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Determination of water activities and osmotic and activity coefficients of the system NaCl–BaCl2–H2O at 298.15 K
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Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381 www.elsevier.com/locate/calphad
Determination of water activities and osmotic and activity coefficients of the system NaCl–BaCl2 –H2 O at 298.15 K Mohamed El Guendouzi∗, Asmaa Benbiyi, Abderrahim Dinane, Rachid Azougen Laboratoire de Chimie Physique, D´epartement de Chimie, Facult´e des Sciences Ben M’sik, Universit´e Hassan II Mohammedia, B.P 7955, Casablanca, Morocco Received 27 October 2003; accepted 16 January 2004
Abstract The water activities of the mixed aqueous electrolyte NaCl–BaCl2 (aq) are measured at total molalities from 0.25 mol kg−1 to about saturation for different ionic strength fractions (y) of NaCl with y = 0.33, 0.50, 0.67. The results allow the deduction of osmotic coefficients. The experimental results are compared with the predictions of the Zdanovskii, Stokes, and Robinson (ZSR), Kusik and Meissner (KM), Robinson and Stokes (RS), Lietzke and Stoughton (LSII), Reilly, Wood, and Robinson (RWR), and Pitzer models. From these measurements, the Pitzer mixing ionic parameters are determined and used to predict the solute activity coefficients in the mixture. The excess Gibbs energy is also determined. © 2004 Elsevier Ltd. All rights reserved.
1. Introduction The investigation of the thermodynamic properties of aqueous mixed electrolyte solutions attracts great interest. Many systems of practical [1], chemical [2–4], biological [5], geological, and atmospheric process [6], as well as chemical interest involve mixed aqueous electrolytes. Study of these systems is interesting for understanding the physical chemistry process. The thermodynamic properties of the mixed aqueous electrolyte continue to be interesting with a common cation or anion. This paper is the continuation of research on binary [7, 8] and ternary solutions [9– 11]. The main objectives are to determine thermodynamic properties of the system {y NaCl–(1 − y)BaCl2 }(aq) at the temperature 298.15 K by the hygrometric method. In this work, the measurements of water activities for the total molality range from 0.25 to the saturation were performed for this electrolyte at different ionic strength fractions y in NaCl (y = INaCl /INaCl + IBaCl2 ) with 0.33, 0.50 and 0.67 at the temperature 298.15 K. The osmotic coefficients are also evaluated for these solutions from the water activities. The results obtained are compared with data from other commonly used thermodynamic models: Zdanovskii, ∗ Corresponding author. Fax: +212-704-675.
E-mail address: [email protected] (M. El Guendouzi). 0364-5916/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2004.01.006
Stokes, and Robinson (ZSR), Kusik and Meissner (KM), Robinson and Stokes (RS), Lietzke and Stoughton (LSII), Reilly, Wood, and Robinson (RWR), and Pitzer. The experimental data are used for the calculation of solute activity coefficients using the Pitzer model with our newly obtained ionic mixing parameters. The results are used to calculate the excess Gibbs energy. 2. Experimental procedure The water activity was determined by a hygrometric method previously described in our earlier work [8]. The apparatus used in this study of mixed electrolyte solutions is essentially the same as that used for the investigation of aqueous single-electrolyte solutions. It is based on the measurement of the relative humidity over an aqueous solution containing non-volatile electrolytes. The apparatus used is a hygrometer in which a droplet of salt solution is maintained on a thin thread. The diameter measurement of the previously calibrated droplet therefore provides the knowledge of the relative humidity of aqueous solutions. The droplets of a reference solution of NaCl(aq) or LiCl(aq) are deposited on the spider-thin thread by pulverisation. This thread is kept tense over a Perspex support, which is fixed to a cup containing the selected studied solution. The cup is then placed in a
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thermostatted box. The droplet diameter is measured by a microscope with an ocular device equipped with the micrometric screw. The relative humidity is equivalent to the water activity aw in our experiments. From measurements of reference droplet diameters D(aw(ref) ) above the reference solution and the same diameters D(aw ) above the solution studied, we calculate the ratio of growth K (K = D(aw )/D(aw(ref) )) and determine graphically the water activity using the variation of the ratio K as a function of the water activity of reference NaCl(aq) or LiCl(aq). Generally, the reference relative humidity is 0.84. For the intermediate dilute solution, the reference is 0.98. The uncertainty in the water activity depends on the accuracy of the diameter measurements and is therefore less than ±0.02% for aw > 0.97, ±0.05% for 0.97 > aw > 0.95, ±0.09% for 0.95 > aw > 0.90 and ±0.2% for 0.90 > aw > 0.85. Also the overall uncertainty of the osmotic coefficient is estimated to be, at most, ±6 × 10−3 . The solutions of NaCl, BaCl2 were prepared from crystalline materials (‘extrapur’-grade chemicals, mass fraction > 0.99) and deionised distilled water.
where y is the ionic strength fraction, and awo ,MX and awo ,NX are the single-component water activities at the same ionic strength as the mixture. 3.3. The Robinson and Stokes (RS) model Robinson and Stokes [13] proposed a form of the vapour pressure of the mixture that involves additively lowering solvent vapour pressures of the solutes and extended it into the following form: (1 − aw(m) ) = {(1 − aw(MX,m) )(m MX /m) + (1 − aw(NX,m) )(m NX /m)}
(3)
where aw(m) , aw(MX,m) , and aw(NX,m) are, respectively, the water activities of the ternary solutions of MX and NX, and the water activities of the binary solutions of MX and NX at the same molality m as the mixed solution. 3.4. The Lietzke and Stoughton (LS) model The Lietzke and Stoughton model (LSII) [16, 17] predicts the osmotic coefficient of a multicomponent solution from
3. Theory and models
(υMX m MX + υNX m NX )φ = υMX m MX φMX + υNX m NX φNX
The Zdanovskii, Stokes, and Robinson (ZSR), Kusik and Meissner (KM), Robinson and Stokes (RS), Lietzke and Stoughton (LS), Reilly, Wood, and Robinson (RWR), and Pitzer models are used to evaluate the performance of our experimental data.
where υMX is the number of ions released by the complete dissociation of one molecule of component MX, m MX is its molality, and φMX is the osmotic coefficient of the binary solution of component MX at the total ionic strength of the multicomponent solution.
