Thermodynamically Traceable Calorimetric Results for Aqueous Sodium Chloride Solutions from T = (273.15 to 373.15) K up to the Saturated Solutions: Part 2 —The Quantities Associated with the Partial Molar Heat Capacity

Sodium chloride is one of the most important electrolytes on our planet. For example, it is the primary salt in seawater and most other natural waters, in addition to being a crucial component in many industrial and biological systems. The thermodynamics of pure NaCl solutions has been thoroughly investigated at the standard pressure of p = 101.325 kPa up to the saturated solutions and summarized in the review articles from Pitzer et al. [1], Clarke and Glew [2], and Archer [3]. The experimental measurements considered in these reviews have served as the basis for many theoretical models, including the Pitzer model [4, 5]. The Pitzer model forms the theoretical foundation for the multiparameter equations presented in the three reviews that can be used to explain and predict the experimental thermodynamical results for pure solutions of several other common electrolytes besides NaCl. The model also serves as the basis for treating more complicated mixed electrolyte systems.

For many practical applications, however, the multiparameter equations are often unnecessarily complicated, containing superfluous parameters associated with the three independent variables of temperature, pressure, and concentration. Despite their complexity, the models still fail to explain all the features of the available experimental data, including, for example, several freezing-point results considered in Ref. [6], the vapor-pressure results at higher temperatures considered in Refs. [7, 8], and some calorimetric results treated in Ref. [9]. In contrast, our recent findings demonstrate that the behavior of these solutions is often accurately described by a much simpler Hückel-type equation where only a handful of estimated parameters are used [8,9,10,11,12].

Up to date, heat capacities for pure aqueous solutions of several univalent electrolytes have been considered in the reviews from Parker [13] and Criss and Millero [14]. Specifically, for NaCl the former review includes results at temperatures 288, 293, 298, and 303 K and the latter at 298 K. The utility of the Pitzer model for the heat capacities of NaCl solutions at various temperatures has also been reviewed by Silvester and Pitzer [15], in addition to Refs. [1,2,3]. Ensor and Anderson [16] have published a wide series of heat-of-dilution data for NaCl(aq) within the range from T = 313 to 353 K with calculated heat capacities. Except for the studies from Refs. [12, 13, 16], the Pitzer formalism [4] has formed the foundation on which the interpretation of the experimental heat capacity data has been based.

As emphasized in Criss and Millero’s review [14], the data considered by Parker [13] were measured before the invention of flow calorimeters, and the results of these flowing-heat-capacity-system measurements (Picker method) are superior to the older data for dilute electrolyte solutions. Several studies have since provided heat capacity data for NaCl solutions using flow calorimeters at various temperatures [17,18,19].

In this study, we investigate the applicability of our three-parameter Hückel equation to the heat capacity quantities of NaCl solutions. In our seminal study [8], we observed that the three-parameter Hückel equation applies for the activity and osmotic coefficients in NaCl(aq) up to saturated solutions in the temperature range from 273 to 373 K. Meanwhile, in part 1 of this calorimetric study [9], we observed that the same equation likely applies in a similar manner to the quantities associated with the partial molar enthalpy. In the present study (part 2), we continue to validate our methodology by comparing its predictions to almost all existing heat capacity data for NaCl(aq).

2.1 The Extended Hückel Model

In aqueous solutions of many salts [7,8,9,10,11,12, 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39], the following normal or extended Hückel equations can be used to predict the mean activity coefficient (γ) of the salt and the osmotic coefficient (ϕ) of water at least up to an ionic strength (I_m) of 1 mol⋅kg⁻¹:

where the electrolyte parameter b₂ is set to zero in the normal Hückel equations. In Eqs. 1 and 2, m is the molality, m^o is 1 mol⋅kg^–1, z₊ is the charge number of the cation and z₋ is that of the anion, and α is the Debye−Hückel parameter. Meanwhile, B and b₁ and b₂ are electrolyte-dependent parameters. For a 1:1 electrolyte such as NaCl,|z₊z₋| = 1 and I_m = m.

