Traceable Thermodynamic Quantities for Dilute Aqueous Sodium

Article pubs.acs.org/jced

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 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, Lappeenranta, FI-53851, Finland ‡ Laboratory of Physical Chemistry, Department of Chemistry, University of Helsinki, P.O. Box 55, A.I. Virtasen aukio 1, Helsinki, FI-00014, Finland S Supporting Information *

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 (0 to 80) °C. These equations apply within experimental error to all thermodynamic data available for these solutions at least up a molality of 0.2 mol·kg−1. In our previous study (J. Chem. Eng. Data 2016, 61, 286−306), these equations were successfully tested against the literature results of electrochemical, isopiestic, and cryoscopic measurements usually in the temperature range from (0 to 25) °C. There, a constant value was employed for B, whereas a linear model with respect to the temperature was utilized for b1. The linear model was determined from the values of b1 at 0 °C and at 25 °C obtained from freezing-point depression data and from isopiestic and cell-potential difference data, respectively. In the present study, these two b1 values are utilized alongside the constant value of parameter B but a new quadratic model is presented for the temperature dependence of b1. The third data point required for this model is obtained from the direct vapor pressure measurements of Gibbard et al. (J. Chem. Eng. Data 1974, 19, 281−288) at 75 °C. The results obtained with this quadratic equation for b1 agree well with the test results of the linear model in the previous paper (see the citation above) up to 25 °C. The most important new test results above that temperature are reported here. Our quadratic model has additionally been tested with all the high-precision calorimetric data available in the literature for NaCl solutions. In this first part (Part 1) of the study, the test results from the thermodynamic quantities associated with partial molar enthalpy are reported. In the forthcoming second part (Part 2) of the study, the results of the quantities associated with the heat capacity of NaCl solutions will be considered. In the tests of these two parts, all calculations dealing with calorimetric data are performed in a new way. Both the calorimetric data and the vapor pressure data (from both direct and isopiestic measurements) can be predicted using the new Hückel equations within experimental error in dilute NaCl solutions from (0 to 80) °C. For comparison, also other Hückel models are considered and at best these apply up to the molality of the saturated NaCl solution at various temperatures. Following the success of the new models, new values for the activity coefficients, osmotic coefficients, relative apparent molar enthalpies, and relative partial molar enthalpies for NaCl solutions at rounded molalities are reported at the end of this Article. We have good reasons to believe that the new values contain the most reliable ones available for the given thermodynamic quantities.

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INTRODUCTION The two-parameter Hückel equation has recently been proven to be very useful when the thermodynamic properties of dilute solutions of pure electrolytes are predicted at 25 °C (see, for example, refs 1 to 3) and also in the temperature range from (0 to 70) °C.4 For example, in ref 4, two new approaches (methods I and II) were presented for the determination of the temperature dependences of the parameters in the Hückel equation. In that study, NaCl and KCl solutions were selected as test cases due to the large amount of thermodynamic data available in literature from solutions of these salts at various temperatures. In method I of ref 4, the following four-step © 2017 American Chemical Society

procedure was used for the determination of the temperature dependences: (1) At 25 °C, the parameter values from ref 1 were utilized. (2) New parameter values for 0 °C were determined from the existing freezing point data. (3) It was observed that parameter B has for NaCl and KCl practically the same values as those obtained in ref 1 for 25 °C. For NaCl, for Special Issue: Memorial Issue in Honor of Ken Marsh Received: January 27, 2017 Accepted: June 6, 2017 Published: June 26, 2017 2617

DOI: 10.1021/acs.jced.7b00091 J. Chem. Eng. Data 2017, 62, 2617−2632

Journal of Chemical & Engineering Data

Article

Table 1. Debye−Hückel Parameters (α and αT)a, the Vapor Pressure of Pure Water (p1*)b, the Molar Volume of Liquid Water [V*m,1(liq)]b and the Second Virial Coefficient of Water Vapor (B1)c as Functions of the Temperature (t)

a

t

α

αT

p1*

Vm,1 * (liq)

B1

°C

(mol·kg−1)−1/2

J·(mol3·kg−1)−1/2

Pa

cm3·mol−1

cm3·mol−1

0 5 10 12.5 15 20 25 30 35 37.5 40 45 50 55 60 65 70 75 80

1.1293 1.1376 1.1462 1.1507d 1.1552 1.1646 1.1744 1.1848 1.1956 1.2012d 1.2068 1.2186 1.2308 1.2436 1.2568 1.2704 1.2846 1.2992 1.3143

1363 1449 1560 1625d 1690 1833 1988 2152 2324

3168.6

18.068

−1194

6310d 7381.2 9589.8 12345

18.139

−966

18.233

−811

18.480

−623

2505 2693 2890 3094 3306 3527 3757 3996 4245

19933

38564 47375

Given by Archer and Wang.27 bGiven by Kell.28 cCalculated from eq 3 in ref 29. dObtained by interpolation from the other values.

example, the value of B is 1.4 (mol·kg−1)−1/2. For NaCl and KCl solutions, therefore, parameter B was regarded as a constant at all temperatures so that only parameter b1 depends on the temperature. (4) On the basis of the values of b1 at 0 and 25 °C for both salts, a linear dependence of this parameter on the temperature was imposed. In method II considered in ref 4, the constant parameter values obtained from method I were accepted for B, and a quadratic temperature dependence for parameter b1 was determined mainly from the all amalgam cell data of Harned and Nims,5 and Harned and Cook6 for NaCl and KCl solutions, respectively. These electrochemical data cover the temperature range from (0 to 40) °C. Because these amalgam electrode data are not as accurate as the data used in the parameter estimations of method I, the resulting models are more utilitarian than the ones based on the linear model for b1. The equations for b1 from both methods were tested in ref 4 with all reliable electrochemical, isopiestic, and freezing point data available in the literature. Apparently, the Hückel parametrization obtained from method I applies very well up to the temperature of 25 °C, whereas the one obtained from method II applies quite well up to 80 °C. In the present study, the B and b1 values obtained through method II were used for NaCl solutions. This parametrization of the Hückel equations shall be designated as PII. Parametrization PI is connected here to the previous calculations of method I except that this method was modified to accommodate a quadratic temperature dependence for parameter b1 while maintaining traceability and transparency. In the determination of its coefficients, the freezing-point depression data of Scatchard and Prentiss7 and the cell potential difference data of Hornibrook et al.8 together with the isopiestic data of Robinson9 were used for temperatures 0 and 25 °C as described in ref 4. The third value required for the quadratic model was determined from the direct vapor pressure measurements from Gibbard et al.10 at 75 °C. From these three values, a traceable polynomial of the third degree in

temperature was obtained and tested against all thermodynamic data available in the literature for dilute NaCl solutions up to 80 °C. The test results obtained by method I in ref 4 agree closely to those obtained with the present parametrization PI in the temperature range (0 to 25) °C. We do not reiterate those results here but rather report the results above 25 °C for the isopiestic method and for the vapor pressure method which are the most accurate approaches available up to high molalities. The corresponding test results for PII have already been presented in ref 4 and are not repeated here. In this article, both PI and PII are thoroughly tested against the heats of dilution available in the literature for NaCl solutions and against the other literature values of quantities associated closely to these heats. In part 2 of this study, the wide amount of the heat-capacity data for NaCl solutions will later be considered. New calculation methods have been developed in these studies for the treatment of both enthalpy and heat capacity data. As the results of these two studies, simple but very accurate equations are presented for the calculation of the enthalpies and heat capacities in dilute aqueous NaCl solutions up to the temperature of 80 °C. These equations can be easily used in many practical applications of these important solutions, and they have a completely traceable theoretical background which is not usually true for the complex multiparameter equations nowadays commonly employed in this context. According to the results of the present study, it seems that PI applies well to dilute NaCl solutions from (0 to 50) °C but issues arise above a molality of 0.5 mol·kg−1 at temperatures higher than 50 °C. For the calculations at high temperatures ranging from (40 to 80) °C many data points are available from the heat-of dilution measurements by Ensor and Anderson11 at intervals of 10 °C. To well reproduce these data also, a cubic equation for b1 was determined from the values of this parameter at 0, 25, 50, and at 75 °C. The Hü c kel parametrization based on the cubic equation is designated as PIII. 2618



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THEORY It has been observed that the Hückel equations ln γ = −

α|z+z −| Im 1 + B Im

+ b1(m /mo)

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 ) (1)

