Traceable Thermodynamic Quantities for Dilute Aqueous NaCl

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

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Traceable Thermodynamic Quantities for Dilute Aqueous NaCl Solutions at Temperatures from (353.15 to 383.15) K and for Dilute Aqueous KCl Solutions from (273.15 to 383.15) K Jaakko I. Partanen,*,† Lauri J. Partanen,‡ and Kari P. Vahteristo† †

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Department of Chemical Technology, LUT School of Engineering Science, Lappeenranta University of Technology, P.O. Box 20, FI-53851 Lappeenranta, Finland ‡ Department of Chemistry and Materials Science, Aalto University of Helsinki, P.O. Box 16100, FI-00076 Aalto, Finland ABSTRACT: We present fully traceable two-parameter Hückel equations with parameters B and b1 for the activity coefficient of sodium chloride and for the osmotic coefficient of water in aqueous NaCl solutions at temperatures from (353.15 to 383.15) K. In our most successful parametrization of these equations, parameter B is treated as a constant whereas b1 is a quadratic function of the temperature. The new calculations extend the tables presented up to 353.15 K in our previous study (Partanen, J. I.; Partanen, L. J.; Vahteristo, K. P. J. Chem. Eng. Data 2017, 62, 2617−2632), showing that our Hückel equations apply up to 383.15 K. We also present fully traceable activity quantities from (273.15 to 383.15) K for dilute KCl solutions. Our equations apply to these solutions at least up to a molality of 0.2 mol·kg−1. This article contains the most important new test results of the Hückel equations, where the literature data have been obtained by direct and isopiestic vapor measurements and boiling-point determinations at various pressures. In forthcoming studies, the test results obtained using these equations from KCl solutions will be published for the existing calorimetric data. On the basis of the extensive testing of our models against the experimental data available, the new tables contain the most reliable values of activity and osmotic coefficients for dilute NaCl and KCl solutions published so far. Finally, we report vapor pressures of water in NaCl and KCl solutions above 383 K up to 473 K. These values are obtained with a new and simplified parametrization of the Hückel equation, and our tests show that the parametrization applies well up to a molality of 6 mol·kg−1 in both cases at the high temperatures.

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INTRODUCTION Because of its role as the main salt component in seawater, in biological solutions, and in many industrially important systems, sodium chloride is one of the most important electrolytes on earth. Therefore, its thermodynamic properties in aqueous solutions have been thoroughly measured in different solute concentrations and in wide ranges of temperature and pressure. (See reviews by Pitzer et al.,1 Clarke and Glew,2 and Archer.3) Similarly, potassium chloride is an important electrolyte both biologically and industrially, and hence the thermodynamics of its solutions under various conditions has been the target of intense experimental study. (See the review by Archer.4) The experimental data of these solutions have then served as the basis for models such as the one proposed by Pitzer5 that help to explain the data and enable the estimation of various thermodynamic properties under conditions where no measurements are yet available. In particular, the Pitzer model serves as the basis in the multiparameter equations used in the thermodynamic treatments of these four extensive reviews.1−4 For practical considerations, these reported equations are often extremely complicated, containing numerous parameters associated with the three fundamental variables (i.e., temperature, pressure, and concentration). However, in recent studies © XXXX American Chemical Society

(e.g., refs 6 and 7) it has been shown that at the normal reference temperature of 298.15 K the simple Hückel equation is equally applicable with the multiparameter equations in the interpretation of the thermodynamic results of dilute solutions of pure electrolytes. In our previous studies,8−10 we have demonstrated that the Hückel equation also applies well to experimental data from various sources in the temperature range of (273.15 to 353.15) K. We have suggested several parametrizations for the temperature dependence of this equation and showed that the best parametrization can predict all calorimetric data reported in the literature up to 353 K for dilute NaCl solutions, even though none of these data were used in the parameter estimation. In the present study, we show that the same parametrization applies to the thermodynamic properties of dilute NaCl solutions from (353 to 383) K. Similar high-quality results are obtained when an analogous parametrization is tested for dilute KCl solutions. The results of the thermodynamic data solved at equilibrium at a constant or at an almost-constant temperature are considered in Received: May 21, 2018 Accepted: December 4, 2018

A

DOI: 10.1021/acs.jced.8b00423 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

Table 1. Debye−Hückel Parameter α,a Vapor Pressure of Pure Water (p1*),b Molar Volume of Liquid Water [Vm,1 * (liq)],b and c Second Virial Coefficient of Water Vapor (β) as Functions of the Temperature (T) in the Range from (273.15 to 373.15) K T/K

α/(mol·kg−1)−1/2

273.15 278.15 283.15 288.15 293.15 294.45 298.15 299.25 303.15 304.05 308.15 308.95 310.65 313.15 313.75 318.15 318.65 323.15 323.55 328.15 328.45 333.15 333.35 335.65 338.15 338.35 343.15 348.05 348.15 352.95 353.15 357.85 358.15 360.65 362.85 363.15 368.05 368.15 372.75 373.15

1.1293 1.1376 1.1462 1.1552 1.1646 1.1668d 1.1744 1.1764d 1.1848 1.1865d 1.1956 1.1972d 1.2011d 1.2068 1.2081d 1.2186 1.2197d 1.2308 1.2317d 1.2436 1.2442d 1.2568 1.2572d 1.2634d 1.2704 1.2709d 1.2846 1.2988d 1.2992 1.3136d 1.3143 1.3289d 1.3299 1.338d 1.345d 1.346 1.362d 1.363 1.378d 1.380

p1*/Pa

Vm,1 * (liq)/cm3·mol−1

β/cm3·mol−1

1227.6 1705.1 2338.4 2524.9e 3168.6 3373.5e 4245.1 4460e

18.048 18.055f 18.068 18.079f 18.095 18.105f

−1382 −1350 −1262 −1238 −1156 −1139

5869e 6458e

18.135f 18.147f

−1050 −1020

7608e

18.168f

−972

9823e 12 345 12 570e

18203f 18.239 18.242f

−901 −841 −837

15 951e 19 933 20 079e 22 361e

18.283f 18.326 18.327f 18.350f

−780 −730 −729 −706

25 196e 31 177 38 312e 38 564 46 875e 47 375 56 992e

18.376f 18.425 18.477f 18.480 18.533f 18.538 18.591f

−681 −640 −602 −601 −567 −565 −535

63 675e 69 140e 70 120 83 993e

18.627f 18.654f 18.662 18.722f

−518 −505 −503 −477

99 794e 101 325

18.789f 18.798

−453 −451

a Defined in eq 14 and the given values presented by Archer and Wang27 for the standard pressure of 101.325 kPa. bGiven by Kell.20 cCalculated from eq 6. dCalculated from eq 9. eCalculated from eq 8. fCalculated from eq 7.

this article, while both enthalpy and heat-capacity results in KCl solutions will be reported in the future analogously to NaCl results in refs 9 and 10. It is important to emphasize that the best parametrization considered here and in the previous studies8−10 is fully traceable and transparent. A critical reader can reproduce all calculations using the results presented in this study or in our previous studies, with the appropriate experimental data given in the literature. In thermodynamic studies of electrolyte solutions, this is not common because the main source of data in these studies is the isopiestic method. Because of its comparative nature, this method requires that the thermodynamic activities in the solution of the reference electrolyte are known. Unfortunately, the thermodynamic methods that give activities for solutions of a pure electrolyte are not as accurate as the isopiestic method over wide ranges of concentration, temperature, and pressure. In ref

6, we have determined for solutions of common reference electrolytes NaCl and KCl fully traceable activity and osmotic coefficients at 298.15 K and have used those in all following isopiestic studies published after that paper.

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THEORY General Equations for the Activity Quantities. In aqueous solutions of many salts, the following Hückel equations can be used to predict the mean activity coefficient (γ) of the salt and the osmotic coefficient (ϕ) of water at least up to an ionic strength (Im) of 1 mol·kg−1 (see refs 6 and 8−16; the whole list of our articles is published in ref 9): ln γ = − B

α|z+z −| Im 1 + B Im

imy + b1jjj o zzz km {

(1)


Journal of Chemical & Engineering Data ÄÅ α|z+z −| ÅÅÅ ÅÅ(1 + B Im ) − 2 ln(1 + B Im ) φ=1− B3Im ÅÅÅÅÇ ÑÉÑ 1 ÑÑÑ + b1 ijj m yzz − Ñ j z 2 k mo { 1 + B Im ÑÑÑÑÖ

Article

α1.0 −1 −1/2

(mol ·kg )

(2)

α5.0

In eqs 1 and 2, m is the molality, z+ is the charge number of the cation and z− is that of the anion, α is the Debye−Hückel parameter, and B and b1 are the electrolyte-dependent parameters. The values of α at 101.325 kPa and at temperatures from (273 to 373) K are given in Table 1. These values have been taken from ref 17. For a 1:1 electrolyte such as NaCl/KCl, | z+z−| = 1 and Im = m. The osmotic coefficient is related to the activity of water (a1, with subscript 1 used here for the solvent) in pure solutions of a uniunivalent electrolyte by the following thermodynamic identity ln a1 = − 2mM1φ

−1 −1/2

where M1 is the molar mass of water (= 0.018015 kg·mol−1). The activity of water is associated with its vapor pressure over the solution (p1) and that over pure water (p1*) through

(4)

where β is the second virial coefficient of water vapor and Vm,1 * (liq) is the molar volume of liquid water. When the temperature is not very high and the solutions are not very concentrated, the second term on the right is small and eq 4 simplifies to a1 =

p1 p*

2 i T − T0 zy zz + 10.3 × 10−6jjj k K {

i T − T0 yz zz = 1.1311 + 1.348 × 10−3jjj k K {

(10)

