Thermodynamically Traceable Calorimetric Results for Aqueous Sodium Chloride Solutions from T = (273.15 to 373.15) K up to the Saturated Solutions: Part 1—The Quantities Associated with the Partial Molar Enthalpy

Abstract

The three-parameter extended Hückel equations with parameters B, b₁, and b₂ have recently been successfully tested against existing vapor pressure, electrochemical, and solubility data for aqueous NaCl solutions at temperatures from (273 to 373) K (Partanen and Partanen in J. Chem. Eng. Data 65:5226–5239, 2020). In the present study, we extend this model to the apparent and partial molar enthalpy data of these solutions. The enthalpy equations were determined using a new calculation method that gives practically the same results as that used in another previous study (Partanen et al. in J. Chem. Eng. Data 62:2617–2632, 2017), but the new method is much simpler. In the previous enthalpy study, dilute NaCl solutions up to m = 0.2 mol⋅kg⁻¹ were considered in the range from T = 273 to 353 K. Following the success of the three-parameter extended Hückel model within the whole concentration range at various temperatures, we tabulate new values for relative apparent and partial molar enthalpies for NaCl solutions at rounded molalities. The resulting values are extensively tested against the literature ones. The best agreement is obtained for temperatures below 288 K and between 313 and 353 K. Elsewhere, at least a reasonable agreement is obtained. As no enthalpy or heat capacity data were used in the estimation of our model’s parameters and as the model has excelled in explaining other high-precision thermodynamic data, we argue that the recommended enthalpy values should be preferred even for the temperatures where the agreement is only reasonable due to potential problems associated with the literature values. These problems are also considered in the study.

Graphical Abstract

1 Introduction

Sodium chloride is the predominant salt in seawater, and Na⁺ and Cl^‒ ions often play decisive roles in the functioning of biological fluids in living organisms. Aqueous NaCl solutions are also essential in various industrial applications. Consequently, the thermodynamic properties of solutions of this electrolyte have been thoroughly examined in a wide range of compositions, temperatures, and pressures [1,2,3].

In 1973, Pitzer presented a versatile mathematical model for the general thermodynamic behavior of electrolyte solutions [4] which the scientific community has widely accepted [1,2,3]. The Pitzer equation contains only three specific parameters for a solution of a pure 1:1 electrolyte like NaCl at each temperature and pressure. Unfortunately, the temperature and pressure dependences of these parameters necessitate the introduction of several additional substance-specific parameters.

Recent studies have indicated that a simpler equation based on the Debye−Hückel theory can accurately describe the thermodynamics of dilute solutions of pure electrolytes of simple charge types at T = 298 K and at the atmospheric pressure [5, 6]. In this two-parameter Hückel equation, the Debye−Hückel term is supplemented with a linear molality term.

In our previous NaCl studies [7,8,9,10], we have demonstrated that the Hückel equation applies to a wide range of experimental data in the temperature range of (273 K, 383 K). Indeed, our best parametrization for this temperature dependence could predict all calorimetric literature data for dilute solutions up to T = 373 K, even though these data were not included in the parameter estimation. Furthermore, when this parametrization is supplemented with a quadratic term in molality, it can predict the equilibrium thermodynamic properties of NaCl solutions up to full saturation at a constant or an almost constant temperature in the 273 to 373 K range [11]. The current study extends the investigation of this promising model to the enthalpy results existing for the less dilute NaCl solutions. Heat-capacity results for these solutions will be reported in the future.

2 Theory

2.1 The Extended Hückel Model

In aqueous solutions of various salts, the extended Hückel equation, Eq. 1, can be used to predict the mean activity coefficient (γ) of the salt at least up to an ionic strength of I_m = 1 mol⋅kg⁻¹ at several temperatures (see, e.g., [5, 7,8,9,10,11,12,13,14,15,16,17] – a comprehensive list of our studies on this topic can be found in Ref. [11]).

$$ \ln \gamma = - \frac{{\alpha \left| {z_{ + } z_{ - } } \right|\sqrt {I_{{\text{m}}} } }}{{1 + B\sqrt {I_{{\text{m}}} } }} + b_{1} \left( {\frac{m}{{m^{{\text{o}}} }}} \right) + b_{2} \left( {\frac{m}{{m^{{\text{o}}} }}} \right)^{2} $$

(1)

With the help of the Gibbs–Duhem equation, the following formula is obtained for the osmotic coefficient of water (ϕ):

$$ \phi = 1 - \frac{{\alpha \left| {z_{ + } z_{ - } } \right|}}{{B^{3} I_{{\text{m}}} }}\left[ {\left( {1 + B\sqrt {I_{{\text{m}}} } } \right) - 2\ln \left( {1 + B\sqrt {I_{{\text{m}}} } } \right) - \frac{1}{{1 + B\sqrt {I_{{\text{m}}} } }}} \right] + \frac{{b_{1} }}{2}\left( {\frac{m}{{m^{{\text{o}}} }}} \right) + \frac{2}{3}b_{2} \left( {\frac{m}{{m^{{\text{o}}} }}} \right)^{2} $$

(2)

In Eqs. 1 and 2, m is the molality, m^o = 1 mol·kg^–1, z₊ and z₋ are the charge numbers of the cation and anion, respectively. For a 1:1 electrolyte such as NaCl, $\left| {z} _{+} {z} _{-} \right|$  = 1 and I_m = m. The electrolyte-dependent parameters in the model are B, b₁, and b₂. It should be noted that in the regular Hückel equation, parameter b₂ is set to zero.

The values of the Debye−Hückel parameter α were obtained using the quadratic equation

$$ \frac{\alpha }{{\alpha_{0} }} = u_{0} + v_{0} \left( {\frac{{T - T_{0} }}{{\text{K}}}} \right) + q_{0} \left( {\frac{{T - T_{0} }}{{\text{K}}}} \right)^{2} $$

(3)

where α₀ = 1 (mol·kg⁻¹)^−1/2. 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. [10] from the α values presented in Ref. [18]. As demonstrated in Table 1, the predicted values show excellent agreement with the original ones.

Table 1 Debye − Hückel parameter α in Eqs. 1 and 2 and the value predicted using Eq. 3, Debye − Hückel parameter (α_T) in Eq. 20, and the specific heat capacity of water (${C}_{\mathrm{sp},1}^{*}$) as functions of temperature T

2.2 General Equations for the Enthalpy Quantities

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

$$ G_{{{\text{ex}}}} = 2w_{1} mRT(1 - \phi + \ln \gamma ) $$

(4)

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

$$ H_{{{\text{app}},2 }} = \frac{{H - n_{1} H_{{{\text{m}},1}}^{*} }}{{n_{2} }} $$

(5)

where n₁ (= w₁/M₁) and n₂ (= mw₁) are the amounts of water and the salt, respectively, H is the enthalpy of the system, ${H}_{\mathrm{m},1}^{*}$ 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,${\Delta H}_{\mathrm{app}}$, is defined with help of the partial molar enthalpy of the salt at infinite dilution ${H}_{\mathrm{m},2}^{\infty }$, as ${\Delta H}_{\mathrm{app} }{=H}_{\mathrm{app},2}-{H}_{\mathrm{m},2}^{\infty }$. It is associated with the excess Gibbs energy through the following thermodynamic identity:

$$ \Delta H_{{{\text{app}} }} = - T^{2} \frac{{\partial \left( {\frac{{G_{{{\text{ex}}}} }}{{n_{2} T}}} \right)}}{\partial T} $$

(6)

Once the relative apparent molar enthalpy is known from Eq. 6, the salt’s relative partial molar enthalpy, ${\Delta H}_{\mathrm{m},2 }={H}_{\mathrm{m},2}-{H}_{\mathrm{m},2}^{\infty }$, can be calculated using

$$ \Delta H_{{{\text{m}},2 }} = \Delta H_{{{\text{app}} }} + m\frac{{\partial \left( {\Delta H_{{{\text{app}}}} } \right)}}{\partial m} $$

(7)

Here, we have modified our notations when compared to those in Refs. [8, 9, 16, 17] to clarify the presentation.

