INTRODUCTION

An important part of thermodynamic modeling is describing the level of reference. The thermodynamic properties of solutions are calculated using the properties of the components that are in selected reference states. With a symmetric normalization, the reference state of the components is the state of a pure substance, and with an asymmetric normalization, it is the state of a pure substance for a solvent and a hypothetical state of an infinitely dilute solution for solutes. The latter normalization is convenient when considering a number of systems, especially poorly soluble compounds, gases, and electrolytes.

In accordance with the established terminology adopted in international publications, states corresponding to the reference level are usually called standard. The properties of substances in the standard state are called standard state properties or standard properties. An example of such a property is the standard chemical potential. Standard properties can also be any thermodynamic properties of a substance in the reference state (e.g., enthalpy, entropy, volume, and heat capacity).

In the context of the theory of solutions, the properties of a substance at infinite dilution are often called standard. In this work, we use the terms standard state, standard property, and infinity symbol ∞ accepted in international publications to describe the state of substance at infinite dilution and its corresponding properties.

The standard properties of substances are used in calculating such thermodynamic properties of solutions as heat capacity (Cp) and volume (V). For gases, the most important quantity associated with the standard is Henry’s constant (kH). Although the thermodynamic apparatus in this case can be used for any solvent, aqueous solutions are of greatest practical interest. In recent years, much attention has been given to describing standard properties in the supercritical region of temperatures and pressures [1–3].

The properties of both pure substances and components at infinite dilution are described using equations of state. At the moment, semi-empirical equations of state are widely used that allow us to interpolate experimental data effectively and, in some cases, successfully predict thermodynamic properties [4]. They have provided a breakthrough in modeling the standard properties of substances over a wide range of temperatures and pressures, including areas where water is in a supercritical state. The most successful equations of state are SOCW [5], AD [6], and POCW [7–9]. However, it should be noted that most popular equations are based on generalizing a number of empirical equations and approximations that do not always have rigorous theoretical substantiation. Although current models are built with limit laws in mind, their use has the potential to produce gross errors when interpolating inconsistent data or extrapolating properties.

We believe a more rational approach is one in which the equation of state is constructed on the basis of equations obtained theoretically. Examples are the equations of state of the SAFT family. However, they are rarely used to describe standard properties due to the mathematical complexity of the procedure for moving from SAFT equations to the component energy in an infinitely dilute solution. For practical implementations, it is therefore advisable to look for simpler models.

Building a model of the standard properties of particles in aqueous solutions is directly related to the theory of the liquid state and is a non-trivial task. Even simplified calculations of the properties of real liquid solutions are currently made by means of molecular dynamics or numerically solving systems of differential equations. Although these methods provide a number of important theoretical results, their complexity and the need for a great many calculations make them unsuitable for solving more practical problems (geochemical or industrial).

In some cases, molecular dynamics allows us to eliminate an experiment and estimate standard properties under conditions that are not available in the experiment, but it is not used for thermodynamic modeling alone [10].

Model analytical equations of state are needed to solve problems of practical modeling. Theoretical simplifications are inevitable when constructing such a model, leading to the introduction of effective parameters whose values are determined empirically. Nevertheless, such models are still based on expressions that are close to physically valid. In our opinion, Hu et al. came closest to solving the problem of constructing a theoretical equation of state [11, 12]. They managed to describe the temperature dependence of Henry’s constant (kH) for a number of substances in a wide range of temperatures and pressures. However, their model is unsuitable for predicting more model-sensitive properties such as the standard volume (V∞) or standard heat capacity (\(C_{p}^{\infty }\)).

The problem of constructing a theoretically substantiated, analytical equation of state capable of describing the standard properties of various substances in a wide range of temperatures and pressures therefore remains relevant. The solution to such a problem can be sought through the sequential consideration of different systems, from simple to complex. The simplest model system for constructing an equation of state is an infinitely dilute solution of an noble gas. Its atom interacts with water weakly and is devoid of internal degrees of freedom (vibrational, rotational), which allows us to exclude these factors from consideration. In this work, we chose the system H2O–Ar as the most studied system with noble gases.