3.1. The Zdanovskii, Stokes, and Robinson (ZSR) model
3.5. The Reilly, Wood, and Robinson (RWR) model
The Zdanovskii, Stokes, and Robinson (ZSR) [12, 13] rule has been used to predict the water activity in mixture solutions. The ZSR equation is expressed as mi = 1, (1) m 0,i (aw )
Reilly, Wood, and Robinson [18] developed a model for mixed electrolyte solutions. For a solution with a molality m MX of an electrolyte MX, and a molality m NX of an electrolyte NX, the osmotic coefficient of the mixed electrolyte solutions according to the RWR model is given by
where m i is the molality of species i in a multicomponent solution with water activity of aw , and m 0,i (aw ) is the molality of the single-component solution at the same aw .
(m MX + m NX )(1 − φ) = m MX (1 − φMX )
i
The Kusik and Meissner (KM) model [14, 15] is based on the Bronsted principle of specific interaction, which asserts that electrolyte solution properties are determined primarily by interactions between pairs of oppositely charged ions. For the mixed electrolytes MX–NX(aq), the corresponding KM model water activity equation is y(2 + y) ln awo ,MX 3 (2 + y)(1 − y) ln awo ,NX − 0.003y(1 − y)I + 2
(5)
where φMX and φNX are respectively the osmotic coefficients of the solutions of electrolyte MX and electrolyte NX at the ionic strength I .
3.2. The Kusik and Meissner (KM) model
ln aw =
+ m NX (1 − φNX )
(4)
(2)
3.6. The Pitzer model The Pitzer model [19–21] is used for calculating the thermodynamic properties of mixed electrolyte solutions. This model requires parameters estimated from a commonion solution in order to characterise binary interactions among different ions of the same sign and ternary interactions between different ions not all of the same sign in a mixed electrolyte solution. The osmotic coefficient of a
M. El Guendouzi et al. / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381 1.00
mixed solution of two salts NaCl and BaCl2 with a common anion is given by Pitzer’s model as I 1/2 1 −0.784 + (2/3)y(2 + y) φ−1 = (1 + y) 1 + 1.2I 1/2 φ
φ
φ
+ (2/9)(1 − y)(2 + y)I [BNX + (2
y = 0.33 0.95
aw
× I [BMX + (1/3)(2 + y)I CMX ] −1/2
377
0.90
/3)
φ
× (2 + y)I CNX ] + (2/3)y(1 − y)I [θMN + (1/3) × (2 + y)I ψMNX ]
0.85
(6)
0.80 0.0
where f φ is the long range electrostatic term, I is the ionic strength of the common ion, θMN is a binary mixing parameter involving unlike ions of the same sign, and ψMNX is a ternary mixing parameter for two unlike ions of the same sign with a third ion of the opposite sign. The second virial coefficient B φ of the ions involved is defined as (7)
2.0
mtot /(mol
3.0
4.0
kg–1)
1.00 y = 0.50 0.95
aw
B φ = β (0) + β (1) exp(−α I 1/2 )
1.0
where β (0) , β (1) and C φ are ion interaction parameters which are functions of temperature and pressure, and α = 2 (mol kg−1 )−1 . The activity coefficients γMX of MX: NaCl and γNX of NX: BaCl2 in a common-anion mixture of NaCl–BaCl2 (aq) are given by Pitzer’s model as
0.90
0.85
0.80 0.0
1.0
2.0
mtot /(mol
ln γMX = f γ + (2/3)(2y + 1)I [BMX + (1/6)(y + 2) φ
3.0
4.0
5.0
kg–1)
1.00
× I CMX ] + (1/3)(1 − y)I [BNX + (21/2/12)
y = 0.67
φ
× (2 + y)I CNX ] + (1/3)y(2 + y)I 2 φ
0.95
× [BMX + (1/2)CMX ] + (1/9)(2 + y)(1 − y)I 2
× [θMN + (1/3)(2y + 1)I ψMNX ]
(8)
aw
φ
× [BNX + (21/2 /4)CNX ] + (1/3)(1 − y)I
0.90
and ln γNX = 2 f γ + (2/9)(4 − y)I [BNX + (21/2 /12)(y + 2) φ
φ
× I CNX ] + (4/3)y I [BMX + (1/6)(2 + y)I CMX ] φ
+ (21/2/3)CNX ] + (1/9)(2 + y)(1 − y)I 2 [2BNX φ
+ (1/3)CMX ] + (2/3)y(2 + y)I 2 [BMX + (2/3)y I [θMN + (2/3)(4 − y)I ψMNX ].
The coefficients BMX and are defined as
0.85
B
MX
(9)
for electrolytes 1–1 and 2–1
(0) (1) + (2βMX /α 2 I )[1 − (1 + α I 1/2 ) BMX = βMX 1/2 × exp(−α I )], 3m tot φ = θNH+ K+ + m SO2− ψNH+ K+ SO2− 4 4 4 4 2m NH+ m K+
(10) (11)
4
From the osmotic coefficients determined from the experimental water activities of the mixture studied at different ionic strength fractions, it is possible to determine the unknowns, the Pitzer mixing ionic θMN and ψMNX . These parameters are used to predict the solute activity coefficients
0.80 0.0
1.0
2.0
3.0
4.0
5.0
mtot /(mol kg–1) Fig. 1. Water activity aw of NaCl–BaCl2 (aq) against total molalities m tot at different ionic strength fractions y of 0.33, 0.50, and 0.67. , experimental points; , KM; ×, ZSR.
in the mixture. θMN and ψMNX are estimated by a graphical procedure. This procedure defines the quantity φ as the difference between the experimental values φexp and that calculated from Eq. (6) φcalc . This yields 2m MX + 3m NX φ = θMN + m X ψMNX 2m MX m NX
(12)
so a plot of φ versus total molality m should give a straight line with intercept θMN and slope ψMNX .
