Experimental measurement and modelling of KBr solubility in water, methanol, ethanol, and its binary mixed solvents at different temperatures

J. Chem. Thermodynamics 2002, 34, 337–360 doi:10.1006/jcht.2001.0856 Available online at http://www.idealibrary.com on

Experimental measurement and modelling of KBr solubility in water, methanol, ethanol, and its binary mixed solvents at different temperatures S. P. Pinhoa and E. A. Macedob Laboratory of Separation and Reaction Engineering, Departamento de Engenharia Qu´ımica, Faculdade de Engenharia, Rua Dr Roberto Frias, 4200 - 465, Porto Portugal

A simple and accurate apparatus has been designed to measure the solubilities of potassium bromide by an analytical method. Salt solubility data have been measured in water, methanol, ethanol, (water + methanol), (water + ethanol), and (methanol + ethanol) solvents in the temperature range between 298.15 K and 353.15 K. A new formulation is presented for the calculation of salt solubility in pure and mixed solvents as a function of the temperature and solvent composition. This formulation is based on the symmetric convention for the normalization of the activity coefficients for all species in solution, and makes possible direct access to the solubility product of the salt in terms of its thermodynamic properties. The new solubility data measured in this work, as well as experimental information from the open literature, are used to estimate the interaction parameters of the two models proposed here. One model combines the original Universal Quasi Chemical (UNIQUAC) equation with a Pitzer–Debye–H¨uckel expression to take into account the long-range interaction forces; the other model only considers the short-range forces through the UNIQUAC equation with linear temperature dependent salt/solvent interaction parameters. Both models correlate satisfactorily the solubility data, although temperature and electrostatic effects are both very important in this type of equilibrium. Finally, some conclusions are drawn concerning the models versatility to represent other type of equilibrium data and prediction capabilities. c 2002 Published by Elsevier Science Ltd.

KEYWORDS: salt; solubility; mixed solvents; UNIQUAC; modelling

1. Introduction The study of phase equilibria in electrolyte systems and more specifically the determination of salt solubilities are extremely important, either from a scientific or an industrial point of view. Besides the many relevant indications that can be given concerning the liquid a Author to whom correspondence should be addressed. Escola Superior de Tecnologia e Gest˜ao, Instituto Polit´ecnico de Braganc¸a, Portugal (E-mail: [email protected]). b E-mail: [email protected]

0021–9614/02

c 2002 Published by Elsevier Science Ltd.

338

S. P. Pinho and E. A. Macedo

phase structure and its thermodynamic properties, it is also very useful as a support for the design and simulation of unit operations such as crystallization, extractive distillation and liquid–liquid extraction. (1, 2) In fact, the solubility of salts can give important information about possible or probable solution structures and have been used along with appropriate classical thermodynamics theory, to deduce Gibbs free energies of transfer from a reference pure solvent to mixed solvents. (3–5) On the other hand, to design processes in order to find the best operational conditions, the variables such as temperature and solvent composition should be included. These data together with other thermodynamic properties may be used to provide a methodology to correlate and/or predict salt solubilities. Generally, modelling processes with electrolytes is not a routine and easy task, and problems that usually arise in the modelling of (solid + liquid) equilibrium (SLE) are numerous. (6, 7) Besides the different concentration scales and standard states possible to adopt, the major restrictive factors to establish an accurate model for SLE calculations include the high complexity of physical and chemical phenomena that might occur in the liquid phase such as solvation or association, (1) the lack of available and accurate data, and the broad range of the salt composition. Therefore, it is not surprising that, up to now, only a few studies have been carried out in this area. Lorimer (5) developed a method based on the unsymmetric convention for the normalization of the activity coefficients, which introduces enormous difficulties in the solubility calculations due to the Gibbs free energy of transfer that must be known. Also during 1993, Chiavone-Filho (8) combined the UNIQUAC equation with temperature dependent water/ion parameters, with a Pitzer–Debye–H¨uckel (PDH) expression to take into account, respectively, the short and long-range interaction forces. Since each system was correlated separately, the results were very satisfactory, with deviations between 1.2 and 7.4 per cent. Finally, Kolker and de Pablo, (9, 10) without using any ternary data, predicted SLE in mixed solvents with a non-random two liquid (NRTL) based model, achieving a reasonable agreement with the experimental data. The accuracy of the prediction varies from one system to another and the effect of temperature on the solubilities is not considered, since the study was only carried out at T = 298.15 K. However, a systematic fundamental study has not been tried in the above mentioned studies. Thus, in this work, a simple and accurate apparatus for the measurement of salt solubility by an analytical method is presented. The solubility of KBr has been measured in the solvents water, methanol, ethanol, (water + methanol), (water + ethanol), and (methanol + ethanol) at different temperatures. Based on literature and measured data, the correlation and prediction capabilities of two models are studied. The solubilities can be calculated with an average absolute deviation around 3.6 per cent, achieving very good results for the prediction of the KBr solubility in ternary mixed solvents.