The values of the Debye−Hückel parameter α were obtained from a quadratic equation of the type

where α₀ = 1 (mol⋅kg⁻¹)^−1/2 and T₀ = 273.15 K. In the temperature range from T = 273 to 373 K at p = 101.325 kPa, the parameter values are u₀ = 1.1296, v₀ = 1.550⋅10⁻³, and q₀ = 9.6⋅10⁻⁶. This equation was determined in Ref. [7] from the α values presented in Ref. [40] that are provided in Table 1. As demonstrated in Table 1 of Ref. [9], the predicted values show excellent agreement with the original ones.

The normal two-parameter Hückel equation for activity and osmotic coefficients with parameters B and b₁ has proven useful in predicting the thermodynamic properties of dilute solutions of pure electrolytes at the normal reference temperature of 298.15 K [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,−37]. More recent studies [7, 11, 12, 38, 39] have shown that this equation applies within experimental error to the thermodynamic properties of both NaCl and KCl solutions at least up to m = 0.2 mol⋅kg^–1 in the range from T = 273 to 373 K. In Ref. [11], the following approach was used to determine the temperature dependence of the electrolyte-dependent parameters B and b₁:

(1) A value of b₁(298.15 K) = 0.0716 was obtained in Ref. [22] at 298.15 K.

(2) A value of b₁(273.15 K) = 0.0077 was determined at 273 K from existing freezing-point-depression data in Ref. [10].

(3) A value of b₁(348.15 K) = 0.105 was calculated from the vapor pressure data of Gibbard et al. [41] in Ref. [11].

(4) A constant value of 1.4 (mol⋅kg⁻¹)^−1/2 was used for the B parameter for the whole temperature range from 273 to 353 K, as this value was utilized for determining the b₁ values at steps 1, 2, and 3.

5) For temperatures other than 273, 298, and 348 K within the 273 to 353 K range, a quadratic temperature dependence was determined for b₁ in Ref. [11] using the three values obtained at steps 1, 2, and 3. This expression had the form

where u₁ = 0.0077, v₁ = 3.1853⋅10⁻³, and q₁ =  − 25.17⋅10⁻⁶. In Eq. 4 and in all subsequent equations, T₀ is 273.15 K.

The three-parameter extended Hückel equation utilizes the same method for determining parameters B and b₁ but the values are extended up to T = 373 K. For the additional b₂ parameter in Eqs. 1 and 2, a second order polynomial was fitted in Ref. [8] from the values obtained at various temperatures using the direct vapor pressure data of Gibbard et al. [41] and Olynyk and Gordon [42]. Additional b₂ values at 273 K and 288 K were determined from the osmotic coefficients of Platford [43] and Childs and Platford [44], respectively. These coefficients are based on the isopiestic data against urea solutions. The resulting expression for \({b}_{2}\) had the following form:

with u₂ = 0.01328, v₂ =  − 364.7⋅10⁻⁶, and q₂ = 2.7⋅10⁻⁶.

Following our notation in Ref. [9], this parametrization of the extended Hückel equation is designated here as PI(con) and the previously determined parametrization for the two-parameter Hückel equation determined for dilute NaCl solutions is referred to as PI(dil).

2.2 General Equations for the Enthalpy and Heat Capacity Quantities

The excess Gibbs energy of aqueous solutions (G_ex) for a pure 1:1 electrolyte on the molality scale is related to the activity and osmotic coefficients through

where R is the universal gas constant and w₁ is the mass of the solvent (symbol 1), i.e., of water. The apparent molar enthalpy of the solute salt (symbol 2) is H_app,2. It is defined by equation

where n₁ = (w₁/M₁) and n₂ (= mw₁) are the amounts of water and the salt, respectively, H is the enthalpy of the system, \({H}_{\text{m},1}^{*}\) is the molar enthalpy of pure water, and M₁ (= 0.018015 kg⋅mol⁻¹) is the molar mass of water. The relative apparent molar enthalpy,\({\Delta H}_{\text{app}}\), is defined with help of the partial molar enthalpy of the salt at infinite dilution (i.e., of \({H}_{\text{m},2}^{\infty }\)) as \({\Delta H}_{\text{app} }{=H}_{\text{app},2}-{H}_{\text{m},2}^{\infty }\). It is associated with the excess Gibbs energy through the following thermodynamic identity:

Analogously, the apparent molar heat capacity of the salt at a constant pressure is C_app,2. For simplicity, we denote it by C_app. It is defined through

where C is the heat capacity of the system and \({C}_{\text{m},1}^{*}\) is the molar heat capacity of pure water. The relative apparent molar heat capacity,\({\Delta C}_{\text{app}}\), is defined similarly by using the partial molar heat capacity of salt at infinite dilution (i.e., by using \({C}_{\text{m},2}^{\infty }\)) as \({\Delta C}_{\text{app} }{=C}_{\text{app}}-{C}_{\text{m},2}^{\infty }\). It is associated with ΔH_app of salt by the following thermodynamic formula:

Once the relative apparent molar heat capacity is known via this equation, the salt’s relative partial molar heat capacity, \({\Delta C}_{\text{m},2 }={C}_{\text{m},2}-{C}_{\text{m},2}^{\infty }\), can be calculated from

As in Ref. [9], it should be noted that we have simplified the notation compared to Refs. [11, 12, 38, 39] to clarify the theoretical background of these enthalpy and heat capacity quantities. For the excess Gibbs energy, for example, we have used the symbol G_ex instead of ΔG_ex because all energy quantities containing Δ are now most often the relative ones with respect to the infinite dilute state.

3.1 Calculation of the Relative Apparent Molar Heat Capacities

In Refs. [11] and [12], parametrization PI(dil) was tested for dilute NaCl solutions using all available high-quality calorimetric data up to T = 353 K. In contrast, [9] and the present study test parametrization PI(con) and extend the temperature range up to T = 373 K. As for the dilute-solutions cases [12, 39], the concentrated-solution calculations were based on a simple quadratic temperature dependence for the quantity G_ex/(n₂T):

where the parameters u, v, and q were determined in Ref. [9] for several rounded molalities from 0.001 to 6.0 mol⋅kg⁻¹. The parameter values were calculated using the following equations, derived by combining using Eqs. 1, 2, 3, 4, 5, 6, and 12 in the present study:

where the \({g}_{0}\) and \({f}_{0}\) functions were defined as:

The parameter values obtained from Eqs. 13–17 can be found in Table 2 of Ref. [9]. Meanwhile, for the dilute-solution case, the PI(dil) parameter values have been collected in Table 2 of Ref. [11] where symbol w was used instead of q for the coefficient of the (T − T₀)² term. The parameter estimation method was more complicated in Ref. [11] than that used in Ref. [9] but these methods lead to practically identical results.

Once the parameter values are known, the relative apparent molar enthalpy of the salt can be calculated for each molality from

Substituting this into Eq. 10 one obtains the following expression for the relative apparent molar heat capacity

For T = 298 K, the resulting \({\Delta C}_{\text{app}}\) values are displayed in Table 2. These heat capacity values can be compared to those in Ref. [12], where only the dilute-solution results up to m = 1 mol⋅kg⁻¹ were included in the estimation of \(u\), \(v\), and q. The two values are virtually identical up to m = 0.5 mol⋅kg⁻¹. In addition, this table gives the real values of the apparent heat capacity, i.e., the values for quantity C_app in Eq. 9. These values were based on the relative PI(con) heat capacities and the value of \({C}_{\text{m},2}^{\infty }\) = – 84 J⋅K^–1⋅mol^–1, suggested by Abraham and Marcus [45] for the partial molar heat capacity of NaCl at infinite dilution.

The relationship between the relative apparent heat capacity and molality at each temperature was determined by fitting the \(\Delta C_{\text{app}}\) values using equation

where the theoretical Debye–Hückel value was accepted for the coefficient of the square root term, i.e., for α_C. Its values are given in Table 1. The estimated values of a_C,1, a_C,2, a_C,3, and a_C,4 for PI(con) are collected in Table 3. For parameter a_C,3, i.e., for the coefficient multiplying the (m/m^o)^3/2 term, the same value that was obtained using PI(dil) in Ref. [12] was accepted when possible. In cases where these values were not available, they were estimated from the PI(dil) results using an identical expression as employed in the original study [12], i.e., using

These new parameter values are given in Table 4.

