Mean Activity Coefficients and Osmotic Coefficients in Dilute Aqueous

Article pubs.acs.org/jced

Mean Activity Coefficients and Osmotic Coefficients in Dilute Aqueous Sodium or Potassium Chloride Solutions at Temperatures from (0 to 70) °C Jaakko I. Partanen* Department of Chemical Technology, LUT School of Engineering Science, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finland ABSTRACT: Two-parameter Hückel equations have been determined for the activity coefficient of the salt and for the osmotic coefficient of water in aqueous NaCl or KCl solutions at 0 °C from the highly accurate freezing point data of Scatchard and Prentiss (J. Am. Chem. Soc. 1933, 55, 4355−4362), and points were taken in the estimation up to a molality of 0.5 mol·kg−1. This molality is the upper limit for the use of the data from this method without any corrections to determine thermodynamic activity values for a constant temperature of 0 °C. The electrolyte parameters of the Hückel equation are B and b1, and the value obtained for parameter B in the case of both salts is closely the same as that determined in a previous NaCl and KCl study (Partanen, J. I.; Covington, A. K. J. Chem. Eng. Data 2009, 54, 208−219) from the isopiestic and cell potential difference (cpd) data at 25 °C. The resulting parameters were tested with the other precise freezing point data existing in the literature for these salts and with the cpd data existing for 0 °C. The tests using the latter data reveal that the new parameters apply up to a molality of 1.0 mol·kg−1. Using the parameter values obtained here for b1 at 0 °C and in the previous study at 25 °C (see the citation above), a linear dependence of the temperature was determined for this parameter for solutions of both salts. The resulting parameters for B and b1 were tested with the cpd and isopiestic data existing for NaCl and KCl solutions at the temperatures other than 0 and 25 °C, and they were observed to be reliable up to a molality of 1.0 mol·kg−1 in this temperature range. Above 25 °C, however, these simple temperature dependences for b1 are probably not sufficient. For a wider range, a quadratic temperature dependence of this parameter was determined for both salt solutions, and these models apply quite well up to 70 °C in these dilute solutions. These b1 models were mainly determined from the concentration cell data of Harned and Nims (J. Am. Chem. Soc. 1932, 54, 423−432) and Harned and Cook (J. Am. Chem. Soc. 1937, 59, 1290−1292) for NaCl and KCl solutions measured on cells containing an alkali metal amalgam electrode in the temperature range from (0 to 40) °C. These data are not as accurate as the other data used in the present parameter estimation, and therefore, the resulting models are more practical than the linear dependences of b1. The activity and osmotic coefficients from the new Hückel equations are tabulated here at rounded molalities in this wider temperature range, and these values are compared to those regarded as the most reliable in the literature in these connections. Also the activity and osmotic coefficients obtained using the Hückel parameters from Rowland and May (J. Chem. Eng. Data 2014, 59, 2030−2039) are considered here in detail.

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In this connection and in two previous studies5,6 where an equation of the Hückel type was also used, it was observed that the most reliable freezing-point set for both salts is the one measured by Scatchard and Prentiss.7 These sets were used in the present parameter estimation. The resulting Hückel parameters determined for 0 °C were tested with the data used in the estimation and with the high-precision freezing point data of Craft and van Hook8 for NaCl solutions and with those of Brown and Prue9 and Garnsey and Prue10 for KCl solutions. (3) It was observed that parameter B has in both cases closely the same value as that obtained in ref 1 for 25 °C. For both salts, therefore, parameter B was regarded as a constant at all temperatures and only parameter b1 depends on the temperature.

INTRODUCTION The two-parameter Hückel equation (with parameters B and b1) 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). In the present study, two new methods (method I and II) are presented for the determination of the temperature dependences of the parameters in this equation. The solutions of NaCl and KCl were selected here as examples because it seems that a sufficient number of thermodynamic data are available in literature for this purpose from solutions of these salts at various temperatures. In method I, the following procedure was used in the determination of these temperature dependences: (1) The parameter values from ref 1 for 25 °C were accepted. (2) New parameter values for 0 °C were determined from the existing freezing point data. The literature freezing points for NaCl and KCl solutions were studied in ref 4 where Pitzer parameters of these salts were determined for this temperature. © 2015 American Chemical Society

Received: July 2, 2015 Accepted: December 7, 2015 Published: December 28, 2015 286

DOI: 10.1021/acs.jced.5b00544 J. Chem. Eng. Data 2016, 61, 286−306

Journal of Chemical & Engineering Data

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(4) On the basis of the values of b1 at 0 °C and at 25 °C, a linear dependence of this parameter on the temperature was assumed for both salts. (5) The constant value of B and the linear dependence of b1 were tested with the experimental data existing in the literature for dilute NaCl or KCl solutions in this temperature range for temperatures other than 25 °C (for this temperature, the test results are presented in ref 1 but the most important results will also be repeated here). Some results that have been measured at temperatures slightly above the upper limit (25 °C) were additionally included in the tests. The new Hückel equations were tested using dilute points from the isopiestic measurements of Platford11 at 0 °C, and the osmotic coefficients reported by Childs and Platford12 for dilute NaCl and KCl solutions at 15 °C. The cell potential difference (cpd) data obtained by Harned and Nims13 using concentration cells with a Na amalgam electrode and two Ag−AgCl electrodes at 0.3, 12.5, 20, 25, 30, 37.5, and at 40 °C for dilute NaCl solutions were also available for the tests as well as the corresponding data measured by Smith14 at 0 °C and by Harned and Cook15 at temperatures from (0 to 40) °C using intervals of 5 °C for dilute KCl solutions. The latter KCl data15 do not contain real experimental points but smoothed cpd values are given at rounded molalities. These KCl data as well as the corresponding NaCl data13 were used in the tests of method I up to 40 °C. Finally, the new Hückel equations of this method were tested with the very precise data obtained using concentration cells with transference by Janz and Gordon16 for NaCl solutions and Hornibrook et al.17 for KCl solutions at 15, 25, 35, and at 45 °C. In the second method (method II), it was accepted the constant values from method I for parameter B, and a quadratic temperature dependence of parameter b1 was determined mainly from the all amalgam cell data of Harned and Nims,13 and Harned and Cook.15 The data from these amalgam electrode sets are not as accurate as the data used in the parameter estimations for method I and, therefore, the resulting models are more practical than the linear ones for b1. In the parameter estimation of method II, it was resorted to these less precise data because useful high-precision data are really available for 0 and 25 °C only. These quadratic models were tested with the data used in the parameter estimations and, additionally, with the data from the following sources: Results from the dilute points of the isopiestic measurements of Platford11 at 0 °C, Robinson18 at 25 °C, Davis et al.19 at 45 °C, Humphries et al.20 at 60 °C, and Moore et al.21 at 80 °C in NaCl and KCl solutions were included in the evaluation. Also the smoothed isopiestic ratios for NaCl and KCl solutions from the data of Hellams et al.22 at 45 °C and the osmotic coefficients reported by Childs and Platford12 for dilute NaCl and KCl solutions at 15 °C were used in the tests as well as the vapor pressures measured by Gibbard et al.23 for less dilute NaCl solutions at 25, 37.5, 50, and at 75 °C. The cpd data measured by Mussini et al.24 and Giordano et al.25 using cells containing a Na and K amalgam electrode, respectively, and a Ag−AgCl electrode at 10, 25, 40, 55, and at 70 °C were included in the consideration. Finally, method II was tested with the concentration cell data of Janz and Gordon16 (NaCl) and Hornibrook et al.17 (KCl). Also the third method (method III) was included in all considerations of the present study. In method III, the parameter values and their temperature derivatives determined by Rowland and May3 for the Hückel equation at 25 °C were used. Here, the parameter values were determined for other temperatures by assuming that the temperature derivatives of the parameters do not depend on the temperature.

Article

THEORY Earlier,1,2,26−40 it was observed that the subsequent Hückel equations 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 ln γ = −

ϕ=1−

α|z+z −| Im 1 + B Im

+ b1(m /mo)

(1)

⎤ α|z+z−| ⎡ 1 ⎢(1 + B Im ) − 2ln(1 + B Im ) − ⎥ 3 ⎢ + 1 B I B Im ⎣ ⎦ m ⎥

1 b1(m /mo) 2

+

(2) o

−1

In these equations, m is the molality, m = 1 mol·kg , z+ is the charge number of the cation and z− that of the anion, α is the Debye−Hückel parameter (its values at a pressure of 101.325 kPa at various temperatures were taken from ref 41 and are given in Table 1 of the present study), and the parameters Table 1. Debye−Hückel Parameter (α)a and the Vapor Pressure of Pure Water (p1*)b as Functions of the Temperature (t) t/°C

α/(mol·kg−1)−1/2

0 0.3 5 10 12.5 15 20 25 30 35 37.5 40 45 50 55 60 70 75 80

1.1293 1.1298c 1.1376 1.1462 1.1507c 1.1552 1.1646 1.1744 1.1848 1.1956 1.2012c 1.2068 1.2186 1.2308 1.2436 1.2568 1.2846 1.2992 1.3143

p1*/Pa 610.7 872.0 1227.6 1705.1 2338.4 3168.6 4245.1 5626.4 6310c 7381.2 9589.8 12345 15752 19933 31177 38564 47375

a

Given by Archer and Wang.41 bGiven by Kell.42 cObtained by interpolation from the other values.

dependent on the electrolyte are B and b1. For a 1:1 electrolyte like NaCl and KCl, |z+z−| is 1 and Im is the same as molality m. The osmotic coefficient is related to the activity of the water (a1, symbol 1 is used here for the solvent) in pure solutions of a uniunivalent electrolyte by the following thermodynamic identity

ln a1 = −2mM1ϕ

(3) −1

where M1 is the molar mass of water (= 0.018015 kg·mol ) and where the activity of water is related to the vapor pressure of water over the solution (p1) and to that of pure water (p*1 ) at the temperature under consideration by p a1 = 1 p* (4) 1

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Table 2. Values of the Parameters of Pitzer and Mayorga44 for eqs 5 to 8 at 25 °C in NaCl and KCl Solutions and Their Temperature Dependencies β0

Cϕ

β1

NaCl

0.0765

0.2664

0.00127

KCl

0.04835

0.2122

−0.00084

104(dβ0/dT)p

104(dβ1/dT)p

105(dCϕ/dT)p

−5.48 7.005d 8.12a 10.71d

−22.9b,c −10.54d −3.36a −5.095d

a

a,b

15.44 7.159d 9.78a 5.794d

Taken from ref 4. bThis value is correct instead of the value of −5.44·10−4 given erroneously in ref 4. cThis value is correct instead of the value of −2.29·10−5 given erroneously in ref 4. dDetermined by Silvester and Pitzer.45 a

up to a molality of 2 mol·kg−1. These derivatives are probably constants at least near to the temperature of 25 °C (298.15 K = T0) and, therefore, the following temperature dependences for parameters BRM and b1,RM can be presented in this case

