Re-evaluation of the Mean Activity Coefficients of Strontium Chloride

Article pubs.acs.org/jced

Re-evaluation of the Mean Activity Coefficients of Strontium Chloride in Dilute Aqueous Solutions from (10 to 60) °C and at 25 °C up to the Saturated Solution Where the Molality Is 3.520 mol·kg−1 Jaakko I. Partanen* Laboratory of Physical Chemistry, Department of Chemical Technology, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finland ABSTRACT: In the present study, two Hückel equations have been determined for the calculation of the mean activity coefficients of SrCl2 in pure aqueous solutions at 25 °C. The two-parameter equation with parameters B and b1 gives a good fit up to an ionic strength (= Im) of 2.0 mol·kg−1, and the three-parameter equation with parameters B, b1, and b2 is probably applicable up to the molality of the saturated solution where Im = 10.56 mol·kg−1. The dilute-solution parameters for SrCl2 solutions were obtained from the isopiestic data of Downes (J. Chem. Thermodyn. 1974, 6, 317−323) against KCl solutions. The parameters resulted from this estimation were tested with all isopiestic and electrochemical data available in the literature for this purpose, and they proved to be reliable in these tests. The galvanic cell data measured by Longhi et al. using strontium amalgam cells (J. Chem. Thermodyn. 1975, 7, 767−776) show that these parameter values apply well up to Im of 0.75 mol·kg−1 in the temperature range of (10 to 55) °C and satisfactorily up to 70 °C. For less dilute solutions at 25 °C, new values for parameters b1 and b2 for the extended Hückel equation were solved from the isopiestic data of Macaskill et al. (J. Solution Chem. 1978, 7, 339− 347) against NaCl solutions, but the dilute-solution value for parameter B was used. These new parameter values were tested with good results with the existing electrochemical, vapor pressure, and isopiestic data. The new values seem to apply up to the molality of the saturated solution. For this reason, these values were also examined with the solubility data available in the best tables of thermodynamic properties (J. Phys. Chem. Ref. Data 1982, 11, Supplement No. 2). Reliable activity and osmotic coefficients for SrCl2 solutions can be obtained using the new Hückel equations. The relevant thermodynamic quantities calculated with these equations are tabulated here at rounded molalities, and the resulting values were compared to the values available in the literature tables.

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of Stokes1 against CaCl2 solutions and those of Macaskill et al.11 against NaCl solutions. The parameters for the Hückel equation of CaCl2 are given in ref 7 and those for NaCl in ref 10. Also, the electrochemical data obtained by Longhi et al.12 using cells with a strontium amalgam electrode and a silver−silver chloride electrode at temperatures from (10 to 70) °C and those measured by Lucasse13 using concentration cells with a Sr amalgam electrode and two Ag-AgCl electrodes at 25 °C were applied in the tests. On the contrary, the cell potential data from Reddy and Ananthaswamy14 for the aqueous mixtures of KCl and SrCl2 cannot be used in this connection because these data were measured using cells containing a potassium-ion selective glass electrode and pure SrCl2 solutions cannot be studied with the cells of this kind. In addition, it is shown here as also in refs 7 and 8 for CaCl2 and MgCl2 solutions that the three-parameter Hückel equation can be used for more concentrated SrCl2 solution. The three parameters are B, b1, and b2, and the resulting equation applies in the SrCl2 case well up to Im of 10.5 mol·kg−1. For this extended

INTRODUCTION Stokes1,2 presented in 1948 well-known tables for activity and osmotic coefficients of electrolytes of the charge types 1:2 and 2:1 for aqueous solution at 25 °C. In these tables, the values of these quantities for SrCl2 solutions were determined from the isopiestic data of Stokes1 and Robinson3 against CaCl2 and KCl reference solutions, respectively. In the tables, these thermodynamic activity quantities are given from a molality of (0.1 up to 4) mol·kg−1. Pitzer and Mayorga also used the tabulated values2 when they solved the parameters of their Pitzer equation for SrCl2 solutions (see refs 4, 5, and 6). In the present study, the same thermodynamic treatment was used for SrCl2 solutions as that in refs 7 and 8 for CaCl2 and MgCl2 solutions, respectively. For dilute solutions, the twoparameter Hückel equation was used, and the parameters which depend on the electrolyte in this equation are B and b1. The values of these parameters were estimated here from the isopiestic set of Downes9 for solutions of KCl and SrCl2 using the Hückel equation given in ref 10 for the solutions of the reference electrolyte. Such points in this set where the SrCl2 ionic strength (= Im) is less than 1.9 mol·kg−1 were possible to take into account in the determination. The resulting parameter values were additionally tested with the dilute points from the isopiestic data © XXXX American Chemical Society

Received: April 21, 2013 Accepted: August 22, 2013

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dx.doi.org/10.1021/je400472v | J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

