- Email: [email protected]
Osmotic and activity coefficients of {yNaH2PO4+(1−y)Na2SO4}(aq) at the temperature 298.15 K
Fluid Phase Equilibria 164 Ž1999. 275–284 www.elsevier.nlrlocaterfluid
Osmotic and activity coefficients of yNaH 2 PO4 q Ž1 y y .Na 2 SO4 4 Žaq . at the temperature 298.15 K V. Pavicevic, ´ ´ R. Ninkovic, ´ M. Todorovic´ ) , J. Miladinovic´ Department of Inorganic Chemical Technology, Faculty of Technology and Metallurgy, UniÕersity of Belgrade, KarnegijeÕa 4, 11001 Belgrade, YugoslaÕia Received 25 March 1999; accepted 30 June 1999
Abstract The osmotic coefficients of the mixed electrolyte solution yNaH 2 PO4 q Ž1 y y .Na 2 SO4 4Žaq. have been measured by the isopiestic method, at the temperature T s 298.15 K. The activity coefficients of NaH 2 PO4 and Na 2 SO4 were calculated by Scatchard’s neutral-electrolyte method and by Pitzer and Kim’s treatment for mixed-electrolyte solutions. The Scatchard interaction parameters are used for calculation of the excess Gibbs energy as a function of ionic strength and ionic-strength fraction of NaH 2 PO4 . Also, the Zdanovskii’s rule of linearity is tested. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Isopiestic data; Mixture; Activity coefficient; Electrolyte
1. Introduction Osmotic and activity coefficients are important properties of mixed-electrolyte solutions for industrial, geochemical and chemical applications. Careful experimental work and extensive data analysis have been combined to produce accurate and useful tables with thermochemical properties. The investigation of the system yNaH 2 PO4 q Ž1 y y .Na 2 SO4 4Žaq. is important for the reason of more adequate and precise leading of NaH 2 PO4 production by the application of computer processing data, and also for the fact that NaH 2 PO4 is widely used as flame retardant of textile and wood. This report deals with isopiestic measurements performed on yNaH 2 PO4 q Ž 1 y y .Na 2 SO4 4Žaq. at the temperature 298.15 K, for which there is a lack of data, with NaCl Ž aq. and Na 2 SO4 Ž aq. as
)
Corresponding author. Tel.: q381-11-3370460; fax: q381-11-3370-387; E-mail: [email protected]
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 2 5 6 - 3
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
276
reference solutions. Scatchard’s neutral-electrolyte method w1x and Pitzer–Kim’s ionic method w2x for mixed electrolyte solutions are applied for calculation of activity coefficients. In addition, the Zdanovskii’s rule of linearity is tested and excess Gibbs energies were calculated.
2. Experimental The isopiestic apparatus in this work consisted of a glass vacuum desiccator mounted on a rocking mechanism, a gold-plated cooper block Ž 17 cm in diameter and 2.5 cm thick. and 12 gold-plated silver dishes. The dishes with hinged lids were 3 cm in diameter and 2 cm deep. To prevent spattering of the solutions during evacuation, the desiccator with dishes was never connected directly to the vacuum pump, but to an empty desiccator of the same volume which acted as an intermediate reservoir between the desiccator with dishes and the pump. When the vapor pressure of water was reached the evacuation was repeated 10 more times. At the beginning of each experiment several drops of water were placed in the central space of the cooper block to help sweep out the air in the vessel during evacuation and to reduce evaporation from the solutions. Also a few drops of water were placed on the block to ensure good thermal contact between the block and each dish. The equilibrium periods in the 25 " 0.018C thermostat lasted from 2 to 11 days. After the allowed time the dishes were closed and the vacuum was then broken. The dishes were weighed and the solution masses were corrected to their ‘‘in vacuo’’ values. Duplicate samples of the mixed and reference solutions together with the single salt solutions of NaH 2 PO4 and Na 2 SO4 were equilibrated in dishes placed crosswise. The molalities at isopiestic equilibrium agreed to "10y3 mol kgy1 or better, with the average value taken for duplicate samples of solutions. Another experiment was begun by adding a few drops of water to each dish and reevacuating. Stock solutions of sodium-dihydrogenphospate, sodium-sulphate and sodium-chloride were prepared from ‘‘Merck’’ suprapur-grade chemicals and deionized distilled water. The molality of NaH 2 PO4 stock solution was determined as the average on three samples by Woy’s method w3x. NaH 2 PO4 was transformed by ŽNH 4 . 2 MoO4 , in the presence of nitric acid, into Ž NH 4 . 3 PO4 P 12 MoO 3, and finally after firing at 1000 K to the constant mass, in the form of P2 O5 P MoO4 , was weighed. The molality of NaH 2 PO4 was Ž1.3170 " 0.0011. mol kgy1. The molality of Na 2 SO4 was determined also on three samples, gravimetrically w3x, by its transformation into BaSO4 , using barium-chloride solution. After firing at 1100 K to the constant mass and weighing it was found that the molality of the Na 2 SO4 stock solutions was Ž 0.9930 " 0.0007. mol kgy1. The molality of NaCl stock solution was determined gravimetrically by precipitated silver-chloride, on three samples, and was Ž2.9869 " 0.0025. mol kgy1. Before that sodium-chloride was dried to constant mass. Salt mixtures were prepared simply by mixing the proper amounts of Na 2 SO4 and NaH 2 PO4 stock solutions.
3. Results The ionic-strength fraction y of NaH 2 PO4 is given by: y s mAr Ž mA q 3m B . ,
Ž1.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
277
where mA and m B are the molalities of NaH 2 PO4ŽA. and Na 2 SO4 ŽB.. The total ionic strength I of mixed solution was calculated by: I s mA q 3m B .
Ž2.
