## Abstract

A robust and efficient procedure is presented for calculating the solubility parameter. An analytical equation for internal pressure is proposed. Through a simple relation reported by Verdier and Andersen (fluid phase equilibrium 231: 125–137, 2005), one can easily find the solubility parameter via our analytical equation for the internal pressure. Also, the radial distribution function (RDF) of a Lennard–Jones LJ (12, 6) fluid, proposed by Xu and Hu (fluid phase equilibrium 30: 221–228, 1986), has been employed to calculate the internal pressure of normal alkanes from methane to decane. Their solubility parameters were evaluated according to the calculated values of the internal pressure. A comparison between the experimental and the estimated values demonstrated a very good agreement between them.

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## References

Gani, R., Muro-Sune, N., Slaes-Cruz, M., Leibvici, C., O’Connell, J.P.: Mathematical and numerical analysis of classes of property models. Fluid Phase Equil.

**250**, 1–32 (2006)Hildebrand, J.H., Scott, R.L.: Solubility of Non-Electrolytes, 3rd edn. Reinhold, New York (1950)

Hildebrand, J.H., Wood, S.E.: The derivation of equations for regular solutions. J. Chem. Phys.

**1**, 817–822 (1933)Panayiotou, C.: Solubility parameter revisited: An equation-of-state approach for its estimation. Fluid Phase Equil.

**131**, 21–35 (1997)Stefanis, E., Tsivintzelis, I., Panayiotou, C.: The partial solubility parameters: an equation-of-state approach. Fluid Phase Equil.

**240**, 144–154 (2006)Eskandari Nasrabad, A., Laghaei, R.: Computational studies on thermodynamic properties, effective diameters, and free volume of argon using an ab initio potential. J. Chem. Phys.

**125**, 84510–84517 (2006)Eskandari Nasrabad, A., Laghaei, R., Eu, B.C.: Modified free volume theory of self diffusion and molecular theory of shear viscosity of liquid carbon dioxide. J. Phys. Chem. B

**109**, 8171–8179 (2005)Laghaei, R., Eskandari Nasrabad, A., Eu, B.C.: Statistical mechanical theory of rheology: Lennard–Jones fluids. J. Chem. Phys.

**123**, 234507–234514 (2005)Eskandari Nasrabad, A., Laghaei, R., Eu, B.C.: Molecular theory of thermal conductivity of the Lennard–Jones fluid. J. Chem. Phys.

**124**, 084506–084510 (2006)Eskandari Nasrabad, A.: Monte carlo simulations of thermodynamic and structural properties of Mie (14, 7) fluids. J. Chem. Phys.

**128**, 154514–154517 (2008)Erkey, C., Rodden, J.B., Akgerman, A.: A correlation for predicting diffusion coefficients in alkanes. Can. J. Chem. Eng.

**68**, 661–665 (1990)Tada, Y., Tamakoshi, A., Kato, Y., Nagatsu, Y.: Generalized Peng–Robinson equation of state with pair potential parameters for liquid

*n*-alkanes. Fluid Phase Equil.**262**, 236–243 (2007)Sheikh, S., Papari, M.M., Boushehri, A.: Equation of state and pressure–volume–temperature properties of refrigerants based on speed of sound data. Ind. Eng. Chem. Res.

**41**, 3274–3281 (2002)Papari, M.M., Razavizadeh, A., Mokhberi, F., Boushehri, A.: Equation of state and

*P*−*V*−*T*−*x*properties of refrigerant mixtures based on speed of sound data. Ind. Eng. Chem. Res.**42**, 3802–3808 (2003)Moghadasi, J., Papari, M.M., Yousefi, F., Haghighi, B.: Transport coefficient of natural gases. J. Chem. Eng. Jpn.

**40**, 697–710 (2007)Nasehzadeh, A., Jamalizadeh, E., Mansoori, G.A.: Prediction of thermodynamics of solution of 15 homologous series of solutes in

*n*-hexadecane via the solubility-parameter based method. J. Mol. Struct. THEOCHEM**629**, 117–126 (2003)Nasehzadeh, A., Mansoori, G.A.: Prediction of gas–hexadecane partition coefficients (L16) via the solubility-parameter based method. J. Mol. Struct.: THEOCHEM

**623**, 135–143 (2003)Parsafar, G.A., Khanpour, M., Mohammadi, A.A.: Calculation of equilibrium and transport properties using modified hard-core potential models. Chem. Phys.

**326**, 527–534 (2006)Goharshadi, E.K., Abbaspour, M.: Prediction of surface tension of HFD-like fluids using the Fowler’s approximation. Chem. Phys.

**328**, 379–384 (2006)Verdier, S., Andersen, S.I.: Internal pressure and solubility parameter as a function of pressure. Fluid Phase Equil.