Hückel equation, the parameters were determined and checked now in the following way (principally it is the same as that in ref 7): The dilute-solution value was used for parameter B. New values of b1 and b2 for more concentrated SrCl2 solutions were first estimated from the isopiestic data of Macaskill et al.,11 and every point in this set was included in the estimation. This set contains points up to a SrCl2 ionic strength of 9.39 mol·kg−1, and it was selected in the estimation instead of the set of Downes9 because the latter set contains points only up Im of 6.54 mol·kg−1 for this electrolyte. For NaCl solutions, the extended Hückel equation was taken from ref 10. The parameter values obtained were, additionally, tested with the isopiestic data of Downes,9 Stokes,1 Robinson,3 and Rard and Miller15 (SrCl2 solutions were measured in this study against solutions of both CsCl and CaCl2), and Clegg et al.16 (against solutions of NaCl); with the electrochemical data of Lucasse;13 with the vapor pressure data of Hepburn;17 and with the solubility data for SrCl2 given in the NBS tables.18 For KCl, CaCl2, and CsCl solutions, the extended Hückel equations are given in refs 10, 7, and 19, respectively. In addition, Apelblat and Korin20 have reported values for the vapor pressure, water activity, and osmotic coefficient in the saturated SrCl2 solution at 25 °C, but these values are different from those of other sources and, therefore, are not considered here. Similarly as in the previous studies, where the thermodynamic properties in solutions of uniunivalent electrolytes (see refs 10, 19, and 21−30) and diunivalent electrolytes (refs 7 and 8) were studied, the present tests were carried out with the real experimental data of various sources. The advantage of this method is that the prediction error was directly compared to the error made in the measurement. In these evaluations, the Hückel equations proved to be very reliable. The activity coefficients of the electrolyte and the osmotic coefficients, activities, and vapor pressures of water were then calculated and tabulated using the resulting Hückel equations at rounded molalities. These tabulated values were compared to the literature values. The results of the activity coefficient comparison are presented as cellpotential deviations for appropriate galvanic cells (as in refs 7, 8, 10, 19, and 21−30) and those of osmotic coefficient comparison as vapor pressure deviations (refs 7, 8, 10, 19, and 22−30). These deviations are plotted as a function of the molality. Exactly in the same way as in refs 7 and 8 for CaCl2 and MgCl2 solutions, the new thermodynamic activity values for SrCl2 solutions are fully traceable because all results from the parameter determinations and evaluations and all sources of data used in these connections are reported in detail here or in the previous studies. For the data from isopiestic technique, this is possible because traceable osmotic coefficients are given in refs 7 and 10 for the standard reference solutions of this technique (for NaCl, KCl, and CaCl2 solutions).

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ϕ=1−

1 + B Im

(2)

In refs 7 and 8, it was shown that eqs 1 and 2 apply also for CaCl2 and MgCl2 (which are 2:1 electrolytes) solutions up to Im = 1.5 mol·kg−1. In those, m is the molality, Im is the ionic strength, z+ is the charge number of the cation and z− that of the anion, α is Debye−Hückel parameter [its values at the pressure of 101.325 kPa and at the temperatures of (10, 20, 25, 30, 40, 50, 55, 60, and 70) °C are (1.1462, 1.1646, 1.1744, 1.1848, 1.2068, 1.2308, 1.2436, 1.2568, and 1.2846) (mol·kg−1)−1/2, respectively, see Archer and Wang31], m° is 1 mol·kg−1, and the parameters that depend on the electrolyte are B and b1. For a 2:1 electrolyte like SrCl2, |z+z−| is 2, and Im is 3m. The osmotic coefficient of the solvent (label 1, water in this case) is defined using the activity of water (a1) as follows ln a1 = −vmM1ϕ

(3) −1

where M1 is the molar mass of water (= 0.018015 kg·mol ) and for a 2:1 electrolyte the stoichiometric number ν is 3. The activity of water is related to the vapor pressure of water over the solution (p1) and that of pure water (p*1 ) by p a1 = 1 p* (4) 1

This relation is not an exact equation, but it is useful now because the difference between the fugacity and vapor pressure is not significant. For the vapor pressure of water at 25 °C, the value of p1* = 3.1686 kPa, suggested by Kell,32 was used as earlier.7,8,10,19,22−30 This value is very close to the experimentally determined high-precision values of 3.1670 and 3.1695 kPa presented by Stimson33 and Pepela and Dunlop,34 respectively. The nowadays often recommended values for this quantity have been based on the multiparameter equations for the properties of water at various temperatures (see refs 35 and 36), but they differ more from the experimental values. In less dilute solutions, as earlier,7,8,10,19,22−30 the extended Hückel equations were applied in the present study to the thermodynamic activity quantities in SrCl2 solutions ln γ = −

ϕ=1− −

2α Im 1 + B Im

+ b1(m /m°) + b2(m /m°)2

(5)

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

⎤ 1 2 1 ⎥ + b1(m /m°) + b2(m /m°)2 2 3 1 + B Im ⎥⎦

(6)

37

Goldberg and Nuttall give the subsequent extended versions of the Hückel equation for the activity and osmotic coefficients of SrCl2 solutions at 25 °C. They can be used up to an ionic strength of 12 mol·kg−1

Previously,10,19,21−30 it was found that the Hückel equations apply well to the mean activity coefficient (γ) and osmotic coefficient (ϕ) in aqueous solutions of many uniunivalent salts at least up to a molality of 1 mol·kg−1 ln γ = −

⎤ 1 1 ⎥ + b1(m /m°) 2 1 + B Im ⎥⎦

−

THEORY

α|z+z −| Im

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

ln(γ ) = − + b1(m /m°)

2A m Im 1 + B* I m

+ E(m /m°)3

(1) B

+ C(m /m°) + D(m /m°)2 (7)


Journal of Chemical & Engineering Data ϕ=1−

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2A m ⎡ ⎢(1 + B* Im ) − 2 ln(1 + B* Im ) (B*)3 Im ⎢⎣

⎤ 1 1 2 ⎥ + C(m /m°) + D(m /m°)2 − 2 3 1 + B* Im ⎥⎦ 3 + E(m /m°)3 4

where k1 = −3b1,yM1/(2m°) and where the quantity f1 is defined by f1 = ln a1,x + 3M1m y −

− 2 ln(1 + By Im,y ) −

(8)

where Am = 1.17625 (mol·kg−1)−1/2 (i.e., the previous value of parameter α) and the parameters that depend on the electrolyte are: B* = 1.4984 (mol·kg−1)−1/2, C = 0.26339, D = 0.079733, and E = −0.005398. For a 2:1 electrolyte, the Pitzer equations4−6 for ln(γ) and ϕ have the forms ln γ = 2f γ +

ϕ=1− +

4 γ B (m /m°) + 23/2C ϕ(m /m°)2 3

Im 2α 4 + (β 0 + β1e−αP 3 1 + b Im 3

Im

⎤ 1 ⎥ 1 + By Im,y ⎥⎦

(14)

All regression equations below will be presented in this way using quantities f i (i = 1, 2, or 3). Some details of the use of eq 13 are described, for example, in ref 25 (see eq 13 in that study). The results from the least-squares fitting using eq 13 are: By = 1.5 (mol·kg−1)−1/2 and b1,y = 0.330 ± 0.003 where the standard deviation is also included. The obtained parameters were tested by predicting the vapor pressures of water over the isotonic pairs of SrCl2 and KCl solutions in the set of Downes9 and comparing the resulting values. The vapor pressures over both solutions were calculated by using eqs 2 to 4 with the new Hückel equations. Figure 1