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

α|z+z −| ⎡ ⎢(1 + B Im ) − 2 ln(1 + B Im ) ϕ=1− B3Im ⎢⎣ ⎤ 1 1 ⎥ + b1(m /mo) 2 1 + B Im ⎥⎦

−

ΔHapp =

(2)

(7)

where subscript 2 refers to the salt and 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 equation ∞ ΔHm,2 = Hm,2 − Hm,2 = ΔHapp + m

∂(ΔHapp) ∂m

(8)

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

(3)

∂(ΔHapp) (9)

∂T

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

where M1 is the molar mass of water (= 0.018015 kg·mol−1). The activity of water is connected to the vapor pressure of water over the solution (p1) and to that over the pure water (p*1 ) by ⎛p ⎞ * (liq))(p − p* ) (B1 − V m,1 1 1 ln a1 = ln⎜⎜ 1 ⎟⎟ + * RT p ⎝ 1⎠

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

H∞ m,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−1.1,2,4,12−26 In eqs 1 and 2, m is the molality, z+ is the charge number of the cation and z− that of the anion, 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 and were taken from ref 27. For a 1:1 electrolyte such as NaCl, |z+z−| is 1 and Im is the same as molality m. In pure solutions of a uniunivalent electrolyte, the osmotic coefficient is related to the activity of water a1, where symbol 1 refers to the solvent, by the following thermodynamic identity

ln a1 = −2mM1ϕ

(6)

Cm,2 = Capp + m

■

(4)

where B1 is the second virial coefficient of water vapor, V*m,1(liq) is the molar volume of pure liquid water, and T is the temperature in Kelvin. The relevant parameter values for the present study for this equation are given in Table 1 where the latter values were determined by Kell.28 According to eq 3 given in the paper of McCullough et al.,29 the value of B1 at 25 °C is −1194 cm3·mol−1 which is very close to the modern value calculated from the complicated equations of Harvey and Lemmon.30 As this correspondence holds at 37.5, 50, and 75 °C, the same eq (eq 3 in ref 29) was used here to obtain B1 values for those. In the previous studies,1,2,13−26 the nonideality correction of water vapor was omitted, and eq 4 was used without the latter term on the right-hand side in the calculation of vapor pressures. It is well-known that the influence of this correction in aqueous solutions at 25 °C is usually very small and has its highest value for the most concentrated solutions. In the isopiestic sets here, as in our previous studies, two almost equal vapor pressures are compared, and this correction is unimportant. Therefore, the simplified equation for the activity of the solvent is p a1 = 1 p* (5)

∂(Capp) (10)

∂m

RESULTS AND DISCUSSION Estimation of Hü ckel Parameters for 75 °C from the Vapor Pressure Data of ref 10 and Determination of a New Quadratic Equation for Parameter b1 for Calculations with Parametrization PI. In ref 10, experimental vapor pressures of NaCl have been reported for 0, 25, 37.5, 50, 75, and for 100 °C. These data were first used to determine a new Hückel parameter for 75 °C. At this temperature, 10 points are given starting from a molality of 1.047 mol·kg−1 and extending up to 6.126 mol·kg−1. In preliminary calculations, it was observed that all of these points could be predicted quite well with the Hückel equation when parameter B has a constant value of 1.4 (mol·kg−1)−1/2 and the value of 0.105 is used for parameter b1. An error plot with these parameter values is given in Figure 1. The vapor pressure errors for this plot have been calculated using ep = p (reported) − p (predicted)

(11)

For the experimental vapor pressures up to 4 mol·kg−1, the errors are smaller than or equal to 30 Pa and the vapor pressures vary in this case inside the approximate range from (31 000 to 37 000) Pa. Also for the five more concentrated solutions at this temperature, all absolute pressure errors are less than 115 Pa. It was observed in the previous study4 that the constant value of 1.4 (mol·kg−1)−1/2 for B in dilute NaCl solution can be used at all temperatures from (0 to 80) °C while the following three

1

For the data obtained by Gibbard et al.10 through direct vapor pressure measurements, this nonideality correction has been made with eq 4. The values of p1* at various temperatures given by Kell28 are reported here in Table 1. 2619



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Figure 1. Plot of ep (eq 11), the deviation between the vapor pressure measured by Gibbard et al.10 at 37.5, 50, and 75 °C and that predicted by the suggested methods for a NaCl solution as a function of molality m. The predicted vapor pressure was calculated by eqs 2, 3, and 4 using the Hückel parametrizations PI and PIII for the osmotic coefficients. Symbols: ●, t = 75 °C, PI; ○, 75 °C, PIII; ▼, 50 °C, PI; △, 37.5 °C, PI.

traceable values are considered to be known for parameter b1: b1(0 °C) = 0.0077, b1(25 °C) = 0.0716, and b1(75 °C) = 0.105. The first value was determined in ref 4 from the freezing point data, the second in ref 1 from the cell potential differences (cpds) measured on concentration cells with transference and from the isopiestic data, and the third here from the direct vapor pressure data. With these three values, a quadratic equation Figure 2. Plot of eip (eq 13), 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 the isotonic solutions measured by Davis et al.31 at 45 °C (graph A, ●), Humphries et al.32 at 60 °C (B, ●), Moore et al.33 at 80 °C (B, ○), and Hellams et al.34 at 45 °C (A, ○) as a function of molality my. The isopiestic molalities for the data reported by Hellams et al.34 were reproduced from the smoothed values reported for the isopiestic ratios (see ref 4). All calculation were made using parametrization PI.

⎛ t ⎞ ⎛ t ⎞2 b1 = 0.0077 + 0.0031853⎜ ⎟ − 0.00002517⎜ ⎟ ⎝ °C ⎠ ⎝ °C ⎠ (12)

where t is the temperature in Celsius, can be deduced for the temperature dependence of parameter b1. This equation was tested in the present study using all important experimental data that have been published for dilute NaCl solutions up to a temperature of 80 °C. As mentioned, the parametrization obtained through the described method is denoted PI. All test results obtained with PI in the temperature range from (0 to 25) °C are very close to those obtained by method I in ref 4 where no calorimetric data have been considered. As described in the Introduction, in method I of ref 4, a linear temperature dependence was determined for parameter b1 from its known values at 0 and 25 °C. Here, PI is first tested with the data used in the parameter estimation for the temperature of 75 °C. The results for 37.5 and 50 °C in ref 10 were also included in the tests. All results of these tests are presented in Figure 1. According to this figure, PI applies quite well also to the results of the concentrated solutions in these three sets. The PI parameters were then tested with the isopiestic data measured by Davis et al.31 at 45 °C, Humphries et al.32 at 60 °C, and Moore et al.33 at 80 °C in NaCl and KCl solutions. The smoothed isopiestic ratios for NaCl and KCl solutions from the data of Hellams et al.34 at 45 °C were also included in the tests. These isopiestic sets were now used as in ref 4: The vapor pressures of water over the isotonic NaCl and KCl solutions of these sets were calculated, and the resulting vapor pressures were compared. NaCl is considered as the reference electrolyte (x) and KCl as the tested electrolyte (y). The two graphs of Figure 2 give the results of the calculations as error plots for

each set. In these plots, the isopiestic vapor pressure error (eip) is defined by e ip = px − py (13) and presented as a function of the KCl molality (= my). The vapor pressures px and py were calculated using eqs 2, 3, and 5. Graph A of Figure 2 shows the results at 45 °C (refs 31 and 34) and graph B those at 60 and 80 °C (refs 32 and 33). For KCl solutions, the previous constant value of 1.3 (mol·kg−1)−1/2 was used for parameter B (ref 4), and the following best values of 0.0396, 0.0486, and 0.044 for parameter b1 at 45, 60, and 80 °C, respectively. In all high-quality isopiestic sets of Figure 2, PI predicts the data excellently almost up to a KCl molality of 4.0 mol·kg−1. Equations of b1 for the Calculations of Parametrizations PII and PIII. The main interest in the present study is to test PI with the existing calorimetric data. However, we also included in these tests the previous quadratic equation for parameter b1 (from ref 4) which was observed to apply sufficiently for many practical purposes for dilute NaCl solutions up to a temperature of 80 °C. This equation has the form 2620



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describe the quality of the fit, the standard deviation about the regression is included in the table for each molality.

