Thermodynamic modelling of near-critical solutions

Fluid Phase Equilibria 144 Ž1998. 1–12

Thermodynamic modelling of near-critical solutions John P. O’Connell

a, )

, Hongqin Liu

b

a

b

Department of Chemical Engineering, UniÕersity of Virginia, CharlottesÕille, VA 22903, USA Department of Chemical Engineering, Beijing UniÕersity of Chemical Technology, 100029 Beijing, China Received 13 January 1997; accepted 1 June 1997

Abstract The challenge of modeling the properties of near-critical systems requires the use of different approaches in order to describe their strong variations with conditions, the variety of different and complex substances that can be present, and the difficulty of obtaining reliable data. This paper describes some approaches that lead to relatively simple descriptions for phase equilibrium and volumetric properties at low solute concentrations as well as suggests methods for solute–solute effects based on Fluctuation Solution Theory. q 1998 Elsevier Science B.V. Keywords: Critical state; Equation of state; Solubility; Dilute solution; Partial molar volume

1. Introduction Fluid mixtures near critical points have continued to be of great fundamental and applied interest w1x. This has been particularly true for separations involving complex natural substances and temperature-sensitive components, but also for aqueous and other extreme pressure and temperature cases. There have been a variety of theoretical, experimental and processing approaches taken, but often the connections among them have been tenuous. Much of this may be due to uncertainties over such aspects as the real influence of nonclassical effects of the critical region which are complicated to describe and implement, the difficulty of obtaining accurate and reliable data and the complexities of the substances and systems for which there is greatest practical interest. The goal of this work is to relate some properties of these systems in a manner that may lead to improved modeling through generalization and consistency. The focus here will be on models for fluids that can be used for both solubility and volumetric properties. The treatment will consider very dilute solutions but also address the importance of solute compositional dependence. To avoid some of the complexities of multiphase )

Corresponding author.

0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 Ž 9 7 . 0 0 2 3 9 - 2

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

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systems w2,3x, discussion will be limited to single fluids or those in phase equilibrium with solids. The view is that understanding adequate for modeling can profitably be pursued by investigating the phases individually and then determining the optimal strategy for describing multiphase cases.

2. Fundamentals 2.1. Thermodynamics The thermodynamic properties of fluids are best described with an equation of state Ž EoS. for the compressibility factor, z, given by z ' PÕr Ž RT . s PVr Ž NRT . s z Ž T ,Õ,  x 4 . s z Ž T ,V  N 4 .

Ž1.

where Õ is the molar volume,  x 4 mole fractions, V total volume, and  N 4 mole numbers. The functional form and parameterization depend on the model requiring from 2 to many parameters to characterize the species and involve parameter mixing and combining rules of varying complexity for solutions w4x. The form with the soundest theoretical basis is the virial expansion. C

zs1q

C

C

C

Ý Ý ri r j Bi j Ž T . q Ý Ý ri r j r k Cijk Ž T . q . . . is1 js1

rr

Ž2.

is1 js1

Here, the density, r ' 1rÕ, and concentrations, r i ' NirV, have been used. The most commonly applied EoS relation is cubic in volume and of the semitheoretical ‘generalized van der Waals form’ w5x. Thermodynamic relations connect the EoS form and parameters to other properties such as enthalpy used for heat and work effects as well as the chemical potential and fugacity used for phase equilibrium relations. For the fugacity the relation is f i Ž T , P ,x . s x i f i Ž T , P ,x . P

Ž3.

where the fugacity coefficient is f i . It is found from `

ln f i z y

HV

E Ž PVrRT . E Ni

/

dV

y1 T ,V , Nj/ i

T , N 4

Vi

r

s V

H0

k T RT

dr

y1

r

Ž4.

T , x 4

where the second integral uses the partial molar volume, Vi , and the solution isothermal compressibility, k T . For condensed phases, the fugacity was traditionally obtained by use of a standard state fugacity and activity coefficient, the precise form depending upon how the pressure dependence is included w6x. The common cases in practice are f i Ž T , P ,x . s x i g i Ž T ,x; f io . f io Ž T . exp f i Ž T , P ,x . s x i g i Ž T , P ,x; f io . f io Ž T .