(9)

)(m /m°)

25/2 ϕ C (m /m°)2 3

⎡ 2αM1 ⎢ (1 + By Im,y ) By3 ⎢⎣

(10)

where ⎤ Im α⎡ 2 fγ = − ⎢ + ln(1 + b Im )⎥ ⎥⎦ 3 ⎢⎣ 1 + b Im b

Bγ = 2β 0 +

2β1 ⎡ −α ⎢1 − e P αP2Im ⎣

Im ⎛ ⎜

⎝

1 + αP Im −

(11)

1 2 ⎞⎟⎤ αP Im ⎥ ⎠⎦ 2 (12)

where the following general parameter values are used now: b = 1.2 (mol·kg−1)−1/2 and αP = 2.0 (mol·kg−1)−1/2. In eqs 9, 10, and 12, β0, β1, and Cϕ are the parameters that depend on the electrolyte. At 25 °C, Pitzer and Mayorga4 have obtained the following values of the parameters for SrCl2: β0 = 0.28575, β1 = 1.6673, Cϕ = −0.001304, and they apply up to Im of 12 mol·kg−1. Using new experimental data, Rard and Miller15 determined the following parameter values for these equations: β0 = 0.28344, β1 = 1.6256, Cϕ = −0.000891. These values can be used up to Im of 11.5 mol·kg−1. Clegg et al.16 presented more complicated Pitzer equations than those in eqs 9 and 10 for SrCl2 solutions, and the activity and osmotic coefficients from the five-parameter equations (useful up to Im of 12 mol·kg−1) will also be considered below. They have been tabulated in Table 4 of ref 16.

Figure 1. Plot of eip (eq 15), the deviation between the vapor pressure of water over the reference solution (x) and that over the tested solution (y) as a function of the molality of the tested solution (my) in the isotonic solutions of KCl or CaCl2 (x) and SrCl2 (y) measured by Downes9 (x = KCl, symbol ●), Stokes1 (CaCl2, ○), and Macaskill et al.11 (NaCl, ▼) . The vapor pressures have been calculated by eqs 3 and 4 using eq 2 with BKCl = 1.3 (mol·kg−1)−1/2, b1,KCl = 0.011, BNaCl = 1.4 (mol·kg−1)−1/2, b1,NaCl = 0.0716, BCaCl2 = 1.55 (mol·kg−1)−1/2, b1,CaCl2 = 0.377, BSrCl2 = 1.50 (mol·kg−1)−1/2, and b1,SrCl2 = 0.330.

shows the results, and there the isopiestic vapor pressure error (eip) is defined by e ip = px − py (15)

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RESULTS AND DISCUSSION Determination of Activity Parameters for Dilute Solutions and Tests of the Resulting Equation Using Isopiestic Data. In ref 10, the parameter values of B = 1.3 (mol· kg−1)−1/2 and b1 = 0.011 were suggested for the Hückel equation of KCl solutions, and they apply well up to 1.5 mol·kg−1. They were used in the estimation of the Hückel equation for SrCl2 in dilute solutions from the isopiestic set of Downes.9 Ten points in this set have an ionic strength for SrCl2 that is less than 2.0 mol· kg−1, and the determination was based on these points. In this estimation, KCl was the reference electrolyte (x) and SrCl2 the tested electrolyte (y). Starting at the equality ln a1,x = ln a1,y, the following equation was derived for this estimation f1 = f0 − 3b1,y M1(m y2 /2m°) = f0 + k1m y2

and presented versus molality my. The highest absolute error for all solutions taken in the estimation is smaller than 0.2 Pa, and the data from Downes thus support well the new parameter values. In this figure are also shown the errors from the six points of Stokes1 for CaCl2 and SrCl2 solutions and those from the four points of Macaskill et al.11 for NaCl and SrCl2 solutions. The Hückel parameters for NaCl and CaCl2 solutions are: BCaCl2 = 1.55 (mol·kg−1)−1/2, b1,CaCl2 = 0.377, BNaCl = 1.4 (mol·kg−1)−1/2, and b1,NaCl = 0.0716, and those were used in the calculations, see refs 7 and 10. The new Hückel model for SrCl2 solutions seems to be useful for all of the points up to Im of 2.0 mol·kg−1. Tests of the Resulting Equation Using Galvanic Cell Data. The new activity coefficient equation obtained was then

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tested with the electrochemical data obtained by Lucasse13 on cell type

where E°(x) is the standard value of cpd and it is dependent on x. In the present Hückel equation tests, these data were predicted, and the resulting error plots are given in Figure 3. In the plots, the

Ag(s)|AgCl(s)|SrCl 2(aq, m1)|Sr(Hg)|SrCl 2(aq, m2) |AgCl(s)|Ag(s)

(16)

where Sr(Hg) is the strontium amalgam electrode, the molality of the reference solution is m1 = 0.01 mol·kg−1, the molality of the tested solution is m2, and eight molalities from (0.02908 to 3.015) mol·kg−1 were studied for m2. The cell potential difference (cpd) for the cells of this kind can be calculated from 3RT 3RT ln(m2 /m1) − ln(γ2/γ1) (17) 2F 2F These eight points were predicted using the new Hückel equation and for dilute solutions, the resulting cpd errors, defined by E=−

eE = E(observed) − E(predicted)

Figure 3. Plot of eE (eq 18), the deviation between the observed and predicted cell potential difference (cpd) from the galvanic cell data measured on cell 19 by Longhi et al.12 at 25 °C as a function of molality m. Symbols: ●, x = 0.0007544; ○, 0.001833; ▼, 0.002098; ▽, 0.005493; ■, 0.008263; □, 0.01025; ◆, 0.01731; ◇, 0.02136. The predicted cpd was calculated by using eq 20 where eq 1 with B = 1.5 (mol·kg−1)−1/2 and b1 = 0.330 was used for the activity coefficients, and the E°(x) values given in Figure 4 were used. The errors of the following points are outside the scale of the figure: (x = 0.001833, m = 0.010089 mol·kg−1, E = 2.21567 V, eE = −2.10 mV), (0.001833, 0.039946, 2.17062, −1.87), (0.001833, 0.2403, 2.11531, 1.33), (0.005493, 0.010089, 2.23540, −1.61), (0.005493, 0.240300, 2.13587, 2.66), and (0.008263, 0.010089, 2.24434, −1.66).