For T = 298 K, the relative apparent molar heat capacities obtained using Eq. 20. are also given in Table 2 with their confidence intervals at the 0.95 probability level. The agreement between the theoretical and predicted heat capacities is always good. The quality of the \({\Delta C}_{\text{app}}\) fit of Eq. 20 is further investigated using temperatures 273, 323, 348, and 373 K in Tables S1, S2, S3, and S4, respectively. These tables are given in Appendix A of the Supplementary Materials of this article, referred to below as “Extra.” Meanwhile, Appendix is abbreviated as “App.” In these tables, as in Table 2, the theoretical apparent heat capacities are predicted using Eq. 20 and the results are shown together with their confidence intervals. The \({\Delta C}_{\text{app}}\) errors in the five tables and the standard deviations in Tables 3 and  4 demonstrate that Eq. 20 applies at least quite accurately to the determined heat capacity values. All statistical indicators suggest that the accuracy obtained from Eq. 20 for PI(con) values seems to be at least on par with what can be obtained using the common experimental heat capacity methods.

3.2 Tests of the New Relative Apparent Molar Heat Capacities Against the Experimental Values from the Literature

The literature datasets used for the experimental C_app values at various temperatures are summarized in Table 5. In addition to measured data, also the smoothed results from Parker [13] were included in the table. Parker’s results have been based on a critical evaluation of the data published in the older literature up to year 1965. The scarce experimental data from Likke and Bromley [54] at T = 353 and 373 K were replaced with the smoothed values obtained from their other tabulated results. Because relative heat capacities are usually not reported for the datasets available, the \({C}_{\text{m},2}^{\infty }\) values needed for our comparison were determined from the equation

This linear relationship was determined from the molar enthalpies of NaCl(aq) at infinite dilution presented in Table 5 of Ref. [8] and was based on aqueous solubility data of NaCl. We use this equation uniformly throughout this study, instead of the \({C}_{\text{m},2}^{\infty }\) values depending on the dataset as in our earlier heat capacity articles [12, 39]. Sample \({C}_{\text{m},2}^{\infty }\) values calculated from Eq. 22 at different temperatures can be found in Table 5.

Another method for obtaining a value for \({C}_{\text{m},2}^{\infty }\) at each temperature would be to use the solution-enthalpy data from Criss and Cobble [55] measured for dilute NaCl solutions up to m = 0.02 mol⋅kg^–1 at the thirteen temperatures from 273.15 to 368.33 K. The solution enthalpies at infinite dilution and the resulting values for the partial molar enthalpies, \({H}_{\text{m},2}^{\infty }\), are listed in Table S5 of App. B. The \({C}_{\text{m},2}^{\infty }\) values calculated from these enthalpies are given in Table S6 at rounded temperatures. Criss and Cobble’s solutions enthalpies were determined from very dilute solutions where the accuracy of experiments is usually not high. This is the reason why the resulting heat capacities are not accurate, as Table S6 shows. Therefore, we prefer the results from Eq. 22 for \({C}_{\text{m},2}^{\infty }\). It is quite possible that the values obtained from this equation are not the best ones available: For example, the value of \({C}_{\text{m},2}^{\infty }= -84~ \text{J}\cdot {\text{K}}^{-1}\cdot {\text{mol}}^{-1}\) recommended by Abraham and Marcus [45] for T = 298 K is likely more realistic than the value of \({C}_{\text{m},2}^{\infty }= -58 ~\text{J}\cdot {\text{K}}^{-1}\cdot {\text{mol}}^{-1}\) obtained from Eq. 22. It is interesting that Parker’s review [13] also presents quite a different value of − 90.40 J⋅K⁻¹ mol⁻¹ for this quantity, whereas the value obtained from the results of Criss and Cobble [55] in App. B is − 96 J⋅K⁻¹ mol⁻¹. However, the choice of only one equation for \({C}_{\text{m},2}^{\infty }\) clarifies the presentation considerably. We discuss the ramifications of using Eq. 22 more fully below.

