Estimation of Hansen Solubility Parameters with an Entropy-Based Solubility Parameter-Translated SRK Equation of State

1. Introduction

Soave-Redlich-Kwong (SRK) equation of state (SRK EoS) [1] is widely used by scientists and engineers as well as Peng-Robinson EoS (PR EoS) [2] to predict high-pressure phase diagrams for manufacturing products. The prediction is mainly based on critical properties not only for pure substances, but also for multiple mixtures. The utility of SRK EoS is very wide for simulation of both super- and sub- critical processes [3,4,5,6] and for petroleum industries [7,8].

However, both SRK EoS and PR EoS have low prediction ability for both liquid and critical densities, especially for high polar substances like ethanol, methanol and water. Therefore, volume-translated cubic EoS has been previously proposed by researchers [9,10,11,12] to improve volumetric representation.

Recently, entropy-based solubility parameter (eSP) to be calculated at arbitrary temperature and pressure conditions based on Hildebrand solubility parameter (SP) obtained at ambient temperature and pressure conditions was applied to development of entropy-based solubility parameter-translated PR EoS (eSPT-PR EoS) [13]. Thus, the prediction accuracy of both pure substance liquid and critical densities was improved, however, it is difficult for both fugacity and entropy to be calculated analytically using departure functions as its present form. From this point of view, a new technique is necessary for addressing these issues.

In the beginning of this work, the SRK EoS was applied to overcome these issues. The added eSP translation correction that treats both fugacity and entropy calculations is applied to the modified SRK EoS and denoted as eSPT-SRK EoS. In the eSPT-SRK EoS, new constants C, D, E and F were added to the EoS and the resulting C and D values were generalized with fundamental properties of chemical species. The other universal constants (E and F) were determined with selected data. Finally, entropy-based Hansen-type solubility parameters (eHSP) were calculated according to the functional form of the eSPT-SRK EoS. Furthermore, we corrected the eSPT-PR EoS with comparison to the developed eSPT-SRK EoS.

2. Materials and Methods

As in the previous study [13], we selected 27 substances generally used in chemistry. We plussed carbon dioxide because carbon dioxide shows gaseous state in the ambient temperature and pressure conditions. Thus, total 28 substances were used in EoS development. Critical properties and acentric factors are listed in Table 1. The Hildebrand solubility parameter (SP) is generally evaluated at ambient conditions [14] and is defined for organic compounds by Equation (1):

$$\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{S}\mathrm{P}\left(\mathrm{S}\mathrm{P}\right)\cong \sqrt{{\displaystyle \frac{{∆}_{\mathrm{V}\mathrm{P}}U}{v_{\mathrm{L}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}}}}}=\sqrt{{\displaystyle \frac{{∆}_{\mathrm{v}\mathrm{p}}H-P\left(v_{\mathrm{G}}-v_{\mathrm{L}}\right)}{v_{\mathrm{L}}}}}$$

(1)

In Equation (1), ${∆}_{\mathrm{V}\mathrm{P}}U$ is the cohesive energy and $v_{\mathrm{L}}$ is the molar volume of the liquid. Usually, the Hildebrand SP is calculated at a SATP condition (298.2 K and 101.3 kPa), where ${∆}_{\mathrm{v}\mathrm{p}}H$ is the enthalpy of vaporization and $v_{\mathrm{G}}$ is the molar volume of gas.

The eSP [15,16] defined in Equation (2) is more widely used not only for multiple phases such as vapor and liquid phases, but also for homogeneous supercritical fluid phases. The eSP is defined as:

$$e\mathrm{SP} \text{≡} \sqrt{{(\partial P/\partial T)}_v}= \sqrt{{(\partial s/\partial v)}_T}$$

(2)

In Equation (2), s is the molar entropy and a Maxwell relationship is used to determine its volumetric dependence.

In the critical point (cp), the thermodynamic relationship is given as follows [17]:

$$\sqrt{{{\displaystyle \frac{dP}{dT}}|}_{\mathrm{c}\mathrm{p}}}=\sqrt{{{\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_v|}_{\mathrm{c}\mathrm{p}}}\equiv {e\mathrm{S}\mathrm{P}}_{\mathrm{c}\mathrm{p}}$$

(3)

In the calculation of ${{\displaystyle \frac{dP}{dT}}|}_{\mathrm{c}\mathrm{p}}$, the end point of saturated vapor pressure curve differentiated by temperature as shown in Equation (4) was used, which is referred to in ref. [18].

$$\sqrt{{{\displaystyle \frac{dP}{dT}}|}_{\mathrm{c}\mathrm{p}}}=\sqrt{P_c\left(-{\displaystyle \frac{B_0}{{T_c}^2}}+{\displaystyle \frac{C_0}{T_c}}+D_0{E}_0{T_c}^{E_0-1}\right)}$$

(4)

In Equation (3), the ${{\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_v|}_{\mathrm{c}\mathrm{p}}$ was calculated for each equation of state. In Equation (5), average relative deviation (ARD) was defined as follows:

$$\mathrm{A}\mathrm{R}\mathrm{D}\left({e\mathrm{S}\mathrm{P}}_{\mathrm{c}\mathrm{p}}\right) [\%]={\displaystyle \frac{100}{n}}{\displaystyle {\sum }_{i=1}^n}\left|{\displaystyle \frac{\sqrt{{\frac{dP}{dT}|}_{\mathrm{c}\mathrm{p}}}-\sqrt{{{\left(\frac{\partial P}{\partial T}\right)}_v|}_{\mathrm{c}\mathrm{p}}}}{\sqrt{{\frac{dP}{dT}|}_{\mathrm{c}\mathrm{p}}}}}\right|$$