DERIVING THE EQUATION OF STATE OF AN INFINITELY DILUTE SOLUTION

Most models of an infinitely dilute solution are based on considering the solvation of the gas of a given substance, based on a description of its ideal gas in the standard state (i.e., at pressure p° = 0.1 MPa). We then consider the immersion of a gas particle in an infinite volume of solvent. Any standard property (Z∞) can be represented as the sum of the properties of an ideal gas at p = p° (\(Z_{{{\text{id}}}}^{^\circ }\)) and changes in the property during solvation (ΔZh):

$${{Z}^{\infty }} = Z_{{{\text{id}}}}^{^\circ } + \Delta {{Z}_{h}}.$$

Since calculating the properties of an ideal gas is a simple task, the models differ in how they describe the solvation process. Some equations of state ignore the ideal gas contribution (\(Z_{{{\text{id}}}}^{^\circ }\)). The Gibbs energy of solvation consists of two contributions [4]:

$$\Delta {{G}_{h}} = \Delta {{G}_{{{\text{in}}}}} + \Delta {{G}_{{ss}}}.$$

The first contribution (ΔGin) describes the actual solvation and is determined by the interaction between the solute and the solvent. The second contribution (ΔGss) is associated with a change in the reference level during the transition from the state of an ideal gas to an infinitely dilute solution. It can be strictly derived by means of statistical thermodynamics [13]:

$$\frac{{\Delta {{G}_{{ss}}}}}{{RT}} = \ln \frac{{RT}}{{p^\circ {{V}_{1}}}},$$

where V1 is the molar volume of the solvent.

Modern equations of state normally consider the energy contribution of interactions (ΔGin) at the macroscopic level. The POCW equation of state deserves attention, since it contains contributions that are more or less duplicated in other equations (SOCW [5], AD [6]). Within POCW, three model approaches are crosslinked to describe standard volume V∞. Other thermodynamic solvation functions (ΔZh) are derived from V∞ via standard thermodynamic transformations. Most of them can be defined up to a constant. In its original form, the model was also unable to adequately describe data in the subcritical region, so Plyasunov et al. introduced an empirical equation for \(\Delta {{C}_{{p,h}}}\) [9] when T < 658 K.

An approximation of the similarity of direct solvent–solvent and solvent–solute correlation functions [7] and the empirical Cooney–O’Connell equation [14, 15] are used to describe standard volume V∞ at high densities (ρ → 1000 kg/m3). Although Cooney and O’Connell initially focused on the expressions obtained within the theory of density fluctuations [14] when choosing the type of the equation, neither it nor subsequent modifications of it received a theoretical substantiation.

The low-density POCW contribution (ρ → 0 kg/m3) is derived strictly by considering the virial equation of state of a gas mixture of the given substances. The expression up to the first mixed virial coefficient (B12) is normally considered. On the one hand, the introduction of a contribution from B12 into the model is its advantage, since it guarantees the correct limit behavior for ρ → 0 kg/m3. On the other hand, it assumes knowledge of temperature dependence B12(T), determined independently on the basis of literature data.

In this work, we obtain a unified description for any densities in statistical thermodynamics by modeling contribution ΔGin at the microscopic level.

The structure of a liquid is determined primarily by repulsive forces. It is therefore advisable to base it on a description of repulsive forces when modeling solvation (especially solvophobic solvation), while attractive forces must be considered as a correction.

Gibbs Energy of Cavitation

The simplest model repulsion potential is the hard sphere potential. The basic model of solvation can therefore be that of the immersion of a hard sphere (diameter σ2) into a liquid of hard spheres (diameter σ1). This model is simple enough to be based on at least approximate analytical equations for the Gibbs energy. At least somewhat more complex model structures of a liquid usually require numerical calculations.

