## Abstract

Solvation Gibbs energies are basically defined as a chemical potential change when transferring a fixed molecule from a perfect gas to a real liquid mixture. This quantity is of special interest for many practical applications as it quantifies the degree of affinity of a solute for its solvent. Few methods are currently available in the literature for the prediction of solvation Gibbs energies. In this article, a new approach is proposed: the use of a predictive cubic equation of state (EoS). The UMR-PRU (Universal Mixing Rule Peng-Robinson UNIFAC) EoS has been selected for its known capacity to semi-predict behaviors of complex systems including polar and associating compounds (by semi-prediction, it is meant that the EoS predicts binary interaction parameters but requires pure-component properties as input parameters). UMR-PRU predictions have been compared to experimental data extracted from the extensive CompSol database (containing around 22 000 pure component data and 70 000 binary data). Accurate predictions were obtained (a mean absolute deviation of 0.36 kcal/mol was obtained for all the binary data). Finally, when using a fully-predictive approach (i.e. pure-component EoS parameters are predicted from group-contribution methods), the prediction accuracy is roughly preserved.

## Introduction

The Gibbs energy of solvation (simply called “solvation energy”, generally) is a convenient indicator of the degree of affinity between a solute and a solvent. It is basically defined as a change of chemical potential when transferring a fixed molecule (a solute) from a perfect gas (pure component or mixture) to a liquid phase (pure component or mixture) [1], [2]. Although the definition of this property involves a rather-unusual quantity, the “pseudo-chemical potential” (i.e. the chemical potential of a pure species devoid of kinetic energy), it has been recently shown that the Gibbs energy of solvation (denoted Δ_{solv}g̅_{i} for a given solute i) can be expressed as a simple function of classical thermodynamic quantities [2]:

where (T, P, **x**) are the temperature, pressure and vector of mole fractions in the liquid phase, respectively; ϕ_{i,liq} is the fugacity coefficient of component i in the liquid phase and ρ_{liq} is the molar density of the liquid phase. R is the gas constant.

Gibbs energies of solvation bear a multitude of miscellaneous applications in various scientific domains. Among these, let us mention:

Their use in pharmacology and, more precisely, in drug-design applications: solvation energies make it possible to predict the distribution of a drug between the lipophilic cell membrane and the intra- and extra-cellular aqueous system [9].

Their use in kinetics: solvation energies were recently utilized to predict thermokinetic constants (i.e. kinetic rate constants and equilibrium constants) of elementary chemical reactions taking place in a liquid phase from their known values in a perfect-gas phase [10], [11]. As a straightforward application, kinetic models (kinetic mechanisms and related kinetic parameters) for a perfect-gas phase were transferred to liquid phase by replacing gas-phase by liquid-phase thermokinetic constants.

Various strategies were experimented in the past to predict solvation Gibbs energies:

molecular-based approaches (including molecular simulations or COSMO-type calculations). These methods are accurate but are sometimes rather time consuming (in comparison with expectations from simulation-software users).

Correlations like the Linear Solvation Energy Relationship (LSER) which is a faster but less accurate method. In particular, only solvation Gibbs energy values at 298.15K are returned by LSER correlations.

In the late 2000s, Hsieh and Lin [12], [13], [14] showed that solvation Gibbs energy data (generated from molecular simulation) could be used as input parameters of cubic equations of state (EoS) for estimating the attractive and co-volume parameters. Recently, Varfolomeev et al. [15] worked on the prediction and correlation of solvation Gibbs energies for six binary systems containing Guaiacol. Different approaches were tested out. Among these, the GC-PPC-SAFT equation of state (EoS) was considered. This model can be seen as partially-predictive, as it involves both predicted and fit parameters.

In this article, a rather unusual strategy is proposed to predict solvation Gibbs energies: the semi-predictive UMR-PRU cubic equation of state (EoS) was selected and benchmarked [16], [17] (the term “semi-predictive” is explicated below). This model is known for providing accurate predictions of fluid-phase equilibria and classical thermodynamic properties (enthalpy, heat capacity, density…) from ideal to complex mixtures. Such an EoS has been chosen as it contains advanced mixing rules for EoS mixture parameters resulting from the combination of the EoS with the UNIFAC activity-coefficient model at a null reference pressure. Such mixing rules make the model capable of reproducing the behavior of mixtures containing polar or associating compounds with a reasonable accuracy [16], [18], [19], [20]. The UNIFAC model being fully predictive, the only input parameters required by the UMR-PRU EoS are (T_{c,i}, P_{c,i}, ω_{i}), the critical temperature, pressure and the acentric factor of each component i, respectively which are always set to their experimental values.