It is checked by calculations with the second virial coefficients of water (see ref 40) that a more complicated equation than eq 4 is needed here in no other connections than in the treatment of vapor pressure data of ref 23. In this case, the activity of water was calculated as in that reference and the resulting values were reported in the present study. The values of p1* at various temperatures given by Kell42 are reported here in Table 1. The Pitzer activity and osmotic coefficient43 for solutions of a uniunivalent electrolyte have the following equations ln γ = f γ + Bγ (m /mo) + (3/2)C ϕ(m /mo)2 ϕ=1−

BRM = BRM (T ) ⎛ dB ⎞ = BRM, T0 + ⎜ RM ⎟ (T − T0) ⎝ dT ⎠T 0

2 ⎛ d2B ⎞ 1 ⎛ d BRM ⎞ 2 2 RM ⎟ T0(T − T0) ⎜ ⎟ ( T T ) + ⎜ − − 0 2 ⎝ dT 2 ⎠ dT 2 ⎠ ⎝ T T

(5)

0

m α 3 1 + 1.2 m /mo

+ (β 0 + β1e−2

m / mo

(9)

)(m /mo) + C ϕ(m /mo)2

b1,RM = b1,RM(T )

(6)

⎛ db1,RM ⎞ = b1,RM, T0 + ⎜ ⎟ (T − T0) ⎝ dT ⎠T

where ⎤ α⎡ m 2 mo o ⎥ f =− ⎢ + ln(1 + 1.2 m / m ) 3 ⎣ 1 + 1.2 m/mo 1.2 ⎦

0

2 ⎛ d2b ⎞ 1 ⎛ d b1,RM ⎞ ⎟ (T 2 − T02) − ⎜⎜ 1,RM ⎟ T0(T − T0) + ⎜⎜ 2 ⎟ 2 ⎟ 2 ⎝ dT ⎠ dT ⎠ ⎝ T T

γ

(7)

Bγ = 2β 0 +

β1mo ⎡ −2 ⎢1 − e 2m ⎣

m / mo

0

These equations form the basis of calculation method III. They will prove to be very useful and apply well to almost all less precise data available for NaCl and KCl solutions at least up to 70 °C.

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RESULTS AND DISCUSSION Determination of Hü ckel Parameters from Freezing Point Data for 0 °C and Tests of the Resulting Equation with the Data for This Temperature. In ref 4, the following equation was derived for the dependence between the activity of water and the freezing point Tf of water in solutions of a single electrolyte ⎡ (T − T *) ⎛1 ⎛ T ⎞⎤ 1 ⎞ f − R ln a1 = ΔH *⎜ − − ln⎜ f ⎟⎥ ⎟ + ΔCp,m⎢ f * ⎢⎣ Tf Tf ⎠ ⎝ Tf ⎝ Tf* ⎠⎥⎦ (11)

where Tf* is the freezing point of pure water (= 273.15 K), ΔH* is the molar enthalpy of water at this temperature (= 6009.5 J·mol−1), and ΔCp,m is the difference between the molar heat capacities of water as liquid and as solid at this temperature (= 37.87 J·K−1·mol−1). The sources of these values are given in ref 4. It was shown there, in addition, that this equation can be used for the determination of the activity parameters for the models describing the nonideality for the constant temperature of 0 °C up to a molality of about 0.5 mol·kg−1. Up to this limit, the fact that the freezing point data are not isothermal has no influence on the results. The freezing points from Scatchard and Prentiss7 were used here in the parameter estimation. For NaCl solutions 20 points up to a molality of 0.53070 mol·kg−1 were included in the

Table 3. Values of the Hückel Parameters (BRM and b1,RM) of Rowland and May3 for eqs 1 and 2 in NaCl and KCl Solutions and Their Temperature Dependencies for eqs 9 and 10 (mo = 1 mol·kg−1) at T0 = 298.15 K NaCl

KCl

1.321 1.34 −6.64 0.0883 1.07 −3.07

1.222 1.77 −5.30 0.0265 0.936 −2.31

0

(10)

⎛ m ⎞⎤ × ⎜1 + 2 m / m o − 2 o ⎟⎥ ⎝ m ⎠⎦ (8)

In these equations, β0, β1, and Cϕ are the parameters being dependent on the electrolyte. Pitzer and Mayorga44 have determined the values shown in Table 2 for NaCl and KCl solutions at 25 °C, and their temperature dependences are also given in the table. These derivatives were taken from refs 4 and 45. The former values4 were determined from the activity parameter values at 25 °C and from those determined using the freezing point data, and the latter values45 were obtained from the calorimetric data. On the basis of the published thermodynamic data (like calorimetric data) at 25 °C, Rowland and May3 suggested the values for the first and second derivatives of parameters B and b1 in eqs 1 and 2 (the parameters from this source are denoted here as parameters BRM and b1,RM, respectively). The resulting values for NaCl and KCl solutions are given in Table 3 and they are valid

BRM/(mo)−1/2 103dBRM/dT/K−1 (mo)−1/2 105d2BRM/dT2/K−2 (mo)−1/2 b1,RM 103db1,RM/dT/K−1 105d2b1,RM/dT2/K−2

0

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determination, and for KCl solutions were included 17 points up to 0.40548 mol·kg−1. The following new method is used here in the estimation, and it is essentially very close to the one used in our previous papers (see, for example, refs 1 and 2) for isopiestic data: From the freezing point data of Scatchard and Prentiss, the activities of water (denoted as a1,exp) were calculated using eq 11. Then, the parameters were estimated using the following equation (derived from eqs 2 and 3): f1 = ln a1,exp + 2M1m − − 2ln(1 + B m ) −

2αM1 ⎡ ⎢(1 + B m ) B3 ⎣ ⎤ 1 ⎥ 1+B m⎦

= f0 − b1M1(m2 /mo) = f0 + km2

(12)

where k = −b1M1/m . When parameter B has been fixed, eq 12 represents an equation of the straight line f1 versus m2. The slope of the straight line is k and parameter b1 can be calculated from this slope. The straight line should go through the origin, and therefore, parameter B must be determined so that the value of intercept f 0 is zero. In these estimations, the following results were obtained: BNaCl = 1.4 (mol·kg−1)−1/2, b1,NaCl = 0.0077 ± 0.0002, BKCl = 1.3 (mol·kg−1)−1/2, and b1,KCl = −0.0515 ± 0.0011 where the standard deviations are also given. The sets used in the parameter estimation and the sets of Craft and van Hook8 (NaCl), Brown and Prue9 (KCl), and Garnsey and Prue10 (KCl) were first used to test the resulting values. The tests were performed here as in ref 4. The freezing point depression (ΔTf) is defined in the following ordinary way o

ΔTf = Tf* − Tf

(13)

Using eqs 3, 11, and 13, the following equation can be derived for ΔTf ΔTf =

2RTf*MA mϕ 2RMA mϕ +

ΔH * Tf*

(

ΔCp,mΔTf + ΔCp,m(Tf* − ΔTf ) ln + 2RMA mϕ +

Tf* − ΔTf Tf*

)

ΔH * Tf*

(14)

This equation requires iterative calculations and it has also been previously used in the thermodynamic considerations of NaCl and KCl solutions based on Hückel equation.5,6 The experimental freezing point depressions were reproduced with the new models using eqs 2 and 14. The errors calculated by e T = ΔTf (observed) − ΔTf (predicted)

Figure 1. Plot of eT (eq 15), the deviation between the observed and predicted freezing-point depressions in NaCl and KCl solutions as a function of molality m. The observed values have been measured by Scatchard and Prentiss,7 Craft and Van Hook,8 Brown and Prue,9 and Garnsey and Prue.10 The predicted values were determined from eq 14 where eq 2 with the suggested Hückel parameters of methods I, II, and III (see text) were used for the osmotic coefficients. Symbols in graph A (all data from ref 7): ●, NaCl, method I; ○, NaCl, II; ▼, KCl, I; △, KCl, II. Symbols in graph B: ■, ref 7, NaCl, method III; □, ref 7, KCl, III; ●, ref 8, NaCl, I; ○, ref 8, NaCl, II; ⧫, ref 8, NaCl, III. Symbols in graph C (all data from KCl sets): ●, ref 10, method I; ▼, ref 9, I.

(15)

are then presented as a function of the molality. The resulting error plots for the solutions of both salts are shown in the three graphs of Figure 1 (see the results of method I in these graphs). Graph A in this figure reveals that the new Hückel models apply well to the experimental data used in the estimation. The largest absolute error for the points included in parameter estimation is 0.00032 K for the NaCl case, and for the KCl case it is 0.00091 K. Thus, both of these models are accurate, but however, the former data support slightly better the Hückel model for NaCl solutions than the latter for KCl solutions. This fact can also be seen in the values of the standard deviations presented above for parameters

b1 of NaCl and KCl solutions (0.0002 and 0.0011, respectively). According to graphs B and C in this figure, the new Hückel models apply also well to the other high-precision freezing-point data used in the tests. 289



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It was also possible to test the new Hückel equations for 0 °C using the electrochemical data. Harned and Nims,13 Smith,14 and Harned and Cook15 measured at low temperatures the concentration cells of the following type:

All points in Tables 4 and 5 were predicted using the new Hückel equations for the activity coefficients (see eq 17). The resulting cpd errors, defined by eE = E (observed) − E (predicted)

Ag(s)|AgCl(s)|MCl(aq, m1)|M(Hg)|MCl(aq, m2)|AgCl(s)|Ag(s) (16)

are also given in the tables as a function of molality m2 (see there the results of method I). The amalgam cell data can be explained within 0.8 mV with the new Hückel parameters and these data thus support quite well the new parameters up to a molality of 1 mol·kg−1. In the evaluation of the accuracy of these results, it is important to remember that the amalgam cells are not comparable in accuracy with the best electrochemical cells. Finally, the new Hückel parameters were tested with the isopiestic data measured by Platford11 at 0 °C for NaCl and KCl solutions. The most dilute points from these data could be used in the present tests, and they are given in Table 6. The vapor

where M(Hg) is the alkali metal amalgam electrode. In the set of Harned and Nims, the temperature t was 0.3 °C, M was Na, the molality of the reference solution was m1 = 0.1 mol·kg−1, the molality of the tested solution (= m2) varied from (0.05 to 4.0) mol·kg−1, and the most dilute points from this set are given in Table 4. In the set of Harned and Cook and in that of Smith, Table 4. Cell Potential Differences (E) Measured by Harned and Nims13 on Concentration Cells of Type 16, for Which m1 = 0.1 mol·kg−1 for NaCl Solutions at 0.3 °C, and the Errors (eE, eq 18) Obtained for the Suggested Calculation Methods of I, II, and III from These Data m2/mol·kg−1