In Table 1 are given experimental results for two sets of experiments: the first one performed with NaCl Žaq. reference solution, and the second one performed with Na 2 SO4 Ž aq. reference solution together with their ionic strength IR and osmotic coefficient f R , calculated using Pitzer’s single-electrolyte equation w4,5x. The osmotic coefficient f of the mixed electrolyte solution is obtained by:
f s n R m R f Rr Ž 2 mA q 3m B . , Ž3. with n R s 2 for NaCl Žaq. and n R s 3 for Na 2 SO4 Žaq. . Eq. Ž3. is valid only at isopiestic equilibrium. In Fig. 1 are presented the experimental values of the osmotic coefficient for the NaH 2 PO4 Ž aq. and Na 2 SO4 Žaq. , together with literature data w6–9x. Osmotic coefficients for the Na 2 SO4 Ž aq. , obtained with NaCl Ž aq. as reference solution, as it can be seen, are in excellent agreement with available smoothed experimental data of Robinson–Stokes w6x and data of Rard and Miller w7x. Also, the osmotic coefficients for the NaH 2 PO4 Žaq. in the concentration range below I s 0.8 mol kgy1, with Na 2 SO4 as reference solution, are in good agreement with literature data w8,9x. In the region of I s 0.8–3.5 mol kgy1 experimental values of f , in both sets of experiments Žwith NaCl Žaq. and Na 2 SO4 Žaq. reference solutions., show the similar tendency of discrepancy, e.g., they are below the data of Robinson and Stokes. Regarding to the fact that experimental values of f for the Na 2 SO4 Žaq. Table 1 Isopiestic ionic strengths I of yNaH 2 PO4 qŽ1y y .Na 2 SO4 4Žaq. for different ionic-strength fractions y of NaH 2 PO4 , and ionic-strengths IR and osmotic coefficients f R of the reference solutions, at the temperature 298.15 K I set of experiment with NaCl (aq) reference solution ys0
0.2500
0.5036
8.2170 7.2870 6.8358 6.2304 5.8329 5.3310 4.9821
6.2506 5.5083 5.1009 4.623 4.3142 3.9615 3.6252
4.5330 4.1430 3.7415 3.4835 3.1644
0.7492
3.5411 3.1914 2.9664 2.6877
1
IR rmol kgy1
fR
3.8688 3.4008 3.2179 2.8913 2.6819 2.4269 2.3663
2.6742 2.3359 2.5164 1.9702 1.8510 1.7038 1.5834
1.0243 1.0034 0.9929 0.9824 0.9758 0.9680 0.9619
II set of experiments with Na2 SO4 (aq) reference solution ys0
0.2527
0.5007
0.7533
1
IR rmol kgy1
fR
4.8246 3.2409 2.8686 2.2877 2.0037 1.7499 1.6269 1.5561 1.5159
3.5737 2.3906 2.1156 1.6784 1.4793 1.2877 1.1905 1.1522 1.0960
2.9124 1.9360 1.7106 1.3439 1.1849 1.0219
2.4633 1.6273 1.4377 1.1347
2.1563 1.4153 1.2524 0.9804 0.8644 0.7419 0.6865 0.6615 0.6491
4.8246 3.2409 2.8686 2.2877 2.0037 1.7499 1.6269 1.5561 1.5159
0.6227 0.6358 0.6430 0.6587 0.6678 0.6793 0.6834 0.6859 0.6884
278
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
Fig. 1. Osmotic coefficients of NaH 2 PO4 Žaq. and Na 2 SO4 Žaq. at 298.15 K.
showed as competent, it seems that Robinson–Stokes data for osmotic coefficient of NaH 2 PO4 Ž aq. in the region of I s 0.8–3.5 mol kgy1 should be revised.
4. Treatments of results and discussion In order to describe the variation of thermodynamic properties of mixture in terms of single-electrolyte contributions and parameters characteristic of the mixture, from a variety of treatments, Scatchard’s neutral electrolyte w1x and Pitzer–Kim’s treatment w2x were chosen. For the system yNaH 2 PO4 q Ž1 y y . Na 2 SO4 4Ž aq. the Scatchard’s neutral electrolyte treatment leads to the following expression: 2 3 Ž 1 q y . f s 2 yfAU q Ž 1 y y . f UB q y Ž 1 y y . ½ b 01Ž Irmo . q b 02 Ž Irmo . q b 03 Ž Irmo . 2
3
2
qb12 Ž 2 y y 1 .Ž Irmo . q b 13 Ž 2 y y 1 .Ž Irmo . q b 23 Ž 2 y y 1 . Ž Irmo .
3
5
Ž4.
where fAU and f UB are the osmotic coefficients of NaH 2 PO4 Žaq. and Na 2 SO4 Žaq. of the same ionic strength as the total ionic strength of the mixed-salt solution, calculated by the Pitzer’s single electrolyte equation w4,5x; m o is a standard molality taken here as 1 mol kgy1. The parameters determined by the least-squares method are: b 01 s y0.0045; b 02 s 0.0140; b 03 s y0.0015; b 12 s y0.0022, with the standard deviation of fit s s 0.0085.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
279
Table 2 Activity coefficients of NaH 2 PO4 and Na 2 SO4 at T s 298.15 K by Scatchard’s equation at different ionic strengths I and ionic-strength fraction y of NaH 2 PO4 Irmol kgy1
y 0
0.2
0.4
0.6
0.8
1
g " (NaH2 PO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.585 0.501 0.450 0.415 0.389 0.368 0.367
0.578 0.493 0.441 0.405 0.378 0.358 0.355
0.573 0.485 0.433 0.396 0.368 0.347 0.344
0.567 0.478 0.424 0.386 0.359 0.337 0.334
0.561 0.470 0.415 0.377 0.349 0.327 0.323
0.555 0.462 0.407 0.368 0.339 0.317 0.301
g " (Na2 SO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.394 0.315 0.272 0.242 0.221 0.205 0.208
0.386 0.306 0.262 0.233 0.212 0.197 0.200
0.378 0.297 0.253 0.224 0.204 0.189 0.193
0.370 0.288 0.244 0.215 0.195 0.181 0.185
0.362 0.279 0.235 0.207 0.187 0.174 0.178
0.355 0.270 0.226 0.198 0.180 0.167 0.171
Activity coefficients according to Scatchard, given in Table 2, were calculated by the equations: 2ln g " Ž A . 4 s 2ln g U" Ž A . 4 q Ž f UB y 1 . y 2 Ž fAU y 1 . 4 Ž 1 y y . 2
3
q b 01Ž Irmo . q b 02 q b 12 4 Ž Irmo . q b 03 Ž Irmo . Ž 1 y y . 2
3
y Ž 1r2 . b 02 q Ž 3r2 . b 12 4 Ž Irmo . q Ž 2r3 . b 03 Ž Irmo . Ž 1 y y .