⎛ t ⎞ ⎛ t ⎞2 b1 = 0.012297 + 0.002712⎜ ⎟ − 0.0000253⎜ ⎟ ⎝ °C ⎠ ⎝ °C ⎠ (14)

Table 2. Parameter Values for the Quadratic Equation of Excess Molar Gibbs Energy Divided by T with Respect to the Temperature (eq 16 in the Text) for NaCl Solutions Obtained by Using Parameterization PI

and the calculations associated with this parametrization are designated here as those of PII. In the tests presented below, it was observed that PI does not apply well to the existing enthalpy data above a temperature of about 60 °C. At temperatures 40, 50, 60, 70, and 80 °C, Ensor and Anderson11 have reported many heat of dilution points for NaCl solutions concerning molalities from 0.007 mol·kg−1 up to the molality of the saturated solutions (see the tables given in the Supporting Information). In preliminary calculations, it was observed that the data above 60 °C can be better predicted if the following cubic equation is employed for parameter b1: ⎛ t ⎞ ⎛ t ⎞2 b1 = 0.0077 + 0.0033673⎜ ⎟ − 0.00003648⎜ ⎟ ⎝ °C ⎠ ⎝ °C ⎠ 3 ⎛ t ⎞ + 0.000000161⎜ ⎟ ⎝ °C ⎠ (15)

This equation was determined by assuming that B = 1.4 (mol· kg−1)−1/2 and that the equation for b1 fulfills the following four conditions: b1(0 °C) = 0.0077, b1(25 °C) = 0.0716, b1(50 °C) = 0.105, and b1(75 °C) = 0.123. The value at 50 °C is almost the same as that for the quadratic equation (i.e., that of eq 12) but the value at 75 °C is different, and the new value results in a better agreement with the heat of dilution data from Ensor and Anderson.11 The calculations based on eq 15 are referred here as calculations of PIII. It is important to realize as shown in Figure 1 that PIII does not predict very well the vapor pressure data of Gibbard et al.10 at 75 °C used as a fixing temperature of eqs 12 and 15. This is a serious shortcoming as the vapor pressure data are likely more accurate than the corresponding heat of dilution data. Determination of a Quadratic Equation for ΔGex/(mT) with Respect to the Temperature at Rounded Molalities from (0 to 1.0) mol·kg−1 for the Calculations of the Relative Apparent Molar Enthalpies. The following simple treatment was observed to give accurate values for the interpretation of the complicated heat results available for dilute NaCl solutions at various temperatures up to 80 °C. We use the results from PI as examples of the new strategy because the calculations of this parametrization lead to a fully transparent treatment. The strategy was based on a simple quadratic equation for a derived quantity associated closely to the excess Gibbs energy, namely ΔGex/(mT), as a function of the temperature. These equations were determined at several rounded molalities from (0.001 to 1.0) mol·kg−1. First, after the Hückel parameters B and b1 were obtained for a certain temperature, the excess Gibbs energies (ΔGex) were calculated for these series of different molalities from eqs 1, 2, and 6. These calculations were made at all temperatures from (0 to 80) °C in intervals of 5 °C. Then we study the ΔGex values at different temperatures using a constant molality. Second, a quadratic equation of the type ΔGex = u + v(T − T0) + w(T − T0)2 mT

a

u

103v

106w

103(sG)a

m/mo

J·mol−1·K−1

J·mol−1·K−2

J·mol−1·K−3

J·mol−1·K−1

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

−0.38323 −0.53484 −0.82421 −1.13329 −1.54245 −2.27019 −2.97959 −3.45901 −3.82636 −4.37843 −4.7908 −5.1200 −5.3936 −5.6273 −5.8307 −6.0107 −6.1716

−0.50229 −0.68510 −1.00504 −1.2995 −1.6000 −1.8123 −1.4711 −0.8128 0.0002 1.8786 3.9496 6.1346 8.3965 10.714 13.073 15.465 17.882

−3.3904 −4.8580 −7.8900 −11.504 −16.998 −29.33 −45.71 −60.18 −73.72 −99.29 −123.70 −147.41 −170.7 −193.6 −216.3 −238.7 −261.1

0.036 0.050 0.077 0.106 0.144 0.21 0.28 0.32 0.36 0.41 0.45 0.48 0.51 0.53 0.55 0.57 0.58

Standard deviation about the regression.

The relative apparent molar enthalpy of the salt can then be calculated from the values of parameters u, v, and w in Table 2 by using ΔHapp = −T 2

∂(ΔGex /mT ) = −T 2[v + 2w(T − T0)] ∂T (17)

As an example, the resulting enthalpies are given in Table 3 for 25 °C. The symbol of these relative apparent molar enthalpies obtained by parametrization PI is ΔHapp,I in the table and below. The relationship between the resulting ΔHapp values and molalities was then determined by fitting these points with a function of the following type ΔHapp = a1 + αT m /mo + a 2(m /mo) + a3(m /mo)3/2 (18)

where a1, a2, and a3 are fitting parameters, and the theoretical value based on the Debye−Hückel theory was used for the coefficient of the square root term (i.e., for αT). These theoretical values were taken from ref 27 and given here in Table 1. The resulting estimated parameters a1, a2, and a3 for this equation at 25 °C and the other temperatures under consideration are collected in Table 4. The standard deviation about the regression is also given for each fit in the table. The outlined empirical approach was adopted instead of the more straightforward one of differentiating Hückel equations with respect to the temperature because of two reasons: (1) The Hückel equation by its nature is already semiempirical, and while the derivatives are mathematically complicated (see ref 3), they offer no theoretical insights and significance. (2) The high-precision calorimetric measurements are very difficult to perform and their accuracies are only seldom comparable to easier isothermal methods such as electrochemical or isopiestic

(16)

where T0 = 273.15 K, was fitted to these results at each of the molalities. The resulting values of parameters u, v, and w are given for PI at the different molalities used in Table 2. To 2621



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Table 3. Values of Excess Molar Enthalpya and Relative Apparent Molar Enthalpy Obtained by Using Parametrization PI or PII for NaCl Solutions at 25 °C and the Errors When the Apparent Values Are Predicted by eq 18 with the Parameter Values Given in Tables 4 and 5 m/m

o

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 a

ΔHapp,I

(eH,I)b

ΔHapp,II

(eH,II)b

−1

−1

−1

J·mol−1

J·mol

59.7 82.5 124.4 166.7 217.8 291.6 334.0 339.9 327.8 274.5 198.9 110.0 12.3 −91.8 −200.7 −313.4 −429.1

J·mol

4.8 4.5 2.7 2.5 0.5 −3.5 −6.3 −6.8 −6.0 −2.8 0.7 3.7 5.3 5.2 3.2 −0.8 −6.9

J·mol

59.9 82.5 125.8 169.7 224.2 308.2 368.0 391.3 396.8 378.7 338.4 284.8 222.3 153.6 80.1 2.7 −77.6

In Table 3, the apparent enthalpies predicted using eq 18 are also given for 25 °C as the errors when compared to the values obtained using PI, and the agreement is good. This table also gives a method to calculate the excess molar enthalpies based on the relative values obtained using parametrization PI or PII with the value of Crist and Cobble35 for the partial molar enthalpy at infinite dilution. This value is 3824 J·mol−1. The calculations described in this section for PI were also performed for PII and PIII. The parameters of the final functions for the molality dependence of apparent molar enthalpies at various temperatures are given in Table 5. For 25

4.8 4.1 3.7 2.3 0.2 −3.8 −6.6 −7.0 −6.1 −2.7 1.1 4.0 5.5 5.4 3.3 −1.0 −7.4

Table 5. Parameter Values for the Equation of the Dependence of Relative Apparent Molar Enthalpy on the Molality (eq 18 in the Text) for NaCl Solutions Obtained by Using Parametrization PII or PIII

The excess molar enthalpy can be calculated from equation

ΔHex,X

∞ where X is I or II and refers to the = ΔHapp,X + Hm,2 parameterization PI or PII and, for example, Criss and Cobble35 have presented the value of 3824 J·mol−1 for the partial molar enthalpy at infinite dilution at 25 °C. bThe difference between the suggested and predicted value. m

Table 4. Parameter Values for the Equation of the Dependence of Relative Apparent Molar Enthalpy on the Molality (eq 18 in the Text) for NaCl Solutions Obtained by Using Parametrization PI

a

t

a1

a2

a3

103(sH)a

°C

J·mol−1

J·mol−1

J·mol−1

J·mol−1

0 5 10 12.5 15 20 25 30 35 40 45 50 55 60 65 70 75 80

−9.67 −6.57 −4.97 −4.82 −4.42 −4.37 −4.93 −5.15 −5.93 −6.66 −7.33 −8.10 −8.86 −9.62 −10.43 −11.36 −12.24 −13.27