2. Experimental work After an extensive literature search based on the compilation books by Stephen and Stephen. (11, 12) Linke and Seidell, (13, 14) and the open literature, it is possible to conclude that aqueous electrolyte systems have received considerable attention. However, for organic solvent/salt or mixed solvent/salt systems, there is a great lack of experimental

KBr solubility in several solvents

339

2.8

KBr solubility/mass per cent

2.6

2.4

2.2

2.0

1.8

1.6 270

280

290

300

310

320

330

340

T/K

FIGURE 1. KBr solubility in methanol plotted against temperature:
, Germuth. (16)

•,

Lloyd et al.; (15)

information. For binary systems, it is possible to determine the influence of temperature on the solubility, but discrepancies like those shown in figure 1 are usually detected. (15, 16) Thus, if the solubility of a salt is needed, even for the most common solvents like methanol or ethanol, reliable information is usually not available, but luckily some rare comprehensive studies like the one by Stenger (17) may be found. Stenger (17) published the data for solubility of several salts in methanol at T = 298.15 K, and concluded that some experimental determinations should be carried out either to check some values or to provide new information. For mixed solvent systems, the available solubility data are also very scarce and most of these were published before 1950. The influence of temperature on the solubility is almost ignored since the majority of the published data are at T = 298.15 K. Therefore, an experimental programme was implemented to carry out the solubility measurements.

MATERIALS

In all experiments, distilled–deionized water was used. All the other chemicals were supplied by Merck; the salt KBr, with a mass fraction purity higher than 0.995, and the solvents, methanol and ethanol, with a minimum purity of 0.998, were employed with no further purification. It should be mentioned that to avoid the water salt contamination, salts were dried before use at T = 393.15 K in a drying oven for a period longer than 2 days.

340

S. P. Pinho and E. A. Macedo

6 7 9

Legend: 1. Solution and salt

4

2. Magnetic bar 3. Magnetic stirrer 1

4. Jacket 5

5. Insulation 6. Thermometer 7. Glass stopper 8. Thermostated water entry

2

9. Thermostated water exit 8 3

FIGURE 2. Cell for solubility measurements.

APPARATUS

The jacketed cell for the equilibrium measurements is based on the one by Chiavone-Filho and Rasmussen. (18) which is appropriate for SLE in electrolyte and non-electrolyte systems. The cell volume is about 120 cm3 with i.d. = 5 cm and height = 6.5 cm, enabling large volume samples without dragging solid salt from the bottom. The top of the cell has two vertical orifices with diameter equal to 1 cm. One is for the mercury thermometer to measure the solution temperature and the other to take the samples. The jacketed glass cell is shown in figure 2. The solution temperature is determined by a mercury thermometer (Amarell Precision) with 0.1 K resolution and calibrated every 4 months, over the temperature range 293 K to 363 K with the estimated accuracy ±0.1 K. The heating water to the cell jacket was controlled at constant temperature within ±0.005 K in a thermostated water bath and fed at a rate of 10 dm3 · min−1 . The phase mixing was achieved using a magnetic stirrer at a speed around 600 rpm, which ensured that the contact between the solid and liquid phases was established, without breaking the crystals, and preventing the formation of micro crystals and subsequent supersaturation.

