Construction and application of physically based equations of state

Fluid Phase Equilibria, 72 (1992) 41-66

41

Elsevier Science Publishers B.V., Amsterdam

Construction of state

and application of physically based equations

Part I. Modification

of the BACK equation

Berthold Saager ‘, Riidiger Hennenberg

2 and Johann Fischer

Znstitutfiir Thermo- und Fluiddynamik, Ruhr-Universitiit, D- W4630 Bochum I (Germany)

(Received May 6, 1991; accepted in final form August 25, 1991)

ABSTRACT Saager, B., Hennenberg, R. and Fischer, J., 1992. Construction and application of physically based equations of state. Part I. Modification of the BACK equation. Fluid Phase Equilibria, 72: 41-66.

In this and a subsequent paper a a class of equations of state with four or five parameters is constructed on the assumption that the Helmholtz energy can be written as F = F, + FA FA the contribution due to the + FP~I3 where F, denotes the hard body contribution, attractive dispersion forces and FPol the contribution of either dipolar or quadrupolar interactions. The present paper concentrates on a reformulation of F, and FA. In comparison with the original BACK equation a different temperature dependence of the hard body size is used which closely follows WCA-type perturbation theory. Consequently, FA had to be redetermined by a simultaneous fit of Maxwell data, pressures, enthalpies of evaporation and isochoric heat capacities to the experimental data of ethane. This substance was chosen to extend the applicability of the equation to lower temperatures. As a by-product, a closed equation of state for ethane is obtained which, except near the critical point, is in good agreement with the NIST tables.

INTRODUCTION

The first equation of state (EOS) suggested by van der Waals in 1873 was based on the physical idea that the hard cores of the molecules move ’ Present address: Ruhrgas AG, Essen, Germany. * Present address: Th. Goldschmidt AG, Essen, Germany. a Saager, B. and Fischer, J., 1992. Construction and application of physically based equations of state. Part II. The dipolar and quadrupolar contributions to the Helmholtz energy. Fluid Phase Equilibria, 72: 67-88.

0378-3812/92/$05.00

0 1992 Elsevier Science Publishers B.V. All rights reserved

42

in a uniform background potential created by their long-range attractive forces. Consequently, the ansatz for the residual Helmholtz energy F was written as

(1)

F=F,+F, with F H,vdW ___ = -ln(l RT

- 45)

and FA,vdW

=

-ap

(3)

where p is the density and 5 the hard body packing fraction, which in the case of hard spheres of diameter d is related to the density via 5 = (V6)pd3. In the last few decades, after computer simulations had become feasible, these ideas were revived and improvements suggested. For a system of hard spheres quite accurate pressures were obtained from molecular dynamics (MD) simulations (Alder and Wainwright, 1960) which could successfully be correlated by Carnahan and Starling (1969) to yield for the free energy FH -=

RT

(45 - 35*) (1 - 6)’

(4)

In order to obtain the attractive force contribution FA, Alder et al. (1972) performed MD simulations for a square-well fluid. The results were fitted by a double polynomial F $

= c Dnm,AT-n[m n,m

(5)

with 24 coefficients Da,,,. Quite soon this concept for the construction of an equation of state was taken up by other authors who suggested modifications or generalizations. Examples are the perturbed hard chain theory (PHCT) of Beret and Prausnitz (19751, the BACK equation of Chen and Kreglewski (1977), the equation of Deiters (1981), the COR equation of Chien et al. (1983) and the equation of Moritz and Kohler (1984). However, the concept of van der Waals is also the basis of modern perturbation theories like BH (Barker and Henderson, 1967a,b) or WCA (Weeks et al., 1971; Andersen et al., 1971) and the extension of the latter to non-spherical molecules (Kohler et al., 1979; Fischer, 1980; Boublik, 1987). The WCA-type theories in particular work quite well and in combi-

43

nation with the Lennard-Jones or n-center Lennard-Jones potentials yield good results for real liquids (Fischer et al., 1984; Bohn et al., 1986; Lustig, 1986, 1987). At this point it is important to state that both the BH and WCA theories use a temperature-dependent hard-core size which according to our experience is crucial for representing caloric properties (Fischer et al., 1987). With regard to the above-mentioned equations of state, we note that only the BACK and Moritz-Kohler equation of state (MKEOS) assume a temperature dependence, each, however, in quite different ways. In the MKEOS it is based on the assumption that the attractive part of the free energy is strictly linear in the density, i.e. that eqn. (3) is rigorously fulfilled, at high densities. On the contrary in the BACK equation the temperature dependence of the hard body size is based on the BH prescription. A problem with the BACK equation arises, however, because a somewhat artificial two-step potential is used there. Comparison with the result of the WCA or the closely related Barker-Henderson-l (BHl) (Smith, 1973) prescription in combination with the LJ potential shows that the temperature dependence of the hard-body size is rather different at low temperatures. That in turn has a strong effect on the division of the Helmholtz energy into the hard body and the attractive force contribution according to eqn. (1). For argon at the triple point, for example, the BACK equation and the WCA/LJ-theory yield nearly the same value for the total value of F, but F, is larger by about l.ORT in the WCA/LJ treatment. As the latter works quite well for spherical and non-spherical non-polar molecules it might be supposed that a better performance of the BACK equation could be obtained if the temperature dependence of the hard-body size was modelled in closer agreement with WCA/LJ. This, as a consequence, would also require a reformulation of FA. Another shortcoming of the above-mentioned equations of state is that they were all constructed for non-polar fluids. Hence, if applied to polar fluids their performance is in general not very satisfactory. An improvement for these has already been attempted by Lee and Chao (1988) who added to the BACK equation an additional pressure contribution pn which accounts for the dipolar interaction. Whilst the basic idea of this augmented BACK equation is appealing the actual construction of pD is not so convincing, as pD was taken from an equation of state for water after subtraction of a hypothetical non-polar BACK contribution. The intermolecular interactions of water, however, are more complicated than a simple dipole-dipole interaction, which makes the properties of this substance so peculiar. Hence, the physical meaning of p,, is not so clear. From the above discussion we may see that there is quite a lot of room left for improving the van der Waals equation of state. It is also seen that

