Correlating Pure Component Properties with MOSCED Solubility Parameters: Enthalpy of Vaporization and Vapor Pressure

Abstract

Tools to predict vapor–liquid phase equilibria are indispensable for the conceptualization and design of separation processes. Modified separation of cohesive energy density (MOSCED) is a solubility-parameter-based method parameterized to make accurate predictions of the limiting activity coefficient. As a solubility-parameter-based method, MOSCED can not only make quantitative predictions, but can shed light on the underlying intermolecular interactions. In the present study, we demonstrated the ability of MOSCED to correlate the enthalpy of vaporization and vapor pressure at a specific temperature using multiple linear regression. With this addition, MOSCED is able to predict vapor–liquid phase equilibria in the absence of reference data. This was demonstrated for the prediction of the Henry’s constant and solvation free energy of organic solutes in water, which was found to be superior to mod-UNIFAC. In addition to being able to make phase equilibrium predictions, the ability to correlate the enthalpy of vaporization and vapor pressure offers the opportunity to include additional properties in the regression of the MOSCED parameters. Given this success, we additionally attempted to correlate a wide range of physical properties using a similar expression. While, in some cases, the results were reasonable, they were inferior to the correlations of the enthalpy of vaporization and vapor pressure. Future efforts will be needed to improve the correlations.

1. Introduction

Methods to predict phase equilibria are crucial for the design of industrial separation processes, which typically constitute 40–70% of both the capital and operating cost of a chemical plant. Separation processes are non-spontaneous processes requiring an external “separating agent”, typically either energy or a solvent, which make them expensive [1]. Here, we considered vapor–liquid separation processes, which include distillation, absorption, and stripping. Distillation alone constitutes approximately 90–95% of the separation processes used in practice and, in the United States, accounts for approximately 11% of all in-plant energy consumed [2,3].

Solubility-parameter-based methods have long been utilized for early-stage process conceptualization and design applications as a result of their ability to both predict the phase equilibrium and shed light on the underlying intermolecular interactions [4,5,6,7,8,9,10,11,12]. When using these methods to compute limiting activity coefficients, we decompose the log limiting activity coefficient into the sum of a combinatorial (COMB) and residual (RES) contribution, where the COMB contribution results from size and shape dissimilarities between the components in the system and the RES contribution results from intermolecular interactions. The COMB contribution may be computed using available athermal solution theories, such as the Flory–Huggins or Flory–Huggins–Staverman–Guggenheim equations [4,13,14].

The earliest solubility parameter method to predict the RES contribution was Hildebrand and Scatchard’s “Regular Solution Theory” (RST). However, RST is limited in that it can only predict positive contributions, as well as being limited to modeling dispersion interactions [4,5]. Solubility parameter methods have been greatly improved by splitting the solubility parameter into a sum of contributions [5,11]. Perhaps the most-well-known of these methods is the Hansen solubility parameter (HSP), which accounts for dispersion, polar, and hydrogen bonding interactions [6]. This results in an expression of the RES contribution of the limiting activity coefficient of the form:

$$ln{\gamma }_2^{\mathrm{RES},\infty }=\frac{v_2}{RT}\left[{\left({\lambda }_2-{\lambda }_1\right)}^2+{\left({\tau }_2-{\tau }_1\right)}^2+{\left({\alpha }_2{\beta }_2-{\alpha }_1{\beta }_1\right)}^2\right]$$

(1)

where ${\gamma }_2^{\mathrm{RES},\infty }$ is the RES contribution of the limiting activity coefficient of Component 2, R is the molar gas constant, T is the absolute temperature, $v_2$ is the pure component molar volume of Component 2, ${\lambda }_i$ is the pure component solubility parameter of component i, which accounts for the dispersion interactions, ${\tau }_i$ is the pure component solubility parameter of component i, which accounts for the polar interactions, and ${\alpha }_i{\beta }_i$ is the pure component solubility parameter of component i, which accounts for hydrogen bonding (or association), where $i=\left\{1,2\right\}$. Within the hydrogen bonding term, ${\alpha }_i{\beta }_i$ corresponds to self-association, where ${\alpha }_i$ accounts for the hydrogen bond donating ability and ${\beta }_i$ accounts for the hydrogen bond accepting ability of component i. However, HSP is still limited in that it can only predict positive RES contributions. Tijssen et al. [12] overcame this limitation by splitting the hydrogen bonding term, resulting in [12].

$$ln{\gamma }_2^{\mathrm{RES},\infty }=\frac{v_2}{RT}\left[{\left({\lambda }_2-{\lambda }_1\right)}^2+{\left({\tau }_2-{\tau }_1\right)}^2+2\left({\alpha }_2-{\alpha }_1\right)\left({\beta }_2-{\beta }_1\right)\right]$$

(2)

This expression forms the basis of modified separation of cohesive energy density (MOSCED), which we utilized here [15,16,17,18,19,20,21,22,23,24]. It should be noted that this splitting has recently been suggested to improve the accuracy of the HSP [10].

The limiting activity coefficient accounts for solute–solvent and solvent–solvent interactions and corresponds to the maximum deviation from the ideal solution behavior. Moreover, the limiting activity can be used to predict a wide range of phase equilibria for early-stage design applications [25]. For example, Henry’s constant for Component 2 in 1 (${\mathcal{H}}_{2,1}$) may be computed as [24]

$$ln{\mathcal{H}}_{2,1}\left(T,P\right)=ln{\gamma }_2^{\infty }\left(T,P\right)+ln{P}_2^{\mathrm{sat}}\left(T\right)$$

(3)

and the fundamental solvation free energy of Component 2 in 1 ($\Delta G_{2,1}^{\mathrm{solv}}$) [24]:

$$\frac{1}{RT}\Delta G_{2,1}^{\mathrm{solv}}\left(T,P\right)=ln{\gamma }_2^{\infty }\left(T,P\right)+ln\frac{v_1\left(T,P\right)P_2^{\mathrm{sat}}\left(T\right)}{RT}$$