2
Ž5.
and ln g " Ž B . 4 s ln g U" Ž B . 4 q 2 Ž fAU y 1 . y Ž f UB y 1 . 4 y q b 01 Ž Irmo . q b 02 q b 12 4 Ž Irmo . 3
2
3
2
qb 03 Ž Irmo . y y Ž 1r2 . b 02 q Ž 3r2 . b 12 4 Ž Irmo . q Ž 2r3 . b 03 Ž Irmo . y 2
Ž6.
where g U"ŽA. and g U"ŽB. are the activity coefficients of NaH 2 PO4 Žaq. and Na 2 SO4 Žaq. of the same ionic strength as the total ionic strength of the mixed-salt solution. The values of f U and g U" of the NaH 2 PO4 Žaq. and Na 2 SO4 Ž aq. were calculated by the corresponding Pitzer’s single-electrolyte equations w4,5x. Pitzer–Kim’s treatment leads to the following expression for the osmotic coefficient applied to the solution with ions Naq, H 2 PO4y and SO42y denoted by M, X, and Y respectively.
f y 1 s Ž1 q y .
y1
2 f f q Ž 2r3 . y Ž 2 q y .Ž Irmo . BAf q Ž 1r3 . CAf Ž 2 q y .Ž Irmo . 4
q Ž 2r9 .Ž 2 q y .Ž 1 y y .Ž Irmo . BBf q Ž 1r3 . 2y1r2 C Bf Ž 2 q y .Ž Irmo . 4 q Ž 2r3 . y Ž 1 y y .Ž Irmo . u XY q Ž 1r3 . c MXY Ž 2 q y .Ž Irmo . 4 .
Ž7.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
280
Table 3 Pitzer’s parameters of the NaH 2 PO4 Žaq., Na 2 SO4 Žaq. and NaCl Žaq., maximum molality, m and standard deviation of fit, s Electrolyte
b Ž0.
b Ž1.
Cf
b
a
m Žmol kgy1 .
s
NaH 2 PO4 Na 2 SO4 NaCl
y0.0533 0.0196 0.0765
0.0396 1.1130 0.2664
0.00795 0.00497 0.00127
1.2 1.2 1.2
2 2 2
6 4 6
0.003 0.003 0.001
The corresponding values of f f , B f and C f for pure electrolytes A and B, were obtained from Pitzer’s expression for the osmotic coefficient w4,5x. f f s yAf
I 1r2 1 q bI 1r2
B f s b Ž0. q b Ž1. exp Ž a I 1r2 . where Af is the Debye–Huckel coefficient for the osmotic function which has the value 0.392 at 298.15 K for water. Parameters b Ž0. and b Ž1. are characteristic for each substance and define the second virial coefficients representing the short-range interactions of pairs of ions while, C f is the third virial coefficient, for short range interactions of triplets of ions, which is usually very small and sometimes completely negligible. In Table 3 are given the Pitzer’s parameters for the NaH 2 PO4 Ž aq. , Na 2 SO4 Žaq. and NaCl Žaq. together with maximum molality, m and standard deviation of fit s w5x. By the known transformation of Eq. Ž7. into a linear form the interaction parameters u and c were determined as the intercept and slope. They have the values: u XY s y0.0928; c MXY s 0.0335, and standard deviation s s 0.0440. The parameter u arises from differences in the short range interactions between like and unlike pairs of ions of the same sign. Similarly, c arises from triple interactions. According to Pitzer, if singly and doubly charged ions are being mixed c is positive or zero, what is confirmed in this study. In this relation is not included the parameter u X Žwhich is predicted to be small. representing a possible dependence of u on ionic strength. The activity coefficients listed in Table 4 were calculated by Pitzer–Kim’s equation in the forms: ln g " Ž A . 4 s f g q Ž 1r3 .Ž Irmo . 2 Ž 2 y q 1 . BA q Ž 1r3 . CA Ž y q 2 .Ž Irmo . q Ž 1r6 . c MXY Ž 1 y y .Ž Irmo . 4 q Ž 1 y y . B B q Ž 1r3 . C B Ž y q 2 .Ž Irmo . q u XY 4 q Ž y q 2 .Ž Irmo . Ž BAX q CA . y q Ž 1r3 .Ž BXB q C B .Ž 1 y y . 4
Ž8.
and ln g " Ž B . 4 s 2 f g q Ž 1r3 .Ž Irmo . Ž 2r3 .Ž 4 y y . B B q Ž 1r3 . C B Ž y q 2 .Ž Irmo . q Ž 1r2 . c MXY y Ž Irmo . 4 q 4 y BA q Ž 1r3 . CAŽ y q 2 .Ž Irmo . q Ž 1r2 . u XY 4 q Ž y q 2 .Ž Irmo . Ž 2 BAX q Ž 4r3 . CA . y q Ž 1r3 . Ž 2 BXB q Ž 4r3 . C B . Ž 1 y y . 4
Ž9.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
281
Table 4 Activity coefficients of NaH 2 PO4 and Na 2 SO4 at T s 298.15 K by Pitzer–Kim’s equation at different ionic strengths I and ionic-strength fraction y of NaH 2 PO4 Irmol kgy1
y 0
0.2
0.4
0.6
0.8
1
g " (NaH2 PO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.595 0.512 0.460 0.423 0.394 0.371 0.367
0.586 0.501 0.448 0.410 0.382 0.360 0.357
0.577 0.490 0.437 0.399 0.371 0.349 0.346
0.569 0.480 0.426 0.388 0.360 0.338 0.335
0.562 0.471 0.416 0.378 0.349 0.328 0.324
0.555 0.462 0.407 0.368 0.339 0.317 0.301
g " (Na2 SO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.394 0.315 0.272 0.242 0.221 0.205 0.208
0.387 0.307 0.262 0.233 0.212 0.196 0.200
0.382 0.300 0.255 0.225 0.204 0.189 0.192
0.376 0.293 0.248 0.218 0.198 0.183 0.186
0.372 0.288 0.242 0.213 0.192 0.178 0.182
0.368 0.283 0.237 0.208 0.188 0.174 0.178
The quantities f g, B, C and BX were calculated from appropriate equations, given below: g
f s yAf
I 1r2 1 q bI
1r2
2 q bln
Ž 1 q bI 1r2 .