The model of hard sphere solvation in a solvent describes cavitation (i.e., the formation of a cavity and the reorganization of the solvent structure around it as a result of the action of repulsive forces [16, 17]). The change in the Gibbs energy (ΔGcav) in the corresponding process is called Gibbs energy of cavitation or even the cavity chemical potential. We shall adhere to the first term.

In this work, our model of the Gibbs energy of cavitation is the equation proposed in [18] for describing the solvation of a rigid sphere with diameter σ2 in an infinite amount of liquid rigid spheres with diameter σ1. Unlike other equations, it has the correct limit behavior as the size of the cavity grows, allowing it to be used for particles that are much larger than water molecules:

$$\begin{gathered} \frac{{{{\Delta }}{{G}_{{{\text{cav}}}}}}}{{RT}} = \frac{{3\eta d}}{{(1 - \eta )}} + \frac{{3\eta (2 - \eta )(1 + \eta ){{d}^{2}}}}{{{{{(1 - \eta )}}^{2}}}} \\ + \;\frac{{\eta (1 + \eta + {{\eta }^{2}} - {{\eta }^{3}}){{d}^{3}}}}{{{{{\left( {1 - \eta } \right)}}^{3}}}} + \ln (1 - \eta )~, \\ \end{gathered} $$

where d = σ2/σ1 is the ratio of effective particle diameters, η = (π/6)ρ1\(\sigma _{1}^{3}\) is the density of solvent particle packing, and \(~{{\rho }_{1}}\) is the concentration of solvent particles (m−3).

A separate problem is choosing adequate values of σi when building the model. There were many estimates of σi earlier in the literature, including corrected crystal diameters [12], experimental effective diameters of hard spheres [16], and estimates based on the Lennard–Jones potential [19]. Estimating σ1 is especially complex.

Water is a structured liquid with pronounced orientational interactions and a tendency to associate. The hard sphere model is therefore not suited for describing it. Molecular dynamics calculations [20] show that regardless of the chosen value of σ1, the hard sphere model is not capable of quantitatively reproducing ΔGcav = f(p, T, σ2) (Fig. 1). At best, it can reproduce only qualitative patterns.

Fig. 1.
figure 1

Reduced Gibbs energy of cavitation of a rigid sphere in water (a) at T = 298.15 K and (b) for σ2 = 3.3 Å. Symbols are results from molecular dynamics calculations using the TIP4P potential ((a) \(\triangledown \) [21], \(\vartriangle \) [22], \(\bigcirc \) [20]; (b) \(\Diamond \) [23], \(\square \) [20]). Lines represent calculations according to the equation of Matyushov and Ladanyi [18]. The solid line is for σ1 = 2.75 Å; the dotted line is for σ1 = 2.85 Å.

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We nevertheless believe that acceptable accuracy in describing ΔGcav can be achieved by selecting diameter σ1 for a specific particle size. Quantity σ1 thus serves as a parameter of optimization within the proposed equation of state. A comparison of the model of hard spheres and molecular dynamics calculations for water shows that the optimum value of diameter σ1 is normally ⁓2.9–3 Å [16, 22]. On the other hand, there is also no clear way of estimating the diameter of a dissolved particle (σ2). We therefore believe it is logical to use σ2 as an optimized parameter.

Gibbs Energy of Attraction

The Gibbs energy of attraction (ΔGatt) is generally described by an integral of the form

$$\Delta {{G}_{{{\text{att}}}}} = 4\pi {{\rho }_{1}}\mathop \smallint \limits_{{{\sigma }_{{12}}}}^\infty {{u}_{{12}}}(r){{g}_{{12}}}(r,p,T){{r}^{2}}dr,$$
(1)

where σ12 = (σ1 + σ2)/2, u12(r), and g12(r, p, T) are the potential of interaction and the pairwise radial distribution function of the solute and solvent particles, respectively.