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Table 1 Ratios of growth K of the NaCl(aq) droplets, water activities aw , and osmotic coefficients φ of NaCl–BaCl2 (aq) at total molalities m tot for different ionic strength fractions y of NaCl m NaCl
m BaCl2
m tot
K
aw
φ
1.258 1.028 1.656 1.515 1.380 1.300 1.183 1.092 1.037 0.985
0.990a 0.981a 0.971 0.960 0.948 0.938 0.914 0.887 0.860 0.827
0.930 0.887 0.908 0.944 0.988 0.987 1.040 1.109 1.163 1.255
1.105 1.638 1.438 1.300 1.216 1.141 1.086 1.036 1.000 0.962
0.985a 0.970 0.955 0.938 0.920 0.902 0.883 0.862 0.840 0.819
0.932 0.940 0.947 0.987 1.029 1.060 1.096 1.145 1.195 1.231
1.748 1.529 1.378 1.300 1.216 1.146 1.103 1.065 1.030 0.983
0.975 0.963 0.950 0.936 0.922 0.905 0.890 0.874 0.857 0.823
0.935 0.928 0.947 0.977 1.000 1.053 1.076 1.105 1.140 1.199
y = 0.33 0.15 0.30 0.45 0.60 0.75 0.90 1.20 1.50 1.80 2.10
0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.20 1.40
0.25 0.50 0.75 1.00 1.30 1.50 2.00 2.50 3.00 3.50 y = 0.50
0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 y = 0.67
0.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00 3.60
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.59
0.70 1.10 1.40 1.80 2.10 2.50 2.80 3.20 3.50 4.20
The reference water activity is 0.84. a The values are for reference water activity of 0.98 (NaCl).
Table 2 Comparison of predictions of the ZSR, KM, RS, LSII, RWR, and Pitzer models with experimental data for different ranges of ionic strength NaCl–BaCl2 (mol kg−1 )
σ (φ)ZSR
σ (φ)KM
σ (φ)RS
σ (φ)LS
σ (φ)RWR
σ (φ)Pitzer
0.45–3.0 3.0–6.3 0.45–6.3
0.0160 0.0360 0.0180
0.0090 0.0060 0.0070
0.0090 0.0070 0.0080
0.0096 0.0060 0.0070
0.0090 0.0060 0.0070
0.0040 0.0060 0.0045
4. Results and discussion 4.1. Water activity and osmotic coefficient In this work, a series of measurements of the water activity were made for the mixture {yNaCl + (1 − y)BaCl2 }(aq) as a function of total molality ranging from 0.25 mol kg−1 to saturation of one of the solutes, with different ionic strength fractions y of NaCl of 0.33, 0.50, and 0.67 at a temperature of 298.15 K. The experimental values
of the water activity are listed in Table 1 and shown in Fig. 1. The KM and ZSR models are used for the predictions of water activity of the mixture. The molalities of NaCl and BaCl2 at different constant water activities of the ternary mixture were evaluated from experimental data. The plots of water activities, isoactivities, dependence of m BaCl2 BaCl2 versus m NaCl , of NaCl are represented in Fig. 2. The ZSR rule has been used to compare with the measurements.
M. El Guendouzi et al. / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381 5
379
1.4 y = 0.33
1.2 3
2
mBaCl /(mol kg–1)
4
2
1.0
1
0 0.0
0.8 0.80 0.5
1.0
1.5
2.0
2.5
3.0
1.0
3.5
1.2
1.4
1.6
1.4
1.6
1.4
1.6
{I/(mol kg–1)}1/2
mNaCl /(mol kg–1) 1.4
y = 0.50
Fig. 2. The dependence of the molality of NaCl versus molality of BaCl2 in mixed NaCl–BaCl2 (aq) at constant water activity aw . , aw = 0.98; , aw = 0.94; , aw = 0.90; , aw = 0.86; •, aw = 0.82; ×, ZSR. 1.2
Pitzer’s model of electrolyte solutions allows parameters determined for single-electrolyte solutions to be applied to multicomponent electrolyte solutions. Parameters were found for Pitzer’s model that characterised the solutions of the individual salts that are components of the mixtures studied here. The osmotic coefficients obtained are given in Table 1. Using the experimental data obtained for the water activity, we have evaluated the osmotic coefficients for different ionic strength fractions y of NaCl. The RS, LS, RWR, and Pitzer models are used to predict the osmotic coefficients. The results are shown in Fig. 3. The experimental data are compared with the predictions of the ZSR, KM, RS, LSII, RWR, and Pitzer models. However, in the Pitzer model the θNa+ Ba2+ and ψNa+ Ba2+ Cl− parameters were adjusted, but the ZSR, KM, RS, LSII, RWR models do not involve any ternary parameters. The deviation between the data and models for different ranges of ionic strength is summarized in Table 2, and it is between 0.004 and 0.036. The corresponding values of the ionic parameters β (0), (1) β , and C φ of the pure electrolytes NaCl(aq) and BaCl2 (aq) were obtained from Pitzer’s expressions by fits of the experimental osmotic coefficients given in our previous work [8]. The corresponding values are presented in Table 3.
Table 3 Pitzer parameters for NaCl and BaCl2 and mixtures at 298.15 K Electrolyte
β (0)
β (1)
Cφ
References
NaCl BaCl2
0.0738 0.2571
0.2712 1.4676
0.00167 −0.01773
[8] [8]
Electrolyte
θNa+ Ba2+
ψNa+ Ba2+ Cl−
Reference
NaCl + BaCl2
0.0069
0.0164
This study
1.0
0.8 0.80
1.0
1.2
{I/(mol kg–1)}1/2 1.4 y = 0.67
1.2
1.0
0.8 0.80
1.0
1.2
{I/(mol kg–1)}1/2 Fig. 3. The osmotic coefficient φ of NaCl–BaCl2 (aq) against the square root of the total ionic strength I 1/2 at different ionic strength fractions y of 0.33, 0.50, 0.67. ×, experimental points; , RS; , LS II; , Pitzer; ◦, RWR.