The results obtained when testing PI(con) against the heat capacity values of Tanner and Lamb [47] at T = 278, 298, 318, 338, and 358 K are illustrated in the error plots of Fig. 1. In all figures, the datasets are abbreviated using the acronyms in Table 5. The errors have been calculated from the reported (repd) and predicted (predd) values using equation

where the predicted values were obtained from Eq. 20. The results in this figure show that the predicted values at all molalities agree very well with experimental ones only at T = 298 K but also in this case there is an absolute systematic error of an order of 25 J⋅K^–1⋅mol^–1. As mentioned, this is due to the use of value \({C}_{\text{m},2}^{\infty }=\boldsymbol{ }-58\) J⋅mol^–1⋅K^–1 instead of a value closer to –84 J⋅mol^–1⋅K^–1, see Ref. [12]. At 278 and 358 K, the errors are large for dilute solutions and result mainly from the use of values − 18 and − 178 J⋅mol^–1⋅K^–1 for \({C}_{\text{m},2}^{\infty }\), respectively (see Table 5). The corresponding values from Tanner and Lamb [47] are − 159 and − 64 J⋅mol^–1⋅K^–1. Original \({C}_{\text{m},2}^{\infty }\) values from different sources at various temperatures are shown in Table 5 of Ref. [12] together with our previously fitted ones. Interestingly, the agreement between with the real molar heat capacities and PI(con) estimates in Fig. 1 is quite good above m = 3 mol⋅kg^–1 for all temperatures considered. Consequently, Eq. 22 seems to work better for these more concentrated solutions where the exact experimental determination of the real heat capacity of a salt in electrolyte solutions is likely easier due to the larger contribution of salt’s heat capacity to that of the whole solution.

Plot of e_C,app (Eq. 23), the deviation between the suggested apparent molar heat capacity of salt in NaCl solutions and the value predicted using parametrization PI(con) for the data of Tanner and Lamb [47] at various temperatures (T) as a function of molality m. The experimental datasets are introduced in Table 5

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The results from the smoothed datasets of Parker [13] for T = 288, 293, 298, and 303 K and those from the experimental ones of Perron et al. [19] for T = 278, 298, and 318 K are shown in Fig. 2 where all datapoints below m = 0.1 mol⋅kg^–1 from Parker’s sets have been omitted for clarity. The data from Perron et al. consist of points for less dilute solutions only. The agreement in the figure is good except at temperatures 278 and 318 K if the possibility of a systematic error stemming from the \({C}_{\text{m},2}^{\infty }\) values is allowed at each temperature. Especially at molalities above 3 mol⋅kg⁻¹, PI(con) seems to work well in every case.

Plot of e_C,app (Eq. 23), the deviation between the suggested apparent molar heat capacity of salt in NaCl solutions and the value predicted using parametrization PI(con) for the data of Parker [13] and Perron et al. [19] at various temperatures as a function of molality m. Datasets are introduced in Table 5, and their abbreviations are explained in that table

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Figure 3 gives the errors for the datasets SiFo25, ASDJ25, OloI25, OloII25, OloIII25, and RaRo25 in Table 5. These sets only included results at T = 298 K. As before, all datapoints below 0.1 mol⋅kg^–1 have been omitted, except one probably erroneous point from the RaRo25 dataset which has been highlighted. When the possibility of a systematic error is again allowed for, all these sets support PI(con) within experimental error up to their highest molality.

Plot of e_C,app (Eq. 23), the deviation between the suggested apparent molar heat capacity of salt in NaCl solutions and the value predicted using parametrization PI(con), for the literature data measured at T = 298.15 K as a function of molality m. Also Figs. 1, 2, 4 contain data measured at this temperature. Datasets are introduced in Table 5, and their abbreviations are explained in that table

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The three graphs of Fig. 4 show the rest of the results for the experimental data at various temperatures. Graph a shows the results from the HeGr25 at T = 298 K and values from other datasets below this temperature. Again, if the systematic error would be eliminated, the results above T = 283 K support parametrization PI(con) well. For temperatures below 283 K, the agreement is not as good but there the data seem not to be sufficiently precise for a conclusive evaluation because mostly dilute solutions were measured. The data from Archer and Carter [46] seem to be quite scattered in dilute solutions but support PI(con) relatively well above 2 mol⋅kg⁻¹.

Plot of e_C,app (Eq. 23), the deviation between the suggested apparent molar heat capacity of salt in NaCl solutions and the value predicted using parametrization PI(con) for the data measured at various temperatures (T) and not considered so far as a function of molality m. Graph a contains results for the HeGr25 dataset and those measured at temperatures below 298.15 K. Graph b contains results measured in the temperature range from T = 298.15 to 318.15 K. Graph c contains the results from the HeGr45 dataset and those measured at temperatures above 318.15 K. Datasets are introduced in Table 5 and their abbreviations are explained in that table

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Graph b of Fig. 4 presents the results from T = 298 to 318 K. With the exception of PFD45 at T = 318 K, all these data support PI(con). The results from HeGr45 also measured at T = 318 K are shown in graph c of Fig. 4 but in this case of the dilute-solution points do not appear very precise. The other datasets considered in graph c are those from Likke and Bromley [54] at elevated temperatures, i.e., LiBr80 and LiBr100. These data do not, however, support PI(con) well but seem to be quite consistent with the data of Tanner and Lamb [47] for 358 K in Fig. 1.

