Traceable Thermodynamic Quantities for Dilute Aqueous Sodium

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX-XXX

pubs.acs.org/jced

Traceable Thermodynamic Quantities for Dilute Aqueous Sodium Chloride Solutions at Temperatures from (0 to 80) °C. Part 2. The Quantities Associated with the Partial Molar Heat Capacity Jaakko I. Partanen,*,† Lauri J. Partanen,‡ and Kari P. Vahteristo† †

Department of Chemical Technology, LUT School of Engineering Science, Lappeenranta University of Technology, P.O. Box 20, FI-53851 Lappeenranta, Finland ‡ Laboratory of Physical Chemistry; Department of Chemistry, University of Helsinki, P.O. Box 55, A. I. Virtasen aukio 1, Helsinki, FI-00014, Finland ABSTRACT: In Part 1 of this two-part study (J. Chem. Eng. Data 2017, 62, 2617−2632), we presented fully traceable two-parameter Hückel equations (with parameters B and b1) for the activity and osmotic coefficients in dilute aqueous NaCl solutions in the temperature range (0 to 80) °C, and these equations apply within experimental error to almost all thermodynamic data existing in the literature and used in the tests at least up to a molality of 0.2 mol·kg−1. These data include also molar enthalpies of the components in the solutions. In our model, parameter B is treated as a constant whereas parameter b1 depends in a quadratic way on the temperature. No calorimetric data were used in the parameter estimation of the model. In this second part (Part 2) of the study, the results of the quantities associated with the heat capacity of NaCl solutions are considered. All heat capacity data available for NaCl solutions at least up to 0.2 mol·kg−1 can be predicted within experimental error using the same Hückel equations as those determined in Part 1 in dilute NaCl solutions from (0 to 80) °C. For comparison, also other parametrization (obtained in Part 1 for higher temperatures) was here considered and it applies better to less dilute solutions in the higher temperatures than the model recommended now primarily. Following the success of the new models, we supplement the thermodynamic tables of Part 1 with the relative apparent and partial molar hear capacities for NaCl solutions. We have good reason to believe that the new tables contain the most reliable values available for the heat capacity quantities in dilute NaCl solutions.

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presented in a previously published part (Part 16) and in the present part (Part 2) of this study, the behavior of these solutions is often accurately described by a much simpler Hückel-type equation containing only a handful of estimated parameters. Up to date, the heat capacity data of pure aqueous solutions of several uniunivalent electrolytes have been considered in the reviews of Parker7 and Criss and Millero.8 The former contains the results for temperatures (15, 20, 25, and 30) °C and the latter only those for 25 °C. At 25 °C, also solutions of other valence-type electrolytes have been considered in another review of the latter researchers.9 The heat capacities of NaCl solutions at various temperatures have also been considered in the review of Pitzer and Silvester10 and in the multiparameter equations of those mentioned above.1−3 Ensor and Anderson11 determined at higher temperatures from (40 to 80) °C the heat capacity values on the basis of their wide heat of dilution data at these temperatures. Except for the studies of Parker7 and Ensor and Anderson,11 Pitzer formalism4 has been used in the

INTRODUCTION Owing to its role as the primary salt in seawater and most other natural waters, within the industry and biological systems, sodium chloride is doubtlessly one of the most important electrolytes on earth. As a consequence, its central thermodynamic properties in aqueous solutions such as activity coefficients, osmotic coefficients, heats of dilution and solution and partial molar heat capacities have been thoroughly measured in various different solute concentrations, temperatures, and pressures (see reviews of Pitzer et al.,1 Clarke and Glew,2 and Archer3). These measurements have then served as the basis for different kind of models like the one proposed by Pitzer4 that help explain the experimental results and enable the calculation of these properties in regions where experimental data are not yet available. This model is also the basis in the multiparameter equations used in the thermodynamic treatments of these three reviews.1−3 For practical considerations, however, these equations are often extremely complicated, containing very many fitting parameters associated with the three fundamental variables (i.e., temperature, pressure, and concentration), while still failing to explain well some features of the available experimental data (see, for example, all freezing point results considered in ref 5). In light of the findings © XXXX American Chemical Society

Received: June 28, 2017 Accepted: November 3, 2017

A

DOI: 10.1021/acs.jced.7b00590 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

properties of these solutions at various compositions and temperatures as accurately as possible. However, it is almost as important to understand the limits of the real accuracies of the calculated values. In the present study, both of these questions will be considered concerning the properties associated with the heat capacities of components in dilute NaCl solutions. In Part 1 of this study,6 we showed that the two-parameter Hückel equation with parameters B and b1 is the most accurate equation to date for the prediction of the thermodynamic properties of dilute NaCl solutions at various temperatures. This also seems true for dilute solutions of other pure electrolytes. For example in this previous part, we observed that the two-parameter Hückel equation applies to the activity and osmotic coefficients and to the quantities associated with the partial molar enthalpy in aqueous NaCl solutions within the temperature range (0 to 80) °C at least up a molality of 0.2 mol·kg−1. In the present part (Part 2), we confirm that the constant value of parameter B (= 1.4 (mol·kg−1)−1/2) and the traceable quadratic equation in temperature determined in Part 1 for parameter b1 (see eq 12 in ref 6) also applies within experimental error to almost all heat capacity data available in the literature for dilute NaCl solutions within (0 to 80) °C. Following our notation in Part 1, this parametrization of the Hückel equations has been designated as PI. The cubic equation for b1 (see eq 15 in ref 6), solved on the basis the heat of dilution measurements of Ensor and Anderson11 and employed in the treatment of higher temperatures in Part 1 is designed also here as PIII. In line with the results of Part 1, we observe that in high temperatures and in less dilute solutions PIII yields superior results for the heat capacities compared to PI.

interpretation of the experimental data in all other studies mentioned above. Unfortunately in addition to relying on complex multiparameter equations, one of the reviews2 does not provide a clear understanding of what is the accuracy of the heat capacity data presented in the literature. This issue is illustrated in Table 1 for the values of the relative partial molar enthalpies of NaCl Table 1. Relative Partial Molar Heat Capacitya (ΔCpart) of Sodium Chloride as a Function of Molality m at 40 °C According to Ensor and Anderson11 and Clarke and Glew.2 m

ΔCpart (ref 11)

Cm,2 (ref 2)

ΔCpart (ref 2)b

mol·kg−1

cal·K−1·mol−1

J·K−1·mol−1

cal·K−1·mol−1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

3.7 5.6 6.9 8.1 9.5 11 12 13 14 14

−49.059 −41.826 −35.799 −30.408 −25.445 −20.808 −16.437 −12.292 −8.346 −4.578

3.6747 5.4512 6.8439 8.1324 9.3186 10.427 11.472 12.462 13.405 14.306

a Defined by equation ΔCpart = Cm,2 − C∞ m,2 where Cm,2 is the partial b molar enthalpy of the salt and C∞ m,2 is its value at infinite dilution. The 2 ∞ −1 −1 value of Cm,2 = −64.434 J·K ·mol was used.

at 40 °C, that is, for ΔCpart = Cm,2 − C∞ m,2, in refs 2 and11. Cm,2 is the partial molar enthalpy of solute 2 (now NaCl) and C∞ m,2 is its value at infinite dilution. The original values of Ensor and Anderson11 are given in unit cal·K−1·mol−1 and this unit is used in the table. The values from the multiparameter equation of Clarke and Glew2 have been changed to the same unit (originally these data are given in unit J·K−1·mol−1) but the number of reported digits is preserved as the same as in the original data.2 The reported accuracy of the values of Ensor and Anderson is close to the present understanding of the precision of the heat capacity measurements at higher molalities at these higher temperatures. The digits given in ref 2 for Cm,2 do not probably correspond to the real accuracy of this quantity. It does not seem possible that the other sources of data used in the parameter estimation of Clarke and Glew2 for their multiparameter equations would yield such accurate values for the heat capacities as those presented in Table 1. Evidently, the values from these two sources are close to each other which can be explained by the fact that the heat of dilution data from the former study11 have been included in the estimation of the model for the latter study.2 As pointed out by Criss and Millero,8 all of the data tabulated by Parker7 were measured before the invention of flow calorimeters, and these flowing-heat capacity system measurements (Picker method) are in dilute electrolyte solutions superior to those used for these older data. For NaCl solutions at various temperatures, the pioneering Picker flow microcalometer data will be considered here later and are given in refs 12 and 13 but even these data cannot result in so accurate values as suggested by Clarke and Glew in Table 1. Nowadays, when the parameter estimation and all calculations are easy to perform, it is very important to know how accurately the calculated results compare with the real experimental data. As NaCl solutions are probably the most important electrolyte solutions, it is especially important to know the values of the

