Research of a Thermodynamic Function (∂p∂x)T, x→0: Temperature Dependence and Relation to Properties at Infinite Dilution

In this work, we propose the idea of considering (∂p∂x)T, x→0 as an infinite dilution thermodynamic function. Our research shows that (∂p∂x)T,x→0 as a thermodynamic function is closely related to temperature, with the relation being simply expressed as: ln(∂p∂x)T, x→0=AT+B. Then, we use this equation to correlate the isothermal vapor–liquid equilibrium (VLE) data for 40 systems. The result shows that the total average relative deviation is 0.15%, and the total average absolute deviation is 3.12%. It indicates that the model correlates well with the experimental data. Moreover, we start from the total pressure expression, and use the Gibbs–Duhem equation to re-derive the relationship between (∂p∂x)T,x→0 and the infinite dilution activity coefficient (γ∞) at low pressure. Based on the definition of partial molar volume, an equation for (∂p∂x)T,x→0 and gas solubility at high pressure is proposed in our work. Then, we use this equation to correlate the literature data on the solubility of nitrogen, hydrogen, methane, and carbon dioxide in water. These systems are reported at temperatures ranging from 273.15 K to 398.15 K and pressures up to 101.325 MPa. The total average relative deviation of the predicted values with respect to the experimental data is 0.08%, and the total average absolute deviation is 2.68%. Compared with the Krichevsky–Kasarnovsky equation, the developed model provides more reliable results.


Introduction
Knowledge of the phase-equilibrium behavior of solutions at infinite dilution is important not only for phase-equilibrium calculations of a dilute solution, but also for highconcentration solutions. Indeed, the phase-equilibrium properties of the dilute region are widely used in the process design of unit operations such as distillation, extraction, and absorption [1][2][3]. Hence, the study of infinite dilution thermodynamic properties is a crucial topic. There are two functions used to describe thermodynamic properties at infinite dilution, i.e., the infinite dilution activity coefficient (γ ∞ ) and Henry's constant (H i ). The former can be used to calculate relative volatility, partition coefficients, and phase-equilibrium data [4][5][6][7]. The latter is used to describe the partitioning capacity of a compound in the gas/water phase, mainly calculating the solubility of the gas in liquids [8][9][10].
Several experimental techniques are available to determine γ ∞ , among which the static differential technique and differential ebulliometry are common means. In the static differential method, the pressure difference between two static units of pure solvent and 2 of 17 dilute solution is measured in a constant temperature bath. Then, the difference is used to plot the composition-pressure diagram of the liquid phase at equilibrium to obtain ∂p ∂x T,x→0 . However, a differential ebulliometer is used to measure the change in the boiling point of a solvent to obtain ∂T ∂x p,x→0 . In these methods, γ ∞ is calculated from the relationship derived from Gautreaux and Coates [11]. Earlier studies have indicated that accurate data from these two methods can be obtained if strictly considering the liquid and vapor hold-ups in various parts of the systems [12,13]. Experimental data for Henry's constant can be calculated using independently measured solubility and partial pressure [10]. Moreover, it can be obtained by the static differential technique. Ayuttaya et al. [14] reported a method for determining H i with a differential static cell. The results suggested that the pressure accuracy influenced the uncertainty of the determined Henry's constant.
As one of the thermodynamic properties at infinite dilution, the value of γ ∞ is closely related to temperature. Currently, the theoretical model of γ ∞ and temperature is derived from the relationship between the activity coefficient and the partial molar excess function, which is expressed as: ln Assuming infinite dilution molar entropy (∆h E,∞ ) and infinite dilution molar enthalpy (∆s E,∞ ) are constants within a certain temperature range, the above relationship expresses a two-parameter model [7,15,16]. The subsequent three-parameter model [17], four-parameter model [18], and five-parameter model with the introduction of molecular descriptors X i [19] were developed from this equation. Notably, although the introduction of X i improves the correlation accuracy, this equation contains five variables, which complicates the relation between γ ∞ and temperature.
The pressure effect has a negligible effect on H i when the pressure is low. In this case, H i is closely related to the temperature. Considerable research efforts have been devoted to the two-parameter model [20,21]. The results of the correlation between H i and temperature were not satisfactory for systems with a wide range of temperatures. When considering the change in dissolution heat with temperature, the relationship between H i and temperature was developed into a three-parameter model and a four-parameter model [22]. Although the correlation accuracy of these two models is high over a wide temperature range, the relationship between H i and temperature becomes complicated due to too many parameters, which is not favorable for engineering applications. However, the effect of pressure on H i at high pressures is not negligible. Hence, it is necessary to study H i at high pressures. Thus far, much work has focused on γ ∞ and H i , where both γ ∞ and H i can be obtained by experimental determination of In this study, we consider ∂p ∂x T,x→0 as an infinite dilution thermodynamic function.
With this viewpoint, we propose a novel correlation model of with temperature.
Based on the calculation results of 40 systems, the novel model is proved to be objective and rational. The outstanding feature of this model is that there are only two adjustable parameters, which can be more easily adopted in engineering applications. Furthermore, as a thermodynamic property, of the mixture at a low pressure, which constitutes a substantial addition to our understanding of an infinitely diluted solution from a thermodynamic viewpoint.
The rest of this paper is organized as follows. Section  For a binary isothermal vapor-liquid equilibrium system, is given by the partial derivative of p with respect to x, where p is the total pressure and x is the molar fraction of solute in the liquid phase. It is notable that x tends to zero. For convenience, the term x in the latter is equivalent to x 1 , and the solute is denoted as component "1", while the solvent is component "2". It can not only describe the nonideality of the solution but also reflect the interaction between the solute and solvent at infinite dilution.