Since it is difficult to calculate the true integral, various simplifications of the model have been proposed in the literature. In this work, we propose considering Eq. (1) in the context of the square well potential model provided by the equation

$${{u}_{{12}}}(r) = \left\{ \begin{gathered} + \infty ,\quad 0 \leqslant r < {{\sigma }_{{12}}} \hfill \\ - {{\varepsilon }_{{12}}},\quad {{\sigma }_{{12}}} \leqslant r < \lambda {{\sigma }_{{12}}} \hfill \\ 0,\quad \lambda {{\sigma }_{{12}}} \leqslant r < + \infty , \hfill \\ \end{gathered} \right.$$
(2)

where ε12 is the potential well depth, and λ is the well width parameter.

With only short-range forces, the square well potential model can satisfactorily describe the main properties of a solution (Fig. 2). Moving to the model potential simplifies the equation for ΔGatt:

$$\begin{gathered} \Delta {{G}_{{{\text{att}}}}} = - {{\varepsilon }_{{12}}}{{N}_{c}}, \\ {{N}_{c}} = 4\pi {{\rho }_{1}}\mathop \smallint \limits_{{{\sigma }_{{12}}}}^{\lambda {{\sigma }_{{12}}}} {{g}_{{12}}}(r,p,T){{r}^{2}}dr. \\ \end{gathered} $$
(3)

In concept, Nc is the average coordination number of a dissolved particle. Unfortunately, it is impossible to obtain a rigorous analytical solution for integral (3) even within such a simple model. However, many semi-empirical analytical equations [25] have been proposed in the literature that were obtained on the basis of the statistical theory of liquids and molecular dynamics. When describing the properties of supercritical fluids over a wide range of temperatures and pressures, equations that have shown their validity at both low and high densities are especially useful.

Fig. 2.
figure 2

Model potential of H2O–Ar interaction. The solid line is the Lennard–Jones potential (parameters from [24]), and the dotted line is the potential of a square well. The parameters were obtained in this work (Table 1).

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Since standard properties are determined by the first and second derivatives of the Gibbs energy (ΔGatt) according to temperature and pressure, the expression for Nc must describe with high accuracy not only the coordination number itself at infinite dilution, but its first and second derivatives as well. We tested several analytical equations and concluded that the best result was obtained with an equation based on the approach of Lee and Chao [26]:

$$\begin{gathered} {{N}_{c}} = \frac{{4\pi }}{3}({{{{\lambda }}}^{3}} - 1){{{{\rho }}}_{1}}{{\sigma }}_{{12}}^{3}\frac{{(1 + 0.57{{{{\rho }}}_{1}}{{\sigma }}_{1}^{3})\exp \left( {\frac{{{{\alpha }}{{\varepsilon }_{{12}}}}}{{RT}}} \right)}}{{1 + {{{{\rho }}}_{1}}{{\sigma }}_{{12}}^{3}\left[ {\exp \left( {\frac{{\alpha {{\varepsilon }_{{11}}}}}{{RT}}} \right) - 1} \right]}}, \\ \alpha = 1 + 0.1044\eta - 2.8469{{\eta }^{2}} + 2.3785{{\eta }^{3}}, \\ \end{gathered} $$
(4)

where ε11 is the energy of interaction among solvent particles in the square well potential approximation. In this work, we chose ε11 = 250R [27].

Gibbs Energy Correction

Using perturbation theory, Neff and McQuarrie showed in [28] that there is a correction contribution ΔGcorr in addition to ΔGatt. In the particle size additivity approximation, the correction contribution is reduced to the equation

$$\begin{gathered} \Delta {{G}_{{{\text{in}}}}} = \Delta {{G}_{{{\text{cav}}}}} + \Delta {{G}_{{{\text{att}}}}} + \Delta {{G}_{{{\text{corr}}}}}, \\ \Delta {{G}_{{{\text{corr}}}}} = {{{{\rho }}}_{1}}{{N}_{1}}\mathop \smallint \limits_{{{\sigma }_{{11}}}}^\infty {\kern 1pt} {{u}_{{11}}}(\vec {r}){{\left[ {\frac{{\partial {{g}_{{11}}}(\vec {r},p,T)}}{{\partial {{N}_{2}}}}} \right]}_{{V,T,{{N}_{1}}}}}dV, \\ \end{gathered} $$
(5)

where Ni denotes the numbers of solvent and solute atoms; u12(\(\vec {r}\)) and g12(\(\vec {r}\), p, T) are the potential of interaction and the pairwise radial function of the distribution of solvent particles, respectively.