An experimentally obtainable calorimetric quantity is the molar heat of dilution, ∆H_m,dil. Theoretically, it is associated with the apparent molar enthalpies of the salt in the initial and final solutions through

$$ \Delta H_{{{\text{m}},{\text{dil}}}} = \Delta H_{{{\text{app}}}} \left( {m_{{\text{f}}} } \right) - \Delta H_{{{\text{app}}}} \left( {m_{{\text{i}}} } \right) $$

(8)

where m_i is the initial and m_f the final molality.

2.3 Temperature Changes Associated with the Dilution and Solution Enthalpies

We considered almost all dilution enthalpies presented in literature for NaCl(aq) in our test calculations. To concretize our findings, we have included the adiabatic dilution temperatures, ΔT_dil(ad), when reporting our ∆H_m,dil values. This is the temperature change upon a fully adiabatic dilution process. It was calculated from the following approximate equation:

$$ \Delta T_{{{\text{dil}}}} \left( {{\text{ad}}} \right) = - \Delta H_{{{\text{dil}}}} /C_{1}^{*} \left( {m_{{\text{f}}} } \right) $$

(9)

where ${C}_{1}^{*}({m}_{\mathrm{f}})$ is the heat capacity of the amount of pure water needed to achieve molality m_f when the amount of salt is 1 mol. The specific heat capacities of pure water were taken from the article of Osborne et al. [19] and can be found in Table 1 of this article. While Eq. 9 is most accurate in dilute solutions, it also provides satisfactory results for the more concentrated solutions considered in this study. This accuracy was verified with the unpublished results of part 2 of this study, which considers the heat capacities of aqueous NaCl solutions. In this study, the lower accuracy of Eq. 9 is accounted for in the number of significant digits for each temperature difference.

The apparent molar enthalpy is closely associated with the solution enthalpy of the solute. Consequently, when analyzing the ΔH_app values in the literature, we also report the adiabatic-solution temperature, ΔT_sol(ad), to provide more concreteness for our results. This is the temperature change when the amount of salt corresponding to molality $m$ dissolves in a fully adiabatic process to a mass of 1 kg of water. It is thus the temperature difference between the final temperature of the solution and the initial one of the two pure components. The values for ΔT_sol(ad) were calculated using the approximate equation

$$ \Delta T_{{{\text{sol}}}} \left( {{\text{ad}}} \right) = - n_{2} \left[ {H_{{{\text{m}},2}}^{\infty } + \Delta H_{{{\text{app}}}} - C_{{{\text{m}},2}}^{*} \left( {T - T_{{{\text{ref}}}} } \right)} \right] /C_{1}^{*} (w_{1} = 1 {\text{kg}})$$

(10)

where T_ref = 298.15 K, ${C}_{1}^{*}({w}_{1}=1\mathrm{ kg})$ is the heat capacity of 1 kg of pure water, n₂ is the amount of salt at molality m, and ${C}_{\mathrm{m},2}^{*}$ is the molar heat capacity of pure solid. As Eq. 9, this equation is most accurate in dilute solutions but retains sufficient accuracy for all molalities considered. However, the uncertainty in the partial molar enthalpy at infinite dilution, ${H}_{\mathrm{m},2}^{\infty }$, poses an additional challenge for obtaining accurate absolute values for ΔT_sol(ad) (see, for example, Table 3 in Ref. [11]). Consequently, we use Eq. 10 only to compare the consistency of the literature values of ΔH_app at high molalities and high temperatures where high-precision data seems to be currently lacking.

2.4 Parametrizations for NaCl Solutions

In our previous studies [5, 7,8,9,10,11], we have observed that a constant value of 1.4 (mol·kg⁻¹)^−1/2 for the B parameter can be used in Eqs. 1 and 2 at all temperatures from 273 to 373 K up to saturated NaCl solutions. Meanwhile, a quadratic equation of form

$$ b_{1} = u_{1} + v_{1} \left( {\frac{{T - T_{0} }}{{\text{K}}}} \right) + q_{1} \left( {\frac{{T - T_{0} }}{{\text{K}}}} \right)^{2} $$

(11)

where u₁ = 0.0077, v₁ = 3.1853·10⁻³ and q₁ =  − 25.17·10⁻⁶ was obtained for b₁ in our most successful parametrization of the Hückel equation in [8]. In Eq. 11 and in all subsequent equations T₀ is 273.15 K. Equation (11) was based on the following three values: b₁(273.15 K) = 0.0077, b₁(298.15 K) = 0.0716, and b₁(348.15 K) = 0.105, which were determined in Refs. [7, 5, 8], respectively. With this choice for b₁ and with b₂ = 0 in Eqs. 1 and 2, the Hückel equation could explain the existing thermodynamic data within experimental error for dilute NaCl solutions at least up to m = 0.2 mol·kg⁻¹ in the temperature range from 273 to 383 K [8,9,10]. When parameter ${b}_{2}$ was allowed to have a similar temperature dependence through

$$ b_{2} = u_{2} + v_{2} \left( {\frac{{T - T_{0} }}{{\text{K}}}} \right) + q_{2} \left( {\frac{{T - T_{0} }}{{\text{K}}}} \right)^{2} $$

(12)

the extended Hückel equation could be used up to saturated solutions at various temperatures [11]. In Eq. 12, we used the values u₂ = 0.01328, v₂ =  − 364.7 × 10⁻⁶ and q₂ = 2.7 × 10⁻⁶ for the parameters. These values were mainly determined using the direct vapor pressure data in the literature [20, 21] as shown in Table 2 of Ref. [11].

3 Results and Discussion

3.1 Determination of a Quadratic Equation for G _ex/(n ₂ T) with Respect to Temperature and the Calculation of Relative Apparent Molar Enthalpies

Throughout the following text and the appendices, two parametrizations are considered for the enthalpy data of NaCl solutions: The first, PI(con), is defined by Eqs. 1, 2, 3, 11, and 12 of this study and encompasses both concentrated and dilute solutions. The second, PI(dil), was the optimal parametrization for dilute solutions in our previous study [8]. In addition to these, the suggested enthalpies from PII and PIII parametrizations are also considered below. These parametrizations were introduced in Ref. [8] alongside PI(dil).