(4)

where T and P are the temperature and pressure, respectively, $P_2^{\mathrm{sat}}$ is the vapor pressure of pure Component 2, R is the molar gas constant, and $v_1$ is the molar volume of Component 1, where we made the standard assumptions that the vapor phase is an ideal gas and the Poynting correction is negligible such that the pure component fugacity is equal to the pure component vapor pressure. Likewise, for the case of isothermal vapor–liquid equilibrium, it can be shown that, for a mono-azeotropic system, for a system to exhibit a minimum boiling azeotrope [26,27]:

$${\gamma }_2^{\infty }\left(T\right)>\frac{P_1^{\mathrm{sat}}\left(T\right)}{P_2^{\mathrm{sat}}\left(T\right)}>\frac{1}{{\gamma }_1^{\infty }\left(T\right)}$$

(5)

and for a system to exhibit a maximum boiling azeotrope [26,27]:

$${\gamma }_2^{\infty }\left(T\right)<\frac{P_1^{\mathrm{sat}}\left(T\right)}{P_2^{\mathrm{sat}}\left(T\right)}<\frac{1}{{\gamma }_1^{\infty }\left(T\right)}$$

(6)

where $P_1^{\mathrm{sat}}$ is the vapor pressure of pure Component 1 and Component 1 corresponds to the most-volatile component ($P_1^{\mathrm{sat}}>P_2^{\mathrm{sat}}$). It is additionally possible to use limiting activity coefficients to parameterize a binary interaction excess Gibbs free energy model, which may then be used to extrapolate to finite concentrations. For the case of Wilson’s equation, we have [4,24,28]

$$\begin{array}{cc}\hfill ln{\gamma }_1& =-ln\left(x_1+{\mathrm{\Lambda }}_{12}{x}_2\right)+x_2\left(\frac{{\mathrm{\Lambda }}_{12}}{x_1+{\mathrm{\Lambda }}_{12}{x}_2}-\frac{{\mathrm{\Lambda }}_{21}}{{\mathrm{\Lambda }}_{21}{x}_1+x_2}\right)\hfill \\ \hfill ln{\gamma }_2& =-ln\left(x_2+{\mathrm{\Lambda }}_{21}{x}_1\right)-x_1\left(\frac{{\mathrm{\Lambda }}_{12}}{x_1+{\mathrm{\Lambda }}_{12}{x}_2}-\frac{{\mathrm{\Lambda }}_{21}}{{\mathrm{\Lambda }}_{21}{x}_1+x_2}\right)\hfill \end{array}$$

(7)

where ${\mathrm{\Lambda }}_{12}$ and ${\mathrm{\Lambda }}_{21}$ are adjustable parameters, which may be related to the binary (intermolecular) interaction parameters (BIPs) of the system ($a_{12}$ and $a_{21}$):

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{12}=\frac{v_2}{v_1}exp\left[-\frac{a_{12}}{RT}\right]\\ \hfill {\mathrm{\Lambda }}_{21}=\frac{v_1}{v_2}exp\left[-\frac{a_{21}}{RT}\right]\end{array}$$

(8)

At infinite dilution, Equation (7) reduces to

$$\begin{array}{cc}\hfill ln{\gamma }_1^{\infty }& =-ln\left({\mathrm{\Lambda }}_{12}\right)+1-{\mathrm{\Lambda }}_{21}\hfill \\ \hfill ln{\gamma }_2^{\infty }& =-ln\left({\mathrm{\Lambda }}_{21}\right)+1-{\mathrm{\Lambda }}_{12}\hfill \end{array}$$

(9)

which can be used to solve for parameters ${\mathrm{\Lambda }}_{12}$ and ${\mathrm{\Lambda }}_{21}$. This may, in turn, be used to model vapor–liquid equilibrium, where

$$\begin{array}{c}\hfill x_1{\gamma }_1{P}_1^{\mathrm{sat}}=y_{1}P\\ \hfill x_2{\gamma }_2{P}_2^{\mathrm{sat}}=y_{2}P\end{array}$$

(10)

where $y_1$ and $y_2$ are the vapor phase mole fraction of Components 1 and 2, respectively, and recall $x_1+x_2=1$ and $y_1+y_2=1$.

Be that as it may, Equations (3)–(6) and (10) all require knowledge of the pure component vapor pressure. While the vapor pressure is available for a wide range of fluids, the ability to predict vapor pressure using MOSCED parameters would be of great value for early-stage process conceptualization and design applications. It has previously been shown that MOSCED parameters may be used to correlate the enthalpy of vaporization [29,30]. Given the relationship between the enthalpy of vaporization and log vapor pressure via the Clapeyron equation, we hypothesized that the correlation of the vapor pressure should also be possible. Likewise, Abraham solute descriptors have been used to correlate a wide range of thermophysical properties, including the enthalpy of vaporization and vapor pressure [31,32,33]. Since MOSCED parameters correspond to physical interactions, we expected that it should likewise be possible to use them as descriptors to correlate and predict vapor pressure.

Recently in 2005, MOSCED was subject to a “revision”, wherein the literature was surveyed and the parameters were regressed for 130 organic solvents, water, 2-room-temperature ionic liquids (ILs), and 5 non-condensable gases using experimental limiting activity coefficients [20]; this was recently expanded to an additional 33 1-n-alkyl-3-methylimidazolium-based ILs [34,35]. The further expansion of MOSCED is limited by the availability of experimental limiting activity coefficients. The ability to use MOSCED to correlate pure component properties could be of additional value in that it would allow one to incorporate additional data into the regression of new parameters. This has been demonstrated previously for MOSCED with the use of the enthalpy of vaporization [36]. Additionally, recent efforts have been made to predict MOSCED parameters devoid of experimental data. This includes the use of molecular simulation and electronic structure calculation to generate reference data [30,36,37,38,39] and traditional group contribution methods [29,36]. The present study is complementary to this work, wherein it presents the opportunity to predict phase equilibrium for early stage process conceptualization and design applications.

