Entropy-Based Solubility Parameter-Translated Peng–Robinson Equation of State (eSPT-PR EoS)

Abstract

Peng–Robinson equation of state (PR EoS) has good prediction accuracy for phase diagrams of pure substances or mixtures, but liquid density, especially for high polar substances, is known to be ~20% lower value compared with experimental data at standard atmospheric temperature and pressure (SATP) conditions. To overcome this issue, translation via entropy-based solubility parameter (eSP) Peng–Robinson EoS (eSPT-PR EoS) is proposed in this work. The technique uses eSP for the liquid phase at SATP conditions and correlates the ideal value and a constant C for each substance as a correction. As a result, the C value can be linearly correlated with critical compressibility factor (Z_C). Finally, the liquid density was improved and gave an average relative deviation (ARD) value of 4.2% for the generally used 27 chemicals selected at SATP condition. Furthermore, critical density was also improved and gave ARD values of 3.9% compared with the original PR EoS of 21.8%. Thus, a universal calculation method based on PR EoS was developed for improving liquid density representation with the eSPT-PR EoS.

1. Introduction

Peng–Robinson equation of state (PR EoS) [1] is widely used by scientists and engineers to predict high-pressure phase diagrams for manufacturing products. Not only pure substances, but also multiple mixtures are studied in the prediction of phase diagrams. The utility of PR EoS for petroleum industries [2,3] is very wide and is commonly applied to simulation of both super- and sub-critical processes in the manufacture of chemical products [4,5,6,7,8].

However, the PR EoS has low prediction ability for liquid density, especially for high polar substances like ethanol, methanol and water. Therefore, volume-translated PR EoS (VT-PR EoS) has been previously proposed by researchers [9,10,11,12] to improve volumetric representation. The VT-PR EoS method has become well established, but always needs experimental density data and a fixed reference condition, which is disadvantageous at times, because experimental density data may be limited or may require estimation when conditions of the simulation are very different from those where density data are available. Therefore, a different type of translation technique is needed to predict phase diagrams reliably, keeping the future development of chemical products in mind.

Hildebrand Solubility Parameter (SP) is widely used for processing extraction and separation systems that involve chemical-dominated phenomena. Entropy-based solubility parameter (eSP) [13,14], which is linearly correlated to the Hildebrand SP [15] at the standard temperature and pressure (SATP) condition, is used not only for vapor–liquid multiple phases but also for a single phase like supercritical fluids for both pure substances and mixtures. From this point of view, eSP has been used for extraction and separation processing of high-pressure systems like subcritical fluid separation (SFS) or supercritical fluid extractions (SFE). In preliminary research, calculated eSP values with the original PR EoS for liquid density at SATP condition deviated from the theoretical values. Therefore, in this study, it was decided to consider a correction to PR EoS as a translation. For adding this eSP correction to PR EoS, a database of chemicals was used as a basis, as described in detail later. These solvents are usually seen in chemistry, for example, in the processing of organic–inorganic hybrid nano particles [16] for dispersion, coating and catalyst preparation.

The added eSP translation correction to the PR EoS is denoted as eSPT-PR EoS. In the eSPT-PR EoS, a newly added constant C is proposed, and the resulting C value is generalized with fundamental properties of chemical species.

2. Materials and Methods

Table 1. Fundamental properties of 27 selected chemicals.