Therefore, the proposed approach takes the advantage to be semi-predictive (i.e. pure-component parameters are not predicted while binary parameters are) and simple to implement (note that Eq. 1 only involves quantities easily estimable from EoS models).

To benchmark the UMR-PRU model, experimental solvation Gibbs energy data were extracted from the comprehensive CompSol database (containing data for pure and mixed solutes for a large number and variety of systems) [2] and compared to UMR-PRU predictions.

Eventually, the UMR-PRU EoS was combined to a group-contribution method enabling the prediction of the three input parameters (T_{c,i}, P_{c,i}, ω_{i}). Doing so, the estimation of solvation Gibbs energies was made fully predictive. The accuracy of such an approach, which could be of interest when considering unconventional components for which (T_{c,i}, P_{c,i}, ω_{i}) were not measured, has been evaluated.

## The CompSol database

For benchmarking the UMR-PRU EoS, reference data are required. In this study, the “Comprehensive Solvation database” (called CompSol) was used. It was recently developed by Moine et al. [2] with the aim of filling the lack of experimental data in terms of solvation Gibbs energies. This databank contains data for 1969 pure components (in this case, the solvation of a solute by itself is called self-solvation) and for 13 966 solute+solvent binary systems, at different temperatures, in infinite-dilution conditions (the solute is infinitely diluted in its solvent).

Starting from the unanimously accepted definition of the solvation process by Ben-Naim [1], Moine et al. have shown how solvation Gibbs energies are related to various kinds of measurable thermodynamic properties. More precisely, for binary systems, solvation Gibbs energy data were generated from three types of experimental data measured at vapor-liquid equilibrium (VLE) conditions: partition-coefficient data, infinite-dilution activity coefficient data and Henry’s law constant measurements.

Claiming that no consistency test can be applied to infinite-dilution data, Moine et al. did not perform any analysis of the data to detect unreliable data (“except in some rare cases” when a comparison between data measured by different authors for the same system was possible and relevant). Consequently, the Δ_{solv}g̅_{i} experimental data used to test the capabilities of the UMR-PRU EoS are affected by uncertainties that cannot be accurately quantified. Depending on protocols, devices and many other factors, the actual uncertainties of experimental data in the CompSol database are certainly of variable magnitude. Nevertheless, as the database contains a considerable number of experimental datapoints, it seems reasonable to postulate that the effect of some outliers is soften by other experimental data.

## The UMR-PRU EoS

The Universal Mixing Rule – Peng Robinson UNIFAC (UMR-PRU) equation of state is developed since 2004 by Voutsas et al. [16], [17]. As mentioned above, the UMR-PRU model combines a volume-translated Peng-Robinson cubic equation of state and a MHV-based mixing rule involving the UNIFAC activity coefficients model. Contrary to the original formulation of the MHV mixing rules, modifications were made to annihilate the negative effects of a double combinatorial term and to improve the representation of complex mixtures involving associating compounds. The development of this EoS was motivated by the following expectations:

to dispose of a semi-predictive cubic EoS, capable of representing the behavior of binary mixtures exhibiting important combinatorial and/or enthalpic effects (it is recalled that combinatorial effects are due to size asymmetry between components while enthalpic effects stem from polarity effects and/or associating phenomena like hydrogen bonding) [16]

Equations 2 and 3 provide the EoS formulation for a given pure component i:

With

where (P, T, v) are the pressure, temperature and molar volume of pure i. The Soave α-function is used in the attractive parameter a_{i}. As highlighted by Eq. 3, three input parameters are required to calculate the pure-component EoS parameters a_{i} and b_{i} (covolume): the experimental critical temperature, critical pressure and acentric factor (T_{c,i}, P_{c,i}, ω_{i}) of i. The constant volume-translation parameter c_{i} involved in Eq. 2 is deduced from a correlation proposed by Magoulas and Tassios (inspired from Péneloux et al. anterior work) [22], [23] involving the EoS input parameters (T_{c,i}, P_{c,i}, ω_{i}):

Experimental values of pure-component input parameters of the EoS (T_{c,i}, P_{c,i}, ω_{i}) were extracted from the DIPPR database [24].