2 i T − T0 yz zz + 10.9 × 10−6jjj (11) k K { where symbol p° in eq 8 unit of pressure (i.e., p° = 1 Pa) and T0 in eqs 7, 9, 10, and 11 is 273.15 K. Equation 6 for the second virial coefficient of water vapor was presented by Keyes et al.,18 and it was taken in the form used by Gibbard et al.19 in the analysis of the vapor pressure data measured at several temperatures in NaCl solutions. Equation 7 for the molar volume of liquid water in the saturated state was determined in the present study from the density values reported by Kell20 from (283.15 to 423.15) K in intervals of 5 K. The standard deviation about the regression of this fit is 0.004 cm3· mol−1. Equation 8 was determined from the 22 vapor pressure points reported by Gardner et al.21 in the temperature range of (395.6 to 543.4) K supplemented with 19 points from the table of Kell20 in the range of (298.15 to 423.15) K. The error plot for this equation is presented in Figure 1 where the relative vapor

(mol ·kg )

(3)

* (liq)](p − p* ) ij p yz [β − V m,1 1 1 ln a1 = lnjjjj 1 zzzz + j p* z RT k 1{

i T − T0 zy zz = 1.1307 + 1.454 × 10−3jjj k K {

(5)

1

We have employed this form in our past studies and in some calculations in this article. In contrast, at temperatures higher than 373 K and in concentrated solutions, the relevant parameter values for the full eq 4 and for eqs 1 and 2 are obtained using the following equations Figure 1. Plot of ew/p* [= (p*1,suggested − p*1,predicted)/p*1,suggested], the relative deviation between the literature value of the vapor pressure of pure water and that predicted by eq 8 as a function of temperature T. The literature values were taken from the data of Kell20 (●) and Gardner et al.21 (○). The point where T = 477.0 K from ref 21 is outside the scale of the figure. Its ew/p* value is −0.0164.

T 2

β 1080870/ ( K ) 34.0 47549 = − · T /K cm 3·mol−1

* (liq) V m,1 3

−1

cm ·mol

i T − T0 yz zz = 17.982 + 2.161 × 10−3jjj k K { 2 i T − T0 zy zz + 59.6 × 10−6jjj k K {

2 ij p* yz iK y iK y lnjjjj 1o zzzz = 23.02129 − 3611.77jjj zzz − 253000jjj zzz jp z kT { kT { k {

α0.1 −1 −1/2

(mol ·kg )

(6)

pressure deviation for pure water is first calculated by dividing the difference between the suggested and predicted values (= ew) by the suggested vapor pressure, and this relative value is then presented as a function of the temperature. As can be seen in the figure, the values from ref 21 are not as precise as those from ref 20. Kell’s vapor pressures20 were included in the fitting to obtain with this equation very reliable values below 373 K. We use eq 8 here mainly for temperatures above 373 K, and at this temperature and below the exact values given in ref 20 were used when possible. When an interpolation was needed at these lower temperatures, eq 8 was also used. Below 373 K, the values reported by Kell20 are listed in Table 1 of the present study. Usually in the technical literature the vapor pressure of water is fitted using either the Antoine equation

(7)

(8)

i T − T0 yz zz = 1.1296 + 1.550 × 10−3jjj k K { 2 i T − T0 zy zz + 9.5 × 10−6jjj k K {

(9) C



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Table 2. Debye−Hückel Parameter αa, the Vapor Pressure of Pure Water (p1*)b, the Molar Volume of Liquid Water [Vm,1 * (liq)]c, d and the Second Virial Coefficient of Water Vapor (β) as Functions of the Temperature (T) above 373.15 K at the Saturated State T/K

α/(mol·kg−1)−1/2

p1*/kPa

378.15 382.05 383.15 393.2 394.3 395.6 398.2 399.0 413.2 413.9 415.0 422.2 423.2 423.8 425.5 427.1 433.2 440.2 445.5 448.2 449.4 450.2 453.2 453.8 454.6 472.6 473.2 474.1 477.0 479.7 480.1 482.2 483.2 483.7 495.6 496.2 504.2 512.4

1.397 1.411 1.415 1.452 1.457 1.462 1.472 1.475 1.534 1.537 1.542 1.573 1.577 1.581 1.588 1.596 1.624 1.659 1.686 1.700 1.705 1.710 1.726 1.729 1.733 1.830 1.834 1.838 1.855 1.871 1.873 1.885 1.891 1.894 1.965 1.969 2.020 2.072

120.8 137.7 143.6 198.3 205.3 213.9 231.9 237.6 361.2 367.9 380.0 463.2 475.9 483.6 506.0 527.8 618.1 736.4 836.4 892.9 918.5 935.9 1003 1017 1036 1537 1556 1584 1684 1779 1794 1871 1909 1928 2428 2456 2847 3295

Vm,1 * (liq)/cm3·mol−1

β/cm3·mol−1

18.924

−412

19.098 19.118 19.138 19.182 19.196 19.451 19.465 19.487 19.625 19.645 19.657 19.691 19.724 19.851 20.005 20.123 20.183 20.210 20.229 20.300 20.313 20.332 20.779 20.794 20.820 20.894 20.970 20.976 21.033 21.060 21.073 21.406 21.423 21.662 21.903

−369 −366 −361 −353 −350 −309 −307 −304 −286 −284 −283 −279 −275 −262 −248 −239 −234 −232 −231 −225 −225 −224 −198 −197 −196 −192 −189 −188 −186 −185 −184 −171 −170 −162 −155

a

Obtained from eq 9, 10, or 11, and interpolation was used when necessary. bCalculated from eq 8. cCalculated from eq 7. dCalculated from eq 6.

ij p* yz B lnjjjj 1o zzzz = A − jp z C+T k {

The source of data for eqs 9, 10, and 11 are the tables of Archer and Wang.17 The theoretical equation for Debye− Hückel parameter α takes the form

(12)

or the Wagner equation

α=

(1 − Tr) (1 − Tr)1.5 (1 − Tr)3 +B +C ln(pr ) = A Tr Tr Tr (1 − Tr) Tr

8π (ε1ε0RT )3/2

(14)

where F is the Faraday constant, R is the gas constant, e is the proton charge, ε0 is the dielectric constant of vacuum, ρ*1 is the density of pure water, and ε1 is the relative dielectric constant of water. The problematic quantity in this equation is ε1, and it does not depend on the temperature or pressure in a simple way. The determination of these dependences has been reviewed in ref 17. Because no simple equation is possible to present for the dielectric constant of water, here the temperature dependences of α at three pressures were estimated using the tables of Archer and Wang.17 Equation 9 was determined from α values at the atmospheric pressure of 0.101325 MPa. Points from (273.15 to 373.15) K at intervals of 10 K were included in its estimation.

6

+D

2ρ1* F 2e

(13)

where A, B, C, and D are constants in both equations and symbols pr and Tr in the Wagner equation refer to the reduced pressure and temperature of water, respectively. For the purposes of the present study, however, eq 8 is better because it is functionally closer to the theoretical Clausius−Clapeyron equation. D



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Figure 2. Graphs A−C show plots of ep (eq 20), the deviation between the literature value of the vapor pressure and that predicted by present parametrizations for NaCl solutions, as a function of molality m. In graph D, the relative deviation ep/p*, where p* (= p*1 ) is the vapor pressure of pure water at the tested temperature, is presented in the same way. The vapor pressures were predicted by eqs 2, 3, and 4 using parametrizations PI or PIII for the osmotic coefficients. Symbols for graph A where the data were measured by Gibbard et al.19 and only PI was used: ●, T = 298.15 K; ○, 310.65 K; ▼, 323.15 K; △, 335.65 K; and ■, 348.15 K. Symbols for graph B where only PI was used: ●, Gibbard et al.,19 T = 298.15 K; ○, Pearce and Nelson,30 298.15 K; ▼, Olynyk and Gordon,31 293.15 K; △, Olynyk and Gordon,31 298.15 K; and ■, Olynyk and Gordon,31 303.15 K. Symbols for graph C where the data were measured by Gibbard et al.:19 ●, T = 348.15 K, PI; ○, 348.15 K, PIII; ▼, 360.65 K, PI; and △, 373.15 K, PI. Symbols for graph D where the values of p* (= p1*) are given in Table 1 and all data were measured by Gibbard et al.:19 ●, T = 298.15 K, PI; ○, 348.15 K, PI; ▼, 360.65, PI; △, 373.15 K, PI; and ■, 348.15 K, PIII.