4.2. Activity coefficient The activity coefficients of NaCl(aq) and BaCl2 (aq) in the mixture were calculated from Pitzer’s equation using the ionic mixing parameters that we obtained; the results are listed in Table 4. The plots of activity coefficients γNaCl and γBaCl2 as a function of the square root of the ionic strength (I 1/2 ) are shown in Figs. 4 and 5. The extremes of composition are
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Table 4 Activity coefficients γNaCl of NaCl(aq) and γBaCl2 of BaCl2 (aq) in NaCl–CaCl2 (aq) at total ionic strength I for different ionic strength fractions y of NaCl y=0 I γNaCl
γBaCl2
y = 0.33 I γNaCl
γBaCl2
y = 0.50 I γNaCl
γBaCl2
y = 0.67 I γNaCl
γBaCl2
y=1 I γNaCl
γBaCl2
0.60 0.90 1.50 2.10 2.40 3.00 4.20 4.50 4.80 6.00
0.447 0.420 0.396 0.389 0.389 0.393 0.415 0.422 0.429 0.464
0.45 0.90 1.35 1.80 2.25 2.70 3.60 4.50 5.40 6.30
0.474 0.431 0.417 0.417 0.417 0.438 0.483 0.549 0.640 0.760
0.60 1.20 1.80 2.40 3.00 3.60 4.20 4.80 5.40 6.00
0.458 0.429 0.431 0.450 0.483 0.530 0.590 0.667 0.764 0.885
0.90 1.35 1.80 2.25 2.70 3.15 3.60 4.05 4.50 5.40
0.444 0.438 0.447 0.466 0.494 0.532 0.580 0.640 0.712 0.907
0.50 1.00 1.50 2.00 2.50 3.00 4.00 5.00 5.50 6.00
0.482 0.458 0.466 0.492 0.535 0.594 0.775 1.076 1.294 1.578
0.680 0.664 0.654 0.657 0.661 0.675 0.713 0.724 0.736 0.788
0.690 0.661 0.654 0.658 0.667 0.682 0.723 0.778 0.845 0.926
0.674 0.654 0.654 0.672 0.697 0.728 0.766 0.811 0.862 0.920
0.658 0.652 0.657 0.668 0.686 0.707 0.733 0.763 0.797 0.877
0.678 0.653 0.652 0.663 0.681 0.706 0.774 0.864 0.918 0.978
0
1.0
–2 0.9
Gex /(kj kg–1)
–4
0.8
0.7
–6 –8 –10 –12
0.6 0.5
1.0
1.5
2.0
2.5
3.0
{I/(mol kg–1)}1/2
–14
0
1
2
3
4
5
6
7
mtot /(mol kg–1)
Fig. 4. Activity coefficients γNaCl of NaCl(aq) in NaCl–BaCl2 (aq) against the square root of the total ionic strength I 1/2 at different ionic strength fractions y of NaCl(aq). ◦, y = 0 (BaCl2 ); , y = 0.33; •, y = 0.50; , y = 0.67; , y = 1.0 (NaCl).
Fig. 6. The excess Gibbs energy of NaCl–BaCl2 (aq) against the total molality m tot at different ionic strength fractions y of NaCl. , y = 0; , y = 0.33; ◦, y = 0.50; , y = 0.67; , y = 1.0.
2.0
4.3. The excess Gibbs energy
1.5
From the osmotic coefficient and activity coefficient, we determined the excess Gibbs energy of NaCl–BaCl2 (aq) using the following expression:
1.0
G ex = RT {ν1 m 1 (1 − φ + ln γ1 ) + ν2 m 2 (1 − φ + ln γ2 )}. (13) The plots of the Gibbs energy as a function of the molality m tot are shown in Fig. 6.
0.5
0.0 0.5
1.0
1.5
2.0
2.5
3.0
5. Conclusion
{I/(mol kg–1)}1/2 Fig. 5. Activity coefficients γBaCl2 of BaCl2 (aq) in NaCl–BaCl2 (aq) against the square root of the total ionic strength I 1/2 at different ionic strength fractions y of NaCl(aq). ◦, y = 0 (BaCl2 ); , y = 0.33; •, y = 0.50; , y = 0.67; , y = 1.0 (NaCl).
illustrated in these figures, i.e., the activity coefficients of pure and trace of components.
The water activities of the system {yNaCl + (1 − y) BaCl2 }(aq) are measured using a hygrometric method at temperature 298.15 K. These measurements were made at total molalities from 0.25 to about saturation for different ionic strength fractions y of NaCl with y = 0.33, 0.50, and 0.67. These results allow the deduction of osmotic coefficients. The experimental data are also compared with the
M. El Guendouzi et al. / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381
predictions of ZSR, KM, RS, LSII, RWR, and Pitzer models. The deviation between the data and models is 0.004 to 0.036. The activity coefficients of NaCl(aq) and BaCl2 (aq) in the mixture were calculated by means of Pitzer’s equation using the ionic mixing parameters that we obtained. The excess Gibbs energy is also determined. References [1] N.A. Gokcen, Report of Investigations. US Department of the interior, Bureau of Mines R.I. 8372 (1979). [2] D.A. Palmer, J.A. Rard, S.L. Clegg, J. Chem. Thermodyn. 34 (2002) 63. [3] F. Malatesta, L. Carbonaro, N. Fanelli, S. Ferrini, A. Giacomelli, J. Solution Chem. 28 (1999) 593. [4] C. Christov, Calphad 26 (2002) 85. [5] M.J. Molina, R. Zhang, P.J. Wooldridge et al., Science 261 (1993) 1418. [6] K.C. Chan, Z. Ha, Y.C. Man, J. Atmos. Environ. 34 (2000) 4795.
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[7] M. El Guendouzi, M. Marouani, J. Solution Chem. 32 (2003) 535. [8] M. El Guendouzi, A. Dinane, A. Mounir, J. Chem. Thermodyn. 33 (2001) 1059. [9] M. El Guendouzi, A. Errougui, J. Chem. Ing. Data. (in press) JE030123. [10] M. El Guendouzi, A. Benbiyi, A. Dinane, R. Azougen, Calphad 27 (2003) 213. [11] M. El Guendouzi, A. Mounir, A. Dinane, Fluid Phase Equi. 201 (2002) 223. [12] A.B. Zdanovskii, Zh. Fiz. Khim. 22 (1948) 1475. [13] R.A. Robinson, R.H. Stokes, Electrolyte Solutions, second ed., fifth Revised Impression, Butterworth, London, 1970. [14] C.L. Kusik, H.P. Meissner, AICHE J. Symp. Ser. 173 (1978) 14. [15] A.W. Stelson, M.E. Bassett, J.H. Seinfeld, Chemistry of Particles, Fogs and Rain, Butterworth, MA, 1984. [16] M.H. Lietzke, R.W. Stoughton, J. Solution Chem. 1 (1972) 299. [17] M.H. Lietzke, R.W. Stoughton, J. Inorg. Nucl. Chem. 37 (1975) 2503. [18] P.J. Reilly, R.H. Wood, R.A. Robinson, J. Phys. Chem 75 (1971) 1305. [19] K.S. Pitzer, J. Phys. Chem. 77 (1973) 268. [20] K.S. Pitzer, G. Mayorga, J. Phys. Chem. 37 (1973) 2300. [21] K.S. Pitzer, J. Kim, J. Am. Chem. Soc. 96 (1974) 5701.