44

among the existing equations the BACK equation is the favourite candidate as a starting point for such improvements, which will be presented in this and a subsequent paper (Saager and Fischer, 1992). In the present paper we will model the temperature dependence of the hard-body size according to results from WCA-type perturbation theory. This, as already mentioned, requires a reformulation of FA which will be made by fitting to the experimental data of ethane. With the choice of ethane, we follow the most recent work of Kreglewski (1984). Compared with argon it has the advantage of a larger liquid range. It could also be treated successfully by WCA-type perturbation theory (Fischer et al., 1984; Bohn et al., 1986a), and with the two-center Lennard-Jones potential thus determined, thermodynamic properties were predicted with good accuracy for the whole fluid region by molecular dynamics (MD) simulations (Fischer et al., 1987; Lustig et al., 1989; Saager et al., 1990). An additional advantage of ethane is that it is a somewhat elongated linear molecule. As the shape dependence of FA is from a theoretical point of view not so clear as for F,, the effect of a slightly improper treatment then will be minimized by using a molecule of “medium” anisotropy as reference substance. The result of this paper will be a straightforward modification of the BACK equation where the attractive force term essentially accounts for the dispersion forces F MOBACK

= FH

+ FA

(6)

This equation is called MOBACK and is thought to be applicable to fluids of non-polar molecules down to T/T, = 0.30. In the second paper of this series (Saager and Fischer, 1992) we present an expression for the additional dipolar contribution F, to the Helmholtz energy. Contrary to Chao’s approach (Lee and Chao, 1988) this expression is not derived from experimental data, but was obtained via extensive molecular dynamics simulations for two-center Lennard-Jones molecules with point dipoles of five different reduced dipole moments p* at 29 state points in a temperature and density grid. A correlation equation for F, will be given as a function of T, p and p*, the partial derivatives of which are fully consistent with the internal energies, pressures and dipole-dipole energies from the simulations. Adding F, to FMOBACKyields an equation of state with five parameters for dipolar molecules F DIBACK

= FFi + FA + FD

(7)

called DIBACK. Similarly, the quadrupolar contribution FQ was constructed yielding an equation of state with five parameters for quadrupolar molecules F QUABACK

= FH

+ FA + FQ

called QUABACK.

(8)

45

In a subsequent paper, the original BACK equation as well as its modifications and extensions MOBACK, DIBACK and QUABACK will be applied competitively to the polar refrigerants R22 and R152a in order to correlate a large number of experimental data. Moreover, the DIBACK equation will also be used to predict thermodynamic data for these two substances using only three experimental vapor pressures and bubble densities in order to fit the five parameters of the equation. (For preliminary results see Saager (1990)). The present paper is organized such that in the next section results from WCA-type perturbation theory will be presented and discussed. In the two subsequent sections the contributions F, and FA to the equation of state will be constructed. Firstly, the hard body contribution F, will be fixed. Then, the attractive dispersion force contribution FA will be derived by fitting to the properly reduced experimental data of ethane. Finally, the quality of the equation of state for ethane, which was obtained as a byproduct, will be discussed. RESULTS FROM WCA-TYPE PERTURBATION THEORY

As WCA-type perturbation theory (PT) turned out to be quite reliable, at least at higher liquid densities, and could also be extended to nonspherical molecules, we decided to inspect more closely the result for F, and FA obtained by one of the authors (Fischer, 1980). Special points of interest are the temperature dependence of the hard body volume, the general behaviour of FA and the relative contributions of F, and FA to F. Calculations with PT were performed for a Lennard-Jones (LJ> fluid and a two-center Lennard-Jones (2CLJ) fluid with molecules of elongation L = l/a = 0.505 in a temperature and density grid at corresponding state points. These are defined with respect to the pseudocritical temperature T, and the pseudocritical density pp which are the critical values obtained from PT (Fischer et al., 1984). In these previous investigations we obtained for the LJ fluid kT,/e = 1.458 and pp~3 = 0.3128 and for the 2CLJ-0.505 fluid kT,/e = 3.078 and ppc3 = 0.2028. The hard body contribution

For the models considered, the hard-body free energy F, depends only on the hard sphere or hard dumb-bell sphere diameter d. Keeping the centre-centre distance, 1, fixed, one has according to the prescription given by Boublik and Nezbeda (1986) L”=;

(9)

46

TABLE 1 Reduced hard sphere diameters d/u for the ~nnard-Jones WCA-type perturbation theory in a grid of T/T, and p/pr T

BHl

WCA (P/P,) 1.0

r, 2.0 1.5 1.0 0.7 0.5 0.3

fhrid according to BHl and

0.9720 0.9845 1.0012 1.0149 1.0268 1.0431

1.0023

1.9

2.3

2.7

3.1

0.9720 0.9844 1.0010 1.0145

0.9711 0.9836 1.0002 1.0138 1.0257

0.9701 0.9826 0.9993 1.0129 1.0249

0.9689 0.9814 0.9981 1.0118 1.0239 1.0405

(1 + bIN2 +&I) *=

fk -Z RT

01)

(2+3&-I;-:,)

(a” - 1) ln(l -5) f

(cy” + 3cr)f - 3acfj2 (1 - Q2

Note, that for spherical molecules the anisotropy parameter Q becomes 1 and hence eqn. (12) reduces to the Carnahan-Starling equation (eqn. (4)). Values of d obtained from our WCA-type theory are compiled in Table 1 for the LJ fluid and in Table 2 for the 2CLJ-0.505 fluid in the temperature and density range 0.3 < T/T, s 2, 1.9 s p/pP I 3.1 and at the pseudoTABLE 2 Reduced hard dumb-bell sphere diameters d/u for the two-center Lennard-Jones fluid of etongation L = 0.505 according to WCA-type perturbation theory in a grid of T/T, and P/Pp T -5

2.0 1.5 1.0 0.7 0.5 0.3

WCA (P/P,)

1.0

0.9626

1.9

2.3

2.7

3.1

0.9339 0.9460 0.9621 0.9753

0.9336 0.9458 0.9619 0.9751 0.9866

0.9333 0.9455 0.9617 0.9749 0.9864

0.9330 0.9452 0.9614 0.9747 0.9862 1.0021

47

critical point. Table 1 also contains the hard sphere diameters according to a modification of the Barker-Henderson (1967a,b) theory called BHl or hBH theory (Smith, 1973)

dBHl=/oz”6”(1 - exp{ -P[uU(r)