The enthalpy expressions of this study were grounded on the following simple quadratic equation for the quantity G_ex/(n₂T) with respect to temperature at a constant molality:

$$ \frac{{G_{{{\text{ex}} }} }}{{n_{2} T}} = u + v \left( {T - T_{0} } \right) + q \left( {T - T_{0} } \right)^{2} $$

(13)

Then, the following equations can be directly derived for the parameters $u$, $v$, and $q$ using Eqs. 1, 2, 3, 4, 11, 12, and 13:

$$ u/\left( {2R} \right) = u_{0} \alpha_{0} (g_{0} + f_{0} ) + \frac{1}{2}u_{1} \left( {m/m^{{\text{o}}} } \right) + \frac{1}{3}u_{2} \left( {m/m^{{\text{o}}} } \right)^{2} $$

(14)

$$ v/\left( {2R} \right) = v_{0} \alpha_{0} (g_{0} + f_{0} ) + \frac{1}{2}v_{1} \left( {m/m^{{\text{o}}} } \right) + \frac{1}{3}v_{2} \left( {m/m^{{\text{o}}} } \right)^{2} $$

(15)

$$ q/\left( {2R} \right) = q_{0} \alpha_{0} (g_{0} + f_{0} ) + \frac{1}{2}q_{1} \left( {m/m^{{\text{o}}} } \right) + \frac{1}{3}q_{2} \left( {m/m^{{\text{o}}} } \right)^{2} $$

(16)

The ${g}_{0}$ and ${f}_{0}$ functions were defined as:

$$ g_{0} = - \frac{\sqrt m }{{1 + B\sqrt m }} $$

(17)

$$ f_{0} = \frac{1}{{B^{3} m}}\left[ {\left( {1 + B\sqrt m } \right) - 2\ln \left( {1 + B\sqrt m } \right) - \frac{1}{1 + B\sqrt m }} \right] $$

(18)

As all parameters in Eqs. 14–16 are known, the values for u, v, and q for Eq. 13 can be calculated. These are listed for various rounded molalities at temperatures from (273 to 373) K in Table 2.

Table 2 Parameter values for the quadratic equation of excess molar Gibbs energy divided by T with respect to temperature, Eq. 13 in the text, for NaCl solutions obtained using Eqs. 14, 15, and 16 at various molalities (m)

Our preliminary calculations based on Eq. 13 show that this procedure gives practically identical results to the previous studies for dilute NaCl and KCl solutions [8, 16], where a different method was used for the calculation of the parameters $u$, $v$, and $q$. For comparison, analogous values, calculated in the previous way, can be found in Table 2 of Ref. [8] up to T = 353 K.

Once the parameters u, v, and q are known, the relative apparent molar enthalpy of the salt can be obtained by combining Eqs. 6 and 13:

$$ \Delta H_{{{\text{app}} }} = - T^{2} \frac{{\partial \left( {\frac{{G_{{{\text{ex}}}} }}{{n_{2} T}}} \right)}}{\partial T} = - T^{2} \left[ {v + 2q\left( {T - T_{0} } \right)} \right] $$

(19)

Sample enthalpies obtained from Eq. 19 at T = 298 K are listed in Table 3. The enthalpy values in Table 3 can be compared to those in Ref. [8], where in Eqs. 1 and 2 parameter b₂ was set to zero, and only the dilute-solution results up to m = 1 mol·kg⁻¹ were included in the estimation of $u$, $v$, and q (the symbol w was used in that study for the last parameter). The two values are virtually identical up to m = 0.3 mol·kg⁻¹. From the relative values in Table 3, the H_app,2 values were calculated using a value of 3824 J·mol⁻¹ for the partial molar enthalpy of NaCl at infinite dilution, as suggested by Criss and Cobble [22].

Table 3 Values of the real and relative apparent molar enthalpies and those of the relative apparent molar heat capacity obtained using Eq. 19 with parametrization PI(con) for NaCl solutions at T = 298.15 K and the errors when the apparent enthalpies are predicted using Eq. 20 with the parameter values given in Table 4 as functions of molality m

To simplify the molality dependence of the relative apparent molar enthalpy, the following equation was fitted to the $\Delta {H}_{\mathrm{app}}$ values at each temperature:

$$ \Delta H_{{{\text{app}} }} = a_{1} + \alpha_{{\text{T}}} \sqrt {\frac{m}{{m^{{\text{o}}} }}} + a_{2} \left( {\frac{m}{{m^{{\text{o}}} }}} \right) + a_{3} \left( {\frac{m}{{m^{{\text{o}}} }} } \right)^{\frac{3}{2}} + a_{4} \left( {\frac{m}{{m^{{\text{o}}} }}} \right)^{2} $$

(20)

where a₁, a₂, and a₄ are the fitting parameters. In this procedure, the following sum of squared residuals was used by the method of linear regression analysis:

$$ {\text{SSR}} = \mathop \sum \limits_{i}^{N} \left[ {y_{i} - y_{i} \left( {{\text{pred}}} \right)} \right]^{2} $$

(21)

where y_i = ΔH_app,i − α_T(m_i/m^o)^1/2 − a₃(m_i/m^o)^3/2 – a₄(m_i/m^o)² and y_i(pred) = a₁ + a₂(m_i/m^o). Parameters a₁ and a₂ were thus obtained using the method of least-squares, while for parameter a₄ the best value was determined at each temperature. The values for the coefficient of the square root term, i.e., for α_T, were based on the Debye−Hückel theory. These are listed in Table 1 and were taken from the tables of Archer and Wang [18]. For parameter a₃, the value obtained for dilute solutions in Ref. [8] was used in the range from T = 273 to 353 K. For higher temperatures, corresponding new values were determined from the dilute-solution results up to m = 1 mol·kg^–1 by setting the parameter a₄ in Eq. 20 equal to zero as in Ref. [8]. The high-temperature results and thus the values for the a₃ coefficient are collected in Table S1 in Appendix A of the Supplementary Materials. These materials are referred below to as “Extra”, and Appendix is abbreviated as “App.” A pre-determined value for parameter a₃ was chosen so that PI(dil) and PI(con) give identical results for dilute solutions.

Table 4 lists the values for the a₁, a₂, a₃, and a₄ parameters in all temperatures considered. The standard deviation about each regression is also displayed. The quality of the ${\Delta H}_{\mathrm{app}}$ fit of Eq. 20 is further investigated for T = 298 K in Table 3 and in Tables S2 to S5 for T = 273, 323, 348, and 373 K in App. B of Extra, respectively. In these tables, the relative apparent enthalpies are predicted using Eq. 20 and displayed as error values (${e}_{\mathrm{H}}$). The ${\Delta H}_{\mathrm{app}}$ errors in these five tables and the standard deviations in Tables 4 and S1 of this study together with Table 2 of Ref. [8] demonstrate that this equation applies quite accurately to the determined enthalpy values. All statistical indicators suggest that the accuracy of the values obtained from Eq. 20 for PI(con) seem to be better than what can be obtained using the common experimental enthalpy methods, i.e., using heat-of-dilution or heat-of-solution determinations.