3.3 Tests of the New Apparent and Partial Molar Heat Capacities Against the Values Presented in the Literature

A detailed comparison between the published relative apparent and partial heat capacities and those predicted by PI(con) is provided in App. C of Extra using Figs. S1 to S4. Here, Figs. 5 and 6 summarize the most important findings. Figure 5 shows the apparent-molar-heat-capacity comparisons against the multiparameter equations of Clarke and Glew [2] at various temperatures. Figure 6 includes the corresponding results for the partial heat capacities. We chose to highlight the results from Clarke and Glew [2], as they provide a good representation of all the literature considered in App. C. In graphs a of Figs. 5 and 6, the apparent and partial heat capacities are respectively presented for temperatures from 273 to 303 K and in graphs b from 313 to 373 K.

Plot of e_C,app (Eq. 23), the deviation between the suggested apparent molar heat capacity of salt in NaCl solutions and the value predicted using parametrization PI(con), for the data of Clarke and Glew [2] at various temperatures (T) as a function of molality m. Graph a shows the results at temperatures from 273.15 to 303.15 K and graph b from 313.15 to 373.15 K

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Plot of e_C,part (Eq. 24), the deviation between the suggested partial molar heat capacity of salt in NaCl solutions and the value predicted using parametrization PI(con), for the data of Clarke and Glew [2] as a function of molality m. Graph a shows the results for temperatures from 273.15 to 303.15 K and graph b from 313.15 to 373.15 K

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In their review [2], Clarke and Glew originally reported values for the apparent and partial enthalpies. In Figs. 5 and 6, these were converted into relative values using Eq. 22 for \({C}_{\text{m},2}^{\infty }\). The values below m = 0.1 mol⋅kg⁻¹ were again omitted for clarity. In Fig. 5, Eq. 23 was used for the deviation, e_C,app, which is represented as a function of molality m. The errors for the partial molar heat capacities in Fig. 6 have been calculated using equation

where the predicted values were obtained with Eqs. 11 and 20 based on PI(con).

The agreement in graph a of Fig. 5 is always quite good except at T = 273 K, but the deviations become larger at the higher temperatures in graph b. In graph a, the agreement at T = 293, 298, and 303 K is excellent for all molalities when the possibility of a systematic error is permitted. There is also widespread agreement for the partial enthalpies in graph a of Fig. 6, and in graph b the agreement is very good up to T = 353 K. If the possibility for a systematic error is allowed, the agreement is at least quite good at molalities above 2.0 mol⋅kg^–1 for all temperatures considered in the four graphs.

The two graphs of Fig. S1 in App. C consider the heat capacity results obtained using the dilution-enthalpy data from Messikomer and Wood [56]. The researchers determined ΔC_app values at rounded molalities up to 5 mol⋅kg^–1 for temperatures from T = 303 to 373 K in intervals of 10 K and at 298 and 348 K. Furthermore, their article included experimental apparent enthalpies for T = 298, 323, 348, and 373 K. Using these values that were directly obtained from the dilution-enthalpy results, we were able to recalculate ΔC_app values in the same way that we used for our parametrization, i.e., through a parabolic temperature dependence of the apparent molar enthalpy at each molality. Details of the calculation method is described in App. D of Extra, and the results are shown in Table S7. The new values agree with the original values reported in Ref. [56] at the rounded molalities in Fig. S1. This calculation demonstrates that a simple quadratic equation like Eq. D.2 in App. D is sufficient for explaining the temperature dependence of the apparent enthalpies at each molality. This conclusion is strengthened in Figs. S2 and S3 of App. C where the heat capacity and enthalpy results from Gibbard et al. [41] and Ensor and Anderson [16] have been similarly treated at various temperatures. The resulting heat capacities are listed in Table S8 in App. D of Extra for the apparent molar enthalpy data of Gibbard et al. [41] and in Table S9 for those of Ensor and Anderson [16].