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THEORY In several recent contributions14−17 (see also refs 12−26 in Part 16), it has been demonstrated that the Hückel equations ln γ = −

ϕ=1− −

α|z+z −| Im 1 + B Im

+ b1(m /mo)

(1)

α|z+z −| ⎡ ⎢(1 + B Im ) − 2 ln(1 + B Im ) B3Im ⎣⎢

1 1 + B Im

⎤ 1 ⎥ + b1(m /mo) ⎥⎦ 2

(2)

apply well to the mean activity coefficient (γ) and osmotic coefficient (ϕ) in aqueous solutions of many salts at least up to an ionic strength (Im) of 1 mol·kg−1. In eqs 1 and 2, m is the molality, z+ and z− are the charge numbers of the cation and anion, respectively, and the parameters dependent on the electrolyte are B and b1. The values of the Debye−Hückel parameter α at 101.325 kPa and at various temperatures are given in Table 1 of Part 1,6 and they have been taken from ref 18. For a 1:1 electrolyte such as NaCl, |z+z−| is 1 and Im is the same as the molality. The excess Gibbs energy of solution (ΔGex) on the molality scale in salt solutions is related to the activity and osmotic coefficients by ΔGex = 2mRT (1 − φ + ln γ )

(3)

For all of the energy quantities below, the molality scale is used and the mass of water is considered to be 1 kg. The excess B



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RESULTS AND DISCUSSION Calculations of the Relative Apparent Molar Heat Capacities. Equation 8 was tested in Part 1 and in the present part using all important experimental data that have been published for dilute NaCl solutions up to a temperature of 80 °C. On the basis of the results in Part 1, the following simple calculation method was observed to give accurate and fully traceable results to interpret the complicated heat capacity data available for dilute NaCl solutions at various temperatures up to 85 °C. First, using eq 3 together with eqs 1 and 2 the excess Gibbs energy of solutions at the temperatures from (0 to 80) °C with intervals of 5 °C has been calculated for the rounded molalities (0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0) mol·kg−1. These results were then fitted to a quadratic equation of the following type ΔGex = u + v(T − T0) + w(T − T0)2 (10) mT where T0 = 273.15 K. For all molalities, the resulting values of parameters u, v, and w are given in Table 2 of Part 1,6 together

molar enthalpy (ΔHex/m) and the relative apparent molar enthalpy (ΔHapp) of salt in these solutions are associated with the Gibbs energy by the following thermodynamic identity ΔHapp =

ΔHex ∂(ΔGex /mT ) ∞ − Hm,2 = −T 2 m ∂T

(4)

where subscript 2 refers to the salt and H∞ m,2 is the partial molar enthalpy of the salt at infinite dilution. The relative partial molar enthalpy (ΔHm,2) of the salt can be calculated from the apparent molar enthalpy using ∞ ΔHm,2 = Hm,2 − Hm,2 = ΔHapp + m

∂(ΔHapp) ∂m

(5)

The apparent molar heat capacity, Capp, of the salt at a constant pressure is connected to the apparent molar enthalpy by ∞ Capp = Cm,2 +

∂(ΔHapp) ∂T

(6)

where C∞ m,2 is the partial molar heat capacity of the salt at infinite dilution. For a certain molality, the partial molar heat capacity of the salt can be calculated from the apparent quantity using Cm,2 = Capp + m

Table 2. Values of Real and Relative Apparent Molar Heat Capacity (Capp and ΔCapp, Respectively) Obtained Using Parametrization PI at 25 °C for NaCl Solutions as a Function of Molality m and the Error (eC,app) when the Relative Value Is Predicted by eq 12 with the Parameter Values Given in Table 3

∂(Capp) ∂m

(7)

−1

mol·kg

(8)

In Part 1, we observed that parametrization PI does not apply well to the existing enthalpy data above the temperature of about 60 °C. In that part, therefore, the following cubic equation was determined for parameter b1 for these higher temperatures:

−1

−1

J·K ·mol

0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

⎛ t ⎞ ⎛ t ⎞2 b1 = 0.0077 + 0.0031853⎜ ⎟ − 0.00002517⎜ ⎟ ⎝ °C ⎠ ⎝ °C ⎠

ΔCapp

(Capp)a

m

We have observed in the previous studies6,14,17 that the constant value of 1.4 (mol·kg−1)−1/2 for parameter B in dilute NaCl solutions can be used at all temperatures from (0 to 80) °C. Additionally, the following three values for parameter b1 have been determined in those studies: b1(0 °C) = 0.0077, b1(25 °C) = 0.0716, and b1(75 °C) = 0.105. With these values, the following quadratic equation was presented in Part 16 for the temperature dependence of parameter b1:

⎛ t ⎞ ⎛ t ⎞2 b1 = 0.0077 + 0.0033673⎜ ⎟ − 0.00003648⎜ ⎟ ⎝ °C ⎠ ⎝ °C ⎠ 3 ⎛ t ⎞ + 0.000000161⎜ ⎟ ⎝ °C ⎠

Article

−83.0 −82.6 −81.8 −80.8 −79.5 −76.8 −73.6 −71.0 −68.7 −64.5 −60.7 −57.1 −53.6 −50.2 −46.9 −43.7 −40.5

−1

(eC,app)b −1

J· K ·mol 1.00 1.42 2.24 3.16 4.48 7.17 10.36 12.97 15.30 19.49 23.32 26.94 30.42 33.79 37.09 40.33 43.52

J·K−1·mol−1 0.098 0.092 0.074 0.050 0.012 −0.062 −0.114 −0.127 −0.116 −0.067 −0.006 0.044 0.075 0.080 0.056 0.001 −0.090

a

The real value calculated from the relative value using a value of −84.0 J·K−1·mol−1 (Marcus19) for the partial molar heat capacity of NaCl at infinite dilution. bThe difference between the suggested and predicted value. (9)

This equation was based on the values of b1(0 °C) = 0.0077, b1(25 °C) = 0.0716, b1(50 °C) = 0.105, and b1(75 °C) = 0.123. The b1 value at 50 °C is almost the same as that from the quadratic equation (i.e., from eq 8) but the value at 75 °C is different and it results in a better fit with the heat of dilution data from Ensor and Anderson11 above 60 °C. It is shown below that also the heat capacity data from less dilute NaCl solutions can be explained better using parametrization PIII based on eq 9 than using parametrization PI based on eq 8 in these higher temperatures.

with the standard deviation about the regression. The apparent molar heat capacity given in eq 6 can be calculated from the parameters of eq 10 by ∞ Capp = Cm,2 +

∂(ΔHapp) ∂T

∞ = Cm,2 − 2T (v + 3wT − 2wT0)

(11)

For 25 °C, the resulting apparent molar heat capacities are reported in Table 2 of the present study. C



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mol−1 recommended by Marcus19 for the partial molar heat capacity of NaCl at infinite dilution at 25 °C. In the tests of parametrizations PI or PIII below, the value of C∞ m,2 at each temperature is used as a fitting parameter because no agreement of their exact values at various temperatures exist in the literature. This point will be clarified by an example later. Tests of the New Values for Relative Apparent Molar Enthalpies with Those Presented in the Literature. The data sets of experimental Capp values taken from the literature and employed in this study are summarized in Table 4. In addition to those, Parker7 has reported derived values at (15, 20, 25, and 30) °C at least up to 2.8 mol·kg−1 based on critical evaluation of the existing literature data, whereas Gibbard et al.27 have calculated derived values for (0, 25, 50, and 75) °C on the basis of their own vapor pressure measurements and some other accurate equilibrium data available in the literature. All of these data were used here to test parametrizations PI and PIII as well as the relative heat capacity values reported by Messikomer and Wood,28 based partially on their own heat of dilution measurements. As can be seen in Table 1, partial molar heat capacities have been reported by Clarke and Glew,2 and this review also contains tables for the apparent values at various temperatures, and these were included in the present tests as well as the relative values reported in the tables of Pitzer et al.1 As mentioned above, the heat capacity tables in both of these reviews are based on the multiparameter models. Because relative values are not given in the review of ref 2, the C∞ m,2 value at each temperature for the present comparison was determined from the reported molar heat capacities in the best case up to 1.0 mol·kg−1 by requiring that at this temperature the sum of all errors obtained using parametrization PI, or PIII above 50 °C, is zero. The resulting ∞ C∞ m,2 values are collected in Table 5 together with the Cm,2 2 values given by Clarke and Glew. The agreement between the two values is always at least satisfactory. The test results obtained at 25 °C are shown in three graphs of Figure 1 where the error plots for the different data sets have been drawn. The data sets are abbreviated in this figure and as in all following figures as described in Table 4. The acronym tells the first three letters of the author’s surname in the case of only one author, first two letters of the authors’ surname in the case of two authors, and just the first letter of the surnames in the case of three or more authors. The number in the acronym represents the data set temperature in Celsius. The errors for the plots have been calculated from