B s b Ž0. q Ž 2 b Ž1.ra 2 I . 1 y Ž 1 q a I 1r2 . exp Ž ya I 1r2 . BX s Ž 2 b Ž1.ra 2 I 2 . y1 q Ž 1 q a I 1r2 q Ž 1r2 . a 2 I . exp Ž ya I 1r2 . C s C fr2 < z M z X < 1r2 In both Tables 2 and 4 at y s 1, the values are related to the activity coefficients for the NaH 2 PO4 Žaq. and to the activity coefficients at infinite dilution for Na 2 SO4 , and vice versa at y s 0. The calculated values of the activity coefficients of Na 2 SO4 and NaH 2 PO4 , for the mixed solutions, by using Scatchard’s and Pitzer–Kim’s treatment exhibit good mutual agreement, but the sensitivity of g " to the mode of calculation has been noted previously w10x. The average difference between them is 0.0034 for g "ŽNaH 2 PO4 . and 0.0040 for g " ŽNa 2 SO4 .. The Zdanovskii’s rule w11x, which is widely used, is based on the experimental fact that linear, or approximately linear, relation exist between the molalities of mixed electrolyte solution in isopiestic equilibrium with the single electrolyte solutions of which it is formed. The Zdanovskii’s rule, for the linear relation, applicable to any type of ternary mixed-electrolyte solutions, is given by: 1rm s x Arm 0A q x Brm 0B ,
Ž 10.
282
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
Fig. 2. Zdanovskii’s isopiestic plots for yNaH 2 PO4 qŽ1y y .Na 2 SO4 4Žaq., at T s 298.15 K, for osmotic coefficients f R : 1, 0.9929; 2, 0.9824; 3, 0.9758; 4, 0.9680; 5, 0.6227; 6, 0.6358; 7, 0.6430; 8, 0.6587.
where m s Ž mA q m B . is the molality of the mixed electrolyte solution in isopiestic equilibrium with the pure electrolyte solutions with molalities m 0A and m 0B respectively, and x A s mArm and x B s m Brm. The deviation from the linearity is given by parameter b, which is calculated from the expression: 1rm s Ž x Arm 0A . q Ž x Brm 0B . q bx A x B .
Ž 11.
In this system the average value < bavg < s 0.102 kg moly1, varying between 0.033 - b - 0.240. According to the parameter b, Chen et al. w12x divided the investigated systems into three groups: group A, with 0 - < bavg < - 0.02 reflecting small deviations from linearity, group B, with 0.02 - < bavg < - 0.1 medium deviations and group C with < bavg < ) 0.1 showing large deviations. It can be seen, that parameter b calculated in this investigated system, belongs to the group C, with large deviations attributed to the formation of complexes w12x. Zdanovskii’s isopiestic plots are illustrated in Fig. 2. The thermodynamic properties of mixed electrolyte solutions are conveniently described in terms of excess quantities. One of these is the excess Gibbs energy. If the non-ideality of the single salt solutions is accepted, then the further departures of the excess Gibbs energy, which occur in forming the mixed solution can be obtained by the Scatchard’s neutral-electrolyte treatment according to equation: 2
g E s y Ž 1 y y . RT Ž Irmo . b 01 q Ž 1r2 . b 02 Ž Irmo . q Ž 1r3 . b 03 Ž Irmo .
½
q Ž 2 y y 1 .Ž 1r2 . b 12 Ž Irmo . .
5
2
Ž 12 .
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
283
Table 5 Dependence of the excess Gibbs energy g E at T s 298.15 K on the ionic strength I and ionic-strength fraction y of NaH 2 PO4 Irmol kgy1 0.5 1.0 1.5 2.0 2.5 3.0 3.5
g E rJ kgy1 ys0
0.2
0.4
0.5
0.6
0.8
1
0 0 0 0 0 0 0
y0.08 1.05 5.23 13.99 28.57 49.90 78.63
y0.15 1.32 6.97 18.89 38.76 67.78 106.73
y0.17 1.24 6.80 18.59 38.25 66.93 105.33
y0.18 1.05 6.08 16.80 34.67 60.72 95.50
y0.14 0.53 3.47 9.80 20.39 35.77 56.19
0 0 0 0 0 0 0
The results of calculation are given in Table 5. 5. List of symbols b b 01, b 02 , b 03 , b 12 B f , B g , BX g C f, Cg f f, f g gE I m mo x
deviation parameter in the Zdanovskii’s formula interaction parameters in the Scatchard’s equations functions obtained from Pitzer’s equations functions obtained from Pitzer’s equations functions obtained from Pitzer’s equations excess Gibbs energy, J kgy1 total ionic strength of mixed electrolyte solution, mol kgy1 molality of mixed electrolyte solution, mol kgy1 standard molality, 1 mol kgy1 molality fraction of the component in a total molality of the mixed electrolyte solution y ionic strength fraction of the component A Subscripts and superscripts A denotes NaH 2 PO4 B denotes Na 2 SO4 M cation Naq R reference solution, NaCl Ž aq. and Na 2 SO4 Ž aq. X anion H 2 PO4y Y anion SO42y Greek letters g" mean activity coefficient g U" mean activity coefficient of the pure electrolyte at the same ionic strength as the total ionic strength of the mixed electrolyte solution f osmotic coefficient fU osmotic coefficient of the pure electrolyte at the same ionic strength as the total ionic strength of the mixed electrolyte solution n total number of ions formed by complete dissociation of electrolyte molecule
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
284
u c
interaction parameter in the Pitzer–Kim’s equation interaction parameter in the Pitzer–Kim’s equation
Acknowledgements This work was supported by the MSTS under Project 02E08.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x
G. Scatchard, J. Am. Chem. Soc. 83 Ž1961. 2636–2642. K.S. Pitzer, J. Kim, J. Am. Chem. Soc. 96 Ž1974. 5701–5707. L.M. Kolthoff, E.B. Sandell, Textbook of Quantitative Inorganic Analysis, Macmillan, New York, 1952. K.S. Pitzer, J. Phys. Chem. 77 Ž1973. 268–277. K.S. Pitzer, G. Mayorga, J. Phys. Chem. 77 Ž1973. 2300–2308. R.A. Robinson, R.H. Stokes, Electrolyte Solutions, Butterworth, London, 1955. J.A. Rard, D.G. Miller, J. Chem. Eng. Data 26 Ž1981. 33–38. G. Scatchard, R.C. Breckenridge, J. Phys. Chem. 58 Ž1954. 596–602. R.F. Platford, J. Chem. Eng. Data 21 Ž1976. 468–469. S. Lindenbaum, R.M. Rush, R.A. Robinson, J. Chem. Thermodyn. 4 Ž1972. 381–390. A.B. Zdanovskii, Trudy Solyanoi Laboratorii, Akad. Nauk. SSSR, No. 6, 1936. H. Chen, J. Sangster, T.T. Teng, F. Lenzi, Can. J. Chem. Eng. 51 Ž1973. 234–242.