(5)

In Table 2, the thermodynamic relationship shown in Equation (3) was assessed for the 28 substances selected in this work. The ARD (${e\mathrm{S}\mathrm{P}}_{\mathrm{c}\mathrm{p}}$) values were 39.3% for the original SRK EoS [1], 4.3% for the original PR EoS [2] and 1.9% for the original eSPT-PR EoS [13], respectively. Due to the high ARD (${e\mathrm{S}\mathrm{P}}_{\mathrm{c}\mathrm{p}}$) value for the original SRK EoS compared with the other EoSs, function forms were examined for its improvement.

By applying the concepts of both SRK EoS and eSPT-PR EoS, the equation denoted as eSPT-SRK EoS was parameterized as follows:

$$P={\displaystyle \frac{RT}{v-Eb}}-CT\left({\displaystyle \frac{1}{v-Eb}}-{\displaystyle \frac{1}{v+Fb}}\right)-{\displaystyle \frac{a_{\mathrm{c}} \alpha \left(T\right)+D}{v\left(v+Fb\right)}}$$

(6)

In Equation (6), C, D, E and F were newly introduced into the eSPT-SRK EoS, though the original SRK EoS has C = D = 0 and E = F = 1.

One can obtain the fugacity coefficient $\phi$ from the departure function (Equation (7)) by substitution of Equation (6):

$$\ln \phi =\ln {\displaystyle \frac{f}{p}}={\displaystyle \frac{ 1}{RT}}{\int }_V^{\infty }\left({\displaystyle \frac{p}{n}}-{\displaystyle \frac{RT}{V}}\right)dV+\left(z-1\right)-\ln z$$

(7)

where f and z are fugacity and compressibility factor, respectively. The resulting expression for the fugacity coefficient $\phi$ can be expressed as follows:

$$\ln \phi =\ln {\displaystyle \frac{v}{v-Eb}}+{\displaystyle \frac{C}{R}}\ln {\displaystyle \frac{v-Eb}{v+Fb}}-{\displaystyle \frac{\left(a_c\alpha \left(T\right)+D\right)}{FbRT}}\ln {\displaystyle \frac{v+Fb}{v}}+\left(z-1\right)-\ln z$$

(8)

Entropy can be calculated from the departure function, (Equation (9)):

$$s\left(T,P\right)-s_{\mathrm{I}\mathrm{G}}\left(T,P\right)=R \ln z+{\int }_{T,v=\infty }^{T,v}\left[{\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_v-{\displaystyle \frac{R}{v}}\right]dv$$

(9)

where IG is ideal gas state at constant T, respectively and ${\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_v$can be calculated by substituting in Equation (6) as follows:

$${\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_v={\displaystyle \frac{R}{v-Eb}}-C\left({\displaystyle \frac{1}{v-Eb}}-{\displaystyle \frac{1}{v+Fb}}\right)-{\displaystyle \frac{a_c{\left(\frac{\partial \alpha \left(T\right)}{\partial T}\right)}_v}{v\left(v+Fb\right)}}$$

(10)

The resulted entropy difference for gas or liquid phase can be expressed as follows:

$$s_G\left(T,P\right)-s_{\mathrm{I}\mathrm{G}}\left(T,P\right)=R \ln z_G+R\ln {\displaystyle \frac{v_G-Eb}{v_G}}+C\ln {\displaystyle \frac{v_G+Fb}{v_G-Eb}}-{\displaystyle \frac{a_c}{Fb}}{\left({\displaystyle \frac{\partial \alpha \left(T\right)}{\partial T}}\right)}_{v_G}\ln {\displaystyle \frac{v_G}{v_G+Fb}}$$

(11)

$$s_L\left(T,P\right)-s_{\mathrm{I}\mathrm{G}}\left(T,P\right)=R \ln z_L+R\ln {\displaystyle \frac{v_L-Eb}{v_L}}+C\ln {\displaystyle \frac{v_L+Fb}{v_L-Eb}}-{\displaystyle \frac{a_c}{Fb}}{\left({\displaystyle \frac{\partial \alpha \left(T\right)}{\partial T}}\right)}_{v_L}\ln {\displaystyle \frac{v_L}{v_L+Fb}}$$

(12)

Thus, both fugacity and entropy calculated at any conditions with the eSPT-SRK EoS, which overcomes issues in the original eSPT-PR EoS as conditions approach the ideal gas state. The corrected form of the eSPT-PR EoS is given in the Appendix.

Table 2. ARD (${e\mathrm{S}\mathrm{P}}_{\mathrm{c}\mathrm{p}}$) values in 28 selected chemicals.

Table 2. ARD (${e\mathrm{S}\mathrm{P}}_{\mathrm{c}\mathrm{p}}$) values in 28 selected chemicals.