Next, the relationship between these values and the molalities was determined by fitting the values with the equation ∞ ΔCapp = Capp − Cm,2

= aC,1 + αC m /mo + aC,2(m /mo) + aC,3(m /mo)3/2

(12)

where the theoretical Debye−Hückel values were used for the coefficient of the square root term [i.e., for αC) at different temperatures. These theoretical values were taken from ref 18 and are collected together with the fitted values αC,1, αC,2, and αC,3 at various temperatures in Table 3 of the present study. Table 3. Parameter Values for eq 12 (i.e., for the Dependence of Relative Apparent Molar Heat Capacity on the Molality) for NaCl Solutions Obtained by Using Parametrization PI (eq 8) and Parameterization PIII (eq 9) as a Function of Temperature t t

aC,1 −1

°C

(αC)

a

J·K ·mol

0 1.5 2 5 7 10 12 12.5 15 20 25 30 35 40 40c 45 50 50c 60 60c 65 65c 70 70c 75 75c 80 80c 85 85c

13.95 15.77 16.37 20.005 21.71 24.254 25.51 25.825 27.400 29.854 31.89 33.68 35.33 36.91 36.91 38.47 40.04 40.04 43.314 43.314 45.054 45.054 46.88 46.88 48.81 48.81 50.85 50.85 53.01 53.01

0.842 0.720 0.680 0.442 0.349 0.202 0.143 0.128 0.055 −0.036 −0.106 −0.124 −0.135 −0.144 −0.151 −0.149 −0.147 −0.148 −0.154 −0.156 −0.164 −0.164 −0.176 −0.177 −0.195 −0.197 −0.223 −0.224 −0.254 −0.255

aC,2 −1

−1

(sC)b

aC,3 −1

J·K ·mol

21.695 18.750 17.776 11.815 9.467 6.116 4.887 4.576 3.060 1.784 1.8305 1.928 2.876 4.066 −10.171 5.346 6.710 −9.291 9.526 −8.304 10.851 −7.924 12.079 −7.660 13.189 −7.535 14.176 −7.552 14.935 −7.817

−1

−1

J·K ·mol −7.6 −5.5 −4.8 −0.50 1.4 4.1 5.3 5.6 7.1 9.0 10 11.3 11.9 12.4 12.4 12.9 13.4 13.4 14.4 14.4 15.0 15.0 15.7 15.7 16.5 16.5 17.4 17.4 18.50 18.50

−1

J·K ·mol−1 0.22 0.25 0.24 0.14 0.11 0.05 0.03 0.03 0.03 0.06 0.08 0.11 0.10 0.11 0.12 0.11 0.11 0.11 0.12 0.12 0.13 0.13 0.13 0.13 0.14 0.14 0.16 0.16 0.17 0.17

eC,app = ΔCapp (observed) − ΔCapp (predicted)

(13)

where the predicted values generally are obtained by PI (or PIII in the forthcoming figures concerning higher temperatures). Graph A of Figure 1 displays the errors for sets AlWo25, FLD25, OloI25, OloII25, and ASDJ25 given in Table 1 with points included up to 1.7 mol·kg−1. These data excellently support the parametrization PI at least up to 1 mol·kg−1. Only two points in the experimental sets have an absolute error larger than 2 J·K−1·mol−1. This high accuracy should be compared, for example, to the uncertainty existing in the literature concerning the C∞ m,2 values: two authoritative reviews by Marcus19 and Parker7 suggest the values of −84.00 and −90.40 J·K−1·mol−1, respectively, based on wide evaluations of the existing experimental data. Graph B of Figure 1 shows the results from the sets of FoSi25, TaLa25, RoRa25, and OloIII25 in Table 4. Furthermore, graph B includes data from more concentrated

a Debye−Hückel parameter for the heat capacity, the values were taken from ref 18 and the unit is J·K−1·mol−1. bStandard deviation about the regression. cParametrization PIII was used.

The apparent molar heat capacities predicted using eq 12 for 25 °C are indirectly given in Table 2 as errors when compared to those obtained using parametrization PI. The agreement between the two values is good. Table 2 also gives the real values of the apparent heat capacities based on the relative −1 values calculated using PI and the value of C∞ m,2 = −84.0 J·K · D



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Table 4. Sets of Experimental Data Available in the Literature for the Apparent Molar Heat Capacity (Capp) in NaCl Solutions at Various Temperatures (t) t °C 1.5 2 5 5 7 10 12 15 25 25 25 25 25 25 25 25 25 35 40 45 45 65 85

mmina mol·kg

−1

0.06845 0.05122 0.02677 0.0763 0.05122 0.05599 0.05122 0.02677 0.0405 0.01000 0.01126 0.03779 0.0411 0.2787 0.06333 0.06662 1.0203 0.02677 0.05858 0.02677 0.0424 0.0763 0.0763

C∞ m,2

mmaxb mol·kg

−1

1.8001 4.5007 0.94710 6.0450 4.5007 0.40704 4.5007 3.01420 2.2687 3.0234 5.325 2.89860 6.0450 5.1053 2.0987 2.8116 3.1947 3.01420 0.34549 2.53330 6.0450 6.0450 6.0450

N

−1

c

JK

mol−1d

−168.74 −151.8g −146.01h −151.18i −148.0g −122.10 −117.5g −103.54j −86.79 −81.19 −84.49 −83.06 −85.18 −82.72 −81.16 −77.67 −81.16k −69.56 −68.44 −64.27l −66.55m (−64.61 or −60.99)n (−68.87 or −67.80)o f

12 10 11 20 10 8 10 16 17 22 22 25 30 10 21 19 13 16 20 15 27 22 19

ref

data sete

13 20 13 21 20 22 20 13 23 12 24 22 21 25 26 26 26 13 22 13 21 21 21

PFD1.5 ArCa2 PFD5 TaLa5 ArCa7 AlWo10 ArCa12 PFD15 RaRo25 FLD25 SiFo25 AlWo25 TaLa25 ASDJ25 OloI25 OloII25 OloIII25 PFD35 AlWo40 PFD45 TaLa45 TaLa65 TaLa85

a

The minimum molality studied. bThe maximum molality studied. cNumber of points. dPartial molar heat capacity at infinite dilution and the values up to 1 mol·kg−1 were included in its determination of not otherwise explained. eIn the data set acronym, the last item shows the temperature measured (t/°C). fFour points up to 0.26 mol·kg−1 were included in the estimation. gTwo points up to 0.99 mol·kg−1 were included in the estimation. hFive points up to 0.26 mol·kg−1 were included in the estimation. iOne point where m = 0.0796 mol·kg−1 was included in the estimation. j Ten points up to 0.66 mol·kg−1 were included in the estimation. kDetermined for set OloI25. lTen points up to 0.66 mol·kg−1 were included in the estimation. mSeven points up to 0.45 mol·kg−1 were included in the estimation. nThree points up to 0.32 mol·kg−1 were included in the estimation, and the former value was determined using parametrization PI and the latter using PIII. oThe former value was estimated using parametrization PI from two points up to 0.18 mol·kg−1 and the latter value was estimated using parametrization PIII from three points up to 0.32 mol·kg−1.