The one-fluid theory is considered to switch from pure components to mixtures: the EoS formulation remains the same (see Eq. 2) but pure-component EoS parameters are replaced by mixture EoS parameters. The following Universal Mixing Rule (UMR) proposed by Voutsas et al. [16] is selected for expressing mixture parameters a, b and c.

The Staverman-Guggenheim (g^{E,SG}) and residual _{mn} (involved in the residual term) was modified with respect to the original UNIFAC formulation and was made temperature dependent according to Hansen et al. [21]:

In this study, interaction parameters A_{nm}, B_{nm} and C_{nm} were taken from Hansen et al. [21] at the exception of those between groups [CO_{2}] (carbon dioxide), [N_{2}] (nitrogen), [CH_{4}] (methane) and [C_{2}H_{6}] (ethane) with [CH_{3}], [CH_{2}], [CH], [C] (alkyl groups), [ACH], [AC], [ACCH_{3}], [ACCH_{2}], [ACCH] (groups for aromatic compounds) and [H_{2}O] (water) for which values published by Petropoulou and Voutsas [20] and Louli et al. [19] were used.

## Semi-prediction of experimental data from the Compsol database and evaluation of the UMR-PRU model performances

In this part, the UMR-PRU EoS is considered. Experimental values are assigned to the input parameters (T_{c,i}, P_{c,i}, ω_{i}) for each pure species i. It is recalled that the approach is said semi-predictive as it requires experimental pure-component parameter values while binary interactions are predicted through the use of the UNIFAC model.

### Practical estimation of solvation Gibbs energies using an EoS

According to Moine et al. [2], solvation Gibbs energies can be expressed using Eq. 1. This equation describes both the solvation of a given solute in a solvent and the self-solvation of a pure species.

Self-solvation data available in the CompSol database were generated in saturation conditions. For each pure component i, the database provides experimental values _{exp}. Equation 1 was used to predict these data from the UMR-PRU EoS (it is worth noting that the UMR-PRU EoS reduces to a translated version of the classical Peng-Robinson EoS for pure species):

where

Binary-system data present in the ComSol database were all estimated in infinite-dilution conditions. For a given solvent, temperature T_{exp} and pressure P_{exp}, the solvation Gibbs energy of solute i was predicted using Eq. 8:

where ϕ_{i,liq} is the fugacity coefficient of the solute i and ρ_{pure solvent,liq} is the molar density of the liquid solvent. In some cases, the UMR-PRU EoS predicted gaseous solvent states at given (T_{exp}, P_{exp}) instead of liquid states. In such cases, pressure P_{exp} was replaced by the saturation pressure of the solvent in order to ensure that the EoS returns a stable liquid phase in the temperature, pressure and mole fraction conditions _{solv}g̅_{i} values due to the incompressible character of a liquid phase). Therefore, the pressure used to perform EoS calculations was denoted P^{used} and determined as the maximum pressure between the one specified in the CompSol database and the saturation pressure of the solvent.

Pressure P^{used} is reported in a file containing experimental data and EoS predictions, provided as a Supplementary Material.

### Semi-prediction of pure-component solvation Gibbs energies

All pure-component data available in the CompSol database were considered first. Among these, data related to compounds that could not be split into elementary UNIFAC groups or such that (T_{c,i}, P_{c,i}, ω_{i}) were not available, were disregarded. Finally, 8223 data associated with 750 pure compounds were kept and compared to UMR-PRU predictions. Deviations between experimental data and EoS predictions are expressed as Absolute Average Deviation, %Absolute Average Deviation and standard deviation (AAD, %AAD, SD) and are shown in Table 1.

Number of data (N_{points}) | SD | ||
---|---|---|---|

8223 | 0.17 kcal·mol^{−1} | 3.46% | 0.28 kcal·mol^{−1} |

These global deviations highlight the capacity of the UMR-PRU EoS to predict solvation data of pure species. This conclusion is reinforced by Figs. 1 and 2. Figure 1 represents the distribution of the data with respect to the absolute deviation (AD). It is observed that more than 90% of the calculated

The parity plot shown in Fig. 2 emphasizes the general good agreement between experimental and calculated _{r,i}=T/T_{c,i} as shown in Fig. 3. This phenomenon can be simply explained: solvation energies were predicted from Eq. 7 involving both saturation pressure _{r,i} values, high deviations are certainly due to the inaccurate prediction of vapor-pressure data by the EoS (while liquid-density data are generally well reproduced in this temperature range). On the contrary, for high T_{r,i} values, high deviations result from inaccurate EoS predictions of

Nevertheless, the overall low absolute deviations observed make the UMR-PRU EoS a choice model for the semi-prediction of pure-component solvation Gibbs energy data.