In addition to the results in Figs. S1, S2, and S3, the heat capacities from PI(con) have been further compared to the values obtained from the multiparameter equations of Pitzer et al. [1] in the two graphs of Fig. S4. In that study, the ΔC_app values are given at some rounded molalities for temperatures from 273 to 373 K in intervals of 10 K and at T = 298 K. The applicability of PI(con) to predict the data in Figs. S1, S2, S3, and S4 is mixed and has been analyzed in detail in App. C.

3.4 Comparison of PI(con) against a Previous Well-Functioning Parametrization for Apparent Molar Enthalpies at Temperatures Close to 298.15 K

In Ref. [9], it was observed that when the b₁ parameter in Eqs. 1 and 2 is represented by

with b₂ = 0 and B = 1.4 (mol⋅kg^–1)^–1/2, the resulting model applies well to existing enthalpy data up to the saturated solutions at temperatures close to 298 K. We denote this parametrization PII. The corresponding the parameter values for the enthalpy equation, Eq. 27 in Ref. [9], are given in Table 5 of Ref. [11], at temperatures from 273 to 313 K.

A summary of the promising enthalpy results for PII at T = 298 K can be found in Fig. 4 of Ref. [9]. As explained therein, the problem with these good results is that PII fails to explain more accurate sources of experimental data at high molalities: For instance, in Fig. 5 of Ref. [9], we show that PII does not predict the apparent experimental heat capacities reported in literature at 298 K. In contrast, PI(con) applies well to these experimental data up the saturated solution as demonstrated in Figs. 1–6, and Figs. S1 − S4 in App. C.

Details and results of the PII calculations are given in App. E of Extra using Tables S10, S11, and S12. The results are also illustrated in Fig. 7, where the apparent-heat-capacity error for the PII values is compared to the PI(con) predictions at T = 288, 293, 298, 303, and 308 K and various molalities. As can be seen, the agreement is not good in concentrated solutions. Thus, it seems possible that the enthalpy data reported in literature for concentrated NaCl solutions at T = 298 K are not fully reliable as pointed out in Ref. [9]. This argument seems to be true also for temperatures close to 298 K.

Plot of e_C,app, the deviation between the relative apparent molar heat capacity of salt for an alternative parametrization, called PII in Refs. [11, 12], and used previously for NaCl solutions and the value predicted using parametrization PI(con) at T = 298.15 K and at temperatures close to this value, as a function of molality m. This alternative parametrization is described in Tables S10 − S12 of App. E

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3.5 Recommended Values for the Apparent and Partial Molar Heat Capacity of NaCl in Aqueous Solutions

Based on the breadth of evidence provided in Refs. [8, 9], and the current study we draw the following conclusion: the existing experimental data for all pure NaCl solutions can be predicted to within experimental error using the new extended Hückel equations in the temperature range from 273 to 373 K. In Ref. [8], we presented recommended values for the activity coefficients of the salt and for osmotic coefficients of water in these solutions based on the new model. In Ref. [9], we extended these tables for the thermodynamic enthalpy quantities of the salt. In these two articles, we argued in detail why the new values could apply all the way up to the saturated NaCl solutions. Here, recommended values are presented for the relative apparent and partial molar heat capacities (i.e., for ∆C_app and ∆C_m,2) of the salt within these molality and temperature ranges using the same PI(con) model.

We have also observed that the upper limit of the original parametrization PI(dil), recommended in Ref. [12] for dilute NaCl solutions up to m = 0.2 mol⋅kg^–1, can be extended to 373 K from the original range of 273 to 353 K. The heat capacity quantities from this model have been tabulated in Ref. [12] up to 323 K in intervals of 5 K and after that in intervals of 10 K. Here, we provide values for the apparent and partial molar heat capacities first at T = 328, 338, and 348 K and then within the range from 358 to 373 K in intervals of 5 K. These values were calculated using the parameters given in Table 3 of Ref. [12] and Table 4 here. Table 6 contains values for ∆C_app and Table 7 for ∆C_m,2, supplementing the PI(dil) tables in Ref. [12].