(E/int. mV)a

[eE,I/mV]b

[eE,II/mV]c

[eE,III/mV]d

0.05 0.2 0.5 1.0 1.5 2.0

29.70 −29.64 −68.76 −98.76 −117.62 −130.99

−0.37 0.08 −0.02 −0.43 −1.46 −2.69

−0.38 0.11 0.06 −0.24 −1.16 −2.28

−0.34 0.09 0.22 0.49 0.26 −0.09

Table 6. Isopiestic Molalities (mx and my) Measured by Platford11 for NaCl (= x) and KCl (= y) Solutions at 0 °C and the Errors (eip, eq 19) of the Vapor Pressures over These Solutions Obtained Using the Calculation Methods of I, II, and III mx/mol·kg−1

my/mol·kg−1

[103eip,I/Pa]a

[103eip,II/Pa]b

[103eip,III/Pa]c

0.4038 0.5011 0.7290 1.018 1.408 2.020

0.4110 0.5126 0.7523 1.059 1.483 2.167

−12.4 −4.6 −8.3 −69.0 −157.5 −415.4

−15.2 −8.9 −17.0 −85.2 −186.4 −468.5

−13.9 −4.9 −0.5 −38.1 −61.5 −115.5

a 1 Int. V = 1.00034 V. bCalculated using method I. cCalculated using method II. dCalculated using method III.

t = 0 °C, M = K, m1 was 0.05 mol·kg−1 or 0.1 mol·kg−1, m2 varied from (0.1 to 4.0) mol·kg−1 or from (0.2 to 3.5) mol·kg−1, respectively, and the most dilute points from these two sets are given in Table 5. The cpd for the cells of this kind can be calculated from E=−

2RT 2RT ln(m2 /m1) − ln(γ2/γ1) F F

a

Calculated using method I. bCalculated using method II. cCalculated using method III.

pressures of water over the isotonic NaCl and KCl solutions of this set were calculated, and the resulting vapor pressures were compared. NaCl is considered here (and also later for all isopiestic sets considered) as the reference electrolyte (x) and KCl as the tested electrolyte (y). Table 6 also gives the results of the calculations (see those of method I) and there the definition of the isopiestic vapor pressure error (eip) is e ip = px − py (19)

(17)

Table 5. Cell Potential Differences (E) Measured by Smith14 or Harned and Cook15 on Concentration Cells of Type 16, for Which m1 was 0.1 mol·kg−1 or 0.05 mol·kg−1, Respectively, for KCl Solutions at 0 °C, and the Errors (eE, eq 18) Obtained for the Suggested Calculation Methods of I, II, and III from These Data m2/mol·kg−1

ref

(E/int. mV)a

[eE,I/mV]b

[eE,II/mV]c

[eE,III/mV]d

0.1 0.2 0.2 0.3 0.35 0.5 0.5 0.7 0.7 1.0 1.0 1.5 1.5 2.0 2.0

15 14 15 15 14 14 15 14 15 14 15 14 15 14 15

−29.60 −29.26 −58.87 −75.78 −52.71 −67.23 −96.91 −81.09 −110.56 −95.79 −125.46 −112.74 −142.41 −125.08 −154.71

0.20 0 0.18 0.14 −0.22 −0.13 −0.01 −0.36 −0.03 −0.81 −0.68 −1.89 −1.76 −3.27 −3.10

0.20 0.01 0.20 0.17 −0.19 −0.07 0.05 −0.28 0.06 −0.68 −0.55 −1.70 −1.56 −3.01 −2.84

0.15 −0.01 0.13 0.13 −0.15 0.09 0.17 0.11 0.39 0.10 0.18 −0.15 −0.06 −0.62 −0.49

(18)

The vapor pressures p x and p y were calculated using eqs 2, 3, and 4. The largest absolute error for all solutions below 1.1 mol·kg−1 is smaller than 0.07 Pa, and the data from ref 11, therefore, support well the new parameter values. Determination of the Hü ckel Parameters for the Temperature Range from (0 to 25) °C. Above, the parameter values of BNaCl = 1.4 (mol·kg−1)−1/2, b1,NaCl = 0.0077, BKCl = 1.3 (mol·kg−1)−1/2, and b1,KCl = −0.0515 were determined for 0 °C. In ref 1, the following parameter values were determined and thoroughly tested for 25 °C: BNaCl = 1.4 (mol·kg−1)−1/2, b1,NaCl = 0.0716, BKCl = 1.3 (mol·kg−1)−1/2, and b1,KCl = 0.011. Thus, the same value was obtained for B for both of these temperatures, and this value can, therefore, be regarded as a constant in the range covering these temperatures for solutions of both NaCl and KCl. The fact, that parameter B does not depend practically at all on the temperature has also been previously observed when the thermodynamic data of aqueous hydrochloric,26,46 hydrobromic,36 and hydriodic36 acids were considered. For dilute solutions of all of these three hydrohalic acids, a great number of high-precision data at various temperatures are available based on measurements using galvanic

a

1 Int. V = 1.00034 V. bCalculated using method I. cCalculated using method II. dCalculated using method III. 290



Article

cells without a liquid junction. In method I, only parameter b1 depends on the temperature and the simplest relationship for this dependence is a linear one. In this way, the following equations were obtained ⎛ t ⎞ b1,NaCl = 0.0077 + 0.002556·⎜ o ⎟ ⎝ C⎠

(20)

⎛ t ⎞ b1,KCl = −0.0515 + 0.0025·⎜ o ⎟ ⎝ C⎠

(21)

In method I, therefore, three electrolyte dependent parameters need to be evaluated. As mentioned above in the present study, also two other calculation methods are considered. For method II, the parameter values of BNaCl = 1.4 (mol·kg−1)−1/2, and BKCl = 1.3 (mol·kg−1)−1/2 were accepted, but the following quadratic equations were used for the temperature dependences of parameter b1 ⎛ t ⎞ ⎛ t ⎞2 b1,NaCl = 0.012297 + 0.002712·⎜ o ⎟ − 0.0000253·⎜ o ⎟ ⎝ C⎠ ⎝ C⎠ (22)

⎛ t ⎞ ⎛ t ⎞2 b1,KCl = −0.048607 + 0.002487⎜ o ⎟ − 0.0000200·⎜ o ⎟ ⎝ C⎠ ⎝ C⎠ (23)

The parameters for these equations were mainly determined from the amalgam cell data of Harned and Nims13 and Harned and Cook15 at temperatures from (0 to 40) °C. In the second method, thus, four electrolyte parameters were estimated. In method III, the parameter values of Rowland and May (see Table 3) were used together with eqs 9 and 10 and in this method, therefore, six electrolyte dependent parameters were used in the calculations. The parameter values for methods II and III at rounded temperatures from (0 to 80) °C are presented in Figure 2 where graph A shows the values of parameter B and graph B those of parameter b1. In the former graph, the values of method II remain thus in constant values for both NaCl and KCl solutions. In the latter graph are also presented the values of parameter b1 for method I in the temperature interval from (0 to 25) °C, and these values are often very close to those of method II. Tests of Methods II and III Using the Freezing Point Data and Isopiestic Data Measured at 0 °C. The results of the tests of method I with the freezing point data are given in the three graphs of Figure 1. Graphs A and B show the similar results obtained with methods II and III. In graph C, very dilute solutions are only considered, and these data do not make any difference between the methods. According to graphs A and B, method I explains these high precision data best but the difference between methods I and II is not very important. Method III does not apply well to these high-precision data above 0.3 mol·kg−1 (especially for NaCl solutions). The test results from the isopiestic data of Platford11 at 0 °C for methods II and III are given in Table 6 in the same way as for method I. The agreement between the methods is good for all solutions considered in the table. Tests of All Methods Using Concentration Cell Data Measured on Cells with Transference. All of these three calculation methods can be critically tested with the data measured by Janz and Gordon16 and Hornibrook et al.17 on concentration cells with transference in dilute NaCl and KCl solutions, respectively, at 15, 25, 35, and at 45 °C. Previously in ref 1, the data from these sources at 25 °C were used in the

Figure 2. Parameter B (graph A) and parameter b1 (graph B) in eqs 1 and 2 as functions of the temperature for the three calculation methods (methods I, II, and III in the text) tested in the present study. Symbols in graph A: ●, NaCl, method II; ○, NaCl, III; ▼, KCl, II; △, KCl, III. For method I, the same values of B are recommended as those for method II but this method applies only up to 25 °C. Symbols in graph B: ●, NaCl, method II; ○, NaCl, III; ▼, KCl, II; △, KCl, III; ■, NaCl, I; □, KCl, I.

parameter estimations because the most reliable technique to determine activity coefficients for alkali metal chlorides in dilute aqueous solutions is to measure on appropriate cells of this kind. The precision of the cpd data obtained in this way can be as high as 0.001 mV. The calculation of activity coefficients from the data of these cells is, however, slightly problematic because this treatment requires that the transference numbers (t+ or t− = 1 − t+) of ions in the electrolyte solutions must be known. The most reliable transference numbers for these calculations can be obtained from data measured using the moving boundary method (see for example ref 47). The cell used in the measurements of NaCl and KCl solutions can be presented as Ag(s)|AgCl(s)|MCl(aq, m1)|MCl(aq, m2)|AgCl(s)|Ag(s) +

(24)

+

where M is Na or K . In the sets measured on cell 24 for the determination of activity coefficients, the molality of solution 1 (= m1) is often exactly, or almost, constant within each set and molality m2 is varied. Theoretically, the cpd of this cell (E) can be expressed by the following equation E=− 291

2RT F

∫1

2

t+ d ln(γm /mo)

(25) DOI: 10.1021/acs.jced.5b00544 J. Chem. Eng. Data 2016, 61, 286−306


Article

where t+,1 is the transference number of M+ at molality m1. To test the suggested Hückel parameters for NaCl and KCl solutions, the data from refs 16 and 17 were predicted by means of these values. The equation used to predict the cpd values has the following form

where t+ is the transference number of cation. In the subsequent calculations, the treatment of Longsworth (see, for example, refs 1 and 47) is followed. Transference number t+ is first divided into two parts: t+ = t+,1 + Δt+

(26)

E=−

Table 7. Parameter Values (ρ1 and A) Presented by Harned and Owen50 for the Equation for Conversion of Concentrations into the Molalities (eq 28) and Parameter Values (t∞, u1, and u2) Determined in the Present Study for the Transference Number Equation (eq 29) for NaCl and KCl Solutions at 15, 35, and 45 °C parameter

15 °C

ρ1 AKCl ANaCl t∞ (K+) u1(K+) u2(K+) t∞ (Na+) u1(Na+) u2(Na+)

0.9991 0.0267 0.0176 0.49254 0 −0.00537 0.39705 0.02480 −0.01015

25 °C

35 °C

45 °C

0.49049b 0.0042b 0.0079b 0.39629b 0.04928b 0.04681b

0.9940 0.0274 0.0188 0.48852 0 0.00258 0.39704 0.01430 −0.03543

0.9901 0.0277a 0.0190a 0.48662 0 0.00571 0.40305 0.03750 0.02125

2RTt+,1

F 2RT − F

ln(m2 /m1) −

2RTt+,1

F m2 2RT Δt+(dm /m) − m1 F

∫

ln(γ2/γ1) ln γ2

∫ln γ

Δt+ d[ln(γ )]

1

(27)