Table 4 Parameter values for the equation of the dependence of the relative apparent molar enthalpy on the molality, Eq. 20 in the text, for NaCl solutions obtained using parametrization PI(con) at various temperatures (T)

3.2 Tests of New Equations for Apparent Molar Enthalpy Against Heat-of-Dilution Data

We have tested PI(con) against almost all existing heat-of-dilution data for NaCl(aq) at various temperatures, following our previous studies on dilute NaCl and KCl solutions [8, 16]. These data form the most important experimental enthalpy source existing in literature for solutions of this salt. The results are shown in Tables S6—S29 in App. C of Extra. For comparison, the values predicted with PI(dil) from Ref. [8] have been included in most of these tests.

When the apparent enthalpies are obtained using Eq. 20, the deviation between the observed and predicted heats of dilution is defined by

$$ e_{{{\text{H}},{\text{PI}}\left( {{\text{con}}} \right)}} = \Delta H_{{{\text{m}},{\text{dil}}}} \left( {{\text{obsd}}} \right) - \Delta H_{{{\text{m}},{\text{dil}}}}\left( {\text{pred}} \right) $$

(22)

where the predicted value was obtained using Eq. 20 and Table 4. The adiabatic dilution temperatures for the experimental points and those predicted using the tested model have been calculated from Eq. 9, and they are given in the dilution-experiment tables using the following symbols:

$$ \Delta T_{{{\text{ad}}}} \left( {\text{p}} \right) = \Delta T_{{{\text{dil}}}}\left( {{\text{ad}},{\text{ pred}}} \right) $$

(23)

$$ \Delta T_{{{\text{ad}}}} \left( {\text{o}} \right) = \Delta T_{{{\text{dil}}}} \left( {{\text{ad}},{\text{ obsd}}} \right) $$

(24)

Only the most relevant observations for the dilution-enthalpy data are outlined here from the exhaustive comparison in App. C between our model and the literature data presented in Refs. [23,24,25,26,27,28,29,30,31,32] in App. C. We argue that the most important datasets are the ones reported by Young and Machin [25] for experiments at T = 273.2, 285.7, and 298.2 K where the data extend almost up to the saturated solution at each of these temperatures. These data stand out from others, because their initial and final molalities are always close to each other. Consequently, even though the observed heats of dilution are relatively small, it may have been easier for Young and Machin to control the flow of heat in the more concentrated solutions resulting in an improved accuracy over other experimentalists like Messikomer and Wood [30], Ensor and Anderson [31], and Mayrath and Wood [32] who utilized larger differences between the m_i and m_f values. For example, Mayrath and Wood [32] criticized Ensor and Anderson [31] that their results suffer from “problems with solvent evaporation at 343 and 353 K”, which could have been caused, at least in part, by the large amount of added water in their experimental setup. Comparison of the results of Young and Machin against our PI(dil) and PI(con) predictions are shown in Tables S11, S12, and S13 of App. C in Extra, while the PI(con) results are further given in Tables 5, 6, and 7, respectively. As shown in the tables, the dilution enthalpies can be excellently predicted using PI(con).

Table 5 Heats of dilution reported by Young and Machin [25] for NaCl solutions at T = 298.2 K, adiabatic dilution temperatures (see Eqs. 9, 23, and 24) and the errors (Eq. 22) that resulted when these heats were predicted using parametrization PI(con) (m^o = 1 mol·kg^–1)

Table 6 Heats of dilution reported by Young and Machin [25] for NaCl solutions at T = 285.7 K, adiabatic dilution temperatures (see Eqs. 9, 23, and 24) and the errors (Eq. 22) that resulted when these heats were predicted using the present parametrization PI(con) (m^o = 1 mol·kg^–1)

Table 7 Heats of dilution reported by Young and Machin [25] for NaCl solutions at T = 273.3 K, adiabatic dilution temperatures (see Eqs. 9, 23, and 24) and the errors (Eq. 22) that resulted when these heats were predicted using the present parametrization PI(con) (m^o = 1 mol·kg^–1)

The small adiabatic dilution temperatures in Tables S6 to S10 of App. C confirm that the precision of the classical datasets from Robinson [23] and Gulbransen and Robinson [24] is high even though their experiments were likely difficult to perform due to the radical dilution of the initial solutions. Unfortunately, these data extend at best only up to m = 0.816 mol·kg⁻¹. Meanwhile, Table S14 shows more recent heat-of-dilution data from flow-micro-calorimeters by Fortier et al. [26] at T = 298 K. In this set, all points up to m = 0.2 mol·kg⁻¹ can be predicted well using the two parametrizations considered. Above this molality, the agreement is not as good, but for PI(con) the absolute errors remain below 0.13 kJ·mol^–1. Even though the errors at the larger molalities are smaller for PI(con) than the corresponding ones in Table S7 for Ref. [24], they are not as small as those for the data of Young and Machin [25] in Table 5 (or in Table S11). It seems to us that the results in Table 5 are the most reliable because the differences between m_i and m_f values again start to increase towards the end of Table S14. Furthermore, the enthalpy values for more concentrated solutions at 298 K are often used as a chemical standard in the calibration of the flow-micro-calorimeters. However, we’ll demonstrate below that these values might not be entirely reliable at this temperature.

Tables S19 to S22 in App. C display the results for the data measured by Messikomer and Wood [30] at T = 298, 323, 348, and 373 K, respectively. The data in Table S19 at T = 298 K yields relatively large errors for PI(con) at higher molalities but likely suffers from the same issues as the dataset from Fortier et al. [24] in Table S14. At T = 323 K in Table S20, the agreement with the new parametrization is always good with all absolute errors below 0.1 kJ·mol^–1. In the more challenging experimental conditions at 348 K in Table S21 and at 373 K in Table S22, the agreement is expectedly not as good as it is closer to the room temperature. Furthermore, significant differences between the experimental and predicted enthalpies are also expected for those temperatures since the activity and osmotic coefficients obtained by parametrization PI(con) were not in complete agreement with the literature values in this region [11].

In the data from Ensor and Anderson [31], separate sets are provided at temperatures ranging from 313 to 353 K in intervals of 10 K as Tables S23 − S27 in App. C show. The points measured at T = 313 K have been considered in Table S23, and they support PI(con) at least satisfactorily up to moderately concentrated solutions. However, the absolute errors once again increase as the molality difference between the initial and final states becomes large. The data measured at 323 K are shown in Table S24, and these data support PI(con) well, with maximum absolute errors at most 0.2 kJ·mol^–1. In Table S25 for 333 K also, all errors remain quite small, i.e., at most 0.6 kJ·mol^–1. In Tables S26 (T = 343 K) and S27 (T = 353 K), agreement below 1.0 kJ·mol^–1 is obtained only for m_i values of about 2.0 mol·kg^–1 and 1.0 mol·kg^–1, respectively. In all experiments of Ensor and Anderson [31], once again, the larger initial molalities were substantially diluted, and this has probably made the calorimetric work challenging especially at higher temperatures. Generally, the present adiabatic-dilution-temperature calculations show good agreement at molalities below 1 mol·kg⁻¹. At T = 323 K (Table S24), the agreement is excellent at all molalities. For the other temperatures, at least satisfactory agreement is obtained also in concentrated solutions.