Relationship between
and γ ∞ For a binary system, the total pressure equation [23] at a low pressure can be described as Equation (1): where p s 1 is the saturated vapor pressure of the solute, p s 2 is the saturated vapor pressure of the solvent. It is important to emphasize that this equation applies to a low pressure. At a constant temperature, the activity coefficients γ 1 and γ 2 are only a function of the composition. For binary systems, the relationship between the two components is described as: where x 1 is the molar fraction of component 1 in the liquid phase, x 2 is the molar fraction of component 2 in the liquid phase. Therefore, taking the partial derivative of x 1 at isothermal conditions, we find: According to the expression of the Gibbs-Duhem equation for the binary system: where h E is the excess enthalpy, V E is the excess volume, R is the universal gas constant, while the temperature remains a constant, dT = 0. The liquids can be considered as an ideal solution at infinite dilution [24,25], V E = 0. It is convenient to rewrite Equation (4) as: Dividing both sides by dx 1 , Equation (5) can be represented as follows: The above equation is equivalent to: From Equation (7), as x 1 approaches zero, x 2 γ 2 dγ 2 dx 1 → 0 . Based on the normalized description of the activity coefficient: as the molar fraction of component 2 approaches 1, its activity coefficient converges to 1. Thus, when x 1 → 0 , ∂γ 1 Notice that Equation (8) is consistent with the formula reported by Gautreaux and Coates at a low pressure [11]. The difference is that the starting point in our work is the expression of the total pressure at a low pressure. The physical meaning of the derivation in our work is clear and its assumptions are accessible. This is because V E RT is usually considered as a negligible factor in many studies, and Equation (5) is a common formula that can be used to examine activity coefficient experimental data for thermodynamic consistency [26][27][28].