−3198.75 −3117.25 −3073.77 −3066.67 −3052.67 −3040.26 −3032.18 −3024.77 −3010.47 −2994.77 −2970.81 −2943.28 −2904.95 −2859.95 −2808.52 −2751.85 −2689.06 −2620.85

519 500 510 526 539 579 627 682 738 800 863 931 998 1068 1141 1218 1299 1384

5.5 4.5 4.1 4.2 4.1 4.3 4.7 5.2 5.6 6.2 6.6 7.3 7.9 8.5 9.1 9.7 10.4 11.2

a

t

a1

a2

a3

103(sH)a

°C

J·mol−1

J·mol−1

J·mol−1

J·mol−1

parametrization

0 5 10 12.5 15 20 25 30 35 40 50 60 70 75 80

−7.84 −5.09 −3.94 −3.96 −3.85 −4.28 −5.19 −6.32 −7.54 −8.87 −8.1 −9.6 −11.34 −12.22 −13.24

−2868.64 −2784.44 −2737.22 −2727.95 −2713.27 −2698.91 −2689.95 −2679.36 −2664.85 −2648.90 −3276.9 −3362.5 −3442.1 −3480.3 −3518.2

492 480 496 515 532 580 637 699 765 837 931 1068 1218 1299 1384

4.8 4.0 3.7 3.8 3.9 4.3 4.9 5.5 6.2 7.0 7.3 8.5 9.8 10.4 11.2

PII PII PII PII PII PII PII PII PII PII PIII PIII PIII PIII PIII


°C, the relative apparent values calculated by PII and those predicted by eq 18 with the parameter values in Table 5 are also given as errors in the same way as for PI in Table 3. The errors in Table 3 and the standard deviations in Tables 4 and 5 are all close to each other for PI and PII or PIII. Tests of the New Equations for the Apparent Molar Enthalpy with the Heat of Dilution Data. The technique of the measurement of heats of dilution of pure electrolytes was developed to a high-precision by Lange and Robinson. The details of this technique and preliminary results for many electrolytes can be found in a review of these researchers.36 In the pioneering high-precision work,37 Robinson applied this technique to dilute NaCl solutions at 25 °C. In this paper, values of molar heat of dilution (= ΔHm,dil) are reported for several initial concentrations of NaCl (denoted here as minitial = mi) and final concentrations of NaCl (mfinal = mf). Actually, Robinson reported the data by using molarities instead of molalities but the correction between these composition variables is not important for the solutions considered in the paper (extending up only to a molarity of 0.1 mol·dm−3). The full data are reported in Table 6 and as can be seen the data contain several repeat determinations. These data were predicted here using PI and PII. The molar heat of dilution can be calculated from the apparent molar enthalpies of the final and initial solutions by


methods. Concurrently, the temperature derivatives of the parameters of the Hückel equation obtained from calorimetric data are usually neither accurate nor theoretically relevant.

ΔHm,dil = ΔHapp(mf ) − ΔHapp(m i ) 2622

(19)



Article

Table 6. Heats of Dilution Reported by Robinson37 for NaCl Solutions at 25 °C and the Resulting Errors (Equation 20) When These Values Are Predicted by Parameterization PI and PII ΔHm,dil o a

o a

(mi/m )

(mf/m )

0.0125 0.0125 0.0125 0.0125 0.0125 0.0125 0.0250 0.0250 0.0250 0.0250 0.0250 0.0250 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.1000 0.1000 0.1000 0.1000

0.000754 0.000754 0.000754 0.000385 0.000385 0.000385 0.001515 0.001515 0.001515 0.00077 0.00077 0.00077 0.00302 0.00302 0.00302 0.00302 0.00154 0.00154 0.00154 0.00154 0.00605 0.00605 0.00308 0.00308

J·mol

−1

−132 −122 −129 −144 −135 −137 −167 −170 −164 −189 −193 −190 −203 −205 −200 −207 −232 −233 −224 −231 −220 −222 −257 −257

eH,I

It is interesting that Gulbransen and Robinson did not extend their high-precision measurements to higher molalities than about 0.8 mol·kg−1. This omission may possibly be because they experienced problems in maintaining a sufficient accuracy in these stronger solutions. Young and Machin39 measured heats of dilution for NaCl solutions at 0, 12.5, and 25 °C, and these data and the results of the ones predicted using PI and PII are presented in Table S3. In their data sets, the molalities of the initial and final solutions were always close to each other, and therefore, the heats are small. At 25 and 12.5 °C, the most dilute pair of solutions has a molality of about 0.2 mol·kg−1, and only the heats of this pair can be predicted well using PI. The results of the second pair around a molality of 0.5 mol·kg−1 and those of the third pair around 0.8 mol·kg−1 are predicted only satisfactorily with this parametrization. On the other hand, all results up to 1 mol·kg−1 follow accurately the predictions of PI at 0 °C. Parametrization PII applies well to all data presented in the table up to 6.0 mol· kg−1 but at 0 °C up to 1 mol·kg−1 PI is slightly better. In Table S4 (see Supporting Information), the results of the tests with more recent heat of dilution data for NaCl solutions are presented. These data have been determined using flow microcalorimeters by Fortier et al.40 at 25 °C. It is interesting to observe that also these data extend only up a molality of 1.0 mol·kg−1. All points in these data up to 0.2 mol·kg−1 can be predicted well by using PI and PII, but PII also applies to the less dilute solutions. Table S5 shows the test results from the data of Langer and Meixner (one point at 25 °C), Vaslow (three points at 25 °C), Craft and van Hook41 (two points at 25 °C and one point at 10 °C), Wood et al.42 (nine points at 25 °C), and Leung and Millero43 (seven points at 30 °C). The data from the first and second study in this list were taken from the paper of Fortier et al.40 Usually the data in this table are from such concentrated solutions that PI does not apply well. On the other hand, PII applies quite well to the data of Vaslow and those of Craft and van Hook41 at 25 °C and satisfactorily to those of Wood et al.42 at this temperature. It also applies excellently to the seven points from Leung and Millero43 at 30 °C. Tables S6 and S7 in the Supporting Information display the heat-of dilution results from Messikomer and Wood44 at 25, 50, and 75 °C based on flow calorimeters. Their data were predicted using PI and PII at 25 °C in Table S6 and using methods PI and PIII at 50 and 75 °C in Table S7. Only the points where the initial molalities are 0.0744 and 0.1492 mol· kg−1 can be predicted well using PI in Table S6. On the other hand, all dilution results at 25 °C in this table can be predicted quite well using PII. According to Table S7, all heat results at 50 °C can predicted well using PI at least up 1.38 mol·kg−1. PI is in this case better than PIII. The opposite is true at 75 °C where PIII explains the heat-of dilution results better than PI. As mentioned above at higher temperatures, the most important source of heat of dilution data is the article of Ensor and Anderson.11 In their data, separate sets are given at temperatures ranging from (40 to 80) °C at intervals of 10 °C. The points measured at 40 °C were predicted here with PI and PII and the results are given in Table S8 (see Supporting Information). The points up to 0.2 mol·kg−1 can be excellently predicted using PI, and the remaining points up to 0.8 mol·kg−1 at least satisfactorily using this parametrization. Up to the same molality, PII also satisfactorily reproduces the points in the set. The points measured at the other temperatures by Ensor and Anderson were predicted using PI and PIII, and the results are

eH,II −1

J·mol

0.8 10.8 4.1 3.5 12.7 10.1 1.2 −2.1 4.6 −1.0 −5.1 −1.8 −2.8 −5.7 −0.7 −7.0 −5.2 −6.1 2.3 −4.0 −11.8 −13.5 −13.1 −12.7

J·mol−1 4.7 14.8 8.1 7.6 16.8 14.3 9.3 6.0 12.7 7.4 3.2 6.5 13.4 10.5 15.5 9.2 11.5 10.6 19.0 12.7 20.7 19.0 20.3 20.8

a

Concentrations are given in the original paper instead of the molalities. The difference between these two is not significant in these dilute solutions.