+ E]}) dr

(13)

Table 1 confirms the well-known fact that the hard sphere diameters from WCA-type theory are somewhat density-dependent for the LJ fluid but also indicates that they become equal to the BHl diameters at some medium density. From Table 2 we learn that for the 2CLJ molecules the density dependence of the hard sphere diameter is considerally smaller. In addition, we see by comparing Tables 1 and 2 that the hard sphere diameters d of the 2CLJ molecules are smaller than those of the LJ molecules, which has already been discussed previously (Bohn et al., 1986b). One may, however, get a more universal behavior by reduction of the diameters to that at the pseudocritical point, i.e. by forming d/d, which yields for the linear molecules values in much closer agreement with the spherical molecules. In constructing an equation of state, one is according to eqn. (12) ultimately interested in the packing fraction 5 into which d enters directly by its third power but also indirectly via L, according to eqn. (9). As previously d/d, shows little dependence on the elongation, one expects the same for t/t, and hence it is rewarding to look at ([/[,)/(p/p,) which is given in Tables 3 and 4 again for the LJ and the 2CLJ-0.505 fluid. An additional reason for scaling with respect to 5, is the fact that the packing fraction at the pseudocritical point is almost constant (Bohn et al., 1986b). From Tables 3 and 4 we learn that the differences in the quantity ([/[,)/(p/p,) between the spherical and the linear molecules are not

TABLE

3

Values of (,$/.$,)(p,/p) for the Lennard-Jones fluid as obtained from BHl and WCA-type perturbation theory. At the pseudocritical point the value for (d/c) is 1.0012 for BHl and 1.0023 for WCA (Bohn et al., 1986b) T

BHl

T, 2.0 1.5 1.0 0.7 0.5 0.3

0.9148 0.9508 1 .oooo 1.0414 1.0785 1.1306

WCA (P/P,) 1.9

2.3

2.7

3.1

0.9120 0.9474 0.9961 1.0370

0.9095 0.9451 0.9937 1.0348 1.0717

0.9067 0.9422 0.9910 1.0321 1.0692

0.9033 0.9387 0.9875 1.0287 1.0661 1.1188

48 TABLE

4

Values of ([/[,Kp,/p) for the two-center Lennard-Jones fluid of elongation L = 0.505 as obtained from WCA-type perturbation theory. At the pseudocritical point the value for d/a is 0.9626 (Bohn et al., 1986b) T

WCA (P/P,,)

T

1.9

2.3

2.7

3.1

2.0 1.5 1.0 0.7 0.5 0.3

0.9226 0.9549 0.9989 1.0357

0.9218 0.9542 0.9983 1.0351 1.0680

0.9209 0.9534 0.9976 1.0346 1.0675

0.9201 0.9526 0.9969 1.0340 1.0670 1.1132

large especially at the lower temperatures. What still remains is the temperature and density dependence. Regarding the density dependence, it is, according to the above observations, larger for the spherical molecules and of the same order of magnitude as the difference between the spherical and the linear molecules. The essential dependence of ((/,!j,)/ (p/p,) is that on the temperature which is an order of magnitude larger than that on the density and the elongation. Hence, as an approximation for the construction of an equation of state we will assume (t/[,)/(p/p,) to be only a function of the temperature. A detailed description will be given in the section on the hard body contribution to the equation of state. The attractive force contribution The attractive dispersion force contribution FA has received less attention in the literature than the hard body contribution F,. The main efforts seem to have been concentrated on the square-well potential (Alder et al., 1972; Ponce and Renon, 1976; Benavides and de1 Rio, 1989). In the liquid, van der Waals expected FA to be proportional to the density as expressed in eqn. (3): Therefore it is interesting to consider F,/p or the dimensionless quantity FA = (p,/p>( F,/RT,). Results obtained from WCA-type perturbation theory for the LJ fluid and the 2CLJ-0.505 fluid at higher densities are given in Tables 5 and 6. In addition, we wanted to get a certain understanding of the behavior of FA at gaseous densities. As WCA-type perturbation theory cannot be applied there reliably, we decided to compute FA at low densities via the Haar-Shenker-Kohler equation (Fischer and Bohn, 1986; Saager et al., 1990)

(14)

49

TABLE 5 Reduced attractive dispersion force contributions F. = (pP /p)(F’ /RT,) for the LennardJones fluid in a grid of T/T, and p/pP. At zero density, the values are from the differences in the second virial coefficients, B - B,, and at liquid densities from WCA-type perturbation theory T T,

P p,

2.0 1.5 1.0 0.7 0.5 0.3

0.0

1.9

2.3

2.1

3.1

-

-

-

-

-

1.390 1.420 1.490 1.603 1.792 2.464

1.580 1.583 1.586 1.590

1.623 1.629 1.637 1.645 1.653

1.655 1.663 1.675 1.686 1.697

1.674 1.683 1.697 1.709 1.721 1.740

where B is the second virial coefficient of the realistic system and B, that of the hard body system; details of the application and the results of this equation have been given in the above references. For the moment it is important that it holds reasonably well from zero density up to $p, and that & = (T/T,)(B - BH)pp, values of which have also been included in Tables 5 and 6 in the column p/pp = 0.0. By inspection of Tables 5 and 6 we observe that FA/p decreases with increasing density up to the very high density p/p, = 3.1 for both the LJ

TABLE 6 Reduced attractive dispersion force contributions $a = (pp /pxF, /RT,) for the LennardJones fluid of elongation 0.505 in a grid of T/ Tp and p/pP. At zero density, the values are from the differences in the second virial coefficients, B - B,, and at liquid densities from WCA-type perturbation theory T