The relationship t+ = t+(m) for eq 27 was determined from the moving boundary sets of Allgood and Gordon48 and Allgood et al.49 for NaCl and KCl solutions, respectively, at the three temperatures considered. These sets contain more than 10 points up to a value of 17 at each temperature. The transference number data have been reported on the concentration (molarity, c) scale and the highest concentration in these sets is about 0.1 mol·dm−3. For the conversion of the data onto the molality (m) scale, the following equation (given by Harned and Owen on page 725 in ref 50) was used cmo m o = ρ1 − A o mc m

Obtained by extrapolation from the values at (0 to 40) °C. b Determined in ref 1. a

(28)

Figure 3. Plot of eE (eq 18), the deviation between the observed and predicted cell potential difference (cpd) from the data measured by Janz and Gordon16 (M = Na+, m1 ≈ 0.05 mol·kg−1) and by Hornibrook et al.17 (M = K+, m1 ≈ 0.05 mol·kg−1) on cell 24 as a function of molality m2. The predicted cpd was calculated by using eq 27 where eq 1 with the suggested Hückel equations of methods I, II, and III were used for activity coefficients. Graph A contains the results for 15 °C, B for 25 °C, C for 35 °C, and D for 45 °C. Symbols in the graphs: ●, NaCl, method I; o, NaCl, II; ■, NaCl, III; ▼, KCl, I; △, KCl, II; □, KCl, III. In graph B in the set of Janz and Gordon, the point (m1 = 0.049826 mol·kg−1, m2 = 0.019978 mol·kg−1, E = 17.112 int. mV) is an outlier and its error has been omitted. 292



Article

Figure 5. Plot of eE (eq 18), the deviation between the observed and predicted cell potential difference (cpd) from the concentration cell data measured on cell 16 by Harned and Cook15 (MCl = KCl, m1 = 0.05 mol·kg−1) as a function of molality m2. The predicted cpd was calculated by using eq 17 where eq 1 with the suggested Hückel parameters of methods I, II, and III (see text) were used for the activity coefficients. Symbols in graph A: ●, 5 °C, method I; ○, 5 °C, II; ⧫, 5 °C, III; ▼, 10 °C, I; △, 10 °C, II; ◊, 10 °C, III; ■, 15 °C, I; □, 15 °C, II; ▲, 15 °C, III. Symbols in graph B: ●, 20 °C, method I; ○, 20 °C, II; ■, 20 °C, III; ▼, 25 °C, I; △, 25 °C, II; □, 25 °C, III. Symbols in graph C: ●, 30 °C, method I; ○, 30 °C, II; ⧫, 30 °C, III; ▼, 35 °C, I; △, 35 °C, II; ◊, 35 °C, III; ■, 40 °C, I; □, 40 °C, II; ▲, 40 °C, III.

Figure 4. Plot of eE (eq 18), the deviation between the observed and predicted cell potential difference (cpd) from the concentration cell data measured on cell 16 by Harned and Nims13 (MCl = NaCl, m1 = 0.1 mol·kg−1) as a function of molality m2. The predicted cpd was calculated by using eq 17 where eq 1 with the suggested Hückel parameters of methods I, II, and III (see text) were used for the activity coefficients. Symbols in graph A: ●, 12.5 °C, method I; ○, 12.5 °C, II; ⧫, 12.5 °C, III; ▼, 15 °C, I; △, 15 °C, II; ◊, 15 °C, III; ■, 20 °C, I; □, 20 °C, II; ▲, 20 °C, III. Symbols in graph B: ●, 25 °C, method I; ○, 25 °C, II; ■, 25 °C, III; ▼, 30 °C, I; △, 30 °C, II; □, 30 °C, III. Symbols in graph C: ●, 37.5 °C, method I; ○, 37.5 °C, II; ■, 37.5 °C, III; ▼, 40 °C, I; △, 40 °C, II; □, 40 °C, III. 293



Article

Figure 6. Plot of eE (eq 18), the deviation between the observed and predicted cpd from the galvanic cell data measured by Mussini et al.24 in NaCl solutions on cell 30 as a function of molality m. The predicted cpd was calculated by using eq 31 in which eq 1 with the suggested Hückel parameters of method II was used for the activity coefficients as well as the Eo values given below. In graph A, however, all three methods were used. Graphs A, B, C, D, and E show the results for 10 °C, 25 °C, 40 °C, 55 °C, and 70 °C, respectively. Symbols for graph A in which x = 0.001520: ●, method I, Eo = 2.01649 V; ○, II, 2.01657 V; ▼, III, 2.01662 V. Symbols for graph B: ●, x = 0.001291, Eo = 2.01196 V; ○, 0.002020, 2.02458 V; ▼, 0.003630, 2.04043 V; △, 0.004268, 2.04577 V; ■, 0.005888, 2.05543 V; □, 0.00795, 2.06595 V; ⧫, 0.01251, 2.08081 V. Symbols for graph C: ●, x = 0.001146, Eo = 2.00710 V; ○, 0.004106, 2.04451 V; ▼, 0.006150, 2.05746 V; △, 0.008440, 2.06840 V; ■, 0.009300, 2.07246 V. Symbols for graph D: ●, x = 0.001146, Eo = 2.00341 V; ○, 0.004106, 2.04210 V; ▼, 0.006150, 2.05573 V; △, 0.00844, 2.06709 V; ■, 0.009300, 2.07153 V. Symbols for graph E: ●, x = 0.001291, Eo = 2.00162 V; ○, 0.002020, 2.01582 V; ▼, 0.003610, 2.03329 V; △, 0.004268, 2.03996 V; ■, 0.00795, 2.06267 V.

where co is 1 mol·dm−3. The parameter values for this equation are given in Table 7. The following functional form was used here as earlier (see ref 1) for relationship t+ = t+(m)

t+ = t+∞ − u1 m /mo + u 2(m /mo)

(29)

The parameter values obtained for this equation from the moving boundary data are given in Table 7. In this table are also 294



Article

Figure 7. Plot of eE (eq 18), the deviation between the observed and predicted cpd from the galvanic cell data measured by Giordano et al.25 in KCl solutions on cell 30 as a function of molality m. The predicted cpd was calculated by using eq 31 in which eq 1 with the suggested Hückel parameters of method II was used for the activity coefficients as well as the Eo values given below. Graphs A, B, C, D, and E show the results for 10 °C, 25 °C, 40 °C, 55 °C, and 70 °C, respectively. Symbols for graph A: ●, x = 0.001433, Eo = 2.02622 V; ○, 0.003517 2.05134 V; ▼, 0.004247, 2.05500 V; △, 0.008832, 2.08372 V; ■, 0.01322, 2.10024 V; □, 0.02001, 2.12198 V; ⧫, 0.02006, 2.12247 V; ◊, 0.02030, 2.12321 V; ▲, 0.02788, 2.14426. Symbols for graph B: ●, x = 0.001433, Eo = 2.03394 V; ○, 0.003279, 2.05640 V; ▼, 0.004148, 2.06419 V; △, 0.007648, 2.08847 V; ■, 0.01172, 2.10408 V; □, 0.01469, 2.11530 V; ⧫, 0.01830, 2.12848 V; ◊, 0.01925, 2.13155 V; ▲, 0.02919, 2.15870 V; ▽, 0.04014, 2.18632 V. Symbols for graph C: ●, x = 0.001433, Eo = 2.03730 V; ○, 0.003032, 2.06308 V; ▼, 0.004247, 2.07133 V; △, 0.008149, 2.09488 V; ■, 0.01026, 2.10788 V; □, 0.01322, 2.11861 V; ⧫, 0.01830, 2.13477 V; ◊, 0.02930, 2.16630 V; ▲, 0.04014, 2.19732 V. Symbols for graph D: ●, x = 0.001433, Eo = 2.03946 V; ○, 0.002986, 2.06468 V; ▼, 0.004634, 2.07710 V; △, 0.01026, 2.11155 V; ■, 0.01322, 2.12161 V; □, 0.01812, 2.13807 V; ⧫, 0.01830, 2.14237 V; ◊, 0.02855, 2.17235 V; ▲, 0.04086, 2.20566 V. Symbols for graph E: ●, x = 0.001433, Eo = 2.03825 V; ○, 0.003517, 2.06661 V; ▼, 0.004847, 2.08001 V; △, 0.005788, 2.08634 V; ■, 0.01026, 2.11127 V; □, 0.01322, 2.11915 V; ⧫, 0.01667, 2.13537 V; ◊, 0.02001, 2.14811 V; ▲, 0.03044, 2.17721 V; ▽, 0.04441, 2.21321 V. In graph C, the results of all four points were omitted in the series where x = 0.007648 as less precise data, as well, in graph D the results of all seven points were omitted in the series where x = 0.005778. Point (x = 0.001433, m = 0.5 mol·kg−1, E = 2.1066 V) in graph D and points (x = 0.004847, m = 0.60 mol·kg−1, E = 2.19985 V), (0.005788, 0.50, 2.15955), (0.01026, 0.50, 2.17644), and (0.01667, 0.20, 2.23249) in graph E were omitted as outliers. 295



Article

Figure 8. Plot of eip (eq 19), 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 Robinson18 at 25 °C. The vapor pressures were calculated by eqs 2, 3, and 4 using the suggested Hückel parameters of methods I, II, and III (see text) for the osmotic coefficients. Graphs A and D show the deviations for method I, graph B shows those for method II, and graphs C and E show those for method III.

included the values presented in ref 1 for 25 °C, and eq 29 reproduces all experimental transference numbers within ±0.0005. When the cpd values were predicted using the activity parameters of the different calculation methods, the integral in the last term on the right-hand side of eq 27 (i.e., in the second activity coefficient term) was evaluated numerically. For methods I, II, and III, the cpd errors defined by eq 18 are shown in four graphs of Figure 3. Graphs A, B, C, and D show the results for 15 °C, 25 °C, 35 °C, and 45 °C, respectively. All errors

in the graphs are small. In graphs C and D, however, there is a trend in the error plots of solutions of both salts for method I. In graph C, however, these trends are only very small. Also in the error plots of graph B, there is a clear trend in both patterns for method III, and the largest error is close to 0.05 mV. It seems important to explain in this connection the following result from ref 1: In that study, the Hückel parameters for KCl were estimated from the cpd results of ref 17. After that, both NaCl parameters B and b1 were estimated from the isopiestic data of 296



Article

Figure 10. Plot of ep (eq 32), the deviation between the reported and predicted vapor pressure of water over NaCl and KCl solutions from the data of Childs and Platford,12 as a function of molality m. The reported vapor pressures were calculated by eqs 3 and 4 from the osmotic coefficients given by Childs and Platford12 for 15 °C, and the predicted ones were calculated by eqs 2, 3, and 4 using the suggested Hückel parameters of methods I, II, and III (see text) for the osmotic coefficients. Graph A shows the results for NaCl solutions and graph B for KCl solutions. Symbols: Methods ●, I; ○, II; and ▼, III.