where the apparent enthalpies were obtained using eq 18 with the parameters given in Table 4 or 5. The deviation between the experimental and predicted values is calculated from eH,X = ΔHm,dil (observed) − ΔHm,dil (predicted using PX ) (20)

with the predicted values generally obtained by PX where X is I, II, or III. According to Table 6 where these errors have been predicted for PI and PII, the heat of dilution values from Robinson can be predicted well using these parametrizations but the former applies slightly better to these heats. Two years later, Gulbransen and Robinson38 reported heat of dilution data for NaCl solutions at 25 °C but also at 20, 15, and 10 °C. These data are presented in Table S1 for 25 °C and in Table S2 for the other temperatures. The two tables are given in the Supporting Information of the present study as also all other tables concerning here the heat of dilution data. The experimental heats in Tables S1 and S2 were predicted using PI and PII, and the results are shown in the two tables. Table S1 contains also results from less dilute solutions than Table 6 below. Both parametrizations predict well these data up to a molality of about 0.2 mol·kg−1, but above this molality only PII gives good results. According to Table S2, calculations associated with PI explain excellently these data at all three temperatures up to 0.2 mol·kg−1 and at 10 °C also up to 0.4040 mol·kg−1. Parametrization PII does not explain these data as well, but it works well at 20 °C in the initial molality range (0.1000 to 0.8160) mol·kg−1 as in the case of 25 °C (Table S1). 2623



Article

given in Table S9. At 50 °C, the calculations based on PI predict well the data up to 2 mol·kg−1 but the data set contains some serious outliers. PIII does not apply to these data as well. At 60, 70, and 80 °C, PI apply satisfactorily only to the most dilute pairs of solutions. PIII works well at 60 °C up to 0.4 mol· kg−1, at 70 °C up to 2.0 mol·kg−1, and finally at 80 °C up to 1 mol·kg−1. The parameter estimation of PIII was based on the results of Ensor and Anderson11 at these high temperatures. However, as stated before, it is not certain whether that these data are fully reliable, as the resulting parameters cannot predict the vapor pressure data of Gibbard et al.10 very well (see Figure 1). Tests of the New Values for Relative Apparent Molar Enthalpies with Those Presented in the Literature. Criss and Cobble35 have measured integral molar heats of solution in very dilute NaCl solutions at 0.02, 5.00, 9.99, 15.00, 19.97, 25.00, 34.96, 45.00, 54.83, 64.86, 75.01, 84.08, and at 95.18 °C. These data were used here to calculate the experimental ΔHapp values at the following rounded temperatures 0, 5, 10, 15, 20, 25, 35, 45, 55, 65, and 75 °C using the values given in Table 7 Table 7. Partial Molar Enthalpies of Sodium Chloride at Infinite Dilution Used in the Calculation of the Observed ΔHapp Values from Integral Heat of Solution Data by Criss and Cobble35 for Figure 3 in the Present Study t/°C

−1 H∞ m,2/Jmol

0 5 10 15 20 25 35 45 55 65 75

7971 6937 6025 5209 4481 3824 2573 1423 335 −837 −2038

Figure 3. Plot of eH,app (eq 21), the deviation between the suggested relative apparent molar enthalpy for NaCl solutions and that predicted using parametrization PI of the present study (see text) as a function of the molality of NaCl (m). The suggested value has been calculated from the experimental integral heats of solution reported by Criss and Cobble35 at various temperatures (graphs A and B) and by Sanahuja and Cesari45 at 25 °C (graph A). Details of the calculation are given in the text. Symbols for graph A: ●, Criss and Cobble,35 25 °C; ○, Sanahuja and Cesari.45 Symbols for graph B: ●, 0 °C; ○, 5 °C; ▼, 10 °C; △, 15 °C; ■, 20 °C; □, 35 °C; ⧫, 45 °C; ◊, 55 °C; ▲, 65 °C; ▽, 75 °C.

for the partial molar enthalpies of NaCl at the infinite dilution in these temperatures. The values in this table were taken from the same paper of Criss and Cobble, and no correction was made to these values because the measured temperature was not exactly in some cases the same as the rounded one. The resulting experimental ΔHapp values were predicted by using PI, and the errors are presented in the two graphs of Figure 3 as a function of molality. Graph A shows the results at 25 °C and graph B at the other temperatures. The errors have been calculated using eH,app = ΔHapp (observed) − ΔHapp (predicted)

heat of dilution data (see Table 6 and Tables S1 and S2). The values determined by Robinson are recommended in the text book of Harned and Owen.46 The error plots for different sets are given in Figure 4. Results of the points where the molality is less than 0.005 mol·kg−1 has been omitted from the figure for all sets because of clarity. Figure 4 shows that PI can well reproduce the data at 15, 20, and 25 °C. When errors in these plots are compared to the experimental errors of the integral heats of solution in Figure 3, one sees the high accuracy of PI also in the case of 10 °C where the errors seem to be larger than for the other temperatures. Figure 4 additionally includes the errors between the PI predictions and the ΔHapp data suggested by Young and Vogel47 on the basis of their own heat of dilution experiments at 25 °C. It also includes the errors obtained from the heat of dilution data at around 25 °C determined by Parker48 on the basis of the evaluation of all literature calorimetric data available for NaCl solutions until the publication of this paper. Also the results from both of the new sets validate the use of PI in these dilute solutions. Finally, the predicted values from PI in dilute solutions were compared to the ΔHapp data that can be obtained from the multiparameter equations in the reviews of Pitzer et al.49 and

(21)

According to this figure, the integral heats of solution of Criss and Cobble support well the suggested parametrization at all temperatures. Sanahuja and Cesari45 have also reported the integral-heat of solution values for dilute NaCl solutions at 25 °C. In the calculation of the observed ΔHapp values, the partial molar enthalpy of the salt at infinite dilution (i.e., H∞ m,2) was taken as 3867 J·mol−1 according to that paper. The resulting relative apparent enthalpies were used here in the tests of PI and the results are shown in graph A of Figure 3. Evidently, the data from this set also support PI. PI was also tested with the ΔHapp values reported by Robinson37 and Gulbransen and Robinson38 based on their 2624



Article

Figure 4. Plot of eH,app (eq 21), the deviation between the suggested relative apparent molar enthalpy and that predicted using parametrization PI of the present study (see text) as a function of the molality of NaCl (m). The suggested value has been presented by Robinson37 at 25 °C (symbol ●), Young and Vogel47 at 25 °C (○), Parker48 at 25 °C (▼), and by Robinson and Gudbransen38 at 10 °C (symbol △), 15 °C (symbol ■), 20 °C (symbol □), and at 25 °C (symbol ◆). The results of the most dilute points have been omitted for clarity.

Clarke and Glew50 for thermodynamic properties of aqueous NaCl solutions. The third wide review on these properties written by Archer51 does not contain enthalpy tables and it is not, therefore, considered here. The multiparameter equations from these three sources apply to large temperature and pressure intervals and were also considered in ref 4. In the two reviews selected for this study, tables are given for the relative apparent molar enthalpies at various pressures and temperatures as a function of the molality. The results of the present comparison are shown in graphs A and B of Figure 5, and the errors are presented in the graphs as a function of the temperature at the rounded molalities used in the original tables. Graph A shows the results up to 0.1 mol·kg−1 from the paper of Clarke and Glew,50 and graph B shows those in the range from (0.1 to 1) mol·kg−1 from the paper of Pitzer et al.49 The starting molality of the values of quantities in the tables of the latter review was 0.1 mol·kg−1. The suggested ΔHapp values in graph A agree well with those obtained using PI at all temperatures except at 0, 70, and 80 °C. In our opinion, however, it is not clear at all that the suggested values in these three temperatures are more reliable than the predicted ones. In graph B, the errors are small at molalities 0.1 and 0.25 mol·kg−1 and quite small at 0.5 mol·kg−1 up to 60 °C. In the temperature range from (15 to 40) °C, this is in agreement with the results obtained above from the heat of dilution data, and the model of Pitzer et al.49 thus follows these experimental data. Graph C displays the errors for the calculations using PII up to 40 °C and PIII after that temperature for the review of Pitzer et al.49 All of these errors are small except those of the three strongest solutions for PII at 0 °C and for PIII at 50 and 60 °C. In the light of the results presented above, PII applies well also to the concentrated NaCl solutions measured around 25 °C. In these temperatures, we chose to further test this observation with the ΔHapp values reported in the literature for these larger molalities up to the saturated solutions. The results at 25 °C are given in graph A of Figure 6. It contains the differences between the values calculated with PII and the apparent heats suggested by Robinson,37 Young and Vogel,47