Finally, the data from Mayrath and Wood [32] at T = 348 and 373 K are considered in Tables S28 and S29 of App. C, respectively. Good agreement with the new model is obtained in Table S28 up to m = 0.6 mol·kg^–1 and in Table S29 up to m = 0.4 mol·kg^–1. At these high temperatures and in concentrated solutions, the discrepancies between experimental results underscore the problems associated with the accuracy of ΔH_m,dil measurements, as discussed in Extra.

3.3 Tests of New Values for the Apparent and Partial Molar Enthalpies Against Literature Values at 298 K

Besides the heats of dilution, the validity of the new PI(con) parametrization can be gauged with the help of the existing apparent and partial molar enthalpies. Here, the first ∆H_app and ∆H_m,2 test values originate from the data of Robinson’s group [23, 24] and were based on their heat-of-dilution data represented in Tables S6 and S7 of App. C. The former dataset [23] includes enthalpies at 298 K for more concentrated solutions than the ones in Table S6 and is partially based on a recalculation of the literature values at the time. These enthalpies have also been reported in the well-known textbook of Harned and Owen [33]. In the error plots of Fig. 1, parametrization PI(con) is tested up to m = 1 mol·kg^–.¹ for all temperatures of Ref. [24], including 298 K. The apparent-molar-enthalpy error in the figure is defined by

$$ e_{{{\text{H}},{\text{app}}}} = \Delta H_{{{\text{app}}}} \left( {{\text{obsd}}} \right) - \Delta H_{{{\text{app}}}}\, (\text{pred})$$

(25)

and the partial-molar-enthalpy error by

$$ e_{{{\text{H}},{\text{part}}}} = \Delta H_{{{\text{m}},2}} \left( {{\text{obsd}}} \right) - \Delta H_{{{\text{m}},2}}\, (\text{pred}) $$

(26)

where the apparent values were predicted using Eq. 20 and the partial values using Eqs. 7 and 20. The errors are presented as functions of the molality. The small errors at molalities less than 0.1 mol·kg⁻¹ have been omitted from the figure for clarity. Figure 1 shows that PI(con) can mostly reproduce the points considered, but the apparent enthalpies show a better agreement than the partial ones. This difference is possibly due to the difficulties associated with the outdated graphical determination of the derivative of the experimental apparent enthalpy values in Eq. 7 that the authors employed [23, 24].

Fig. 1

Plot of e_H,app (Eq. 25) or e_H,part (Eq. 26), the deviation between the relative apparent or partial molar enthalpy suggested in the literature for NaCl solutions at various temperatures (T) and that predicted using parametrization PI(con), as a function of molality m. The suggested values were taken from the articles of Robinson [23] and Gulbransen and Robinson [24]. The symbols are given in the legend of the figure

Figure 2 shows the results for the other dilute solutions measured at T = 298 K. It contains the enthalpy data determined by Young and Vogel [34] using their heat-of-dilution measurements up to m = 0.8 mol·kg^–1. It also displays the errors from the heat-of-dilution data determined by Parker [35] based on a critical evaluation of all calorimetric data available for NaCl solutions at around T = 298 K up to year 1965. As in Fig. 1, several points with small errors have been omitted from this figure for the most dilute solutions of Refs. [34, 35] to improve readability. The remaining points demonstrate that the data from Young and Vogel and from Parker support parametrization PI(con) similarly to those from Robinson’s group in Fig. 1.

Fig. 2

Plot of e_H,app (Eq. 25) or e_H,part (Eq. 26), the deviation between the relative apparent or partial molar enthalpy suggested in the literature for NaCl solutions at 298.15 K and that predicted using parametrization PI(con), as a function of molality m. The suggested values were taken from the articles of Messikomer and Wood [30], Young and Vogel [34], Parker [35], Lipsett et al. [37], and Benjamin [38]. Symbols are given in the legend of the figure, and the values of 3824 and 3866 J·mol^–1 have been used for the molar solution enthalpy of NaCl at infinite dilution for the data of Refs. [37, 38]. The former value originates from the article of Criss and Cobble [22] and was used for [37] while the latter was used for [38]. It was determined in Ref. [16] from the calculations of the dilute solutions results of this dataset [38]

Lipsett et al. [36, 37] determined apparent molar enthalpies using solubility measurements. In Ref. [37], they report values up to m = 6 mol·kg^–1 at T = 293 and 298 K. As the enthalpies in Ref. [36] at 298 K are very similar to those in Ref. [37], only the latter data have been included in our analysis. Details of the calculations are given in the caption of Fig. 2. In addition to the data from [37], Fig. 2 also includes errors for the smoothed apparent enthalpies from Messikomer and Wood [30]. These data were obtained from the results of the heat-of-dilution experiments shown in Table S19 of App. C. Finally, Fig. 2 contains results from the solution-enthalpy data of Benjamin [38]. The latter high-precision data support PI(con) well, as do the other sets in Fig. 2. It should be noted that of the datasets in Fig. 2, only the results from Benjamin and Lipsett et al. consist entirely of experimental points. The rest are derived from pre-existing measurements and should thus be treated with a measure of caution.

Error plots for the experimental apparent enthalpy data from very dilute solutions at T = 298 K measured by Criss and Cobble [22] and Sanahuja and Cesari [39] are considered in graph a of Fig. 3. Both datasets support PI(con) well. This graph also contains results from Criss and Cobble [22] at temperatures below 298 K. Meanwhile, all their results above 298 K are presented in graph b of Fig. 3. It should be noted that most of these experimental results have already been covered in our previous research [8] using PI(dil), and the predictions of PI(con) are identical to those results. Additional computational details can be found in the caption of the figure and in Table 8.

Fig. 3

Plot of e_H,app (Eq. 25), the deviation between the suggested relative apparent molar enthalpy for NaCl solutions and that predicted using parametrization PI(con), as a function of molality m. In graphs a and b, the suggested value has been calculated from the experimental heats of solution reported by Criss and Cobble [22] at various temperatures and by Sanahuja and Cesari [39] at 298.15 K (only in graph a). In graph c, the suggested enthalpy value is the one reported by Ensor and Anderson [31] for the final molality in their dilution experiments. In graph d, the suggested enthalpy value is the one obtained for the initial molality from these heat-of-dilution data (see Tables S23 to S27 in App. C of Extra) and from the PI(con) value for the enthalpy of the final molality. Details of the calculation are given in the text, and the symbols are shown in the legends of the graphs. The value of 3867 J·mol^–1 from Sanahuja and Cesari was used in graph a for the molar solution enthalpy of NaCl at infinite dilution for the data of Ref. [39]. For the data of Criss and Cobble [22], the original values given here in Table 8 were used for this quantity at various temperatures. Several results from graphs a and b for parametrization PI(dil) have been considered in Fig. 3 of Ref. [8]

Table 8 Standard molar solution enthalpy of sodium chloride at infinite dilution in water (i.e., the quantity of $\Delta {H}_{\mathrm{m}}^{\infty }(\mathrm{aq}))$ presented as a function of temperature T, taken from the article of the Criss and Cobble [22], and used in the present study for graphs a and b of Fig. 3

For the more concentrated solutions at T = 298 K, the ΔH_app and ΔH_m,2 results are first considered directly instead errors in App. D of Extra. The outcomes of these tests obtained using PI(con) are presented in graphs a and b of Fig. S1, respectively. These graphs include the high-molality results for the same datasets as in Figs. 1 and 2. Remarkably, the calorimetric data do not support this parametrization at molalities higher than about 0.5 mol·kg^–1. Furthermore, graph b in Fig. S1 also includes the ΔH_m,2 values presented by Harned and Cook [40] and Smith and Hirtle [41]. These sets were chiefly determined from electrochemical data, but for [41] were also used freezing-point-depression and boiling-point-elevation data. Similarly to the calorimetric data, neither of these two studies supports parametrization PI(con) for NaCl solutions in concentrated solutions.