KBr solubility in several solvents

341

PROCEDURE

Desired amounts of each solvent, starting with the less volatile one, were weighed in a balloon-flask, using a 0.1 mg precision electronic balance (model A 200 S, Sartorius) to prepare approximately 110 g of solvent mixture. The dried salt was quickly introduced to the cell with a small excess over the expected solubility limit. Immediately, the cell was charged with at least 75 g of the prepared solvent, the magnetic bar introduced, and the thermometer placed in one of the top orifices. The cell was then closed and the stoppers wrapped with laboratory film to prevent solvent evaporation. The solution in the cell was heated and its temperature controlled by circulating thermostated water in the jacket of the cell. To avoid the formation of micro crystals, normally the temperature was first set slightly above that of the equilibrium temperature. A cooling cycle was used for the isotherms at 323.15 K and 298.15 K, that is, the temperature was first set at T = 325.15 K and then the solubilities were measured at T = 323.15 K, and T = 298.15 K. It should be noticed that when doing a cooling cycle after the first sampling (at T = 323.15 K) fresh solvent mixture already prepared was added to the cell in order to decrease the amount of salt in the solid phase as well as the volume of the vapour phase over the liquid. The solubilities at T = 348.15 K were measured separately to avoid large solvent composition changes during the cooling cycle. Stirring lasted for 3 hours at the working temperature. The magnetic stirrer was then turned off and the equilibrium saturated solution allowed to settle for at least 1/2 h before sampling. For each determination, usually 3 to 4 samples of approximately 5 cm3 each were withdrawn from the saturated solution using a stainless steel heated jacket, where the syringe was fitted, in order to preheat it at the desired temperature. The samples of the saturated solution were then inserted into glass vessels (25 cm3 ) with a ground-in glass stopper and immediately weighed. The total solvent evaporation was achieved in two stages. Initially, the samples were placed on a heating plate, at a temperature lower than the boiling temperature of the most volatile component, avoiding the dragging of small particles of salt. The process enhanced the formation of salt crystals in the glass vessel, which were then completely dried in a drying oven (Memmert) at T = 393.15 K. The glass vessels remained in the drying oven for periods longer than 3 days, and then cooled in a drier with silica gel for one day. Finally, they were weighed and the process regularly repeated until a constant value was achieved. Each experimental data point is an average of at least three different measurements obeying one of the following criteria. If the solubility is higher than 10 per cent by mass, the quotient 2s/solubility ∗100, should be lower than 0.1. The standard deviation (s), within a set of different experimental results is defined as, " #1/2 n X S = 1/(n − 1) (xi − x)2 , (1) i=1

where xi is the experimental solubility of sample i and x is the arithmetical mean of n experimental results. If the experimental solubility is less than 10 per cent, this criterion is difficult to attain and in this case an equivalent criterion is that the standard deviation should be lower than 0.005.

342

S. P. Pinho and E. A. Macedo TABLE 1. Solubility (grams of salt per 100 grams of saturated solution) of KBr in water, methanol, and ethanol at different temperatures T/K

Water

Methanol

Ethanol

298.15

40.713

2.063

0.135

303.15

41.670

2.150

313.15

43.359

2.324

323.15

44.932

2.503

333.15

46.360

2.672

343.15

47.725

348.15

48.349

353.15

48.961

0.185

0.232

TABLE 2. Solubility of KBr (grams of salt per 100 grams of saturated solution) in (water + methanol), and (methanol + ethanol) solvents. The mixed solvent composition 0 0 is expressed as water (wwater ) or methanol (wmethanol ) mass fraction on a salt free basis Water + methanol

Methanol + ethanol

Salt solubility

Salt solubility

0 wwater

T = 298.15 K

T = 323.15 K

0.1000

3.124

4.258

0.2000

5.206

7.303

0.3000

8.350

11.484

0.4000

12.291

16.326

0.5000

16.662

21.424

0.6000

21.334

26.506

0.7000

26.131

31.435

0.8000

31.033

36.210

0.9000

35.833

40.660

0 wmethanol

T = 298.15 K

T = 323.15 K

0.2060

0.286

0.369

0.3999

0.513

0.652

0.6025

0.892

1.124

0.8001

1.386

1.721

SOLUBILITY DATA

The measured solubilities in single solvent systems are reported in table 1. The water/KBr system was studied over the temperature range between 298.15 K and 353.15 K, while for methanol/KBr, the maximum temperature was 333.15 K. The ethanol/KBr system was studied at three different temperatures (298.15, 323.15, and 348.15) K since the salt solubilities are too low and the temperature dependency is not pronounced. The measured solubilities of KBr in (water + methanol), and (methanol + ethanol) mixed solvents at T = 298.15 K, and T = 323.15 K, are given in table 2. Table 3

KBr solubility in several solvents

343

TABLE 3. Solubility of KBr (grams of salt per 100 grams of saturated solution) in (water + ethanol) solvents. The mixed solvent composition is expressed as water mass fraction on a salt 0 free basis (wwater ) Water + ethanol Salt solubility

Salt solubility

0 wwater

T = 298.15 K

T = 323.15 K

0 wwater

T = 348.15 K

0.1000

0.734

1.141

0.1000

1.593

0.2000

2.678

4.010

0.2002

5.430

0.3000

6.112

8.671

0.3000

11.060

0.4000

10.374

14.043

0.3997

17.262

0.5000

14.997

19.542

0.4995

23.317

0.5999

19.370

24.908

0.6000

29.023

0.7000

24.574

30.016

0.6992

34.124

0.8000

29.656

35.015

0.8006

39.238

0.9000

35.121

39.991

0.9000

43.929

reports the solubilities of KBr in (water + ethanol) mixed solvents at the temperatures (298.15, 323.15, and 348.15) K. The solubility as for the binary systems, is expressed as 0 mass per cent, while the solvent composition is expressed as water (wwater ) or methanol 0 (wmethanol ) mass fraction on a salt free basis.