P

V ŽT , P , x4 .

o i

RT

HP

/

dP

Ž5.

T , x 4

Ž6.

Eq. Ž 5. puts the pressure dependence into a Poynting Correction w7x while Eq. Ž6. puts it into the

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

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activity coefficient w8x. For low pressure systems of condensed phases there is no difference because the effects of pressure are small in any case. For subcritical or slightly supercritical fluids, the standard state fugacity is normally that of the saturated phase Ž extrapolated if necessary. and the vapor pressure is the standard state pressure, Pio s PiS Ž T .. The partial molar volume is often approximated as the pure molar volume. For dilute solutions, Henry’s constant has often been used for the standard state fugacity, f io Ž T . s Hi Ž T ., and the standard state pressure is the solvent vapor pressure, Pio s PsS Ž T ., and the partial molar volume can be approximated by the infinite dilution value, Vi`Ž T . in the solvent. The relationships among g i , f i , and f io are g i s f irf io, and f io s f io Pio, where f io is the fugacity coefficient of component i as a real or hypothetical pure component at the standard state conditions of Ž T, Pio .. The integrand of Eq. Ž4. also appears in other relationships of importance. For example, connections among composition derivatives are N

E ln g i E Nj

/

sN

E ln f i E Nj

T , P , Nk / j

/

sr

E ln g i

T , P , Nk / j

Er j

/

y T , rk / j

Vi

ž / k T RT

r Vj

Ž7.

and the Gibbs–Duhem equation gives C

Vi

k T RT

s

Ý rj

E ln g i

js1

Er j

/

Ž8. T , rk/ j

Finally, the reduced bulk modulus is

E Ž PrRT . Er

/

' T , x 4

1

rk T RT

1 s

r

C

Ý ri k

is1

Vi T

RT

1 s

r

C

C

Ý Ý ri rj is1 js1

E ln g i Er j

/

Ž9. T , rk/ j

For a pure component, the reduced bulk modulus becomes zero with ‘weak’ nonclassical variation w9x. At infinite dilution, Vi`rk T0 RT, which is called the generalized Krichevskii Parameter, A`12 , is also well behaved even in the solvent critical region where both Vi` and k T0 become divergent; their ratio remains finite. For substances ‘lighter’ than the solvent the sign of A`12 is always positive; for substances ‘heavier’ than the solvent, it is negative from less than one-half the critical density to about twice the critical density, being positive elsewhere. In the critical region of the solvent, A`12 is extremely important in characterizing the composition variations of very dilution solutions w9x. For solids of pure components with low sublimation pressure, Pisub, a good approximation to Eq. Ž5. is s Hpure i

ŽT , P .

s Pisub

Ž T . exp

Õis Ž T . P RT

Ž 10.

Multiphase equilibrium is found by equating the fugacities for the components appearing in the phases. For example for vapor–liquid equilibria, the vapor mole fractions,  y4 , can be found with the same EoS for both phases yi f i Ž T , r Õ  y 4 . s x i f i Ž T , r l ,  x 4 .

Ž 11.

In Eq. Ž11. the value of for the phase to be used in Eq. Ž 4. is found by solving Eq. Ž 1. for the proper

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

4

phase. Eq. Ž 11. has no discontinuity in behavior at the solution critical point as the use of Eq. Ž 5. or Eq. Ž6. with Eq. Ž3. would inevitably have when models are used. However, for separation purposes, it is often intended that the compositions of the phases are sufficiently different that this advantage may not be relevant. Further, since the EoS forms commonly used do not have proper critical behavior w9x, proper descriptions may not be easily obtained in any case. For systems at what can be considered infinite dilution, the EoS is that of the pure solvent, z 0 s P r r 0 RT and Ž 4. becomes, in a binary of solute Ž 1. in solvent Ž 2. ln f 1` Ž T , P . z 0 s

H0r

0

A`12 y 1

dr

Ž 12.

r

For the virial equation of state 2

A`12 s 1 q 2 r 0 B12 q 3 Ž r 0 . C112 q . . .