**231**, 125–137 (2005)McQuarrie, D.A.: Statistical mechanics. Harper and Row, New York (1976)

Matteoli, E., Mansoori, G.A.: A simple expression for radial distribution functions of pure fluids and mixtures. J. Chem. Phys.

**103**, 4672–4677 (1995)Morsali, A., Goharshadi, E.K., Mansoori, G.A.: An accurate expression for radial distribution function of the Lennard–Jones fluid. Chem. Phys.

**310**, 11–15 (2005)Xu, Y.N., Hu, Y.: Prediction of Henry’s constants of gases in electrolyte solutions. Fluid Phase Equil.

**30**, 221–228 (1986)Chicurel-Uziel, E.: Dirac delta representation by exact parametric equations: Application to impulsive vibration systems. J. Sound Vibr.

**305**, 134–150 (2007)Smith, W.R., Henderson, D.: Analytical representation of the Percus Yevick handsphere radial distribution function. Mol. Phys.

**19**, 411–415 (1970)Li, Z., Lu, B.C.-Y.: A molecular model for representing surface tension for polar liquids. Chem. Eng. Sci.

**56**, 6977–6987 (2001)Ben-Amotz, D., Herschbach, D.R.: Estimation of effective diameters for molecular fluids. J. Phys. Chem.

**94**, 1038–1047 (1990)Silva, C.M., Liu, H., Macedo, E.A.: Models for self diffusion coefficients of dense fluids, including hydrogen-bonding substances. Chem. Eng. Sci.

**53**, 2423–2429 (1998)Mousavi, S.M., Pazuki, G.R., Pakizeseresht, M., Dashtizadeh, A., Pakizeseresht, M.: Surface tension correlation for pure polar fluids by a new molecular model and SRK equation of state. Fluid Phase Equil.

**255**, 24–30 (2007)Yaws, C.L.: Chemical Properties Handbook. McGraw–Hill, New York (1999)

Barton, A.F.: Solubility parameters. Chem. Rev.

**75**, 731–753 (1975)Gomez, H.H., Aroche, J.F.B., Rueda, E.R.B.: A modification to the Peng–Robinson-fitted equation of state for pure substances. Fluid Phase Equil.

**298**, 12–23 (2010)Nicolas, J.J., Gubbins, K.E., Streett, W.B., Tildesley, D.J.: Equation of state for the Lennard–Jones fluid. Mol. Phys.

**37**, 1429–1454 (1979)Mecke, M., Muller, A., Winkelmann, J., Vrabec, J., Fisher, J., Span, R., Wagner, W.: An accurate van der Waals-type equation of state for the Lennard–Jones fluid. Int. J. Thermophys.

**17**, 391–404 (1996). errata,**19**, 1493–1498 (1998)

## Acknowledgments

The authors are indebted to Prof. A. Boushehri for his stimulating suggestions, incisive comments, important criticisms, skillful technical assistance, instructive and useful discussions, and sharing of information. We acknowledge helpful conversations with him. We are much indebted to Prof. A. Maghari for his generous encouragement, his provocative insights, and many enticing vistas opened up by his classic studies of statistical mechanics (both equilibrium and non-equilibrium).

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This paper is dedicated to Professor G. A. Mansoori.

## Appendices

### Appendix A. Some Well-Known Expressions for the RDF of a LJ Fluid

### 1.1 A. 1 The Matteoli and Mansoori Expression [22]

See Appendix Table 2

Matteoli and Mansoori [22] have derived an explicit expression for the RDF of a LJ fluid as follows:

where \( y = r/h\sigma \) is the dimensionless inter-particle distance and \( h,\,m,\,\lambda ,\,\alpha \) and \( \beta \) are adjustable parameters that are functions of both temperature and density. These parameters have been expanded in terms of \( \rho^{*} \) and \( T^{*} \), using \( 21 \) constants. This expression is valid within the ranges \( 0.6 \le T^{*} \le 3.7 \) and \( 0.35 \le \rho^{*} \, \le \;0.9 \).

### 1.2 A.2 The Morsali et al. Expression [23]

Morsali et al. [23] derived the following expressions for the RDF of a LJ fluid:

where \( a,b,c,d,g,h,k,\,l,s,m, \) and \( n \) are adjustable parameters, being functions of both temperature and density. These parameters have been expanded in terms of \( \rho^{*} \) and \( T^{*} \), using \( 65 \) constants. This expression is valid within the ranges \( 0.5 \le T^{*} \, \le \;\,5.1 \) and \( 0.35 \le \rho^{*} \, \le \;1.1 \).

### Appendix B. Equations of State Employed in this Study

### 2.1 B.1 Peng–Robinson Equation of State [33]

The Peng–Robinson EOS [33] for pure fluids can be written as

where *p* denotes pressure, *T* temperature, *v* molar volume and *R* the gas constant.