3. Analysis of the data The quality of the measured data may be investigated comparing it with literature values reported in the compilation books published by Stephen and Stephen. (11, 12) Linke and Seidell, (13, 14) and in the open literature: for the water/KBr system the comparison can be easily done. However, for the other systems, data at temperatures different from 298.15 K are not available which makes difficult or even impossible the confrontation of the obtained experimental results. So, a comparison is only possible for the solubility of KBr in water, and in (water + ethanol) mixed solvent at T = 298.15 K. In figure 3 it is possible to observe the good agreement between the solubility of KBr in water obtained in this work and those reported by Linke and Seidell. (13) As shown in figure 4, for the solubility of KBr in (water + ethanol) at T = 298.15 K a good resemblance between the data obtained in this work, and those reported by Delesalle and Heubel (19) is observed in all the solvent composition range.

4. Solid–liquid equilibrium modelling The common practice in electrolyte thermodynamics in the use, for the ions, of the unsymmetric convention for the normalization of the activity coefficients, which for the

344

S. P. Pinho and E. A. Macedo 50

KBr solubility/mass per cent

48

46

44

42

40

38 290

300

310

320

330

340

350

360

T/K

FIGURE 3. KBr solubility in water plotted against temperature: work.

•, Linke and Seidell; (13)
, this

45

KBr solubility/mass per cent

40 35 30 25 20 15 10 5 0 0.0

0.2

0.4

0.6

0.8

1.0

Water mass fraction (salt free basis)

FIGURE 4. Plot of KBr solubility in (water + methanol) plotted against mass fraction of water at T = 298.15 K: , Delesalle and Heubel; (19)
, this work.

•

KBr solubility in several solvents

345

representation of (vapour + liquid) equilibrium (VLE) does not introduce any difficulties, since the equilibrium criteria only involves the solvent species. For SLE calculations, using this unsymmetric convention, a major difficulty arises from the fact that the salt standard chemical potential in the liquid phase, which depends on the mixed solvent composition, should be known. However, no suitable models for that property exist yet and so, to overcome that problem, a new formulation based on the symmetric convention for all species in solution and in the ionized mole fraction basis (20, 21) for the concentration scale is presented here. The salt solubility product (K salt ) is defined as K salt = (Q f ± xsalt /v)v .

(2)

In equation (2) xsalt is the salt mole fraction on ionized basis, f ± the mean ionic rational activity coefficient, v the sum of the anion and cation stoichiometric coefficients, and Q a constant related to those coefficients. Thus, the salt solubility calculation is possible if an activity coefficient model and K salt are available. In order to obtain K salt , a thermodynamic cycle was idealized between the pure solid and liquid salt phase states which, after some assumptions, leads to the following expression: (22) K salt = exp{1Hf /R(1/Tf − 1/T ) + 1c p f /R[Tf /T − 1 + ln(T /Tf )]}.

(3)

The solubility product is related to some thermodynamic properties of the salt such as the melting temperature (Tf ), the enthalpy of fusion (1Hf ) at Tf , and the change of heat capacity (1C p f ), between the liquid and solid state also at Tf , defined by: 1C p f = C p (Tf )liquid − C p (Tf )solid ,

(4)

Equation (3) has been widely applied for the representation of solid–liquid equilibrium of non-electrolyte systems, namely of binary hydrocarbon mixtures, (23) hydrocarbons in mixed solvents, (22, 24) and sugars in pure and mixed solvents, (25–28) Kolker (29, 30) has been well succeeded in the calculation of salt solubility in water, using this procedure, but has pointed out the high sensitivity on the heat of fusion for the obtained results. Concerning the activity coefficient models the two approaches proposed are following presented. UNIQUAC + PDH MODEL

The most widely used models to represent VLE in mixed solvents electrolyte systems semi-empirical. Thus, in this work a semi-empirical model based on the assumption that the symmetric molar excess Gibbs free energy (G E ) of the system on mole fraction basis is a linear combination of two terms is proposed: G E = G EUNIQUAC + G EPDH .