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Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381 www.elsevier.com/locate/calphad
Determination of water activities and osmotic and activity coefficients of the system NaCl–BaCl2 –H2 O at 298.15 K Mohamed El Guendouzi∗, Asmaa Benbiyi, Abderrahim Dinane, Rachid Azougen Laboratoire de Chimie Physique, D´epartement de Chimie, Facult´e des Sciences Ben M’sik, Universit´e Hassan II Mohammedia, B.P 7955, Casablanca, Morocco Received 27 October 2003; accepted 16 January 2004
Abstract The water activities of the mixed aqueous electrolyte NaCl–BaCl2 (aq) are measured at total molalities from 0.25 mol kg−1 to about saturation for different ionic strength fractions (y) of NaCl with y = 0.33, 0.50, 0.67. The results allow the deduction of osmotic coefficients. The experimental results are compared with the predictions of the Zdanovskii, Stokes, and Robinson (ZSR), Kusik and Meissner (KM), Robinson and Stokes (RS), Lietzke and Stoughton (LSII), Reilly, Wood, and Robinson (RWR), and Pitzer models. From these measurements, the Pitzer mixing ionic parameters are determined and used to predict the solute activity coefficients in the mixture. The excess Gibbs energy is also determined. © 2004 Elsevier Ltd. All rights reserved.
1. Introduction The investigation of the thermodynamic properties of aqueous mixed electrolyte solutions attracts great interest. Many systems of practical [1], chemical [2–4], biological [5], geological, and atmospheric process [6], as well as chemical interest involve mixed aqueous electrolytes. Study of these systems is interesting for understanding the physical chemistry process. The thermodynamic properties of the mixed aqueous electrolyte continue to be interesting with a common cation or anion. This paper is the continuation of research on binary [7, 8] and ternary solutions [9– 11]. The main objectives are to determine thermodynamic properties of the system {y NaCl–(1 − y)BaCl2 }(aq) at the temperature 298.15 K by the hygrometric method. In this work, the measurements of water activities for the total molality range from 0.25 to the saturation were performed for this electrolyte at different ionic strength fractions y in NaCl (y = INaCl /INaCl + IBaCl2 ) with 0.33, 0.50 and 0.67 at the temperature 298.15 K. The osmotic coefficients are also evaluated for these solutions from the water activities. The results obtained are compared with data from other commonly used thermodynamic models: Zdanovskii, ∗ Corresponding author. Fax: +212-704-675.
E-mail address: [email protected] (M. El Guendouzi). 0364-5916/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2004.01.006
Stokes, and Robinson (ZSR), Kusik and Meissner (KM), Robinson and Stokes (RS), Lietzke and Stoughton (LSII), Reilly, Wood, and Robinson (RWR), and Pitzer. The experimental data are used for the calculation of solute activity coefficients using the Pitzer model with our newly obtained ionic mixing parameters. The results are used to calculate the excess Gibbs energy. 2. Experimental procedure The water activity was determined by a hygrometric method previously described in our earlier work [8]. The apparatus used in this study of mixed electrolyte solutions is essentially the same as that used for the investigation of aqueous single-electrolyte solutions. It is based on the measurement of the relative humidity over an aqueous solution containing non-volatile electrolytes. The apparatus used is a hygrometer in which a droplet of salt solution is maintained on a thin thread. The diameter measurement of the previously calibrated droplet therefore provides the knowledge of the relative humidity of aqueous solutions. The droplets of a reference solution of NaCl(aq) or LiCl(aq) are deposited on the spider-thin thread by pulverisation. This thread is kept tense over a Perspex support, which is fixed to a cup containing the selected studied solution. The cup is then placed in a
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thermostatted box. The droplet diameter is measured by a microscope with an ocular device equipped with the micrometric screw. The relative humidity is equivalent to the water activity aw in our experiments. From measurements of reference droplet diameters D(aw(ref) ) above the reference solution and the same diameters D(aw ) above the solution studied, we calculate the ratio of growth K (K = D(aw )/D(aw(ref) )) and determine graphically the water activity using the variation of the ratio K as a function of the water activity of reference NaCl(aq) or LiCl(aq). Generally, the reference relative humidity is 0.84. For the intermediate dilute solution, the reference is 0.98. The uncertainty in the water activity depends on the accuracy of the diameter measurements and is therefore less than ±0.02% for aw > 0.97, ±0.05% for 0.97 > aw > 0.95, ±0.09% for 0.95 > aw > 0.90 and ±0.2% for 0.90 > aw > 0.85. Also the overall uncertainty of the osmotic coefficient is estimated to be, at most, ±6 × 10−3 . The solutions of NaCl, BaCl2 were prepared from crystalline materials (‘extrapur’-grade chemicals, mass fraction > 0.99) and deionised distilled water.
where y is the ionic strength fraction, and awo ,MX and awo ,NX are the single-component water activities at the same ionic strength as the mixture. 3.3. The Robinson and Stokes (RS) model Robinson and Stokes [13] proposed a form of the vapour pressure of the mixture that involves additively lowering solvent vapour pressures of the solutes and extended it into the following form: (1 − aw(m) ) = {(1 − aw(MX,m) )(m MX /m) + (1 − aw(NX,m) )(m NX /m)}
(3)
where aw(m) , aw(MX,m) , and aw(NX,m) are, respectively, the water activities of the ternary solutions of MX and NX, and the water activities of the binary solutions of MX and NX at the same molality m as the mixed solution. 3.4. The Lietzke and Stoughton (LS) model The Lietzke and Stoughton model (LSII) [16, 17] predicts the osmotic coefficient of a multicomponent solution from
3. Theory and models
(υMX m MX + υNX m NX )φ = υMX m MX φMX + υNX m NX φNX
The Zdanovskii, Stokes, and Robinson (ZSR), Kusik and Meissner (KM), Robinson and Stokes (RS), Lietzke and Stoughton (LS), Reilly, Wood, and Robinson (RWR), and Pitzer models are used to evaluate the performance of our experimental data.
where υMX is the number of ions released by the complete dissociation of one molecule of component MX, m MX is its molality, and φMX is the osmotic coefficient of the binary solution of component MX at the total ionic strength of the multicomponent solution.