Osmotic and activity coefficients of yNaH 2 PO4 q Ž1 y y .Na 2 SO4 4 Žaq . at the temperature 298.15 K V. Pavicevic, ´ ´ R. Ninkovic, ´ M. Todorovic´ ) , J. Miladinovic´ Department of Inorganic Chemical Technology, Faculty of Technology and Metallurgy, UniÕersity of Belgrade, KarnegijeÕa 4, 11001 Belgrade, YugoslaÕia Received 25 March 1999; accepted 30 June 1999
Abstract The osmotic coefficients of the mixed electrolyte solution yNaH 2 PO4 q Ž1 y y .Na 2 SO4 4Žaq. have been measured by the isopiestic method, at the temperature T s 298.15 K. The activity coefficients of NaH 2 PO4 and Na 2 SO4 were calculated by Scatchard’s neutral-electrolyte method and by Pitzer and Kim’s treatment for mixed-electrolyte solutions. The Scatchard interaction parameters are used for calculation of the excess Gibbs energy as a function of ionic strength and ionic-strength fraction of NaH 2 PO4 . Also, the Zdanovskii’s rule of linearity is tested. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Isopiestic data; Mixture; Activity coefficient; Electrolyte
1. Introduction Osmotic and activity coefficients are important properties of mixed-electrolyte solutions for industrial, geochemical and chemical applications. Careful experimental work and extensive data analysis have been combined to produce accurate and useful tables with thermochemical properties. The investigation of the system yNaH 2 PO4 q Ž1 y y .Na 2 SO4 4Žaq. is important for the reason of more adequate and precise leading of NaH 2 PO4 production by the application of computer processing data, and also for the fact that NaH 2 PO4 is widely used as flame retardant of textile and wood. This report deals with isopiestic measurements performed on yNaH 2 PO4 q Ž 1 y y .Na 2 SO4 4Žaq. at the temperature 298.15 K, for which there is a lack of data, with NaCl Ž aq. and Na 2 SO4 Ž aq. as
)
Corresponding author. Tel.: q381-11-3370460; fax: q381-11-3370-387; E-mail: [email protected]
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 2 5 6 - 3
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
276
reference solutions. Scatchard’s neutral-electrolyte method w1x and Pitzer–Kim’s ionic method w2x for mixed electrolyte solutions are applied for calculation of activity coefficients. In addition, the Zdanovskii’s rule of linearity is tested and excess Gibbs energies were calculated.
2. Experimental The isopiestic apparatus in this work consisted of a glass vacuum desiccator mounted on a rocking mechanism, a gold-plated cooper block Ž 17 cm in diameter and 2.5 cm thick. and 12 gold-plated silver dishes. The dishes with hinged lids were 3 cm in diameter and 2 cm deep. To prevent spattering of the solutions during evacuation, the desiccator with dishes was never connected directly to the vacuum pump, but to an empty desiccator of the same volume which acted as an intermediate reservoir between the desiccator with dishes and the pump. When the vapor pressure of water was reached the evacuation was repeated 10 more times. At the beginning of each experiment several drops of water were placed in the central space of the cooper block to help sweep out the air in the vessel during evacuation and to reduce evaporation from the solutions. Also a few drops of water were placed on the block to ensure good thermal contact between the block and each dish. The equilibrium periods in the 25 " 0.018C thermostat lasted from 2 to 11 days. After the allowed time the dishes were closed and the vacuum was then broken. The dishes were weighed and the solution masses were corrected to their ‘‘in vacuo’’ values. Duplicate samples of the mixed and reference solutions together with the single salt solutions of NaH 2 PO4 and Na 2 SO4 were equilibrated in dishes placed crosswise. The molalities at isopiestic equilibrium agreed to "10y3 mol kgy1 or better, with the average value taken for duplicate samples of solutions. Another experiment was begun by adding a few drops of water to each dish and reevacuating. Stock solutions of sodium-dihydrogenphospate, sodium-sulphate and sodium-chloride were prepared from ‘‘Merck’’ suprapur-grade chemicals and deionized distilled water. The molality of NaH 2 PO4 stock solution was determined as the average on three samples by Woy’s method w3x. NaH 2 PO4 was transformed by ŽNH 4 . 2 MoO4 , in the presence of nitric acid, into Ž NH 4 . 3 PO4 P 12 MoO 3, and finally after firing at 1000 K to the constant mass, in the form of P2 O5 P MoO4 , was weighed. The molality of NaH 2 PO4 was Ž1.3170 " 0.0011. mol kgy1. The molality of Na 2 SO4 was determined also on three samples, gravimetrically w3x, by its transformation into BaSO4 , using barium-chloride solution. After firing at 1100 K to the constant mass and weighing it was found that the molality of the Na 2 SO4 stock solutions was Ž 0.9930 " 0.0007. mol kgy1. The molality of NaCl stock solution was determined gravimetrically by precipitated silver-chloride, on three samples, and was Ž2.9869 " 0.0025. mol kgy1. Before that sodium-chloride was dried to constant mass. Salt mixtures were prepared simply by mixing the proper amounts of Na 2 SO4 and NaH 2 PO4 stock solutions.
3. Results The ionic-strength fraction y of NaH 2 PO4 is given by: y s mAr Ž mA q 3m B . ,
Ž1.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
277
where mA and m B are the molalities of NaH 2 PO4ŽA. and Na 2 SO4 ŽB.. The total ionic strength I of mixed solution was calculated by: I s mA q 3m B .
Ž2.