### Semi-prediction of solvation Gibbs energies for solutes infinitely diluted in solvents

When a solute is (infinitely) diluted in a solvent, solvation Gibbs energies are influenced by various factors, mainly: the structural effects on the one side and temperature, on the other side. These latter can be basically sorted in combinatorial and enthalpic effects:

Combinatorial (or entropic) effects result from shape and size-asymmetries in the binary system.

Residual (or enthalpic) effects are observed when molecular interactions A-A, B-B and A-B of a binary system {A+B} are of different natures and strengths. These effects often result from the polarity of the species or their capacity to associate (through, e.g. hydrogen bonding).

When the two components A and B of a binary system are sufficiently similar in terms of size, shape and interactions, combinatorial and residual effects vanish and the binary system {A+B} behaves as an ideal mixture.

On the contrary, the presence of residual and/or combinatorial effects reveals the complexity of a binary system. Therefore, the more such effects are present, the more an EoS shows difficulties to reproduce accurately thermodynamic properties (including solvation Gibbs energies). Results presented in this section are analyzed trough this interpretation grid.

To do so, the 9793 binary mixtures at infinite dilution extracted from the CompSol database were sorted into four different families:

Ideal systems,

Non-ideal systems resulting from residual effects,

Non-ideal systems resulting from combinatorial effects,

Non-ideal systems resulting from mixed effects (residual and combinatorial).

The assignment of a given system to one out of these four families was established using an approximate method aimed at putting in evidence general trends. The temperature was fixed at 298.15 K and an arbitrary equimolar composition was adopted. Two simple models were considered to discuss the presence of combinatorial or enthalpic effects:

For roughly quantifying combinatorial effects, a Flory-Huggins model was chosen:

Parameter r_{i} is the Van der Waals volume of component i; it was taken from the DIPPR database.

By analyzing how *n*-alkane systems (for n from 4 to 12), it was decided to consider that combinatorial effects are present if

For roughly quantifying residual effects, a Scatchard-Hildebrand model was considered.

where v_{i} and δ_{i} are the pure liquid molar volume and solubility parameter of component i. Both of them were extracted from the DIPPR database. Danner and Gess [27] proposed a similar approach but considered a 1-suffix Margules activity-coefficient model. They used as criterion to discriminate ideal/non-ideal systems the condition:

As previous, binary systems involving components for which at least one of the input parameters (T_{c,i}, P_{c,i}, ω_{i}) was missing in the DIPPR database were disregarded. Similarly, systems involving components that could not be split into elementary UNIFAC groups were also disregarded. The numbers of datapoints and systems for each of the four families are reported in Table 2. Absolute Average Deviations (AAD) and standard deviations (SD) between model predictions and experimental solvation Gibbs energy data are provided in this table.

Binary system family | Number of datapoints used to calculate AAD and SD | Number of binary systems considered to calculate AAD and SD | AAD (kcal·mol^{−1}) | SD (kcal·mol^{−1}) | %AAD^{a} | Number of datapoints used to calculate %AAD |
---|---|---|---|---|---|---|

Ideal | 20 793 | 3861 | 0.15 | 0.25 | 3.67 | 20 793 |

Residual | 18 949 | 3928 | 0.33 | 0.64 | 13.6 | 18 726 |

Combinatorial | 3489 | 691 | 0.18 | 0.22 | 6.36 | 3301 |

Residual and combinatorial | 10 006 | 1313 | 0.89 | 1.04 | 42.4 | 9127 |

Total | 53 237, (≈77 % of the CompSol databank) | 9793, (≈70 % of the CompSol databank) | 0.36 | 0.67 | 14.2 | 51 947 |

^{a}%AAD were calculated by removing experimental solvation Gibbs energies too close to zero (to avoid infinite %AAD values); a boundary value of ±0.5 kcal/mol was adopted.

These statistical results show that:

with an overall AAD of 0.36 kcal·mol

^{−1}for 77% of the data included in the CompSol database, it can be claimed that the UMR-PRU EoS provides quite reasonable estimations of the property of interest (∆_{solv}g̅_{i}).As expected, the predictive capacity of the UMR-PRU EoS depends on the thermodynamic complexity of binary systems. It is observed that for highly non-ideal systems, the UMR-PRU EoS should be used with caution when predicting ∆

_{solv}g̅_{i}values.