Know that many excellent methods exist to predict pure component properties from molecular structures. For example, the group contribution method of Gani and co-workers can be used to predict a wide range of pure component properties [40,41]. This includes the critical temperature, pressure, and acentric factor, which, in turn, could be used to predict vapor pressure using a cubic equation of state. Likewise, Gharagheizi et al. [42] recently developed a group contribution/machine learning method to predict vapor pressure directly. As mentioned earlier, Abraham solute descriptors have been used to correlate a wide range of thermophysical properties including vapor pressure [32]. The goal of the present study was to assess the ability to use MOSCED parameters to correlate and predict pure component properties, specifically the enthalpy of vaporization and vapor pressure. This would allow for the prediction of phase equilibrium using only MOSCED parameters for early-stage process conceptualization and design applications and, additionally, provides the opportunity to incorporate additional data into the parameterization of MOSCED. Moreover, in general, the ability to predict pure component properties and to provide molecular-level insight via the solubility parameter would be of great utility.

In the present study, we demonstrated the ability to successfully correlate the enthalpy of vaporization and vapor pressure at a specific temperature. Furthermore, we demonstrated the ability to use the predicted enthalpy of vaporization to extrapolate the predicted vapor pressure to additional temperatures. This was applied to predict ${\mathcal{H}}_{2,1}$ and $\Delta G_{2,1}^{\mathrm{solv}}$. We also summarize our results, wherein we attempted to correlate additional pure component properties, but with limited success.

2. Method

2.1. Theory

Using MOSCED, the log limiting activity coefficient of Component 2 (solute) in Component 1 (solvent), $ln{\gamma }_2^{\infty }$, is computed as [15,20]

$$\begin{array}{c}ln{\gamma }_2^{\infty }=ln{\gamma }_2^{\mathrm{COMB},\infty }+ln{\gamma }_2^{\mathrm{RES},\infty }\\ ln{\gamma }_2^{\mathrm{COMB},\infty }=ln{\left(\frac{v_2}{v_1}\right)}^{a{a}_2}+1-{\left(\frac{v_2}{v_1}\right)}^{a{a}_2}\\ ln{\gamma }_2^{\mathrm{RES},\infty }=\frac{v_2}{RT}\left[{\left({\lambda }_1-{\lambda }_2\right)}^2+\frac{q_1^2{q}_2^2{({\tau }_1^{\left(T\right)}-{\tau }_2^{\left(T\right)})}^2}{{\psi }_1}+\frac{({\alpha }_1^{\left(T\right)}-{\alpha }_2^{\left(T\right)})({\beta }_1^{\left(T\right)}-{\beta }_2^{\left(T\right)})}{{\xi }_1}\right]\\ a{a}_2=0.953-0.002314\left[{\left({\tau }_2^{\left(T\right)}\right)}^2+{\alpha }_2^{\left(T\right)}{\beta }_2^{\left(T\right)}\right]\\ {\alpha }_i^{\left(T\right)}={\alpha }_i{\left(\frac{293\mathrm{K}}{T}\right)}^{0.8},{\beta }_i^{\left(T\right)}={\beta }_i{\left(\frac{293\mathrm{K}}{T}\right)}^{0.8},{\tau }_i^{\left(T\right)}={\tau }_i{\left(\frac{293\mathrm{K}}{T}\right)}^{0.4},\mathrm{where}i=\left\{1\mathrm{or}2\right\}\\ {\psi }_1=\mathrm{POL}+0.002629{\alpha }_1^{\left(T\right)}{\beta }_1^{\left(T\right)}\\ {\xi }_1=0.68\left(\mathrm{POL}-1\right)+{\left[3.4-2.4exp\left(-0.002687{\left({\alpha }_1{\beta }_1\right)}^{1.5}\right)\right]}^{{\left(293\mathrm{K}/T\right)}^2}\\ \mathrm{POL}=q_1^4\left[1.15-1.15exp\left(-0.002337{\left({\tau }_1^{\left(T\right)}\right)}^3\right)\right]+1\end{array}$$

(11)

where $ln{\gamma }_2^{\mathrm{COMB},\infty }$, $ln{\gamma }_2^{\mathrm{RES},\infty }$ correspond to the combinatorial and residual contribution to $ln{\gamma }_2^{\infty }$, respectively, $v_i$ is the (liquid) molar volume, ${\lambda }_i$, ${\tau }_i$, ${\alpha }_i$, and ${\beta }_i$ are the solubility parameters due to dispersion, polarity, and hydrogen bond acidity and basicity, respectively, and the induction parameter, $q_i$, reflects the ability of the non-polar part of a molecule to interact with a polar part, where $i=\{1,2\}$. The terms ${\psi }_1$ and ${\xi }_1$ are (solvent-dependent) asymmetry terms; these terms are not adjustable, but are a function of the solvent solubility parameters. The asymmetry terms are important for the accuracy of MOSCED and are intended to account for the fact that descriptors are solute solubility parameters, and the interactions of a component as a solute at infinite dilution are different than a component as a solvent (in the pure component limit) [43]. The superscript $\left(T\right)$ is used to indicate temperature-dependent parameters, where the temperature dependence is computed using the empirical correlations provided in Equation (11). As suggested by the equations, MOSCED adopts a reference temperature of 293 K (20 ${}^{°}$C). The COMB term is modeled using a modified athermal Flory/Huggins equation, where $a{a}_2$ is an empirical (solute-dependent) term to modify the size dissimilarity for polar and hydrogen bonding interactions. The term $a{a}_2$ is not adjustable, but is a function of the solubility parameters of the solute. For all cases, $a{a}_2\le 0.953$, which reduces the effective size dissimilarity and magnitude of the combinatorial contribution, with the value smaller for polar and associating compounds. An equivalent expression for the the limiting activity coefficient for Component 1 in 2 ($ln{\gamma }_1^{\infty }$) can be written by switching the subscript indices in Equation (11).

