Reference materials for phase equilibrium studies. 2. Solid–liquid equilibria (IUPAC Technical Report)

2.1 Urea (carbonic diamide) + water system

One of the best-studied systems with consistent experimental SLE data in this category is urea + water. The selected literature sources associated with the SLE data for this system are listed in Supplement 2A. It should be noted that there were also a few series of works on ternary SLE diagrams at fixed temperatures, where urea was one of the solutes and binary SLE data for urea + water were reported [11], [12], [13], [14]; however, the binary mixture was not the primary focus of the works, and the solubilities were similar within each series. To avoid any bias of the generated recommendation toward the data from those laboratories, only representative cases were included in the assessment.

Most of the selected sources are consistent, with the prominent exception of all data points from Refs. [14], [15], [16] as well as one data point from Ref. [17] and two points from Ref. [12] (Figure 1). The value in Ref. [16] is accompanied by the sign “+,” which probably indicates that the actual solubility is above that reported value. Some researchers (e.g., [12]) have claimed three different states of solid urea. These claims are doubtful given that reliable heat capacity measurements, by adiabatic and triple thermal bridge calorimetries, found no solid-to-solid phase transitions up to the melting point of urea [18], [19], [20].

Fig. 1:

Experimental mole-fraction SLE data (dots) for urea (1) + water (2) vs the NRTL evaluation with the parameters from Table 1 (blue line). Rejected points from Refs. [12, 14], [15], [16], [17] are shown as red dots.

The selected SLE data were smoothed with the NRTL (Non-Random Two-Liquid) excess Gibbs energy model [21] by the ThermoData Engine software [8]. For a binary mixture, this model gives the following expression for the excess molar Gibbs energy G^E:

with the temperature-dependent parameters c_i defined as follows:

where α is an empirical fitting parameter called the “non-randomness parameter”; A_i through D_i are the empirical fitting parameters for component i; T is the temperature in K; x_i is the mole fraction of component i; and R is the molar gas constant.

From basic thermodynamic relationships, the mole-fraction solubility of component i (x_i) at temperature T can be shown to be:

where T_tp,i is the triple-point temperature (crystal + liquid + gas) for component i, Δ_fusH_m,i is the enthalpy of fusion of the crystalline form of component i, and Δ_fusC_p,m,i is the heat-capacity difference between the liquid and crystalline form of component i, and γ_i is the activity coefficient of component i, which can be derived from eq. (1) as described in Ref. [21]:

The parameters of the resulting NRTL fit for urea (1) + water (2) are presented in Table 1. This table also contains the melting parameters (for both mixture components) used in the model. For urea, these parameters were obtained from the ThermoData Engine [8] as weighted averages of the values reported in the literature (Supplement 2A), while for water they were adopted from the water and ice recommendations from IAPWS (International Association for the Properties of Water and Steam) [22, 23]. Since T_tp, Δ_fusH_m, and Δ_fusC_p,m were assigned fixed values during the NRTL fitting, any inaccuracy in them was absorbed by the fitting parameters α and A_i through D_i, so their uncertainties were not needed for the purposes of the present project and thus were not evaluated.

The performance of the NRTL model relative to the selected experimental data is shown in Figure 1 and as deviation plots in Supplement 2B. Although the NRTL model was based only on the SLE data, available experimental VLE pressures for the mixture (given in Supplement 2B) are also in fair agreement with it, which can serve as additional verification for the selected SLE data and model.

Smoothed solubility data are listed in Table 2. The eutectic point, according to the model, is at T = (262.30 ± 0.29) K and x₁ = 0.1283 ± 0.0061 with expanded uncertainties U at the 0.95 level of confidence (coverage factor 2) based on the data scatter. The uncertainty assessment procedure is described in Supplement 1C of the LLE part of this project [10]. Experimental uncertainties of 2 K and 0.03 x_guest (where the “guest component” for SLE is the solute) were assumed unless individual uncertainties were assigned to particular data sets. Minimum uncertainties were taken to be 0.25 K and 0.01 x_guest. The uncertainties for SLE mole fractions and temperatures are approximated by the equations reported in Supplement 2C.