Relationship between
and H i Henry's law is usually described as a limit law, that is: When considering an ideal solution based on Raoult's law to define the activity coefficient, the liquid phase fugacity can be expressed as: where f 1 is the fugacity of the pure liquid at solution temperature and pressure. Thus, combining Equations (9) and (10), that is: At low pressures, f 1 is replaced by the saturated vapor pressure of the liquid p s 1 . Hence, Equation (11) becomes [7,29]: Thus, the expression of the relationship between ∂p ∂x T,x→0 and H 1 is obtained directly from Equations (8) and (12): Obviously, Equation (13) indicates that there is a simple relationship between ∂p ∂x T,x→0 and H 1 . In particular, when the solvent saturation vapor pressure p s 2 is low, Equation (10) becomes: ∂p This reveals that is Henry's coefficient in some special conditions. Therefore, it is appropriate to consider it as a thermodynamic property alone.

Relationship between
∂p ∂x T,x→0 and Solubility at High Pressures The above consideration represents the case of a low pressure. However, the influence of the pressure effect on Henry's constant cannot be ignored when the pressure is high. Therefore, it is necessary to extend the expressions of and H 1 under high pressures.
From strict thermodynamic relations, we can obtain: where v 1 is the partial molar volume of the solute in liquids. Substituting this result into Equation (9), we get: where v ∞ 1 is the partial molar volume of the solute at infinite dilution. Assuming that f 1 is proportional to x 1 at constant temperature and pressure, a more general form of Henry's law can be obtained: where p 0 is the reference pressure, p 0 = 101.325 kPa, H is the Henry's constant at 101.325 kPa. By definition, v 1 can be written as: Thus, the partial molar volume, as defined by Equation (18), can be evaluated using the following transformation: where ∂p ∂V T,n 1 ,n 2 describes the properties of solvents, which can be obtained by the equation of state. Since V changes near the saturated volume at infinite dilution, ∂p ∂n 1 T,V,n 2 can be written as . Thus: Substituting Equation (19) to Equation (20) gives: represents the properties of the solute at a low pressure. Generally, it is difficult to determine v ∞ 1 by experiment. In this work, v ∞ 1 is expressed as the properties of the solvent at a high pressure and the solute at a low pressure, which provides a new idea for obtaining v ∞ 1 .

Hence, an equation for
and solubility at high pressures can be written as: Considering the variation in the solute activity coefficient with molar composition, we added the activity coefficient for correction. Therefore, the final equation can be written as follows: .
According to the Gibbs phase rule, pressure is a function of temperature and composition for a binary system: When dp is equal to zero at a constant pressure, we get: Equation (27)

Source of
We select the isothermal vapor-liquid equilibrium (VLE) data [33][34][35][36][37] with the pressure as a function of the molar fraction for regression at low concentration ranges (x 1 < 0.05), typically 3-7 points. As shown in Figure 1, the VLE data of water in the tert-amyl alcohol system are regressed using the least squares method, where x represents the molar fraction of water in tert-amyl alcohol. It was found that p is linearly related to x, which can be written as Equation (28). More interestingly, Equation (28) can also be applied to other systems. Thus, the slope a is at a constant temperature.
Int. J. Mol. Sci. 2022, 23, x FOR PEER REVIEW Figure 1. Relationship between total pressure and liquid phase composition x of water in a alcohol system at 303.32 K.

Correlation of ( ) , → with Temperature
As previously noted, there is a simple relationship between ( ) , → and i tions (13) and (14). In particular, when is very low, ( ) , → is equal to . Several excellent studies which describe the relationship between and t ture are all two-parameter models [7,20,38], with the relations being expressed as ln = + Therefore, we propose a relationship between ( ) , → and temperature wh be written as follows:  and H i in Equations (13) and (14). In particular, when p s 2 is very low, Several excellent studies which describe the relationship between H i and temperature are all two-parameter models [7,20,38], with the relations being expressed as follows: Therefore, we propose a relationship between ∂p ∂x T,x→0 and temperature which can be written as follows: To prove the correctness of Equation (29), acetaldehyde-water and water-tert-amylalcohol systems are correlated by this relationship, as illustrated in Figure 2. The results show that the correlation coefficient R 2 > 0.99, which indicates a good linear relationship between ln ∂p ∂x T, x→0 and the inverse of temperature.