B the derived results at the smoothed molalities. According to graph A of Figure 2, the heat capacities up 0.4 mol·kg−1 can be predicted within experimental error using parametrization PI. Above that molality, parametrization PI seems to predict relative heat capacities that are too low but the error is not very significant when the internal variation of the heat capacity data is taken into account. In this graph, the values from the sets of Archer and Carter20 seem to be less precise than the other data probably because the measurements have been made using differential scanning calorimetry which is not generally regarded as accurate a method as the flow calorimetry on which the other data have been determined. In graph B close to the temperature of 0 °C, the heat capacities from Gibbard et al.27 agree slightly better than the values of Clark and Glew2 with the experimental values of Perron et al.13 (see set PFD1.5 in Table 4). At 5 and 10 °C, the heat capacities from the multiparameter equations1,2 follow exactly the experimental values. The heat capacities of dilute NaCl solutions in the temperature range (15 to 30) °C excluding 25 °C (see Figure 1) are considered in Figure 3, where graph A contains the experimental results measured by Perron et al.13 at 15 °C (set PFD15 in Table 4) and the smoothed results suggested in the critical review of Parker7 for (15, 20, and 30) °C. All of these data can be predicted at least relatively well with parametrization PI up to a molality of 2 mol·kg−1. In graph B, the smoothed values from Messikomer and Wood,28 from Clarke

solutions incorporating the higher molality points of sets FLD25, ASDJ25, and AlWo25 up to 2.5 mol·kg−1. According to these error plots, parametrization PI applies well up to 1.5 mol· kg−1 but falters as the solutions became stronger. The results in these concentrated solutions seem to be quite consistent with each other for the different data sets. Derived heat capacity data at 25 °C are considered in graph C of Figure 1 which contains errors from the sets of Parker,7 Gibbard et al.,27 Messikomer and Wood,28 Clarke and Glew,2 and Pitzer et al.1 The heat-capacity data from Messikomer and Wood were partially determined from the derived values of Parker at this temperature but, according to this graph, do not agree with the heat capacities given by Parker. They do agree better with the values suggested in the present study and we believe that these heat capacities were calculated from the enthalpy data of Parker in the same way as Messikomer and Wood used their own enthalpy data. In the new calculations to reproduce the heat capacities from Clarke and Glew,2 the value given in Table 5 was used for the molar heat capacity at infinite dilution. The apparent heat capacities from Messikomer and Wood, Clarke and Glew, and Pitzer et al. agree well with those obtained using parametrization PI but the values from Parker7 and Gibbard et al.27 are slightly different. The heat capacities from dilute NaCl solutions in low temperatures up to 12 °C are considered in Figure 2 where graph A contains the results from experimental data and graph E



Article

−1 −1 Table 5. Partial Molar Heat Capacity of the Salt at Infinite Dilution (C∞ Suggested by Criss and Cobble,30 m,2) in Unit J·K ·mol 2 That Obtained from the Tables Reported by Clarke and Glew for the Apparent Molar Heat Capacity (Capp), and That Obtained from eq 14 Suggested by Desnoyers et al.29 for Quantity Capp in NaCl Solutions at Various Temperatures (t) c C∞ m,2(CC)

d C∞ m,2(CG)

e C∞ m,2(CG)

−169.5

−183.08

−183.80

−87.4 −79.1 −72.0

−153.36 −129.60 −110.48 −95.58 −83.70 −76.53

−153.92 −129.79 −110.58 −95.53 −83.95 −75.248

6

−64.0

−65.07

−64.434

6 6 5 5 5

−62.3

−60.35 −60.12 −58.89 −59.96 −62.48

−59.70 −59.70 −58.65 −59.80 −62.40

t/°C

mmaxa

Nb

0 1.5 5 10 15 20 25 30 35 40 45 50 50h 60h 70h 80h

0.0100

4

0.0200 0.0200 0.0500 0.1000 1.0000 0.6000

5 5 6 7 16 12

0.0500 0.0500 0.0500 0.0200 0.0200 0.0200

−121.3

−63.6 −66.1 −69.0

f C∞ m,2(DVPP)

g C∞ m,2(DVPP)

−172.8 −152.3

−176.6 −155.4

−110.3

−109.1

−84.26

−84.4

−69.81

−69.2

−64.78

−63.7

a

The maximum molality included in the estimation from Capp values of ref 2. bNumber of points included in the estimation from Capp values of ref 2. Determined by Criss and Cobble30 from their own data. dThe resulting value for partial molar heat capacity at infinite dilution obtained in this study using parametrization PI from heat capacity values of Clarke and Glew.2 eGiven by Clarke and Glew.2 fThe resulting value for partial molar heat capacity at infinite dilution obtained in this study using parametrization PI from apparent heat capacities calculated from the equation of Desnoyers et al.29 (eq 14) at molalities 0.02, 0.04, 0.06, 0.08, 0.10, and 0.15 mol·kg−1. gGiven by Desnoyers et al.29 hObtained using parametrization PIII instead of PI. c

Figure 1. Plot of eC,app (eq 13), the deviation between the suggested relative apparent molar heat capacity for NaCl solutions and that predicted using parametrization PI of the present study (see text) at 25 °C as a function of molality m. The suggested values are experimental in graphs A and B and derived in graph C. The experimental data sets are introduced in Table 4. Symbols for graph A: ●, AlWo25 (see Table 4); ○, FLD25; ▼, OloI25; △, OloII25; ■, ASDJ25. Symbols for graph B: ●, SiFo25; ○, TaLa25; ▼, RaRo25; △, OloIII25; ■, ASDJ25; □, FLD25; ⧫, AlWo25. Symbols for graph C: ●, Parker;7 ○, Gibbard et al.;27 ▼, Messikomer and Wood;28 △, Clarke and Glew;2 ■, Pitzer et al.1 In the calculations of the data from Clarke and Glew and from Parker, the results of the most dilute points have been omitted from the figure and the values of −83.70 and −90.396 J· K−1·mol−1 was used for C∞ m,2, respectively. The former value is also given in Table 5.

F



Article

Figure 2. Plot of eC,app (eq 13), the deviation between the suggested relative apparent molar heat capacity for NaCl solutions and that predicted using parametrization PI of the present study (see text) from (0 to 12) °C as a function of molality m. The suggested values are experimental in graph A and derived in graph B. The experimental data sets are introduced in Table 4. Symbols for graph A: ●, PFD1.5 (see Table 4); ○, PFD5; ▼, TaLa5; △, AlWo10, ■, ArCa2; □, ArCa7; ⧫, ArCa12. Symbols for graph B: ●, Gibbard et al.,27 0 °C; ○, Clarke and Glew,2 0 °C; ▼, Clarke and Glew,2 5 °C; △, Clarke and Glew,2 10 °C; ■, Pitzer et al.,1 0 °C; □, Pitzer et al.,1 10 °C. In the calculations of the data from Clarke and Glew, the results of the most dilute points have been omitted from the figure and the values given in Table 5 were used for C∞ m,2. The points from sets ArCa2, ArCa7, and ArCa12 at molality 0.1662 mol·kg−1 are outside the scale of graph A. These errors are (76.8, 82.1, and 54.7) J·K−1·mol−1, respectively.

Figure 3. Plot of eC,app (eq 13), the deviation between the suggested relative apparent molar heat capacity for NaCl solutions and that predicted using parametrization PI of the present study (see text) at (15, 20, and 30) °C as a function of molality m. The suggested values are experimental or critically evaluated values by Parker7 in graph A and derived values in graph B. The experimental data sets are introduced in Table 4. Symbols for graph A: ●, PFD15 (see Table 4); ○, Parker,7 15 °C; ▼, Parker,7 20 °C; △, Parker,7 30 °C. Symbols for graph B: ●, Messikomer and Wood,28 30 °C; ○, Clarke and Glew,2 15 °C; ▼, Clarke and Glew,2 20 °C; △, Clarke and Glew,2 30 °C; ■, Pitzer et al.,1 20 °C; □, Pitzer et al.,1 30 °C. In the calculations of the data from Clarke and Glew, the values given in Table 5 were used for C∞ m,2 and in those from Parker the corresponding values of (−104.94, −97.551, and −84.747) J·K−1·mol−1 at (15, 20, and 30) °C, respectively. In the presentation of the data from Clarke and Glew and from Parker, the results of the most dilute points have been omitted from the figure.