(18)

are given in Figure 2 as a function of molality m2. The amalgam cell data can be explained well with the new Hückel parameters

cpd errors in eq 18 are illustrated as a function of the molality. In the calculation of the predicted cpd values for these plots using eq 20, the best value for E°(x) was used for each series, and they are given in Figure 4. Also the amalgam data at 25 °C reveal the quality of the new Hückel equation. Figure 2. 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 Lucasse13 (m1 = 0.01 mol·kg−1) as a function of molality m2. The predicted cpd was calculated by using eq 17 where eq 1 with B = 1.50 (mol·kg−1)−1/2 and b1 = 0.330 (symbol ●) and eq 5 with B = 1.50 (mol·kg−1)−1/2, b1 = 0.3046, and b2 = 0.0534 (○) were used for the activity coefficients.

up to Im of 3 mol·kg−1. Lucasse13 also measured cpd values using a concentration cell with transference, but these data cannot be used in the present tests because no sufficiently reliable transference numbers for SrCl2 solutions are available for these calculations. The quality of the estimated Hückel parameters for SrCl2 solutions were finally evaluated using the data from galvanic cells containing no liquid junction. Longhi et al.12 measured SrCl2 solutions on amalgam cells of the subsequent type Sr(Hg, x)|SrCl 2(aq, m)|AgCl(s)|Ag(s)

Figure 4. Plot of the best values of the standard cell potential differences E°(x) in eq 20 as a function of the mole fraction (x) of strontium in the amalgam from the data of Longhi et al.12 at temperatures of 25 °C (symbol ●), 10 °C (○), 40 °C (▼), 55 °C (▽), and 70 °C (■). These values were calculated by using eq 1 with B = 1.5 (mol·kg−1)−1/2 and b1 = 0.330 for the activity coefficients of SrCl2 (see text).

(19)

where again Sr(Hg, x) is the strontium amalgam electrode. At 25 °C, these data contained of eight series of measurements. In each series, the mole fraction (x) of strontium in the amalgam was constant, and the molality of SrCl2 (m) varied from (0.010089 to 0.24030) mol·kg−1. For the direct cells of this type, the cpd is given by E = E°(x) −

3RT ln(41/3γm /m°) 2F

Longhi et al.12 reported similar strontium amalgam cell data for temperatures (10, 40, 55, and 70) °C. Preliminary calculations revealed that the new Hückel parameters apply to these data without any changes. The error plots obtained from the data using these parameters are described in graphs A, B, C, and D of Figure 5, respectively, and the E°(x) values are given in Figure 4. Also these plots support well the new parameters. To

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Figure 5. Plot of eE (eq 18), the deviation between the observed and predicted cell potential difference (cpd) from the galvanic cell data measured on cell 19 by Longhi et al.12 at temperatures of 10 °C (graph A), 40 °C (B), 55 °C (C), and 70 °C (D) as a function of molality m. Symbols: ●, x = 0.0003764 (A), 0.000861 (B), 0.0008585 (C), 0.0009496 (D); ○, 0.001827 (A), 0.001933 (B), 0.001942 (C), 0.001984 (D); ▼, 0.001878 (A), 0.002047 (B), 0.002041 (C), 0.002040 (D); ▽, 0.005493 (A), 0.005336 (B), 0.005287 (C), 0.005493 (D); ■, 0.008388 (A), 0.008263 (B), 0.008263 (C), 0.008263 (D); □, 0.009967 (A), 0.009240 (B), 0.009268 (C), 0.009268 (D); ◆, 0.01654 (A), 0.01709 (B), 0.01731 (C), 0.01731 (D); ◇, 0.02136 (B). The predicted cpd was calculated by using eq 20 where eq 1 with B = 1.5 (mol·kg−1)−1/2 and b1 = 0.330 was used for the activity coefficients and the E°(x) values given in Figure 4 were used. The errors of the following points are outside the scales of the graphs: (graph B, x = 0.01709, m = 0.010089 mol·kg−1, E = 2.27046 V, eE = −3.11 mV), (C, 0.005287, 0.010089, 2.24274, −2.52), (C, 0.005287, 0.2403, 2.13507, 2.67), (C, 0.01731, 0.2403, 2.1673, 2.47), (D, 0.005493, 0.010089, 2.24553, −2.83), and (D, 0.005493, 0.2403, 2.13397, 2.78). Additionally, the following points at t = 70 °C and x = 0.009268 have been omitted as outliers: (m = 0.15959 mol·kg−1, E = 2.14965 V) and (0.19906, 2.15785).

show additionally that all of these amalgam cell data behave very regularly with the Hückel parameters, the following calculations were carried out: First it was assumed as in the original paper12 that the logarithm of the (mole fraction) activity coefficient f of strontium in the amalgam depends linearly on the mole fraction of this metal, i.e., as

ln(f ) = Q ·x

Table 1. Results from Least-Squares Fitting Using eq 22 from the Strontium Amalgam Data Measured by Longhi et al.12 on Cell 19 at Different Temperatures

(21)

It is thermodynamically possible to show that the subsequent equation is valid at each temperature used in that study for the values of quantity E°(x) given in Figure 4 (see also eq 20).

t/°C

Na

E°/V

s(E°)/mVb

Q

s(Q)c

(s/mV)d

10 25 40 55 70

7 8 8 6e 7

2.12217 2.12449 2.12496 2.12279 2.1189

0.05 0.10 0.03 0.04 0.4

130.9 108.4 98.6 91.6 82

0.4 0.7 0.2 0.4 3

0.08 0.17 0.05 0.07 0.7

(22)

a Number of points. bStandard deviation of parameter Eo. cStandard deviation of parameter Q. ds is the standard deviation about the regression. eThe point where x = 0.009268 and E°(x) = 2.066445 V was omitted from the analysis as an erroneous point.