Equation 10 was determined from the α values at a pressure of 1.0 MPa with points in the range from (273 to 443) K over the same intervals. Finally, eq 11 was determined in the same way from the α values at 5.0 MPa using points from (273 to 473) K. The standard deviations of the regressions of these fits are 0.00015, 0.0012, and 0.0029 (mol·kg−1)−1/2, respectively. These equations are used in the present study for the interpolations or extrapolations of α values above 373 K to various pressures. In our calculations, the α values do not depend sensitively on the pressure, and our evaluations for these calculations are probably completely reliable. The resulting values for β, Vm,1 * (liq), p1*, and α at temperatures higher than 373 K are reported in Table 2. Parametrizations for NaCl Solutions. We have observed in our previous studies6,8−10 that a constant value of 1.4 (mol· kg−1)−1/2 for parameter B can be used at all temperatures from (273 to 353) K for dilute NaCl solutions. In our most successful parametrization (PI) of the Hückel equation, the following quadratic equation was determined for parameter b1 (refs 9 and 10) i T − T0 yz zz b1,NaCl = 0.0077 + 3.1853 × 10−3jjj k K { 2 i T − T0 zy zz − 25.17 × 10−6jjj k K {

where T0 = 273.15 K. This equation was based on the following three values for parameter b1: b1(273.15 K) = 0.0077 (determined in ref 8), b1(298.15 K) = 0.0716 (ref 6), and b1(348.15 K) = 0.105 (ref 9). This equation explains within experimental error the existing thermodynamic data at least up to 0.2 mol·kg−1 in the temperature interval of (273 to 353) K for NaCl solutions. In the present study, we will show that this equation also applies well up to temperatures slightly above 373 K. In ref 9, we observed that PI does not apply well to the existing enthalpy data above 333 K in less dilute solutions. Therefore, the following cubic equation was determined there for parameter b1 for these higher temperatures i T − T0 yz zz b1,NaCl = 0.0077 + 3.3673 × 10−3jjj k K {

2 3 i T − T0 yz i T − T0 yz zz + 161 × 10−9jjj zz − 36.48 × 10−6jjj k K { k K {

(16)

It was based on the values of b1(273.15 K) = 0.0077, b1(298.15 K) = 0.0716, b1(323.15 K) = 0.105, and b1(348.15 K) = 0.123. The b1 value at 323.15 K was selected to be almost the same as that obtained from eq 15 at this temperature, but at 348.15 K, the values from eqs 15 and 16 differ and the value from eq 16

(15) E



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Figure 3. Plot of ep (eq 20), the deviation between the vapor pressure from Hubert et al.32 and that predicted by parametrization PI for NaCl solutions as a function of molality m. The vapor pressures were predicted by eqs 2, 3, and 4 using PI for the osmotic coefficients. Symbols for graph A: ●, T = 294.45 K; ○, 299.25 K; ▼, 304.05 K; △, 308.95 K; ■, 313.75 K; □, 318.65 K; ⧫, 323.55 K; and ◇, 328.45 K. Symbols for graph B: △, T = 308.95 K; ■, 313.75 K; □, 318.65 K; ●, 333.35 K; ○, 343.15 K; ▼, 348.05 K; and ▲, 352.95 K. Symbols for graph C: ⧫, T = 323.55 K; ◇, 328.45 K; ●, 338.35 K; ○, 343.15 K; ▲, 352.95 K; ▼, 357.85 K; ■, 362.85 K; and □, 368.05 K. The point where m = 5.423 mol·kg−1 at 362.85 K and that where m = 4.086 mol· kg−1 at 368.05 K are outside the scale of graph C. These errors are −893 and −708 Pa, respectively.

results in a better fit with the heat-of-dilution data from Ensor and Anderson22 above 333 K. We named the parametrization based on eq 16 in ref 9 PIII. As observed in ref 10, the heat capacity data from less dilute NaCl solutions could also be explained better using parametrization PIII than PI at high temperatures. Thus, PIII will also be considered below. From eq 15, it follows that the value of parameter b1 is zero when T = 402 K. As around this region, the value of b1 can expected to be small, here we also tested the following simplified parametrization of eq 2 where B = 1.4 (mol·kg−1)−1/2 and b1 = 0 for the osmotic coefficients in NaCl solutions above 383 K ÅÄ α ÅÅÅÅ φ = 1 − 3 ÅÅ(1 + B Im ) − 2 ln(1 + B Im ) B Im ÅÅÅÇ ÉÑ ÑÑ 1 ÑÑ − Ñ 1 + B Im ÑÑÑÑÖ (17)

al.27 at 353.15 K. In addition, the smoothed isopiestic ratios for NaCl and KCl solutions have been published by Hellams et al.28 at 318.15 K. These isopiestic data were included in the tests of ref 9. At 353.15 K for KCl solutions, the parameter values of BKCl = 1.3 (mol·kg−1)−1/2 and b1,KCl = 0.044 were successfully used to predict the isopiestic data of ref 27. For this value of BKCl and with the values of b1(353.15 K) = 0.0440, b1(273.15 K) = −0.0515 (determined in ref 8), and b1(298.15 K) = 0.0110 (ref 6) for KCl, the following quadratic equation was determined here for the temperature dependence for dilute KCl solutions i T − T0 yz zz b1,KCl = −0.0515 + 3.09375 × 10−3jjj k K { 2 i T − T0 yz zz − 23.75 × 10−6jjj k K {

(18)

This equation corresponds to eq 15 for NaCl solutions, and the calculations associated here with this equation are thus called the calculations of parametrization PI. For temperatures above 373 K, a simple equation (akin to eq 17 for NaCl solutions) was also considered in the KCl case. This estimation was based on eq 17 for NaCl solutions, on the value of BKCl = 1.3 (mol·kg−1)−1/2, and on the isopiestic data measured by Holmes et al.29 for NaCl and KCl solutions. At these high temperatures, the data set consists of measurements starting from 382 K and extending up to 474 K. The final equation for b1,KCl in eq 2 for high temperatures is

As will be shown, this equation applies much better than parametrization PI to the high-temperature vapor pressure data of Gardner et al.,21 Gardner,23 and Liu and Lindsay24 for temperatures above 383 K. In addition, this simple parametrization works quite well in concentrated NaCl solutions. The parametrization represented by eq 17 is called PIVNaCl, and eq 17 is used here only for temperatures above this 383 K. Parametrizations for KCl Solutions. As discussed in ref 9, the following three wide isopiestic sets have been reported in the literature for NaCl and KCl solutions at temperatures higher than the normal reference temperature of 298.15 K: Davis et al.25 at 318.15 K, Humphries et al.26 at 333.15 K, and Moore et

i T − T0 yz zz b1,KCl = −0.05121 + 158.5 × 10−6jjj k K {

F

(19)



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Figure 4. Graphs A and B plot ep (eq 20), the deviation between the literature values of the vapor pressure measured by Liu and Lindsay24 and that predicted by the present parametrizations for NaCl solutions, as a function of molality m. In graph C, the relative deviation ep/p* where p* (= p1*) is the vapor pressure of pure water at the tested temperature is presented as a function of temperature T from the vapor pressures measured by Gardner et al.21,23 The values of the predicted vapor pressure were calculated using eqs 2, 3, and 4. Symbols for graphs A and B, where graph A was obtained using PI and graph B was obtained using PIV: ●, T = 398.2 K; ○, 423.2 K; ▼, 448.2 K; and △, 473.2 K. Symbols for graph C, where the values of p*1 are given in Table 2 and only PIV was used: ●, m = 0.5 mol·kg−1, ref 23; ○, 1 mol·kg−1, ref 21; ▼, 1 mol·kg−1, ref 21; △, 2 mol·kg−1, ref 21; and ■, 3 mol·kg−1, ref 21. In ref 23, the point where m = 1.005 mol·kg−1 and T = 415.0 K was omitted in graph C as an outlier.

Details of this estimation are presented below, and we symbolize the calculations associated with eq 19 by PIVKCl. Equation 19 is used here only for temperatures above 393 K.

■

RESULTS AND DISCUSSION Tests of Hückel Parametrizations PI, PIII, and PIV for NaCl Solutions with the Vapor Pressure Data Presented in the Literature. Gibbard et al.19 give experimental osmotic coefficients for NaCl solutions at 298.15, 310.65, 323.15, 335.65, 348.15, 360.65, and 373.15 K. Previously, we used their data for 348.15 K to determine the Hü ckel parameters at this temperature9 and tested the parametrizations with the data at this temperature and below it. In contrast, the data of all temperatures were employed here to test parametrization PI. The resulting error plots are shown in Figure 2. The vapor pressure error has been calculated using ep = p(reported) − p(predicted)

Figure 5. Plot of ep (eq 20), the deviation between the literature value of the vapor pressure measured by Lovelace et al.,33 Pearce and Nelson,30 and Herrington and Jackson34 and that predicted by parametrization PI for KCl solutions as a function of molality m. The values of the predicted vapor pressure were calculated using eqs 2, 3, and 4. Symbols: ●, T = 293.15 K, ref 33; ○, 298.15 K, ref 30; ▼, 323.15 K, ref 34; and △, 343.15 K, ref 34.

(20)

and is presented as a function of molality m. When comparing the present error plots to those reported in ref 9 in the overlapping cases, the errors differ slightly from each other because the nonideality correction to water vapor was made in a different way. This deviation is always very small and has no influence on the conclusions of ref 9. In Figure 2, graph A shows the results at 298.15, 310.65, 323.15, 335.65, and 348.15 K. The errors at 298.15 K are also illustrated in graph B, which additionally contains the errors for the other experimental vapor pressure sets in the literature at or near this temperature.

Specifically, the data from Pearce and Nelson30 at 298.15 K and those from Olynyk and Gordon31 at 293.15, 298.15, and 303.15 K are included in this graph. Parametrization PI applies well to these vapor pressure data because almost all absolute errors are below 10 Pa at molalities smaller than 3 mol·kg−1. As seen in graph B, above this molality the absolute values of the errors G



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Figure 6. Graphs A and B show plots of eip (eq 21), the deviation between the vapor pressure of water over the reference solution (NaCl = x) and that over the tested solution (KCl = y) as a function of the molality of the tested solution (my). In graph C, the relative deviation ep/p*, where p* (= p*1 ) is the vapor pressure of pure water at the tested temperature, is presented in the same way. The isopiestic sets from Amado-González and Blanco,35 Patterson et al.,36 Soldano and Patterson,37 and Holmes et al.29 were used. Symbols for graph A where all data were taken from ref 35 and parametrization PI was used for both salts: ●, T = 283.15 K; ○, 288.15 K; ▼, 293.15 K; and △, 298.15 K. Symbols for graph B: ●, parametrization PINaCl and PIKCl, T = 372.75 K, ref 36; ○, PINaCl and PIKCl, 382.0 K, ref 29; ▼, PINaCl and PIKCl, 394.3 K, ref 37; △, PINaCl and PIKCl where b1,KCl = 0.0127 (see the text), 382.0 K, ref 29; and ■, PIVNaCl and PIVKCl, 394.3 K, ref 37. Symbols for graph C where all data were taken from ref 29, the values of p1* are given in Table 2, and PIV was used for both salts: ●, T = 413.8 K; ○, 445.5 K; and ▼, 474.0 K. The point where mx = 0.676 mol·kg−1 in the set of ref 36 was omitted as an outlier from graph B.