Are our PI(con) enthalpies for concentrated solutions at T = 298 K simply wrong? To investigate this issue, the literature values are compared to the predictions of a previous parametrization, PII, which was observed to be successful up to the saturated solution [7, 8]. In this parametrization, the apparent molar enthalpy was represented by

$$ \frac{{\Delta H_{{{\text{app}}}} }}{{{\text{J}} \cdot {\text{mol}}^{{ - 1}} }} = a_{1} + \alpha _{{\text{T}}} \sqrt {\frac{m}{{m^{{\text{o}}} }}} + a_{2} \left( {\frac{m}{{m^{{\text{o}}} }}} \right) + a_{3} \left( {\frac{m}{{m^{{\text{o}}} }}} \right)^{{\frac{3}{2}}} $$

(27)

This equation was determined in Ref. [8]. using a method analogous to that used for PI(dil). The parameter values for a₁, a₂, and a₃ were determined for the temperature interval (273 K, 313 K) and are given in Table 5 of Ref. [8]. Details of this parametrization are included in App. E of Extra. At T = 298 K, the parameter values are a₁ =  − 5.19, α_T = 1988, a₂ =  − 2689.95, and a₃ = 637. The PII results are shown in the text in Fig. 4, the apparent molar enthalpy errors are given in graph a, and this error is defined by

$$ e_{{{\text{H}},{\text{app}}}} = \Delta H_{{{\text{app}}}} \left( {{\text{obsd}}} \right) - \Delta H_{{{\text{app}}}}\, \text{(pred)} $$

(28)

The partial molar enthalpy error in graph b of Fig. 4 defined by

Fig. 4

The plots of e_H,app of Eq. 28 and e_H,part in Eq. 29, i.e., the deviations between the relative apparent or partial molar enthalpy suggested in the literature and the prediction of parametrization PII from Ref. [8], as a function of molality m for NaCl solutions at T = 298.15 K. In graphs a and b the suggested values were taken from the studies of Pitzer et al. [1], Clarke and Glew [2], Gibbard et al. [20], Robinson [23], Craft and Van Hook [27], Messikomer and Wood [30], Ensor and Anderson [31], Parker [35] (smoothed values given in [31] were used), and Hubert et al. [42]. The last set [42] was measured at 297.55 K. In this figure, the deviations for all dilute points up to 0.6 mol·kg⁻¹ have been omitted. The symbols are given in the legends of the graphs. The following error in graph b is outside the scale on the y axis: Ref. [23], m = 6.12 mol·kg^–1, e_H,part = –942 J·mol^–1. For the molar solution enthalpy of NaCl at infinite dilution in set [42], the value of 3824 J·mol^–1 was used. It was determined by Criss and Cobble [22]

.

$$ e_{{{\text{H}},{\text{part}}}} = \Delta H_{{{\text{m}},2}} \left( {{\text{obsd}}} \right) - \Delta H_{{{\text{m}},2}}\, \text{(pred)}$$

(29)

In Eqs. 28 and 29 the predicted values were obtained from Eq. 27 and from Eqs. 7 and 27, respectively.

Figure 4 compares the errors defined by Eqs. 28 and 29 against several important datasets: First, it includes the apparent and partial enthalpies of Robinson [23], which are also shown directly using two graphical presentations a and b in Fig. S1 of Extra. Second, this figure includes the ∆H_app data from the multiparameter equations of Pitzer et al. [1] and the ∆H_app and the ∆H_m,2 data from the multiparameter equations of Clarke and Glew [2]. Third, it shows the results for the calorimetric apparent enthalpies from Messikomer and Wood [30], the thermodynamically determined ∆H_app data from Gibbard et al. [20], and the partial enthalpies from Parker [35] which we took from the article of Ensor and Anderson [31]. Finally, it contains the calorimetric partial enthalpies of Craft and Van Hook [27] and the solution-enthalpy data of Hubert et al. [42] in graph a.

While the smaller enthalpy scale of the graphs makes the differences between the dataset more tangible, Fig. 4 primarily shows that almost all apparent enthalpies in graph a can be excellently predicted using PII up to the saturated solution. Graph b further demonstrates that the partial enthalpies are also well explained using PII. In this evaluation, it should be borne in mind that even the best experimental solution-enthalpy data have an inherent accuracy that is not much better than about 100 J·mol^–1 as shown, for example, here in graphs a and b of Fig. 3 or in graph A of Fig. 1 in Ref. [16] where we consider the international reaction-enthalpy standards.

A crucial problem associated with these superb results is that several experimental thermodynamic quantities that are more accurately known than the enthalpy cannot be predicted by PII up to high concentrations. We demonstrate this in two ways:

• 1) In Table S33 in App. F of Extra, the vapor pressures from Gibbard et al. [20] at T = 298 K have been predicted using PI(con) and PII. While the experimental vapor pressures do not support PII in the more concentrated solutions, they do support PI(con).

• 2) In Fig. 5 of the main text, the relative apparent molar heat capacities (= ΔC_app) at T = 298 K for the most important datasets [30, 35, 43,44,45] are compared with the predictions obtained using PII. Some details of the heat-capacity calculations for PII are given in App. E of Extra, and Table S32 displays the numerical ΔC_app values obtained by these determinations. As shown in Fig. 5, the literature heat-capacity values consistently do not agree with the PII predictions above m = 2 mol⋅kg^–1. In contrast, when these ΔC_app data are predicted using the apparent heat-capacity equation from PI(con), all absolute errors are very small as can be seen from Fig. 5 or more clearly by comparing the values in Table 3 to the literature values.