For eSPT-SRK EoS parameters a_c and b in Equation (6), a constant C was introduced as follows:

$$a_{\mathrm{c}}=a_0{\displaystyle \frac{R^2{\left(1-C/R\right)}^2{ T}_C^2}{P_C}}$$

(13)

$$b=b_0{\displaystyle \frac{R\left(1-C/R\right) T_C}{P_C}}$$

(14)

The a(T) function of original SRK EoS was adopted and is described by Equation (15):

$$\alpha \left(T\right)=1+\left(K_{C1}+K_{C2}\omega +K_{C3}{\omega }^2\right)\left(1-T_{\mathrm{r}}^{0.5}\right)$$

(15)

In Eq (15), T_r is a reduced temperature (= T/T_C). For eSPT-SRK EoS, pure parameters $a_0$ in Equation (13), $b_0$ in Equation (14) and $K_{C1}$, $K_{C2}$, $K_{C3}$ in Equation (15) were redetermined by fitting to experimental data for the compounds in the database.

The objective function for data fitting was chosen to be dimensionless as follows:

$$\begin{gathered} \mathrm{O}.\mathrm{F}.=\sum {[w_1\left\{{\displaystyle \frac{\left({\rho }_{L,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}-{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}\right)}{{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}}}\right\}}^2+w_2{\left\{{\displaystyle \frac{\left({\rho }_{C,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}-{\rho }_{C,\mathrm{E}\mathrm{x}\mathrm{p}}\right)}{{\rho }_{C,\mathrm{E}\mathrm{x}\mathrm{p}}}}\right\}}^2+w_3{\left\{{\displaystyle \frac{\sqrt{{\frac{dP}{dT}|}_{\mathrm{c}\mathrm{p}}}-\sqrt{{{\left(\frac{\partial P}{\partial T}\right)}_v|}_{\mathrm{c}\mathrm{p}}}}{\sqrt{{\frac{dP}{dT}|}_{\mathrm{c}\mathrm{p}}}}}\right\}}^2+ \\ {w}_4{\left\{{\displaystyle \frac{\sqrt{{\frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}|}_{\mathrm{S}\mathrm{A}\mathrm{T}}\left(\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right(17\left)\right)}-\sqrt{{\frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}|}_{\mathrm{S}\mathrm{A}\mathrm{T}}\left(\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right(18\left)\right)}}{\sqrt{{\frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}|}_{\mathrm{S}\mathrm{A}\mathrm{T}}\left(\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right(17\left)\right)}}}\right\}}^2] \end{gathered}$$

(16)

where ${{\displaystyle \frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}}|}_{\mathrm{S}\mathrm{A}\mathrm{T}}$ is ${\displaystyle \frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}}$ value at a standard temperature (T₀ = 298.2 K) as calculated with Equation (17) or Equation (18), respectively. Equation (18) is the Clapeyron equation and $\sqrt{{\displaystyle \frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}}}$ value can be calculated with the eSPT-SRK EoS as follows:

$$\sqrt{{{\displaystyle \frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}}|}_{\mathrm{S}\mathrm{A}\mathrm{T}}}=\sqrt{P_{\mathrm{v}\mathrm{p}}\left(-{\displaystyle \frac{B_0}{{T_0}^2}}+{\displaystyle \frac{C_0}{T_0}}+D_0{E}_0{T_0}^{E_0-1}\right)}$$

(17)

$$\sqrt{{\displaystyle \frac{d{P}_{\mathrm{v}\mathrm{p}}}{dT}}}=\sqrt{{\displaystyle \frac{{∆}_{\mathrm{v}\mathrm{p}}s}{{∆}_{\mathrm{v}\mathrm{p}}v}}}=\sqrt{{\displaystyle \frac{s_{\mathrm{G}}-s_{\mathrm{L}}}{v_{\mathrm{G}}-v_{\mathrm{L}}}}}$$

(18)

In Equation (16), arbitrary weights were used to adjust each term to have the same order of magnitude.

Average relative deviation (ARD) of the calculated properties was defined as follows.

$$ARD\left({\rho }_L^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}\right)={\displaystyle \frac{100}{n}}{\displaystyle {\sum }_{i=1}^n}\left|{\displaystyle \frac{{\rho }_{L,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}-{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}}{{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}}}\right|$$

(19)

$$ARD\left({\rho }_c\right)={\displaystyle \frac{100}{n}}{\displaystyle {\sum }_{i=1}^n}\left|{\displaystyle \frac{{\rho }_{c,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}-{\rho }_{c,\mathrm{E}\mathrm{x}\mathrm{p}}}{{\rho }_{c,\mathrm{E}\mathrm{x}\mathrm{p}}}}\right|$$

(20)

3. Results and Discussion

Substance-dependence C and D parameters in Equation (6) were correlated against the critical compressibility factor (z_c) as shown in Equations (21) and (22), respectively:

$$C=C_1{z}_{\mathrm{c}}+C_0$$

(21)

$$D=D_1{z}_{\mathrm{c}}+D_0$$

(22)

The five pure parameters ($a_0$ in Equation (13), $b_0$ in Equation (14) and $K_{C1}$, $K_{C2}$, $K_{C3}$ in Equation (15)) and constant C₀, C₁, D₀ and D₁ in Equations (21) and (22) were determined by properties of the selected 28 chemicals (Table 1) using the objective function in Equation (16) by a least-squares method. Namely, a total 11 parameters including E and F in Equation (6) were simultaneously fitted. The resulting parameters are listed in Table 2 and detailed results are given in Table 3.