Table 3. Molar Enthalpy of Vaporization of Water, ΔH*vap, Difference between the Molar Heat Capacity of Water as a Vapor and as a Liquid, ΔCp,vap, and Parameters u1, u2, and u3 in eq 26a as Functions of the Boiling Point of Pure Water (Tb*)b Tb*/K

ΔHvap * /(J·mol−1)c

ΔCp,vap/(J·K−1 mol−1)d

u1

u2/K

103u3/K2

333.15 343.15 353.15 363.15 373.15

42 488.9 42 043.1 41 590.5 41 127.6 40 655.8

−40.37 −40.43 −39.56 −38.76 −38.48

295.6377 285.2212 275.3535 265.7926 256.9587

1.36659 1.25139 1.14276 1.01500 0.95652

5.5272 4.7772 4.0915 3.2606 2.9994

a

This equation is needed here for the calculation of the boiling-point elevations from the osmotic coefficients reported by Smith and Hirtle40 above a molality of 1 mol·kg−1, and it is presented by Smith.39 bThe corresponding pressures are given in Table 1. cGiven by Smith.39 dCalculated from ΔCp,vap = Cp,m(g) − Cp,m(l). The former values were obtained here from the smoothed values of ref 44, and the latter were taken from ref 43.

light, parametrization PI applies slightly better to more concentrated solutions at 373.15 K than at 298.15 K. At 348.15 and 360.65 K, it applies well to all solutions considered in the graph. Thus, one cannot be sure that PIII is superior to PI even in these more concentrated solutions. Consequently, parametrization PIII is not considered here any further. Below 373 K, Hubert et al.32 also reported data sets from the static vapor pressure measurements in NaCl solutions. These sets usually contain 9 molalities from (1.245 to 5.423) mol·kg−1, and altogether 15 temperatures were studied from (294.40 to 368.00) K. We used these data here to test parametrization PI. The results of the calculations are reported in the three graphs of Figure 3. The error plots in these graphs are analogous to those in graphs A−C of Figure 2. Graph A here shows the results up to

gradually increase, indicating the breakdown of PI that is common to all error plots in the graph. In graph C of Figure 2, we consider temperatures that were not included in our previous studies9,10 on the thermodynamics of NaCl solutions. This graph shows that parametrization PI applies well to 360.65 K and reasonably well to 373.15 K below 3.0 mol·kg−1. The graph also reveals that parametrization PIII (eq 16) is not as good as PI for the interpretation of the results at 348.15 K. The fact that the direct vapor pressure errors (ep) at high temperatures are larger than those at low temperatures follows mostly from the increase in the vapor pressure with increasing temperature. One possibility to eliminating this effect is to consider relative errors, as is done in graph D, where quantity ep/p1* is presented as a function of the molality. In this H



Article

Figure 7. Plot of ebp (eq 24), the deviation between the observed and predicted boiling-point elevation in dilute solutions (graph A) and in less dilute solutions (graph B) of NaCl and KCl as a function of molality m. The experimental values have been reported by Smith39 and Smith and Hirtle40 (NaCl solutions) at the boiling points of pure water (= T*b ) given in Table 3 and at T*b = 373.15 K reported by Saxton and Smith41 (KCl solutions, symbol □ in both graphs). The predicted values were determined from eq 23, where eq 2 with the suggested Hückel parameters of parametrization PI were used for the osmotic coefficients (see text). Symbols in graph A where the data from dilute solutions given in refs 39 and 41 are used: ●, T*b = 333.15 K; ○, 343.15 K; ▼, 353.15 K; △, 363.15 K; and ■, 373.15 K. Symbols in graph B are the same as in graph A. For NaCl solutions, these data were taken from refs 39 and 40 (see text). The errors not shown in the graph are all positive and outside the scale of this graph.

Table 4. Boiling Point Elevations (ΔTb)a Calculated Using Equations 25 and 26 with the Parameter Values Given in Table 3 from the Osmotic Coefficients Reported by Smith and Hirtle40 ffor NaCl Solutions as a Function of the Boiling Point of Pure Water (Tb*)b (Tb*/K)b

ΔTb(1.5)/K

ΔTb(2.0)/K

ΔTb(2.5)/K

ΔTb(3.0)/K

ΔTb(3.5)/K

ΔTb(4.0)/K

333.15 343.15 353.15 363.15 373.15

1.141 1.223 1.307 1.394 1.484

1.572 1.684 1.798 1.916 2.036

2.031 2.174 2.322 2.474 2.628

2.513 2.690 2.876 3.065 3.259

3.023 3.236 3.454 3.681 3.916

3.581 3.832 4.079 4.337 4.601

The value of ΔTb is reported at the molality/(mol·kg−1) given. bThe corresponding pressures are given in Table 1.

a

Table 5. Recommended Activity Coefficients (γ) of Salt in Aqueous Sodium Chloride Solutions at Temperatures from (358.15 to 383.15) K as a Function of Molality ma m/mol·kg−1

γ(358.15)

γ(363.15)

γ(368.15)

γ(373.15)

γ(378.15)b

γ(383.15)c

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9184 0.8907 0.8564 0.8332 0.8155 0.8012 0.7892 0.7788 0.7697 0.7616 0.7544 0.7418 0.7312 0.7265 0.7221 0.7142 0.7072

0.9175 0.8894 0.8546 0.8312 0.8133 0.7988 0.7866 0.7761 0.7668 0.7586 0.7513 0.7385 0.7277 0.7229 0.7185 0.7104 0.7032

0.9164 0.8881 0.8528 0.8291 0.8109 0.7962 0.7838 0.7732 0.7638 0.7555 0.7480 0.7350 0.7240 0.7191 0.7145 0.7063 0.6990

0.9154 0.8867 0.8510 0.8269 0.8085 0.7936 0.7810 0.7702 0.7606 0.7522 0.7446 0.7313 0.7201 0.7151 0.7105 0.7020 0.6945

0.9143 0.8852 0.8491 0.8246 0.8060 0.7908 0.7781 0.7670 0.7574 0.7488 0.7410 0.7275 0.7161 0.7109 0.7062 0.6975 0.6898

0.9132 0.8837 0.8471 0.8223 0.8034 0.7880 0.7750 0.7638 0.7540 0.7452 0.7373 0.7235 0.7118 0.7066 0.7017 0.6928 0.6849

a

The values have been calculated using parametrization PI (see text) and the values at T/K are given. bThe pressure is 120.8 kPa instead of 101.325 kPa. cThe pressure is 143.6 kPa instead of 101.325 kPa.

support the tested parametrization well at least in less

333 K, graph B, up to 358 K, and graph C, the rest of the results. Graphs B and C also contain the errors of more concentrated solutions when they are outside the scale of graph A or B. In Figure 3, the same symbol is used in both graphs for each temperature when the results of this temperature are shown in two graphs. In all graphs in this figure, the experimental data

concentrated solutions. Using PI, it is possible to reproduce these data almost within their precision. However, the precision of the data is not as good as that of the data sets considered in Figure 2. I



Article

Table 6. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Sodium Chloride Solutions at Temperatures from (358.15 to 383.15) K as a Function of Molality ma m/mol·kg−1

ϕ(358.15)

ϕ(363.15)

ϕ(368.15)

ϕ(373.15)

ϕ(378.15)b

ϕ(383.15)c

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9731 0.9641 0.9533 0.9462 0.9410 0.9369 0.9336 0.9308 0.9285 0.9265 0.9247 0.9219 0.9197 0.9188 0.9181 0.9167 0.9157

0.9727 0.9636 0.9526 0.9454 0.9401 0.9360 0.9326 0.9297 0.9273 0.9253 0.9235 0.9205 0.9183 0.9173 0.9165 0.9151 0.9140

0.9723 0.9631 0.9520 0.9446 0.9392 0.9350 0.9315 0.9286 0.9261 0.9240 0.9221 0.9191 0.9167 0.9157 0.9148 0.9133 0.9121

0.9720 0.9626 0.9513 0.9438 0.9383 0.9339 0.9303 0.9273 0.9248 0.9226 0.9206 0.9175 0.9149 0.9139 0.9129 0.9113 0.9100

0.9716 0.9621 0.9505 0.9429 0.9373 0.9328 0.9291 0.9260 0.9234 0.9211 0.9191 0.9158 0.9131 0.9120 0.9110 0.9092 0.9078

0.9712 0.9616 0.9498 0.9420 0.9362 0.9316 0.9278 0.9247 0.9219 0.9196 0.9175 0.9140 0.9112 0.9100 0.9089 0.9070 0.9054

a

The values have been calculated using parametrization PI (see text) and the value at T/K are given. bThe pressure is 120.8 kPa instead of 101.325 kPa. cThe pressure is 143.6 kPa instead of 101.325 kPa.