MOSCED is based on the theory that the cohesive energy may be separated into individual contributions (due to specific intermolecular interactions), which are additive. MOSCED is a “modified” separation in that empirical asymmetry terms are introduced, which improve the accuracy of the model, in addition to the splitting of the association term. Both modifications are physically, but not theoretically justified. We, therefore, expected that we can relate the MOSCED parameters (partial solubility parameters) to the total cohesive energy ($\Delta U^{\mathrm{coh}}$). Following the work of [12], which, like MOSCED, splits the association parameter, we expected an expression of the form:

$$\frac{\Delta U^{\mathrm{coh}}}{RT}=\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta \right)$$

(12)

where $a_1$, $a_2$, and $a_3$ are constants that have been introduced to account for the empiricism introduced by MOSCED. In Equation (12), the cohesive energy is written as the sum of the contributions due to the dispersion ($v{\lambda }^2$), polar ($v{\tau }^2$), and association ($v\alpha \beta$) interactions. Assuming the vapor phase is an ideal gas and that the molar volume of the vapor phase is much greater than the liquid phase, the cohesive energy may be related to the enthalpy of vaporization ($\Delta H^{\mathrm{vap}}$) as

$$\frac{\Delta H^{\mathrm{vap}}}{RT}=\frac{\Delta U^{\mathrm{coh}}+RT}{RT}=\frac{\Delta U^{\mathrm{coh}}}{RT}+1=a_0+\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta \right)$$

(13)

where, in theory, $a_0=1$ is a constant. In this light, we view $a_0$ as being equal to a reference value of $\Delta H^{\mathrm{vap}}/\left(RT\right)$, our property of interest, and the term $v/\left(RT\right)\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta \right)$ as the deviation between the reference and actual value. Previously, it has been shown that using $a_0=1$ and taking $a_1$ to be either adjustable or $a_1=1$, Equation (13) can be used to accurately correlate $\Delta H^{\mathrm{vap}}$ at 20 ${}^{°}$C [29,30]. In this expression, the term $\alpha \beta$ corresponds to self-association interactions. Here, we also considered the following modified expression:

$$\frac{\Delta H^{\mathrm{vap}}}{RT}=a_0+\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(14)

where the self-association interaction term ($\alpha \beta$) is complemented with separate terms accounting for the hydrogen bond donating ($\alpha \alpha$) and accepting ($\beta \beta$) strength. In addition to correlating $\Delta H^{\mathrm{vap}}$ at 20 ${}^{°}$C, we also investigated the ability to correlate $\Delta H^{\mathrm{vap}}$ at 0 and 40 ${}^{°}$C. With the ability to predict $\Delta H^{\mathrm{vap}}$ at three temperatures for a given compound, one could extrapolate and interpolate to other temperatures, for example using a modified Watson’s equation of the form:

$$H^{\mathrm{vap}}=A{\left(1-\frac{T}{B}\right)}^n$$

(15)

where A, B, and n are all adjustable constants [44,45]. Note that, while in our previous work, we correlated $\Delta H^{\mathrm{vap}}$ in units of kJ/mol, here we divided through by the term $RT$ to form dimensionless terms.

Furthermore, via the Clapeyron equation, $ln{P}^{\mathrm{sat}}\propto \Delta H^{\mathrm{vap}}$. We, therefore, investigated the ability to correlate $ln{P}^{\mathrm{sat}}$ at a given temperature as

$$ln{P}^{\mathrm{sat}}=a_0+\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta \right)$$

(16)

and

$$ln{P}^{\mathrm{sat}}=a_0+\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(17)

We again considered the case of 0, 20, and 40 ${}^{°}$C. The intention was again that the ability to predict $ln{P}^{\mathrm{sat}}$ at three temperatures for a given compound would allow one to extrapolate and interpolate to other temperatures, for example using an Antoine equation of the form:

$$ln{P}^{\mathrm{sat}}=A-\frac{B}{T+C}$$

(18)

where A, B, and C are all adjustable constants. Within Equations (16) and (17), the second term on the right-hand side is dimensionless, and we took $a_0=ln{P}_0^{\mathrm{sat}}$ to be equal to a reference value of $ln{P}^{\mathrm{sat}}$, such that $ln{P}^{\mathrm{sat}}-a_0=ln{P}^{\mathrm{sat}}/P_0^{\mathrm{sat}}$ is dimensionless.

2.2. Data Compilation and Regression

In the most-recent MOSCED revision in 2005, the parameters were regressed for 130 organic solvents and water using experimental limiting activity coefficients [20]. Here, we considered only the organic solvents. Additionally we excluded acetic acid because of the difficulties in measuring $\Delta H^{\mathrm{vap}}$ and $ln{P}^{\mathrm{sat}}$ experimentally; monocarboxylic acids are known to associate strongly in the vapor phase with the measured values of $\Delta H^{\mathrm{vap}}$ and $ln{P}^{\mathrm{sat}}$ corresponding to a vapor phase consisting of a mixture of monomers and dimers [46]. In Equation (12), we assumed the vapor phase is an ideal gas, so the monomer value is needed. Reference values of $\Delta H^{\mathrm{vap}}$ and $ln{P}^{\mathrm{sat}}$ at 0, 20, and 40 ${}^{°}$C were sourced from Yaws’ Handbook of Thermodynamic and Physical Properties of Chemical Compounds [45] for as many of the organic solvents as possible at each temperature. The compiled reference data are provided in the Supporting Information accompanying the electronic version of this manuscript.

For Equations (13), (14), (16) and (17), the parameters were regressed using multiple linear regression with MATLAB [47]. The problem was formulated and solved using linear algebra and required the use of only basic matrix operations [48]. To quantify the goodness of fit, we subsequently computed the Pearson correlation coefficient ($R^2$), the root-mean-squared error (RMS), and the average absolute percent error (AAPE).

To further assess the accuracy with which we can predict $ln{P}^{\mathrm{sat}}$ and the sensitivity of the resulting phase equilibrium calculations, we predicted ${\mathcal{H}}_{2,1}$ and $\Delta G_{2,1}^{\mathrm{solv}}$ for all possible organic compounds in water using Equations (3) and (4) for which MOSCED parameters existed and reference data were available. The reference values for ${\mathcal{H}}_{2,1}$ were taken from Yaws’ Handbook of Properties for Aqueous Systems [49], and the reference values for $\Delta G_{2,1}^{\mathrm{solv}}$ were taken from the the FreeSolv database [50,51,52]. These are the same reference sets we used previously to assess MOSCED [34].