Fig. 5

Plot of the relative apparent molar heat capacity (ΔC_app) of the salt in NaCl solutions at T = 298.15 K as a function of molality m from various sources. The heat capacities based on PII are given in Table S32 in App. E of Extra (see text). The literature values have been taken from the studies of Tanner and Lamb [43], Parker [35], Simard and Fortier [44], Alary et al. [45], and Messikomer and Wood [30]. The symbols are presented in the legend of the figure. For Refs. [43, 35, 44, 45], the values of (–85.18, –90.40, –84.49, and –82.72) J·K^–1·mol^–1 from Ref. [9] were used for the partial molar heat capacity of NaCl at infinite dilution, respectively

In conclusion, our extended Hückel-equation model at T = 298 K can explain quantities associated with the excess Gibbs energy, like vapor pressures. It can also explain data associated with the second derivative of G_ex, i.e., the apparent molar heat capacity. It fails, however, to predict data associated with the first derivative of G_ex i.e., the apparent molar enthalpy. As the quantities associated with G_ex, can be substantially more accurately measured than enthalpies, this counterintuitive result forces us to question the reliability of the existing enthalpy data for concentrated NaCl solutions at T = 298 K. This disconcerting conclusion rests only on the thermodynamic facts. We do not yet have a complete explanation that covers this unexpected result. Some speculations have been, however, presented in connection of the existing dilution-enthalpy data in App. C.

3.4 Tests of New Values for the Relative Apparent and Partial Molar Enthalpies Against Literature Values at Temperatures Other Than 298 K

In addition to the T = 298 K results, Fig. 1 displays the apparent and partial molar enthalpy errors for the Gulbransen and Robinson study [24] at T = 283, 288, and 293 K. At the three temperatures, the agreement with the predictions of PI(con) is relatively good. Similarly, graph a of Fig. 3 contains the experimental apparent enthalpies from Criss and Cobble [22] at and below T = 298 K while their results above this temperature are presented in graph b of Fig. 3. The agreement with PI(con) is always good at the thirteen temperatures ranging from 273 to 363 K in these graphs.

In graphs c and d of Fig. 3, apparent enthalpies of Ensor and Anderson [31] are considered for temperatures from 313 to 353 K. In connection to their heat-of-dilution data (see App. C), these researchers also reported apparent enthalpies for both the initial and final molalities. In graph c, these apparent enthalpies for the final molalities were tested against PI(con). These data can be predicted otherwise well, but the accuracy with which these enthalpies are reported is substantially greater than, for example, the experimental solution-enthalpy data from Criss and Cobble considered in graph b of this figure. As it is unlikely that Ensor and Anderson were able to achieve such accuracy compared to the other researchers in this field, the authors have likely reported too many digits for their values. In graph c, the most significant inconsistency is observed at T = 343 K.

Graph d of Fig. 3 shows the PI(con) errors for the apparent enthalpies given for the initial molalities from Ensor and Anderson [31] but only up to m = 1.5 mol·kg⁻¹. Because of the precision problem shown in graph c of this figure, we have drawn graph d so that the initial enthalpies were recalculated from the dilution-enthalpy data shown in Tables S23 to S27 of App. C using the enthalpy values that were obtained with PI(con) for the final solutions. All resulting errors in graph d are quite small but a very good agreement is observed only for T = 313, 323, and 333 K. The original apparent enthalpies for all initial solutions reported in Ref. [31] are considered in detail in the figures of App. D in Extra. The revised results in Graph 3d are, however, only slightly better than the results in App. D.

The rest of the PI(con)-test results for temperatures other than 298 K are presented in Figs. S2 to S16 and discussed in App. D of Extra. As in Fig. S1, relative apparent and partial molar enthalpies, instead of errors, are directly shown in these figures in addition to the values recommended in this study. Each figure shows results for a single temperature, and the apparent enthalpies are again considered in graph a of the figure while the partial enthalpies are in graph b. A table of content for the figures in this appendix can be found in Table S30. Some key results from App. D are summarized in Figs. 6 and 7. The four graphs of Fig. 6 show the apparent-molar-enthalpy comparison with the values obtained from the multiparameter equations of Clarke and Glew [2] at various temperatures. Meanwhile, Fig. 7 contains the corresponding comparison with the experimentally-obtained-solubility data from Lipsett et al. [37] at T = (293 and 298) K and those from Hubert et al. [42] at T = 298, 318, and 333 K.

Fig. 6

Plot of the relative apparent molar enthalpy (ΔH_app) suggested for NaCl solutions by Clarke and Glew [2] and that obtained by parametrization PI(con) as a function of molality m for temperatures from T = 273.15 to 288.15 K in graph a, from 293.15 to 313.15 K in graph b, from 323.15 to 343.15 K in graph c, and from 353.15 to 373.15 K in graph d. The symbols are given in the legends of the graphs. The open symbols refer to the values from Ref. [2], and the filled ones to those from the present study

Fig. 7

Plot of the relative apparent molar enthalpy (ΔH_app) based on the solution-enthalpy data measured by Lipsett et al. [37] at T = 293.15 and 298.15 K and by Hubert et al. [42] at T = 298.15, 318.15, and 333.15 K for NaCl solutions and the corresponding values obtained by parametrization PI(con) as a function of molality m. The symbols are given in the legend of the figure. The filled symbols refer to experimental values and the open ones to those of the present study. The exact temperatures for the datasets of Ref. [42] are 297.55, 317.45, and 332.35 K. Details of the calculations are given in App. D of Extra

The reason why we only compare our apparent enthalpies with the values from Clarke and Glew [2] in Fig. 6 can be seen in Figs. S2–S16 of App. D: the values from Ref. [2] provide a decent representation of the existing experimental data at each temperature. According to graph a of Fig. 6, the PI(con)-enthalpy values agree well with the values from Ref. [2] in the range T = 273 to 283 K in the same way as the dilution-enthalpy data at T = 273 K in Table 7. At 288 K, the agreement is not as good. However, since the dilution-enthalpy data from Young and Machin [25] support our model well in Table 6 at 285.65 K, the results at 288 K might already suffer from the problems identified above for the 298 K data. Perhaps for the same reason, the agreement in graph b, which covers the four temperatures from 293 to 313 K, seems to be relatively good only at the highest temperature. In graph c, the agreement is excellent at T = 323 K but is also relatively good at 333 K. Finally, at the high temperatures of graph d, the agreement again drops. This is to be expected, as the dilution-enthalpy results presented here in App. C and our previous activity-and-osmotic-coefficient results do not support the literature values in these cases [11]. All solution-enthalpy results in Fig. 7 support the conclusions presented for Fig. 6.

To further clarify the issues associated with the enthalpy data at higher temperatures, Table 9 shows the adiabatic solution temperature obtained from Eq. 10 at m = 5 mol·kg⁻¹ and T = 348, 353, and 373 K for the existing datasets [1, 2, 20, 30,31,32]. The corresponding apparent enthalpy data are represented in App. D using Fig. S13 for 348 K, Graph S14a for 353 K, and Graph S16a for 373 K. Provided that the accuracy of our ΔT_sol(ad) values from Eq. 10 is sufficient, we raise the following questions about the values in Table 9:

• 1) The experimental data from Messikomer and Wood [30] and Gibbard et al. [20] give practically the same value for ΔT_sol(ad) at T = 348 K. Have these studies been fully independent?

• 2) The multiparameter equations from Pitzer et al. [1] and from Clarke and Glew [2] and the experimental data of Mayrath and Wood [32] give the same value within the likely accuracy for ΔT_sol(ad) at T = 373 K. This might indicate that, due to overparameterization, these multiparameter equations generally follow the enthalpy values of Ref. [32] more strongly than they should.