At the temperature considered, the relationship between quantity f 2 versus x represents an equation of a straight line. This straight line has a slope of (RT/2F)·Q, and the parameter Q for the activity coefficient equation can thus be solved from this slope. The intercept of the straight line with the f 2 axis is the standard potential (E°) at the infinite dilution of strontium in the amalgam. The results from the regression analysis with eq 22 at the five temperatures used are collected in Table 1. The very small values for the standard deviation about regression (s) at the other temperatures than 70 °C confirm in this table that the model for strontium in the amalgam in eq 21 is a good one, and the amalgam results behave very regularly with the suggested

Hückel parameters for the SrCl2 solutions at these four temperatures. Determination of Activity Parameters for More Concentrated Solutions. In less dilute solutions, the parameter values for the extended Hückel equation of SrCl2 were determined in the present study from the isopiestic data of Macaskill et al.11 These data contain 14 points up to a SrCl2 ionic strength 9.3924 mol·kg−1. In the determination, NaCl was regarded as the reference electrolyte (x). The activity of water in the reference solutions was calculated from the isopiestic molality of this salt (mx) using eqs 3 and 6 with B = 1.4 (mol·

E°(x) − (RT /2F )ln(x) = f2 = E° + (RT /2F )Qx

E



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kg−1)−1/2, b1 = 0.0699, and b2 = 0.0062.10 The molality of the SrCl2 solution is the response variable (my) and parameter By has the value of 1.50 (mol·kg−1)−1/2 based above on dilute solution data. As in ref 7 for CaCl2 solutions, the subsequent equation was used for the estimation of the values of b1 and b2 for SrCl2 solutions f3 = ln a1,x + 3M1m y −

points were included where the CaCl2 molality is smaller than 3.2 mol·kg−1. The reason for this is that the upper limit of use of the extended Hückel equation for CaCl2 (for which7 B = 1.55 (mol· kg−1)−1/2, b1 = 0.3486, and b2 = 0.0667) is not much higher than 3 mol·kg−1. In the latter set, all points could be included in the tests. For CsCl solutions, the parameter values of B = 0.84 (mol· kg−1)−1/2, b1 = 0.03234, and b2 = 0.00057 (see ref 19) were used. The results from Downes and the CaCl2 results from Rard and Miller and those from Stokes are presented in Figure 6 and the CsCl results from Rard and Miller and the results from the four points of Clegg et al. are presented similarly in Figure 7. The

⎡ 2αM1 ⎢ (1 + By Im,y ) By3 ⎢⎣

− 2 ln(1 + By Im,y ) −

⎤ 2M1b2,y m y3 1 ⎥+ 1 + By Im,y ⎥⎦ (mo)2

= f0 − 3b1,y M1(m y2 /2mo) = f0 + k 3m y2

(23)

where k3 = −3b1,yM1/2m°. The estimations with the equation of this kind are described in detail in ref 25, see eq 25 in that study. The present regression analysis leads to the following results: b2,y = 0.0534 and b1,y = 0.3046 ± 0.0004 where the standard deviation is also given. The goodness of the estimated parameters was first evaluated by predicting the vapor pressures of all points used in the determination and comparing the results. For each isotonic point, the vapor pressures were calculated using eqs 3, 4, and 6 with the suggested parameter values. Figure 6 shows the results, Figure 7. Plot of eip (eq 15), the deviation between the vapor pressure of water over the reference solution (x) and that over the tested solution (y) as a function of the molality of the tested solution (my) in the isotonic solutions of KCl or CsCl (x) and SrCl2 (y) measured by Robinson3 (x = KCl, symbol ●), Rard and Miller15 (CsCl, ○), and Clegg et al.16 (NaCl, ▼); and plot of ep (in eq 24), the deviation between the reported and predicted vapor pressure of water over SrCl2 solutions in the set of Hepburn17 of as a function of molality m (symbol ▽). The vapor pressures have been predicted by eqs 3 and 4 using eq 6 with BKCl = 1.3 (mol·kg−1)−1/2, b1,KCl = 0.01324, b2,KCl = 0.0036, BCsCl = 0.84 (mol· kg−1)−1/2, b1,CsCl = 0.03234, b2,CsCl = 0.00057, BNaCl = 1.4 (mol·kg−1)−1/2, b1,NaCl = 0.0699, b2, NaCl = 0.0062, BSrCl2 = 1.50 (mol·kg−1)−1/2, b1, SrCl2 = 0.3046, and b2,SrCl2 = 0.0534. The errors of the points (m = 3.245 mol· kg−1, p = 17.7 mmHg), (3.403, 17.27), and (3.517, 16.9) in the set of Hepburn17 lay outside the scale of the figure; their values are 18.0, 21.4, and 16.9 Pa, respectively.

Figure 6. Plot of eip (eq 15), the deviation between the vapor pressure of water over the reference solution (x) and that over the tested solution (y) as a function of the molality of the tested solution (my) in the isotonic solutions of NaCl, KCl, or CaCl2 (x) and SrCl2 (y) measured by Macaskill et al.11 (x = NaCl, symbol ●), Downes9 (KCl, ○), Stokes1 (CaCl2, ▼), and Rard and Miller15 (CaCl2, ▽). The vapor pressures have been calculated by eqs 3 and 4 using eq 6 with BNaCl = 1.4 (mol· kg−1)−1/2, b1,NaCl = 0.0699, b2,NaCl = 0.0062, BKCl = 1.3 (mol·kg−1)−1/2, b1,KCl = 0.01324, b2,KCl = 0.0036, BCaCl2 = 1.55 (mol·kg−1)−1/2, b1,CaCl2 = 0.3486, b2,CaCl2 = 0.0667, BSrCl2 = 1.50 (mol·kg−1)−1/2, b1,SrCl2 = 0.3046, and b2,SrCl2 = 0.0534. The error of the point (mx = 1.2070 mol·kg−1, my = 0.7416 mol·kg−1) in the set of Downes9 lies outside the scale of the figure; it is −3.63 Pa.