3. Results and Discussion

3.1. Regression

A summary of the regressed model parameters and goodness of fit is provided in

Table 1. Summary of the regressed model parameters using Equations (13) and (16) along with the Pearson correlation coefficient ($R^2$), the root-mean-squared error (RMS), and the average absolute percent error (AAPE) to assess the goodness of fit, where N corresponds to the number of organic compounds included in the fit. In the correlations, $P^{\mathrm{sat}}$ is in units of bar and $\Delta H^{\mathrm{vap}}$ corresponds to $\Delta H^{\mathrm{vap}}/\left(RT\right)$, which is dimensionless. The temperature indicates the temperature at which the regression was performed.

Property	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{R}}^2$	RMS	AAPE (%)	N
$ln{P}^{\mathrm{sat}}\left(0{}^{°}\mathrm{C}\right)$	4.94	−0.63	−1.65	−1.48	0.93	0.88	18.14	102
$ln{P}^{\mathrm{sat}}\left(20{}^{°}\mathrm{C}\right)$	5.14	−0.61	−1.61	−1.38	0.94	0.80	90.14	113
$ln{P}^{\mathrm{sat}}\left(40{}^{°}\mathrm{C}\right)$	5.34	−0.59	−1.59	−1.32	0.93	0.74	40.79	116
$\Delta H^{\mathrm{vap}}\left(0{}^{°}\mathrm{C}\right)$	3.38	0.94	2.20	2.92	0.91	1.65	6.86	95
$\Delta H^{\mathrm{vap}}\left(20{}^{°}\mathrm{C}\right)$	2.76	0.93	2.27	2.88	0.92	1.46	6.59	107
$\Delta H^{\mathrm{vap}}\left(40{}^{°}\mathrm{C}\right)$	2.04	0.93	2.35	2.81	0.93	1.33	6.67	113

for the use of Equations (13) and (16) to correlate $ln{P}^{\mathrm{sat}}$ and $\Delta H^{\mathrm{vap}}/\left(RT\right)$, respectively. In general, the correlation takes the form:

$$X=a_0+\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta \right)$$

(19)

where X is the property of interest and $a_0$ can be considered the reference value of property X.

In general, we found the correlations to be very good; in all cases, $R^2>0.9$. Considering first the case of $\Delta H^{\mathrm{vap}}/\left(RT\right)$, we found that $a_0$ is not equal to 1, as suggested by Equations (12) and (13). Comparing to our previous work [29], there, we set $a_0=a_1=1$ at 20 ${}^{°}$C and obtained $R^2=0.933$, $a_2=2.8373$, and $a_3=3.1152$ for 81 organic compounds (which also excluded acetic acid). Likewise, correlating the cohesive energy density, the work of Gnap and Elliott [30] suggested that $a_0=a_1=1$, $a_2=2.5$, and $a_3=3$, where acetic acid was also excluded. We found that here, the value of $a_1$ was close to 1, as well as $a_3>a_2$ with similar values.

While this previous work suggested that we could achieve a similar accuracy of $\Delta H^{\mathrm{vap}}/$$\left(RT\right)$ with $a_0=a_1=1$, here, we made the parameters adjustable in an attempt to generalize the ability of MOSCED to correlate arbitrary physical properties comparable to a traditional group contribution method (see, for example, the work of Gani and co-workers [40,41]). In this light, the terms ${\lambda }^2$ (dispersion), ${\tau }^2$ (polar), and $\alpha \beta$ (self-association) correspond to physical descriptors. For $\Delta H^{\mathrm{vap}}/\left(RT\right)$, we found that physical interactions increase the value of $\Delta H^{\mathrm{vap}}/\left(RT\right)$ with respect to the reference value, with the effect of dispersion less than polar interactions and polar slightly less than self-association interactions. This agrees with physical intuition.

Considering $ln{P}^{\mathrm{sat}}$, we again found that the correlation was very good. Compared to the reference value, we found that the physical interactions decreased the value of $ln{P}^{\mathrm{sat}}$, where, here, the effect of dispersion was less than self-association interactions and self-association slightly less than polar interactions.

Next, a summary of the regressed model parameters and goodness of fit is provided in

Table 2. Summary of the regressed model parameters using Equations (14) and (17) along with the Pearson correlation coefficient ($R^2$), the root-mean-squared error (RMS), and the average absolute percent error (AAPE) to assess the goodness of fit, where N corresponds to the number of organic compounds included in the fit. In the correlations, $P^{\mathrm{sat}}$ is in units of bar, and $\Delta H^{\mathrm{vap}}$ corresponds to $\Delta H^{\mathrm{vap}}/\left(RT\right)$, which is dimensionless. The temperature indicates the temperature at which the regression was performed.

Property	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{R}}^2$	RMS	AAPE (%)	N
$ln{P}^{\mathrm{sat}}\left(0{}^{°}\mathrm{C}\right)$	4.94	−0.63	−1.64	−1.59	0.11	0.00	0.93	0.87	18.25	102
$ln{P}^{\mathrm{sat}}\left(20{}^{°}\mathrm{C}\right)$	5.13	−0.61	−1.64	−1.34	−0.04	0.01	0.94	0.80	100.47	113
$ln{P}^{\mathrm{sat}}\left(40{}^{°}\mathrm{C}\right)$	5.30	−0.58	−1.56	−1.26	−0.03	−0.01	0.93	0.74	40.35	116
$\Delta H^{\mathrm{vap}}\left(0{}^{°}\mathrm{C}\right)$	3.40	0.94	2.18	3.11	−0.20	0.00	0.91	1.65	6.85	95
$\Delta H^{\mathrm{vap}}\left(20{}^{°}\mathrm{C}\right)$	2.78	0.93	2.28	2.82	0.04	0.00	0.92	1.45	6.54	107
$\Delta H^{\mathrm{vap}}\left(40{}^{°}\mathrm{C}\right)$	2.07	0.90	2.34	2.74	0.04	0.01	0.93	1.32	6.62	113

for the use of Equations (14) and (17) to correlate $ln{P}^{\mathrm{sat}}$ and $\Delta H^{\mathrm{vap}}/\left(RT\right)$, respectively. In general, the correlation takes the form:

$$X=a_0+\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(20)

where we added two additional terms corresponding to the hydrogen bond donating ($\alpha \alpha$) and accepting ($\beta \beta$) ability.