Figure 5. Plot of eH,app (eq 21), the deviation between the suggested relative apparent molar enthalpy and that predicted by using parametrization PI, PII, or PIII of the present study (see text) as a function of the temperature (t) at various rounded molalities. In graph A, the suggested value has been presented by Clarke and Glew50 and PI was used. In graphs B and C, the suggested value has been presented by Pitzer et al.49 and PI and PII or PIII were used, respectively. Symbols for graph A: ●, m = 0.01 mol·kg−1; ○, 0.02 mol· kg−1; ▼, 0.05 mol·kg−1; △, 0.1 mol·kg−1. Symbols for graph B: ●, m = 0.1 mol·kg−1; ○, 0.25 mol·kg−1; ▼, 0.5 mol·kg−1; △, 0.75 mol·kg−1; ■, 1.0 mol·kg−1. Symbols for graph C: ●, m = 0.1 mol·kg−1; ○, 0.25 mol· kg−1; ▼, 0.5 mol·kg−1; △, 0.75 mol·kg−1; ■, 1.0 mol·kg−1. In graph C, PII was used up to t = 40 °C and after that PIII was used. In graph B, the errors in molalities m = 0.75 mol·kg−1 and m = 1.0 mol·kg−1 at 80 °C and the error in m = 1.0 mol·kg−1 at 70 °C are outside the scale of the graph. The values of these errors are −673, − 928, and −545 J· mol−1, respectively. 2625



Article

solution. The trends in this graph are clear because the heat of dilution data from different sources seem to be consistent with each other. These data were used in the calculations of the suggested apparent molar enthalpies by the research groups considered in the graph. In graph B of Figure 6 are shown the eH,app values obtained using PII and the set of Lipsett et al.53 for 20 °C and those obtained using PI and PIII and the sets of Gibbard et al.10 at 0, 50, and 75 °C. The values for 0 and 50 °C have been calculated using PI and those for 75 °C using both PI and PIII. In the calculations of the calorimetric data from the former set at 20 °C, the value of 4481 J·mol−1 from Criss and Cobble35 was used for the Hm,2 at infinite dilution (see Table 7). According to this graph, the data corroborate the results of the calculations using PII. Also PI performs quite well with the enthalpy values recommended by Gibbard et al. for 0 and 50 °C up to a molality of 2 mol·kg−1. On the other hand, the thermodynamically derived enthalpy values of this study cannot be predicted well at 75 °C using PI but are predicted much better using PIII. Tests of the New Values for Relative Partial Molar Enthalpies with Those Presented in the Literature. In thermal literature, the heat of dilution or heat of solution values are often expressed also as the partial molar enthalpies of the solute. These values can be easily calculated using eq 8 from the polynomials presented in eq 18 and the parameter values given in Tables 4 or 5. Parametrizations PI, PII, and PIII were also tested here against the relative partial molar enthalpies suggested in the literature. In these tests, the reported ΔHm,2 values are compared to the predicted ones, and the resulting errors defined by Figure 6. Plot of eH,app (eq 21), the deviation between the suggested relative apparent molar enthalpy and that predicted using parametrization PI, PII, or PIII of the present study (see text) as a function of the molality (m) at various temperatures. In graph A, the suggested values are associated with the temperature 25 °C and only PII was used. In graph B, the suggested values of other temperatures are used. Symbols for graph A: ●, Robinson;37 ○, Young and Vogel;47 ▼, Lipsett et al.;52 △, Lipsett et al.;53 ■, Gibbard et al.;10 □, Parker.48 Symbols for graph B: ●, Lipsett et al.,53 PII, 20 °C; ○, Gibbard et al.,10 PI, 0 °C; ▼, Gibbard et al.,10 PI, 50 °C; △, Gibbard et al.,10 PI, 75 °C; ■, Gibbard et al.,10 PIII, 75 °C. In graph B, the errors at 75 °C for method PI above 0.6 mol·kg−1 are outside the scale of the graph.

eH,part = ΔHm,2 (reported) − ΔHm,2 (predicted)

(22)

are 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 PII was used in the calculations because of quite concentrated solutions only being considered. The following data have been included in this graph: Robinson,37 Messikomer and Wood,44 Craft and Van Hook,41 Parker48 (the data smoothed by Ensor and Anderson11 were used), Harned and Cook,54 and Smith and Hirtle.55 The corresponding heat of dilution data of refs 37, 41, 44, and 48 have been considered above. Harned and Cook54 determined thermodynamically their partial molar enthalpies from results of the amalgam cell measurements at various temperatures by Harned and Nims.5 Smith and Hirtle55 determined their values from the amalgam cell results of Harned and Nims and their own boiling-point elevation data. All of these data support PII as well as the corresponding ΔHapp data in graph A of Figure 6. The small clear trends observed in the graph have the same background as that described above in connection with Figure 6A because partial molar enthalpies are functionally related to the apparent ones. The amalgam cell data considered by Harned and Cook,54 however, did not give good enthalpy values above 1.0 mol·kg−1 and these values were omitted from the graph. Also the ΔHm,2 value of Smith and Hirtle55 at m = 0.05 mol·kg−1 shows a large error, and the enthalpy values of the sets from this source are not considered here further. In graph B of Figure 7 are given the error plots for the lower temperatures and the calculations of PI and PII have been used to predict these suggested data. The data were taken from the following sources: Craft and Van Hook41 (5, 15, and 40 °C), Messikomer and Wood44 (30 °C), and Harned and Cook54 (0, 5, 10, 15, 20, and 30 °C). Only solutions up to 1.0 mol·kg−1

Lipsett et al.,52,53 Gibbard et al.,10 and Parker48 starting at a molality of 0.16 mol·kg−1. The values suggested by Robinson37 for these less dilute solutions were based on the recalculation of the existing literature data. In the calculations of the calorimetric data from Lipsett et al., the value of 3824 J· mol−1 from Criss and Cobble35 was used for the partial molar enthalpy at infinite dilution (see Table 7). The values reported by Gibbard et al.10 were not determined calorimetrically but rather were calculated thermodynamically from the available data obtained using static vapor pressure and freezing-point depression measurements. The small errors in this graph lend to evidence for PII up to the saturated solutions despite the fact that the same small trend appears in the pattern of errors for all data sets. Like parametrization PI, PII is very simple and contains only four parameters depending on the electrolyte (see eq 14). Unlike PI, this parametrization is not completely traceable and PII was not able to explain very well all highprecision data presented in the literature for 25 °C (see ref 4). It would be very strange if PII applied within experimental error to all of these calorimetric data up to the saturated NaCl 2626



Article

Figure 7. continued Anderson,11 PIII, 60 °C; ◆, Messikomer and Wood,44 PIII, 70 °C; ◇, Ensor and Anderson,11 PIII, 70 °C; ▲, Messikomer and Wood,44 PIII, 80 °C; ▽, Ensor and Anderson,11 PIII, 80 °C. In graph A, all errors of the set of Harned and Cook54 above 1.0 mol·kg−1 are outside the scale of the graph as are some errors from this source at 1.0 mol·kg−1 in graph B.

were included in the graph and some results of very dilute solutions were omitted from it for clarity. Parameterization PI was used for temperatures of 0, 5, 10, and 40 °C, and PII was used for the other temperatures. The amalgam cell results of Harned and Cook at 40 °C cannot be predicted satisfactorily using PI and these results were omitted from the graph. Other data support at least satisfactorily the used calculation methods. In graph C of Figure 7 are shown the error plots for the higher temperatures and these plots have been calculated using PI and PIII. At temperatures 40 and 50 °C PI and at the other temperatures PIII was used in the calculations. The reported partial molar enthalpies from Messikomer and Wood44 and Ensor and Anderson11 were predicted using these parametrizations. All of these data except those at 80 °C at higher molalities than 1.2 mol·kg−1 can be at least satisfactorily explained with the calculation methods used. Errors for partial molar enthalpies increase as the molality increases, because the accuracy of the heat-of dilution data of refs 11 and 44 depends on the molality. These data were of course used in the calculation of the suggested partial molar enthalpies of the two research groups. Recommended Values for the Activity Coefficient, Relative Apparent Molar Enthalpy and Relative Partial Molar Enthalpy of NaCl and for the Osmotic Coefficient of Water in NaCl Solutions. On the basis of the wide evidence provided by the test results given here (see Figures 1 to 7 and Table 6 and Tables S1 to S9 in the Supporting Information) and in refs 1 and 4, the experimental data available in the literature for dilute NaCl solutions from (0 to 80) °C can often be predicted within experimental error using the new Hückel equations at least up to a molality of 1 mol· kg−1. Completely traceable thermodynamic quantities can be obtained by using parametrization PI at least up to 0.2 mol·kg−1 in all of these temperatures. Therefore, new values are now presented for the relative apparent and partial molar enthalpies (i.e., for ΔHapp and ΔHm,2) for these dilute solutions based on the model. Also the activity coefficient of NaCl and osmotic coefficient of water in NaCl solutions can be reliably calculated using PI. Reference 4 gives the values of these quantities up to 25 °C, and now these tables have been continued up to 80 °C. The activity coefficients obtained by PI are presented in Table 8 up to 25 °C, and the values in this table are very close to those presented in ref 4 for method I. Table 9 displays the new traceable activity coefficients in the range (30 to 55) °C, and Table 10 shows these values from (60 to 80) °C. The corresponding values for the osmotic coefficients are given in Tables 11, 12, and 13, and the values at 25 °C and below this temperature in Table 11 agree very well with those recommended on the basis of method I in ref 4. We collected the values for apparent molar enthalpies (ΔHapp) at rounded molalities in graphs A, B, and C of Figure 8 at the temperatures in the range (0 to 25) °C, (30 to 55) °C, and (60 to 80) °C, respectively. We also give the corresponding partial molar enthalpies (ΔHm,2) accordingly in graphs A, B, and C of Figure

