Prediction of Solubility Parameters Based on the Explicit Expression of Statistical Thermodynamics

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.

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

Table 2 The parameters $x_{1} , \ldots .,x_{32}$ in the Nicolas EoS [34] below for $\gamma = 3.0$

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

$$ \begin{aligned} g(y) & = 1 + y^{ - m} [g(d) - 1 - \lambda ] + [(y - 1 + \lambda )/y]\; \times \;\{ \exp [ - \alpha (y - 1)]\cos [\beta (y - 1)]\} \\ & \quad \quad m \ge 1,\;y \ge 1 \\ g(y) & = g(d)\exp [ - \theta (y - 1)],\;y < 1 \\ \end{aligned} $$

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:

$$ \begin{aligned} g(r^{*} ) & = 1 + (r^{*} )^{ - 2} \exp [ - (ar^{*} \, + \;b)]\sin [(cr^{*} \, + \;d)] + (r^{*})^{ - 2} \exp [ - (gr^{*} \, + \;h)]\cos [(kr^{*} \, + \;l)] \\ &\quad\, r^{*} \, > \;1 \\ g(r^{*} ) & = s\exp [ - (mr^{*} \, + \;n)^{4} ],\;r^{*} \, \le \;1 \\ \end{aligned} $$

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

$$ p = \frac{RT}{v - b} - \frac{\alpha (T)}{v(v + b) + b(v - b)} $$

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

The temperature-independent repulsive parameter b is:

$$ b = 0.077796\left( {\frac{{RT_{c} }}{{p_{c} }}} \right) $$

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:

$$ \alpha (T) = a_{c} \left[ {1 + m\left( {1 - \sqrt {\frac{T}{{T_{c} }}} } \right)} \right]^{2} $$

where

$$ \begin{aligned} a_{c} & = 0.457235\frac{{(RT_{c} )^{2} }}{{p_{c} }} \\ m & = 0.378893 + 1.4897153\omega - 0.17131848\omega^{2} + \;0.0196554\omega^{3} \\ \end{aligned} $$

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:

$$ T^{\prime}_{c} (T) = j_{1} + j_{2} T + j_{3} T^{2} $$

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:

$$ \begin{aligned} p^{\prime }_{c} & = \eta_{p} + \mu_{p} T^{\prime}_{c} \\ \omega^{\prime} & = \eta_{\omega } + \mu_{\omega } T^{\prime}_{c} \\ \end{aligned} $$

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:

$$ p = \frac{RT}{v - b} - \frac{{\alpha (T_{c}^{\prime } )}}{v(v + b) + b(v - b)} $$

where b is

$$ b = 0.077796\frac{{RT_{c}^{\prime } (T)}}{{P_{c} }} $$

and

$$ \alpha (T_{c}^{\prime } ) = a_{c} \left[ {1 + m\left( {1 - \sqrt {\frac{T}{{T_{c}^{\prime } }}} } \right)} \right]^{2} $$

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:

$$ \begin{aligned} p^{*} = \,&\rho^{*} T^{*} \, + \;\rho^{*2} (x_{1} T^{*} \, + \;x_{2} T^{*1/2} \, + \;x_{3} + x_{4} T^{* - 1} \, + \;x_{5} T^{* - 2} ) + \rho^{*3} (x_{6} T^{*} \, + \;x_{7} + x_{8} T^{* - 1} \, + \;x_{9} T^{* - 2} ) \\ & \rho^{*4} (x_{10} T^{*} + \;x_{11} + x_{12} T^{* - 1} ) + \rho^{*5} (x_{13} ) + \rho^{*6} (x_{14} T^{* - 1} \, + \;x_{15} T^{* - 2} ) + \rho^{*7} (x_{16} T^{* - 1} ) \\ & \quad + \rho^{*8} (x_{17} T^{* - 1} \, + \;x_{18} T^{* - 2} ) + \rho^{*9} (x_{19} T^{* - 2} ) + \rho^{*3} (x_{20} T^{* - 2} \, + \;x_{21} T^{* - 3} )exp( - \gamma \rho^{*2} ) \\ & \quad + \rho^{*5} (x_{22} T^{* - 2} \, + \;x_{23} T^{* - 4} )exp( - \gamma \rho^{*2} ) + \rho^{*7} (x_{24} T^{* - 2} \, + \;x_{25} T^{* - 3} )exp( - \gamma \rho^{*2} ) \\ & \quad + \rho^{*9} (x_{26} T^{* - 2} \, + \;x_{27} T^{* - 4} )exp( - \gamma \rho^{*2} ) + \rho^{*11} (x_{28} T^{* - 2} \, + \;x_{29} T^{* - 4} )exp( - \gamma \rho^{*2} ) \\ & \rho^{*11} (x_{30} T^{* - 2} \, + \;x_{31} T^{* - 3} )exp( - \gamma \rho^{*2} ) + \rho^{*13} (x_{30} T^{* - 2} \, + \;x_{31} T^{* - 3} \, + \;x_{32} T^{* - 4} )exp( - \gamma \rho^{*2} ) \\ \end{aligned} $$

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

$$ U^{*} = \int\limits_{0}^{{\rho^{*} }} {d\rho^{*}\frac{1}{{\rho^{*} }}} \left[ {p^{*} \, - T^{*} (\frac{{dp^{*}}}{{dT^{*} }})_{{\rho^{*} }} } \right] $$

and the second virial coefficient is given by

$$ B_{2}^{*} = \frac{3}{2\pi }\left( {\frac{\partial Z}{{\partial \rho^{*} }}} \right)_{{\rho^{*} = 0}} = \frac{3}{2\pi }(x_{1} + x_{2} T^{* - 1/2} + x_{3} T^{* - 1} + x_{4} T^{* - 2} + x_{5} T^{* - 3} ) $$

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

$$ F_{H}^{*} /T^{*} = (4\xi - 3\xi^{2} )/(1 - \xi )^{2} $$

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

$$ \xi = 0.1617(\rho^{*} /\rho_{C}^{*} )[0.689 + 0.311(T^{*} /T_{C}^{*} )^{0.3674} ]^{ - 1} $$

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:

$$ F_{A}^{*} /T^{*} = \sum\limits_{i} {c_{i} } (T^{*} /T_{C}^{*} )^{{m_{i} }} (\rho^{*} /\rho_{C}^{*} )^{{n_{i} }} \exp [p_{i} (\rho^{*} /\rho_{C}^{*} )^{{q_{i} }} ] $$

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:

$$ g(r) = H(r - r^{**} ) + (r^{**3} - \sigma^{3} )\delta (r - r^{*} )/3r^{*2} $$

where H is the Heaviside function:

$$ \left\{ {\begin{array}{*{20}c} {H(x - a) = 1} & {} & {x > a} & {} \\ {H(x - a) = 0} & {} & {x < a} & {} \\ {H(x - a) = \frac{1}{2}} & {} & {x = a} & {} \\ \end{array} } \right. $$

and $\delta$ is the Dirac delta function:

$$ \left\{ {\begin{array}{*{20}c} {\delta (x - a) = 0} & {} & {x \ne a} & {} \\ {\delta (x - a) = \infty } & {} & {x = a} & {} \\ \end{array} } \right. $$

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

$$ \begin{gathered} \pi = - 2\pi \rho^{2} \int\limits_{0}^{\infty } {u(r)\left\{ {g(r) + \rho \left( {\frac{\partial g}{\partial \rho }} \right)_{r,T} } \right\}} r^{2} dr \hfill \\ \hfill \\ \end{gathered} $$

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:

$$ \pi_{T} = 5.08585588\varepsilon \sigma^{3} \rho^{2} $$

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:

$$ \pi_{T}^{*} = 5.08585588\rho^{*2} $$