The eSPT-SRK EoS compared favorably with the original eSPT-PR EoS (Table 3). In Table 4, four pure substances (n-hexane, CO₂, methanol and water) were predicted with the eSPT-SRK EoS given by Equations (6) at several (T, P) conditions and compared with the pc-SAFT EoS [19]. For water, the pc-SAFT EoS was fit to an intermediate isotherm of NIST data (523.15 K) and then was applied to calculate densities at the other stated conditions.

In the calculation of ARD [%], values from the NIST chemistry webbook [20] were used for reference as follows:

$$ARD\left({\rho }_G\right)={\displaystyle \frac{100}{n}}{\displaystyle {\sum }_{i=1}^n}\left|{\displaystyle \frac{{\rho }_{G,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}-{\rho }_{G,\mathrm{N}\mathrm{I}\mathrm{S}\mathrm{T}}}{{\rho }_{G,\mathrm{N}\mathrm{I}\mathrm{S}\mathrm{T}}}}\right|$$

(23)

$$ARD\left({\rho }_L\right)={\displaystyle \frac{100}{n}}{\displaystyle {\sum }_{i=1}^n}\left|{\displaystyle \frac{{\rho }_{L,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}-{\rho }_{L,\mathrm{N}\mathrm{I}\mathrm{S}\mathrm{T}}}{{\rho }_{L,\mathrm{N}\mathrm{I}\mathrm{S}\mathrm{T}}}}\right|$$

(24)

ARDs of liquid densities of the eSPT-SRK EoS (Table 4) were comparable with those of both the eSPT-PR EoS and pc-SAFT EoS for two non-polar substances (n-hexane, CO₂), but were higher than those of pc-SAFT EoS for two polar substances (methanol, water). On the other hand, ARDs of gas densities were much lower for the eSPT-SRK EoS than the eSPT-PR EoS (Table 4). The pc-SAFT EoS is neither a cubic equation nor a fixed-degree polynomial; instead, it incorporates statistical-mechanical integral terms that enable accurate description of substance properties in the critical region. All cubic EoS have a cubic critical isotherm and a quadratic coexistence curve [17] unlike that of the pc-SAFT EoS. Nevertheless, the generalized form of the eSPT-SRK EoS in this work improves gaseous density representation compared with the eSPT-PR EoS (Table 4).

Table 2. Pure parameters in orignial SRK EoS and eSPT-SRK EoS given by Equation (6).

	a0		b0	KC1	KC2	KC3
Original SRK EoS [1]	0.42748		0.08664	0.48	1.574	-0.176
eSPT-SRK EoS (This work)	0.21407		0.044749	-0.050191	2.48002	-1.29804
	C1	C0	D1	D0	E	F
Original SRK EoS [1]	0	0	0	0	1	1
eSPT-SRK EoS (This work)	-39.9554	1.56665	-1.27049	0.31888	0.679395	2.38633

Table 2. Pure parameters in orignial SRK EoS and eSPT-SRK EoS given by Equation (6).

	a0		b0	KC1	KC2	KC3
Original SRK EoS [1]	0.42748		0.08664	0.48	1.574	-0.176
eSPT-SRK EoS (This work)	0.21407		0.044749	-0.050191	2.48002	-1.29804
	C1	C0	D1	D0	E	F
Original SRK EoS [1]	0	0	0	0	1	1
eSPT-SRK EoS (This work)	-39.9554	1.56665	-1.27049	0.31888	0.679395	2.38633

28 substances selected in this work. The results showed that eSPT-SRK EoS was compared well with the original eSPT-PR EoS, although the deviations in eSPT-SRK EoS were slightly higher than those in the original eSPT-PR EoS. However, eSPT-SRK EoS improved the reliability compared with the original SRK EoS.

For comparison of the entropy-based Hansen-type solubility parameter with the eSPT-SRK EoS, Equation (6) was differentiated with temperature to give:

$${{e\mathrm{S}\mathrm{P}}^2=\left(\partial P/\partial T\right)}_v={\displaystyle \frac{R}{v-Eb}}-C\left({\displaystyle \frac{1}{v-Eb}}-{\displaystyle \frac{1}{v+Fb}}\right)-{\displaystyle \frac{a_{\mathrm{c}} {\left(\frac{\partial \alpha \left(T\right)}{\partial T}\right)}_v}{v\left(v+Fb\right)}}$$

(25)

In the eSPT-SRK EoS, polar and hydrogen bond terms were defined by the following equations:

$${{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{o}\mathrm{S}}}^2={\displaystyle \frac{C}{v+Fb}}$$

(26)

$${{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{E}\mathrm{o}\mathrm{S}}}^2=-{\displaystyle \frac{a_{\mathrm{c}} {\left(\frac{\partial \alpha \left(T\right)}{\partial T}\right)}_v}{v\left(v+Fb\right)}}$$

(27)

where subscript EoS is the calculation with the eSPT-SRK EoS in this case.