• 3) The experimental data of Messikomer and Wood [30] and Ensor and Anderson [31] give practically identical values for ΔT_sol(ad) at T = 353 K. Have these studies been fully independent?

• 4) The experimental uncertainty in these challenging measurements can be seen in the variability of the ΔT_sol(ad) values for T = 373 K. This uncertainty is not far from what is observed for ΔT_dil(ad) at that temperature in App. C (see Tables S22 and S29), which implies that in these conditions, the accuracy of our enthalpy results is close to the experimental one.

Table 9 Rough estimates for the adiabatic solution temperature, i.e., for ΔT_sol(ad) in Eq. 10, at the molality of 5 mol·kg⁻¹ for the datasets containing values for the apparent molar enthalpies at T = 348.15, 353.15, and 373.15 K

3.5 Recommended Values for Relative Apparent and Partial Molar Enthalpies of NaCl in Aqueous Solutions

Judging from the extensive evidence provided in this study and in Ref. [11], it appears that most existing experimental data associated with the excess Gibbs energy and its first derivative can be predicted within experimental error using the new extended Hückel equations for NaCl solutions in the temperature range (273 K, 373 K). Furthermore, the suggested parametrization PI(con) is both traceable and transparent.

For dilute NaCl solutions where the molality is less than about 0.2 mol·kg^–1, we have extended the application range of the previous PI(dil) parametrization for enthalpy quantities up to the limit of T = 373 K. The range for this simpler method presented in Ref. [8] was from T = 273 to 353 K. The new values are displayed in Tables 10 and 11 from T = 358 to 373 K. Table 10 contains the apparent molar enthalpies and Table 11 the partial ones. These tables thus supplement the graphical representations of Figs. 8 and 9 in Ref. [8] where these enthalpies were presented as functions of the molality at various temperatures. We also provide the numerical values for ∆H_app and ∆H_m,2 at rounded molalities for the range from T = 273 to 353 K in App. G of Extra.

Table 10 Recommended value for the relative apparent molar enthalpy (∆H_app) of salt (symbol 2) in dilute aqueous NaCl solutions at temperatures from 358.15 to 373.15 K as a function of molality m

Table 11 Recommended value for the relative partial molar enthalpies (∆H_m,2) of salt (symbol 2) in dilute aqueous NaCl solutions at temperatures from 358.15 to 373.15 K as a function of molality m

In Ref. [11], we presented tables for the activity coefficients of salt and for the osmotic coefficients of water in NaCl(aq) using parametrization PI(con). Here, analogous values are tabulated for the relative apparent and partial molar enthalpies. The ∆H_app values at rounded molalities can be found in Tables 12, 13 and 14 and those for ∆H_m,2 in Tables 15, 16 and 17. The temperature ranges are as follows: Tables 12 and 15, 273–298 K; Tables 13 and 16, 303–328 K; Tables 14 and 17, 333–373 K. Up to T = 333 K, the enthalpy values are reported in intervals of 5 K and after that in those of 10 K. For the apparent enthalpies, Tables 12, 13 and 14 also list the values for the solutions at the saturated state. The molalities for this state were taken from Table 4 of Ref. [11]. It should be noted that the enthalpies in these saturated conditions do not differ substantially from those at m = 6.0 mol·kg^–1.

Table 12 Recommended value for the relative apparent molar enthalpy (∆H_app) of salt (symbol 2) in aqueous NaCl solutions at temperatures from 273.15 to 298.15 K as a function of molality m

Table 13 Recommended value for the relative apparent molar enthalpy (∆H_app) of salt (symbol 2) in aqueous NaCl solutions at temperatures from 303.15 to 328.15 K as a function of molality m

Table 14 Recommended value for the relative apparent molar enthalpy (∆H_app) of salt (symbol 2) in aqueous NaCl solutions at temperatures from 333.15 to 373.15 K as a function of molality m

Table 15 Recommended value for the relative partial molar enthalpies (∆H_m,2) of salt (symbol 2) in aqueous NaCl solutions at temperatures from 273.15 to 298.15 K as a function of molality m

Table 16 Recommended value for the relative partial molar enthalpies (∆H_m,2) of salt (symbol 2) in aqueous NaCl solutions at temperatures from 303.15 to 328.15 K as a function of molality m

Table 17 Recommended value for the relative partial molar enthalpies (∆H_m,2) of salt (symbol 2) in aqueous NaCl solutions at temperatures from 333.15 to 373.15 K as a function of molality m

In Tables S40 to S43 of App. H in Extra, we have included the confidence intervals at the 0.95 significance level to the apparent molar enthalpies recommended in Tables 12, 13 and 14. For comparison, the appendix also contains the values suggested in Ref. [8] using previous parametrizations PI(dil), PII, and PIII for solutions with molalities above 0.2 mol·kg⁻¹. The previous values generally agree well with the current ones only for small molalities. Above m = 1.5 mol·kg^–1 from those parametrizations, only PI(dil) gives even satisfactory results (with the exception of PII for T = 288 K in Table S40). We argue that the values predicted by PI(con) are always more reliable in these less dilute solutions and are the only ones that should be employed. In the range from T = 333 to 353 K, for example, parametrization PIII for which the values are given up to m = 3.0 mol·kg^–1, compares poorly with the currently recommended enthalpies. We think that the PI(con) values should be preferred, as PIII was based on the enthalpy data from Ensor and Anderson [31], which we have demonstrated to be of problematic quality through Graph 3c of the main text, Tables S25, S26, and S27 in App. C, and through Figs. S11, S12, and S14 in App. D of Extra.

4 Conclusions

The models and chemicals considered in this study are summarized in the chemical sample table of Table 18. This study proposes a new way to calculate the relative apparent and partial molar enthalpies of sodium chloride in aqueous NaCl solutions. The extended Hückel equation for the activity coefficient that provides the foundation for our calculations complements the Debye-Hückel term with linear and quadratic terms in molality. The coefficients of these terms depend quadratically on the temperature, while the ion-size parameter of the Debye−Hückel term has a fixed value.

Table 18 Chemical compounds and models

It is important to understand that all parameters in our model have been estimated from high precision experimental data, including vapor-pressure and cell-potential-difference measurements, and no enthalpy or heat capacity data were included in the parameter estimation. Despite this, our extensive test calculations indicate that good agreement with the literature values up the saturated solutions is observed in the temperature ranges from both 273 to 283 K and 313 to 333 K.

In the 288−303 K, temperature range, the agreement is good only for dilute solutions, whereas in the range 363–373 K, it is good only in very dilute solutions. The model agrees excellently with all existing enthalpy data up to m = 1.0 kg mol^‒1 and almost up to T = 373 K. As our model can accurately predict multiple thermodynamic properties that have not been included in the parameter estimations, we believe that our values should be preferred even when they disagree with the literature. Especially the solubility results in connection to Eq. 24 in Ref. [11] offer convincing support to our predictions. Consequently, we argue that our recommended relative apparent and partial molar enthalpies for NaCl(aq) may represent the most accurate values up to date. However, we call for new high-precision enthalpy measurements to elucidate the observed discrepancies especially at high temperatures and molalities.