40 and 50 °C. Especially, the equations of Clarke and Glew2 at these temperatures follow the smoothed values of Messikomer and Wood so accurately that the errors from these calculations are indiscernible in the graph. Altogether, all smoothed results in graph B can be predicted quite well with parametrization PI or PIII (at 50 °C). Finally, the errors for apparent molar heat capacities from dilute NaCl solutions in the temperature range (60 to 85) °C are illustrated in Figure 5, where graph A contains the results from the experimental data measured by Tanner and Lamb21 at (65 and 85) °C (sets TaLa65 and TaLa85 in Table 4). These results concur with the heat capacities predicted by parametrization PIII up to 1 mol·kg−1 and relatively well with those predicted by PI up to 0.5 mol·kg−1. In graph B, only the results obtained using parametrization PIII are presented. The heat capacities from Gibbard et al.27 at 75 °C deviate somewhat from the values predicted by parametrization PIII but the other values in the graph agree better. Again the equations of Clarke and Glew2 and those of Pitzer et al.1 predict almost exactly the

and Glew,2 and from Pitzer et al.1 agree completely at 30 °C. Also the heat capacities from the multiparameter equations1,2 follow exactly the values from Parker 20 °C (see graph A) as do also the values from Clarke and Glew2 follow those suggested by Parker at 15 °C (graph A). The heat capacities from dilute NaCl solutions in the temperature range (35 to 50) °C are covered in Figure 4 where graph A contains the experimental results by Perron et al.13 at (35 and 45) °C (sets PFD35 and PFD45 in Table 4), by Allred and Woolley22 at 40 °C (set AlWo40), and by Tanner and Lamb21 at 45 °C (set TaLa45). According to this graph, all of these data up to 1 mol·kg−1 can be well predicted using parametrization PI. At 50 °C in graph B, the results from the calculations with parametrization PIII are presented in addition to those with PI. The smoothed heat capacity results from Gibbard et al.27 at this temperature can be explained much better when parametrization PIII is used instead of PI. Again the heat capacities suggested by Messikomer and Wood,28 Clarke and Glew,2 and Pitzer et al.1 agree almost completely at G



Article

Figure 4. Plot of eC,app (eq 13), the deviation between the suggested relative apparent molar heat capacity for NaCl solutions and that predicted using parametrization PI or PIII of the present study (see text) from (35 to 50) °C as a function of molality m. The suggested values are experimental in graph A and derived in graph B. In graph A only parametrization PI was used and the experimental data sets are introduced in Table 4. Symbols for graph A: ●, PFD35 (see Table 4); ●, AlWo40; ▼, PFD45; △, TaLa45. Symbols for graph B: ●, Gibbard et al.,27 50 °C, parametrization PI; ○, Gibbard et al.,27 50 °C, PIII; ▼, Messikomer and Wood,28 40 °C, PI; △, Messikomer and Wood,28 50 °C, PI; ■, Messikomer and Wood,28 50 °C, PIII; □, Clarke and Glew,2 40 °C, PI; ⧫, Clarke and Glew,2 50 °C, PI; ◊, Clarke and Glew,2 50 °C, PIII; ▲, Pitzer et al.,1 40 °C, PI; ▽, Pitzer et al.,1 50 °C, PI; ⬢, Pitzer et al.,1 50 °C, PIII. In the calculations of the data from Clarke and Glew, the results of the most dilute points have been omitted from the figure and the values given in Table 5 were used for C∞ m,2.

Figure 5. Plot of eC,app (eq 13), the deviation between the suggested relative apparent molar heat capacity for NaCl solutions and that predicted using parametrization PI or PIII of the present study (see text) from (60 to 85) °C as a function of molality m. The suggested values are experimental in graph A and derived in graph B. The experimental data sets are introduced in Table 4, and in graph B only parametrization PIII was used. Symbols for graph A: ●, TaLa65 (see Table 4), parametrization PI; ○, TaLa65, PIII; ▼, TaLa85, PI; △, TaLa85, PIII. Symbols for graph B: ●, Gibbard et al.,27 t = 75 °C; ○, Messikomer and Wood,28 60 °C ▼, Messikomer and Wood,28 70 °C; △, Messikomer and Wood,28 75 °C; ■, Messikomer and Wood,28 80 °C; □, Clarke and Glew,2 60 °C; ⧫, Clarke and Glew,2 70 °C; ◊, Clarke and Glew,2 80 °C; ▲, Pitzer et al.,1 60 °C; ▽, Pitzer et al.,1 70 °C; ⬢, Pitzer et al.,1 80 °C. In the calculations of the data from Clarke and Glew, the results of the most dilute points have been omitted from the figure and the values given in Table 5 were used for C∞ m,2.

heat capacities reported by Messikomer and Wood at (60, 70, and 80) °C. In Table 4 are introduced the heat capacity data measured by Perron et al.13 using the original prototype of the Picker flow microcalorimeter at (1.5, 5, 15, 35, and 45) °C and by Fortier et al.12 at 25 °C. In a later paper,29 this group observed that a small systematic difference exists between the heat capacities per unit volume determined with the commercial instruments and this prototype. Through various tests, the group concluded that commercial instruments give more reliable values. In ref 29, the correction method for the original data12,13 has been given. In the present study, the influence of this correction was investigated in the following way. The smoothed values for the apparent molar heat capacity of NaCl at rounded molalities obtained after the correction were used in this investigation. In ref 29, the following equation has been determined for this quantity up to a molality of 1 mol·kg−1 ∞ Capp = Cm,2 (DVPP) + qC ,1 m /mo + qC ,2(m /mo)

The values of the parameter representing the partial molar heat capacity at infinite dilution (i.e., C∞ m,2 (DVPP)) in this equation are given in Table 5 and the values of other parameters from refs 13 and 29 at the relevant temperatures are given in Table 6. Parameter qC,1 is a general parameter and qC,2 depends on the salt. In the present tests, the values from eq 14 are predicted with the parameter values obtained for eq 12 using paraTable 6. Parameter Values for eq 14 Determined by Perron et al.13 and Desnoyers et al.29 as a Function of Temperature t

(14) H

t/°C

qC,1/J·K−1·mol−1

qC,2/J·K−1·mol−1

1.5 5 15 25 35 45

21.23 22.52 25.84 28.95 32.13 35.56

57.8 50.0 27.8 15.6 7.8 3.1 DOI: 10.1021/acs.jced.7b00590 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

metrization PI. The results are shown in Figure 6 where eC,app errors (see eq 13) are presented as a function of the molality at

are compared to the predicted ones. The resulting error is defined by eC,part = ΔCpart(reported) − ΔCpart(predicted)

and this error is presented as a function of the molality. The results of these tests are displayed in Figure 7. Those for 25 °C are given in graph A where parametrization PI was used in the calculations. The following data have been included in this graph: Gulbransen and Robinson,31 Harned and Cook,32 Clarke and Glew,2 and Parker7 (where the data smoothed by Ensor and Anderson11 were used). Gulbransen and Robinson determined the partial molar heat capacities thermodynamically from their heat of dilution data, whereas Harned and Cook32 determined their values from results of the amalgam cell measurements by Harned and Nims33 at various temperatures. Up to 1.5 mol·kg−1, almost all of these data corroborate the PI calculations as well as the corresponding ΔCapp data in Figure 1. The amalgam cell data set considered by Harned and Cook32 is, however, an exception and does not give good heat capacity values even in dilute solutions. The review of Clarke and Glew,2 also reports partial molar heat capacities at various temperatures (see for example Table 1). These values are considered in graph A (t is 25 °C) and B (other temperatures up to 50 °C) where these values are predicted using parametrization PI. According to these graphs, the agreement between these two calculation methods is best in the temperature range from (15 to 40) °C. Parametrization PIII is tested in graph C of Figure 7 with the suggested partial molar heat capacities from the equations of Clarke and Glew2 and with the suggested values of Ensor and Anderson11 in the temperature range from (40 to 80) °C. The latter values were determined thermodynamically from the comprehensive heat of dilution data of these researchers. According to the graph, the partial heat capacities from refs 2 and 11 are quite close to each other at each temperature, implying that equations of Clarke and Glew 2 follow meticulously the values given by Ensor and Anderson.11 The problem in the error plots of graph C is that the best agreement with literature values and those resulting from parametrization PIII is obtained at 40 °C and not at the higher temperatures where PIII works better for the apparent molar heat capacities in graph B of Figure 5 and for enthalpy data in general in Part 16 of this study. In this light, it appears that the partial molar heat capacities tabulated by Ensor and Anderson in these higher temperatures are not completely correct. Recommended Values for the Relative Apparent Molar Heat Capacity and Relative Partial Molar Heat Capacity of NaCl in Aqueous Solutions. Relative apparent molar heat capacities for dilute NaCl solutions are given at rounded molalities in Table 7 up to 25 °C, and Table 8 displays these values in the range (30 to 50) °C. The corresponding values for the partial heat capacities have been collected in Tables 10 and 11, respectively. Almost all values in these four tables have been obtained using parametrization PI and are therefore fully traceable and transparent. The exception is the set of the values obtained by parametrization PIII for 50 °C in Tables 8 and 11. In Tables 9 and 12, we report traceable relative apparent and partial molar heat capacities for NaCl(aq) obtained also using parametrization PI in dilute aqueous solutions from (60 to 80) °C. Without question, the traceable values obtained by parametrization PI at the rounded molalities of Tables 7 to 12 are the most reliable ones determined so far