Table 7. Recommended Activity Coefficients (γ) of Salt in Aqueous Potassium Chloride Solutions at Temperatures from (273.15 to 298.15) K as a Function of Molality ma m/mol·kg−1

γ(273.15)

γ(278.15)

γ(283.15)

γ(288.15)

γ(293.15)

γ(298.15)

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9292 0.9044 0.8729 0.8511 0.8342 0.8202 0.8082 0.7978 0.7885 0.7801 0.7724 0.7589 0.7472 0.7418 0.7368 0.7275 0.7192

0.9288 0.9039 0.8723 0.8505 0.8336 0.8196 0.8077 0.7973 0.7881 0.7797 0.7721 0.7587 0.7472 0.7419 0.7370 0.7278 0.7196

0.9284 0.9033 0.8716 0.8498 0.8329 0.8190 0.8071 0.7967 0.7875 0.7792 0.7717 0.7584 0.7470 0.7418 0.7369 0.7279 0.7198

0.9279 0.9027 0.8709 0.8491 0.8321 0.8182 0.8063 0.7960 0.7868 0.7786 0.7711 0.7579 0.7466 0.7415 0.7366 0.7278 0.7198

0.9274 0.9021 0.8701 0.8482 0.8313 0.8173 0.8055 0.7952 0.7860 0.7778 0.7704 0.7573 0.7460 0.7409 0.7361 0.7274 0.7195

0.9268 0.9014 0.8693 0.8473 0.8303 0.8164 0.8045 0.7942 0.7851 0.7769 0.7694 0.7564 0.7452 0.7402 0.7354 0.7268 0.7190

a

The values have been calculated using parametrization PI (see the text), and the values at T/K are given.

parametrization PIV from all data of Liu and Lindsay24 up to 473 K are shown in graph B of Figure 4. Surprisingly, all of the data in graph B can be predicted accurately with simple eq 17 for osmotic coefficients. In addition to Liu and Lindsay, Gardner et al.21 have measured vapor pressures from (396 to 543) K and at the three molalities of about 1.0, 2.0, and 3.0 mol·kg−1, and Gardner23 has measured vapor pressures at the two molalities of about 0.5 and 1.0 mol·kg−1 in the range from (415 to 539) K. Preliminary considerations already revealed that these data are not as accurate as the vapor pressures measured below 373 K, i.e., those considered in Figures 2 and 3, or even those reported by Liu and Lindsay24 (Figure 4A,B). However, we used the data from refs 21 and 23 up to 513 K in the tests of parametrization PIV. It is important to remember that eqs 6 to 11 needed in the calculations might not be very accurate at such high temper-

Vapor pressure data for NaCl solutions are also available at temperatures higher than 373 K. In dilute solutions, the measurements have been performed by Liu and Lindsay24 from (398 to 573) K with intervals of 25 K. At each temperature, the researchers usually reported vapor pressures at molalities of about 0.1, 0.25, 0.5, and 1.0 mol·kg−1. Parametrization PI was used here to predict these data up to 473 K. Graph A in Figure 4 illustrates the results using error plots obtained from eq 20. Table 2 provides the values of the parameters and quantities needed in these calculations. All experimental results at 398.15 and 423.15 K are predicted within ±1 kPa using PI. Of course, the vapor pressures of pure water at these temperatures are above atmospheric pressure and are 232 and 476 kPa, respectively. As described in the Theory section for temperatures above 383 K, a constant value of b1 = 0 was tested in eq 2 (i.e., eq 17 was used). The error plots obtained using this J



Article


γ(303.15)

γ(308.15)

γ(313.15)

γ(318.15)

γ(323.15)

γ(328.15)

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9262 0.9006 0.8684 0.8463 0.8292 0.8152 0.8034 0.7931 0.7839 0.7758 0.7683 0.7553 0.7442 0.7392 0.7345 0.7259 0.7182

0.9256 0.8998 0.8674 0.8452 0.8281 0.8140 0.8021 0.7918 0.7827 0.7745 0.7671 0.7541 0.7430 0.7380 0.7333 0.7247 0.7171

0.9250 0.8990 0.8663 0.8440 0.8268 0.8127 0.8008 0.7904 0.7813 0.7731 0.7657 0.7527 0.7416 0.7366 0.7319 0.7234 0.7157

0.9243 0.8981 0.8652 0.8427 0.8254 0.8113 0.7993 0.7889 0.7797 0.7715 0.7640 0.7510 0.7399 0.7349 0.7303 0.7217 0.7141

0.9236 0.8972 0.8640 0.8414 0.8240 0.8097 0.7977 0.7872 0.7780 0.7697 0.7623 0.7492 0.7381 0.7331 0.7284 0.7199 0.7123

0.9228 0.8962 0.8628 0.8400 0.8224 0.8081 0.7959 0.7854 0.7761 0.7678 0.7603 0.7472 0.7360 0.7310 0.7263 0.7178 0.7101

a

The values have been calculated using parametrization PI (see text), and the values at T/K are given.


γ(333.15)

γ(338.15)

γ(343.15)

γ(348.15)

γ(353.15)

γ(358.15)

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9221 0.8952 0.8614 0.8384 0.8207 0.8063 0.7941 0.7835 0.7741 0.7658 0.7582 0.7450 0.7338 0.7287 0.7240 0.7154 0.7077

0.9213 0.8941 0.8600 0.8368 0.8190 0.8044 0.7921 0.7814 0.7720 0.7636 0.7560 0.7427 0.7313 0.7262 0.7215 0.7128 0.7051

0.9204 0.8930 0.8586 0.8351 0.8171 0.8024 0.7900 0.7792 0.7697 0.7612 0.7535 0.7401 0.7287 0.7235 0.7187 0.7100 0.7022

0.9195 0.8918 0.8570 0.8334 0.8152 0.8003 0.7878 0.7769 0.7673 0.7587 0.7509 0.7374 0.7258 0.7206 0.7158 0.7069 0.6991

0.9186 0.8906 0.8555 0.8315 0.8131 0.7981 0.7854 0.7744 0.7647 0.7560 0.7482 0.7344 0.7228 0.7175 0.7126 0.7037 0.6957

0.9177 0.8893 0.8538 0.8296 0.8110 0.7958 0.7829 0.7718 0.7620 0.7532 0.7452 0.7314 0.7195 0.7142 0.7092 0.7001 0.6920

a

The values have been calculated using parametrization PI (see the text), and the values at T/K are given.

The results of these tests are shown in the two graphs in Figure 2 of the previous article.9 These graphs show the isopiestic errors defined by the difference between vapor pressures px and py

atures. The results of the tests are shown in graph C of Figure 4, where the relative vapor pressure error plots are reported as in graph D of Figure 2. In contrast to previous graphs, we present in this graph the relative errors as a function of the temperature, whereas molalities are distinguished using different symbols. Exact molalities in the articles21,23 were employed in the calculations, but only rounded values are given in the caption of the figure. Apparently, all points reported in refs 21 and 23 below 513 K support at least satisfactorily the simple Debye−Hückel term as the osmotic coefficient (eq 17). Estimation of Hü ckel Parameters for Potassium Chloride from the Existing Isopiestic Data for NaCl and KCl Solutions at Temperatures Higher than 298 K. As mentioned above, the following four extensive isopiestic sets have been included in the tests of ref 9 at temperatures higher than 298 K: Davis et al.25 and Hellams et al.28 at 318.15 K, Humphries et al.26 at 333.15 K, and Moore et al.27 at 353.15 K.

eip = px − py = p1, x − p1, y

(21)

and presented as a function of the KCl molality (= my). Symbol x refers to the reference solution, which is the NaCl solution (an arbitrary choice in ref 6), and symbol y refers to the tested solution, which is the KCl solution. These notations are utilized later when isopiestic data are considered. For KCl solutions in ref 9, the previous constant value of 1.3 (mol·kg−1)−1/2 was used for parameter B (see ref 6), and the following best estimates of 0.0396, 0.0486, and 0.0440 were employed for parameter b1 at 318.15, 333.15, and 353.15 K, respectively. In all high-quality isopiestic sets of Figure 2 in ref 9, the new parameter values for KCl together with parametrization PI for NaCl solutions K



Article

Tests of the Hückel Parameters of Parametrization PI for KCl Solutions with the Vapor Pressure Data. In refs 33 and 30, Lovelace et al. and Pearce and Nelson reported vapor pressures of KCl solutions at 293.15 and 298.15 K, respectively. The former data extend up to 4.0 mol·kg−1, and the latter extend up to full saturation (i.e., to 4.81 mol·kg−1). Herrington and Jackson34 measured points for 323.15 and 343.15 K up to 3.25 mol·kg−1 in both cases. We used these data to test eq 18. The results are shown as error plots in Figure 5 (eq 20). This figure reveals that the data can be predicted very well using parametrization PI at least up to 3 mol·kg−1. The largest absolute error of 14 Pa is obtained at T = 343.15 K, where the vapor pressure of pure water is 31 177 Pa (Table 1). Tests of the Hückel Parameters of Parametrization PI and PIV for NaCl and KCl Solutions with the Isopiestic Data. In addition to the isopiestic data used in refs 6,8, and 9, parametrization PI was tested against the data of AmadoGonzález and Blanco35 measured at 283.15, 288.15, 293.15, and at 298.15 K up to a molality of about 1.0 mol·kg−1. At temperatures close to or slightly above 373 K, the data sets of Patterson et al.36 at 372.8 K, Holmes et al.29 at 382.0 K, and Soldano and Patterson37 at 394.3 K were also employed. Two graphs of Figure 6 present the results from these sets as isopiestic error plots. In these plots, the isopiestic vapor pressure error (eip) defined by eq 21 is presented as a function of the KCl molality (= my). The vapor pressures px and py for this equation were corrected with the nonideality of the water vapor through eqs 2, 3, and 4 for temperatures close to or above 373 K. This correction is not important in dilute solutions, and thus eq 5 was used instead of eq 4 in the calculations for the sets of AmadoGonzález and Blanco.35 The results of the sets in ref 35 are shown in graph A, and these data well support parametrization PI. Graph B shows the error plots at the other temperatures, and these plots support PI up to 3.0 mol·kg−1. The largest absolute error below this molality is obtained at T = 394.3 K, and this error is about 0.3 kPa. Thus, it is small when compared to the vapor pressure of pure water at this temperature (i.e., to 198 kPa).