No.	Substance	Mw 1 [-]	TC 2 [K]	PC 3 [MPa]	ρC 4 [kg/m3]	ω 5 [-]	ρL at SATP [kg/m3]	Hildebrand SP [MPa0.5]
1	Water	18.015	647.1	22.064	321.98	0.345	997.0	48.0
2	Ethylene Glycol	62.068	719	8.2	331.91	0.521	1110.1	34.5
3	N-Methyl-2-pyrrolidone	99.131	721.6	4.52	319.78	0.373	1026.1	30.3
4	Methanol	32.042	512.5	8.084	273.86	0.566	789.6	29.5
5	Ethanol	46.068	514	6.137	274.21	0.644	785.9	26.4
6	Dimethyl Sulfide	62.134	503.04	5.53	309.12	0.194	842.6	26.3
7	1-Propanol	60.095	536.8	5.169	274.41	0.621	799.5	24.6
8	N,N-Dimethylformamide	73.094	649.6	4.42	279.00	0.318	944.5	24.0
9	1-Butanol	74.122	563.1	4.414	271.51	0.588	804.0	23.3
10	2-Butanol	74.122	535.9	4.188	274.53	0.581	802.3	22.6
11	Pyridine	79.1	619.95	5.63	311.42	0.239	978.0	21.8
12	Cyclopentanone	84.116	624.5	4.6	326.03	0.288	944.2	21.1
13	Acetophenone	120.149	709.6	4.01	311.27	0.383	1023.5	20.9
14	Dichloromethane	84.933	510	6.08	459.10	0.199	1318.2	20.4
15	Dimethyl Carbonate	90.078	548	4.5	358.88	0.385	1063.2	20.3
16	Cyclohexanone	98.143	653	4	315.57	0.299	942.7	20.1
17	Acetone	58.079	508.1	4.7	272.67	0.307	785.6	19.8
18	Tetrahydrofuran	72.106	540.15	5.19	321.90	0.225	880.0	19.1
19	Benzene	78.112	562.05	4.895	305.13	0.21	873.0	18.8
20	Toluene	92.138	591.75	4.108	291.58	0.264	863.9	18.3
21	1-Decanol	158.281	688	2.308	245.40	0.607	821.0	17.8
22	Trans-Decahydronaphthalene	138.25	687	3.2	288.02	0.299	866.7	17.0
23	Cyclohexane	84.159	553.8	4.08	273.24	0.208	773.1	16.7
24	Tetradecane	198.388	693	1.57	221.17	0.643	759.3	15.9
25	1-Decene	140.266	616.6	2.223	240.18	0.48	738.2	15.9
26	Decane	142.282	617.7	2.11	230.60	0.492	726.6	15.5
27	Hexane	86.175	507.6	3.025	232.28	0.301	656.0	15.0

¹ Mw: Molecular weight, ² T_c: Critical temperature, ³ P_c: Critical pressure, ⁴ ρ_c: Critical density, ⁵ ω: Acentric factor.

shows the fundamental properties for the 27 selected substances that are generally used in chemistry. Critical properties and acentric factors are listed. The Hildebrand solubility parameter (SP) is generally evaluated at ambient conditions [15] and is defined for organic compounds by Equation (1):

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

(1)

In Equation (1), ${∆}_{\mathrm{V}\mathrm{P}}U$ is the cohesive energy and $v_{\mathrm{L}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}}$ 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 ${∆}_{\mathrm{v}\mathrm{a}\mathrm{p}}H$ is the enthalpy of vaporization and $v_{\mathrm{G}\mathrm{a}\mathrm{s}}$ is the molar volume of gas.

The eSP [13,14] 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

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

(2)

In Equation (2), s is the molar entropy, and a Maxwell relationship is used to determine its volumetric dependence. The relationship between SP and eSP is estimated with the following thermodynamic relationships:

$${\left({\displaystyle \frac{\partial u_L}{\partial v_L}}\right)}_T\cong {\mathrm{S}\mathrm{P}}^2=T{\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_v-P=T{{e\mathrm{S}\mathrm{P}}_{\mathrm{I}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}}^2-P$$

(3)

where eSP_Ideal is the theoretical eSP, which was plotted as a function of Hildebrand SP, as shown in Figure 1 for 124 substances from ref. [17] at SATP conditions (298.2 K and 101.3 kPa). In Figure 1, the results calculated with the original PR EoS were added, which shows a large deviation from the theoretical line. The slope of theoretical line was 1.17 (= 57.9/49.5) times larger than the slope of the original PR EoS. Therefore, it is clear that a correction in form of eSP can be added to improve liquid density representation compared with the original PR EoS.

For the liquid density corrections with eSPT-PR EoS, the original PR EoS in Equation (4):

$$P={\displaystyle \frac{RT}{v-b}}-{\displaystyle \frac{a_{\mathrm{c}}\alpha \left(T\right)}{v\left(v+b\right)+b\left(v-b\right)}}$$

(4)

was modified as given by Equation (5):

$$P={\displaystyle \frac{RT}{v-b}}-{\displaystyle \frac{C{T}^{\beta }}{f\left(v\right)}}-{\displaystyle \frac{a_{\mathrm{c}}\alpha \left(T\right)}{v\left(v+b\right)+b\left(v-b\right)}}$$

(5)

A constant C was newly introduced into the eSPT-PR EoS, even though the original PR EoS has only two constants. In the eSPT-PR EoS model, Equation (5), f (v) = v − b and β = 1 were adopted.