We again found that we were able to well correlate $ln{P}^{\mathrm{sat}}$ and $\Delta H^{\mathrm{vap}}/\left(RT\right)$. We found that the parameters changed little, with the values of $a_4$ and $a_5$ close to 0, further demonstrating that the additional terms had little effect. Nonetheless, moving forward in this work, we focused on the use of Equation (20). While the two additional parameters had little effect, we still had a model with only six adjustable parameters that may readily be regressed.

The ability of Equation (17) to correlate $ln{P}^{\mathrm{sat}}$ is illustrated in Figure 1 and Figure 2, a parity and residual plot, respectively. In Figure 1, we see that the values of $ln{P}^{\mathrm{sat}}$ were well correlated. In Figure 2, the error appears random. However, it does appear that the smaller the value of $ln{P}^{\mathrm{sat}}$, the larger the deviation between the calculated and reference value was. As noted earlier, the effect of intermolecular interactions was to decrease the value of $ln{P}^{\mathrm{sat}}$ as compared to the reference value. Therefore, the larger the deviation from the reference value, the larger the observed error was.

The ability of Equation (14) to correlate $\Delta H^{\mathrm{vap}}/\left(RT\right)$ is illustrated in Figure 3 and Figure 4, a parity and residual plot, respectively. In Figure 3, we likewise see that the values of $\Delta H^{\mathrm{vap}}/\left(RT\right)$ were well correlated. In Figure 4, the error appears random. However, it did appear that the larger the value of $\Delta H^{\mathrm{vap}}/\left(RT\right)$, the larger the deviation between the calculated and reference value was. As noted earlier, the effect of intermolecular interactions was to increase the value of $\Delta H^{\mathrm{vap}}/\left(RT\right)$ as compared to the reference value. Therefore, the larger the deviation from the reference value, the larger the observed error was. This was exactly the same as found for $ln{P}^{\mathrm{sat}}$.

3.2. Temperature-Dependent $P^{\mathrm{sat}}$

Having correlated $ln{P}^{\mathrm{sat}}$ at 0, 20, and 40 ${}^{°}$C, we next sought to use these values to obtain Antoine constants (see Equation (18)), which would allow one to interpolate and extrapolate to other temperatures. For the 130 organic solvents and water for which MOSCED parameters exist, we used the correlations to predict $ln{P}^{\mathrm{sat}}$ at 0, 20, and 40 ${}^{°}$C. These values were then used to solve for the Antoine constants A, B, and C.

Unfortunately, this approach was not successful for phenol, m-cresol, acetic acid, and water. As temperature increases, $P^{\mathrm{sat}}$ increases and is equal to the critical pressure at the critical point. The issue was that our correlations at 0, 20, and 40 ${}^{°}$C were independently correlated, and this requirement was, therefore, not enforced.

Given this limitation, we alternatively employed the Clausius–Clapeyron equation of the form [53]:

$$ln{P}^{\mathrm{sat}}=ln{P}^{\mathrm{sat,0}}-\frac{\Delta H^{\mathrm{vap,0}}}{R}\frac{1}{T}$$

(21)

where $P^{\mathrm{sat}}$ is the vapor pressure at the temperature of interest, T, and $P^{\mathrm{sat,0}}$ and $\Delta H^{\mathrm{vap,0}}$ are the vapor pressure and enthalpy of vaporization at the reference temperature, $T^0$. Within this expression, we assumed that the change in compressibility upon vaporization was approximately 1, $\Delta Z^{\mathrm{vap}}\approx 1$, and the the enthalpy of vaporization was constant and equal to the value at $T^0$. Since $P^{\mathrm{sat,0}}$ and $\Delta H^{\mathrm{vap,0}}$ are constants and $\Delta H^{\mathrm{vap,0}}>0$, Equation (21) ensures that as T increases, $P^{\mathrm{sat}}$ will increase.

Our assessment of the use of the Clausius–Clapeyron equation is provided in

Table 3. Summary of the goodness of fit of using the correlation (Equation (17)) for $ln{P}^{\mathrm{sat}}$ at the temperature of interest as compared to the Clausius–Clapeyron equation (Equation (21)) with the indicated reference temperature. The goodness of fit is quantified using the standard deviation ($\sigma$), the average absolute percent error (AAPE), and the root-mean-squared error (RMS), where N corresponds to the number of organic compounds for which reference value of $ln{P}^{\mathrm{sat}}$ are available at the temperature of interest. The goodness of fit is computed for $ln{P}^{\mathrm{sat}}$, where $P^{\mathrm{sat}}$ is in units of bar.

		Correlation			Clausius–Clapeyron
$\mathit{T}\left({}^{°}\mathbf{C}\right)$	N	$\mathit{\sigma }$	AAPE (%)	RMS	${\mathit{T}}^0\left({}^{°}\mathbf{C}\right)$	$\mathit{\sigma }$	AAPE (%)	RMS
0	102	0.64	18.25	0.87	20	0.70	70.80	0.90
					40	0.64	42.40	0.83
20	114	0.58	99.75	0.79	0	0.64	18.68	0.88
					40	0.54	41.86	0.74
40	117	0.54	40.10	0.74	0	0.67	14.55	0.88
					20	0.79	337.73	1.48

. Let us walk through the table so that we can better understand the result. In the first two columns, we indicate the temperature and the number of reference values of $ln{P}^{\mathrm{sat}}$ available at this temperature. In Columns 3 to 5, we report the standard deviation of the difference between the predicted and reference values ($\sigma$), AAPE, and RMS error for the correlation (Equation (17)) at the indicated temperature to quantify the goodness of fit. This information is the same as reported in Table 2, where we additionally included $\sigma$. We took the results of the correlation as a measure of the best we can hope to achieve at that temperature. Next, we made two sets of predictions using the Clausius–Clapeyron equation. In Column 6, we report the reference temperature used, and Columns 7 to 9 report the goodness of fit.