The relationship between ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{O}\mathrm{S}}$ and ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$ of which value was taken consideration of corresponding to the same contribution of ${\delta }_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}$ in the ${\delta }_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ and that between ${e\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{E}\mathrm{o}\mathrm{S}}$ and ${\delta }_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$ of which value was taken consideration of corresponding to the same contribution of ${\delta }_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}$ in the ${\delta }_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ were respectively investigated as shown in Figure 2 and Figure 3. In Figure 2, the ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$ was quadratically linked to the ${e\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{o}\mathrm{S}}$ in HSP (R² = 0.4546), although some deviations between them were observed. In Figure 3, the ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$ was found to be strongly correlated to the ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{E}\mathrm{o}\mathrm{S}}$ in HSP (R² = 0.9543). Thus, we summarized that an eHSP concept was developed as shown in the following predictive forms

$${e\mathrm{S}\mathrm{P}}^2={({e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}^2=){{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}}^2+{{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}}^2+{{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}}^2$$

(28)

From the positive correlations observed in both Figure 2 and Figure 3, we found the next relationships:

$${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}= -0.7181 {\left({e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{o}\mathrm{S}}\right)}^2+59.323 {e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{o}\mathrm{S}}- 450.10(R^2=0.4546)$$

(29)

$${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}=2.3679\times {e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{E}\mathrm{o}\mathrm{S}}-949.95(R^2=0.9543)$$

(30)

$${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n},\mathrm{E}\mathrm{o}\mathrm{S}}=\sqrt{{{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l},\mathrm{E}\mathrm{o}\mathrm{S}}}^2-{{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{o}\mathrm{S}}}^2-{{e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{E}\mathrm{o}\mathrm{S}}}^2}$$

(31)

Table 5 shows the eHSP values for the selected 28 chemicals. The prediction with the eSPT-SRK EoS was well correlated with the original HSP data [21]. Figure 5 shows relationship between the predicted eHSP and original HSP at 298.2 K and 0.1013 MPa. Each term denoted “Dispersion”, “Polar”, “Hydrogen bond”, respectively showed linearity between predicted eHSPs and the original HSP. Tus, the method to calculate eHSP with the eSPT-SRK EoS was developed. The present prediction method with the eSPT-SRK EoS is applicable to the calculation of HSP values under arbitrary temperature, pressure, and mixture composition conditions. Results with the corrected form of the eSPT-PR EoS gave similar results to the eSPT-SRK EoS.

Table 3. ARDs calculated with eSPT-SRK EoS from experimental data values.

No.	Substance	Group	ARD (${\mathit{\rho }}_{\mathit{L}}^{\mathbf{S}\mathbf{A}\mathbf{T}\mathbf{P}}$) [%]		ARD (${\mathit{\rho }}_{\mathit{C}}$) [%]
1	Water	Water	1.3	(2.4)1	1.2	(1.4)1
2	Ethylene Glycol	Alcohol	4.1	(2.7)1	0.9	(2.2) 1
3	N-Methyl-2-pyrrolidone	Cyclic Compound	2.3	(0.9)1	9.0	(1.5) 1
4	Methanol	Alcohol	0.3	(6.5)1	5.8	(1.2) 1
5	Ethanol	Alcohol	5.4	(7.5)1	3.6	(1.8) 1
6	Dimethyl Sulfide	Sulfur compound	1.3	(4.6)1	3.1	(2.4)1
7	1-Propanol	Alcohol	7.9	(6.5)1	0.1	(2.1) 1
8	N,N-Dimethylformamide	Nitrogen compound	10.7	(6.1)1	18.6	(0.9) 1
9	1-Butanol	Alcohol	8.9	(5.6)1	1.4	(2.2) 1
10	2-Butanol	Alcohol	7.0	(5.7)1	0.0	(2.1) 1
11	Pyridine	Aromatic	2.3	(2.5)1	7.9	(2.6) 1
12	Cyclopentanone	Ketone	4.8	(8.0)1	10.5	(1.4) 1
13	Acetophenone	Ketone	4.3	(2.5)1	3.8	(2.3) 1
14	Dichloromethane	Halide	2.8	(0.1)1	2.7	(2.4) 1
15	Dimethyl Carbonate	Carbonate	2.1	(2.8)1	2.1	(1.9) 1
16	Cyclohexanone	Ketone	3.7	(5.8)1	11.0	(1.4) 1
17	Acetone	Ketone	3.4	(2.2)1	5.6	(1.6) 1
18	Tetrahydrofuran	Aromatic	6.2	(7.9)1	1.6	(2.2) 1
19	Benzene	Aromatic	4.6	(4.0)1	4.8	(2.4) 1
20	Toluene	Aromatic	5.1	(2.8)1	3.9	(2.3) 1
21	1-Decanol	Alcohol	4.3	(2.8)1	3.2	(2.2) 1
22	Trans-Decahydronaphthalene	Aromatic	11.8	(4.9)1	6.5	(2.4) 1
23	Cyclohexane	Cyclic Compound	5.5	(4.5)1	6.7	(2.5) 1
24	Tetradecane	Alkane	0.4	(4.6)1	4.2	(1.8) 1
25	1-Decene	Alkene	6.6	(2.9)1	0.1	(2.1) 1
26	Decane	Alkane	4.3	(0.6)1	0.2	(2.1) 1
27	Hexane	Alkane	2.8	(3.3)1	4.6	(2.4) 1
28	Carbon Dioxide	Inorganic Carbon	-	-	7.8	(2.6) 1
-	-	Average	5.2	(4.0)1	4.7	(2.0)1

¹ Values obtained with the original eSPT-PR EoS (predictive form) [13] (in parentheses).

Table 3. ARDs calculated with eSPT-SRK EoS from experimental data values.