γ(363.15)

γ(368.15)

γ(373.15)

γ(378.15)b

γ(383.15)c

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9167 0.8880 0.8521 0.8276 0.8087 0.7934 0.7804 0.7691 0.7591 0.7502 0.7422 0.7281 0.7161 0.7107 0.7056 0.6964 0.6882

0.9157 0.8867 0.8503 0.8255 0.8064 0.7908 0.7776 0.7662 0.7561 0.7471 0.7389 0.7246 0.7124 0.7069 0.7018 0.6924 0.6840

0.9146 0.8852 0.8484 0.8233 0.8040 0.7882 0.7748 0.7632 0.7530 0.7438 0.7355 0.7210 0.7086 0.7030 0.6978 0.6882 0.6797

0.9136 0.8838 0.8465 0.8211 0.8015 0.7855 0.7719 0.7602 0.7498 0.7404 0.7320 0.7173 0.7046 0.6990 0.6936 0.6839 0.6752

0.9124 0.8823 0.8445 0.8187 0.7989 0.7826 0.7689 0.7569 0.7464 0.7369 0.7284 0.7133 0.7005 0.6947 0.6893 0.6793 0.6704

a

The values have been calculated using parametrization PI (see text), and the values at T/K are given. bThe pressure is 120.8 kPa instead of 101.325 kPa. cThe pressure is 143.25 kPa instead of 101.325 kPa.

excellently predict all data almost up to a KCl molality of 4.0 mol·kg−1. In the present study, the following previously determined values were used in the determination of the quadratic equation for parameter b1 of KCl: b1(273.15 K) = −0.0515, b1(298.15 K) = 0.011, and b1(353.15 K) = 0.0440 together with B = 1.3 (mol· kg−1)−1/2. The resulting temperature dependence of parameter b1 for KCl solutions up to 353 K is given in eq 18 of the present study. It is important to note that the previously9 determined values of b1,KCl = 0.0396 and b1,KCl = 0.0486 are also roots of eq 18 at temperatures 318.15 and 333.15 K, respectively. This, together with the positive test results below, confirms the reliability of eq 18.

Table 11. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Potassium Chloride Solutions at Temperatures from (273.15 to 298.15) K as a Function of Molality ma m/mol·kg−1

ϕ(273.15)

ϕ(278.15)

ϕ(283.15)

ϕ(288.15)

ϕ(293.15)

ϕ(298.15)

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9766 0.9684 0.9582 0.9513 0.9460 0.9416 0.9379 0.9347 0.9319 0.9293 0.9270 0.9230 0.9196 0.9180 0.9166 0.9139 0.9115

0.9764 0.9683 0.9581 0.9512 0.9459 0.9415 0.9379 0.9347 0.9320 0.9295 0.9273 0.9234 0.9201 0.9186 0.9172 0.9147 0.9124

0.9763 0.9681 0.9579 0.9510 0.9457 0.9414 0.9378 0.9347 0.9320 0.9296 0.9274 0.9236 0.9204 0.9190 0.9177 0.9153 0.9131

0.9761 0.9679 0.9577 0.9508 0.9456 0.9413 0.9377 0.9347 0.9320 0.9296 0.9275 0.9238 0.9207 0.9193 0.9180 0.9157 0.9137

0.9760 0.9677 0.9575 0.9506 0.9453 0.9411 0.9376 0.9345 0.9319 0.9295 0.9275 0.9239 0.9208 0.9195 0.9183 0.9161 0.9141

0.9758 0.9675 0.9572 0.9503 0.9451 0.9409 0.9374 0.9343 0.9317 0.9294 0.9274 0.9238 0.9209 0.9196 0.9184 0.9163 0.9144

a

The values have been calculated using parametrization PI (see text), and the values at T/K are given. L



Article


ϕ(303.15)

ϕ(308.15)

ϕ(313.15)

ϕ(318.15)

ϕ(323.15)

ϕ(328.15)

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9756 0.9673 0.9569 0.9500 0.9448 0.9406 0.9371 0.9341 0.9315 0.9292 0.9272 0.9237 0.9208 0.9196 0.9184 0.9163 0.9146

0.9754 0.9670 0.9566 0.9497 0.9444 0.9402 0.9367 0.9337 0.9311 0.9289 0.9269 0.9234 0.9206 0.9194 0.9183 0.9163 0.9145

0.9752 0.9667 0.9563 0.9493 0.9440 0.9398 0.9363 0.9333 0.9308 0.9285 0.9265 0.9231 0.9203 0.9191 0.9180 0.9160 0.9144

0.9749 0.9664 0.9559 0.9489 0.9436 0.9393 0.9358 0.9329 0.9303 0.9280 0.9260 0.9226 0.9199 0.9187 0.9176 0.9157 0.9140

0.9747 0.9661 0.9555 0.9484 0.9431 0.9388 0.9353 0.9323 0.9297 0.9275 0.9255 0.9221 0.9194 0.9182 0.9171 0.9152 0.9136

0.9744 0.9658 0.9551 0.9479 0.9426 0.9383 0.9347 0.9317 0.9291 0.9268 0.9248 0.9214 0.9187 0.9175 0.9164 0.9145 0.9129

a



ϕ(333.15)

ϕ(338.15)

ϕ(343.15)

ϕ(348.15)

ϕ(353.15)

ϕ(358.15)

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9742 0.9654 0.9546 0.9474 0.9420 0.9376 0.9341 0.9310 0.9284 0.9261 0.9241 0.9207 0.9179 0.9167 0.9156 0.9137 0.9121

0.9739 0.9650 0.9541 0.9468 0.9413 0.9370 0.9334 0.9303 0.9276 0.9253 0.9233 0.9198 0.9170 0.9158 0.9147 0.9128 0.9112

0.9736 0.9646 0.9536 0.9462 0.9407 0.9362 0.9326 0.9295 0.9268 0.9244 0.9224 0.9189 0.9160 0.9148 0.9137 0.9117 0.9101

0.9733 0.9642 0.9530 0.9456 0.9399 0.9355 0.9317 0.9286 0.9259 0.9235 0.9214 0.9178 0.9149 0.9137 0.9125 0.9105 0.9088

0.9730 0.9638 0.9524 0.9449 0.9392 0.9346 0.9308 0.9276 0.9249 0.9224 0.9203 0.9166 0.9137 0.9124 0.9112 0.9092 0.9074

0.9726 0.9633 0.9518 0.9442 0.9384 0.9337 0.9299 0.9266 0.9238 0.9213 0.9191 0.9154 0.9123 0.9110 0.9098 0.9077 0.9058

a


apply quite well to the existing rough vapor pressure data up to m = 3 mol·kg−1 and T = 513 K (graph C in Figure 4). The corresponding simple equation for KCl at high temperatures was determined from the isopiestic data of Holmes et al.29 at 413.8, 445.4, and 474.0 K. All three sets contain data points from less dilute solutions starting at a molality of about 1.0 mol·kg−1 and extending at least up to mNaCl = 5.9 mol·kg−1. The minimum number of points is 17 at each temperature. The resulting estimates of parameter b1,KCl are −0.0293, −0.0231, and −0.0198 at temperatures 413.8, 445.4, and 474.0 K, respectively. These values were obtained by minimization of the square sum containing deviations between the predicted pressures of py and px. The estimated linear dependence between these b1 values and the temperature is presented in eq 19, and it is used here together with the value of b1,NaCl = 0 in the interpretation of these three isopiestic sets. The calculated values of parameter b1,KCl are

As can be seen in graph B of Figure 6, parametrization PI does not apply very well to the isopiestic data measured by Holmes et al.29 at 382.0 K because this set contains only measurements from less dilute solutions. According to the results presented above, it seems probable that eq 15 can be used at least for the less concentrated NaCl solutions in this set. The problematic equation now is eq 18 for KCl solutions. From the experimental molalities of Holmes et al.29 at this temperature, it is possible to obtain a better value of b1,KCl = 0.0127 than that obtained from eq 18 (b1,KCl = 0.004). The error plot obtained with the new value is also presented in graph B of this figure. This value together with the value of b1,NaCl = −0.0562 from eq 15 well explains all experimental data at 382.0 K in this graph at molalities up to 6.2 mol·kg−1 for NaCl solutions and at molalities up to 7.17 mol·kg−1 for KCl solutions. Parametrization PIV was used for the osmotic coefficients of NaCl solutions at high temperatures. This equation is seen to M



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Table 14. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Potassium Chloride Solutions at Temperatures from (363.15 to 383.15) °C as a Function of Molality ma m/mol·kg−1

ϕ(363.15)

ϕ(368.15)

ϕ(373.15)

ϕ(378.15)b

ϕ(383.15)c

0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.12 0.14 0.15 0.16 0.18 0.20

0.9723 0.9629 0.9512 0.9434 0.9375 0.9328 0.9289 0.9255 0.9226 0.9201 0.9178 0.9140 0.9109 0.9095 0.9082 0.9060 0.9041

0.9719 0.9624 0.9505 0.9426 0.9366 0.9318 0.9278 0.9244 0.9214 0.9188 0.9165 0.9125 0.9093 0.9079 0.9066 0.9042 0.9023

0.9716 0.9619 0.9498 0.9417 0.9356 0.9307 0.9266 0.9231 0.9201 0.9174 0.9150 0.9109 0.9076 0.9061 0.9047 0.9023 0.9002

0.9712 0.9613 0.9491 0.9409 0.9346 0.9296 0.9254 0.9218 0.9187 0.9160 0.9135 0.9093 0.9058 0.9042 0.9028 0.9003 0.8981

0.9708 0.9608 0.9483 0.9400 0.9336 0.9285 0.9242 0.9205 0.9173 0.9144 0.9119 0.9075 0.9039 0.9022 0.9008 0.8981 0.8957

a The values have been calculated using parametrization PI (see text), and the values at T/K are given. bThe pressure is 120.8 kPa instead of 101.325 kPa. cThe pressure is 143.6 kPa instead of 101.325 kPa.