Figure 9. Plot of eip (eq 19), 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.19 at 45 °C, Humphries et al.20 at 60 °C and Moore et al.21 at 80 °C, and Hellams et al.22 at 45 °C. The isopiestic molalities for the data reported by Hellams et al.22 were reproduced from the smoothed values reported for the isopiestic ratios (see text). Symbols in graph A (all data measured at 45 °C): ●, ref 19, method II; ○, ref 19, III; ▼, ref 22, II; △, ref 22, III. Symbols in graph B: ●, ref 20, t = 60 °C, method II; ○, ref 20, 60 °C, III; ▼, ref 21, 80 °C, II; △, ref 21, 80 °C, III.

those of the other temperatures in graphs A (5, 10, and 15 °C), B (20 and 25 °C), and C (30, 35, and 40 °C) of Figure 5. Table 5 contains also the errors from the data of Smith14 for methods II and III. In the error plots of Figures 4 and 5, the errors for different sets are presented as a function of molality m2. According to these figures, method I applies to these data quite well up to a molality of 1.0 mol·kg−1 at temperatures from (0 to 25) °C. For KCl solutions, it also applies well to 30 °C in the molality range from (0 to 2) mol·kg−1 (see graph C of Figure 5). Method II applies quite well to all temperatures used in the figures up to 1 mol·kg−1. Method III applies very well to these data in all temperatures and molalities used in the figure. It seems in this case, however, that the errors are even smaller than the precision of these amalgam cell measurements. This point will be discussed in more detail later. The quality of the estimated Hückel parameters for NaCl and KCl solutions at various temperatures were then tested with the galvanic cell data measured on cells containing no liquid junction. Mussini et al.24 and Giordano et al.25 measured

Robinson.18 Using these new NaCl parameters, all cpd values from ref 16 could be predicted within ±0.01 mV. Thus, no parameter estimation was needed for this excellent fit that can also be seen in graph B. Tests of All Methods Using Amalgam Cell Data. Above, method I was tested with the data measured on concentration cells containing an amalgam electrode (see cell 16) at 0.3 °C for NaCl solutions13 and at 0 °C for KCl solutions.14,15 The data from all temperatures of Harned and Nims13 for NaCl solutions and of Harned and Cook15 for KCl solutions were used in the estimation of the quadratic temperature dependence of parameter b1 (see eqs 22 and 23). All of these data were then used to test the new calculation methods. The results for the NaCl solutions at 0.3 °C are shown in Table 4, and those of the other temperatures are presented as error plots in graphs A (12.5, 15, and 20 °C), B (25 and 30 °C), and C (35 and 40 °C) of Figure 4. For KCl solutions, the results at 0 °C are shown in Table 5 and 297



Article

Table 9. Recommended Activity Coefficients (γ) of Sodium Chloride in Aqueous Solutions at Temperatures from (30 to 80) °C as a Function of Molality ma γ

m mol·kg 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 11. Plot of ep (eq 32), the deviation between the vapor pressure measured by Gibbard et al.23 at 25, 37.5, 50, and 75 °C and that predicted by the suggested methods for the NaCl solution as a function of molality m. The predicted vapor pressure was calculated by eqs 2, 3, and 4 using the suggested Hückel parameters of methods I, II, and III for the osmotic coefficients. Symbols: ●, t = 25 °C, method I; ○, 25 °C, method II; ▼, 25 °C, III; △, 37.5 °C, III; ■, 50 °C, II; □, 50 °C, III, ⧫, 75 °C, III.

solutions of these salts using direct amalgam cells of the following type M(Hg, x)|MCl(aq, m)|AgCl(s)|Ag(s)

(30)

where M again refers to sodium or potassium ions or the corresponding alkali metal and x is the mole fraction of alkali metal in the amalgam. The former paper24 contains the results for NaCl solutions and the latter25 for KCl solutions. Some sets in the latter paper at 25 °C were also used in ref 1. The former data consisted of one series (in each series the mole fraction of alkali metal in amalgam (= x) is constant and the molality of salt varies) at 10 °C, seven series at 25 °C, and five series at 40, 55, and at 70 °C. The latter data contain nine series at 10 °C and eight series at 25, 40, 55, and at 70 °C. All data in these series were used in the

−1

30 °C

40 °C

50 °C

60 °C

70 °C

80 °C

0.902 0.871 0.850 0.833 0.820 0.809 0.799 0.791 0.784 0.777 0.751 0.732 0.707 0.691 0.680 0.672 0.666 0.661 0.658 0.655

0.900 0.869 0.847 0.831 0.818 0.806 0.797 0.788 0.781 0.774 0.748 0.729 0.705 0.689 0.678 0.670 0.664 0.660 0.657 0.655

0.898 0.866 0.844 0.828 0.814 0.803 0.793 0.785 0.777 0.770 0.743 0.725 0.700 0.685 0.674 0.666 0.660 0.656 0.654 0.652

0.896 0.864 0.841 0.824 0.811 0.799 0.789 0.780 0.773 0.766 0.739 0.720 0.695 0.678 0.667 0.659 0.653 0.649 0.646 0.644

0.894 0.861 0.838 0.821 0.807 0.795 0.785 0.776 0.768 0.761 0.733 0.713 0.687 0.671 0.659 0.650 0.644 0.639 0.636 0.633

0.891 0.856 0.833 0.815 (0.817) 0.800 (0.802) 0.788 (0.790) 0.777 (0.780) 0.768 (0.770) 0.760 (0.762) 0.752 (0.755) 0.723 (0.726) 0.703 (0.706) 0.676 (0.679) 0.659 (0.661) 0.647 0.639 0.633 0.629 (0.626) 0.626 (0.622) 0.624 (0.619)

The values at 80 °C have been calculated using method III but the other values using method II. At 80 °C, the values from method II are in the parentheses (see text).

a

present Hückel parameter tests. The cpd for these cells can be calculated from the following equation E = Eo −

2RT ln(γm /mo) F

(31)

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

m

a

mol·kg−1

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9058 0.8753 0.8545 0.8385 0.8254 0.8142 0.8045 0.7960 0.7883 0.7813 0.7539 0.7341 0.7063 0.6868 0.6720 (0.674) 0.6603 (0.662) 0.6507 (0.653) 0.6425 (0.645) 0.6356 (0.638) 0.6295 (0.632)

0.9052 0.8747 0.8539 0.8378 0.8247 0.8136 0.8040 0.7954 0.7878 0.7809 0.7538 0.7344 0.7071 0.6884 0.6744 (0.676) 0.663 (0.665) 0.654 (0.657) 0.647 (0.649) 0.641 (0.643) 0.635 (0.638)

0.9046 0.8740 0.8532 0.8371 0.8240 0.8130 0.8034 0.7949 0.7873 0.7804 0.7536 0.7345 0.7080 0.6899 0.6766 0.666 (0.668) 0.658 (0.660) 0.651 (0.653) 0.646 (0.648) 0.641 (0.644)

0.9040 0.8733 0.8524 0.8364 0.8233 0.8123 0.8027 0.7942 0.7867 0.7800 0.7534 0.7345 0.7087 0.6914 0.6788 0.669 0.662 0.656 0.651 0.647

0.9034 0.8726 0.8516 0.8356 0.8225 0.8115 0.8019 0.7935 0.7860 0.7793 0.7530 0.7345 0.7094 0.6927 0.6808 0.672 0.665 0.660 0.656 0.653 (0.651)

0.9028 0.8718 0.8508 0.8347 0.8217 0.8107 0.8011 0.7928 0.7853 0.7786 0.7526 0.7344 0.7099 (0.708) 0.6940 (0.692) 0.6828 (0.680) 0.675 (0.672) 0.669 (0.665) 0.664 (0.660) 0.661 (0.657) 0.659 (0.654)

The values have been calculated using method I and the values in parentheses using method II (see text). 298



Article

Table 10. Recommended Activity Coefficients (γ) of Potassium Chloride in Aqueous Solutions at Temperatures from (0 to 25) °C as a Function of Molality ma γ

m

a

mol·kg−1

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9044 0.8729 0.8511 0.8342 0.8202 0.8082 0.7978 0.7885 0.7801 0.7724 0.7418 0.7192 0.6861 0.6620 0.6429 0.6270 0.6135 (0.615) 0.602 0.591 0.581 (0.583)

0.9039 0.8723 0.8504 0.8335 0.8195 0.8076 0.7972 0.7879 0.7796 0.7719 0.7416 0.7193 0.6868 0.6634 0.6449 0.630 0.617 0.606 0.596 0.586 (0.588)

0.9033 0.8716 0.8497 0.8328 0.8188 0.8069 0.7965 0.7873 0.7790 0.7714 0.7414 0.7193 0.6875 0.6647 0.6469 0.632 0.620 0.609 0.600 0.592

0.9027 0.8709 0.8490 0.8320 0.8180 0.8062 0.7958 0.7866 0.7783 0.7708 0.7411 0.7193 0.6881 0.6659 0.6488 0.635 0.623 0.613 0.605 0.597

0.9021 0.8701 0.8482 0.8312 0.8172 0.8054 0.7950 0.7859 0.7776 0.7702 0.7407 0.7192 0.6886 0.6671 0.6506 (0.649) 0.637 (0.635) 0.626 (0.624) 0.617 (0.614) 0.609 (0.606) 0.602 (0.599)

0.9014 0.8693 0.8473 0.8303 0.8164 0.8045 0.7942 0.7851 0.7769 0.7694 0.7402 0.7190 0.689 (0.687) 0.668 (0.666) 0.652 (0.649) 0.640 (0.636) 0.629 (0.625) 0.621 (0.616) 0.613 (0.608) 0.607 (0.601)

The values have been calculated using method I and the values in parentheses using method II (see text).

The best value of Eo was used for each method at this mole fraction of amalgam when the errors (see eq 18) for this series was calculated, and these Eo values are given in the caption of the figure. As can be seen in the error plots of this graph, there is no appreciable difference between the methods to predict these less precise data. In the latter graphs of Figures 6 and 7, therefore, only the results for method II are given. In Figure 6, these results for 25 °C are given in graph B, for 40 °C in graph C, for 55 °C in graph D, and for 70 °C in graph E. In Figure 7, the results from KCl solutions are presented for temperatures 10, 25, 40, 55, and 70 °C in graphs A, B, C, D, and E, respectively. The Eo values used in the calculations of the errors for Figures 6 and 7 are given in the captions of these figures. As can been seen, all data presented in the graphs of Figures 6 and 7 up to 70 °C (graph E in both figures) can be predicted quite well using method II. The results for method III from these data24,25 correspond closely in quality to those of method II as well as the results of method I up to 25 °C. Tests of All Methods Using Vapor Pressure Data. Isopiestic data of Robinson18 at 25 °C for NaCl and KCl solutions were used to test methods I, II, and III in the same way as above the data of Platford for 0 °C in Table 6. At 25 °C, several isopiestic sets are available for NaCl and KCl solutions, but the set of Robinson is the most reliable. The results for method I from this set18 have been previously presented in ref 1 (see there graph B in Figure 3), and in that connection also the other isopiestic sets were considered. In the present study for molalities less than 2 mol·kg−1, the error plots are given in graphs A, B, and C of Figure 8 for methods I, II, and III, respectively. In these graphs, the isopiestic error, eip in eq 19, is presented as a function of molality my (= mKCl). As can be seen in graph B, method II applies to these data only up to a molality of 1 mol·kg−1, and the other two methods to all molalities used in the graphs. The validity of methods I and III were also tested with all isotonic points given by Robinson, and the data extend up to the saturated KCl solution at 25 °C, that is, up to 4.81 mol·kg−1.