(5)

The original UNIQUAC model has been chosen due to the accurate results obtained for the description of the short-range forces in non-electrolyte systems, (22, 31) as well as for the representation of VLE in mixed solvent electrolyte systems. (32–34) Furthermore, it is possible to extend it to multicomponent systems or to a group-contribution method. To account for the long-range forces a PDH equation is used. Pitzer, (21) and Pitzer and Simonson, (35) and Simonson and Pitzer (36) have developed proper equations to calculate

346

S. P. Pinho and E. A. Macedo

the activity coefficients of all species in the symmetric convention and ionized basis mole fraction scale, as used in this work. Though the published studies concerned only aqueous systems, those authors suggest the study with appropriate modifications of the capabilities of this model for mixed solvent systems. From equation (5), the rational symmetric activity coefficient for any species, ion or solvent, can be derived as: UNIQUAC

ln f i = ln f i

+ ln f iPDH .

(6) UNIQUAC

The UNIQUAC model (32) calculates the activity coefficient of species i( f i the mole fraction scale and symmetric convention, according to: UNIQUAC

ln f i

= ln f iC + ln f iR .

) in (7)

The combinatorial term f iC , representing the differences in size and shape of the species in the system, is the same as in the original equation: Nspec

ln f iC = ln(φi /xi ) + 5qi ln(θi /φi ) + li − φi /xi

X

x jl j

(8)

j=1

where, li = 5(ri − qi ) − (ri − 1),

(9)

the volume fraction (φi ) and surface area fraction (θi ) of component i are calculated with the relationships: Nspec

φi = ri xi /

X

rjxj,

(10)

qjxj.

(11)

j=1 Nspec

θi = qi xi /

X j=1

In these equations, Nspec refers to the total number of species in the solution, and pure molecular or ion parameters ri and qi are, respectively, measures of molecular van der Waals volumes and molecular surface areas. The residual term f iR , takes short-range molecular energetic interactions into account, and is expressed as: " ! Nspec !# Nspec Nspec X X X R ln f i = qi 1 − ln θ j τ ji − θi τi j / θk τk j . (12) j=1

j=1

k=1

The parameter τ is given by: τi j = exp(−ai j /T ),

(13)

where ai j is the UNIQUAC interaction parameter between the species i and j. Since the symmetric convention is used for all the components, a conversion is necessary when using the UNIQUAC model. In fact, for the salt, as the ion activity coefficients are

KBr solubility in several solvents

347

first evaluated, the calculation of their activity coefficients in a pure salt system should be performed. The conversion can be expressed as: UNIQUAC

ln f i (x, T ) = ln f i

UNIQUAC

(x, T ) − ln f i

(vi /v, T ).

(14)

This last equation requires some attention. For an ionic species i, the normalized activity coefficient, f i , will be equal to one when xsalt is unity or in terms of ions composition when xi is equal to the ratio vi /v. So, for a multicomponent mixture with a mole fraction vector x and temperature T , the ion activity coefficient should be calculated using equation (14). The PDH contribution presented in equation (6) is given as: p 1/2 1/2 ln f iPDH = −z i2 A D H,x {2/b ln[(1+bI x )/(1+b(I x∇ )1/2 )]+ I x (1− I x /I x∇ )/(1+bI x )}, (15) where, A D H,x = (2∗ NA )1/2 /(24π)(e2 /εo k)3/2 ρ 1/2 /(εT )3/2 = 4.4316 · 104 ρ 1/2 /(εT )3/2 . (16) In equation (15) and (16), b is a model parameter related to the closest approach of the ions. NA is Avogadro’s number, e is the electronic charge (C), εo , is the vacuum permittivity (C2 · J−1 · m−1 ), k is the Boltzmann constant (J−1 · K−1 ). Finally, ρ and ε are, respectively, the molar density (mol · m−3 ) and the dielectric constant of the solvent. The ionic strength (I x ) is defined according to: I x = 0.5

Nion X

zl2 xl ,

(17)

l=i

where Nion is the number of different ionic species in solution. For electrolytes, like the ones studied in this work, with unitary charge for each ion: I x∇ = z i2 /2.

(18)

For a solvent m, the charge number is zero and equation (15) reduces to 3/2

1/2

ln f mPDH = A D H,x 2I x /(1 + bI x ).