For EoS parameters a_c and b in Equation (5), the constant C was also introduced as follows:

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

(6)

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

(7)

The $\alpha \left(T\right)$ function of original PR EoS was adopted and is described by Equation (8):

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

(8)

In Equation (8), T_r is a reduced temperature (= T/T_C). For eSPT-PR EoS, pure parameters $a_0$ in Equation (6), $b_0$ in Equation (7) and $K_{C1}$, $K_{C2}$, $K_{C3}$ in Equation (8) 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:

$$\mathrm{O}.\mathrm{F}.=\sum {[w_1\left\{{\displaystyle \frac{\left({\rho }_{L,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}-{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}\right)}{{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}}}\right\}}^2+w_2{\left\{{\displaystyle \frac{\left({\rho }_{C,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}-{\rho }_{C,\mathrm{E}\mathrm{x}\mathrm{p}}\right)}{{\rho }_{C,\mathrm{E}\mathrm{x}\mathrm{p}}}}\right\}}^2+w_3{\left\{{\displaystyle \frac{\left({e\mathrm{S}\mathrm{P}}_{L,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}-{e\mathrm{S}\mathrm{P}}_{L,\mathrm{I}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}\right)}{{e\mathrm{S}\mathrm{P}}_{L,\mathrm{I}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}}}\right\}}^2]$$

(9)

In Equation (9), arbitrary weights were used to adjust each term to have the same order of magnitude. Average Relative Deviation (ARD) was defined as follows:

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

(10)

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

(11)

3. Results

The five pure-component parameters ($a_0$ in Equation (6), $b_0$ in Equation (7) and $K_{C1}$, $K_{C2}$, $K_{C3}$ in Equation (8)) and each constant C in Equation (5), which is substance-dependent due to the 27 selected chemicals, were determined using the objective function in Equation (9) by a least-squares method. Namely, a total of 32 (=5 + 27) parameters were simultaneously determined by data fitting. The resulting parameters are listed in

Table 2. Pure-component parameters in original PR EoS given by Equation (4) and eSPT-PR EoS given by Equation (5).

	a0	b0	KC1	KC2	KC3	C
Original PR EoS [1]	0.45724	0.07780	0.37464	1.54226	−0.26990	0
eSPT-PR EoS	0.47024	0.08085	0.32183	1.84761	−0.18613	Substance-dependent

and detailed results are given in

Table 3. Obtained constant C of 27 selected chemicals with eSPT-PR EoS and deviations from experimental data values.