2.2 Sulfolane (1λ⁶-thiolane-1,1-dione) + water system

The sulfolane + water system was selected as a reserve system due to its interesting behavior with regard to the presence of two consecutive stable crystalline modifications of sulfolane [24, 25] and a possible supercooling of its high-temperature crystal (form I), giving rise to a metastable SLE [26, 27]. The latter may be used for testing the ability of experimental methods to determine stable equilibrium. Though reported in only two sources [24, 25], the experimental SLE data for the system are very consistent, as seen in Figure 2. A tentative evaluation for this system is described below.

Fig. 2:

Experimental mole-fraction SLE data (dots) for sulfolane (1) + water (2) vs the Wilson model with the parameters from Table 3 (blue line). Data from Refs. [24, 25] are shown as black and gray dots, respectively.

The selected SLE data were smoothed with the Wilson-type excess Gibbs energy model [28] by the ThermoData Engine software [8]. For a binary mixture, the model gives the following expressions for the excess molar Gibbs energy, G^E, and activity coefficient of components 1 and 2, γ₁ and γ₂, respectively:

with the temperature-dependent parameter a_i defined as follows:

where A_i through C_i are the empirical fitting parameters for component i; T is the temperature in K; x_i is the mole fraction of component i; and R is the molar gas constant.

Equation (3) is valid for the solubility curve of component i between its melting and its solid-to-solid phase transition temperatures. Below the solid-to-solid phase transition temperature, eq. (3) is modified to:

where T_{tp(crII–crI),i} is the triple-point temperature (crystal II + crystal I + gas) for component i, Δ_trsH_m,i is the enthalpy of the solid-phase transition of component i, and Δ_trsC_p,m,i is the heat-capacity difference between crystal I and crystal II for component i.

The parameters of the resulting Wilson fit for sulfolane (1) + water (2) are presented in Table 3. This table also contains the phase-transition parameters for both mixture components used in the modeling. For water, these parameters were as described above for the urea + water system. For sulfolane, the melting temperature and the enthalpy of the solid-to-solid phase transition were obtained from the ThermoData Engine [8] as weighted averages of the values reported in the literature (Supplement 2D). As described in Ref. [24], the liquidus curve of crystal I (in an orientationally disordered phase) of sulfolane for the sulfolane + water system deviates from the one expected from its enthalpy of fusion, which may indicate the presence of small quantities of water in the solid phase. Hence, the melting enthalpy for sulfolane was modified from the experimental value of pure sulfolane, from 1.4 kJ mol⁻¹ [29], to 2.2 kJ mol⁻¹ for the model fitting. This increase is consistent with the expectation of the formation of a solid solution. As a result, the solid-to-solid transition temperature used (287.4 K) was also modified from the experimental value of 288.6 K [29] for a better fit. It should be emphasized that the magnitude of the modifications is small, leading to a conclusion that the Wilson model can still be applied, even though a solid solution may be present. At the same time, no presence of water in the ordered phase (crystal II) was detected [24]. There are no heat-capacity data for crystalline sulfolane available to derive Δ_fus/trsC_p,m directly; however, an indirect analysis in Ref. [30] gave Δ_fusC_p,m = 0 and Δ_trsC_p,m = 45.5 J K⁻¹ mol⁻¹.

The performance of the Wilson model with the parameters from Table 3 relative to the selected experimental data is shown in Figure 2 and as deviation plots in Supplement 2E. Since the model was obtained solely to fit the selected SLE data, it cannot be used for deriving other properties (e.g., excess enthalpy or vapor pressures). Additional verification of the selected SLE data was done by simultaneously fitting the Wilson model to the available literature SLE, excess enthalpy, and vapor pressure data between 260 and 323 K. The results of that more generic fit are provided in Supplement 2E, where it is clearly seen that the selected SLE data are in fair agreement with other properties. It must be emphasized that the parameters from Table S2 are not recommended for SLE calculations, since their only purpose is to prove consistency with other properties.