T, 2.0 1.5 1.0 0.7 0.5 0.3

-

P

PP 0.0

1.9

-

-

1.417 1.454 1.548 1.705 1.987 3.161

1.663 1.664 1.667 1.668

2.3 -

1.724 1.730 1.737 1.743 1.748

2.7 -

1.776 1.784 1.796 1.806 1.814

3.1 -

1.817 1.828 1.843 1.856 1.867 1.884

50

and the 2CLJ fluid. Obviously there is a difference between these two fluids with realistic potentials in one case and the square-well potential in the other, for which F,/p had been found to pass through a minimum (Alder et al., 1972). This fact was pointed out some time ago by Kohler (1977) and makes the use of the square-well F, in equations of state for real fluids somewhat questionable. Regarding the low density, p = 0, we note that at the higher temperatures FA is above the high density values and can nearly be obtained from them by linear hextrapolation. With decreasing temperature, however, the low density FA falls considerably below a linear extrapolation from high densities. Regarding the effect of the mo)ecular shape, an important finding from Tables 5 and 6 is the fact that FA becomes more negative for the linear molecules than for the spherical ones. The relative contributions of FH and FA to F

It is also rewarding to compare the contributions of F,/RT and F,/RT to the sum F/RT = F,/RT + F,/RT. A crucial difference between both contributions is their dependence on temperature. Neglecting for the moment the temperature dependence of the hard body diameter, it is the combination F,/RT which is constant, whilst the attractive force contribution FA by itself is nearly temperature independent, which means that F,/RT is inversely proportional to T. This implies that the relative contribution of F,/RT is larger at low temperatures and decreases with increasing temperature. Regarding now the effect of the molecular shape, we have already noted above that at corresponding state points F,/RT becomes more negative with increasing molecular anisotropy. However, as can be verified from Tables 1 and 2 in combination with eqn. (121, F,/RT becomes more positive with increasing molecular anisotropy. As a further illustration, we have compiled in Table 7 the contributions F,/RT and F,/RT for the LJ fluid and the 2CLJ-0.505 fluid at some corresponding state points. Finally, in order to understand the effect of the anisotropy on the total free energy we have to combine the two observations from above. At high temperatures, the quantity F,/RT is small as it scales with l/T, and so is its decrease with elongation. Consequently, it cannot compensate for the increase of F,/RT with elongation and hence the total F/RT increases with increasing anisotropy at high temperatures. At low temperatures the situation is just the opposite, as F,/RT takes on a large negative value and also its decrease with elongation becomes relatively large. Hence the total F/RT decreases with increasing anisotropy at low temperatures. These considerations are illustrated in Table 7.

51 HARD

BODY

CONTRIBUTION

TO THE EQUATION

OF STATE

The hard body contribution to the class of physically based equations of state will be given by eqn. (12) which is repeated here for convenience &I RT = (a2 - 1) ln(1 -S)

(a3 + 3a)5 - 3a52 +

(12)

(1 - Q2

It requires the anisotropy parameter (Yand the packing fraction 5, which according to WCA-type perturbation theory should both be state dependent. For convenience we assume that in the following (Yis a constant for a given substance and is one of the parameters to be determined by fitting to the experimental data. Regarding 5 we have found by PT that the quantity (,$/,$,)/(p/p,) depends essentially on temperature. As the WCA-type PT values also depend somewhat on density, there is some arbitrariness in the choice of that temperature dependence. In order to find a simple and easily reproducible prescription, we decided to neglect in the EOS any density dependence of (t/[,)/(p/p,) and to formulate the temperature dependence according to BHl. This choice was already made by Nezbeda and Aim (1984) but we also believe that the reason for it is now clearer, especially with respect to non-spherical molecules on the basis of the results presented in the previous section, Before proceeding to the equation for 5 we change for reasons of simplicity from the pseudocritical constants pp and T, to the parameters p,, and To, which in the construction of the EOS for non-polar substances are identified with the critical density pc and the critical temperature T,. As was shown earlier (Fischer et al., 1984) pp is for non-polar substances

TABLE

7

Hard body contributions F, /RT, attractive dispersion force contributions FA /RT and total residual free energies F/RT for the Lennard-Jones fluid and the two-center Lennard-Jones fluid of elongation 0.505 at corresponding state points according to WCAtype perturbation theory T

P

T,

p,

2.0 1.5 1.0 0.7 0.5 0.3

1.9 1.9 1.9 2.3 2.7 3.1

LJ

2CLJ-0.505

F,

F*

F

FH

FA

F

RT

RT

RT

RT

RT

RT

1.76 1.87 2.02 3.00 4.46 7.13

- 1.50 - 2.00 - 3.01 -5.41 -9.17 - 17.98

0.26 -0.14 - 0.99 - 2.40 - 4.70 - 10.85

1.97 2.07 2.22 3.28 4.85 7.65

- 1.58 -2.11 -3.17 - 5.73 - 9.80 - 19.46

0.39 0.04 0.95 2.45 4.94 - 11.81 -

52

within the experimental uncertainty equal to the critical density pc whilst T, turned out to be always higher than T, by about 15%. For the following, if we do not speak about a specific real substance, we fix p0 = pP and To = T,/1.15 so as to have clearly defined relations. After this change in variables we are looking now for the expression ([/[O>/(p/pO), which for a system of spherical particles is given by

d3(T)

(15)

w5o)/(P/Po) = d”o

where the density dependence of the hard sphere diameters has been neglected according to our above assumption. For the following we assume a LJ system for which kT,,/e = 1.458 and hence we get by the above convention kTO/e = 1.268. The values dBHl at any temperature can then be obtained easily by numerical evaluation of eqn. (13). The results can be correlated with good accuracy by the expression

~5/5o)/W~,) =

[a+Cl- W’/7dY]-’

(16)

with a = 0.689 and y = 0.3674. Finally, we still have to specify to. One should note that in the application of the EOSs to be constructed here, to is not explicitly needed in the non-polar parts as it always occurs there in the combination pO/SO. In the polar contributions to the Helmholtz energy, however, the situation is different as p0 occurs there by itself and hence to has to be specified. Again, there is some arbitrariness in the choice of to. We could either think of taking 5, = 0.165 which is an average value for the small elongations (Bohn et al., 1986b) and then obtain with T,/T, = 1.15 and pP =p,, the values to = 0.1677. Alternatively, one could take, for example, the values for ethane pP/pO = 0.982, T,/T, = 1.161, 5, = 0.1561 (Fischer et al., 1984; Bohn et al., 1986b), from which one obtains to = 0.1617. In view of the fact that the expression for the attractive dispersion forces will be fitted to ethane, we decided to take the latter value. Hence, the complete expression to be used in the following is 5 = O.l617(p/p,)[

a + (1 - a)(T/T,JY]