We can see from Eq. (5) that the correction contribution considers the change in the energy of interaction between solvent particles caused by changes in the structure around a dissolved particle. With the solvophobic hydration of small particles, we may assume that the structure of the solvent undergoes minor perturbations, so correction ΔGcorr can be ignored and was not considered in this work.

Calculations

Hydration parameters (ΔZh) were calculated using standard thermodynamic transformations of the Gibbs energy of hydration (ΔGh). Henry’s constant (kH) was recalculated to the Gibbs energy using the relations

$$\Delta {{G}_{h}} = RT\ln ({{k}_{{\text{H}}}}{\text{/}}p^\circ ),$$
$${{k}_{{\text{H}}}} = \mathop {\lim }\limits_{{{x}_{2}} \to 0} \frac{{{{p}_{2}}}}{{{{x}_{2}}}}.$$

The standard properties of argon were calculated by summing the corresponding standard properties of an ideal gas and parameters of hydration. The standard volume was calculated separately. Contribution \(V_{{{\text{id}}}}^{^\circ }\) was calculated as the pressure derivative of the Gibbs energy of an ideal gas in the standard state (\(G_{{{\text{id}}}}^{^\circ }\)). The derivative was zero because \(G_{{{\text{id}}}}^{^\circ }\) was fixed at 1 bar [4]. The standard volume at infinite dilution is thus strictly equal to the volume of hydration:

$$V_{{{\text{id}}}}^{^\circ } = \frac{\partial }{{\partial p}}{{\left[ {{{{\left. {{{G}_{{{\text{id}}}}}} \right|}}_{{p = p^\circ }}}} \right]}_{T}} = 0,$$
$${{V}^{\infty }} = \Delta {{V}_{h}}.$$

Our calculations were made in the MATLAB® R2021b software environment. The model parameters were optimized by minimizing objective function τ according to least squares, using the Levenberg–Marquardt algorithm [29]:

$$\tau = \mathop \sum \limits_Z \,{{w}_{Z}}{\kern 1pt} \mathop \sum \limits_{i = 1}^{{{N}_{Z}}} {{\left( {\frac{{Z_{i}^{{{\text{exp}}}} - Z_{i}^{{{\text{calc}}}}}}{{Z_{i}^{{{\text{exp}}}}}}} \right)}^{2}},$$

where Zi represents values of property Z = (V∞, \(C_{p}^{\infty }\), ΔGh, ΔHh, Nc), NZ is the number of values, wZ is the statistical weight of data according to property Z, and superscripts calc and exp correspond to the calculated and experimental values of property Z.

When optimizing the model over the data set, data on the parameters of hydration (ΔGh, ΔHh) were given statistical weight wZ = 3. Data on standard properties (V∞, \(C_{p}^{\infty }\)) were given weight wZ = 1. It was decided to raise the importance of the first data, since they were collected from various sources. Data on V∞ and \(C_{p}^{\infty }\) were taken from one and two works by Biggerstaff et al., respectively. The literature also suggested [8] that the high-temperature data of Biggerstaff et al. might be inaccurate, so data collected at T > 650 K were not considered in optimization. Information about coordination numbers was also not considered.

All statistical weights were assumed to be equal when determining values of the parameters of the extrapolation model according to data on ΔGh, ΔHh, V∞, and Nc.

Literature data were used as an initial approximation for the parameters of the hard sphere model: σ1 = 3.0 [22, 27] and σ2 = 3.2 [16], and for energy parameters, λ = 1.5 and ε12 = 1000 J/mol.