The temperature-independent repulsive parameter *b* is:

where \( T_{c} \) and \( p_{c} \) are the critical temperature and pressure, respectively. The temperature-dependent attractive parameter \( \alpha (T) \) is given by the expression:

where

### 2.2 B.2 Modified Peng–Robinson Equation of State \( ({\text{PR - f}} - mod) \) [33]

Modification of the Peng–Robinson (PR) equation of state (EOS) gives a useful method that enhances the equation of state pure component property predictions through simple temperature dependences such as:

where *T* is the temperature and \( j_{1} ,j_{2} \) and \( j_{3} \) are fitting parameters. The \( p_{c}^{\prime } \) and \( \omega^{\prime } \) values are correlated as follows:

where \( \mu_{p} ,\mu_{m} ,\eta_{p} ,\eta_{m} \) are linear fitting parameters. These equations indicate that only one of the regressed parameters is independent and the other two are correlated with it. Then, the \( {\text{PR - f}} - mod \) EOS can be written as:

where *b* is

and

### 2.3 B.3 The Nicolas et al. EOS [34]

Molecular dynamics calculations of the pressure and configurational energy of a LJ fluid have been reported for 108 state conditions in the density range \( 0.35 \le \rho^{*} \, \le \;1.20 \) and temperature range \( 0.5 \le T^{*} \, \le \;6 \) (where \( \rho^{*} = \rho \sigma^{3} ,\;T^{*} \, = \;k_{B} T/\varepsilon \)) [34]. Particular attention is paid to the dense fluid region \( (\rho^{*} \, \ge \;0.9) \), including state conditions in the sub-cooled liquid region These simulation results for \( p \) and \( U \) were combined with results from previous workers, together with low density values calculated from a virial series, to derive an equation of state for the LJ fluid that is valid over a wide range of temperatures and densities. The equation used by Nicolas et al. in dimensionless form is as follows:

The equation for the reduced configurational energy corresponding to the above equation is obtained from

and the second virial coefficient is given by

Values of the resulting parameters \( x_{1} , \ldots .,x_{32} \) and \( \gamma \) are listed in Table 2.

### 2.4 B.4 The Mecke et al. EOS [35]

An equation of state (EOS) was proposed [35] for the Helmholtz energy \( F \) of the LJ fluid, which represents the thermodynamic properties over a wide range of temperature and densities. The EOS was written in the form of a generalized vdW equation \( F = F_{H} + F_{A} \), where \( F_{H} \) accounts for the hard-body interaction and \( F_{A} \) for the attractive dispersion forces.

For a system of hard spheres with a packing fraction \( \xi \) the residual Helmholtz energy \( F_{H} \) is given according to Carnahan and Starling as

where \( F_{H}^{*} = F_{H} /N\varepsilon \)and \( \xi \) by

where \( \rho_{C}^{*} = 0.3107 \) and \( T_{C}^{*} = 1.328 \) are the critical density and temperature, respectively.

The crucial point is now represented by an equation for\( F_{A} \). \( F_{A} \) is given by the following equation:

where \( F_{A}^{*} = F_{A} /N\varepsilon \) and the powers \( m_{i} ,\;n_{i} ,\;p_{i} \), and \( q_{i} \), as well as the coefficients \( c_{i} \), are to be determined by an optimization procedure. The EOS expressed above covers the whole fluid region up to the highest densities in the temperature range \( 0.7 \le T^{*} \, \le \;10.0 \) with high accuracy. For higher temperatures, the EOS follows a physically correct behavior by using simulation data up to \( T^{*} \, = \;100.0 \) in the density region \( \rho^{*} \, \le \;1.0 \).

### Appendix C Derivation of the Theoretically-Based Relationship Between Internal Pressure and Density

As mentioned in the text, the explicit RDF expression of Xu and Hu [24] is as follows:

where *H* is the Heaviside function:

and \( \delta \) is the Dirac delta function:

The internal pressure and inter-particle pair potential, \( u(r) \), are related by the following equation:

where \( g(r) \) is the RDF, and \( \rho \) is the density.

Insertion of the RDF expression of Xu and Hu [24] into the aforesaid equation and then integration gives:

For the sake of brevity we define the reduced expressions for internal pressure and density, respectively:\( \pi_{T}^{*} \equiv \pi_{T} \sigma^{3} /\varepsilon \) and \( \rho^{*} \equiv \rho \sigma^{3} \),

The final result in terms of reduced parameters is:

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Oghaz, N.M., Haghighi, B., Alavianmehr, M.M. *et al.* Prediction of Solubility Parameters Based on the Explicit Expression of Statistical Thermodynamics.
*J Solution Chem* **42**, 544–554 (2013). https://doi.org/10.1007/s10953-013-9978-9

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DOI: https://doi.org/10.1007/s10953-013-9978-9