No.	Substance	Group	ARD (${\mathit{\rho }}_{\mathit{L}}^{\mathbf{S}\mathbf{A}\mathbf{T}\mathbf{P}}$) [%]		ARD (${\mathit{\rho }}_{\mathit{C}}$) [%]
1	Water	Water	1.3	(2.4)1	1.2	(1.4)1
2	Ethylene Glycol	Alcohol	4.1	(2.7)1	0.9	(2.2) 1
3	N-Methyl-2-pyrrolidone	Cyclic Compound	2.3	(0.9)1	9.0	(1.5) 1
4	Methanol	Alcohol	0.3	(6.5)1	5.8	(1.2) 1
5	Ethanol	Alcohol	5.4	(7.5)1	3.6	(1.8) 1
6	Dimethyl Sulfide	Sulfur compound	1.3	(4.6)1	3.1	(2.4)1
7	1-Propanol	Alcohol	7.9	(6.5)1	0.1	(2.1) 1
8	N,N-Dimethylformamide	Nitrogen compound	10.7	(6.1)1	18.6	(0.9) 1
9	1-Butanol	Alcohol	8.9	(5.6)1	1.4	(2.2) 1
10	2-Butanol	Alcohol	7.0	(5.7)1	0.0	(2.1) 1
11	Pyridine	Aromatic	2.3	(2.5)1	7.9	(2.6) 1
12	Cyclopentanone	Ketone	4.8	(8.0)1	10.5	(1.4) 1
13	Acetophenone	Ketone	4.3	(2.5)1	3.8	(2.3) 1
14	Dichloromethane	Halide	2.8	(0.1)1	2.7	(2.4) 1
15	Dimethyl Carbonate	Carbonate	2.1	(2.8)1	2.1	(1.9) 1
16	Cyclohexanone	Ketone	3.7	(5.8)1	11.0	(1.4) 1
17	Acetone	Ketone	3.4	(2.2)1	5.6	(1.6) 1
18	Tetrahydrofuran	Aromatic	6.2	(7.9)1	1.6	(2.2) 1
19	Benzene	Aromatic	4.6	(4.0)1	4.8	(2.4) 1
20	Toluene	Aromatic	5.1	(2.8)1	3.9	(2.3) 1
21	1-Decanol	Alcohol	4.3	(2.8)1	3.2	(2.2) 1
22	Trans-Decahydronaphthalene	Aromatic	11.8	(4.9)1	6.5	(2.4) 1
23	Cyclohexane	Cyclic Compound	5.5	(4.5)1	6.7	(2.5) 1
24	Tetradecane	Alkane	0.4	(4.6)1	4.2	(1.8) 1
25	1-Decene	Alkene	6.6	(2.9)1	0.1	(2.1) 1
26	Decane	Alkane	4.3	(0.6)1	0.2	(2.1) 1
27	Hexane	Alkane	2.8	(3.3)1	4.6	(2.4) 1
28	Carbon Dioxide	Inorganic Carbon	-	-	7.8	(2.6) 1
-	-	Average	5.2	(4.0)1	4.7	(2.0)1

¹ Values obtained with the original eSPT-PR EoS (predictive form) [13] (in parentheses).

Table 4. Prediction of properties for four pure chemicals with eSPT-SRK EoS given by Equation (6) at several (T, P) conditions. For water, the pc-SAFT EoS [19] was fit to an intermediate isotherm (523.15 K) for parameter determination.

n-Hexane	eSPT-SRK EoSARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoSARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoSARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	9.1	-	1.8	-	2.2	-
(Tr = 1.15, Pr = 1.2)	-	10.5	-	23.9	-	2.6
CO2	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	12.3	-	1.1		6.0	-
(Tr = 1.2, Pr = 1.2)	-	3.9	-	19.0	-	5.1
Methanol	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	12.6	-	3.5	-	0.53	-
(Tr = 1.2, Pr = 1.2)	-	0.5	-	43.7	-	1.7
Water	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoS*ARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoS*ARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	18.0	-	3.9	-	0.09	-
(Tr = 1.2, Pr = 1.2)	-	5.3	-	42.3	-	1.5

* m_i 1.59364, σ [Å] 2.46841,ε_i/kT [K] 235.681, k_AB 0.071007,ε_AB/kT [K] 3387.68.

Table 4. Prediction of properties for four pure chemicals with eSPT-SRK EoS given by Equation (6) at several (T, P) conditions. For water, the pc-SAFT EoS [19] was fit to an intermediate isotherm (523.15 K) for parameter determination.

n-Hexane	eSPT-SRK EoSARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoSARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoSARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	9.1	-	1.8	-	2.2	-
(Tr = 1.15, Pr = 1.2)	-	10.5	-	23.9	-	2.6
CO2	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	12.3	-	1.1		6.0	-
(Tr = 1.2, Pr = 1.2)	-	3.9	-	19.0	-	5.1
Methanol	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	12.6	-	3.5	-	0.53	-
(Tr = 1.2, Pr = 1.2)	-	0.5	-	43.7	-	1.7
Water	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-SRK EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	Original eSPT-PR EoSARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoS*ARD(${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoS*ARD(${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	18.0	-	3.9	-	0.09	-
(Tr = 1.2, Pr = 1.2)	-	5.3	-	42.3	-	1.5

* m_i 1.59364, σ [Å] 2.46841,ε_i/kT [K] 235.681, k_AB 0.071007,ε_AB/kT [K] 3387.68.