In Table 1 are given experimental results for two sets of experiments: the first one performed with NaCl Žaq. reference solution, and the second one performed with Na 2 SO4 Ž aq. reference solution together with their ionic strength IR and osmotic coefficient f R , calculated using Pitzer’s single-electrolyte equation w4,5x. The osmotic coefficient f of the mixed electrolyte solution is obtained by:
f s n R m R f Rr Ž 2 mA q 3m B . , Ž3. with n R s 2 for NaCl Žaq. and n R s 3 for Na 2 SO4 Žaq. . Eq. Ž3. is valid only at isopiestic equilibrium. In Fig. 1 are presented the experimental values of the osmotic coefficient for the NaH 2 PO4 Ž aq. and Na 2 SO4 Žaq. , together with literature data w6–9x. Osmotic coefficients for the Na 2 SO4 Ž aq. , obtained with NaCl Ž aq. as reference solution, as it can be seen, are in excellent agreement with available smoothed experimental data of Robinson–Stokes w6x and data of Rard and Miller w7x. Also, the osmotic coefficients for the NaH 2 PO4 Žaq. in the concentration range below I s 0.8 mol kgy1, with Na 2 SO4 as reference solution, are in good agreement with literature data w8,9x. In the region of I s 0.8–3.5 mol kgy1 experimental values of f , in both sets of experiments Žwith NaCl Žaq. and Na 2 SO4 Žaq. reference solutions., show the similar tendency of discrepancy, e.g., they are below the data of Robinson and Stokes. Regarding to the fact that experimental values of f for the Na 2 SO4 Žaq. Table 1 Isopiestic ionic strengths I of yNaH 2 PO4 qŽ1y y .Na 2 SO4 4Žaq. for different ionic-strength fractions y of NaH 2 PO4 , and ionic-strengths IR and osmotic coefficients f R of the reference solutions, at the temperature 298.15 K I set of experiment with NaCl (aq) reference solution ys0
0.2500
0.5036
8.2170 7.2870 6.8358 6.2304 5.8329 5.3310 4.9821
6.2506 5.5083 5.1009 4.623 4.3142 3.9615 3.6252
4.5330 4.1430 3.7415 3.4835 3.1644
0.7492
3.5411 3.1914 2.9664 2.6877
1
IR rmol kgy1
fR
3.8688 3.4008 3.2179 2.8913 2.6819 2.4269 2.3663
2.6742 2.3359 2.5164 1.9702 1.8510 1.7038 1.5834
1.0243 1.0034 0.9929 0.9824 0.9758 0.9680 0.9619
II set of experiments with Na2 SO4 (aq) reference solution ys0
0.2527
0.5007
0.7533
1
IR rmol kgy1
fR
4.8246 3.2409 2.8686 2.2877 2.0037 1.7499 1.6269 1.5561 1.5159
3.5737 2.3906 2.1156 1.6784 1.4793 1.2877 1.1905 1.1522 1.0960
2.9124 1.9360 1.7106 1.3439 1.1849 1.0219
2.4633 1.6273 1.4377 1.1347
2.1563 1.4153 1.2524 0.9804 0.8644 0.7419 0.6865 0.6615 0.6491
4.8246 3.2409 2.8686 2.2877 2.0037 1.7499 1.6269 1.5561 1.5159
0.6227 0.6358 0.6430 0.6587 0.6678 0.6793 0.6834 0.6859 0.6884
278
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
Fig. 1. Osmotic coefficients of NaH 2 PO4 Žaq. and Na 2 SO4 Žaq. at 298.15 K.
showed as competent, it seems that Robinson–Stokes data for osmotic coefficient of NaH 2 PO4 Ž aq. in the region of I s 0.8–3.5 mol kgy1 should be revised.
4. Treatments of results and discussion In order to describe the variation of thermodynamic properties of mixture in terms of single-electrolyte contributions and parameters characteristic of the mixture, from a variety of treatments, Scatchard’s neutral electrolyte w1x and Pitzer–Kim’s treatment w2x were chosen. For the system yNaH 2 PO4 q Ž1 y y . Na 2 SO4 4Ž aq. the Scatchard’s neutral electrolyte treatment leads to the following expression: 2 3 Ž 1 q y . f s 2 yfAU q Ž 1 y y . f UB q y Ž 1 y y . ½ b 01Ž Irmo . q b 02 Ž Irmo . q b 03 Ž Irmo . 2
3
2
qb12 Ž 2 y y 1 .Ž Irmo . q b 13 Ž 2 y y 1 .Ž Irmo . q b 23 Ž 2 y y 1 . Ž Irmo .
3
5
Ž4.
where fAU and f UB are the osmotic coefficients of NaH 2 PO4 Žaq. and Na 2 SO4 Žaq. of the same ionic strength as the total ionic strength of the mixed-salt solution, calculated by the Pitzer’s single electrolyte equation w4,5x; m o is a standard molality taken here as 1 mol kgy1. The parameters determined by the least-squares method are: b 01 s y0.0045; b 02 s 0.0140; b 03 s y0.0015; b 12 s y0.0022, with the standard deviation of fit s s 0.0085.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
279
Table 2 Activity coefficients of NaH 2 PO4 and Na 2 SO4 at T s 298.15 K by Scatchard’s equation at different ionic strengths I and ionic-strength fraction y of NaH 2 PO4 Irmol kgy1
y 0
0.2
0.4
0.6
0.8
1
g " (NaH2 PO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.585 0.501 0.450 0.415 0.389 0.368 0.367
0.578 0.493 0.441 0.405 0.378 0.358 0.355
0.573 0.485 0.433 0.396 0.368 0.347 0.344
0.567 0.478 0.424 0.386 0.359 0.337 0.334
0.561 0.470 0.415 0.377 0.349 0.327 0.323
0.555 0.462 0.407 0.368 0.339 0.317 0.301
g " (Na2 SO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.394 0.315 0.272 0.242 0.221 0.205 0.208
0.386 0.306 0.262 0.233 0.212 0.197 0.200
0.378 0.297 0.253 0.224 0.204 0.189 0.193
0.370 0.288 0.244 0.215 0.195 0.181 0.185
0.362 0.279 0.235 0.207 0.187 0.174 0.178
0.355 0.270 0.226 0.198 0.180 0.167 0.171
Activity coefficients according to Scatchard, given in Table 2, were calculated by the equations: 2ln g " Ž A . 4 s 2ln g U" Ž A . 4 q Ž f UB y 1 . y 2 Ž fAU y 1 . 4 Ž 1 y y . 2
3
q b 01Ž Irmo . q b 02 q b 12 4 Ž Irmo . q b 03 Ž Irmo . Ž 1 y y . 2
3
y Ž 1r2 . b 02 q Ž 3r2 . b 12 4 Ž Irmo . q Ž 2r3 . b 03 Ž Irmo . Ž 1 y y .
