arzola
1.6
I
T= 4.47T,
1
I
bo 9.66
-
-
bo=9. 56 b0=9.55 b,=9.54 2.0
2.5 9r
3.0
Figure 2. Detailed behavior of the mean-field parameter. In this work b was chosen such that a is constant at p = 2 . 8 5 ~ The ~ . same behavior is observed at lower temperatures.
2.04 2.47 3.80 4.47
1.0. 1.5 KVE
a,
lo6 atm cm6/mo12 1.654 f 1.563 f 1.388 f 1.355 i
0.011 0.014 0.015 0.017
b, cm3/mol 10.91 f 10.56 i 9.80 f 9.54 i
0.01 0.01 0.01 0.01
procedure is carried out at a given temperature value, but that it can be repeated at different temperatures.
111. Results The application of the procedure described in section I1 gives as a result the values of b and a listed in Table I at four temperatures. The value of b decreases monotonically with increasing temperature, in accordance with physical intuition, and fulfilling requirement (11). These values were fitted within their uncertainty to a temperature function of the form b( 2’) = Bo/Trm (7) with Bo = 12.30 cm3/mol and m = 0,1688. Equation 7 was suggested by Rowlinson’s perturbation treatment of repulsive particles,22and the agreement between the data of Table I and an equation such as eq 7 is an indication of the appropriateness of the values obtained at high temperatures. Equation 7 may be used to extrapolate b(T) to lower temperatures. Chosing for argon the LennardJones parameters t/k = 119.8 K and CT = 3.405 A, the values of the temperature-dependent hard-sphere diameters may be compared with those obtained from the Barker-Henderson and the Weeks-Chandler theories for the same case. This comparison is shown in Figure 3. It is apparent that b(T) predicts values of the right order of magnitude and with an adequate temperature dependence, but that corresponds to a model sphere which is softer than
2.2 V,, cm3/mol Tc, K P,, atm
0.333 147 449.0 83.7
-
those obtained from perturbation theories. This may be due to the particular form of eq 7, which agrees with a Rowlinson type of behavior for soft spheres with a l / r ” repulsion if n = 17.77, thus producing a very “hard” molecule at high temperatures, but much “softer” than the 12-6 Lennard-Jones molecules at low temperatures. Thus, the behavior in eq 7 should change radically for real molecules at low temperatures, where eq 7 predicts an infinite diameter. The values of a(p, 7‘)have been plotted in Figure 4 against the inverse reduced temperature. The linear behavior is in agreement with the last requirement (111). Thus we write
~(2’) = a0 + al*/T,
(8)
and fitting of the data produces a. = 1.156 X lo6 (atm cm6/mo12)and al* = 0.990 X lo6 (atm cm6/mo12). The value obtained for a. agrees within 5% with that obtained from the calculation of a. in eq 4 for a square well with Elk = 119.8 K and p = 2 . 8 5 ~Even ~ if this small departure should be considered as a coincidence, it stresses the fact that a. is of the physical order of magnitude that could be expected.
IV. Critical Parameters and Discussion One further point is concerned with the prediction of the critical properties P,,po and T,from eq 1,2, 7, and 8. Table I1 shows these values in comparison with those obtained from other equations of state. The empirical Redlich-Kwong equation has been included on account of its wide application.
TABLE 11: Critical Properties of Argon as Predicted by Several Equations of State‘ vdW RK CSRK CSvdWb 0.375 115 128.3 34.5
I
20
Flgure 3. Effective molecular diameter as a function of temperature. The points marked 0 are values obtained from Table I and eq 7. The curves correspond to a LJ 12-6 system according to: Weeks-Chandler-Andersen,’ Mansoorl-Canfleld’ both at a density corresponding to 2 . 8 5 ~-.-a~ ; and - - Barker-Henderson’ values with the potential cut at its minimum and at its zero values, respectively.
----
TABLE I Tr
I
I
0.5
0.316 115 503.5 113.8
0.359 73.2 163.2 65.8
CSvdWC
Exptd
0.359 94.3 200.9 62.7
0.292 75.2 150.7 48.0
a vdW, van der Waals; RK, Redlich-Kwong; CSRK, equation obtained from a Carnahan-Starling repulsive term and a RK Using temperature-independent attractive term; CSvdW, Carnahan-Starling equation with a van der Waals attractive term. Experimental values. values of a and b at T = 4.47TC. Using eq 7 and 8.