Smoothed solubility data are given in Table 4. The coordinates of the eutectic and phase-transition points derived from the model are x₁ = 0.3079 ± 0.0034 at T = (262.38 ± 0.36) K and x₁ = 0.9557 ± 0.0011 at T fixed at 287.4 K, respectively, with expanded uncertainties at the 0.95 level of confidence (coverage factor 2) based on the data scatter.

3.1 Potassium chloride + water system

The solubility of potassium chloride in water was discussed in SDS-47 [31] (hereafter, the abbreviation SDS-X-Y is used for “Solubility Data Series, Volume X, Part Y”; if Volume X does not have separate parts, its abbreviation is contracted to SDS-X) and in Krumgalz’s review [32]. Additional data sources available from the NIST/TRC Source database [8] (Supplement 2F) do not affect their evaluations, since they are consistent with the previous data except for two points from Refs. [33, 34]. The experimental data cited in Refs. [31, 32] (excluding the points consistently rejected in both reviews) as well as all additional data from Supplement 2F are shown in Figure 3.

Fig. 3:

Experimental mole-fraction solubility data for potassium chloride (1) in water (2) cited in Refs. [31, 32] (excluding rejected points) as well as all additional data from Supplement 2F (dots) vs the SDS eq. (11) with the parameters from Table 5 (line: dark blue — within the temperature range accepted for the purpose of this report as indicated in Table 5, light blue — outside of that temperature range). Two deviating points from additional sources [33, 34] are highlighted in red.

The two solubility evaluations [31, 32] are comparable, but the SDS one was selected here due to its better performance at lower temperatures, which is of interest for the current project. Interested readers are referred to the supporting information spreadsheet file from Ref. [32] for the plot showing deviations of available literature data from Krumgalz’s evaluation. The mole-fraction solubility of potassium chloride in water between 262 and 1044 K was smoothed in Ref. [31] by the following equation:

where A through J are the empirical fitting parameters; T is the temperature in K; x₁ is the mole fraction of the salt; m₁ is the molality of the salt in mol kg⁻¹; and M₂ is the molar mass of water in kg mol⁻¹. The equation parameters for the system are listed in Table 5.

It is clear from Figure 3 and Figure S5 in Supplement 2G that the SDS evaluation has a bias above 570 K. Accordingly, we restrict our recommendation in Table 5 to T/K ≤ 570. It is also suggested in Ref. [32] that an unstable form of KCl·xH₂O may be formed below 268 K. Though this claim is made in very few works, some care should be taken below 268 K to confirm the solid phase identity.

Smoothed solubility data calculated from the equation from Ref. [31] are given in Table 6. The eutectic point is at T = 262.505 K with calculated x₁ = 0.0558 as reported in Ref. [31]. The SDS-47 [31] relative uncertainty estimate for the whole temperature range was 2.5 %. Based on the scatter analysis (Figure S5), we estimated the uncertainties in Table 6 for both temperature and composition representations.

Thermodynamic evaluation of potassium chloride + water that corroborated the solubility data by models consistent with other property data have been done by Pabalan and Pitzer [35] and Archer [36]. The most recent publication Ref. [37] cites those and re-applies the Pitzer model. We additionally verified the consistency of the solubility data with other measured thermodynamic properties with the Pitzer model using the parameters and procedures implemented in the PHREEQC program [38, 39] designed for aqueous geochemical calculations, as shown in Figure S6 in Supplement 2G. It should be noted that the comparison is limited to T/K ≤ 523.

3.2 Sodium chloride + water system

The solubility of sodium chloride (1) in water (2) has been evaluated in NIST Technical Note 1387 [40], in SDS-47 [31], by Krumgalz [32], and discussed briefly by one of us [41]. The last included some key experimental details, such as the correct drying of solid sodium chloride, that are often overlooked. It should be noted that a hydrate NaCl⋅2H₂O is formed at T < 273.15 K: the peritectic point is T = 273.15 K and x₁ = 0.0990 according to SDS-47 [31]. However, it appears that there were not enough consistent experimental data to develop a reliable solubility equation for the hydrate region, especially due to difficulties in handling NaCl⋅2H₂O. Consequently, we considered only the solubility profile for anhydrous sodium chloride (i.e., above 273.15 K). Additional data sources available from the NIST/TRC Source database [8] for that phase diagram portion (Supplement 2H) are generally consistent with the previous data. The experimental data cited in Refs. [31, 32] (excluding the points consistently rejected in both reviews) as well as all additional data from Supplement 2H are shown in Figure S7 in Supplement 2H.