latter contains also the error plot evaluated from the isopiestic points of Robinson3 for KCl and SrCl2 solutions and that obtained from the vapor pressure data of Hepburn.17 For KCl solutions, the Hückel parameters of B = 1.3 (mol·kg−1)−1/2, b1 = 0.01324, and b2 = 0.0036 (see ref 10) were used. For the set of Hepburn, the predicted vapor pressures were obtained by using eqs 3, 4, and 6 and in Figure 7 the following vapor pressure error ep ep = p(observed) − p(predicted)

(24)

is presented as a function of the molality. The data sets from Downes and Clegg et al. and the CsCl set from Rard and Miller support well and the CaCl2 sets from Rard and Miller and Stokes support quite well in Figures 6 and 7 the new SrCl2 parameters. The isopiestic data of Robinson above an ionic strength of 3 mol· kg−1 seem to be (according to Figure 7) erroneous, and the vapor pressure data of Hepburn are not high-precision data and cannot thus be used very critically in this connection. The validity of the new extended Hückel equation was then evaluated with the cpd data of Lucasse13 on cells of type 16. The error plot that resulted from these data is given in Figure 2, and it

and the error plots of this figure correspond exactly to those in Figure 1. In the plot for the points used in the estimation, the absolute errors are less than 1.2 Pa, and no clear trends can be seen in the error plot. Therefore, the data support well the suggested parameters. The suggested SrCl2 parameter values were then checked similarly with all isopiestic results of Downes9 and Clegg et al.,16 with those of Stokes1 up to the molality of the saturated solution, and with those of Rard and Miller15 against CaCl2 solutions and CsCl solutions. In the former set of Rard and Miller only such F



Article

Table 2. Mean Activity Coefficient Obtained from the Solubility Data for the Saturated Solution of SrCl2 at 25 °C, γsat(obsd), and That Obtained by the Extended Hückel Equation with the Recommended Parameter Values, γsat(pred), for This Solution substance SrCl2 H2O(l) SrCl2·H2O(s) SrCl2·2H2O(s) SrCl2·6H2O(s)

ΔfG°a

ΔfG°(aq)a,b

kJ·mol−1

kJ·mol−1

(msat/m°)c

−821.910

3.52

γsat(obsd)

γsat(pred) 1.544

−237.129 −1036.300 −1281.800 −2240.920

4.28 1.559 0.60

a

Taken from ref 18. bThe standard molar Gibbs energy of formation for aqueous solution at infinite dilution. cThe molality of the saturated solution (m° = 1 mol·kg−1).

Table 3. Recommended Activity Coefficients (γ) of the Electrolyte and Osmotic Coefficients (ϕ), Activities (a1), and Vapor Pressures of Water (p) in Aqueous Strontium Chloride Solutions at 25 °C as Functions of the Molality (m)a m/mol·kg−1

γ

ϕ

a1

p/Pa

0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 3.52

0.726 0.661 0.571 (0.572) 0.509 (0.510) 0.459 (0.460) 0.439 (0.440) 0.430 (0.431) 0.428 (0.428) 0.430 (0.428) 0.435 (0.431, −0.32b) 0.442 (0.436, −0.53b) 0.451 0.462 0.489 0.522 0.562 0.609 0.665 0.849 1.120 1.524 1.544

0.906 0.884 0.858 0.846 (0.847) 0.849 (0.850) 0.863 (0.864) 0.881 (0.880) 0.900 (0.897) 0.921 (0.915) 0.942 (0.934) 0.965 (0.952) 0.988 1.012 1.062 1.114 1.169 1.227 1.287 1.450 1.630 1.826 1.835

0.99951 0.99904 0.99768 0.99543 0.99086 0.9861 0.9811 (0.9812) 0.9760 0.9706 (0.9708) 0.9650 (0.9653) 0.9592 (0.9597) 0.9531 0.9468 0.9335 0.9192 0.9038 0.8875 0.8701 0.8221 0.7678 0.7079 0.7054

3167.0 3165.6 3161.3 3154.1 3139.6 3124.6 3108.9 3092.5 (−0.2c) 3075.4 (−0.5c) 3057.6 (−1.0c) 3039.2 (−1.7c) 3020.0 3000.0 2957.8 2912.4 2863.9 2812.1 2757.0 2604.8 2432.8 2243.0 2235.1

The activity values were calculated using the extended Hückel equation with B = 1.5 (mol·kg−1)−1/2, b1 = 0.3046, and b2 = 0.0534 (see text). Galvanic cell deviation in mV calculated using the equation: eE,GC = −(3RT/2F) ln[γ(eq 5)/γ(eq 1)], where the γ(eq 1) values were calculated using the Hückel equation with B = 1.50 (mol·kg−1)−1/2 and b1 = 0.330, and the γ(eq 5) values using the recommended parameter values (see footnote a). cVapor pressure deviation in Pa calculated using the equation: ep,VPW = p(ϕ from eq 6) − p(ϕ from eq 2), where the ϕ(eq 2) values were calculated using the recommended Hückel equation and the ϕ(eq 6) values using the extended Hückel equation (see footnotes a and b). a b

corresponds exactly to that of the Hückel equation for dilute solutions in this figure. These electrochemical data support rather well the suggested parameters also in less dilute solutions. The new extended Hückel equation for SrCl2 solutions at 25 °C apply probably up to the molality of the saturated solution. Therefore, the goodness of the parameter values of this equation was additionally investigated using the solubility data. The NBS tables of thermodynamic properties18 report values at 25 °C for the standard molar Gibbs energy of formation for crystalline state [ΔfG°(cr)] of salts and hydrates of salts, for the standard molar Gibbs energy of formation for aqueous solutions of salts at infinite dilution [ΔfG°(aq)], and for the standard molar Gibbs energy of formation for liquid water [ΔfG°(H2O)]. For SrCl2, the values are given for the following three hydrates: SrCl2· nH2O(s) where n = 1, 2, and 6. All of these thermodynamic values are collected in Table 2. In the saturated solution of a hydrate of the salt under consideration, the chemical potential of the hydrate is the same as the sum of the chemical potentials of

the dissolved salt and of the water released from the hydrate. Therefore, the following equation is valid in the saturated solution of the certain hydrate of SrCl2 that contains the amount of n moles of water. Δf G°(cr) = Δf G°(aq) + 3RT ln(41/3γsatmsat /m°) + nΔf G°(H 2O) + nRT ln a1,sat