Table 15. Recommended Vapor Pressure (p) of Water in Aqueous Sodium Chloride Solutions at Temperatures from (393.2 to 473.2) K as a Function of Molality ma m/mol·kg−1 0 0.1 0.2 0.5 1.0 2.0 3.0 4.0 5.0 6.0

p(393.2)b 0.1983 0.1977 0.1970 0.1951 0.1920 0.1859 0.1799 0.1741 0.1684 0.1629

c

p(413.2)b c

0.3612 0.3599 0.3588 0.3553 0.3496 0.3384 0.3275 0.3168 0.3065 0.2964

p(423.2)b 0.4759 0.4743 0.4727 0.4681 0.4606 0.4459 0.4314 0.4173 0.4036 0.3903

c

p(433.2)b 0.6181 0.6160 0.6140 0.6080 0.5983 0.5790 0.5602 0.5419 0.5240 0.5067

c

p(453.2)b 1.003 1.000 0.997 0.987 0.971 0.939 0.909 0.879 0.850 0.821

c

p(473.2)b 1.556c 1.551 1.545 1.530 1.505 1.456 1.408 1.361 1.315 1.271

a The vapor pressure of water has been calculated using eqs 3, 4, and 17, and the values at T/K are given. bThe unit is MPa. cThe vapor pressure of pure water has been calculated using eq 8.

Table 16. Recommended Vapor Pressure (p) of Water in Aqueous Potassium Chloride Solutions at Temperatures from (393.2 to 473.2) K as a Function of Molality ma m/mol·kg−1 0 0.1 0.2 0.5 1.0 2.0 3.0 4.0 5.0 6.0

p(393.2)b 0.1983 0.1977 0.1970 0.1952 0.1922 0.1865 0.1811 0.1760 0.1712 0.1668

c

p(413.2)b c

0.3612 0.3599 0.3588 0.3554 0.3499 0.3394 0.3295 0.3201 0.3113 0.3030

p(423.2)b 0.4759 0.4743 0.4727 0.4683 0.4610 0.4472 0.4340 0.4215 0.4098 0.3986

c

p(433.2)b 0.6181 0.6160 0.6140 0.6082 0.5988 0.5807 0.5635 0.5471 0.5317 0.5171

c

p(453.2)b 1.003 1.000 0.997 0.987 0.972 0.942 0.914 0.887 0.861 0.837

c

p(473.2)b 1.556c 1.551 1.546 1.531 1.506 1.460 1.415 1.372 1.331 1.293

a

The vapor pressure of water has been calculated using eqs 2, 3, 4, and 19, and the values at T/K are given. bThe unit is MPa. cThe vapor pressure of pure water has been calculated using eq 8.

applies up to molalities greater than 5 mol·kg−1 at this temperature. Graph C in this figure displays the error plots based on the relative errors for the sets measured at 413.8, 445.4, and 474.0 K. These relative errors are also small, and the equations are seen to apply at least up to a KCl molality of 6.0 mol·kg−1. Thus, simple parametrizations PIVNaCl and PIVKCl can

−0.0289, −0.0239, and −0.0194 at the three experimental temperatures, respectively. The existing isopiestic data above 393 K were predicted using PIV for both NaCl and KCl. The results from data at 394.3 K (taken from ref 37) are shown in graph B of Figure 6. All errors in the graph are small, and thus the simple parametrization PIV N



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Table 17. Difference (ep)a and the Relative Differenceb in Parentheses Obtained in the Comparison of the Vapor Pressure of Water Calculated from the Osmotic Coefficient Given by Pitzer et al.1 or Gardner23 to That Obtained Using Equation 17 in the Present Study for NaCl Solutions as a Function of Molality m m/mol·kg−1

ep(393.2)c

ep(413.2)c

ep(423.2)c

ep(433.2)c

ep(453.2)c

ep(473.2)c

0.1d 0.5d 0.5e 1.0d 1.0e 2.0d 2.0e 3.0d 3.0e 4.0d 5.0d 6.0d

0.003 (0.01) 0.085 (0.43) 0.110 (0.55) 0.324 (1.63) 0.381 (1.92) 1.26 (6.35) 1.32 (6.67) 2.86 (14.4) 2.77 (14.0) 4.96 (25.0) 7.39 (37.3) 10.0 (50.5)

0.006 (0.02) 0.141 (0.39) 0.207 (0.57) 0.545 (1.51) 0.675 (1.87) 2.15 (5.96) 2.25 (6.24) 4.82 (13.5) 4.54 (12.5) 8.32 (23.0) 12.3 (34.2) 16.6 (45.6)

0.006 (0.01) 0.174 (0.37) 0.280 (0.59) 0.666 (1.40) 0.873 (1.83) 2.68 (5.62) 2.84 (5.97) 6.03 (12.7) 5.65 (11.9) 10.4 (21.8) 15.4 (32.4) 20.7 (43.5)

0.006 (0.01) 0.203 (0.33) 0.364 (0.59) 0.804 (1.30) 1.12 (1.81) 3.28 (5.31) 3.50 (5.66) 7.32 (11.8) 6.88 (11.1) 12.7 (20.5) 18.9 (30.5) 25.2 (40.8)

0.003 (0) 0.245 (0.24) 0.568 (0.57) 1.01 (1.01) 1.65 (1.64) 3.38 (3.37) 3.95 (3.94) 10.1 (10.1) 9.41 (9.38) 17.9 (17.8) 26.5 (26.1) 35.5 (35.4)

−0.003 (0) 0.200 (0.13) 0.772 (0.50) 1.02 (0.66) 2.21 (1.42) 5.11 (3.29) 6.25 (4.02) 12.5 (8.01) 11.4 (7.33) 22.5 (14.4) 33.9 (21.8) 45.4 (29.2)

a The difference is calculated from ep = p1(present study) − p1(literature), and the values at T/K are given. bThe value given is 103ep/p1*. (See also footnote a.) cThe unit is kPa. dReference 1. eReference 33.

Table 18. Difference (ep)a and the Relative Differenceb in Parentheses Obtained in the Comparison of the Vapor Pressure of Water Calculated from the Osmotic Coefficient Given by Liu and Lindsay24 to That Obtained Using Equation 17 in the Present Study for NaCl Solutions m/mol·kg−1

ep(398.2)/kPa

0.2458 0.5309 1.0039 0.1013 0.2463 0.5328 0.9967 1.0061 0.1021 0.2472 0.5360 0.9590 1.0045 0.1025 0.2486 0.5409 0.9641 1.0149 1.0170

0.049 (0.21) 0.037 (0.16) 0.336 (1.45)

ep(424.2)/kPa

ep(448.2)/kPa

ep(473.2)/kPa

−0.041 (−0.09) 0.096 (0.20) 0.052 (0.11) 0.526 (1.11) 0.446 (0.94) 0.008 (0.01) 0.117 (0.13) −0.115 (−0.13) 0.257 (0.29) 0.309 (0.35) −0.006 (0) 0.048 (0.03) −0.017 (−0.01) 0.450 (0.29) 0.701 (0.45) 0.348 (0.22)

The difference is calculated from ep = p1(present study) − p1(literature), and the values at T/K are given. bThe value given is 103ep/p*1 . (See also footnote a.) a

points using molalities from (0.1 to 7.8) mol·kg−1. The boilingpoint elevation (ΔTb) is defined by

be quite safely used up to 474 K. In the literature, such simple equations are not available for solutions of NaCl or KCl at such high temperatures. This is also true for pure electrolyte solutions in general. Tests of the Hückel Parameters of Parametrization PI for NaCl and KCl Solutions with Boiling-Point Elevation Data. Accurate boiling points in aqueous NaCl solutions have been measured at and below atmospheric pressure by Smith38,39 and Smith and Hirtle40 at temperatures of close to 333, 343, 353, 363, and 373 K. Reference 39 gives the values of boiling-point elevations at 12 molalities in the range of (0.05 to 1.00) mol·kg−1 in the five cases. In ref 40, the boiling points are considered in more concentrated solutions from (1.5 to 4.0) mol·kg−1 at intervals of 0.5 mol·kg−1. KCl solutions at atmospheric pressure have been studied by Saxton and Smith.41 They reported 20

ΔTb = Tb − Tb*

(22)

where Tb and Tb* are the boiling point of water in the solution and that of pure water, respectively, at the pressure considered. The following equation is utilized in the present study for the calculation of ΔTb ΔTb =

2RT b*M1mφ * Δ H vap Tb*

− 2RM1mφ

(

ΔCp,vapΔTb − ΔCp,vap(T b* + ΔTb)ln +

* Δ H vap Tb*

O

− 2RM1mφ

Tb* + Δ Tb Tb*

) (23)