ln( ) , → = +
To prove the correctness of Equation (29), acetaldehyde-water and water-te alcohol systems are correlated by this relationship, as illustrated in Figure 2. Th show that the correlation coefficient > 0.99, which indicates a good linear rela between ln( ) , → and the inverse of temperature.   can be obtained as x approaches 0, when the correlation between ∂p ∂x T,x→0 and temperature improves. can be calculated from γ ∞ and H i as shown in Equations (8) and (13). In this section, these two infinite dilution thermodynamic functions are calculated by ∂p ∂x T,x→0 from isothermal VLE data. Moreover, the accuracy of Equations (22) and (23) are verified using high-pressure solubility data for different systems.

Using
to Calculate γ ∞ We use isothermal VLE data [33][34][35] with the pressure as a linear function of the molar fraction to calculate ∂p ∂x T,x→0 , and then obtain γ ∞ from Equation (8). The results are shown in Table 3. According to the results of the calculations, most systems are well adapted to the literature, while a few systems such as nitromethane-water show significant differences. This result may be related to the accuracy of the γ ∞ experimental value; many researchers reported that γ ∞ measured by different methods showed significant differences [18,41]. Moreover, Sherman et al. [42] evaluated the database of γ ∞ for nonelectrolytes in water and found that the accuracy of the database for measured values is estimated at 10% for γ ∞ < 1000. from the isothermal VLE data [33][34][35] with the pressure as a linear function of the molar fraction, and we use Equation (13) to obtain H i . The calculation results are shown in Table 4 . This proves that is reasonable as a thermodynamic property. to Calculate the Solubility of the Gas at High Pressure As a traditional model, the Krichevsky-Kasarnovsky (K-K) equation is often used to express the effect of high pressures on gas solubility [46], given by: where p 0 is the reference pressure, and H is Henry's constant at p 0 .
In this work, we deduce formulas for solubility and ∂p ∂x T,x→0 at high pressure, as shown in Equations (22) and (23). In both formulae, ∂v ∂p T is the property of pure water, which can be calculated by the equations of IAPWS-IF97 [47]. In Figure 3, we compared the relationship between solubility and fugacity using different models for four aqueous systems. The blue solid lines indicate the best fits to the experimental data ("this work 2"), while pink dotted lines represent K-K equation results. It can be seen that both the proposed model and the K-K equation can describe the solubility of hydrogen and carbon dioxide well in water at high pressures. In particular, the model in this work is superior to the K-K equation for the nitrogen and methane systems. Moreover, Figure 3 shows that the model with the activity coefficient in this work, as illustrated Equation (23)  In addition, four typical systems are selected for comparison with the proposed model, with temperatures ranging from 273.15 K to 398.15 K and pressures up to 101.325 MPa. The average deviations of the calculated results concerning experimental data in this work are listed in Table S1.
The results show that the A.A.D. (%) of the model with the activity coefficient for 13 groups is as low as 2.68%, and the model prediction of this paper is more accurate compared with that of the K-K equation. In this study, we find that the activity coefficient of the solute is closely related to the composition. The parameters between the activity coefficient and the composition are shown in Table 5. It is worth noting that for hydrogen and nitrogen systems, the relationship between the activity coefficient and the composition can be simply expressed as a linear relationship. However, for systems with relatively strong interactions with water, such as carbon dioxide and methane systems, it can be expressed as a quadratic functional relationship.

Conclusions
In this paper, we propose the idea of considering In addition, we re-derive the relationship between ∂p ∂x T,x→0 and γ ∞ at low pressures, in terms of the Gibbs-Duhem equation. We also describe the relationship between 08%. Compared to the classical K-K model, the new model can correlate the solubility data more accurately. Overall, these results give a comprehensive understanding of infinite dilution properties, which can facilitate improvements in promising applications for chemical process design.

Informed Consent Statement: Not applicable.
Data Availability Statement: All data generated or analyzed during this study are available within the article or upon request from the corresponding author.