The Journal of Physical Chemistry, Vol. 81, No. 9 , 1977
Equation of State of Argon
885
shows deviations of 8,53,238, and 140% for Z, V,, T,, and P,, respectively. The prediction of the augmented van der Waals equation with a constant a calculated at T = 4.47TC has 23,3, 10,and 39% for Z, V,, T,,and P,, respectively. One may notice from the data in Table I1 that a
0.2
0.3
0.4
0.5
I/T, Flgure 4. Mean-field parameter a as a function of reduced temperature. The straight line corresponds to eq 8.
somewhat best prediction is obtained with the CSvdW equation and temperature-independent parameters. Even if the prediction of individual critical properties is not as good as that obtained from virial-coefficient determinations of a and b, the values shown indicate the general adequacy of the method followed in this work. Obviously, to obtain unambiguously good values of the critical properties, one should use a still-unknown equation of state predicting good values of 2,. Further, since the regularities in the PVT behavior of argon that have been made explicit in this work do not depend on the precise form of the intermolecular potential, the same behavior should be observed in other simple systems. Indeed, preliminary work indicates that such is the case for krypton, xenon, and nitr~gen.'~
References and Notes The evaluation of the parameters for the different equations of state was carried out by the same procedure explained in section 111. One may notice the gross inadequacy of the equations of state considered here in predicting the position of the critical point. The failure in obtaining a good value of 2, is seen to be unevenly distributed among the variables. This is particularly true for the empirical Redlich-Kwong equation, which predicts the best values of Z, but presents great deviations in the isolated variables T,, P,, and p,. We have shown that the PVT behavior of argon at high temperature and density agrees approximately with an augmented van der Waals model as defined for a hardsphere plus square-well potential. The agreement is sufficient for a self-consistent, boots-strap-like evaluation of the parameters. The values so obtained behave with changing temperature in a smooth manner in accordance with their physical interpretation. Nevertheless, the uncertainty introduced by the procedure is quite large and particularly responsible for too large a ratio of al/ao,when the best fit is obtained. This produces an excessive dependence of the attractive energy on the temperature. The procedure used here has the advantage that it allows for the prediction of the critical variables. In this respect, the equation of state used here deviates 24,31, and 30% from the critical volume, temperature, and pressure, respectively. As the deviation in 2, is 29% from the experimental value, this represents a uniform lack of agreement for each of the variables. This uniformity is not shown by the empirical RK equation of state, which
(1) H. C. Longuet-Higgins and B. Widom, Mol. Phys., 8, 849 (1964). (2) E. A. Guggenheim, Mol. Phys., 9, 43, 199 (1965); N. F. Carnahan and K. E. Starling, J. Chem. Phys., 53, 472, 600 (1970); Phys. Rev. A, 1, 1672 (1970); E. Wilhelm and R. Battino, J . Chem. Phys., 55, 4012 (1971); E. Wilhelm, bid., 58, 3558 (1973); 60, 3896 (1974). (3) H. Relss, H. L. frisch, and J. L. Lebowitz, J. Chem. Phys., 50, 2160 (1959); M. S. Wertheim, Phys. Rev. Lett., 10, E501 (1963). (4) F. H. Ree and W. 0. Hoover, J. Chem. Phys., 40, 939 (1964). (5) N. F. Carnahan and K. E. Starling, J. Chem. Phys., 51, 635 (1969). (6) J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys., 54, 5237 (1971). (7) J. A. Barker and D. Henderson, J. Chem. Phys., 47, 4714 (1969). (8) G. A. Mansoori and F. B. Canfield, J. Chem. Phys., 51, 4958, 4967, 5295 (1969); 53, 1618 (1970). (9) R. Zwanrig, J. Chem. Phys., 22, 1420 (1954). (10) P. Clippe and R. Evrard, Mol. Phys., 29, 645 (1975). (11) A. J. B. Crvlukshank, D. E. Goodwin, R. N. Mercer, and A. J. Terry, Presented at the 3rd International Conference on Chemical Thermodynamics, IUPAC, Baden, Austria, 3-973. (12) N. F. Carnahan and K. E. Starllng, AIChE J., 18, 1184 (1972). (13) 0. Redlich and J. N. S.Kwong, Chem. Rev., 44, 233 (1949). (14) A. Kreglewskl, R. C. Wilhoit, and B. Zwollnsky, J. Chem. Eng. Data, 18, 432 (1973). (15) J. A. Barker and D. Henderson, J. Chem. Phys., 47, 2856, 4714 (1967). (16) H. Renon and L. Ponce, prlvate communication. (17) B. J. Alder, D. A. Young, and M. A. Mark, J. Chem. Phys., 56, 3013 (1972). (18) A. Michels, J. M. Levek, and W. de Graaf, Physica, 24, 659 (1958); S. L. Robertson, S.E. Babb, Jr., and G. E. Scott, J. Chem. Phys., 50, 2160 (1969). (19) H. L. Frisch, J. L. Kate, E. Praestgaard, and J. L. Lebowitz, J. Phys. Chem., 70, 2016 (1966). (20) J. H. Dymond and B. Alder, J. Chem. Phys., 45, 2061 (1966). (21) J. H. Vera and J. M. Prausnitz, Can. J. Chem., 49, 2037 (1971). (22) J. S.Rowlinson, Mol. Phys., 8, 107 (1964). (23) F. del RIo and C. Arzola, presented at the 5th IUPAC Conference on Chemical Thermodynamics, Montpellier, France, Vol. 2, 1975, p 103.
The Journal of Physical Chemistry, Vol. 81,No. 9 , 1977