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for example, this agreement in the less dilute solutions: the boiling point of water at the NaCl molality of 3.0 mol·kg−1 and at 31 177 Pa is 345.840 K, whereas parametrization PI predicts a value of 345.829 K. This is good correspondence considering the theoretical problems associated, for example, with the heat capacities of the superheated liquid phase needed in this evaluation. Recommended Values for the Activity Coefficient of NaCl and KCl and for the Osmotic Coefficient of Water in Aqueous NaCl and KCl Solutions. On the basis of the abundant evidence provided by Figures 2−7 here and by the results in refs 6 and 8, the new Hückel equations apply well to the experimental data available in the literature for dilute NaCl and KCl solutions from (273.15 to 383.15) K. Using these equations, the data can very often be predicted within the experimental error at least up to a molality of 1 mol·kg−1. Traceable and transparent thermodynamic quantities can be obtained using parametrization PI at least up to 0.2 mol·kg−1 at all of these temperatures. Therefore, we tabulate here the activity coefficients of NaCl and KCl together with the osmotic coefficients of water in these solutions at rounded molalities using parametrization PI. In ref 9, we have already presented the values of these quantities for NaCl solutions up to 353.15 K, and now these tables are continued up to 383.15 K. Table 5 presents the new activity coefficients based on PI from (358.15 to 383.15) K, and Table 6 shows the new osmotic coefficients. Table 7 displays the traceable activity coefficients of KCl in the range of (273.15 to 298.15) K. In Table 8 are reported the values from (303.15 to 328.15) K, in Table 9, from (333.15 to 358.15) K, and finally in Table 10, from (363.15 to 383.15) K. Tables 11−14 give the corresponding values for the osmotic coefficients. Table 10 of ref 8 shows the recommended activity coefficients in dilute KCl solutions from (273.15 to 298.15) K based on a linear representation of parameter b1 with temperature. Table 14 in that reference gives the new osmotic coefficients obtained in the same way. The agreement between the corresponding activity coefficients in Table 7 here and in Table 10 of ref 8 and the corresponding osmotic coefficients in Table 11 here and in Table 14 of ref 8 is always very good. Tables 15 and 16 provide at temperatures between 393 and 473 K the vapor pressures of water in NaCl and KCl solutions on the basis of eq 17 (NaCl) and eq 2 with eq 19 (KCl), respectively. The new vapor pressures of NaCl solutions are compared to the literature values in Tables 17 and 18. The literature values were obtained for Table 17 from the tabulated osmotic coefficients of Gardner23 and Pitzer et al.1 The latter values are based on the multiparameter Pitzer equations. In Table 18, the corresponding comparison is made for the osmotic coefficients reported by Liu and Lindsay.24 Both absolute and relative differences between the new values and the ones obtained from the literature osmotic coefficients are given in these two tables. The relative difference is calculated by dividing the absolute difference by the vapor pressure of pure water at the temperature considered. The new vapor pressures agree well with the literature values in Table 18. In contrast, in Table 17 the agreement is not as good. Above 2 mol·kg−1, the values in Pitzer et al. and Gardner agree poorly with the new values. The relative errors in Table 17 can be as high as 5%, but in Table 18, they are always less than 0.2%. The vapor pressures given Table 15 that were utilized in the comparison are supported both by the experimental vapor pressure measurements at molalities up to 3 mol·kg−1 and by the isopiestic measurements at least up to 6

and it can be derived analogously to the corresponding equation in ref 42 for the freezing-point depression, ΔTf. (See eq 13 in that reference.) In eq 23, ΔHvap * is the molar enthalpy of vaporization, and its values at the boiling points considered were taken from the second paper39 by Smith. ΔCp,vap is the difference between the molar heat capacities of water vapor and liquid water. The heat capacities of liquid water at the boiling points considered were taken from the tables of Osborne et al.,43 and the heat capacities of water vapor were evaluated here from the values reported by Vestfálová and Š afařic44 at rounded pressures and at various temperatures. On the other hand, these values under rounded conditions in ref 44 were determined from the general water-property tables of Wagner and Pruss.45 The values of all parameters needed in the calculations with eq 23 are collected here in Table 3. The use of eq 23 requires iterative evaluations. The corresponding equation for the freezing-point depression was employed previously in the thermodynamic consideration of NaCl and KCl solutions based on the Hückel equation for describing the nonideality (e.g., ref 8). The experimental ΔTb values were reproduced with the new Hückel model using eq 2 with parametrization PI in eq 23. The boiling-point errors calculated by ebp = ΔTb(reported) − ΔTb(predicted)

(24)

are presented here as a function of the molality. The errors in the boiling-point data up to 0.5 mol·kg−1 are reported in graph A of Figure 7. All absolute errors in this graph are less than 0.002 K except the ones for the three highest molalities (two for NaCl solutions and one for KCl solution) at the pressure where the boiling point of pure water is 373.15 K. Therefore, the boilingpoint data well support parametrization PI. Graph B in this figure shows the errors in the three points not included in graph A. Additionally, this graph contains the results of ref 39 above 0.5 mol·kg−1 and the results of the more concentrated solutions in refs 40 and 41. The boiling points for ref 40 were not directly available in the original paper, but they could be calculated from the given experimental osmotic coefficients. The following equations were used in the calculations ΔTb =

2(Tb*)2 M1mφ * f (ΔTb) ΔH vap

f (ΔTb) J·K−1·mol−1

=1−

(25)

u (ΔTb)3 u1ΔTb u (ΔTb)2 + 2 − 3 * * * 2ΔH vap 3ΔH vap 4ΔH vap (26)

For function f, the values of parameters u1, u2, and u3 are given in Table 3 and must be used in the calculation of its values. The values of u1, u2, and u3 were taken from the original paper.39 As for eq 23, the use of eq 25 requires iterative calculations. Table 4 gives the resulting values of the boiling-point elevations in the less dilute solutions considered in ref 40. When the boiling-point data were reproduced using parametrization PI, the temperature dependences of eqs 15 and 18 for parameter b1 must be taken into account, especially in the calculations of these less dilute solutions. Graph B of Figure 7 shows the results obtained from the data in Table 4. According to this graph, all absolute errors are less than 0.011 K up to a molality of 3 mol·kg−1 when Tb* is 333.15 or 343.15 K. All absolute errors are less than 0.016 K up to 1.5 mol·kg−1 when this quantity is 353.15 or 363.15 K. When Tb* = 373.15 K, similar agreement is observed only up to 1 mol· kg−1. The result concerning an individual data point illustrates, P



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mol·kg−1 at these higher temperatures. Therefore, these values are probably more reliable.

(4) Archer, D. G. Thermodynamic properties of the KCl + H2O system. J. Phys. Chem. Ref. Data 1999, 28, 1−16. (5) Pitzer, K. S. Thermodynamics of electrolytes. I. Theoretical basis and general equations. J. Phys. Chem. 1973, 77, 268−277. (6) 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. (7) 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. (8) 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. (9) 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. (10) 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 2. The quantities associated with the partial molar heat capacity. J. Chem. Eng. Data 2017, 62, 4215− 4227. (11) Partanen, J. I.; Juusola, P. M.; Vahteristo, K. P.; de Mendonça, A. J. G. Re-evaluation of the activity coefficients of aqueous hydrochloric acid solutions up to a molality of 16.0 mol·kg−1 using the Hückel and Pitzer equations at temperatures from 0 to 50 °C. J. Solution Chem. 2007, 36, 39−59. (12) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous solutions of silver nitrate, alkali metal fluorides and nitrites, and dihydrogen phosphate, dihydrogen arsenate, and thiocyanate salts with sodium and potassium ions at 25 °C. J. Chem. Eng. Data 2011, 56, 2044−2062. (13) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous alkali metal bromide solutions at 25 °C. J. Chem. Eng. Data 2010, 55, 2202−2213. (14) Partanen, J. I.; Covington, A. K. Re-evaluation of the thermodynamic activity quantities in aqueous solutions of uni-univalent alkali metal salts of aliphatic carboxylic acids and thallium acetate at 25 °C. J. Chem. Eng. Data 2011, 56, 4524−4543. (15) 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. (16) Partanen, J. I. Traceable activity and osmotic coefficients in pure aqueous solutions of alkaline earth metal bromides and iodides at 25 °C. J. Chem. Eng. Data 2014, 59, 2530−2540. (17) 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. (18) Keyes, F. G.; Smith, L. B.; Gerry, H. T. The specific volume of steam in the saturated and superheated condition together with derived values of the enthalpy, entropy, heat capacity and Joule Thomson coefficients. Part IV. Steam research program. Proc. Am. Acad. Arts Sci. 1936, 70, 319−364. (19) 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. (20) Kell, G. S. Density, thermal expansivity, and compressibility of liquid water from 0° to 150 °C: correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. J. Chem. Eng. Data 1975, 20, 97−105. (21) Gardner, E. R.; Jones, P. J.; de Nordwall, H. J. Osmotic coefficients of some aqueous sodium chloride solutions at high temperatures. Trans. Faraday Soc. 1963, 59, 1994−2000. (22) Ensor, D. D.; Anderson, H. L. Heats of dilution of NaCl: temperature dependence. J. Chem. Eng. Data 1973, 18, 205−212.

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SUMMARY AND CONCLUSIONS In this study, we have extended our previously reported,9 fully traceable, two-parameter Hückel equations for the activity coefficients of sodium chloride and for the osmotic coefficients of water in aqueous NaCl solutions to temperatures from (353.15 to 383.15) K. We have also demonstrated that a similar model applies to these activity quantities from (273.15 to 383.15) K for dilute KCl solutions at least up to a molality of 0.2 mol·kg−1. In our most successful parametrization of the Hückel equations (eqs 1 and 2), parameter B is treated as a constant whereas parameter b1 is a quadratic function of the temperature. On the basis of extensive testing of our models against existing experimental data, we believe that the activity and osmotic coefficients tabulated here are the most reliable values for dilute NaCl and KCl solutions published so far. For NaCl and KCl solutions at temperatures above 383 K, we tested a simplified parametrization of the Hückel equations. In these parametrizations, parameter b1 in the Hückel equation is set to zero for NaCl, while a linear temperature dependence is assumed for b1 of KCl. Our test results demonstrate that these parametrizations apply almost within experimental error up to a temperature of 500 K and up to full saturation for the thermodynamic data available from solutions of both salts. Interestingly, however, the high-temperature vapor pressures reported here for the concentrated NaCl solutions do not agree well with the values obtained from the osmotic coefficients tabulated by Pitzer et al.1 on the basis of their multiparameter equations. In the near future, we report for KCl solutions the results of the testing of the present equations against the calorimetric data.

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AUTHOR INFORMATION

Corresponding Author

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

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We dedicate this article to the memory of Professor Arthur K. Covington (University of Newcastle, U.K.) in recognition of his many invaluable contributions to a broad range of analytical and physical chemistry. He passed away in his hometown of Ponteland on December 29, 2017. We feel privileged to have known him and fortunate to have been able to enjoy his advice and guidance.

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REFERENCES

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