(19)

It should be noticed that expressions (15) and (19) are obtained by proper differentiation of the excess Gibbs energy, G EPDH , which in the original paper was developed for single solvent systems. Similarly to Koh et al. (37) Chen and Evans, (38) and Achard et al. (39) in this work, the differentiation of the parameter A D H,x which is solvent composition dependent through the density and the dielectric constant of the solvent has been neglected. The values of that parameter are, however, evaluated as a function of the solvent composition. While this procedure has some degree of inconsistency, the quality of the results obtained for the correlations, predictions and hence industrial applications, support the approach taken. The use of this activity coefficient model requires the knowledge of the temperature and composition dependence of the PDH parameter A D H,x according to equation (16), the distance of closest approach parameter b, the structural parameters ri and qi , of solvent and ions, and finally the interaction parameters between the different molecules or ions present in the solution.

348

S. P. Pinho and E. A. Macedo

The temperature and composition dependency of the A D H,x parameter is considered via equation (16), using different methods to estimate the density and the dielectric constant of the solvent. (A) DENSITY OF THE PURE AND MIXED SOLVENT MIXTURES

The molar density of the mixed solvent ρms can be calculated based on the density of its constituents pure solvents densities (ρm∗ ), which may be found on the DIPPR Tables. (40) according to the empirical equation: ρms = 1/

N solv X

xm0 /ρm∗ ,

(20)

m=1

where xm0 is the salt-free mole fraction of solvent m in the liquid phase. (B) DIELECTRIC CONSTANT OF THE PURE AND MIXED SOLVENT MIXTURES

The dielectric constant of a solvent mixture εms is obtained from the pure solvent values (εm ) given by Maryott and Smith. (41) For a binary mixed solvent mixture, constituted by solvents 1 and 2, the Oster’s mixing rule (42) is adopted: εms = ε1 + {(ε2 − 1)(2ε2 + 1)/(2ε2 ) − (ε1 − 1)}ϕ2 ,

(21)

where ϕ2 is the solvent salt free basis volumetric fraction, defined for any solvent m: ϕm = xm ρms /ρm∗ .

(22)

For a mixture with more than two solvents, the dielectric constant of the mixed solvent is estimated according to: (34) εms =

N solv X

εm ϕm .

(23)

m=1

The parameter b depends on the short-range model used and on the solvent composition, and may be considered as a parameter to estimate during the experimental data fitting. However, in this work a constant value of 14.9 was used. This value was also set by several authors for the study of phase equilibria in aqueous, (29, 35) and mixed solvent electrolyte systems with the NRTL equation. (38, 43) Achard et al. (39) using the Universal Functional Activity Coefficient (UNIFAC) model by Larsen et al. (44) have given a slightly higher value for this parameter (b = 17.1). The structural parameters ri and qi are the measures of the van der Waals volumes and surface areas, respectively. For molecules and organic ions, they are usually estimated with the method proposed by Bondi, (45) and for the other ions, based on the molecular size of the ions. However, the very small values of the ionic radii for the cations lead to q values, in the range 0.1–0.5. Sander et al. (32) have concluded that values of this order of magnitude reduce the fitting capabilities of the UNIQUAC equation, and therefore r and q can be treated as adjustable parameters. In order to reduce the number of parameters to estimate in this work, the fixed values used by Macedo et al. (33) and Kikic et al. (46) with, respectively, the UNIQUAC and UNIFAC models for VLE in mixed solvent are applied.

KBr solubility in several solvents

349

UNIQUAC MODEL

In order to test the capabilities of the UNIQUAC equation and to perform comparisons with the previous model, the use of the UNIQUAC equation alone, considering the salt in the molecular form is also suggested. Thus, for this case the rational symmetric activity coefficient for any species, salt or solvent, can be defined as UNIQUAC

ln f i = ln f i

.

(24)

UNIQUAC

The equations used to calculate f i already presented for the UNIQUAC + PDH model still hold for this case. The only change is the introduction of a temperature dependency for the interaction parameters between the salt (l) and the solvent (m) in accordance to: o t alm = alm + alm (T − 298.15),

(25)

o t aml = aml + aml (T − 298.15),

(26)

and where the superscripts o and t refer, respectively, to the reference interaction parameter and to the temperature dependent parameter. This linear temperature dependence is applied only if the temperature interval of the available experimental data is at least 50 K. The structural parameters ri and qi , of solvents and salts, and finally the interaction parameters between the different species present in the solution are the only requirements for the application of this model. While for the solvents the structural parameters are the same as indicated before for the UNIQUAC + PDH model, (33, 46) for the salts they were obtained from Dahl and Macedo. (47)