Figure 6. Plot of eC,app (eq 13), the deviation between the suggested relative apparent molar heat capacity for NaCl solutions by Desnoyers et al.29 and that predicted using parametrization PI of the present study (see text) at (1.5, 5, 15, 25, 35, and 45) °C as a function of molality m. The suggested values at rounded molalities have been based on the corrected heat capacities calculated from sets PFD1.5, PFD5, PFD15, FLD25, PFD35, and PFD45 in Table 4 and they have been calculated using eq 14 with the parameter values in Tables 5 and 6. The background of this correction is given in the text. The errors from the experimental values of Figure 2A have been included for sets PFD1.5 and PFD5. Symbols: ●, t = 1.5 °C; ○, PFD1.5; ▼, 5 °C; △, PDF5; ■, 15 °C; □, 25 °C; ⧫, 35 °C; ◊, 45 °C.

rounded molalities. In the calculations of these errors, the values for quantity C∞ m,2 are determined here as described in Table 5. In this figure, the errors for temperatures (15, 25, 35, and 45) °C are small and needs no comments. The errors for (1.5 and 5) °C are larger but they correspond closely to the errors from the original data of sets PFD1.5 and PFD5 in Figure 2A. This is illustrated in Figure 6 including also in this figure the errors of these experimental values. For the main conclusions of the present study, the experimental errors in the original sets of Fortier et al.12 and Perron et al.13 have no influence because the resulting corrections are small when compared to the accuracy of the heat capacity data at various temperatures. Because the corrected values suggested by Desnoyers et al.29 are, however, more reliable than the original values, they are considered here in Table 5 where partial molar heat capacities at infinite dilution are collected at different temperatures. This table contains also the values determined by Criss and Cobble30 on the basis of their heat of solution experiments. The agreement between the values is very satisfactory when overlapping occurs. However, the values from the last study30 seem to differ most from the other values. Tests of the New Values for Relative Partial Molar Heat Capacities with Those Presented in the Literature. In thermal literature, heat capacities in solutions are often expressed as the partial molar heat capacities of solute. These values can be easily calculated using eq 7 from the polynomials presented in eq 12 and the parameter values given in Table 3. Parametrizations PI and PIII were further tested here against the relative partial molar heat capacities suggested in the literature. In these tests, the reported ΔCpart values, defined by ∞ ΔCpart = ΔCm,2 = Cm,2 − Cm,2

(16)

(15) I



Article

Figure 7. Plot of eC,part (eq 16), the deviation between the suggested relative partial molar heat capacity for NaCl solutions and that predicted using parametrizations PI or PIII of the present study (see text) as a function of molality m. The suggested values are those presented by Gulbransen and Robinson,31 (graph A), Harned and Cook,32 (graph A), Clarke and Glew,2 (graphs A, B, and C), and Ensor and Anderson11 (graphs A and C). In graph A, the temperature is 25 °C and only parametrization PI was used in the calculations. In graph B, the temperatures are in the range from (0 to 50) °C and only parametrization PI was used in the calculations. In graph C, the temperature range is from (40 to 80) °C and only parametrization PIII was used in the calculations. Symbols for graph A: ●, Gulbransen and Robinson;31 ○, Harned and Cook;32 ▼, Clarke and Glew;2 △, Ensor and Anderson.11 Symbols for graph B (contains only data from the sets of Clarke and Glew2): ●, t = 0 °C; ○, 5 °C; ▼, 10 °C; △, 15 °C; ■, 20 °C; □, 30 °C; ⧫, 40 °C; ◊, 50 °C. Symbols for graph C: ●, Clarke and Glew,2 t = 40 °C; ○, Ensor and Anderson,11 40 °C; ▼, Clarke and Glew,2 50 °C; △, Ensor and Anderson,11 50 °C; ■, Clarke and Glew,2 60 °C; □, Ensor and Anderson,11 60 °C; ⧫, Clarke and Glew,2 70 °C; ◊, Ensor and Anderson,11 70 °C; ▲, Clarke and Glew,2 80 °C; ▽, Ensor and Anderson,11 80 °C. In the calculations of the data from Clarke and Glew, the results of the most dilute points have been omitted from the figure and the values given in Table 5 were used for C∞ m,2.

■

CONCLUSIONS On the basis of extensive testing against the existing highquality data presented in Part 16 and the present part (Part 2) of this study and in our previous work,14,17 we conclude that the experimental data for dilute NaCl solutions can often be predicted within experimental errors up to 1.0 mol·kg−1 in the temperature range (0 to 80) °C with a simple reparametrization of the Hückel equation. Completely traceable thermodynamic quantities can be obtained by using the new Hü ckel parametrization at least up to 0.2 mol·kg−1 in all of these temperatures. In Part 1 of this study, new values have been presented for the relative apparent and partial molar enthalpies, that is, for ΔHapp and ΔHm,2 where 2 refers to the solute, for these dilute solutions based on our simple model.6 Activity coefficient of NaCl and osmotic coefficient of water in aqueous NaCl solutions can also be reliably calculated using PI in all of these solutions. These values have been reported in Part 1. In this second part of the article, we report the values for the relative apparent and partial molar heat capacities, that is, for ΔCapp and ΔCpart. The tabulated values are congruent with all available high-quality experimental data and represent the most accurate values determined for this system up to date. For concentrated solutions, previously suggested alternative parametrizations in higher molalities yield better agreement

for NaCl solutions and no high-quality experimental data in the literature contradict these values. At temperatures above 40 °C, we have also examined here more concentrated solutions than those for which recommended values in the basis of parametrization PI are given in Tables 7, 8, 10, and 11. For these solutions in several cases, parametrization PIII gives a better agreement than PI in the interpretation of the existing calorimetric data. Tables 8 and 9 give at 50 °C the relative apparent and partial heat capacities for NaCl solutions at rounded molalities for these less dilute solutions based on PIII, respectively, and Tables 11 and 12 do so for temperatures (60, 70, and 80) °C. However, the heat capacity values in the higher molalities are not as reliable as those obtained for more dilute solutions on the basis of PI. At these molalities, furthermore, the values from PIII are not fully traceable like those from PI. It seems, however, that the heat capacity values for more concentrated solutions can also be useful for many applications because the existing experimental data are often well explained with these values, and the calculation of new values at different temperatures is simple when compared with calculation of the heat capacity values obtained from multiparameter equations.1−3 J



Article

Table 7. Recommended Values of the Relative Apparent Molar Heat Capacity (ΔCapp)a of Salt (Symbol 2) in Aqueous Sodium Chloride Solutions at Temperatures from (0 to 25) °C as a Function of Molality mb

Table 9. Recommended Values of the Relative Apparent Molar Heat Capacity (ΔCapp)a of Salt (Symbol 2) in Aqueous Sodium Chloride Solutions at Temperatures from (60 to 80) °C as a Function of Molality m

ΔCapp

ΔCapp

mc

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

mb

60 °Cc

60 °Cd

70 °Cc

70 °Cd

80 °Cc

80 °Cd

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.20 0.30 0.40 0.50 0.70 1.00 1.50

1.93 2.45 3.23 3.87 4.44 4.96 5.45 5.91 6.35 6.77 7.18 9.06 10.74

1.92 2.56 3.51 4.26 4.91 5.50 6.04 6.55 7.03 7.49 7.93 9.93 11.71 14.86

1.95 2.69 3.77 4.61 5.33 5.98 6.57 7.12 7.64 8.14 8.61 10.75 12.64 15.99 19.03

2.01 2.83 4.01 4.93 5.71 6.41 7.05 7.65 8.21 8.74 9.25 11.54 13.56 17.15 20.40 23.47

2.09 2.98 4.25 5.24 6.08 6.83 7.52 8.15 8.75 9.32 9.87 12.32 14.48 18.33 21.84 25.15 31.46 40.60 55.74

2.16 3.11 4.47 5.52 6.43 7.23 7.96 8.64 9.29 9.90 10.48 13.10 15.42 19.55 23.33 26.89 33.71 43.61 60.07

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.30 0.40 0.50 0.70 1.0 1.5

3.0 4.3 6.2 7.7 9.0 10.2 11.2 12.2 13.2 14.1 14.9 18.9 22.4

2.9 4.1 5.8 7.2 8.3 9.3 10.2 11.0 11.8 12.5 13.2 16.2 18.8 23.4 27.6 31.4 38.7 49.3 66.9