3.1. The Zdanovskii, Stokes, and Robinson (ZSR) model
3.5. The Reilly, Wood, and Robinson (RWR) model
The Zdanovskii, Stokes, and Robinson (ZSR) [12, 13] rule has been used to predict the water activity in mixture solutions. The ZSR equation is expressed as mi = 1, (1) m 0,i (aw )
Reilly, Wood, and Robinson [18] developed a model for mixed electrolyte solutions. For a solution with a molality m MX of an electrolyte MX, and a molality m NX of an electrolyte NX, the osmotic coefficient of the mixed electrolyte solutions according to the RWR model is given by
where m i is the molality of species i in a multicomponent solution with water activity of aw , and m 0,i (aw ) is the molality of the single-component solution at the same aw .
(m MX + m NX )(1 − φ) = m MX (1 − φMX )
i
The Kusik and Meissner (KM) model [14, 15] is based on the Bronsted principle of specific interaction, which asserts that electrolyte solution properties are determined primarily by interactions between pairs of oppositely charged ions. For the mixed electrolytes MX–NX(aq), the corresponding KM model water activity equation is y(2 + y) ln awo ,MX 3 (2 + y)(1 − y) ln awo ,NX − 0.003y(1 − y)I + 2
(5)
where φMX and φNX are respectively the osmotic coefficients of the solutions of electrolyte MX and electrolyte NX at the ionic strength I .
3.2. The Kusik and Meissner (KM) model
ln aw =
+ m NX (1 − φNX )
(4)
(2)
3.6. The Pitzer model The Pitzer model [19–21] is used for calculating the thermodynamic properties of mixed electrolyte solutions. This model requires parameters estimated from a commonion solution in order to characterise binary interactions among different ions of the same sign and ternary interactions between different ions not all of the same sign in a mixed electrolyte solution. The osmotic coefficient of a
M. El Guendouzi et al. / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381 1.00
mixed solution of two salts NaCl and BaCl2 with a common anion is given by Pitzer’s model as I 1/2 1 −0.784 + (2/3)y(2 + y) φ−1 = (1 + y) 1 + 1.2I 1/2 φ
φ
φ
+ (2/9)(1 − y)(2 + y)I [BNX + (2
y = 0.33 0.95
aw
× I [BMX + (1/3)(2 + y)I CMX ] −1/2
377
0.90
/3)
φ
× (2 + y)I CNX ] + (2/3)y(1 − y)I [θMN + (1/3) × (2 + y)I ψMNX ]
0.85
(6)
0.80 0.0
where f φ is the long range electrostatic term, I is the ionic strength of the common ion, θMN is a binary mixing parameter involving unlike ions of the same sign, and ψMNX is a ternary mixing parameter for two unlike ions of the same sign with a third ion of the opposite sign. The second virial coefficient B φ of the ions involved is defined as (7)
2.0
mtot /(mol
3.0
4.0
kg–1)
1.00 y = 0.50 0.95
aw
B φ = β (0) + β (1) exp(−α I 1/2 )
1.0
where β (0) , β (1) and C φ are ion interaction parameters which are functions of temperature and pressure, and α = 2 (mol kg−1 )−1 . The activity coefficients γMX of MX: NaCl and γNX of NX: BaCl2 in a common-anion mixture of NaCl–BaCl2 (aq) are given by Pitzer’s model as
0.90
0.85
0.80 0.0
1.0
2.0
mtot /(mol
ln γMX = f γ + (2/3)(2y + 1)I [BMX + (1/6)(y + 2) φ
3.0
4.0
5.0
kg–1)
1.00
× I CMX ] + (1/3)(1 − y)I [BNX + (21/2/12)
y = 0.67
φ
× (2 + y)I CNX ] + (1/3)y(2 + y)I 2 φ
0.95
× [BMX + (1/2)CMX ] + (1/9)(2 + y)(1 − y)I 2
× [θMN + (1/3)(2y + 1)I ψMNX ]
(8)
aw
φ
× [BNX + (21/2 /4)CNX ] + (1/3)(1 − y)I
0.90
and ln γNX = 2 f γ + (2/9)(4 − y)I [BNX + (21/2 /12)(y + 2) φ
φ
× I CNX ] + (4/3)y I [BMX + (1/6)(2 + y)I CMX ] φ
+ (21/2/3)CNX ] + (1/9)(2 + y)(1 − y)I 2 [2BNX φ
+ (1/3)CMX ] + (2/3)y(2 + y)I 2 [BMX + (2/3)y I [θMN + (2/3)(4 − y)I ψMNX ].
The coefficients BMX and are defined as
0.85
B
MX
(9)
for electrolytes 1–1 and 2–1
(0) (1) + (2βMX /α 2 I )[1 − (1 + α I 1/2 ) BMX = βMX 1/2 × exp(−α I )], 3m tot φ = θNH+ K+ + m SO2− ψNH+ K+ SO2− 4 4 4 4 2m NH+ m K+
(10) (11)
4
From the osmotic coefficients determined from the experimental water activities of the mixture studied at different ionic strength fractions, it is possible to determine the unknowns, the Pitzer mixing ionic θMN and ψMNX . These parameters are used to predict the solute activity coefficients
0.80 0.0
1.0
2.0
3.0
4.0
5.0
mtot /(mol kg–1) Fig. 1. Water activity aw of NaCl–BaCl2 (aq) against total molalities m tot at different ionic strength fractions y of 0.33, 0.50, and 0.67. , experimental points; , KM; ×, ZSR.
in the mixture. θMN and ψMNX are estimated by a graphical procedure. This procedure defines the quantity φ as the difference between the experimental values φexp and that calculated from Eq. (6) φcalc . This yields 2m MX + 3m NX φ = θMN + m X ψMNX 2m MX m NX
(12)
so a plot of φ versus total molality m should give a straight line with intercept θMN and slope ψMNX .