2
Ž5.
and ln g " Ž B . 4 s ln g U" Ž B . 4 q 2 Ž fAU y 1 . y Ž f UB y 1 . 4 y q b 01 Ž Irmo . q b 02 q b 12 4 Ž Irmo . 3
2
3
2
qb 03 Ž Irmo . y y Ž 1r2 . b 02 q Ž 3r2 . b 12 4 Ž Irmo . q Ž 2r3 . b 03 Ž Irmo . y 2
Ž6.
where g U"ŽA. and g U"ŽB. are the activity coefficients of NaH 2 PO4 Žaq. and Na 2 SO4 Žaq. of the same ionic strength as the total ionic strength of the mixed-salt solution. The values of f U and g U" of the NaH 2 PO4 Žaq. and Na 2 SO4 Ž aq. were calculated by the corresponding Pitzer’s single-electrolyte equations w4,5x. Pitzer–Kim’s treatment leads to the following expression for the osmotic coefficient applied to the solution with ions Naq, H 2 PO4y and SO42y denoted by M, X, and Y respectively.
f y 1 s Ž1 q y .
y1
2 f f q Ž 2r3 . y Ž 2 q y .Ž Irmo . BAf q Ž 1r3 . CAf Ž 2 q y .Ž Irmo . 4
q Ž 2r9 .Ž 2 q y .Ž 1 y y .Ž Irmo . BBf q Ž 1r3 . 2y1r2 C Bf Ž 2 q y .Ž Irmo . 4 q Ž 2r3 . y Ž 1 y y .Ž Irmo . u XY q Ž 1r3 . c MXY Ž 2 q y .Ž Irmo . 4 .
Ž7.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
280
Table 3 Pitzer’s parameters of the NaH 2 PO4 Žaq., Na 2 SO4 Žaq. and NaCl Žaq., maximum molality, m and standard deviation of fit, s Electrolyte
b Ž0.
b Ž1.
Cf
b
a
m Žmol kgy1 .
s
NaH 2 PO4 Na 2 SO4 NaCl
y0.0533 0.0196 0.0765
0.0396 1.1130 0.2664
0.00795 0.00497 0.00127
1.2 1.2 1.2
2 2 2
6 4 6
0.003 0.003 0.001
The corresponding values of f f , B f and C f for pure electrolytes A and B, were obtained from Pitzer’s expression for the osmotic coefficient w4,5x. f f s yAf
I 1r2 1 q bI 1r2
B f s b Ž0. q b Ž1. exp Ž a I 1r2 . where Af is the Debye–Huckel coefficient for the osmotic function which has the value 0.392 at 298.15 K for water. Parameters b Ž0. and b Ž1. are characteristic for each substance and define the second virial coefficients representing the short-range interactions of pairs of ions while, C f is the third virial coefficient, for short range interactions of triplets of ions, which is usually very small and sometimes completely negligible. In Table 3 are given the Pitzer’s parameters for the NaH 2 PO4 Ž aq. , Na 2 SO4 Žaq. and NaCl Žaq. together with maximum molality, m and standard deviation of fit s w5x. By the known transformation of Eq. Ž7. into a linear form the interaction parameters u and c were determined as the intercept and slope. They have the values: u XY s y0.0928; c MXY s 0.0335, and standard deviation s s 0.0440. The parameter u arises from differences in the short range interactions between like and unlike pairs of ions of the same sign. Similarly, c arises from triple interactions. According to Pitzer, if singly and doubly charged ions are being mixed c is positive or zero, what is confirmed in this study. In this relation is not included the parameter u X Žwhich is predicted to be small. representing a possible dependence of u on ionic strength. The activity coefficients listed in Table 4 were calculated by Pitzer–Kim’s equation in the forms: ln g " Ž A . 4 s f g q Ž 1r3 .Ž Irmo . 2 Ž 2 y q 1 . BA q Ž 1r3 . CA Ž y q 2 .Ž Irmo . q Ž 1r6 . c MXY Ž 1 y y .Ž Irmo . 4 q Ž 1 y y . B B q Ž 1r3 . C B Ž y q 2 .Ž Irmo . q u XY 4 q Ž y q 2 .Ž Irmo . Ž BAX q CA . y q Ž 1r3 .Ž BXB q C B .Ž 1 y y . 4
Ž8.
and ln g " Ž B . 4 s 2 f g q Ž 1r3 .Ž Irmo . Ž 2r3 .Ž 4 y y . B B q Ž 1r3 . C B Ž y q 2 .Ž Irmo . q Ž 1r2 . c MXY y Ž Irmo . 4 q 4 y BA q Ž 1r3 . CAŽ y q 2 .Ž Irmo . q Ž 1r2 . u XY 4 q Ž y q 2 .Ž Irmo . Ž 2 BAX q Ž 4r3 . CA . y q Ž 1r3 . Ž 2 BXB q Ž 4r3 . C B . Ž 1 y y . 4
Ž9.
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
281
Table 4 Activity coefficients of NaH 2 PO4 and Na 2 SO4 at T s 298.15 K by Pitzer–Kim’s equation at different ionic strengths I and ionic-strength fraction y of NaH 2 PO4 Irmol kgy1
y 0
0.2
0.4
0.6
0.8
1
g " (NaH2 PO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.595 0.512 0.460 0.423 0.394 0.371 0.367
0.586 0.501 0.448 0.410 0.382 0.360 0.357
0.577 0.490 0.437 0.399 0.371 0.349 0.346
0.569 0.480 0.426 0.388 0.360 0.338 0.335
0.562 0.471 0.416 0.378 0.349 0.328 0.324
0.555 0.462 0.407 0.368 0.339 0.317 0.301
g " (Na2 SO4 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.394 0.315 0.272 0.242 0.221 0.205 0.208
0.387 0.307 0.262 0.233 0.212 0.196 0.200
0.382 0.300 0.255 0.225 0.204 0.189 0.192
0.376 0.293 0.248 0.218 0.198 0.183 0.186
0.372 0.288 0.242 0.213 0.192 0.178 0.182
0.368 0.283 0.237 0.208 0.188 0.174 0.178
The quantities f g, B, C and BX were calculated from appropriate equations, given below: g
f s yAf
I 1r2 1 q bI
1r2
2 q bln
Ž 1 q bI 1r2 .