In contrast to the potassium chloride + water system, the two solubility evaluations [31, 32] are not fully consistent, especially at the low-temperature and high-temperature ends. Moreover, they both have some fitting issues leading to offsets relative to the available experimental data in some temperature regions. Interested readers are referred to the supporting information spreadsheet file from Ref. [32] for the plot showing deviations of available literature data from Krumgalz’s evaluation. Since no similar deviation can be found for the SDS-47 evaluation in Ref. [31], the corresponding deviation plot is shown in Figure S8a in Supplement 2I. Consequently, we decided to refit equation (11) to the mole-fraction solubilities of anhydrous sodium chloride in water shown in Figure S7 but to limit the data to temperatures below the critical temperature of water (Figure 4) as the current report does not cover supercritical SLE measurements. Accordingly, we restrict our recommendations to 273 ≤ T/K ≤ 635. The fitting was done by the least-squares method with enforcement of the peritectic-point coordinate from Ref. [31] within the uncertainty of the fit. The resulting parameters are listed in Table 5. The performance of the evaluation relative to the available experimental data is shown in Figure 4 and Figure S8 in Supplement 2I.

Fig. 4:

Experimental mole-fraction solubility data for sodium chloride (1) in water (2) cited in Refs. [31, 32] (excluding rejected points) as well as all additional data from Supplement 2H (dots) vs the SDS-type eq. (11) with the parameters from Table 5 (blue line).

The thus-calculated smoothed solubility data are given in Table 7. Because of a steep temperature rise with increasing concentration of sodium chloride, the representation of the SLE temperature as a function of the mole-fraction solubility would not be practical and so is not provided. Based on the scatter analysis (Figure S8), we estimated the uncertainties in Table 7 for the composition representation.

Thermodynamic evaluations of sodium chloride + water that corroborated the solubility data by models consistent with other property data have been done by Pitzer et al. [42], Archer [43], and Krumgalz [32]. We additionally verified the consistency of the solubility data with other measured thermodynamic properties with the Pitzer model using the parameters and procedures implemented in the PHREEQC program [38, 39], as shown in Figure S9 in Supplement 2I. It should be noted that the comparison is limited to T/K ≤ 523. In contrast to the present model, the Krumgalz polynomial equation [32] and the Pitzer model from PHREEQC [38, 39] show a minimum-solubility temperature around 285 K, which may be a fitting artifact that cannot be reliably proven because of the experimental data scatter in that temperature region.

It should be also noted that the solubility equations provided in Refs. [31, 32] for the hydrate region of the NaCl + water phase diagram are fairly consistent (the maximum difference in mole-fraction compositions reaches 0.7 %), so researchers interested in working with salt hydrates at sub-zero temperatures can use any of them, but we believe that we do not have sufficient information at the present time to recommend that region for method testing.

3.3 Additional note: reserve system

The solubility of ammonium chloride in water was discussed in SDS-47 [31] and assessed in Ref. [44]. The system ammonium chloride + water may be used as a reserve test mixture for SLE measurements. Additional data sources for it are listed in Supplement 2J.

4.1 Naphthalene (bicyclo[4.4.0]deca-1,3,5,7,9-pentaene) + toluene (methylbenzene) system

One of the best studied organic systems with consistent data is naphthalene + benzene [45], which unfortunately was excluded from consideration in the current project due to the toxicity of benzene and restrictions on its use. A similar but less toxic system is naphthalene + toluene discussed in SDS-59 [45] and SDS-98-3 [46]. Two additional sources [47, 48] were not mentioned in those reviews. The selected literature sources associated with the SLE data for this system are listed in Supplement 2K. The experimental SLE data are generally consistent; only two points from Ref. [17] deviate noticeably from the other data (Figure 5).