(25)

where msat is the molality of the saturated solution, γsat is the mean activity coefficient of salt, and a1,sat is the activity of water in that solution. In this case, the solubility product Ksp is defined by equation Ksp = 4(γsatmsat/m°)3, and the following equation can be derived from eq 25 −RT ln K sp = −3RT ln(41/3γsatmsat /m°) = Δf G°(aq) + nΔf G°(H 2O) + nRT ln a1,sat − Δf G°(cr) G

(26)



Article

Table 4. Recommended Activity Coefficients (γ)a of Strontium Chloride in Aqueous Solutions at Some Rounded Molalities (m° = 1 mol·kg−1) as a Function of the Temperature (t)

a

t/°C

γ(0.01m°)

γ(0.02m°)

γ(0.05m°)

γ(0.10m°)

γ(0.20m°)

γ(0.30m°)

10 20 30 40 50 60

0.732 0.728 0.724 0.720 0.715 0.710

0.668 0.663 0.658 0.653 0.648 0.642

0.580 0.575 0.569 0.563 0.556 0.549

0.519 0.513 0.507 0.500 0.493 0.485

0.470 0.464 0.457 0.450 0.442 0.434

0.450 0.444 0.437 0.429 0.421 0.413

Calculated by using the Hückel equation with B = 1.5 (mol·kg−1)−1/2 and b1 = 0.330.

comparisons are given in Figure 8 where graph A gives the results for the activity coefficients and graph B for the osmotic coefficients. The comparison was carried out as in ref 7 for CaCl2 solutions using cell potential deviations (eE,GC) and vapor pressure deviations (ep,VPW) defined by the following equations, respectively,

This equation is used in the present study. From the latter identity, the experimental value of the activity coefficient in the saturated solution can be calculated for all three hydrates using the thermodynamic data given. These values are shown in Table 2, and they are denoted as γsat(obsd). The activity of water (a1,sat) for these calculations was obtained from the osmotic coefficient with the new activity parameters using eqs 3 and 6. The γsat(obsd) values in the table are not very sensitive for this choice of the osmotic coefficient. The molality of the saturated solution (msat) was taken from ref 1. The predicted activity coefficient, γsat(pred), was calculated using eq 5 with the suggested parameters, and it is also given in Table 2. The observed and predicted activity coefficients correspond quite well to each other in the case where the dihydrate salt is in equilibrium with the ions in the solution. This result is not the same as suggested that by, for example, Rard and Miller15 that the hexahydrate is in equilibrium with the ions at the molality of 3.52 mol·kg−1. At the moment, unfortunately, this question remains open because we cannot be sure that the thermodynamic values for the hydrates of SrCl2 in NBS tables are fully reliable. Recommended Activity and Osmotic Coefficients. Based on the evidence provided by the tests of the present study (see Figures 1 to 7 and Tables 1 and 2), the suggested Hückel and extended Hückel equations explain well the experimental data available for SrCl2 solutions. A new table (Table 3) of the values of thermodynamic properties in solutions of this salt at 25 °C was based on these equations. The threeparameter extended Hückel equation with the new parameter values was used in the calculation of the values of the activity quantities in Table 3. In dilute solutions (i.e., when m ≤ 0.8 mol· kg−1), the values from the two-parameter Hückel equation are reported in Table 3 in parentheses in case they are different from those obtained with the three-parameter equation. The absolute value for the difference between these two thermodynamic values is in every case quite small. It is smaller than 0.6 mV for the galvanic cell deviation for γ (the definition is shown in the table) and smaller than 1.7 Pa for the vapor pressure deviation for ϕ. In dilute solutions, the new Hückel equation at 25 °C also apply to all other temperatures from (10 to about 60) °C. The activity coefficients at rounded molalities up to 0.3 mol·kg−1 are given in Table 4 for the other temperatures. Comparison of the New Activity and Osmotic Coefficients to the Values Presented in the Literature. The values of the relevant thermodynamic quantities in Table 3 for 25 °C were compared to the corresponding values given by Robinson and Stokes,2 Rard and Miller,15 and Clegg et al.16 In this comparison were, additionally, considered the values that resulted from the Pitzer equations (eqs 9 and 10) with the parameters given by Pitzer and Mayorga4 and Rard and Miller15 and from the extended Hückel equations determined by Goldberg and Nuttall37 (eqs 7 and 8). The results from these

Figure 8. Deviation, expressed as galvanic cell error eE,GC (eq 27), between the literature activity coefficients and those recommended in this study (eq 5, graph A) and deviation, expressed as vapor pressure error ep,VPW (eq 28), between the literature osmotic coefficients and those recommended in this study (eq 6, graph B) for SrCl2 solutions (see Table 3) as functions of molality m. Symbols: ●, Robinson and Stokes;2 ○, eqs 7 and 8, Goldberg and Nuttall;37 ▼, eqs 9 and 10, Pitzer and Mayorga;4 ▽, eqs 9 and 10, Rard and Miller;15 ■, Table 5 in ref 15, Rard and Miller; □, Table 4 in ref 16, Clegg et al. The error of the point (m = 3.500 mol·kg−1, ϕ = 1.802) from Robinson and Stokes,2 that of the point (3.500 mol·kg−1, 1.796) from Pitzer equation of Pitzer and Mayorga,4 that of the point (3.500 mol·kg−1, 1.795) from Pitzer equation of Rard and Miller,15 those of the points (3.000, 1.616) and (3.500, 1.795) from Goldberg and Nuttall,37 those of the points (3.4, 1.7653) and (3.52, 1.8045) from Clegg et al.,16 and those of the points (3.2, 1.6946), (3.400, 1.7612), and (3.5195, 1.8001) from Rard and Miller15 lay outside the scale of graph B; these errors are (10.3, 12.8, 13.5, 5.4, 13.2, 8.6, 12.9, 4.8, 10.3, and 14.6) Pa, respectively. H