No.	Substance	Group	C [J·mol−1·K−1]	ARD (${\mathit{\rho }}_{\mathit{L}}^{\mathbf{S}\mathbf{A}\mathbf{T}\mathbf{P}}$) [%]		ARD (${\mathit{\rho }}_{\mathit{C}}$) [%]
1	Water	Water	1.7068	3.4	(17.5) 1	3.8	(28.6) 1
2	Ethylene Glycol	Alcohol	1.1163	1.4	(10.1) 1	1.2	(20.2) 1
3	N-Methyl-2-pyrrolidone	Cyclic Compound	1.6422	2.7	(17.1) 1	3.0	(27.3) 1
4	Methanol	Alcohol	1.7852	5.1	(17.5) 1	5.8	(30.9) 1
5	Ethanol	Alcohol	1.2025	6.4	(6.7) 1	6.0	(24.9) 1
6	Dimethyl Sulfide	Sulfur compound	0.5601	5.5	(2.4) 1	5.0	(17.3) 1
7	1-Propanol	Alcohol	0.8655	6.2	(2.0) 1	5.6	(21.1) 1
8	N,N-Dimethylformamide	Nitrogen compound	2.3108	3.0	(37.7) 1	1.0	(33.3) 1
9	1-Butanol	Alcohol	0.7835	5.8	(1.2) 1	5.3	(19.9) 1
10	2-Butanol	Alcohol	0.8906	5.8	(2.7) 1	5.3	(21.0) 1
11	Pyridine	Aromatic	0.4891	1.7	(0.5) 1	1.7	(13.7) 1
12	Cyclopentanone	Ketone	1.5600	6.2	(11.7) 1	6.2	(28.9) 1
13	Acetophenone	Ketone	0.9394	1.7	(7.0) 1	1.4	(18.4) 1
14	Dichloromethane	Halide	0.7571	3.2	(2.4) 1	2.7	(17.5) 1
15	Dimethyl Carbonate	Carbonate	1.1720	4.3	(7.8) 1	3.8	(22.9) 1
16	Cyclohexanone	Ketone	1.6131	5.1	(13.8) 1	5.2	(28.7) 1
17	Acetone	Ketone	1.5010	3.8	(13.4) 1	3.6	(26.3) 1
18	Tetrahydrofuran	Aromatic	0.6463	6.9	(2.5) 1	6.4	(19.4) 1
19	Benzene	Aromatic	0.5147	5.1	(2.5) 1	4.7	(16.6) 1
20	Toluene	Aromatic	0.6947	4.5	(0.5) 1	4.0	(17.9) 1
21	1-Decanol	Alcohol	1.0130	1.4	(8.7) 1	1.2	(19.0) 1
22	Trans-Decahydronaphthalene	Aromatic	0.4547	5.4	(3.3) 1	5.2	(16.3) 1
23	Cyclohexane	Cyclic Compound	0.3409	5.3	(4.8) 1	5.1	(15.1) 1
24	Tetradecane	Alkane	1.5126	0.0	(18.4) 1	0.4	(23.9) 1
25	1-Decene	Alkene	1.0125	4.4	(5.5) 1	3.9	(21.2) 1
26	Decane	Alkane	1.0892	3.2	(7.9) 1	2.8	(21.1) 1
27	Hexane	Alkane	0.6060	4.8	(0.8) 1	4.4	(17.3) 1
-	-		Average	4.2	(8.4) 1	3.9	(21.8) 1

¹ Results obtained from original PR EoS (in parentheses).

.

4. Discussion

At first, the resulting constant C was plotted against many fundamental properties including compressibility factor Z_C. As a result, it was found that the constant C was proportional to the value of Z_C as follows:

$$C=D{Z}_{\mathrm{C}}+E$$

(12)

In Equation (12), D = −28.537 and E = 8.2377 (R² = 0.9285). Then, the calculated data were refit to the experimental data to redetermine constant D and E in Equation (12), as well as five pure parameters ($a_0$ in Equation (6), $b_0$ in Equation (7) and $K_{C1}$, $K_{C2}$, $K_{C3}$ in Equation (8)). According to this treatment, the number of fitting parameters was reduced from a total of 32 parameters to a total of 7 parameters, and universal constants were determined as D = −27.6704 and E = 8.73306 (R² = 1.000).

In

Table 4. Comparison of generalized eSPT-PR EoS given by Equations (5) and (12) with other equations of state.

	a0	b0	KC1	KC2	KC3	D	E	ARD (${\mathit{\rho }}_{\mathit{L}}^{\mathbf{S}\mathbf{A}\mathbf{T}\mathbf{P}}$) [%] 1	ARD (${\mathit{\rho }}_{\mathit{C}}$) [%] 1
Original PR EoS [1]	0.45724	0.07780	0.37464	1.54226	−0.26990	-	-	8.4	21.8
VT-PR EoS [10] (Predictive form)	0.45724	0.07780	-	-	-	-	-	7.3	21.8
eSPT-PR EoS (Predictive form)	0.51119	0.09079	0.34687	1.93487	−0.25698	−27.6704	8.73306	4.1	2.0

¹ 27 chemicals selected in this research as shown in Table 1.