Table 11. Recommended Activity Coefficients (γ) of Potassium Chloride in Aqueous Solutions at Temperatures from (30 to 80) °C as a Function of Molality ma γ

m mol·kg−1

30 °C

40 °C

50 °C

60 °C

70 °C

80 °C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.901 0.868 0.846 0.829 0.815 0.803 0.792 0.783 0.775 0.767 0.738 0.716 0.686 0.665 0.649 0.636 0.625 0.617 0.609 0.602

0.899 0.866 0.844 0.826 0.812 0.800 0.790 0.780 0.772 0.764 0.735 0.714 0.684 0.663 0.647 0.635 0.625 0.616 0.609 0.603

0.897 0.864 0.841 0.823 0.809 0.797 0.786 0.777 0.768 0.761 0.731 0.710 0.680 0.659 0.644 0.632 0.622 0.614 0.607 0.601

0.895 0.861 0.838 0.820 0.805 0.793 0.782 0.773 0.764 0.757 0.727 0.705 0.675 0.654 0.638 0.626 0.617 0.608 0.602 0.596

0.893 0.858 0.835 0.816 0.802 0.789 0.778 0.768 0.760 0.752 0.721 0.699 0.668 0.647 0.632 0.619 0.609 0.601 0.594 0.588

0.890 0.854 0.830 0.811 0.796 0.783 0.772 0.762 0.753 (0.755) 0.745 (0.747) 0.714 0.691 (0.693) 0.660 0.638 0.623 0.610 0.601 0.593 (0.591) 0.586 (0.584) 0.580 (0.577)


a

where Eo is the standard cpd and it depends on the temperature and on x. For NaCl data the results are shown in five graphs of Figure 6 and for KCl data in five graphs of Figure 7. Graph A in Figure 6 gives the results for the only NaCl series measured at 10 °C, and all of the three calculation methods were used to predict these data. 299



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Table 12. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Sodium Chloride Solutions at Temperatures from (0 to 25) °C as a Function of Molality m.a ϕ

m

a

mol·kg−1

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9692 0.9596 0.9532 0.9484 0.9446 0.9414 0.9386 0.9363 0.9342 0.9324 0.9255 0.9210 0.9155 0.9123 0.9104 0.9092 (0.911) 0.9086 (0.910) 0.9082 (0.910) 0.9081 (0.910) 0.9082 (0.911)

0.9690 0.9594 0.9531 0.9483 0.9445 0.9413 0.9386 0.9363 0.9343 0.9325 0.9259 0.9217 0.9167 0.9142 0.9129 0.912 (0.914) 0.912 (0.914) 0.913 (0.915) 0.913 (0.915) 0.914 (0.916)

0.9688 0.9592 0.9529 0.9481 0.9444 0.9413 0.9386 0.9364 0.9344 0.9326 0.9263 0.9224 0.9180 0.9161 0.9154 0.9155 0.9161 0.9170 0.918 (0.920) 0.920

0.9686 0.9590 0.9527 0.9480 0.9442 0.9412 0.9386 0.9364 0.9344 0.9327 0.9267 0.9230 0.9192 0.9179 0.9179 0.919 0.920 0.921 0.923 0.925

0.9684 0.9588 0.9525 0.9478 0.9441 0.9411 0.9385 0.9363 0.9345 0.9328 0.9270 0.9236 0.9204 0.9197 0.9203 0.922 0.924 0.926 0.928 0.931

0.9682 0.9586 0.9523 0.9476 0.9439 0.9409 0.9384 0.9363 0.9345 0.9329 0.9273 0.9242 0.9216 0.9215 (0.920) 0.9227 (0.921) 0.9246 (0.922) 0.9271 (0.925) 0.930 (0.927) 0.933 (0.930) 0.936 (0.933)


Table 13. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Sodium Chloride Solutions at Temperatures from (30 to 80) °C as a Function of Molality ma

Table 14. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Potassium Chloride at Temperatures from (0 to 25) °C as a Function of Molality ma

ϕ

m

ϕ

m

mol·kg−1

30 °C

40 °C

50 °C

60 °C

70 °C

80 °C

mol·kg−1

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.968 0.958 0.952 0.947 0.943 0.940 0.938 0.936 0.934 0.932 0.927 0.923 0.921 0.921 0.922 0.924 0.926 0.929 0.932 0.935

0.967 0.958 0.951 0.946 0.943 0.940 0.937 0.935 0.933 0.931 0.926 0.923 0.920 0.921 0.922 0.925 0.928 0.931 0.934 0.938

0.967 0.957 0.950 0.945 0.941 0.938 0.936 0.934 0.932 0.930 0.925 0.922 0.919 0.920 0.921 0.924 0.927 0.930 0.934 0.938

0.966 0.956 0.949 0.944 0.940 0.937 0.934 0.932 0.930 0.929 0.923 0.920 0.917 0.917 0.919 0.922 0.925 0.928 0.932 0.936

0.965 0.955 0.948 0.943 0.939 0.935 0.933 0.930 0.928 0.927 0.920 0.917 0.914 0.914 0.915 0.918 0.920 0.923 0.927 0.930

0.964 0.953 0.946 0.940 0.936 (0.937) 0.932 (0.934) 0.929 (0.931) 0.927 0.925 0.923 0.916 (0.918) 0.913 0.910 0.910 0.912 (0.910) 0.915 (0.912) 0.918 (0.914) 0.922 (0.917) 0.926 (0.919) 0.931 (0.922)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9684 0.9582 0.9513 0.9460 0.9416 0.9379 0.9347 0.9319 0.9293 0.9270 0.9180 0.9115 0.9022 0.8955 0.8901 0.8856 0.8817 0.878 0.875 0.872

0.9683 0.9581 0.9511 0.9458 0.9415 0.9378 0.9347 0.9319 0.9294 0.9271 0.9184 0.9122 0.9034 0.8973 0.8926 0.889 0.885 0.882 0.880 0.877

0.9681 0.9579 0.9510 0.9457 0.9414 0.9377 0.9346 0.9319 0.9294 0.9272 0.9187 0.9128 0.9046 0.8991 0.8949 0.8917 0.8889 0.8866 0.885 0.883

0.9679 0.9577 0.9508 0.9455 0.9412 0.9376 0.9345 0.9318 0.9294 0.9273 0.9191 0.9134 0.9058 0.9008 0.8973 0.895 0.893 0.891 0.889 0.888

0.9677 0.9575 0.9505 0.9453 0.9411 0.9375 0.9345 0.9318 0.9294 0.9273 0.9193 0.9139 0.9069 0.9025 0.8996 (0.898) 0.898 (0.896) 0.896 (0.894) 0.895 (0.893) 0.894 (0.892) 0.894 (0.891)

0.9675 0.9572 0.9503 0.9451 0.9409 0.9374 0.9343 0.9317 0.9294 0.9274 0.9196 0.9144 0.9080 (0.906) 0.904 (0.902) 0.902 (0.899) 0.900 (0.897) 0.900 (0.896) 0.899 (0.895) 0.899 (0.894) 0.899 (0.894)


a

a

The results for methods I and III from all points are given in graphs D and E of this figure, respectively. Method I explains all of these data very well, but this fact does not mean that the vapor pressure can be calculated accurately using the Hückel parameters of method I in the concentrated solutions also. It means, however, that the isopiestic ratio mKCl/mNaCl can be calculated

with these parameters up to the saturated KCl solution. This result is not true for method III in graph E. Methods II and III were then tested using the isopiestic NaCl/KCl data from Davis et al.19 at 45 °C, Humpries et al.20 at 60 °C, and Moore et al.21 at 80 °C. The data were used as the isopiestic data for Figure 8. The results are shown in graphs A (ref 19) and B (refs 20 and 21) of Figure 9. Graph A shows that


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and are presented in Figure 10 in which graph A shows the NaCl results and graph B the KCl results. The error plots in graphs A and B indicate for solutions of both salts, that methods I and II predict better these smoothed data than method III in the molality range from (0.5 to 1.0) mol·kg−1. Hellams et al.22 reported isopiestic ratios for KCl and NaCl solutions at 45 °C from a molality of (0.5 to 3.5) mol·kg−1. These my/mx (= mKCl/mNaCl) values were used to calculate the isopiestic molalities. The vapor pressures were then predicted using these molalities and the isopiestic vapor pressure errors (eq 19) are then shown in the error plot. This plot is presented in graph A of Figure 9. This graph confirms that both method II and III apply well to the experimental data up to 1.5 mol·kg−1. Figure 11 shows the errors of the vapor pressure points (calculated using eq 32) measured by Gibbard et al.23 at 25, 37.5, 50, and at 75 °C in the case they are sufficiently dilute for the present consideration (m is less than or equal to about 3 mol·kg−1). At 25 °C method I applies well up to a molality of 2 mol·kg−1, at 25 and 50 °C method II applies up 1 mol·kg−1, and at 25, 37.5, and 50 °C method III applies up to 3 mol·kg−1. At 75 °C, even method III does not predict well these vapor pressures so far because at m = 1.047 mol·kg−1 its error ep is 2.6 Pa, at 1.840 mol·kg−1 it is 5.7 Pa, and at 2.511 and 3.036 mol·kg−1 the errors are −12.8 and −27.3 Pa, respectively. The first two errors are shown in Figure 11 but the latter cannot be discerned well because of overlapping of errors. Recommended Activity and Osmotic Coefficients. On the basis of the wide evidence provided by the test results presented here (see Figure 1 and Figures 3 to 11 and Tables 4, 5, and 6), the experimental data available in the literature for NaCl and KCl solutions from (0 to 70) °C can very often be predicted within experimental error using the new Hückel equations at least up to a molality of 1 mol·kg−1. Therefore, new activity and osmotic coefficients are recommended in the present study on the basis of the tested calculation methods. The new activity values up to this molality are given in Tables 8 to 15. In method I, only three electrolyte-dependent parameters were estimated, this estimation was made from high-precision data, and the resulting parameters apply from (0 to 25) °C. Therefore, activity and osmotic coefficients of this method were recommended in the temperature range. These activity coefficients are given in Tables 8 and 10 and these osmotic coefficients in Tables 12 and 14 for NaCl and KCl solutions, respectively. In the temperature range from (30 to 70) °C, method II seems to be the most reliable. In this method, four electrolytedependent parameters were estimated but all data used in the

Table 15. Recommended Osmotic Coefficients (ϕ) of Water in Aqueous Potassium Chloride Solutions at Temperatures from (30 to 80) °C as a Function of Molality ma ϕ

m mol·kg

−1

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

30 °C

40 °C

50 °C

60 °C

70 °C

80 °C

0.967 0.957 0.950 0.945 0.940 0.937 0.934 0.931 0.929 0.927 0.919 0.913 0.907 0.903 0.900 0.899 0.898 0.897 0.897 0.896