-I

(17)

Summarizing, we state that the hard body contribution to the Helmholtz energy will be given by eqn. (12) with 5 as given in eqn. (17). In applications, pO, T,, and cx are parameters to be obtained by fitting to experimental data, whilst a and y are two constants that were given above. At this point it may be interesting to compare the temperature dependence of the packing fraction 5 given in eqn. (17) with that of the original BACK equation (Chen and Kreglewski, 1977), the Nezbeda-Aim prescrip-

53

tion (1984) and the MKEOS (Moritz and Kohler, 1984). In the original BACK equation 5 is given by 5 BACK = 0.74048( Voo/u)[ 1 - C exp( - 3u”/( kT))13 (18) For the case of spherical molecules, one has to assume u” = kTo and with C = 0.12 one gets [(~/~O)/b/PO)lBACK=

1.0181[1 - C exp( -3T,/T)]’

(19)

Nezbeda and Aim (1984) arrived at the correlation exp[ -0.03125

ln(T/T,)

- 0.00540 ln*(T/T,)]

(20) where T, is the triple point temperature. Assuming the values for argon, Tt = 83.80 K and To = 150.86 K, one obtains [d(T)/d(T,)].A=

1.0626 exp[ - 0.09375 ln( @T/T,)

[@/~O)/(dpO)]NA=

-0.0162

ln*(@T/T,)]

(21) with 0 = 1.8002. For the MKEOS, no explicit expression is given in the original paper (Moritz and Kohler, 1984), but from a recent publication (Iglesias-Silva and Kohler, 1990) one may derive [(5/50)/(P/P0)1MK

= 1.2209( [ 1 + k,(T,‘To)k’] + [ 1 + k3(T,To)k”]

-N

-N)1’N

(22)

12-

a

wPO

1. \

-

BH1

----

Wezbedo- Alrn

-o-

11 -

-

09. 0

05

10

BACK -

HYEOS

15

TIT0

20

Fig. 1. Comparison of the reduced hard body packing fraction ([/.$O)/(p/pO) as a function present work (BHl); - of the reduced temperature (T/ T,) for four approaches: Nczbeda-Aim (1984); .- .-. MKEOS (Iglesias-Silva and Kohler, 1990); -o- BACK (Chen and Kreglewski, 1977).

54

with k, = 0.23833208, k, = 1.2849269, k, = 0.23760582, k, = 0.51973232 and N = 50. The values for ([/[O>/(p/pO> of all approaches are compared in Fig. 1 as a function of the temperature. As is seen from Fig. l., the results of the present approach agree very well with those of Nezbeda and Aim (19841, as was to be expected, since both use the BHl prescription for the LJ system. The slight difference at low temperatures is presumably caused by the fact that the correlation eqn. (20) is not very accurate there. Compared with these results, the BACK prescription shows a considerably weaker temperature dependence especially at lower temperatures. This is obviously caused by the assumption of the step-square-well potential which we do not believe to be a suitable model. Finally, the MKEOS yields a stronger temperature dependence, which results from the requirement that the attractive part of the free energy should be strictly linear in the density at high densities.

ATTRACTIVE DISPERSION CONTRIBUTION TO THE EQUATION OF STATE

In dealing with the attractive dispersion contribution FA to the Helmholtz energy we encounter two problems. The first is to arrive at a closed mathematical form for FA as a function of p and T for some geometrical shape of the molecules and the second is to take account of different shapes. The situation here is considerably more difficult than for the hard body contribution where eqn. (12) already yields the essential expression which has to be supplemented only by the temperature dependence of the hard body volume. The first step in the solution of these problems by Chen and Kreglewski (1977) was to fit F,/RT by a polynomial expression in the variables 5 and T/T,, to the experimental values of argon. Then the dependence of F,/RT on the molecular shape was taken into account by resealing the temperature such that T/T, is replaced by (T/TJ(l + q/kT)-’ which follows a previous result of Rowlinson (1969). In a later version of the BACK equation (Kreglewski, 1984) F,/RT was fitted to the experimental data of ethane, but that formulation was rarely used. To begin with we decided to follow the route of Kreglewski. Hence, what is required here as a consequence of the change in the temperature dependence of the hard body volume is a new fit of FJRT in the variables 6 and T/T, to experimental data. Like Kreglewski (1984) in his second study, we are also fitting to ethane for two reasons: the first is that the liquid range of ethane goes down to rather low reduced temperatures; and the second is that as the dependence of FA on the molecular shape is not yet fully understood a molecule of “medium” anisotropy offers some

55

advantages. The form of F,/RT will again be a double polynomial in 5 and T/T, with 24 terms using the same powers as Chen and Kreglewski. Regarding the parameter 7, we simply put it equal to zero in obtaining F,/RT from ethane, but in applying the equation to other substances the transformation T/T,, + (T/T&l + ~/IcT>-~ will again be used and may then lead to negative values of n for molecules which are less anisotropic than ethane. We should also mention that our fitting technique is more refined than that used by Kreglewski and will be described below in more detail. The temperature resealing by (T/T,)(l + q/kT)-l means that the F,/RT values for non-spherical molecules should be obtained from the values for spherical molecules at the same packing fraction but at the resealed temperature. Looking now at Tables 5 and 6 we see that such a transformation yields qualitatively correct results. Closer inspection of these tables suggest that at least in the liquid range a transformation of the type (T/T,) + (T/T&l + yp) is worth investigation at a later stage. Before proceeding further, it may be helpful to repeat the full set of equations which constitute MOBACK. The residual free energy is given by F MOBACK

= F”

(6)

+ FA

with

FH

RT = (CX’- 1) ln(1 - 6) +

(ay” + 3a)5 - 3at2 (1 -t)’

(12)

and

(23) where -1

5 = 0.1617(~/‘~,)[ a + (I-

a)(T/T,#]