Ž 13.

giving an infinite dilution fugacity coefficient expression of 2

ln f 1` Ž T , P . z 0 s 2 r 0 B12 Ž T . q 3r2 Ž r 0 . C122 Ž T . q . . .

Ž 14.

2.2. Fluctuation solution theory The statistical mechanics of concentration fluctuations leads to relationships between thermodynamic derivatives and integrals of molecular correlation functions w10–12x. The formulations chosen here are in terms of integrals Ž DCFI. of the molecular direct correlation function defined by Ornstein and Zernike. The basic relation is

r

E lng i Er j

/

s 1 y Ci j Ž T ,  r 4 .

Ž 15.

T , rk/ j

where ™ Ci j Ž T ,  r 4 . s rHc i j Ž r ,T ,  r 4 . dr

Ž 16.

This also yields V1

k T RT

1 s

r

C

Ý r j Ž 1 y Ci j .

Ž 17.

is1

The connection of DCFI and the virial equation is C

1 y C i j s r 2 Bi j Ž T . q 3

Ý r k Ci jk Ž T . q . . .

Ž 18.

ks1

With the near-critical region behavior of the derivatives being only weakly divergent, the DCFI are also well-behaved. In particular, the condition of the spinodal Žboundary of the thermodynamically unstable region which includes the critical point. is for a binary

Ž 1 y x 1C11 .Ž 1 y x 2 C22 . y x 1 x 2 C12 s 0

Ž 19.

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

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It is also possible to relate the isothermal variation of solubility with pressure of a solid component at equilibrium with a fluid to quantities considered in this section.

E ln x 1 EP

/

s T ,s

ž Õ1o y V1 / rk T RT r RT < Ž 1 y x 1C11 .Ž 1 y x 2 C22 . y x 1 x 2 C12 <

Ž 20a.

The variation of solubility shows sharp increases with pressure near the pure component critical due to the denominator becoming near zero rather than the numerator becoming large w9x. However, as is well-known Žsee for example, w13x., the solubility variation with density is not strong as shown by the derivative.

E ln x 1 E ln r

/

s

ž Õ1s y V1 / rk T RT

Ž 20b.

1 y x 1Ž C11 y C22 .

T ,s

The presence of DCFI in Eq. Ž15., Eq. Ž17. and Eq. Ž 19. suggests that modeling them could be quite useful in describing near-critical properties. The infinitely dilute solution results for a binary with solute Ž1. and solvent Ž2. can be summarized as 1 o s 1 y C22 ŽT , r o . Ž 21. o o r k T RT ` A`12 s 1 y C12 ŽT , r o .

E ln x 1 EP E ln x 1 E ln r

`

/

s

s T ,s

y

RT

T ,s `

/

Õ 1s

Õ 1s

k T RT

Ž 22. ` 1 y C12

Ž 23a.

o r o RT Ž 1 y C22 . ` y Ž 1 y C12 .

Ž 23b.

Again, the rapid change in solubility with pressure near a solvent’s critical arises from the denominator of the second term on the right of Eq. Ž 23a. becoming small and the weak variation with density is shown in Eq. Ž23b.. Besides infinite dilution behavior, the initial composition dependence of the DCFI can be obtained by expansions about infinite dilution which are reliable because of the DCFI are nondivergent in the critical region. The expansion of the fugacity coefficient is found in the work of Debenedetti and Kumar w14x and Chialvo w15x.

Ž24. In terms of effect in terms of infinite dilution DCFI ` K f 1 y 1 y C11 y

r 0 Ž Õ`1 . k T0 RT

2 ` s 1 y C11 y

` Ž 1 y C12 . 0 1 y C22

2

Ž 25.