3.2 4.6 6.7 8.4 9.8 11.1 12.3 13.4 14.4 15.4 16.4 20.7 24.6

3.1 4.5 6.3 7.8 9.0 10.1 11.1 12.0 12.8 13.6 14.4 17.7 20.7 25.8 30.4 34.7 42.9 54.7 74.6

3.4 5.0 7.3 9.1 10.7 12.1 13.3 14.5 15.7 16.8 17.8 22.6

3.3 4.8 6.9 8.4 9.8 11.0 12.0 13.0 13.9 14.8 15.7 19.3 22.6 28.2 33.3 38.1 47.2 60.5 82.7

a Defined by equation ΔCapp = Capp − C∞ m,2 where Capp is defined in eq 6 and C∞ m,2 is the partial molar enthalpy at infinite dilution and the unit is J·K−1·mol−1. bThe values have been calculated using parametrization PI (see text). cUnit is mol·kg−1.

a Defined by equation ΔCapp = Capp − C∞ m,2 where Capp is defined in eq 6 and C∞ m,2 is the partial molar enthalpy at infinite dilution and the unit is J·K−1·mol−1. bUnit is mol·kg−1. cParametrization PI was used (see text). dParametrization PIII was used (see text).

Table 8. Recommended Values of the Relative Apparent Molar Heat Capacity (ΔCapp)a of Salt (Symbol 2) in Aqueous Sodium Chloride Solutions at Temperatures from (30 to 50) °C as a Function of Molality mb

Table 10. Recommended Values of the Relative Partial Molar Heat Capacity (ΔCpart)a of Salt (Symbol 2) in Aqueous Sodium Chloride Solutions at Temperatures from (0 to 25) °C as a Function of Molality mb

ΔCapp

ΔCpart

mc

30 °C

35 °C

40 °C

45 °C

50 °C

50 °Cd

mc

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.20 0.30 0.40 0.50 0.70 1.0 1.5

2.27 3.27 4.71 5.83 6.78 7.63 8.41 9.13 9.81 10.46 11.08 13.87 16.33 20.76 24.81 28.65 36.02 46.78 64.78

2.38 3.44 4.95 6.13 7.14 8.04 8.87 9.63 10.36 11.05 11.70 14.67 17.31 22.04 26.37 30.49 38.41 49.97 69.31

2.49 3.60 5.19 6.44 7.50 8.45 9.32 10.14 10.90 11.63 12.33 15.48 18.28 23.33 27.96 32.37 40.85 53.23

2.60 3.77 5.44 6.74 7.86 8.87 9.79 10.64 11.45 12.22 12.96 16.30 19.28 24.65 29.58 34.29 43.33

2.7 3.9 5.7 7.1 8.2 9.3 10.3 11.2 12.0 12.8 13.6 17.2 20.3 26.0

2.6 3.8 5.4 6.6 7.6 8.5 9.3 10.0 10.7 11.4 12.0 14.7 17.1 21.2 24.8 28.3 34.7 44.0 59.6

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.20 0.30 0.40 0.50 0.70 1.00 1.50

2.53 3.35 4.62 5.67 6.61 7.48 8.29 9.06 9.80 10.51 11.20 14.35 17.18

2.68 3.68 5.15 6.34 7.38 8.32 9.19 10.01 10.79 11.54 12.25 15.54 18.48 23.76

2.84 3.97 5.62 6.92 8.05 9.06 10.00 10.87 11.70 12.49 13.25 16.72 19.84 25.48 30.70

3.00 4.24 6.04 7.45 8.66 9.75 10.75 11.69 12.57 13.42 14.23 17.92 21.25 27.32 32.99 38.45

3.16 4.50 6.43 7.94 9.24 10.41 11.48 12.48 13.42 14.33 15.19 19.15 22.72 29.26 35.41 41.37 53.11 70.81 101.50

3.30 4.74 6.80 8.42 9.81 11.05 12.20 13.27 14.28 15.25 16.18 20.42 24.25 31.30 37.94 44.39 57.12 76.39 109.90

a Defined by equation ΔCapp = Capp − C∞ m,2 where Capp is defined in eq 6 and C∞ m,2 is the partial molar enthalpy at infinite dilution and the unit is J·K−1·mol−1. bThe values have been calculated using parametrization PI (see text). cUnit is mol·kg−1. dParametrization PIII was used (see text).

a Defined by equation ΔCpart = Cm,2 − C∞ m,2 where Cm,2 is the partial molar enthalpy of the salt and C∞ m,2 is its value at infinite dilution and the unit is J·K−1·mol−1. bThe values have been calculated using parametrization PI (see text). cUnit is mol·kg−1.

K



Article

with experiments than the parametrization that resulted to completely traceable values of these thermodynamic quantities. Concurrently, these alternative parametrizations were employed in the prediction of the recommended thermodynamic properties of NaCl solutions in Parts 1 and 2. Even though these parametrizations lack traceability and the values they predict are generally less reliable than the ones recommended for more dilute solutions, they are still useful for applications because the existing experimental data is often well reproduced. Furthermore, with all of the tested parametrizations, the calculation of new values at different temperatures is simple when compared with the calculation of enthalpy or heat capacity values from alternative multiparameter equations.1−3 Owing to the ubiquitous nature of NaCl solutions on earth and its wide industrial and medical applications, we believe that the availability of simple and accurate equations is important for the determination of these properties at different temperatures and molalities even if these equations lack the complete traceability.

Table 11. Recommended Values of the Relative Partial Molar Heat Capacity (ΔCpart)a of Salt (Symbol 2) in Aqueous Sodium Chloride Solutions at Temperatures from (30 to 50) °C as a Function of Molality mb ΔCpart mc

30 °C

35 °C

40 °C

45 °C

50 °C

50 °Cd

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.20 0.30 0.40 0.50 0.70 1.0 1.5

3.48 4.99 7.18 8.89 10.36 11.68 12.90 14.04 15.11 16.14 17.13 21.66 25.77 33.35 40.52 47.51 61.39 82.50 119.43

3.65 5.25 7.56 9.37 10.93 12.34 13.63 14.84 15.99 17.09 18.14 22.98 27.38 35.51 43.21 50.74 65.66 88.37 128.06

3.82 5.50 7.94 9.85 11.50 12.99 14.36 15.65 16.87 18.03 19.16 24.32 29.02 37.71 45.97 54.03 70.03 94.35

4.00 5.76 8.32 10.34 12.08 13.65 15.10 16.47 17.76 19.00 20.19 25.68 30.68 39.96 48.78 57.40 74.50

4.2 6.0 8.7 10.8 12.7 14.3 15.9 17.3 18.7 20.0 21.2 27.1 32.4 42.3

4.0 5.7 8.1 9.9 11.4 12.7 13.9 15.1 16.1 17.1 18.0 22.3 26.0 32.7 38.9 44.9 56.7 74.8 107.1

■

Corresponding Author

*E-mail: jpartane@lut.fi. ORCID

Jaakko I. Partanen: 0000-0002-4519-0958 Notes

a Defined by equation ΔCpart = Cm,2 − C∞ m,2 where Cm,2 is the partial molar enthalpy of the salt and C∞ m,2 is its value at infinite dilution and the unit is J·K−1·mol−1. bThe values have been calculated using parametrization PI (see text). cUnit is mol·kg−1. dParametrization PIII was used (see text).

The authors declare no competing financial interest.