(17)

and T = (T/T,,)(l

+ q/kT)-’

(24)

Now in order to determine FA by fitting to the experimental data of ethane, we have to form FMOBACK- F, and to specify the parameters pO, T,, and (Yappearing in F, for ethane. According to our convention, p0 and To have to be identified in the construction of the equation with the critical quantities pc and T,; from the experimental data of ethane it follows that p,, = 6.8 mol dm -3 and To = 305.33 K. The value of CI is fixed

by taking the real critical point as reference

point and using results from

56

perturbation theory (Fischer et al., 1984; Bohn et al., 1986b). It was shown that ethane can be described as 2CLJ with L = Z/o = 0.67 yielding a pseudocritical temperature T,, = 1.161 T, with the hard dumb-bell sphere diameter being at the pseudocritical point d, = 0.9706~. Using eqn. (16) we therefore obtain at the critical point d, = 0.97620, and consequently L H,c = l/d, = 0.6863. Hence, according to eqn. (11) we find (Y= 1.2126. Having fixed the parameters pa, TO and (Y,FH is completely determined for ethane. Consequently, we can separate for any thermodynamic quantity that is linear in F or its derivatives an attractive part which can be obtained from experimental data; the “attractive pressure”, for example, is PA =Pexp --PH. In the following, we will use the attractive parts of pressures, enthalpies of vaporizations, Maxwell data and isochoric heat capacities to fit FA in a functional form given by eqn. (23). The selection and treatment of the data as well as the result of FA and its discussion will be presented in the next two subsections. Selection and treatment of the ethane data

In order to obtain an optimized expression for F,/RT for ethane in the form of eqn. (23) we used the following data and global weights G: 970 pvT data from the homogeneous region, G = 1; 100 phase equilibrium or Maxwell data, G = 10; 197 isochoric heat capacities, G = 5; and 100 enthalpies of vaporization, G = 10. The different types of data are weighted by the G-values which were chosen such that the products of G and the number of data are nearly the same. The function to be minimized is the weighted sum STA2 of the squared standard deviations STA, of the data sets of type (Y STA’ = c G,M,STA2,

(25)

a

with

where the x1? denote the values of type (Y, the Axp the combined experimental uncertainties resulting from Gauss’ law of error propagation, and M, the number of data of type CLAll the input data were taken from a compilation of Goodwin et al. (1976). In total, 970 experimental pvT data were used and for the estimation of the Api only the uncertainties in the measured densities were taken into

57

TC

100

0

10

rl

PC

x \ II

1

+ ++

+

PS

: O.lL

/

’

’ 50

too

I’

ISO zoo

’

’

’

’

’

’

’

’

’

250

300

350

400

450

500

550

600

650

T

/

700

K

Fig. 2. p-T diagram showing the distribution of the ethane puT data included in the fit: IJ Michels et al. (1954); A, Pal (1971); o, Douslin et al. (1973); f, Straty et al. (1976).

,

account. It was assumed that the relative uncertainties Ap/p in the densities were as follows: for the 101 data of Michels et al. (1954) 0.37%; for the 298 data of Douslin and Harrison (1973) 0.43%; for the 141 data of Pal (1971) 0.11%; and for the 430 data of Straty and Tsumura (1976) 0.09%. The distribution of the data points in a p-T diagram is shown in Fig. 2. The phase equilibrium or Maxwell data are defined as R = P,(

l/p”

-

VP’)

(27)

where ps denotes the vapour pressure, p” the dew-point and p’ the bubble-point density, all at the same temperature. Because of the equality of the Gibbs energy at phase equilibrium, R is also equal to R = F(T,

p’) - F(T,

p”) + RT In p”/p’

(28)

According to Wagner (1972) use of R in the fitting procedure greatly improves the representation of the coexistence curve. In that process the experimental data should be taken for p,, p” and p’, and for F the ansatz to be optimized. In practice few of the required experimental data are available for the same temperature, and so we used the correlation equations for ps, p” and p’ given by Goodwin et al. (1976). From these, 100 values of R were calculated in the temperature range from 107 to 305 K. The uncertainties ARi were taken as the combined uncertainties of all quantities, with ApJp, = 0.001, Ap’/p’ = 0.0011 for p’ > 17.5 mol dmp3, Ap’/p’ = 0.0025 for p’ < 17.5 mol dmp3 and Ap”/p” = 0.002.

58

Also included in the fitting procedure were 197 isochoric heat capacity data c, measured by Roder (1976). As it was not so easy to estimate their uncertainty, we simply took the deviation between the experimental data and the correlated values of Goodwin et al. (1976) as a measure for Ac,i if these were larger than 1%; otherwise we fixed them to be 1%. Finally, for the enthalpies.of vaporization we took again the values from a correlation of Goodwin et al. (1976) and assigned to all values an uncertainty of 2%. The enthalpies were taken at just the same 100 temperatures as the Maxwell data. Result for FA

From the data listed above the hard body parts resulting from F, according to eqn. (12) with the auxiliary eqn. (17) and p0 = 6.8 mol dmV3, 7’, = 305.33 K and (Y= 1.2126 were subtracted. The “attractive parts” were fitted by a polynomial for F,/RT according to eqn. (23) with the auxiliary eqns. (17) and (24) and 17= 0. The objective function was STA2 according to eqn. (25). The resulting coefficients D,, of eqn. (23) are presented in Table 8. With these coefficients, the standard deviations defined by eqn. (26) are 0.71 for the puT data, 0.14 for the Maxwell data, 0.81 for the isochoric heat capacities and 0.35 for the enthalpies of vaporization. At this point it is interesting to compare the above defined FA term with the expression. from the original BACK equation. As in a previous section we use as a convenient reduced form FA = (p,/p)/(F,/RT,). This quantity is shown for the BACK equation in Fig. 3 and for the new MOBACK expression in Fig. 4 for several isotherms as a function of the density. The left-hand side of these figures gives an overall view; the right-hand side focuses on the high densities. For a fair comparison, both figures include the same reduced temperatures and the same range of reduced densities. An essential difference occurs at the high densities, where the isotherms in BACK are crossing whilst MOBACK yields a clear ordering of the isotherms. This difference is obviously caused by the different temperature dependence of the hard body volume assumed in BACK and MOBACK. This ordering of the isotherms seems to be important with respect to the temperature resealing by (T/T,,)(l + ~//CT)-’ in applying the equation to other substances. In Fig. 5 the MOBACK result for FA is shown for the full temperature and density range where the results are thought to be applicable. At the lowest temperatures, i.e. for T/T, = 0.30 and 0.35, the ordering of the isotherms no longer holds at medium densities. However, as the figure shows, these parts of the isotherms are far inside the two-phase region of ethane and it does not seem very likely that one arrives there by the temperature resealing. At the gas side, the isotherms are still well