Fig. 5:

Experimental mole-fraction SLE data (dots) for naphthalene (1) + toluene (2) vs the NRTL evaluation with the parameters from Table 8 (line: dark blue — within the temperature range accepted for the purpose of this report as indicated in the text, light blue — outside of that temperature range). Two rejected points from Ref. [17] are shown as red dots.

The selected SLE data were smoothed with the NRTL model given by eqs. (1)–(5) with the use of the ThermoData Engine software [8]. Since there is only one experimental work at T < 247 K, naphthalene solubilities below that temperature were not included in the fit. The parameters of the resulting NRTL fit for naphthalene (1) + toluene (2) are presented in Table 8. This table also contains the melting parameters for both mixture components used in the modeling. For naphthalene, the melting parameters were adopted from Ref. [49] (note: the selected melting temperature and the enthalpy of fusion are in excellent agreement with the recommendations of Ref. [50]); for toluene, they were taken from Ref. [51]. Since T_tp, Δ_fusH_m, and Δ_fusC_p,m were assigned fixed values during the NRTL fitting, any inaccuracy in them was absorbed by the fitting parameters α, A_i, and B_i, so their uncertainties were not needed for the purposes of the present project and thus were not evaluated.

The performance of the NRTL model relative to the selected experimental data is shown in Figure 5 and as deviation plots in Supplement 2L. Although the present model was developed using only the SLE data, available experimental VLE pressures for this system (shown in Supplement 2L) are in good agreement with it, which can serve as additional verification for the selected SLE data and model. Since naphthalene + aromatic hydrocarbon systems are in general nearly ideal, an additional verification became possible, namely a comparison of the naphthalene solubility profiles in the homologous solvent series: benzene, toluene, and ethylbenzene. The comparison over the range of validity of the developed NRTL model is shown in Figure S12 in Supplement 2L, where all the experimental data for the three mixtures are in excellent agreement.

Smoothed solubility data are given in Table 9. The coordinates of the eutectic point cannot be reliably found from the developed NRTL model, since the fit was limited to temperatures above 247 K, as discussed above.

4.2 Additional note: reserve systems

Reserve non-aqueous systems can be naphthalene + tetrachloromethane, discussed in SDS-59 [45] and SDS-98-3 [46], and benzoic acid + cyclohexane, discussed in SDS-99 [52] with additional data sources listed in Supplement 2M.

5.1 Benzoic acid (benzenecarboxylic acid) + water system

A convenient and well-studied system is benzoic acid + water. The available experimental SLE and LLE data are listed in Supplement 2N. Both SLE and LLE data (the latter can apparently be supercooled) are shown in Figure S13 in Supplement 2N, where all rejected points are identified. Only the side corresponding to the low solubility of benzoic acid in water was considered for this category (Figure 6, given in the composition-stretched representation [53] for convenience) – from the eutectic temperature to the monotectic point.

Fig. 6:

Experimental mole-fraction SLE data (dots) for benzoic acid (1) + water (2) vs the NRTL evaluation with the parameters from Table 10 (blue line) in the composition-stretched representation [53]. The rejected points from Figure S13 are excluded.

The selected SLE data have been smoothed with the NRTL model given by eqs. (1)–(5) with the use of the ThermoData Engine software [8]. The parameters of the resulting NRTL fit for benzoic acid (1) + water (2) are presented in Table 10. The table also contains the melting parameters for both mixture components used in the modeling. For water, these parameters were as described in the urea + water section. For benzoic acid, the melting temperature and enthalpy of fusion were taken from the recommendations in [50], which are based on a precise experimental work [54]; the Δ_fusC_p,m value was taken as the average of two values obtained by adiabatic calorimetry in the vicinity of the melting point [55, 56]. Since T_tp, Δ_fusH_m, and Δ_fusC_p,m were assigned fixed values during the NRTL fitting, any inaccuracy in them was absorbed by the fitting parameters α and A_i through D_i, so their uncertainties were not needed for the purposes of the present project and thus were not evaluated.