Consider the case of 0 ${}^{°}$C. At 0 ${}^{°}$C, reference values of $ln{P}^{\mathrm{sat}}$ were available for 102 compounds with the results of the correlation reported. For these 102 compounds, we then used Equations (17) and (14) to predict $ln{P}^{\mathrm{sat,0}}$ and $\Delta H^{\mathrm{vap,0}}$ at the reference temperature (20 or 40 ${}^{°}$C), as indicated. These reference values were then used in the Clausius–Clapeyron equation to predict $ln{P}^{\mathrm{sat}}$ at the temperature of interest, here 0 ${}^{°}$C. These predicted values were then compared to the reference values of $ln{P}^{\mathrm{sat}}$.

While we did not expect the predictions using the Clausius–Clapeyron equation to be as accurate as the direct correlation at that temperature, we found that they did appear reasonable. We subsequently evaluated their accuracy for phase equilibria predictions by next predicting the values of ${\mathcal{H}}_{2,1}$ and $\Delta G_{2,1}^{\mathrm{solv}}$ in water.

3.3. ${\mathcal{H}}_{2,1}$ and $\Delta G_{2,1}^{\mathrm{solv}}$ in Water

To assess the use of the Clausius–Clapeyron equation (CC) to predict $ln{P}^{\mathrm{sat}}$ for phase equilibria calculation, we next predicted the log Henry’s constant ($ln{\mathcal{H}}_{2,1}$, Equation (3)) and the solvation free energy ($\Delta G_{2,1}^{\mathrm{solv}}$, Equation (4)) for organic solutes (2) in water (1). The results are summarized in

Table 4. Summary of the error in the predicted log Henry’s constant ($ln{\mathcal{H}}_{2,1}$, Equation (3)) for organic solutes in water, where ${\mathcal{H}}_{2,1}$ is in units of bar. The error is quantified using the standard deviation computed for the difference between the calculated and reference value ($\sigma$) and the root-mean-squared error (RMS), the average absolute percent error (AAPE), the average absolute error (AAE), and $R^2$, where N is the number of organic solutes for which reference data were available. The majority of the data were at 25 ${}^{°}$C. Predictions were made for MOSCED with $ln{P}_2^{\mathrm{sat}}$ computed using the Clausius–Clapeyron equation with a reference temperature of 20 and 40 ${}^{°}$C, as indicated, plus the use of the reference vapor pressure for comparison. Predictions were additionally made using mod-UNIFAC with the reference vapor pressures.

${\mathit{P}}_2^{\mathbf{sat}}$ Calc	$\mathit{\sigma }$	RMS	AAPE (%)	AAE	${\mathit{R}}^2$	N
CC (20 ${}^{°}$C)	1.08	1.20	64.72	0.94	0.96	81
CC (40 ${}^{°}$C)	1.10	1.20	68.98	0.94	0.96	81
reference	0.97	1.07	10.61	0.74	0.97	81
mod-UNIFAC	2.12	3.73	27.77	3.08	0.63	60

and

Table 5. Summary of the error in the predicted solvation free energy ($\Delta G_{2,1}^{\mathrm{solv}}$, Equation (4)) for organic solutes in water, where $\Delta G_{2,1}^{\mathrm{solv}}$ is in units of kJ/mol. The error is quantified using the standard deviation computed for the difference between the calculated and reference value ($\sigma$) and the root-mean-squared error (RMS), the average absolute percent error (AAPE), the average absolute error (AAE), and $R^2$, where N is the number of organic solutes for which reference data were available. All of the data were at 25 ${}^{°}$C. Predictions were made for MOSCED with $ln{P}_2^{\mathrm{sat}}$ computed using the Clausius–Clapeyron equation with a reference temperature of 20 ${}^{°}$C plus the use of the reference vapor pressure for comparison. Predictions were additionally made using mod-UNIFAC with the reference vapor pressures.

${\mathit{P}}_2^{\mathbf{sat}}$ Calc	$\mathit{\sigma }$	RMS	AAPE (%)	AAE	${\mathit{R}}^2$	N
CC (20 ${}^{°}$C)	2.54	2.71	22.07	2.18	0.96	73
reference	2.52	2.73	22.89	2.04	0.96	73
mod-UNIFAC	4.88	9.31	229.80	7.95	0.70	60

. To assess the use of the CC for phase equilibria predictions, a comparison was made to the use of MOSCED with reference vapor pressures and the use of mod-UNIFAC to predict $ln{\gamma }_2^{\infty }$ with reference vapor pressure values [54,55,56,57,58,59]; the data for this comparison were taken from our previous work [24].

For the case of $ln{\mathcal{H}}_{2,1}$, we found that, with the use of the CC, all of the errors increase with respect to the use of the reference vapor pressure values. Nonetheless, the predictions remained very well correlated. Moreover, the predictions remained superior to the use of mod-UNIFAC. We note that the AAPE was smaller with mod-UNIFAC. However, the value of the AAPE is sensitive to deviations when the property of interest is close to zero. A residual plot for $ln{\mathcal{H}}_{2,1}$ predicted using the CC is provided in Figure 5. The errors appeared random, suggesting there were no systematic errors in the predictions.

The results were similar for $\Delta G_{2,1}^{\mathrm{solv}}$. Here, all of the reference data were a 25 ${}^{°}$C, so we used only the CC with a reference temperature of 20 ${}^{°}$C. Interestingly, the difference between the use of the CC and the reference values of $ln{P}_2^{\mathrm{sat}}$ was very small. Additionally, we again found that the MOSCED-based predictions were superior to the use of mod-UNIFAC. A residual plot for $\Delta G_{2,1}^{\mathrm{solv}}$ predicted using the CC is provided in Figure 6. The errors appeared random, suggesting there were no systematic errors in the predictions.