The properties of water were calculated using the IAPWS equation of state [30, 31] implemented [32] by Voskov for the MATLAB environment.

RESULTS AND DISCUSSION

Solubility of Ar

The proposed equation of state is in excellent agreement with the literature data on Henry’s constant (kH) at moderate temperatures. Substantial deviations are observed only at high temperatures (T > 450 K), for which information was given in only two works. At the same time, only one of these works can be considered reliable, since the data of the authors of [33] do not agree with other works. Some reasons for their erroneous results were given in [34].

Deviations of calculation from experiment at high temperatures can be explained by the low accuracy of determining kH. The correctness of the model is also supported by its excellent agreement with data on \(C_{p}^{\infty }\) in this range of temperatures. This is indirectly confirmed by other equations of state (AD [6], POCW [8]) exhibiting similar behavior of kH for the same data (Fig. 3).

Fig. 3.
figure 3

(a) Gibbs energy of hydration, (b) Henry’s constant for Ar in H2O at p = 0.1 MPa (T < 373.15 K) and p = psat (T > 373.15 K). The solid line represents calculations using parameters obtained via optimization for all data; the dotted line, model predictions based on data at 298.15 K. The symbols are experimental data (\(\square \) [34], \(\bigcirc \) [15], \(\Diamond \) [36], × [37], + [33]).

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The model also adequately describes data on the enthalpy of hydration (ΔHh) within the experimental error (Fig. 4). However, there are few data on ΔHh at different temperatures.

Fig. 4.
figure 4

(a) Gibbs energy of the transfer of Ar from the gas phase to an equal volume of H2O at T = 973 and 1273 K; (b) enthalpy of hydration at p = 0.1 MPa. The solid line represents calculations using parameters obtained via optimization for all data; the dotted line, the predictions of the model based on data at 298.15 K. The symbols are (a) calculated results (\(\bigcirc \) [8, 38, 39]) and (b) experimental data (\(\bigcirc \) [40], \(\square \) [41], \(\vartriangle \) [42], \(\Diamond \) [43]).

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Like Plyasunov et al. [8], we tested the ability of the model to predict the energy properties of a system in the supercritical region (T > Tc). To do so, we compared the predicted Gibbs energy of transferring Ar from the gas phase to an equal volume of water (\(\Delta G_{h}^{V}\)) with calculations according to the SUPERFLUID equation of state [38, 39]:

$$\Delta G_{h}^{V} = \Delta {{G}_{h}} - \Delta {{G}_{{ss}}}.$$

The SUPERFLUID equation of state was built on the basis of results from describing mixtures of supercritical fluids obtained via molecular dynamics, and can therefore be used to verify the model independently. Allowing for the error of molecular dynamics calculations, our proposed equation of state predicts \(\Delta G_{h}^{V}\) satisfactorily up to 1273 K.

Standard Properties of Ar

In addition to limited measurements at 0.1 MPa [44, 45], the standard properties of Ar (V∞, \(C_{p}^{\infty }\)) were studied in detail only by one team of authors [46–48] in a wide range of temperatures and pressures. Unfortunately, there is no independent verification of their results. At the same time, it was noted in [8] that the real measuring error in [46–48] in the supercritical region is probably much higher than the one declared.

The model we propose describes the data in [46–48] with high accuracy up to temperatures of ~660 K in terms of both V∞ and \(C_{p}^{\infty }\) (Figs. 5, 6). It also generally reproduces the behavior of the curves at higher temperatures. The observed deviations of calculations from experiments in sign are the same as for the AD [6] and POCW [8] equations of state. Allowing for the high experimental error in [46–48] when T > Tc, we may conclude the model describes the available data satisfactorily.

Fig. 5.
figure 5

Standard heat capacity of Ar in H2O at p = 17 and 32 MPa. The solid line represents calculations using parameters obtained via optimization for all data; the dotted line, model predictions based on data at 298.15 K. The symbols are experimental data (\(\bigcirc \) [46, 47]).