These statements are illustrated in Fig. 4 showing the distribution of (∆_{solv}g̅_{I}) with respect to Absolute Deviations (AD) between EoS predictions and data. In particular, for ideal systems and systems showing a non-ideal behavior resulting from combinatorial effects, nearly 5% of the data exhibit an AD higher than 0.5 kcal/mol. This proportion increases to 20% when considering all data or when considering data from systems associated with residual effects. Finally, as expected above, the prediction of ∆_{solv}g̅_{i} using the UMR-PRU EoS when mixed effects take place is, by far, less accurate. In this class of systems, nearly 50% of the considered datapoints show an AD higher than 0.5 kcal/mol. It is worth noting that aqueous systems all belong to this class of systems.

### How accurate is the prediction of solvation Gibbs energies when using a fully-predictive version of the UMR-PRU EoS?

In this section, it is proposed to investigate whether solvation Gibbs energies can be accurately predicted with a model combining the UMR-PRU EoS and a group-contribution (GC) method to estimate the three pure-component input parameters (T_{c,i}, P_{c,i}, ω_{i}). This practice is especially of interest for studying systems containing complex or uncommon molecules, for which no experimental value of at least one input parameters is available. As previously mentioned, GC methods take as input parameter the chemical structure of the components expressed in terms of group occurrences. Three different GC methods were considered in the present study: Marrero and Gani [28], Constantinou and Gani [29] and Joback and Reid [30]. For a given component and a given property, the most accurate method among these three ones was selected (this accuracy depends both on the nature of the component and the property). The ICAS software (originally developed by Prof. Rafiqul Gani’s team from the Danish Technical University) was used to perform this selection automatically. The GC methods used for each pure component EoS input parameter and their corresponding estimated values of (T_{c,i}, P_{c,i}, ω_{i}) are reported in a file provided as supplementary materials.

The results obtained from this approach are compared to both experimental and the semi-predictive approach detailed in the previous section (in which, three EoS input parameters (T_{c,i}, P_{c,i}, ω_{i}) were experimental data issued from the DIPPR database).

In this section, the superscript GC-input is used to designate a fully-predictive approach [i.e. when (T_{c,i}, P_{c,i}, ω_{i}) are guesstimated from GC methods] while superscript EXP-input is used for referring to the approach relying on experimental (T_{c,i}, P_{c,i}, ω_{i}) parameters.

### Full prediction of pure-component solvation Gibbs energy of solvation

In the CompSol database, 7821 experimental data at different temperatures for 731 components were extracted to benchmark the fully-predictive approach. Note that all the molecules that could be split into elementary groups were considered here. Let us mention that the critical temperature _{exp} higher than

Comparison between | AAD | %AAD | SD |
---|---|---|---|

0.17 kcal·mol^{−1} | 3.12% | 0.27 kcal·mol^{−1} | |

0.34 kcal·mol^{−1} | 6.94% | 0.42 kcal·mol^{−1} |

Statistical results (AAD, %AAD and SD) are calculated over 7821 experimental datapoints.

As expected, _{c,i}, P_{c,i}, ω_{i})^{GC-input} are used instead of (T_{c,i}, P_{c,i}, ω_{i})^{EXP-input}. However, deviations remain acceptable regardless of the chosen approach. Figure 5 showing the parity plot between calculated and experimental _{c,i}, P_{c,i}, ω_{i})^{EXP-input} to (T_{c,i}, P_{c,i}, ω_{i})^{GC-input}.

The details of our calculations of _{c,i}, P_{c,i}, ω_{i}) can be found in a file provided as supplementary materials.

### Full prediction of Gibbs energy of solvation for solutes infinitely-diluted in solvents

The study can now be performed on 44 952 datapoints at different temperatures belonging to 9003 binary mixtures {solute+solvent}. As previously explained, systems containing molecules that could not be split into elementary groups were disregarded. Results are shown in Table 4. A good agreement between UMR-PRU EoS predictions and experimental data is found which is more or less independent of the approach used to estimate (T_{c,i}, P_{c,i}, ω_{i}). Consequently, the same conclusions as previous can be drawn: calculations of solvation Gibbs energies for infinitely-diluted solutes remain accurate and little depend on input EoS parameter estimation methods.