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Table 1 Ratios of growth K of the NaCl(aq) droplets, water activities aw , and osmotic coefficients φ of NaCl–BaCl2 (aq) at total molalities m tot for different ionic strength fractions y of NaCl m NaCl
m BaCl2
m tot
K
aw
φ
1.258 1.028 1.656 1.515 1.380 1.300 1.183 1.092 1.037 0.985
0.990a 0.981a 0.971 0.960 0.948 0.938 0.914 0.887 0.860 0.827
0.930 0.887 0.908 0.944 0.988 0.987 1.040 1.109 1.163 1.255
1.105 1.638 1.438 1.300 1.216 1.141 1.086 1.036 1.000 0.962
0.985a 0.970 0.955 0.938 0.920 0.902 0.883 0.862 0.840 0.819
0.932 0.940 0.947 0.987 1.029 1.060 1.096 1.145 1.195 1.231
1.748 1.529 1.378 1.300 1.216 1.146 1.103 1.065 1.030 0.983
0.975 0.963 0.950 0.936 0.922 0.905 0.890 0.874 0.857 0.823
0.935 0.928 0.947 0.977 1.000 1.053 1.076 1.105 1.140 1.199
y = 0.33 0.15 0.30 0.45 0.60 0.75 0.90 1.20 1.50 1.80 2.10
0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.20 1.40
0.25 0.50 0.75 1.00 1.30 1.50 2.00 2.50 3.00 3.50 y = 0.50
0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 y = 0.67
0.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00 3.60
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.59
0.70 1.10 1.40 1.80 2.10 2.50 2.80 3.20 3.50 4.20
The reference water activity is 0.84. a The values are for reference water activity of 0.98 (NaCl).
Table 2 Comparison of predictions of the ZSR, KM, RS, LSII, RWR, and Pitzer models with experimental data for different ranges of ionic strength NaCl–BaCl2 (mol kg−1 )
σ (φ)ZSR
σ (φ)KM
σ (φ)RS
σ (φ)LS
σ (φ)RWR
σ (φ)Pitzer
0.45–3.0 3.0–6.3 0.45–6.3
0.0160 0.0360 0.0180
0.0090 0.0060 0.0070
0.0090 0.0070 0.0080
0.0096 0.0060 0.0070
0.0090 0.0060 0.0070
0.0040 0.0060 0.0045
4. Results and discussion 4.1. Water activity and osmotic coefficient In this work, a series of measurements of the water activity were made for the mixture {yNaCl + (1 − y)BaCl2 }(aq) as a function of total molality ranging from 0.25 mol kg−1 to saturation of one of the solutes, with different ionic strength fractions y of NaCl of 0.33, 0.50, and 0.67 at a temperature of 298.15 K. The experimental values
of the water activity are listed in Table 1 and shown in Fig. 1. The KM and ZSR models are used for the predictions of water activity of the mixture. The molalities of NaCl and BaCl2 at different constant water activities of the ternary mixture were evaluated from experimental data. The plots of water activities, isoactivities, dependence of m BaCl2 BaCl2 versus m NaCl , of NaCl are represented in Fig. 2. The ZSR rule has been used to compare with the measurements.
M. El Guendouzi et al. / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381 5
379
1.4 y = 0.33
1.2 3
2
mBaCl /(mol kg–1)
4
2
1.0
1
0 0.0
0.8 0.80 0.5
1.0
1.5
2.0
2.5
3.0
1.0
3.5
1.2
1.4
1.6
1.4
1.6
1.4
1.6
{I/(mol kg–1)}1/2
mNaCl /(mol kg–1) 1.4
y = 0.50
Fig. 2. The dependence of the molality of NaCl versus molality of BaCl2 in mixed NaCl–BaCl2 (aq) at constant water activity aw . , aw = 0.98; , aw = 0.94; , aw = 0.90; , aw = 0.86; •, aw = 0.82; ×, ZSR. 1.2
Pitzer’s model of electrolyte solutions allows parameters determined for single-electrolyte solutions to be applied to multicomponent electrolyte solutions. Parameters were found for Pitzer’s model that characterised the solutions of the individual salts that are components of the mixtures studied here. The osmotic coefficients obtained are given in Table 1. Using the experimental data obtained for the water activity, we have evaluated the osmotic coefficients for different ionic strength fractions y of NaCl. The RS, LS, RWR, and Pitzer models are used to predict the osmotic coefficients. The results are shown in Fig. 3. The experimental data are compared with the predictions of the ZSR, KM, RS, LSII, RWR, and Pitzer models. However, in the Pitzer model the θNa+ Ba2+ and ψNa+ Ba2+ Cl− parameters were adjusted, but the ZSR, KM, RS, LSII, RWR models do not involve any ternary parameters. The deviation between the data and models for different ranges of ionic strength is summarized in Table 2, and it is between 0.004 and 0.036. The corresponding values of the ionic parameters β (0), (1) β , and C φ of the pure electrolytes NaCl(aq) and BaCl2 (aq) were obtained from Pitzer’s expressions by fits of the experimental osmotic coefficients given in our previous work [8]. The corresponding values are presented in Table 3.
Table 3 Pitzer parameters for NaCl and BaCl2 and mixtures at 298.15 K Electrolyte
β (0)
β (1)
Cφ
References
NaCl BaCl2
0.0738 0.2571
0.2712 1.4676
0.00167 −0.01773
[8] [8]
Electrolyte
θNa+ Ba2+
ψNa+ Ba2+ Cl−
Reference
NaCl + BaCl2
0.0069
0.0164
This study
1.0
0.8 0.80
1.0
1.2
{I/(mol kg–1)}1/2 1.4 y = 0.67
1.2
1.0
0.8 0.80
1.0
1.2
{I/(mol kg–1)}1/2 Fig. 3. The osmotic coefficient φ of NaCl–BaCl2 (aq) against the square root of the total ionic strength I 1/2 at different ionic strength fractions y of 0.33, 0.50, 0.67. ×, experimental points; , RS; , LS II; , Pitzer; ◦, RWR.