Figure 7. Plot of eH,part (eq 22), the deviation between the suggested relative partial molal enthalpy of NaCl and that predicted by using parametrization PI, PII, or PIII of the present study (see text) as a function of molality m at various temperatures. In graph A, the suggested values are associated with the temperature 25 °C and only PII was used. In graph B, the suggested enthalpy values at the lower temperatures were predicted using PI and PII. Graph C contains the enthalpy data from higher temperatures and these data were predicted using PI and PIII. Symbols for graph A: ●, Robinson;37 ○, Messikomer and Wood;44 ▼, Craft and Van Hook;41 △, Harned and Cook;54 ■, Smith and Hirtle;55 □, Parker.48 Symbols for graph B: ●, Craft and Van Hook,41 PI, t = 5 °C; ○, Graft and Van Hook,41 PII, 15 °C; ▼, Graft and Van Hook,41 PI, 40 °C; △, Messikomer and Wood,44 PII, 30 °C; ■, Harned and Cook,54 PI, 0 °C; □, Harned and Cook,54 PI, 5 °C; ◆, Harned and Cook,54 PI, 10 °C; ◇, Harned and Cook,54 PII, 15 °C; ▲, Harned and Cook,54 PII, 20 °C; ▽, Harned and Cook,54 PII, 30 °C. Symbols for graph C: ●, Messikomer and Wood,44 PI, 40 °C; ○, Ensor and Anderson,11 PI, 40 °C; ▼, Messikomer and Wood,44 PI, 50 °C; △, Ensor and Anderson,11 PI, 50 °C; ■, Messikomer and Wood,44 PIII, 60 °C; □, Ensor and 2627



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Table 8. Recommended Activity Coefficients (γ) of Salt in Aqueous Sodium Chloride Solutions at Temperatures from (0 to 25) °C as a Function of Molality ma γ

m mol·kg

−1

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 a

Table 10. Recommended Activity Coefficients (γ) of Salt in Aqueous Sodium Chloride Solutions at Temperatures from (60 to 80) °C as a Function of Molality ma

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

mol·kg

0.9295 0.9052 0.8747 0.8539 0.8379 0.8248 0.8137 0.8041 0.7956 0.7880 0.7811 0.7690 0.7587 0.7541 0.7498 0.7418 0.7347

0.9291 0.9047 0.8741 0.8533 0.8373 0.8242 0.8131 0.8036 0.7951 0.7876 0.7807 0.7688 0.7586 0.7541 0.7498 0.7420 0.7350

0.9286 0.9041 0.8734 0.8525 0.8365 0.8235 0.8124 0.8029 0.7945 0.7870 0.7802 0.7684 0.7583 0.7538 0.7496 0.7419 0.7351

0.9281 0.9034 0.8726 0.8517 0.8357 0.8226 0.8116 0.8021 0.7937 0.7862 0.7795 0.7677 0.7578 0.7533 0.7492 0.7416 0.7349

0.9276 0.9028 0.8718 0.8508 0.8347 0.8217 0.8107 0.8011 0.7928 0.7853 0.7786 0.7669 0.7570 0.7526 0.7485 0.7410 0.7344

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 a

Table 9. Recommended Activity Coefficients (γ) of Salt in Aqueous Sodium Chloride Solutions at Temperatures from (30 to 55) °C as a Function of Molality m.a

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 a

60 °C

65 °C

70 °C

75 °C

80 °C

0.9228 0.8966 0.8640 0.8420 0.8253 0.8117 0.8003 0.7905 0.7819 0.7743 0.7675 0.7556 0.7457 0.7413 0.7372 0.7298 0.7232

0.9220 0.8955 0.8626 0.8404 0.8235 0.8098 0.7983 0.7885 0.7798 0.7721 0.7652 0.7533 0.7432 0.7388 0.7346 0.7271 0.7206

0.9212 0.8944 0.8611 0.8387 0.8217 0.8078 0.7962 0.7862 0.7775 0.7697 0.7627 0.7507 0.7405 0.7360 0.7318 0.7243 0.7176

0.9203 0.8932 0.8596 0.8370 0.8197 0.8057 0.7940 0.7839 0.7750 0.7672 0.7601 0.7479 0.7376 0.7331 0.7288 0.7212 0.7144

0.9194 0.8920 0.8580 0.8351 0.8177 0.8035 0.7916 0.7814 0.7725 0.7645 0.7573 0.7450 0.7345 0.7299 0.7256 0.7178 0.7109

The values have been calculated using parametrization PI (see text).

Table 11. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Sodium Chloride Solutions at Temperatures from (0 to 25) °C as a Function of Molality ma

γ

m mol·kg

−1

0.9300 0.9058 0.8753 0.8545 0.8385 0.8254 0.8142 0.8045 0.7960 0.7883 0.7813 0.7691 0.7587 0.7539 0.7495 0.7414 0.7341

The values have been calculated using parametrization PI (see text) .

−1

γ

m

ϕ

m −1

30 °C

35 °C

40 °C

45 °C

50 °C

55 °C

mol·kg

0.9270 0.9020 0.8709 0.8498 0.8337 0.8206 0.8096 0.8000 0.7917 0.7842 0.7775 0.7659 0.7561 0.7517 0.7476 0.7402 0.7336

0.9264 0.9012 0.8699 0.8487 0.8325 0.8194 0.8083 0.7988 0.7904 0.7830 0.7763 0.7647 0.7549 0.7505 0.7464 0.7391 0.7326

0.9257 0.9004 0.8689 0.8476 0.8313 0.8181 0.8070 0.7974 0.7890 0.7816 0.7749 0.7633 0.7535 0.7491 0.7451 0.7378 0.7313

0.9251 0.8995 0.8677 0.8463 0.8299 0.8167 0.8055 0.7959 0.7875 0.7800 0.7733 0.7617 0.7519 0.7475 0.7435 0.7361 0.7297

0.9243 0.8986 0.8666 0.8450 0.8285 0.8151 0.8039 0.7943 0.7858 0.7783 0.7715 0.7599 0.7500 0.7457 0.7416 0.7343 0.7278

0.9236 0.8976 0.8653 0.8435 0.8269 0.8135 0.8022 0.7925 0.7839 0.7764 0.7696 0.7578 0.7480 0.7436 0.7395 0.7321 0.7257

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 a


9. Without question, the values presented at these rounded molalities in the graphs are the most reliable ones determined so far for NaCl solutions and no high-quality experimental data in the literature contradict these values. In the present study, we have also examined more concentrated solutions than those considered in Tables 8 to 13. For these solutions, in several cases parametrizations PII and PIII give a better agreement than PI in the interpretation of the existing calorimetric data. Graphs A and B in Figure 10 show the relative apparent molar enthalpies for NaCl solutions at rounded molalities for these less dilute solutions in the same way as a function of the temperature as Figure 8. However, the values on these higher molalities are not as reliable as those

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

0.9769 0.9692 0.9596 0.9532 0.9484 0.9446 0.9414 0.9386 0.9363 0.9342 0.9324 0.9292 0.9266 0.9255 0.9245 0.9226 0.9210

0.9768 0.9690 0.9594 0.9531 0.9483 0.9445 0.9414 0.9387 0.9364 0.9344 0.9326 0.9296 0.9272 0.9261 0.9251 0.9234 0.9219