Journal of Chemical & Engineering Data eE,GC = −

3RT γ(literature) ln 2F γ(recd)

ep,VPW = p(literature) − p(recd)

Article

(11) Macaskill, J. B.; White, D. R., Jr.; Robinson, R. A.; Bates, R. G. Isopiestic measurements on aqueous mixtures of sodium chloride and strontium chloride. J. Solution Chem. 1978, 7, 339−347. (12) Longhi, P.; Mussini, T.; Vaghi, E. Standard potentials of the strontium amalgam electrode, and thermodynamics of dilute strontium amalgams. J. Chem. Thermodyn. 1975, 7, 767−776. (13) Lucasse, W. W. Activity coefficients and transference numbers of the alkaline earth chlorides. J. Am. Chem. Soc. 1925, 47, 743−754. (14) Reddy, D. C.; Ananthaswamy, J. Thermodynamic properties of aqueous electrolyte solutions: an e.m.f. study of {KCl(mA) + SrCl2(mB)}(aq) at the temperatures 298.15 K, 308.15 K, and 318.15 K. J. Chem. Thermodyn. 1990, 22, 1015−1023. (15) Rard, J. A.; Miller, D. G. Isopiestic determination of the osmotic and activity coefficients of aqueous CsCl, SrCl2, and mixtures of NaCl and CsCl at 25 °C. J. Chem. Eng. Data 1982, 27, 169−173. (16) Clegg, S. L.; Rard, J. A.; Miller, D. G. Isopiestic determination of the osmotic and activity coefficients of NaCl + SrCl2 + H2O at 298.15 K and representation with an extended ion-interaction model. J. Chem. Eng. Data 2005, 50, 1162−1170. (17) Hepburn, J. R. I. The vapour pressure of water over aqueous solutions of the chlorides of the alkaline-earth metals. Part 1. Experimental, with a critical discussion of vapour-pressure data. J. Chem. Soc. 1932, 550−566. (18) Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nuttall, R. L. The NBS tables of chemical thermodynamic properties. Selected values for inorganic and C1 and C2 organic substances in SI units. J. Phys. Chem. Ref. Data 1982, 11, Supplement No. 2. (19) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous rubidium and cesium chloride solutions at 25 °C. J. Chem. Eng. Data 2010, 55, 249−257. (20) Apelblat, A.; Korin, E. Vapour pressures of saturated aqueous solutions of ammonium iodide, potassium iodide, potassium nitrate, strontium chloride, lithium sulphate, sodium thiosulphate, magnesium nitrate, and uranyl nitrate from T = (278 to 323) K. J. Chem. Thermodyn. 1998, 30, 459−471. (21) Partanen, J. I.; Juusola, P. M.; Vahteristo, K. P.; de Mendonça, A. J. G. Re-evaluation of the activity coefficients of aqueous hydrochloric acid solutions up to a molality of 16.0 mol·kg−1 using the Hückel and Pitzer equations at temperatures from 0 to 50 °C. J. Solution Chem. 2007, 36, 39−59. (22) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous lithium chloride solutions at 25 °C up to a molality of 6.0 mol·kg−1. J. Chem. Eng. Data 2009, 54, 882−889. (23) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous solutions of silver nitrate, alkali metal fluorides and nitrites, and dihydrogen phosphate, dihydrogen arsenate, and thiocyanate salts with sodium and potassium ions at 25 °C. J. Chem. Eng. Data 2011, 56, 2044−2062. (24) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in pure aqueous solutions of chlorate, perchlorate, and bromate salts with lithium, sodium or potassium ions at 298.15 K. J. Solution Chem. 2012, 41, 271−293. (25) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous alkali metal bromide solutions at 25 °C. J. Chem. Eng. Data 2010, 55, 2202−2213. (26) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous alkali metal iodide solutions at 25 °C. J. Chem. Eng. Data 2010, 55, 3708−3719. (27) Partanen, J. I. Re-evaluation of the thermodynamic activity quantities in aqueous alkali metal nitrate solutions at T = 298.15 K. J. Chem. Thermodyn. 2010, 42, 1485−1493. (28) Partanen, J. I.; Covington, A. K. Re-evaluation of the thermodynamic activity quantities in aqueous solutions of uni-univalent alkali metal salts of aliphatic carboxylic acids and thallium acetate at 25 °C. J. Chem. Eng. Data 2011, 56, 4524−4543. (29) Partanen, J. I. Mean activity coefficients and osmotic coefficients in aqueous solutions of salts of ammonium ions with univalent anions at 25 °C. J. Chem. Eng. Data 2012, 57, 2654−2666.

(27) (28)

where GC refers to galvanic cell without a liquid junction (like cell 19) and VPW refers to the vapor pressure of water. Vapor pressures were calculated for eq 28 from the osmotic coefficients using eqs 3 and 4. For the recommended values, the values obtained from eqs 5 and 6 in Table 3 were used. In general, the literature activity coefficients considered in graph A of Figure 8 are in a quite good agreement with the values recommended up to Im of 9.0 mol·kg−1. The best agreement is obtained with the activity coefficients calculated by the Pitzer equations of Rard and Miller15 and Clegg et al.16 At the saturated solution (Im = 10.56 mol·kg−1), the absolute deviation for these literature values is the highest, and it is only about 1 mV for both models. The osmotic coefficients in graph B do not agree as well. At least a satisfactory agreement is obtained for all literature values up to Im = 7.5 mol·kg−1. For all literature values near to the saturated solution, the vapor pressure deviations are as large as about 10 Pa (see the caption of the figure). It seems, however, that the recommended values in Table 3 are in these cases slightly more reliable owing to the experimental evidence presented especially in Figures 6 and 7.

■

AUTHOR INFORMATION

Corresponding Author

*Fax: +358 5 621 2350. E-mail: jpartane@lut.fi. Funding

The author is indebted to the Research Foundation of Lappeenranta University of Technology for financial support. Notes

The authors declare no competing financial interest.

■

REFERENCES

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Re-evaluation of the Mean Activity Coefficients of Strontium Chloride

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