It is clear that the last term of Eq. Ž 25. is large in the neighborhood of, and diverges at, the solvent 0 . ` . critical point because the value of Ž 1 y C22 becomes small while Ž 1 y C12 is finite. Thus, the

6

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

composition effect on the near-critical solute fugacity coefficient can be large. This is consistent with the simulation results of Li et al. w16x in the near-critical region. The most appropriate expansion for the properties influencing volumetric behavior, V1 and q T are from differentiating the DCFI to yield an expression with infinite dilution pair and triplet DCFI w11x. This expansion is well-behaved because it involves only quantities with weak zeros

Ž26.

Ž27.

Ž28.

Ž29.

Ž30. Eqs. Ž28. and Ž29. show that not only will V1` and k T0 be large near the solvent critical, but the effect 0 . y2 of small amounts of solute can be very significant. In these cases, the variation includes Ž 1 y C22 0 . y1 which is stronger than that of ln f 2 which varies as Ž1 y C22 . If the numerators in the linear 0 coefficients are of the order of unity, even if C22 K 0.9 Žwhich occurs over a significant range of conditions near the solvent critical point., the effects would be important at x 2 s 10y3. Also, note that the coefficients have terms of different sign, which is consistent with the variable composition effects found by O’Connell et al. w17x for aqueous systems.

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

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3. Modeling The virial form Ž14. in Eq. Ž3. was used by Quiram et al. w18x to correlate solubilities of solids described by Eq. Ž10. for a wide variety of systems with CO 2 as the solvent including those with cosolvents Žentrainers. . Their basic equation is ln

P1sub

ž / x1 P

q

Õ 1s P RT

rr 0 s 2 B12 q 3 r 0 C112

Ž 31.

suggesting that the left-hand-side of Eq. Ž 31. plotted versus r 0 will give a straight line. Fig. 1 shows representative data for naphthalene Ž1. in carbon dioxide Ž2. at 308 K w19,20,32,34x. The intercept of the best fit to the several sets of high density data w19,20x is the same as given by the low density data of Najor and King w21x though there are other scattered data at intermediate densities. This figure indicates how difficult it is to obtain reliable and consistent in this range of conditions. Besides giving a simple 2-parameter form for correlating good data, Eq. Ž 31. allows a direct method for determining consistency among measurements similar to the work of Prausnitz and Keeler w22x. Though it is likely that the fitted quadratic coefficients are effective, but not rigorous, values of C122 w23x, the B12 values found by Quiram et al. are consistent with low density solubility data and estimated values. As shown in Table 1, the temperature dependence is somewhat more reliable than from a generalized cubic equation of state such as the Peng–Robinson EoS.

Fig. 1. Solubility Data and Correlations for Naphthalene Ž1. in CO 2 at 308 K.

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

8

Table 1 Second cross virial coefficients for the phenanthrene Ž2.-CO 2 system T ŽK.

Bradley and King w33x

Quiram et al.

Estimateda

PR EoS

308.15 323.15 343.15

y695 y610 y546

y692 y598 y491

y677 y607 y532

y660 y634 y588

a

Method of Hayden and O’Connell w24x.

An alternative expression to Eq. Ž 13. for A`12 for infinitely dilute solutes in aqueous systems over wide ranges of conditions has been found empirically by Cooney and O’Connell w25x for salts and by O’Connell et al. w17x for solutes such as methane, ammonia, hydrogen sulfide and carbon dioxide. Its basic form is A`12 s 1 q 2 b 12 r 0 q c12 exp Ž d 2 r 0 . y 1 Ž 32. where b 12 and c12 depend on the solute and may be temperature-dependent while d 2 appears to be a universal constant for all solutes in water. To describe experimental data for V1` and C`p1 , within their uncertainty, small corrections have also been added to Eq. Ž32. . Used in Eq. Ž12., this relation gives ln Ž f 1`z 0 . s Ž 2 b 12 y c12 . r 0 q Ž c12rd1 . exp Ž d 2 r 0 . y 1 Ž 33. Ž . Ž . Both the virial form Eq. 14 and the new form Eq. 33 have leading terms linear in the solvent density and an expansion of the second term of Eq. Ž33. begins with a quadratic. However, whether

Fig. 2. Generalized Krichevskii Parameters for Naphthalene Ž1. in CO 2 at 308 K.