Comparison between | AAD | %AAD^{a} | SD |
---|---|---|---|

0.23 kcal·mol^{−1} | 8.17% | 0.44 kcal·mol^{−1} | |

0.34 kcal·mol^{−1} | 11.6% | 0.49 kcal·mol^{−1} |

Statistical results (AAD, %AAD and SD) are calculated over 44 952 experimental datapoints.

^{a}%AAD were calculated by removing experimental solvation Gibbs energies too close to zero (to avoid infinite %AAD values); a boundary value of ±0.5 kcal/mol was adopted. Eventually, %AAD were calculated over 44 537 datapoints.

This result is confirmed by Fig. 6 showing a similar distribution of the calculated solvation Gibbs energy with respect to AD when using (T_{c,i}, P_{c,i}, ω_{i})^{GC-input} or (T_{c,i}, P_{c,i}, ω_{i})^{EXP-input} parameters. Eventually, both of these input-parameter sets lead to mean absolute deviations lower than 1 kcal/mol for 95% of the considered points of the CompSol database.

## Conclusion

In this article, the cubic UMR-PRU EoS was used to predict solvation Gibbs energies. This EoS requires three input parameters per component: (T_{c,i}, P_{c,i}, ω_{i}). To benchmark the proposed approach:

8223 solvation Gibbs energy data of pure species (at VLE conditions) at different temperatures were extracted from the CompSol database and predicted with the UMR-PRU model (the three input parameters were estimated from experimental data; such an approach was said semi-predictive); note that these data are associated with 750 pure components.

53 237 solvation Gibbs energies of infinitely-diluted solutes were extracted from the same database and estimated using the semi-predictive approach. These data cover no less than 9793 binary systems.

Absolute Average Deviations on all these points are 0.17 kcal/mol and 0.36 kcal/mol for pure-component and binary-system solvation Gibbs energies, respectively.

It can be concluded that the UMR-PRU EoS approach makes it possible to safely semi-predict such quantities. Nevertheless, the UMR-PRU EoS shows some defects that were put in evidence in this study:

the calculation of pure-component solvation Gibbs energies at low reduced temperatures or at reduced temperatures close to one could be tarnished by a significant error.

For binary systems, predictions of solvation Gibbs energies from an EoS deteriorate with the binary system complexity. For mixtures exhibiting a pronounced non-ideal behavior (mixing combinatorial and residual effects), the use of the UMR-PRU EoS may induce more significant deviations.

Eventually, an additional study was performed to investigate whether the good performances of the UMR-PRU model are maintained when switching from experimental to predicted input parameters (T_{c,i}, P_{c,i}, ω_{i}). Deviations observed for pure and mixed solutes from experimental data were found very close to the ones associated with experimental (T_{c,i}, P_{c,i}, ω_{i}) input parameters.

As a conclusion, we recommend the use of a semi-predictive UMR-PRU EoS for estimating solvation Gibbs energies but encourage users to remain cautious when considering highly non-ideal mixtures. If a semi-predictive approach cannot be used, it can be safely replaced by a fully-predictive approach.

## Article note

A collection of invited papers based on presentations at the 18^{th} International Symposium on Solubility Phenomena and Related Equilibrium Processes (ISSP-18), Tours, France, 15–20 July 2018.

## Acknowledgment

The French Petroleum Company TOTAL (and more particularly Dr. Laurent AVAULLEE, expert in thermodynamics) is gratefully acknowledged for sponsoring this research.

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## Appendix A: Procedure to estimate the solvation Gibbs energy from a cubic EoS

At given temperature, pressure and mole fraction vector (T, P_{0}, **x**), the general expression for ∆_{solv}g̅_{i}(T, P_{0}, **x**) is given by eq. 1. The expression of a pressure-explicit EoS can be written in terms of intensive phase variable P(T, v, **x**) or in terms of extensive phase variables P(T, V, **n**) where (V, **n**) denote the total volume and mole vector, respectively. After rearranging eq. 1, the following expression is obtained, making it possible to deduce the quantity of interest from the mere knowledge of the pressure-explicit EoS P(T, V, **n**):

For a pure component in VLE, the mole vector reduces to the mole number of the pure component and the total volume is determined by solving the VLE conditions.

## Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/pac-2018-1112).

**Published Online:**2019-02-09

**Published in Print:**2019-08-27

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