5. Parameter estimation To estimate the energy interaction parameters for the two proposed models, a modified Levenberg–Marquardt method (48, 49) was used to minimize the following objective function (OBJ): OBJ =

N SLE X

exp

exp

calc ωd [(xsalt,p − xsalt,p )/xsalt,p ]2 +

p=1

NX VLE

exp

exp

ωd [(φpcalc − φp )/φp ]2 ,

(27)

p=1

where calc and exp mean the experimental and calculated values according to the model, NSLE and NVLE are, respectively, the total number of experimental SLE and VLE data points, φ is the osmotic coefficient, ωd is a weighting factor for data set number d, and p refers to the experimental point. To calculate the salt solubility using the developed methodology, it is useful to rewrite equation (2) in a more explicit way: 1/v

calc xsalt = v K salt /(Q f ± ).

(28)

From equation (28), the salt solubility calculation involves an iterative procedure for

350

S. P. Pinho and E. A. Macedo

the system composition, since it depends on the activity coefficients, which in turn are composition dependent. To be able to perform the calculations, the KBr thermodynamic properties, that must be known, were taken from the JANAF thermochemical tables. (50) A reliable database on salt solubility and osmotic coefficient data was built to allow the estimation of the relevant energy interaction parameters needed. The solubility data used includes the experimental results obtained in this work, as well as those selected from the open literature. To accept a solubility data set, since there is no consistency test for SLE, different procedure can be implemented. In this work, two coherence tests were used. One involves a comparison between the accepted binary salt solubility values with those reported for mixed solvent sources. The second involves the estimation of the quality of the salt solubility isotherms along the solvent composition making a curve fit with an empirical equation and analyse the deviations from the curve. For (water + KBr), 26 solubility data points, and 118 osmotic coefficients (51, 52) data points in the temperature range between 273.15 K and 373.15 K, covering the concentration range from zero up to saturation, were used to estimate the interaction parameters for each model proposed here: four t t o o parameters (awater,KBr , aKBr,water , awater,KBr , aKBr,water ) for the UNIQUAC model, and for the UNIQUAC + PDH model, 4 water/ion parameters (awater,K+ , awater,Br− , aK+ ,water , and aBr− ,water ) and 2 cation/anion parameters (aK+ ,Br− , aBr− ,K+ ). It is important to mention that to fit the data points selected, the weighting factors ωd have been set equal to 1 for salt solubility data, and 0.2 for the osmotic coefficients. This was done because the main purpose of this work is the correlation and prediction of salt solubility in mixed solvents. Accordingly, for the experimental VLE data used in the parameter estimation, for which the number of experimental data points is much higher than for the solubilities, a low weight factor was used. For the (water + methanol + KBr), 43 solubilities data points were collected and used o o to estimate the parameters (amethanol,KBr , aKBr,methanol ) for the UNIQUAC model (no temperature dependency was introduced, because of the narrow temperature range), and for the UNIQUAC + PDH model, 4 methanol/ion parameters (amethanol,K+ , amethanol,Br− , aK+ ,methanol , aBr− ,methanol ). Additionally, the two interaction parameters between the solvents water and methanol were also estimated for both models. The other relevant parameters to represent the mixed solvent systems involving ethanol were obtained regressing the 72 data points collected for the (water + ethanol) and (methanol + ethanol) mixed solvents: besides the four interaction parameters between the solvents (awater,ethanol , aethanol,water , amethanol,ethanol , aethanol,methanol ) needed for both models, 4 interaction t t o o parameters (aethanol,KBr , aKBr,ethanol , aethanol,KBr , aKBr,ethanol ) were also estimated for the UNIQUAC model, and 4 ethanol/ion parameters (aethanol,K+ , aethanol,Br− , aK+ ,ethanol , aBr− ,ethanol ) were regressed for the UNIQUAC + PDH model. As a matter of convenience, the experimental data for the solubility of KBr in pure methanol and ethanol were included, respectively, in the (water + methanol) and (water + ethanol) mixed solvents data sets. For systems involving organic solvents, no experimental VLE information was included in the database, since it turned out during the minimization process that it was difficult to represent simultaneously, with high accuracy, both types of data.

KBr solubility in several solvents

351

TABLE 4. New interaction parameters a o (K) and a t for the UNIQUAC model Water Water

Methanol

Ethanol

1013

134.9

0.0

KBr −131.5 0.7010t

Methanol

−258.7

0.0

Ethanol

−346.7

57.13

−131.6

−113.2

0.0

144.4 1.878t

KBr

66.67

1002

104.3

−0.2137a

0.0

−1.038a

a According to equations (25), and (26).