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

9

Eq. Ž33. gives the linear relation of Eq. Ž31. depends upon the parameter values for a particular system. Thus, Fig. 1 also shows a fit to Eq. Ž 33. ; there is a slight curvature which may or may not be consistent with the data. Fig. 1 also shows a fit of the Peng–Robinson EoS w26x to the same data; its line cannot be distinguished from the virial form because its higher density effects are not significant at the conditions of the plot. To further test equations for describing all of the available data, V1` measurements can be examined. Fig. 2 shows some available data for naphthalene Ž1. in carbon dioxide Ž 2. at 308 K obtained by both composition dependence of density w27x and from superfluid chromatography Ž SFC. w13x plotted as the generalized Krichevskii parameter. There are three lines on the plot: a fit to Eq. Ž32. of the data; and calculations of A`12 from both Eq. Ž13. and Eq. Ž 32. using the parameters found from the solubility data of Fig. 1. In the density range from 12 to 20 mol ly1 there is consistency between the properties. However, at both near-critical and higher densities, the volumetric measurements are above the predictions. This has been noted in the past Ž See, for example, w28–30x. The high density differences are understandable since the virial equation would not be applicable at such densities. However, those at near-critical conditions can only be explained by an inconsistency between the density and solubility results. The framework developed here allows a sound basis for investigating this discrepancy.

Fig. 3. Solute Terms for Naphthalene Ž1. in CO 2 at 308 K.

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

10

Though a thorough treatment is in process w31x, it is likely that the difficulty is in the assumptions used to extrapolate the V1 data to infinite dilution. In the case of density data, the variations need to be done with Eq. Ž26. while the SFC results assume that very low solute concentrations require no extrapolation. Eq. Ž28. suggests that this may not be correct. For example, preliminary calculations of the various terms in Eqs. Ž 25. and Ž28. and Eq. Ž29. using the Peng–Robinson EoS are shown in Fig. 3. They indicate that the contribution of K Õ1 for x 1 s 0.00005 is as large as the base term from the SFC partitioning. It is likely that these EoS estimates are not quantitatively reliable, but certainly can be indicative of the large influence of solute concentration. Work in progress w31x involves estimating the triplet DCFIs of Eq. Ž 30. to obtain the composition dependence of the partial molar volume in a manner consistent with the solubility data. This will be able to be tested with the data of Wood and coworkers as well as with SFC measurements that vary the solute amounts.

4. Conclusions Fundamental relations and simple correlating equations are given for the solubility and partial molar volume of solutes at low concentrations in near-critical fluids. Fluctuation solution theory direct correlation function integrals are closely tied to the important properties and property variations in this region such as the pressure dependence of the solubility and the generalized Krichevskii parameter. The apparent discrepancy between solubility and V1` may originate in extrapolating data to obtain the partial molar volume since solute concentration effects on it are expected to be very large.

5. List of symbols A`12 B12 , C112 b 12 , c12 , d 2 Ci j , Ci jk Hi

generalized Krichevskii parameter, Vi`rk T0 RT 2nd, 3rd cross virial coefficients among components ij and ijk coefficients in Eq. Ž32., Eq. Ž33. pair, triplet direct correlation function integrals ŽDCFI., Eq. Ž16., Eq. Ž30. Henry’s Constant for component i, lim Ž f irx i ..

K f 1, K Õ1, Kk 1 N Ni Pio Pisub V, Õ Vi Õ 1s x, y z

1st-order coefficients in component expansions Eq. Ž 24. , Eq. Ž 28. , Eq. Ž 29. total number of moles in system total number of moles of component i standard state pressure for component i sublimation pressure of pure solid component i at system temperature total, molar volume of system partial molar volume of component i molar volume of pure solid solute component 1 at system temperature mole fractions in liquid, vapor phases compressibility factor, PvRT, Eq. Ž 1.

x i™0

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

Greek symbols kT r , ri fi

isothermal compressibility, Ž E ln rrE P . T molar density of fluid, molar concentrations of component i fugacity coefficient of component i, Eq. Ž3.