59 TABLE 8 The 24 coefficients D,,, occuring in the expression for the attractive dispersion contribution FA /RT = CD,,,,,-“y’ obtained from a fit to ethane n

m

D

1 1 1 1 1 1 2 2 2 2 2 2 2

1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 1 2 3 4 5 1 2 3 4

-“tG63031455509 X 102 0.236050623433 x 10’ - 0.202780263621 x lo3 0.708094241795 x lo3 - 0.111419155539 x 104 0.617745238907 x lo3 0.116598345999~ 10’ -0.198345701422 x lo* - 0.123560229535 x lo2 0.512656570463 x lo3 - 0.402302186654 x lo4 0.137248475700 x lo5 - 0.233748173692 x lo5 0.203815498538 x 10’ -0.763817381071 x lo4 - 0.930674564526 x 10’ 0.112366881230 x lo* 0.105838007455 x lo3 - 0.330441036286 x lo3 0.271037994941 x lo3 0.177215789449 x 10’ - 0.768101660142 x 10’ 0.149849547238 x lo* -0.116361036713x lo*

2 3 3 3 3 3 4 4 4 4

ordered except at the two lowest temperatures; this is caused by the fact that the gas data were fitted only down to T/T, = 0.35. However, this has no practical consequences, as at these low densities the contribution of the residual Helmholtz energy to the thermodynamic properties is negligible in comparison to the ideal gas part. Finally, we compare the FA/p function of MOBACK with the results from perturbation theory which we;e shown in Tables 5 and 6. At the highest densities the PT results for FA show a dependence on the temperature which is comparable to MOBACK. With decreasing-density, however, the FA isotherms from PT are converging whilst the FA isotherms from MOBACK are nearly parallel straight lines. We admit that we do not yet understand this discrepancy. In any case, the MOBACK values are still further away from the square-well behavior of FA than the PT results; this yields an additional justification for fitting directly to the experimental data.

60 BACK, origlnai

I

I

0.5

0

1.0

1.5

2.0

2.5

2.00

3.0

2.25

2.50

2.75

Ph

P&k--

3.00

-

Fig. 3. Reduced attractive dispersion force contribution &=((p,/p)/(FA /RT,) as obtained in the original BACK equation by fitting to argon. The left-hand side gives an overall view, the right-hand side focuses on high densities. -1.5

-1.98

t

I-

&

2.02

* -2.06 -2.0 -2.10 -2.lL - 2.18

-2.5

-2.22 -2.26 -3.0 -2.30

0

0.5

1.0

1.5

2.0

2.5

30

2.00

2.25

2.50

2.75

3.00

P'Pc -

P'Pc-

Fig. 4. Reduced attractive dispersion force contribution &=(pc/p)/(FA tained in this work (MOBACK) by fitting to ethane. This figure includes isotherms and the same range of reduced densities as Fig. 3. THE EQUATION

OF STATE

/RT,) as obthe same reduced

FOR ETHANE

Using the fitting procedure of the last section in order to obtain FA, we also obtained an equation of state for ethane in the MOBACK form, This

61

0

0.5

10

15 2D P/PC

2.5

3.0

200

225

250

275

P’(! -

300

Fig. 5. Reduced attractive dispersion force contribution g*= (p,/p)/(F, /RT,) as obtained in this work (MOBACK) by fitting to ethane. This figure includes the full temperature and density range where the results are thought to be applicable. The dew-point densities of ethane are marked by filled circles, the bubble-point densities by open circles and the isotherms in the two-phase region are dashed.

equation is given by eqns. (6), (12) and (23) with the auxiliary equations (17) and (24) and the universal D,, coefficients from Table 8. The parameters specific for ethane are pa = 6.8 mol dmP3, T, = 305.33 K, cy= 1.2126 and 77= 0. It should b e noticed here that very recently Friend et al. (1991) used a 32-term modified Benedict-Webb-Rubin equation of state to represent the experimental data of ethane. Firstly, vapor pressures ps, bubble-point densities p’ and dew-point densities P” from MOBACK are compared with the correlations of Goodwin et al. (1976) in Table 9. We learn that the calculated vapor pressures always agree with the experimental values within the uncertainties of the latter. The calculated bubble-point densities agree with the experimental values within their uncertainties up to 298 K, which is 0.976 of T,. Hence the overall representation of the coexistence properties can be termed as very good with the exception of the critical region. We did not, however, try to improve the behavior of MOBACK in the critical region, because we believe that such an attempt would create some peculiarities in FA which could worsen its quality in applications for other substances.

Pg”” (MPaI

0.000053 0.000266 0.001015 0.003126 0.008128 0.018465 0.037626 0.07016 0.12161 0.19837 0.30756 0.45684 0.65436 0.90861 1.22849 1.62331 2.10300 2.67853 3.36289 4.17407 4.35407 4.44662 4.54100 4.63734 4.73588 4.83709

T(K)

108 118 128 138 148 158 168 178 188 198 208 218 228 238 248 258 268 278 288 298 300 301 302 303 304 305

0.000053 0.000266 0.001015 0.003125 0.008129 0.018471 0.037641 0.07019 0.12164 0.19840 0.30757 0.45682 0.65429 0.90848 1.22829 1.62308 2.10282 2.67858 3.36350 4.17589 4.35625 4.44900 4.54357 4.64007 4.73867 4.83956

(MPa)

PS

cd

-

-

0.00 0.00 0.00 0.03 0.01 0.03 0.04 0.04 0.02 0.02 0.00 0.00 0.01 0.01 0.02 0.01 0.01 0.00 0.02 0.04 0.05 0.05 0.06 0.06 0.06 0.05

s

St%) dm-3)