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Fig. 6.
figure 6

Standard volume of Ar in H2O at p = (a) 21 and 29 and (b) 26 and 32 MPa. The solid line represents calculations using parameters obtained via optimization for all data; the dotted line, model predictions based on the data at 298.15 K. The symbols are experimental data (\(\square \) [44], \(\Diamond \) [45], \(\bigcirc \) [48]).

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Equation of State Parameters

Optimized values of the model parameters are given in Table 1. The effective diameter of molecular H2O as a rigid sphere is in good agreement with other estimates from the literature, but the effective diameter of an Ar atom turns out to be underestimated by 30–40%, due probably to the estimates being made on the basis of the properties of pure Ar. The effective size of an atom in a solvent can differ from it.

Table 1.   Parameters of the equation of state for an infinitely dilute solution of Ar in H2O (I and II are optimization over the entire set and for a limited set of data, respectively; III is recommended values from the literature
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Indirect confirmation of the high accuracy of the model and the parameters of attraction (λ, ε12) in particular is the excellent agreement between the calculated and experimental values of virial coefficient B12 of a gaseous H2O–Ar mixture, for which the model of a rectangular potential well allows us to obtain the explicit expression

$${{B}_{{12}}} = \frac{{2\pi }}{3}{{N}_{{\text{A}}}}\sigma _{{12}}^{3}\left[ {1 - \left( {{{\lambda }^{3}} - 1} \right)\left( {\exp \left( {\frac{{{{\varepsilon }_{{12}}}}}{{RT}}} \right) - 1} \right)} \right],$$

where NA is Avogadro’s number.

As can be seen from Fig. 7, the model also predicts the values of coordination number Nc satisfactorily, allowing for high measuring errors.

Fig. 7.
figure 7

(a) Virial coefficient B12 of a gaseous H2O–Ar mixture; (b) average coordination number of Ar in H2O. The solid line represents calculations using the parameters obtained via optimization for all data; the dotted line, model predictions based on data at 298.15 K. The symbols are experimental data ((a) \(\bigcirc \) [50]; (b) \(\bigcirc \) [51], \(\vartriangle \) [52], \(\square \) [53]) and quantum-chemical calculations ((а) \(\bullet \) [50]).

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Let us recall that information about the coefficient B12 and coordination number Nc were not used in the optimization of the model, however, the resulting description is in good agreement with experimental data and quantum-chemical calculations. The ability of the model to predict B12 can be considered additional confirmation of the correctness and predictive ability of the proposed model.

Extrapolating Ability of the Model

Data on the standard properties and hydration parameters of various substances are sometimes very limited. An important indicator of the quality of the equation of state of an infinitely dilute solution is therefore its ability to predict properties on the basis of a small data set.

To test the model, we optimized its parameters for a limited set of data on ΔGh, ΔHh, V∞, and Nc at low temperatures (Table 2). The obtained values of the parameters are given in Table 1.

Table 2.   Data set used to test the predictive power of the model
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As can be seen from Figs. 3–7, the model parametrized using the reduced data set is virtually the same as the main model. The relative discrepancy between the properties predicted by the two models does not exceed 1.5% in the investigated range of temperatures and pressures. We may therefore conclude that the proposed equation of state has high extrapolating power and can correctly predict the behavior of the system in a wide range of temperatures and pressures, including the supercritical region of water, using only data at 298.15 K and 0.1 MPa.

CONCLUSIONS

Despite the theoretical simplicity of the model and the small number of parameters, the proposed equation of state for an infinitely dilute solution is able to satisfactorily describe the parameters of dissolution (hydration) and the standard properties of an inert gas in water. It is suitable for describing standard properties in the supercritical region at both low and high fluid densities. The proposed equation of state can also be used to predict standard properties over a wide range of temperatures and pressures, based on a limited set of data.

To prove the universality of the proposed approach, we plan to consider in the future possible improvements to the model using more complex systems as an example.