4.2. Activity coefficient The activity coefficients of NaCl(aq) and BaCl2 (aq) in the mixture were calculated from Pitzer’s equation using the ionic mixing parameters that we obtained; the results are listed in Table 4. The plots of activity coefficients γNaCl and γBaCl2 as a function of the square root of the ionic strength (I 1/2 ) are shown in Figs. 4 and 5. The extremes of composition are
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Table 4 Activity coefficients γNaCl of NaCl(aq) and γBaCl2 of BaCl2 (aq) in NaCl–CaCl2 (aq) at total ionic strength I for different ionic strength fractions y of NaCl y=0 I γNaCl
γBaCl2
y = 0.33 I γNaCl
γBaCl2
y = 0.50 I γNaCl
γBaCl2
y = 0.67 I γNaCl
γBaCl2
y=1 I γNaCl
γBaCl2
0.60 0.90 1.50 2.10 2.40 3.00 4.20 4.50 4.80 6.00
0.447 0.420 0.396 0.389 0.389 0.393 0.415 0.422 0.429 0.464
0.45 0.90 1.35 1.80 2.25 2.70 3.60 4.50 5.40 6.30
0.474 0.431 0.417 0.417 0.417 0.438 0.483 0.549 0.640 0.760
0.60 1.20 1.80 2.40 3.00 3.60 4.20 4.80 5.40 6.00
0.458 0.429 0.431 0.450 0.483 0.530 0.590 0.667 0.764 0.885
0.90 1.35 1.80 2.25 2.70 3.15 3.60 4.05 4.50 5.40
0.444 0.438 0.447 0.466 0.494 0.532 0.580 0.640 0.712 0.907
0.50 1.00 1.50 2.00 2.50 3.00 4.00 5.00 5.50 6.00
0.482 0.458 0.466 0.492 0.535 0.594 0.775 1.076 1.294 1.578
0.680 0.664 0.654 0.657 0.661 0.675 0.713 0.724 0.736 0.788
0.690 0.661 0.654 0.658 0.667 0.682 0.723 0.778 0.845 0.926
0.674 0.654 0.654 0.672 0.697 0.728 0.766 0.811 0.862 0.920
0.658 0.652 0.657 0.668 0.686 0.707 0.733 0.763 0.797 0.877
0.678 0.653 0.652 0.663 0.681 0.706 0.774 0.864 0.918 0.978
0
1.0
–2 0.9
Gex /(kj kg–1)
–4
0.8
0.7
–6 –8 –10 –12
0.6 0.5
1.0
1.5
2.0
2.5
3.0
{I/(mol kg–1)}1/2
–14
0
1
2
3
4
5
6
7
mtot /(mol kg–1)
Fig. 4. Activity coefficients γNaCl of NaCl(aq) in NaCl–BaCl2 (aq) against the square root of the total ionic strength I 1/2 at different ionic strength fractions y of NaCl(aq). ◦, y = 0 (BaCl2 ); , y = 0.33; •, y = 0.50; , y = 0.67; , y = 1.0 (NaCl).
Fig. 6. The excess Gibbs energy of NaCl–BaCl2 (aq) against the total molality m tot at different ionic strength fractions y of NaCl. , y = 0; , y = 0.33; ◦, y = 0.50; , y = 0.67; , y = 1.0.
2.0
4.3. The excess Gibbs energy
1.5
From the osmotic coefficient and activity coefficient, we determined the excess Gibbs energy of NaCl–BaCl2 (aq) using the following expression:
1.0
G ex = RT {ν1 m 1 (1 − φ + ln γ1 ) + ν2 m 2 (1 − φ + ln γ2 )}. (13) The plots of the Gibbs energy as a function of the molality m tot are shown in Fig. 6.
0.5
0.0 0.5
1.0
1.5
2.0
2.5
3.0
5. Conclusion
{I/(mol kg–1)}1/2 Fig. 5. Activity coefficients γBaCl2 of BaCl2 (aq) in NaCl–BaCl2 (aq) against the square root of the total ionic strength I 1/2 at different ionic strength fractions y of NaCl(aq). ◦, y = 0 (BaCl2 ); , y = 0.33; •, y = 0.50; , y = 0.67; , y = 1.0 (NaCl).
illustrated in these figures, i.e., the activity coefficients of pure and trace of components.
The water activities of the system {yNaCl + (1 − y) BaCl2 }(aq) are measured using a hygrometric method at temperature 298.15 K. These measurements were made at total molalities from 0.25 to about saturation for different ionic strength fractions y of NaCl with y = 0.33, 0.50, and 0.67. These results allow the deduction of osmotic coefficients. The experimental data are also compared with the
M. El Guendouzi et al. / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375–381
predictions of ZSR, KM, RS, LSII, RWR, and Pitzer models. The deviation between the data and models is 0.004 to 0.036. The activity coefficients of NaCl(aq) and BaCl2 (aq) in the mixture were calculated by means of Pitzer’s equation using the ionic mixing parameters that we obtained. The excess Gibbs energy is also determined. References [1] N.A. Gokcen, Report of Investigations. US Department of the interior, Bureau of Mines R.I. 8372 (1979). [2] D.A. Palmer, J.A. Rard, S.L. Clegg, J. Chem. Thermodyn. 34 (2002) 63. [3] F. Malatesta, L. Carbonaro, N. Fanelli, S. Ferrini, A. Giacomelli, J. Solution Chem. 28 (1999) 593. [4] C. Christov, Calphad 26 (2002) 85. [5] M.J. Molina, R. Zhang, P.J. Wooldridge et al., Science 261 (1993) 1418. [6] K.C. Chan, Z. Ha, Y.C. Man, J. Atmos. Environ. 34 (2000) 4795.
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[7] M. El Guendouzi, M. Marouani, J. Solution Chem. 32 (2003) 535. [8] M. El Guendouzi, A. Dinane, A. Mounir, J. Chem. Thermodyn. 33 (2001) 1059. [9] M. El Guendouzi, A. Errougui, J. Chem. Ing. Data. (in press) JE030123. [10] M. El Guendouzi, A. Benbiyi, A. Dinane, R. Azougen, Calphad 27 (2003) 213. [11] M. El Guendouzi, A. Mounir, A. Dinane, Fluid Phase Equi. 201 (2002) 223. [12] A.B. Zdanovskii, Zh. Fiz. Khim. 22 (1948) 1475. [13] R.A. Robinson, R.H. Stokes, Electrolyte Solutions, second ed., fifth Revised Impression, Butterworth, London, 1970. [14] C.L. Kusik, H.P. Meissner, AICHE J. Symp. Ser. 173 (1978) 14. [15] A.W. Stelson, M.E. Bassett, J.H. Seinfeld, Chemistry of Particles, Fogs and Rain, Butterworth, MA, 1984. [16] M.H. Lietzke, R.W. Stoughton, J. Solution Chem. 1 (1972) 299. [17] M.H. Lietzke, R.W. Stoughton, J. Inorg. Nucl. Chem. 37 (1975) 2503. [18] P.J. Reilly, R.H. Wood, R.A. Robinson, J. Phys. Chem 75 (1971) 1305. [19] K.S. Pitzer, J. Phys. Chem. 77 (1973) 268. [20] K.S. Pitzer, G. Mayorga, J. Phys. Chem. 37 (1973) 2300. [21] K.S. Pitzer, J. Kim, J. Am. Chem. Soc. 96 (1974) 5701.