0.967 0.956 0.949 0.944 0.939 0.936 0.933 0.930 0.928 0.926 0.918 0.913 0.907 0.903 0.901 0.900 0.899 0.899 0.900 0.900

0.966 0.955 0.948 0.943 0.938 0.935 0.932 0.929 0.927 0.925 0.917 0.912 0.906 0.902 0.901 0.900 0.900 0.900 0.900 0.901

0.965 0.954 0.947 0.942 0.937 0.933 0.930 0.928 0.925 0.923 0.915 0.910 0.904 0.901 0.899 0.899 0.898 0.899 0.899 0.900

0.965 0.953 0.946 0.940 0.936 0.932 0.929 0.926 0.923 0.921 0.913 0.908 0.902 0.898 0.897 0.896 0.896 0.896 0.896 0.897

0.963 0.952 0.944 0.938 0.934 0.930 0.926 0.923 0.921 0.918 0.910 0.905 0.898 0.895 0.894 0.893 0.894 (0.891) 0.894 (0.891) 0.896 (0.891) 0.897 (0.892)


a

both method II and III apply well to the experimental data up to 1.5 mol·kg−1. Graph B shows that both methods predict accurately the vapor pressures up to 2.0 mol·kg−1 at 60 °C, and method III applies well to the isopiestic mKCl/mNaCl ratio up to 3.5 mol·kg−1 at this temperature and at as high temperature as 80 °C. On the basis of isopiestic data against urea and sulfuric acid solutions, Childs and Platford12 reported osmotic coefficients at rounded molalities for 15 °C from 0.1 to 6.0 or 4.0 mol·kg−1 for NaCl or KCl solutions, respectively. These data up to 1 mol·kg−1 were used in the present Hückel equation tests. For these data, the vapor pressures were first calculated using eqs 3 and 4 from the reported osmotic coefficients and then the resulting vapor pressures were predicted using eqs 2, 3, and 4 with the new Hückel equations. The errors were finally calculated using equation ep = p (reported) − p (predicted)

(32)

Table 16. Confidence Intervals for the Activity Coefficients (γ) of Sodium and Potassium Chloride at the Significance Level of 0.95 in Aqueous Solutions at Temperatures 0, 10, and 25 °C as a Function of Molality m m

γNaCl

γKCl

γNaCl

γKCl

γNaCl

γKCl

mol·kg−1

0 °C

0 °C

10 °C

10 °C

25 °C

25 °C

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.0

(0.8253, 0.8254) (0.7813, 0.7814) (0.7340, 0.7343) (0.7061, 0.7064) (0.6866, 0.6870) (0.6718, 0.6723) (0.6600, 0.6606) (0.6503, 0.6510) (0.6422, 0.6430) (0.6352, 0.6360) (0.6291, 0.6300)

(0.8201, 0.8202) (0.7723, 0.7725) (0.7189, 0.7194) (0.6858, 0.6864) (0.6616, 0.6623) (0.6424, 0.6433) (0.6265, 0.6276) (0.6128, 0.6141) (0.6008, 0.6023) (0.590, 0.592) (0.580, 0.582)

(0.8240, 0.8241) (0.7803, 0.7806) (0.7343, 0.7347) (0.7077, 0.7083) (0.6896, 0.6904) (0.6762, 0.6772) (0.6658, 0.6670) (0.6575, 0.6589) (0.651, 0.652) (0.645, 0.647) (0.640, 0.642)

(0.8188, 0.8189) (0.7713, 0.7715) (0.7191, 0.7195) (0.6872, 0.6878) (0.6643, 0.6651) (0.6464, 0.6474) (0.6318, 0.6329) (0.6194, 0.6207) (0.6087, 0.6102) (0.5992, 0.6009) (0.591, 0.593)

(0.8216, 0.8217) (0.7785, 0.7787) (0.7342, 0.7346) (0.7096, 0.7102) (0.6935, 0.6944) (0.6823, 0.6833) (0.6741, 0.6753) (0.6680, 0.6694) (0.663, 0.665) (0.660, 0.662) (0.657, 0.659)

(0.8161, 0.8165) (0.7690, 0.7698) (0.7181, 0.7196) (0.688, 0.690) (0.667, 0.669) (0.650, 0.654) (0.637, 0.641) (0.627, 0.631) (0.618, 0.623) (0.610, 0.615) (0.603, 0.609)

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Table 17. Confidence Intervals for the Osmotic Coefficients (ϕ) of Water in Aqueous Sodium and Potassium Chloride Solutions at the Significance Level of 0.95 at Temperatures 0, 10, and 25 °C as a Function of Molality m m

ϕNaCl

ϕKCl

ϕNaCl

ϕKCl

ϕNaCl

ϕKCl

mol·kg−1

0 °C

0 °C

10 °C

10 °C

25 °C

25 °C

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.0

(0.9445, 0.9446) (0.9323, 0.9324) (0.9209, 0.9211) (0.9154, 0.9156) (0.9122, 0.9125) (0.9102, 0.9106) (0.9090, 0.9095) (0.9083, 0.9088) (0.9080, 0.9086) (0.9078, 0.9085) (0.9078, 0.9086)

(0.94155, 0.94163) (0.9270, 0.9271) (0.9114, 0.9117) (0.9020, 0.9024) (0.8952, 0.8958) (0.8898, 0.8905) (0.8852, 0.8861) (0.8812, 0.8822) (0.8775, 0.8787) (0.8742, 0.8755) (0.8710, 0.8725)

(0.9443, 0.9444) (0.9326, 0.9327) (0.9222, 0.9225) (0.9178, 0.9183) (0.9158, 0.9164) (0.9151, 0.9158) (0.9151, 0.9160) (0.9157, 0.9167) (0.9165, 0.9177) (0.9176, 0.9190) (0.9189, 0.9204)

(0.9413, 0.9414) (0.9272, 0.9273) (0.9126, 0.9129) (0.9044, 0.9048) (0.8988, 0.8994) (0.8946, 0.8953) (0.8912, 0.8921) (0.8884, 0.8895) (0.8860, 0.8872) (0.8839, 0.8852) (0.8820, 0.8835)

(0.9439, 0.9440) (0.9328, 0.9329) (0.9241, 0.9244) (0.9214, 0.9218) (0.9212, 0.9218) (0.9223, 0.9131) (0.9242, 0.9251) (0.9266, 0.9277) (0.9294, 0.9306) (0.9324, 0.9337) (0.9355, 0.9370)

(0.9407, 0.9410) (0.9271, 0.9276) (0.9138, 0.9148) (0.9071, 0.9086) (0.9030, 0.9050) (0.900, 0.903) (0.899, 0.902) (0.897, 0.901) (0.897, 0.901) (0.896, 0.901) (0.896, 0.901)

Hückel parameters for NaCl solutions were determined from the isopiestic data of Robinson18 for NaCl and KCl solutions. The details of these calculations are also given in ref 1, and in those the KCl value of b1 = 0.011 was accepted. From the standard deviation given in ref 1 for parameter b1 of NaCl solutions, the confidence interval of (0.070, 0.073) can be deduced by assuming also a small variation of parameter b1,KCl around the value of 0.011. With the former value and with the value of 0.0716 recommended in ref 1 and used here, the largest absolute vapor pressure errors of 0.31 and 0.19 Pa, respectively, are obtained when only the points used in the parameter estimation in ref 1 were included in the evaluation. With the latter value, the corresponding error of 0.40 Pa was obtained from these data. The confidence intervals determined in the previous paragraph were then used to calculate the lower and upper limits of the activity and osmotic coefficients at rounded molalities at temperatures at 0 and 25 °C. These values are given in Tables 16 and 17. In these tables are also included the values at 10 °C by assuming that the confidence intervals of this temperature correspond to those of 25 °C for NaCl solutions and to those of 0 °C for KCl solutions. For b1,NaCl the resulting interval is (0.032, 0.035) and for b1,KCl it is (−0.025, − 0.028). In my opinion, no other choice for this evaluation can be made because no high-quality data are available at this temperature. The reason for this choice is that the confidence interval for NaCl solutions is wider in 25 °C than that in 0 °C, and the reverse is true for KCl solutions. At the higher temperatures than 25 °C also considered in the present study, the Hückel equations have not been connected so directly to the experimental data, and the uncertainty considerations at these temperatures cannot be presented in a fully traceable way in these connections. Comparison of the New Mean Activity Coefficients to Literature Values. The recommended activity coefficients for NaCl and KCl solutions at different temperatures are finally compared to the values that probably are at the moment regarded in the literature as the most reliable. These values include the activity coefficients from the textbook of Harned and Owen50 for solutions of both salts, and those were mainly based on the electrochemical sets of Harned and Nims13 and Harned and Cook,15 see above. Also in the comparison were included the activity coefficients for NaCl solutions from the multiparameter equations of Pitzer et al.,51 Clarke and Glew,52 and Archer,53 as well as those for KCl solutions from the multiparameter equation of Archer.54 All of these complicated equations apply to high temperatures and pressures, and the activity coefficients from these equations are always very close to those of Harned and

estimation were not as precise as those used in method I. The activity coefficients from this method are given in Tables 9 and 11 and the osmotic coefficients in Tables 13 and 15 for NaCl and KCl solutions, respectively. Values are tabulated in this temperature range at intervals of 10 °C, but these values are not as accurate as the values based on method I. The hypothetical activity values at 80 °C are also presented in Tables 9, 11, 13, and 15, and those have been calculated using method III where the values of six electrolyte-dependent parameters from Rowland and May (given here in Table 3) were used together with eqs 9 and 10. All resulting Hückel parameters from these equations for method III at various temperatures are given in Figure 2. The activity and osmotic coefficients at 80 °C are hypothetical because the KCl results could not be tested alone with the real experimental data measured at this temperature (no data are available) and the vapor pressure data23 for NaCl solutions at 75 °C cannot be explained very well with the Hückel equations (see text above). In the tables concerning the temperature interval from (0 to 25) °C, the values obtained using method II are also included in the tables when they differ from those of method I by more than ±0.002. These values are presented in parentheses, but the values from method I are more reliable. At 80 °C, the values of method II have been similarly included in the tables for the temperature interval from (30 to 80) °C. First, the accuracy of the Hückel equations from method I was studied with the data used in the parameter estimations of this method. At 0 °C, the confidence interval of parameter b1 with the significance level of 0.95 for NaCl solutions is close to (0.0070, 0.0085) based on the standard deviations of this parameter given above. The corresponding interval for KCl solutions is close to (−0.050, −0.053). When the lower and upper limits of these intervals are applied to predict the freezing point data7 used in the parameter estimation here, largest absolute errors from these data are for NaCl solutions 0.00064 and 0.00057 K, respectively. For KCl solutions, both of these errors are close to 0.00114 K. At 25 °C for method I, the Hückel parameters for KCl solutions were determined from the cpd data measured by Hornibrook et al.17 on cells containing a liquid junction (cell 24). The details of these calculations have been presented in ref 1. From the standard deviation given in that reference for parameter b1 of KCl solutions the confidence interval of (0.005, 0.015) can be obtained. With the value of b1 = 0.011 recommended in ref 1 (and also used here), the largest absolute cpd error of 0.021 mV resulted from these data (see graph B in Figure 2 of the present study). With the value b1 = 0.005, this error is 0.025 mV, and with the value of b1 = 0.015, it is also 0.021 mV. At 25 °C for method I, the 302