3.4. Other Correlations

Finally, given the success in correlating $ln{P}^{\mathrm{sat}}$ and $\Delta H^{\mathrm{vap}}$ using multiple linear regression, we attempted to correlate a range of additional properties. For surface tension ($\gamma$), this led to

$$ln\gamma =a_0+\frac{v^{1/3}}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(22)

for (dynamic) viscosity ($\mu$):

$$ln\mu =a_0+\frac{v}{RT}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(23)

the normal melting point ($T_m$):

$$T_m=a_0+\frac{v^{1/2}}{R}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(24)

the normal boiling point ($T_b$):

$$T_b=a_0+\frac{v}{R}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(25)

the acentric factor ($\omega$):

$$\omega =a_0+\frac{v}{R}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(26)

the normal enthalpy of fusion ($\Delta H^{\mathrm{fus}}$):

$$\frac{\Delta H^{\mathrm{fus}}}{R}=a_0+\frac{v}{R}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(27)

the critical pressure ($P_c$):

$$ln{P}_c=a_0+\frac{v}{R}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(28)

and the critical temperature ($T_c$):

$$T_c=a_0+\frac{v^{1/2}}{R}\left(a_1{\lambda }^2+a_2{\tau }^2+a_3\alpha \beta +a_4\alpha \alpha +a_5\beta \beta \right)$$

(29)

In general, we eliminated T in the right-hand-side of the equation when the temperature of the property was not the same for each compound, and we manually varied the power of v in an attempt to improve the fit for $ln\gamma$ and $T_c$. We note that the scaling of $v^{1/3}$ used here for $ln\gamma$ was also used by Koenhen and Smolders [60] for $\gamma$. The results are provided in

Table 6. Summary of the regressed model parameters using Equations (22)–(29) along with the Pearson correlation coefficient ($R^2$), the root-mean-squared error (RMS), and the average absolute percent error (AAPE) to assess the goodness of fit, where N corresponds to the number of organic compounds included in the fit. In the correlations, $\gamma$ is in units of dyn/cm, $\mu$ is in cP, $T_m$, $T_b$, and $T_c$ are all in K, $\omega$ is dimensionless, $\Delta H^{\mathrm{fus}}/R$ is in units of K, and $P_c$ is in units of bar. When applicable, the temperature indicates the temperature at which the regression was performed.

Property	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{R}}^2$	RMS	AAPE (%)	N
$ln\gamma \left(0{}^{°}\mathrm{C}\right)$	−2.23	49.65	95.21	38.17	2.15	−1.23	0.82	3.28	3.03	96
$ln\gamma \left(20{}^{°}\mathrm{C}\right)$	−4.72	54.03	102.96	45.08	−0.82	−2.92	0.82	3.31	9.07	112
$ln\gamma \left(40{}^{°}\mathrm{C}\right)$	−6.82	57.61	105.70	45.82	−0.63	−2.36	0.84	3.18	9.56	118
$ln\mu \left(0{}^{°}\mathrm{C}\right)$	−2.65	0.15	0.51	0.62	0.05	−0.03	0.81	0.42	204.22	77
$ln\mu \left(20{}^{°}\mathrm{C}\right)$	−2.40	0.13	0.41	0.64	−0.03	−0.01	0.79	0.43	342.72	112
$ln\mu \left(40{}^{°}\mathrm{C}\right)$	−0.49	0.25	0.77	1.05	0.06	0.00	0.80	6.51	96.06	117
$T_m$	33.80	0.45	0.65	0.40	0.06	0.02	0.56	35.27	13.95	124
$T_b$	214.45	0.04	0.11	0.08	0.00	0.00	0.88	24.00	4.33	127
$\omega$	0.06	0.00	0.00	0.00	0.00	0.00	0.61	0.09	19.98	122
$\Delta H^{\mathrm{fus}}/R$	−792.98	0.55	0.42	0.55	−0.09	0.00	0.56	603.18	86.51	114
$ln{P}_c$	4.21	−0.05	0.02	0.01	0.01	0.00	0.64	0.19	4.21	121
$T_c$	220.99	0.97	1.44	0.82	0.01	0.02	0.85	33.86	7.91	121

.

Unfortunately, these predictions are inferior to those for $ln{P}^{\mathrm{sat}}$ and $\Delta H^{\mathrm{vap}}$. Further efforts are necessary to improve their predictive abilities. The results are nonetheless interesting. Consider the case of surface tension; the correlations suggested that dispersion, polar, and self-association interactions increased the observed surface tension.

4. Summary and Conclusions

In the present study, we demonstrated and assessed the ability of MOSCED to correlate the enthalpy of vaporization and vapor pressure at a specific temperature using multiple linear regression. MOSCED is a solubility-parameter-based method, wherein the parameters correspond to specific, physical intermolecular interactions. In this way, MOSCED can be used to both predict phase equilibria and shed light on the underlying intermolecular interactions, which is advantageous for early-stage process development and design.

MOSCED is based on the theory that the cohesive energy may be separated into individual contributions (due to specific intermolecular interactions), which are additive. The cohesive energy is directly related to the enthalpy of vaporization, and MOSCED has, in turn, been shown previously to well correlate the enthalpy of vaporization. Here, we showed that MOSCED is additionally able to correlate the log vapor pressure. While MOSCED can be used to predict limiting activity coefficients, the additional ability to predict vapor pressure extends MOSCED to be able to predict vapor–liquid phase equilibrium devoid of reference data. Here, this was demonstrated by using MOSCED to predict the Henry’s constant and solvation free energy of organic solutes in water. While the errors in the predicted Henry’s constant were slightly larger when MOSCED was used to additionally predict the vapor pressure, the difference was small and still superior to the use of mod-UNIFAC. The same was true with the solvation free energy; only the difference between the use of reference and MOSCED predicted vapor pressures was insignificant. In the most-recent MOSCED revision, parameters were regressed for 130 organic compounds. This was limited due to the availability of reference limiting activity coefficients. In addition to being able to make phase equilibrium predictions, the ability to correlate the enthalpy of vaporization and vapor pressure offers the opportunity to include additional properties in the regression of MOSCED parameters.

Given the success in correlating the enthalpy of vaporization and vapor pressure, we attempted to correlate a wide range of physical properties using similar expressions. While, in some cases, the results were reasonable, they were inferior to the correlations of the enthalpy of vaporization and vapor pressure. Future efforts will be needed to improve the correlations.