, the eSPT-PR EoS was compared with the original PR EoS and the VTPR-EoS. For calculation with the eSPT-PR EoS, the objective function was changed to Equation (13) to account for all terms. In Equation (13), the saturated pressure was added to the standard atmospheric temperature ($P_{\mathrm{E}\mathrm{q}}^{\mathrm{S}\mathrm{A}\mathrm{T}}$) as follows:

$$\mathrm{O}.\mathrm{F}.=\sum {[w_1\left\{{\displaystyle \frac{\left({\rho }_{L,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}-{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}\right)}{{\rho }_{L,\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}}}\right\}}^2+w_2{\left\{{\displaystyle \frac{\left({\rho }_{C,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}-{\rho }_{C,\mathrm{E}\mathrm{x}\mathrm{p}}\right)}{{\rho }_{C,\mathrm{E}\mathrm{x}\mathrm{p}}}}\right\}}^2+w_3{\left\{{\displaystyle \frac{\left({e\mathrm{S}\mathrm{P}}_{L,\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}-{e\mathrm{S}\mathrm{P}}_{L,\mathrm{I}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}\right)}{{e\mathrm{S}\mathrm{P}}_{L,\mathrm{I}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}^{\mathrm{S}\mathrm{A}\mathrm{T}\mathrm{P}}}}\right\}}^2+w_4{\left\{{\displaystyle \frac{\left(P_{\mathrm{E}\mathrm{q},\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathrm{S}\mathrm{A}\mathrm{T}}-P_{\mathrm{E}\mathrm{q},\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}}\right)}{P_{\mathrm{E}\mathrm{q},\mathrm{E}\mathrm{x}\mathrm{p}}^{\mathrm{S}\mathrm{A}\mathrm{T}}}}\right\}}^2]$$

(13)

The arbitrarily weighted calculation method was also applied to Equation (13) because of the different magnitudes of the terms. The eSPT-PR EoS compared favorably with the original and volume-translated PR-EoS (Table 4). In

Table 5. Prediction of properties for four pure chemicals with eSPT-PR EoS given by Equation (5) according to Equation (12) at several (T, P) conditions. For water, the pc-SAFT EoS [18] was fit to an intermediate isotherm (523.15 K) for parameter determination.

n-Hexane	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoS ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoS ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	1.8	-	2.2	-
(Tr = 1.15, Pr = 1.2)	-	23.9	-	2.6
CO2	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoS ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoS ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	1.1		6.0	-
(Tr = 1.2, Pr = 1.2)	-	19.0	-	5.1
Methanol	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoS ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoS ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	3.5	-	0.53	-
(Tr = 1.2, Pr = 1.2)	-	43.7	-	1.7
Water	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	eSPT-PR EoS ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]	pc-SAFT EoS * ARD (${\mathit{\rho }}_{\mathit{L}}$) [%]	pc-SAFT EoS * ARD (${\mathit{\rho }}_{\mathbf{G}}$) [%]
(Tr = 0.8, Pr = 0.8)	3.9	-	0.09	-
(Tr = 1.2, Pr = 1.2)	-	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.

, four pure substances (n-hexane, CO₂, methanol and water) were predicted with the eSPT-PR EoS given by Equations (5) and (12) at several (T, P) conditions and compared with the pc-SAFT EoS [18]. 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 [19] were used for reference as follows:

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

(14)

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

(15)

ARDs of liquid densities of the eSPT-PR EoS (Table 5) were generally comparable with those of the pc-SAFT EoS for two non-polar substances (n-hexane, CO₂) but were higher than those of the pc-SAFT EoS for two polar substances (methanol, water). Furthermore, ARDs of gas densities were much lower for the pc-SAFT EoS than the eSPT-PR EoS (Table 5). The pc-SAFT EoS is neither a cubic equation nor any fixed-degree polynomial, but includes statistical mechanic integral terms that capture the trend of liquid and vapor densities in the critical region better than cubic EoS. All cubic EoSs have a cubic critical isotherm and a quadratic coexistence curve as conditions [20], unlike that of the pc-SAFT EoS. Nevertheless, generalized form of the eSPT-PR EoS improves liquid density representation compared with the PR-EoS (Table 4) and provides comparable representation of liquid densities calculated with the pc-SAFT EoS.

5. Conclusions

A translation type of correction (eSP) was added to the PR EoS. The resulting eSPT-PR EoS showed good prediction ability for both liquid molar volumes and critical molar volumes. In the eSPT-PR EoS, the newly added constant C could be well correlated with compressibility factor Z_C and thus, a predictive form was produced based on the original PR EoS.

For the use of predictive eSPT-PR EoS, if experimental density data are available, the data can be used for data-fitting to redetermine the more accurate C value. If data are unavailable, then the generalized values of D and E given in this work can be used to estimate C.