Figure 1. Thermodynamic relationship, Equation (3), assessed for four types of equations of state for 28 substances (Table 1).

Figure 1. Thermodynamic relationship, Equation (3), assessed for four types of equations of state for 28 substances (Table 1).

Table 5. Entropy-based Hansen solubility parameter (eHSP) of 28 selected chemicals (Table 1).

No.	Substance	${\mathit{e}\mathbf{H}\mathbf{S}\mathbf{P}|}_{\mathbf{D}\mathbf{i}\mathbf{s}\mathbf{p}\mathbf{e}\mathbf{r}\mathbf{s}\mathbf{i}\mathbf{o}\mathbf{n}}$ [(Pa/K)0.5]	${\mathit{e}\mathbf{H}\mathbf{S}\mathbf{P}|}_{\mathbf{P}\mathbf{o}\mathbf{l}\mathbf{a}\mathbf{r}}$ [(Pa/K)0.5]	${\mathit{e}\mathbf{H}\mathbf{S}\mathbf{P}|}_{\mathbf{H}\mathbf{y}\mathbf{d}\mathbf{r}\mathbf{o}\mathbf{g}\mathbf{e}\mathbf{n}\mathbf{b}\mathbf{o}\mathbf{n}\mathbf{d}}$ [(Pa/K)0.5]
1	Water	682.0 (514.5)1	704.0 (710.2)2	1861 (1912)3
2	Ethylene Glycol	748.4 (880.2)1	484.3 (414.2)2	1145 (1076)3
3	N-Methyl-2-pyrrolidone	781.3 (891.3)1	533.9 (229.0)2	312.5 (381.7)3
4	Methanol	669.5 (588.8)1	545.4 (488.6)2	988.8 (1067)3
5	Ethanol	706.9 (650.7)1	393.7 (413.6)2	867.9 (901.8)3
6	Dimethyl Sulfide	604.9 (746.5)1	539.1 (392.9)2	335.3 (238.8)3
7	1-Propanol	714.6 (731.6)1	303.7 (355.9)2	777.1 (737.9)3
8	N,N-Dimethylformamide	649.6 (845.0)1	511.5 (240.9)2	421.9 (298.4)3
9	1-Butanol	703.0 (778.0)1	250.4 (298.5)2	694.2 (586.5)3
10	2-Butanol	690.9 (758.2)1	249.3 (294.5)2	634.1 (528.7)3
11	Pyridine	859.5 (852.7)1	398.1 (349.2)2	266.9 (345.1)3
12	Cyclopentanone	744.1 (832.0)1	494.7 (268.0)2	216.2 (283.8)3
13	Acetophenone	848.3 (859.1)1	372.2 (212.9)2	160.1 (315.7)3
14	Dichloromethane	796.2 (769.4)1	341.9 (418.1)2	332.5 (307.3)3
15	Dimethyl Carbonate	705.2 (783.5)1	391.3 (304.9)2	441.3 (371.4)3
16	Cyclohexanone	795.8 (818.4)1	281.7 (218.6)2	228.0 (215.7)3
17	Acetone	677.2 (751.4)1	454.4 (330.7)2	305.8 (290.4)3
18	Tetrahydrofuran	771.0 (778.4)1	261.6 (353.1)2	367.1 (258.3)3
19	Benzene	866.8 (779.5)1	0.0 (329.5)2	94.22 (209.3)3
20	Toluene	833.9 (777.3)1	64.86 (264.8)2	92.66 (183.1)3
21	1-Decanol	644.3 (765.0)1	189.3 (83.38)2	422.8 (191.9)3
22	Trans-Decahydronaphthalene	793.1 (770.0)1	0.0 (163.8)2	0.0 (93.80)3
23	Cyclohexane	790.8 (731.0)1	0.0 (282.6)2	9.413 (104.1)3
24	Tetradecane	667.0 (666.2)1	0.0 (0.0)2	0.0 (14.71)3
25	1-Decene	700.6 (698.7)1	44.34 (95.09)2	97.55 (66.88)3
26	Decane	695.1 (687.9)1	0.0 (82.85)2	0.0 (50.07)3
27	Hexane	696.2 (659.4)1	0.0 (219.0)2	0.0 (39.11)3
28	Carbon Dioxide	18.42 (-)1	1.706 (-)2	1.408 (-)3

¹ Prediction with Equation (31), ² Prediction with Equation (29) and ³ Prediction with Equation (30) (in parentheses).

Table 5. Entropy-based Hansen solubility parameter (eHSP) of 28 selected chemicals (Table 1).