The recommended PI(con) values are collected in Tables 8, 9, 10, and 11 for the apparent molar heat capacity, and in Tables 12, 13, 14, and 15 for the partial molar heat capacity at rounded molalities up to the saturated solutions. Tables 8 and 12 cover the temperature range (273 K; 293 K), Tables 9 and 13 cover the range (303 K; 328 K), Tables 10 and 14 cover the range (323 K; 348 K), and finally Tables 11 and 15 the range (353 K; 373 K). At each temperature, the molalities for the saturation state were taken from Table 4 of Ref. [8]. The heat capacities for these states do not differ much from those obtained at m = 6.0 mol⋅kg^–1.

In Tables S13 − S16 of App. F, we have included the 0.95 significance level confidence intervals for the apparent molar heat capacities recommended in Tables 8, 9, 10, and 11. For comparison, the tables of this appendix also contain the values obtained by previous parametrizations PI(dil) and PIII in Ref. [12] for solutions which have a molality equal to or above 0.2 mol⋅kg⁻¹ in parentheses. The PI(dil) values in these tables agree generally well with the current ones for almost all given molalities. We think, however, that the PI(con) value is slightly more reliable above m = 0.2 mol⋅kg⁻¹. In Ref. [12], we also tabulated the heat capacities based on the more complicated PIII parametrization up to m = 1.5 mol⋅kg^–1 for temperatures from 323 to 353 K. These values are presented in Tables S15 and S16, and the agreement with our PI(con) values is not very good. Because the PIII parametrization is not traceable and was based on the not-fully-reliable enthalpies from Ensor and Anderson [16], we recommend the PI(con) values for the heat capacities. The corresponding heat capacity comparison using partial values are presented in Tables S17 − S19 of App. F. Concerning to the reliability of PI(dil) and PIII values in these tables, similar conclusions can be drawn as for the apparent values.

The models and chemicals considered here are summarized in the chemical sample table of Table 16. In this study, we extend our previously proposed extended Hückel equation to cover the heat capacities of NaCl in water [8]. The three-parameter activity coefficient model that serves as a backdrop for our calculations consists of the Debye − Hückel term together with linear and quadratic terms with respect to the molality. The coefficients of these terms depend quadratically on the temperature, while the ion-size parameter of the Debye-Hückel term is held constant.

The recommended relative apparent and partial molar heat capacities based on the extended-Hückel model are tabulated up to the saturated NaCl solutions at temperatures from 273 to 373 K in Tables 8, 9, 10, 11, 12, 13, 14, and 15. We argue that the values in these tables represent the most reliable values determined so far even though they sometimes disagree with literature ones: In general, good agreement is observed in the temperature range (283 K; 323 K) up the saturated solutions. Meanwhile, in the more challenging temperature ranges (273 K; 283 K) and (323 K; 373 K), the agreement is satisfactory only for less dilute solutions, where the experimental determination of the heat capacity of the solute at each temperature is easiest.

The failure of our model to explain the heat capacity data at temperatures close to 373 K not a surprise in the light of the discrepancies observed for the enthalpies [9] and the activity and osmotic coefficients in Ref. [8] at these temperatures. At the same time, our activity and osmotic coefficients successfully explain existing vapor pressures, isopiestic data against KCl solutions, and a wide series of solubility data of NaCl solutions at these high temperatures [8, 11]. Consequently, we claim that the enthalpies and heat capacities obtained with PI(con) are the most reliable values even in the problematic cases.

Finally, it is important to emphasize the traceable and transparent nature of the best parametrizations PI(dil) and PI(con) used here and in our previous studies, see Refs. [7,8,9,10,11,12, 21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39] in addition to their ability to predict experimental results. A critical reader can reproduce all calculations using the presented results with the appropriate literature. This is not common in thermodynamic studies of electrolyte solutions, as the best data often comes from the isopiestic method. Due to its comparative nature, this method requires knowledge of the thermodynamic activities in the reference electrolyte solution. However, the methods that provide activities for solutions of a pure electrolyte are not as accurate as the isopiestic method over wide ranges of composition, temperature, and pressure. To tackle this issue, we have previously determined Hückel equations for the common reference electrolyte solutions of NaCl and KCl at 298 K which give traceable and highly accurate activity and osmotic coefficients. These values have been utilized by us in all following studies containing isopiestic data, see Refs. [7, 10, 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37].