Smoothed solubility data are given in Table 11. Because of a steep temperature rise with increasing concentrations of benzoic acid, the representation of the SLE temperature as a function of the mole-fraction solubility would not be practical and, hence, is not provided. The eutectic and monotectic points evaluated from the model are as follows: T = (273.137 ± 0.005) K and x₁ = 0.000 232 ± 0.000 010; T = (370.5 ± 2.0) K and x₁(side 1) = 0.011 ± 0.002 and x₁(side 2) = 0.37 ± 0.06, respectively (with expanded uncertainties at the 0.95 level of confidence, coverage factor 2). Note that extra care should be taken regarding the solubilities at high temperatures, since LLE in benzoic acid + water can be supercooled below the monotectic temperature, according to several sources listed in Supplement 2N.

The performance of the NRTL model relative to the selected experimental data is shown in Figure 6 and as a deviation plot (Figure S14) in Supplement 2O. Although the NRTL model was developed using only the SLE data, it predicts the existence and position of the LLE reasonably well, as shown in Figure S15 in Supplement 2O, which can serve as additional verification for the selected SLE data and model.

5.2 Naphthalene (bicyclo[4.4.0]deca-1,3,5,7,9-pentaene) + water system

Another well-studied system featuring low solubilities is naphthalene + water. The solubilities for this system have been compiled and critically evaluated in SDS-81-9 [57]. The available literature sources associated with the SLE data for this system are listed in Supplement 2P. Fourteen sources in addition to those cited in SDS-81-9 have been included in the list. Because of these additions, it was decided to update the solubility equation. Most of the experimental SLE data are in fair agreement with each other, except for seven values from Refs. [58], [59], [60], [61], [62], [63], which look obviously erroneous (Figure 7) and thus were rejected.

Fig. 7:

Experimental mole-fraction SLE data (dots) for naphthalene (1) + water (2) vs the smoothing eq. (12) (blue line) in the composition-stretched representation [53]. Rejected data from [58], [59], [60], [61], [62], [63] are highlighted in red.

The mole-fraction solubility data of naphthalene (x₁) in water were smoothed by the least-squares method with the following empirical equation:

where T is the temperature in K; x₁ is the mole fraction of the solute. This equation is valid from 273.15 to 350 K.

The performance of the evaluation relative to the available experimental data is shown in Figure 7 and Figure S16 in Supplement 2P. It should be noted that the current evaluation by eq. (12) is generally consistent with that of SDS-81-9 [57] – the relative difference is not more than 6 %, which is within the claimed uncertainties.

Smoothed solubility data are given in Table 12. Due to the extremely small solubility, the representation of the SLE temperature as a function of the mole-fraction solubility would not be practical and, hence, is not provided. Since the solubility of naphthalene in water is very small, the eutectic temperature cannot significantly depart from the melting temperature of pure water and was estimated to be T = (273.15 ± 0.01) K. The eutectic composition thus derived from eq. (12) is x₁ = (1.89 ± 0.09) × 10⁻⁶. The uncertainties are expanded uncertainties at the 0.95 level of confidence (coverage factor 2).

5.3 Additional notes: excluded system and reserve systems

We considered including the anthracene + water system as a reserve system due to its extremely low solubility and the large number of fairly consistent measurements available. The solubilities for this system have been compiled and critically evaluated in SDS-81-11 [64]. The available literature sources associated with the SLE data for this system are collected in Supplement 2Q and are shown in Figure S17. Unfortunately, two distinct clusters of the SLE data are apparent. A possible explanation is the presence of phenanthrene as an impurity in commercial anthracene [65]. Phenanthrene is about 20 times more soluble in water than in anthracene [64], and it forms a solid solution with anthracene [66], which potentially explains the higher solubilities reported in some works. However, the absence of detailed information about the sample purities precludes a definite conclusion.

Other reserve test systems for this category of solubility data could be water + 2-hydroxybenzoic acid (SDS-90-1 [67]) and water + 4-hydroxybenzoic acid (SDS-90-1 [67]). However, special care should be taken with the solubilities of such systems, since hydroxybenzoic acids may form metastable LLE with water [68], [69], [70].