No.	Substance	${\mathit{e}\mathbf{H}\mathbf{S}\mathbf{P}|}_{\mathbf{D}\mathbf{i}\mathbf{s}\mathbf{p}\mathbf{e}\mathbf{r}\mathbf{s}\mathbf{i}\mathbf{o}\mathbf{n}}$ [(Pa/K)0.5]	${\mathit{e}\mathbf{H}\mathbf{S}\mathbf{P}|}_{\mathbf{P}\mathbf{o}\mathbf{l}\mathbf{a}\mathbf{r}}$ [(Pa/K)0.5]	${\mathit{e}\mathbf{H}\mathbf{S}\mathbf{P}|}_{\mathbf{H}\mathbf{y}\mathbf{d}\mathbf{r}\mathbf{o}\mathbf{g}\mathbf{e}\mathbf{n}\mathbf{b}\mathbf{o}\mathbf{n}\mathbf{d}}$ [(Pa/K)0.5]
1	Water	682.0 (514.5)1	704.0 (710.2)2	1861 (1912)3
2	Ethylene Glycol	748.4 (880.2)1	484.3 (414.2)2	1145 (1076)3
3	N-Methyl-2-pyrrolidone	781.3 (891.3)1	533.9 (229.0)2	312.5 (381.7)3
4	Methanol	669.5 (588.8)1	545.4 (488.6)2	988.8 (1067)3
5	Ethanol	706.9 (650.7)1	393.7 (413.6)2	867.9 (901.8)3
6	Dimethyl Sulfide	604.9 (746.5)1	539.1 (392.9)2	335.3 (238.8)3
7	1-Propanol	714.6 (731.6)1	303.7 (355.9)2	777.1 (737.9)3
8	N,N-Dimethylformamide	649.6 (845.0)1	511.5 (240.9)2	421.9 (298.4)3
9	1-Butanol	703.0 (778.0)1	250.4 (298.5)2	694.2 (586.5)3
10	2-Butanol	690.9 (758.2)1	249.3 (294.5)2	634.1 (528.7)3
11	Pyridine	859.5 (852.7)1	398.1 (349.2)2	266.9 (345.1)3
12	Cyclopentanone	744.1 (832.0)1	494.7 (268.0)2	216.2 (283.8)3
13	Acetophenone	848.3 (859.1)1	372.2 (212.9)2	160.1 (315.7)3
14	Dichloromethane	796.2 (769.4)1	341.9 (418.1)2	332.5 (307.3)3
15	Dimethyl Carbonate	705.2 (783.5)1	391.3 (304.9)2	441.3 (371.4)3
16	Cyclohexanone	795.8 (818.4)1	281.7 (218.6)2	228.0 (215.7)3
17	Acetone	677.2 (751.4)1	454.4 (330.7)2	305.8 (290.4)3
18	Tetrahydrofuran	771.0 (778.4)1	261.6 (353.1)2	367.1 (258.3)3
19	Benzene	866.8 (779.5)1	0.0 (329.5)2	94.22 (209.3)3
20	Toluene	833.9 (777.3)1	64.86 (264.8)2	92.66 (183.1)3
21	1-Decanol	644.3 (765.0)1	189.3 (83.38)2	422.8 (191.9)3
22	Trans-Decahydronaphthalene	793.1 (770.0)1	0.0 (163.8)2	0.0 (93.80)3
23	Cyclohexane	790.8 (731.0)1	0.0 (282.6)2	9.413 (104.1)3
24	Tetradecane	667.0 (666.2)1	0.0 (0.0)2	0.0 (14.71)3
25	1-Decene	700.6 (698.7)1	44.34 (95.09)2	97.55 (66.88)3
26	Decane	695.1 (687.9)1	0.0 (82.85)2	0.0 (50.07)3
27	Hexane	696.2 (659.4)1	0.0 (219.0)2	0.0 (39.11)3
28	Carbon Dioxide	18.42 (-)1	1.706 (-)2	1.408 (-)3

¹ Prediction with Equation (31), ² Prediction with Equation (29) and ³ Prediction with Equation (30) (in parentheses).

Figure 2. Relationship in Equation (29) between ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{O}\mathrm{S}}$ and ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$.

Figure 2. Relationship in Equation (29) between ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{E}\mathrm{O}\mathrm{S}}$ and ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$.

Figure 3. Relationship in Equation (30) between ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{E}\mathrm{o}\mathrm{S}}$ and ${e\mathrm{H}\mathrm{S}\mathrm{P}}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$.

Figure 3. Relationship in Equation (30) between ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{E}\mathrm{o}\mathrm{S}}$ and ${e\mathrm{H}\mathrm{S}\mathrm{P}}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d},\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$.

Figure 4. Relationship between the predicted eHSP and original HSP. (a) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}$ (R² = 0.993), (b) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}$ (R² = 0.783), (c) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}$ (R² = 0.969), (d) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ (Hildebrand SP) (R² = 0.995).

Figure 4. Relationship between the predicted eHSP and original HSP. (a) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}$ (R² = 0.993), (b) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}$ (R² = 0.783), (c) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{H}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}$ (R² = 0.969), (d) ${e\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ vs. ${\mathrm{H}\mathrm{S}\mathrm{P}|}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ (Hildebrand SP) (R² = 0.995).

4. Conclusions

A translation type of correction (eSP) was applied to develop an alternative form of the popular Soave-Redlich-Kwong equation. The resulting eSPT-SRK EoS showed good correlation results for both liquid molar volumes and critical molar volumes. Thermodynamic expressions of the eSPT-SRK EoS and the corrected eSPT-PR EoS are simple and have closed forms. In the eSPT-SRK EoS, the additional constants C and D correlate with the compressibility factor Z_C which gives a convenient predictive form. The eHSP values estimated with either the eSPT-SRK EoS or the eSPT-PR EoS can be used to estimate conditions required for extractions and separations that are at conditions far removed from atmospheric pressure and ambient temperature.