■

ΔCpart m

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.30 0.40 0.50 0.70 1.0 1.5

60 °C 4.5 6.6 9.5 11.9 13.9 15.7 17.4 19.0 20.6 22.0 23.4 30.0 35.9

c

60 °C

d

4.4 6.2 8.8 10.8 12.5 13.9 15.3 16.5 17.7 18.8 19.9 24.6 28.8 36.4 43.4 50.2 63.7 84.2 121

70 °C 4.9 7.1 10.4 12.9 15.2 17.2 19.1 20.8 22.5 24.2 25.7 33.0 39.6

c

70 °Cd

80 °Cc

80 °Cd

4.7 6.7 9.6 11.7 13.6 15.2 16.7 18.1 19.4 20.6 21.8 27.0 31.7 40.2 48.1 55.8 70.9 94.1 135

5.3 7.7 11.3 14.1 16.5 18.7 20.8 22.7 24.6 26.4 28.1 36.1

5.1 7.3 10.4 12.8 14.8 16.6 18.2 19.7 21.1 22.5 23.8 29.6 34.8 44.2 53.0 61.5 78.5 104 150

REFERENCES

(1) Pitzer, K. S.; Peiper, J. C.; Busey, R. H. Thermodynamic properties of aqueous sodium chloride solutions. J. Phys. Chem. Ref. Data 1984, 13, 1−102. (2) Clarke, E. C. W.; Glew, D. N. Evaluation of the thermodynamic functions for aqueous sodium chloride from equilibrium and calorimetric measurements below 154 °C. J. Phys. Chem. Ref. Data 1985, 14, 489−610. (3) Archer, D. G. Thermodynamic properties of the NaCl+H2O system. II. Thermodynamic properties of NaCl(aq), NaCl·2H2O(cr), and phase equilibria. J. Phys. Chem. Ref. Data 1992, 21, 793−829. (4) Pitzer, K. S. Thermodynamics of electrolytes. I. Theoretical basis and general equations. J. Phys. Chem. 1973, 77, 268−277. (5) Hasan, M.; Partanen, J. I.; Vahteristo, K. P.; Louhi-Kultanen, M. Determination of the Pitzer interaction parameters at 273.15 K from the freezing-point data available for NaCl and KCl solutions. Ind. Eng. Chem. Res. 2014, 53, 5608−5616. (6) Partanen, J. I.; Partanen, L. J.; Vahteristo, K. P. Traceable thermodynamic quantities for dilute aqueous sodium chloride solutions at temperatures from (0 to 80) °C. Part 1. Activity coefficient, osmotic coefficient, and the quantities associated with the partial molar enthalpy. J. Chem. Eng. Data 2017, 62, 2617−2632. (7) Parker, V. B. Thermal properties of aqueous uni-univalent electrolytes. Natl. Stand. Ref. Data Ser. − Natl. Bur. Stand. 2 1965, 1− 66. (8) Criss, C. M.; Millero, F. J. Modeling the heat capacities of aqueous 1−1 electrolyte solutions with Pitzer’s equations. J. Phys. Chem. 1996, 100, 1288−1294. (9) Criss, C. M.; Millero, F. J. Modeling the heat capacities of high valence−type electrolyte solutions with Pitzer’s equations. J. Solution Chem. 1999, 28, 849−864. (10) Silvester, L. F.; Pitzer, K. S. Thermodynamics of electrolytes. 8. High-temperature properties, including enthalpy and heat capacity, with application to sodium chloride. J. Phys. Chem. 1977, 81, 1822− 1828. (11) Ensor, D. D.; Anderson, H. L. Heats of dilution of NaCl: temperature dependence. J. Chem. Eng. Data 1973, 18, 205−212.

Table 12. Recommended Values of the Relative Partial Molar Heat Capacity (ΔCpart)a of Salt (Symbol 2) in Aqueous Sodium Chloride Solutions at Temperatures from (60 to 80) °C as a Function of Molality m. b

AUTHOR INFORMATION

Defined by equation ΔCpart = Cm,2 − C∞ m,2 where Cm,2 is the partial molar heat capacity of the salt and C∞ m,2 its value at infinite dilution and the unit is J·K−1·mol−1. bUnit is mol·kg−1. cParametrization PI was used (see text). dParametrization PIII was used (see text). a

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Article

(12) Fortier, J. L.; Leduc, P. A.; Desnoyers, J. E. Thermodynamic properties of alkali halides. II. Enthalpies of dilution and heat capacities in water at 25 °C. J. Solution Chem. 1974, 3, 323−349. (13) Perron, G.; Fortier, J. L.; Desnoyers, J. E. The apparent molar heat capacities and volumes of aqueous NaCl from 0.01 to 3 mol·kg−1 in the temperature range 274.65 to 318.15 K. J. Chem. Thermodyn. 1975, 7, 1177−1184. (14) Partanen, J. I.; Covington, A. K. Re-evaluation of the thermodynamic activity quantities in aqueous sodium and potassium chloride solutions at 25 °C. J. Chem. Eng. Data 2009, 54, 208−219. (15) Rowland, D.; May, P. M. Thermodynamics of strong aqueous electrolyte solutions at t = 25 °C described by the Hückel equations. J. Chem. Eng. Data 2014, 59, 2030−2039. (16) Partanen, J. I. Traceable mean activity coefficients and osmotic coefficients in aqueous calcium chloride solutions at 25 °C up to a molality of 3.0 mol·kg−1. J. Chem. Eng. Data 2012, 57, 3247−3257. (17) Partanen, J. I. Mean activity coefficients and osmotic coefficients in dilute aqueous sodium or potassium chloride solutions at temperatures from (0 to 70) °C. J. Chem. Eng. Data 2016, 61, 286− 306. (18) Archer, D. G.; Wang, P. The dielectric constant of water and Debye−Hückel limiting law slopes. J. Phys. Chem. Ref. Data 1990, 19, 371−411. (19) Marcus, Y. Ion Properties; Marcel Dekker Inc.: New York, U.S.A., 1977. (20) Archer, D. G.; Carter, R. W. Thermodynamic properties of the NaCl + H2O system. 4. Heat capacities of H2O and NaCl(aq) in coldstable and supercooled states. J. Phys. Chem. B 2000, 104, 8563−8584. (21) Tanner, J. E.; Lamb, F. W. Specific heats of aqueous solutions of NaCl, NaBr, and KCl: comparisons with related thermal properties. J. Solution Chem. 1978, 7, 303−316. (22) Allred, G. C.; Woolley, E. M. Heat capacities of aqueous HCl, NaOH, and NaCl at 283.15, 298.15 and 313.15 K: ΔCpo for ionization of water. J. Chem. Thermodyn. 1981, 13, 147−154. (23) Randall, M.; Rossini, F. D. Heat capacities in aqueous salt solutions. J. Am. Chem. Soc. 1929, 51, 323−345. (24) Simard, M. A.; Fortier, J. L. Heat capacity measurements of liquids with a Picker mixing flow microcalorimeter. Can. J. Chem. 1981, 59, 3208−3211. (25) Alary, J. F.; Simard, M. A.; Dumont, J.; Jolicoeur, C. Simultaneous flow measurement of specific heats and thermal expansion coefficients of liquids: aqueous t-BuOH mixtures and neat alkanols and alkanediols at 25 °C. J. Solution Chem. 1982, 11, 755− 776. (26) Olofsson, I. V. Apparent molar heat capacities and volumes of aqueous NaCl, KCl, and KNO3 at 298.15 K. Comparison of Picker flow calorimeter with other calorimeters. J. Chem. Thermodyn. 1979, 11, 1005−1014. (27) Gibbard, H. F., Jr.; Scatchard, G.; Rousseau, R. A.; Creek, J. L. Liquid-vapor equilibrium of aqueous sodium chloride, from 298 to 373 K and from 1 to 6 mol·kg−1, and related properties. J. Chem. Eng. Data 1974, 19, 281−288. (28) Messikomer, E. E.; Wood, R. H. The enthalpy of dilution of aqueous sodium chloride at 298.15 to 373.15 K, measured with a flow calorimeter. J. Chem. Thermodyn. 1975, 7, 119−130. (29) Desnoyers, J. E.; de Visser, C.; Perron, G.; Picker, P. Reexamination of the heat capacities obtained by flow microcalorimetry. Recommendation for the use of a chemical standard. J. Solution Chem. 1976, 5, 605−618. (30) Criss, C. M.; Cobble, J. W. The thermodynamic properties of high temperature aqueous solutions. I. Standard partial molal heat capacities of sodium chloride and barium chloride from 0 to 100 °C. J. Am. Chem. Soc. 1961, 83, 3223−3228. (31) Gulbransen, E. A.; Robinson, A. L. The integral heats of dilution, relative partial molal heat contents and heat capacities of dilute aqueous sodium chloride solutions. J. Am. Chem. Soc. 1934, 56, 2637−2641.

(32) Harned, H. S.; Cook, M. A. The thermodynamics of aqueous sodium chloride solutions from 0 to 40 °C from electromotive force measurements. J. Am. Chem. Soc. 1939, 61, 495−497. (33) Harned, H. S.; Nims, L. F. The thermodynamic properties of aqueous sodium chloride solutions from 0 to 40°. J. Am. Chem. Soc. 1932, 54, 423−432.

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Traceable Thermodynamic Quantities for Dilute Aqueous Sodium

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