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Table 18. Results of the Comparison of the Activity Coefficients Recommended in This Study to the Values Presented in the Literature for NaCl Solutions at Various Temperatures up to a Molality of 1.0 mol·kg−1 0 °C m = 0.1 mo (γrecd)a (γHO)b (γArc)c (γClGl)d (γPPB)e (γPical)f (γPifrp)g m = 0.2 mo (γrecd)a (γHO)b (γClGl)d (γPical)f (γPifrp)g m = 0.25 mo (γrecd)a (γPPB)e m = 0.5 mo (γrecd)a (γHO)b (γArc)c (γClGl)d (γPPB)e (γPical)f (γPifrp)g m = 0.7 mo (γrecd)a (γClGl)d (γPical)f (γPifrp)g m = 0.75 mo (γrecd)a (γPPB)e m = 1.0 mo (γrecd)a (γHO)b (γArc)c (γClGl)d (γPPB)e (γPical)f (γPifrp)g

10 °C

20 °C

30 °C

40 °C

50 °C

60 °C

70 °C

80 °C

0.766 0.766

0.761 0.762

0.752 0.757

0.766 0.767

0.762 0.762

0.758 0.757

0.7813 0.781 0.780 0.780 0.780 0.781 0.781

0.7804 0.781

0.7793 0.779

0.777 0.777

0.774 0.774

0.780 0.780 0.780 0.780

0.779 0.779 0.778 0.778

0.776 0.777 0.776 0.776

0.774 0.774 0.773 0.773

0.770 0.770 0.770 0.770 0.770 0.770 0.770

0.7341 0.731 0.733 0.736 0.734

0.7345 0.734 0.734 0.735 0.734

0.7345 0.733 0.734 0.733 0.733

0.732 0.731 0.732 0.731 0.731

0.729 0.728 0.730 0.729 0.730

0.725 0.725 0.726 0.726 0.727

0.720 0.721 0.722

0.713 0.717 0.717

0.703 0.711 (−0.7)h 0.711 (−0.7)h

0.7187 0.717

0.7198 0.720

0.7205 0.720

0.718 0.719

0.715 0.716

0.711 0.713

0.706 0.708

0.699 0.703

0.688 0.697 (−0.8)h

0.6720 0.671 0.671 0.672 0.672 0.679 (−0.5)h 0.672

0.6766 0.677

0.6808 0.679

0.680 0.679

0.678 0.678

0.667 0.671

0.659 0.667

0.647 0.660 (−1.2)h

0.678 0.678 0.679 0.676

0.681 0.681 0.680 0.678

0.681 0.681 0.679 0.681

0.680 0.680 0.679 0.682

0.674 0.675 0.676 0.677 0.676 0.677 0.683 (−0.7)h

0.672 0.672

0.667 0.667

0.661 (−1.4)h 0.660 (−1.2)h

0.6507 0.652 0.661 (−0.7)h 0.651

0.658 0.661 0.663 0.657

0.665 0.666 0.665 0.663

0.666 0.668 0.666 0.668

0.664 0.667 0.666 0.672 (−0.6)h

0.660 0.664 0.665 0.675 (−1.3)h

0.653 0.660

0.644 0.655

0.633 0.648 (−1.4)h

0.6464 0.649

0.655 0.658

0.662 0.663

0.663 0.665

0.662 0.665

0.658 0.662

0.651 0.658

0.641 0.652

0.631 0.645 (−1.3)h

0.6295 0.635 0.634 0.635 0.635 0.647 (−1.3)h 0.632

0.641 0.649 (−0.6)h

0.653 0.654

0.655 0.657

0.655 0.657

0.644 0.654

0.633 0.648

0.624 0.641 (−1.6)h

0.648 0.648 0.651 (−0.8)h 0.642

0.656 0.655 0.654 0.651

0.659 0.659 0.657 0.660

0.660 0.659 0.659 0.668 (−1.1)h

0.652 0.656 0.657 0.658 0.657 0.660

0.654 0.653

0.648 0.647

0.641 (−1.6)h 0.640 (−1.5)h

a

Calculated using the Hückel equations recommended in the present study, see Tables 8 and 9. bRecommended by Harned and Owen50 and based on the amalgam cell data of Harned and Nims.13 cCalculated using the multiparameter equation of Archer.53 dCalculated using the multiparameter equation of Clarke and Glew.52 eCalculated using the multiparameter equation of Pitzer et al.51 fCalculated using the Pitzer equation where the temperature dependences of the parameters were determined using the calorimetric data (cal) by Silvester and Pitzer,45 see Table 2. gCalculated using the Pitzer equation where the temperature dependences of the parameters were determined using the freezing point data (frp) by Hasan et al.,4 see Table 2. hHarned cell error in mV calculated by using eq 33.

Owen50 in the dilute solutions considered now. Therefore, the amalgam cell data from Harned and Nims13 and Harned and Cook15 have also played an important role in the parameter estimations of these equations in dilute solutions. The activity coefficients from these equations are thus comparable in accuracy with those obtained using method II in the present study. We have recently presented4,55 that the temperature dependence of the Pitzer parameters for solutions of single uniunivalent electrolytes can be estimated from the freezing point data together with the parameter values at 25 °C. The dependences that resulted from these considerations for NaCl and KCl

solutions are given in Table 2 of the present study, and the activity coefficients calculated using these derivatives (denoted here as γPifrp) were also tested with those recommended now for various temperatures. The most common way to determine the temperature derivatives of the Pitzer parameters is to use the existing calorimetric data. Silvester and Pitzer45 have determined in this way these derivatives for solutions of many electrolytes of different charge types. For NaCl and KCl solutions, the values are given in Table 2. The activity coefficients calculated using the calorimetric derivatives (denoted here as γPical) in Table 2 were also included in the tests of the present study. 303



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Table 19. Results of the Comparison of the Activity Coefficients in This Study to the Values Presented in the Literature for KCl Solutions at Various Temperatures up to a Molality of 1.0 mol·kg−1 0 °C m = 0.1 mo (γrecd)a (γHO)b (γArc)c (γPical)d (γPifrp)e m = 0.2 mo (γrecd)a (γHO)b (γArc)c (γPical)d (γPifrp)e m = 0.5 mo (γrecd)a (γHO)b (γArc)c (γPical)d (γPifrp)e m = 1.0 mo (γrecd)a (γHO)b (γArc)c (γPical)d (γPifrp)e

10 °C

20 °C

30 °C

40 °C

50 °C

60 °C

70 °C

0.7724 0.768 0.769 0.772 0.771

0.7714 0.769 0.769 0.770 0.770

0.7702 0.770 0.768 0.768 0.768

0.767 0.768 0.766 0.766 0.767

0.764 0.765 0.763 0.764 0.765

0.761

0.757

0.752

0.760 0.761 0.762

0.757

0.753

0.7192 0.717 0.715 0.718 0.717

0.7193 0.718 0.716 0.718 0.717

0.7192 0.718 0.716 0.717 0.717

0.716 0.718 0.714 0.715 0.716

0.714 0.715 0.712 0.713 0.714

0.710

0.705

0.699

0.709 0.711 0.713

0.705

0.701

0.6429 0.642 0.640 0.648 (−0.4)f 0.642

0.6469 0.648 0.645 0.648 0.645

0.6506 0.651 0.647 0.649 0.648

0.649 0.651 0.647 0.648 0.649

0.647 0.646 0.647 0.648 0.651

0.644

0.638

0.632

0.644 0.646 0.651

0.640

0.636

0.581 0.588 0.586 0.596 (−1.2)f 0.586

0.592 0.598 0.595 0.599 (−0.6)f 0.593

0.602 0.604 0.601 0.602 0.600

0.602 0.604 0.604 0.604 0.607

0.603 0.603 0.605 0.606 0.612 (−0.8)f

0.601

0.596

0.588

0.604 0.607 0.618 (−1.55)f

0.601

0.596 (−0.8)f

a Calculated by using the Hückel equations recommended in the present study, see Tables 10 and 11. bRecommended by Harned and Owen50 and based on the amalgam cell data of Harned and Cook.15 cCalculated using the multiparameter equation of Archer.54 dCalculated using the Pitzer equation where the temperature dependences of the parameters were determined using the calorimetric data (cal) by Silvester and Pitzer,45 see Table 2. eCalculated using the Pitzer equation where the temperature dependences of the parameters were determined using the freezing point data (frp) by Hasan et al.,4 see Table 2. fHarned cell error in mV calculated by using eq 33.

at 40 and 50 °C. The former case can be commented in the same way as in refs 4 and 54: the calorimetric activity coefficients do not apply well to the freezing point data on which the recommended values were based here and in ref 4 at this temperature. The latter case results probably from the fact that the temperature dependence of the Pitzer parameters shows also clear curvature at temperatures higher than 25 °C in the same way as observed for Hückel parameter b1 above.

The results of all activity coefficient comparisons are shown in Tables 18 and 19, and the former gives the results for NaCl solutions and the latter for KCl solutions. In general in these tables, the agreement between the literature activity coefficients and the new ones is good. In the tables are presented the Harned cell errors for the cases where these errors are absolutely close to or larger than 1 mV. As in our previous papers (see for example ref 1), the Harned cell error for the activity coefficient comparison was calculated using eE,GC = −

γ (literature) 2RT ln F γ (recd)

■

AUTHOR INFORMATION

Corresponding Author

(33)

*E-mail: jpartane@lut.fi. Fax: +358 5 411 7201.

where GC refers to galvanic cell without a liquid junction (like cell 30 in the present study). An absolute error of the order of 1 mV seem to be an acceptable upper limit for the activity coefficients determined using amalgam cells as mainly in the present study. The error plots obtained with these less precise experimental data confirm here this suggestion. The largest absolute error in Tables 18 and 19 is 1.6 mV. This error is observed for t = 80 °C as well as almost all absolute errors which are larger than 1 mV. The larger errors at 80 °C are expected because the experimental techniques at this high temperature are not very accurate and the extrapolation from temperature 40 °C (highest temperature included in the parameter estimation for method II) is long. The other literature activity coefficients, for which the absolute Harned cell errors are larger than or close to 1 mV, are the γPical values at m = 1 mol·kg−1 and at 0 °C for both NaCl and KCl solutions and the γPifrp values at m = 0.7 and 1.0 mol·kg−1 and

Notes

The authors declare no competing financial interest.

■

REFERENCES

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Mean Activity Coefficients and Osmotic Coefficients in Dilute Aqueous

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