Superscripts 0 ` o

pure component infinite dilution standard state

11

References w1x E. Kiran, J.M.H. Levelt Sengers ŽEds.., Supercritical Fluids; Fundamentals for Application, Kluwer, Dordrecht, 1994. w2x Th.W. de Loos, in: E. Kiran, J.M.H. Levelt Sengers ŽEds.., Supercritical Fluids; Fundamentals for Application, Kluwer, Dordrecht. 1994, p. 65. w3x C.J. Peters, in: E. Kiran, J.M.H. Levelt Sengers ŽEds.., Supercritical Fluids; Fundamentals for Application, Kluwer, Dordrecht, 1994, p. 117. w4x H. Orbey, in: E. Kiran, J.M.H. Levelt Sengers, ŽEds.., Supercritical Fluids; Fundamentals for Application, Kluwer, Dordrecht, 1994, p. 177. w5x S.A. Sandler, in: E. Kiran, J.M.H. Levelt Sengers, ŽEds.., Supercritical Fluids; Fundamentals for Application, Kluwer, Dordrecht, 1994, p. 147. w6x J.P. O’Connell, in: E. Kiran, J.M.H. Levelt Sengers, ŽEds.., Supercritical Fluids; Fundamentals for Application, Kluwer, Dordrecht, 1994, p. 191. w7x J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, PrenticeHall, Englewood Cliffs, NJ, 1986. w8x E.A. Campanella, P.M. Mathias, J.P. O’Connell, AIChE J. 33 Ž1987. 2057. w9x J.M.H. Levelt Sengers, in: E. Kiran, J.M.H. Levelt Sengers ŽEds.., Supercritical Fluids; Fundamentals for Application, Kluwer, Dordrecht, 1994, p. 10. w10x J.G. Kirkwood, F.P. Buff, J. Chem. Phys. 19 Ž1951. 774. w11x J.P. O’Connell, Mol. Phys. 20 Ž1971. 27. w12x E. Matteoli, G.A. Mansoori ŽEds.., Fluctuation Theory of Mixtures, Taylor and Francis, NY, 1990. w13x J.-J. Shim, K.P. Johnston, J. Phys. Chem. 95 Ž1991. 353. w14x P.G. Debenedetti, S.K. Kumar, AIChE J. 32 Ž1986. 1253. w15x A.A. Chialvo, J. Phys. Chem. 97 Ž1993. 2740. w16x T.W. Li, E.H. Chimowitz, F. Munoz, AIChE J. 41 Ž1995. 2300. w17x J.P. O’Connell, A.V. Sharygin, R.H. Wood, IEC Research 35 Ž1996. 2808. w18x D.J. Quiram, J.P. O’Connell, H.D. Cochran, J. Supercr. Fluids 7 Ž1994. 194. w19x J.M. Dobbs, J.M. Wong, K.P. Johnston, J. Chem. Eng. Data 31 Ž1986. 303. w20x M. McHugh, M.E. Paulaitis, J. Chem. Eng. Data 25 Ž1980. 326. w21x C.C. Najor, A.D. King Jr., J. Chem. Phys. 52 Ž1970. 5206. w22x J.M. Prausnitz, R.N. Keeler, AIChE J. 7 Ž1961. 349. w23x A.H. Harvey, Fluid Phase Equil. 130 Ž1997. 87. w24x J.G. Hayden, J.P. O’Connell, Ind. Eng. Chem. Proc. Des. Dev. 14 Ž1975. 221. w25x W.J. Cooney, J.P. O’Connell, Chem. Eng. Comm. 56 Ž1987. 341. w26x D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 Ž1976. 59. w27x C.A. Eckert, D.H. Ziger, K.P. Johnston, S. Kim, J. Phys. Chem. 90 Ž1986. 2738. w28x G.C. Nielson, J.M.H. Levelt Sengers, J. Phys. Chem. 91 Ž1987. 4078.

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w29x w30x w31x w32x w33x w34x

J.P. O’Connell, H. Liu r Fluid Phase Equilibria 144 (1998) 1–12

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