B s b Ž0. q Ž 2 b Ž1.ra 2 I . 1 y Ž 1 q a I 1r2 . exp Ž ya I 1r2 . BX s Ž 2 b Ž1.ra 2 I 2 . y1 q Ž 1 q a I 1r2 q Ž 1r2 . a 2 I . exp Ž ya I 1r2 . C s C fr2 < z M z X < 1r2 In both Tables 2 and 4 at y s 1, the values are related to the activity coefficients for the NaH 2 PO4 Žaq. and to the activity coefficients at infinite dilution for Na 2 SO4 , and vice versa at y s 0. The calculated values of the activity coefficients of Na 2 SO4 and NaH 2 PO4 , for the mixed solutions, by using Scatchard’s and Pitzer–Kim’s treatment exhibit good mutual agreement, but the sensitivity of g " to the mode of calculation has been noted previously w10x. The average difference between them is 0.0034 for g "ŽNaH 2 PO4 . and 0.0040 for g " ŽNa 2 SO4 .. The Zdanovskii’s rule w11x, which is widely used, is based on the experimental fact that linear, or approximately linear, relation exist between the molalities of mixed electrolyte solution in isopiestic equilibrium with the single electrolyte solutions of which it is formed. The Zdanovskii’s rule, for the linear relation, applicable to any type of ternary mixed-electrolyte solutions, is given by: 1rm s x Arm 0A q x Brm 0B ,
Ž 10.
282
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
Fig. 2. Zdanovskii’s isopiestic plots for yNaH 2 PO4 qŽ1y y .Na 2 SO4 4Žaq., at T s 298.15 K, for osmotic coefficients f R : 1, 0.9929; 2, 0.9824; 3, 0.9758; 4, 0.9680; 5, 0.6227; 6, 0.6358; 7, 0.6430; 8, 0.6587.
where m s Ž mA q m B . is the molality of the mixed electrolyte solution in isopiestic equilibrium with the pure electrolyte solutions with molalities m 0A and m 0B respectively, and x A s mArm and x B s m Brm. The deviation from the linearity is given by parameter b, which is calculated from the expression: 1rm s Ž x Arm 0A . q Ž x Brm 0B . q bx A x B .
Ž 11.
In this system the average value < bavg < s 0.102 kg moly1, varying between 0.033 - b - 0.240. According to the parameter b, Chen et al. w12x divided the investigated systems into three groups: group A, with 0 - < bavg < - 0.02 reflecting small deviations from linearity, group B, with 0.02 - < bavg < - 0.1 medium deviations and group C with < bavg < ) 0.1 showing large deviations. It can be seen, that parameter b calculated in this investigated system, belongs to the group C, with large deviations attributed to the formation of complexes w12x. Zdanovskii’s isopiestic plots are illustrated in Fig. 2. The thermodynamic properties of mixed electrolyte solutions are conveniently described in terms of excess quantities. One of these is the excess Gibbs energy. If the non-ideality of the single salt solutions is accepted, then the further departures of the excess Gibbs energy, which occur in forming the mixed solution can be obtained by the Scatchard’s neutral-electrolyte treatment according to equation: 2
g E s y Ž 1 y y . RT Ž Irmo . b 01 q Ž 1r2 . b 02 Ž Irmo . q Ž 1r3 . b 03 Ž Irmo .
½
q Ž 2 y y 1 .Ž 1r2 . b 12 Ž Irmo . .
5
2
Ž 12 .
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
283
Table 5 Dependence of the excess Gibbs energy g E at T s 298.15 K on the ionic strength I and ionic-strength fraction y of NaH 2 PO4 Irmol kgy1 0.5 1.0 1.5 2.0 2.5 3.0 3.5
g E rJ kgy1 ys0
0.2
0.4
0.5
0.6
0.8
1
0 0 0 0 0 0 0
y0.08 1.05 5.23 13.99 28.57 49.90 78.63
y0.15 1.32 6.97 18.89 38.76 67.78 106.73
y0.17 1.24 6.80 18.59 38.25 66.93 105.33
y0.18 1.05 6.08 16.80 34.67 60.72 95.50
y0.14 0.53 3.47 9.80 20.39 35.77 56.19
0 0 0 0 0 0 0
The results of calculation are given in Table 5. 5. List of symbols b b 01, b 02 , b 03 , b 12 B f , B g , BX g C f, Cg f f, f g gE I m mo x
deviation parameter in the Zdanovskii’s formula interaction parameters in the Scatchard’s equations functions obtained from Pitzer’s equations functions obtained from Pitzer’s equations functions obtained from Pitzer’s equations excess Gibbs energy, J kgy1 total ionic strength of mixed electrolyte solution, mol kgy1 molality of mixed electrolyte solution, mol kgy1 standard molality, 1 mol kgy1 molality fraction of the component in a total molality of the mixed electrolyte solution y ionic strength fraction of the component A Subscripts and superscripts A denotes NaH 2 PO4 B denotes Na 2 SO4 M cation Naq R reference solution, NaCl Ž aq. and Na 2 SO4 Ž aq. X anion H 2 PO4y Y anion SO42y Greek letters g" mean activity coefficient g U" mean activity coefficient of the pure electrolyte at the same ionic strength as the total ionic strength of the mixed electrolyte solution f osmotic coefficient fU osmotic coefficient of the pure electrolyte at the same ionic strength as the total ionic strength of the mixed electrolyte solution n total number of ions formed by complete dissociation of electrolyte molecule
V. PaÕiceÕic ´ ´ et al.r Fluid Phase Equilibria 164 (1999) 275–284
284
u c
interaction parameter in the Pitzer–Kim’s equation interaction parameter in the Pitzer–Kim’s equation
Acknowledgements This work was supported by the MSTS under Project 02E08.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x
G. Scatchard, J. Am. Chem. Soc. 83 Ž1961. 2636–2642. K.S. Pitzer, J. Kim, J. Am. Chem. Soc. 96 Ž1974. 5701–5707. L.M. Kolthoff, E.B. Sandell, Textbook of Quantitative Inorganic Analysis, Macmillan, New York, 1952. K.S. Pitzer, J. Phys. Chem. 77 Ž1973. 268–277. K.S. Pitzer, G. Mayorga, J. Phys. Chem. 77 Ž1973. 2300–2308. R.A. Robinson, R.H. Stokes, Electrolyte Solutions, Butterworth, London, 1955. J.A. Rard, D.G. Miller, J. Chem. Eng. Data 26 Ž1981. 33–38. G. Scatchard, R.C. Breckenridge, J. Phys. Chem. 58 Ž1954. 596–602. R.F. Platford, J. Chem. Eng. Data 21 Ž1976. 468–469. S. Lindenbaum, R.M. Rush, R.A. Robinson, J. Chem. Thermodyn. 4 Ž1972. 381–390. A.B. Zdanovskii, Trudy Solyanoi Laboratorii, Akad. Nauk. SSSR, No. 6, 1936. H. Chen, J. Sangster, T.T. Teng, F. Lenzi, Can. J. Chem. Eng. 51 Ž1973. 234–242.