0.9767 0.9688 0.9593 0.9529 0.9482 0.9445 0.9414 0.9388 0.9365 0.9346 0.9328 0.9299 0.9276 0.9266 0.9257 0.9241 0.9227

0.9765 0.9687 0.9591 0.9528 0.9481 0.9443 0.9413 0.9387 0.9365 0.9346 0.9329 0.9301 0.9279 0.9270 0.9261 0.9246 0.9234

0.9763 0.9685 0.9588 0.9525 0.9479 0.9442 0.9411 0.9386 0.9364 0.9346 0.9329 0.9302 0.9281 0.9272 0.9264 0.9250 0.9239

0.9762 0.9682 0.9586 0.9523 0.9476 0.9439 0.9409 0.9384 0.9363 0.9345 0.9329 0.9302 0.9282 0.9273 0.9266 0.9252 0.9242


given in Figure 8 and the values from PII and PIII are not fully traceable like those from PI. It seems, however, that the enthalpy 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 enthalpy values obtained from multiparameter equations.49−51

■

CONCLUSIONS We present a new and fully traceable reparametrization of the two-parameter Hückel equations shown in eqs 1 and 2 for the 2628



Article

Table 12. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Sodium Chloride Solutions at Temperatures from (30 to 55) °C as a Function of Molality ma ϕ

m mol·kg

−1

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 a

30 °C

35 °C

40 °C

45 °C

50 °C

55 °C

0.9760 0.9680 0.9583 0.9520 0.9473 0.9436 0.9407 0.9382 0.9361 0.9343 0.9327 0.9301 0.9281 0.9273 0.9266 0.9253 0.9244

0.9758 0.9677 0.9580 0.9517 0.9470 0.9433 0.9403 0.9379 0.9358 0.9340 0.9324 0.9299 0.9280 0.9272 0.9265 0.9253 0.9244

0.9756 0.9675 0.9577 0.9513 0.9466 0.9429 0.9399 0.9375 0.9354 0.9336 0.9321 0.9296 0.9277 0.9269 0.9262 0.9251 0.9242

0.9753 0.9672 0.9573 0.9509 0.9462 0.9425 0.9395 0.9370 0.9349 0.9331 0.9316 0.9291 0.9273 0.9265 0.9258 0.9248 0.9239

0.9751 0.9669 0.9569 0.9504 0.9457 0.9420 0.9390 0.9365 0.9344 0.9326 0.9311 0.9286 0.9268 0.9260 0.9253 0.9243 0.9235

0.9748 0.9665 0.9565 0.9499 0.9451 0.9414 0.9384 0.9359 0.9338 0.9320 0.9304 0.9280 0.9261 0.9253 0.9247 0.9236 0.9228


Table 13. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Sodium Chloride Solutions at Temperatures from (60 to 80) °C as a Function of Molality m.a ϕ

m mol·kg

−1

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 a

60 °C

65 °C

70 °C

75 °C

80 °C

0.9746 0.9662 0.9560 0.9494 0.9446 0.9408 0.9377 0.9352 0.9331 0.9313 0.9297 0.9272 0.9253 0.9246 0.9239 0.9228 0.9221

0.9743 0.9658 0.9555 0.9489 0.9440 0.9401 0.9370 0.9345 0.9323 0.9305 0.9289 0.9264 0.9245 0.9237 0.9230 0.9219 0.9211

0.9740 0.9654 0.9550 0.9482 0.9433 0.9394 0.9363 0.9337 0.9315 0.9296 0.9280 0.9254 0.9235 0.9227 0.9220 0.9208 0.9200

0.9737 0.9650 0.9545 0.9476 0.9426 0.9386 0.9354 0.9328 0.9306 0.9286 0.9270 0.9244 0.9223 0.9215 0.9208 0.9196 0.9187

0.9734 0.9646 0.9539 0.9469 0.9418 0.9378 0.9345 0.9318 0.9296 0.9276 0.9259 0.9232 0.9211 0.9202 0.9195 0.9183 0.9173

Figure 8. Plot of ΔHapp (= ΔHex/m − H∞ m,2), the recommended value for the relative apparent molar enthalpy of NaCl of as a function of molality m according to the parametrization PI at various temperatures. Symbols for graph A: ●, t = 0 °C; ○, 5 °C; ▼, 10 °C; △, 15 °C; ■, 20 °C; □, 25 °C. Symbols for graph B: ●, t = 30 °C; ○, 35 °C; ▼, 40 °C; △, 45 °C; ■, 50 °C; □, 55 °C. Symbols for graph C: ●, t = 60 °C; ○, 65 °C; ▼, 70 °C; △, 75 °C; ■, 80 °C.


activity coefficient and the osmotic coefficient, respectively. In our model, the B parameter is treated as a constant, whereas the b1 parameter is obtained from a quadratic polynomial with respect to temperature. After exhaustive testing of this parametrization both in our previous publications1,4 and within this article in terms of the relative apparent molar enthalpies and partial molar enthalpies, we conclude that the thermodynamic quantities of dilute NaCl solutions up to 0.2 mol·kg−1 in the temperature range (0 to 80) °C are reproduced to within experimental error. On the basis of our calculations, we offer recommended values for osmotic coefficients, activity coefficients, relative apparent molar enthalpies, and relative partial molar enthalpies within this region that are completely

congruent with all the high-quality experimental data available, and thus represent the most accurate and reliable values reported for this system up to date. In addition to dilute NaCl solutions, the behavior of the suggested model was also examined in more concentrated ones. In several cases, another new and a previously suggested alternative parametrization yielded better agreement with experiments than the current one. Concurrently, these alternative parametrizations were employed in the prediction of the recommended relative apparent molar enthalpies. Even though these parametrizations lack traceability and are generally 2629



Article

Figure 10. Plot of ΔHapp (= ΔHex/m − H∞ m,2), the recommended value for the relative apparent molar enthalpy of NaCl of as a function of molality m in less dilute solutions at various temperatures. Parametrization I, II, or III was used in the calculations. Symbols for graph A: ●, t = 0 °C, parametrization PI; ○, 5 °C, PI; ▼, 10 °C, PI; △, 15 °C, PII; ■, 20 °C, PII; □, 25 °C, PII; ◆, 30 °C, PII; ◇, 35 °C, PII. Symbols for graph B: ●, t = 40 °C, PI; ○, 45 °C, PI; ▼, 50 °C, PI; △, 55 °C, PI; ■, 60 °C, PIII; □, 70 °C, PIII; ◆, 80 °C, PIII. The recommended values of ΔHapp at 25 °C in the larger molalities of 4.0, 5.0, and 6.0 mol·kg−1 are −1700, −1890, and −1910 J·mol−1, respectively. At t = 30 °C and at m = 4.0 mol·kg−1, the value is −780 J· mol−1.

performed in a new way. We show also in this second part that all existing heat capacity data can be predicted using the present parametrization PI for the Hückel equations within experimental error in dilute NaCl solutions from (0 to 80) °C.

H∞ m,2),

Figure 9. Plot of ΔHm,2 (= Hm,2 − the recommended value for the relative partial molar enthalpy of NaCl as a function of molality m according to the parametrization PI at various temperatures. Symbols for graph A: ●, t = 0 °C; ○, 5 °C; ▼, 10 °C; △, 15 °C; ■, 20 °C; □, 25 °C. Symbols for graph B: ●, t = 30 °C; ○, 35 °C; ▼, 40 °C; △, 45 °C; ■, 50 °C; □, 55 °C. Symbols for graph C: ●, t = 60 °C; ○, 65 °C; ▼, 70 °C; △, 75 °C; ■, 80 °C.

■

less reliable than the values obtained at more dilute solutions, these values are still useful for applications because the existing experimental data are often well reproduced and the calculation of new values at different temperatures is simple when compared with calculation of the enthalpy values obtained from alternative multiparameter equations.49−51 This article is the first of a two-part study. In the forthcoming second part (Part 2), we will focus on the quantities associated with heat capacity of NaCl solutions. Following the approach of this article, all calculations dealing with calorimetric data are

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b00091. Heat of dilution data existing in the literature for NaCl solutions at various temperatures and the results obtained with these data using the thermodynamic parametrization considered in the present study (PDF)

AUTHOR INFORMATION

Corresponding Author

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

Jaakko I. Partanen: 0000-0002-4519-0958 2630



Article

Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We dedicate this paper to the memory of Professor Kenneth N. Marsh in recognition of this many invaluable contributions to a broad range of physical chemistry.

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Traceable Thermodynamic Quantities for Dilute Aqueous Sodium

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