21.0376 20.6677 20.2952 19.9189 19.5383 19.1524 18.7597 18.3581 17.9453 17.5185 17.0742 16.6087 16.1171 15.5930 15.0278 14.4089 13.7161 12.9140 11.9271 10.5283 10.1335 9.9063 9.6509 9.3554 8.9983 8.5290

21.0361 20.6687 20.2985 19.9243 19.5451 19.1596 18.7663 18.3635 17.9491 17.5206 17.0751 16.6087 16.1165 15.5922 15.0266 14.4069 13.7125 12.9072 11.9151 10.5000 10.0927 9.8540 9.5794 9.2485 8.8120 8.0473

(mol

I

Pd

(mol dm-3)

p=P

I

0.00 0.02 0.03 0.03 0.04 0.04 0.03 0.02 0.01 0.01 - 0.00 -0.00 - 0.01 - 0.01 - 0.01 - 0.03 - 0.05 -0.10 - 0.27 - 0.40 - 0.53 - 0.75 - 1.15 -2.11 - 5.99

- 0.01

-3)

0.00006 0.00027 0.00095 0.00273 0.00664 0.01419 0.02740 0.04869 0.08095 0.12748 0.19209 0.27930 0.39459 0.54498 0.73988 0.99269 1.3242 1.7706 2.4083 3.4746 3.8130 4.0171 4.2568 4.5517 4.9491 4.6585

d

I,

EC%) Pexp (mol 0.00006 0.00027 0.00095 0.00273 0.00664 0.01419 0.02739 0.04866 0.08088 0.12735 0.19190 0.27900 0.39411 0.54415 0.73836 0.98996 1.3195 1.7629 2.3962 3.4522 3.7832 3.9803 4.2077 4.4781 4.8154 5.2743

(mol dm-3)

Peal

I,

0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.06 0.09 0.10 0.10 0.11 0.12 0.15 0.20 0.27 0.36 0.43 0.50 0.64 0.78 0.91 1.15 1.62 2.70 6.79

$$(%I

Vapor pressures p,, bubble-point densities p’ and dew-point densities p” for ethane from the MOBACK equation (Cal) in comparison with the correlations (exp) of Goodwin et al. (1976)

TABLE 9

o\ h)

63 TABLE

10

Pressures for ethane as obtained from the MOBACK equation compared with experimental values (Pal, 1971) on a liquid isochore. The last column shows the relative density difference corresponding to the difference in the pressure

T(K)

p(mol dme3)

p,,,(MPa)

p,JMPa)

220.889 243.045 260.378 278.130 293.608

16.7476 16.7233 16.7044 16.6847 16.6681

5.7842 25.7383 40.8368 55.9695 68.8810

5.7890 25.4641 40.5428 55.7180 68.7606

- 0.00 0.07 0.06 0.05 0.02

Pressures on a liquid isochore are compared in Table 10 with the experimental values of Pal (1971). This table also shows the relative density differences corresponding to the differences between the experimental and the calculated pressures. It is seen that the maximum deviation in Ap/p of 0.07% is smaller than the estimated uncertainty of the experimental value of 0.11%. Table 11 shows isochoric heat capacities on several liquid isochores compared with the experimental data of Roder (1976). With the exception

TABLE

11

Isochoric heat capacities for ethane son with experimental data (Roder,

T(K)

p(mol dmp3)

289.038 283.089 277.105 271.085 204.510 201.281 198.035 195.613 172.795 169.533 168.242 166.260 116.167 114.288 112.401 110.494

14.580 14.592 14.604 14.616 17.927 17.938 17.950 17.959 18.885 18.901 18.908 18.919 20.957 20.969 20.981 20.996

as obtained from the MOBACK 1976) on several liquid isochores

5.985 5.908 5.847 5.794 5.377 5.352 5.325 5.303 5.313 5.317 5.315 5.310 5.558 5.557 5.541 5.387

5.977 5.917 5.861 5.805 5.366 5.340 5.313 5.292 5.300 5.263 5.246 5.218 5.573 5.523 5.466 5.408

equation

0.13 -0.17 - 0.24 - 0.20 0.23 0.23 0.21 0.21 0.24 1.03 1.30 1.73 - 0.28 0.61 1.35 - 0.40

in compari-

64 TABLE

12

Enthalpies comparison

of vaporization h, for ethane as obtained from the MOBACK with the correlation (exp) of Goodwin et al. (1976)

T(K)

hyP(J mol-‘)

hy’(J mol-‘1

109 119 129 139 149 159 169 179 189 199 209 219 229 239 249 259 269 279 289 299 301 303 305

17117 16878 16614 16327 16015 15678 15316 14926 14508 14058 13575 13052 12484 11863 11173 10395 9495 8409 7004 4846 4181 3292 1592

17279 16940 16614 16288 15959 15623 15278 14915 14529 14112 13658 13158 12604 11984 11283 10481 9543 8412 6968 4818 4171 3311 1650

equation

(call in

- 0.95 - 0.37 - 0.00 0.24 0.35 0.35 0.25 0.07 -0.15 - 0.39 - 0.62 - 0.82 - 0.96 - 1.02 - 0.99 - 0.82 - 0.50 0.04 0.51 0.57 0.25 - 0.57 - 3.68

of the critical region the calculated results generally agree with the experimental data within their estimated uncertainty. Finally, in Table 12 enthalpies of vaporization are compared with the correlation of Goodwin et al. (1976). The uncertainties of the correlated values were estimated to be 2% and all the calculated values are far within that limit except again very close to the critical point. ACKNOWLEDGMENTS

Die Autoren danken Professor T. Boublik (Prag), Dr. N. Gadalla (Texas A&M), Professor F. Kohler (Bochum) und Dozent Dr. I. Nezbeda (Prag) fur zahlreiche fruchtbare Diskussionen. Die Arbeit wurde von der Deutschen Forschungsgemeinschaft im Schwerpunktprogramm “Thermophysikalische Eigenschaften neuer Arbeitsmedien der Energie- und Verfahrenstechnik”, Az. Fi 287/5, gefordert.

65

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