TABLE 5. New interaction parameters (K) for the UNIQUAC + PDH model Water Water

Methanol

K+

Br−

−693.5

−444.1

Ethanol

0.0

−648.3

Methanol

−169.3

0.0

Ethanol

−274.7

−59.68

K+

−355.8

1892

2049

Br−

−340.5

−505.6

8568

−1073 −42.38 0.0

1281

−1336

311.4

−1501 0.0

14.77 0.0

−1650

6. Results and discussion The new estimated interaction parameters for the UNIQUAC and the UNIQUAC + PDH models are listed, respectively, in tables 4 and 5. The quality of the correlations and a comparison between the performance of the proposed models can be made by calculating the absolute average deviation (AAD) AAD = 100/Ndata

N data X

exp

exp

|(ypcalc − yp )/yp |,

(29)

p=1

where y represents the thermodynamic property under study, the salt solubility or the osmotic coefficient, and Ndata is the total number of experimental points for each property. Table 6 summarizes the AAD values obtained in the correlation of both salt solubility and osmotic coefficients with the UNIQUAC and the UNIQUAC + PDH models. Concerning the (water + KBr), from the results shown in table 6, it is possible to conclude that while the UNIQUAC model is slightly superior in the representation of salt

352

S. P. Pinho and E. A. Macedo TABLE 6. AAD, and number of parameters estimated in the correlation of salt solubility, and osmotic coefficients with the UNIQUAC and UNIQUAC + PDH models UNIQUAC AAD Parameters Per cent φ

12.20

Solubility

2.22

UNIQUAC + PDH AAD Parameters Per cent 2.61

Water

4

Water + methanol

4.03

Water + ethanol

3.55

Methanol + ethanol

5.06

6 2.60

4

3.02

6

3.98 8

8 2.51

55

KBr solubility/mass per cent

50

45

40

35

30 270

290

310

330 T/K

350

370

FIGURE 5. Solubility of KBr in water plotted against temperature: comparison between the experimental data and model curves. , Linke and Seidell; (13)
, this work: ——, UNIQUAC; ······, UNIQUAC + PDH.

•

solubility, the UNIQUAC + PDH model is much more accurate for the correlation of osmotic coefficient data. Despite the fact that two more parameters have been estimated for the UNIQUAC + PDH model, it should be pointed out that the temperature influence on the solubility and osmotic coefficients is considered directly in the UNIQUAC model through the a t parameters, while for the other model the temperature influence on the

KBr solubility in several solvents

353

1.5 1.4

Osmotic coefficient

1.3 1.2 1.1 1.0 0.9 0.8 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Molality/mol . Kg –1

FIGURE 6. Correlated and experimental osmotic coefficients in (water + KBr) plotted against molality at T = 298.15 K: , Hamer and Wu; (52) ——, UNIQUAC; · · · · ··, UNIQUAC + PDH.

•

properties is only considered in the change of the solvent density and dielectric constant. The good quality for the salt solubility correlation can be observed in figure 5. On the other hand, the importance of a PDH type expression in the representation of the electrostatic forces in electrolyte systems is completely shown. For very diluted solutions these forces are dominant (22, 51) and the UNIQUAC + PDH model can, as shown in figure 6, accurately represent the osmotic coefficients in that concentration region. For the mixed solvent systems from table 6, it is possible to conclude that both models represent with similar accuracy the salt solubility data, although for (water + methanol) and (methanol + ethanol) mixed solvents, the UNIQUAC + PDH model is superior. The average accuracy is around 3.9 per cent. In figure 7 for the (water + methanol) mixed solvent, the experimental solubility of KBr is compared with the calculated values with the models at the two different temperatures studied. The UNIQUAC + PDH model presents a more reliable description in all the solvent composition range. Shown in figure 8 is a comparison between the experimental solubilities obtained in this work and those calculated using the models for the (water + ethanol) mixed solvent. Again, both models represent well the experimental curves. The same number of parameters were estimated for both models. Actually, without using any direct temperature dependency, the UNIQUAC + PDH model describes with an accuracy similar to the UNIQUAC model, the temperature influence on the solubility for (water + ethanol) mixed solvent. This is in part due to the fact that the PDH term accounts for the electrostatic interactions with the correct temperature and composition dependency. Moreover, it should be remembered that to represent the solubility in (water + ethanol), data were correlated simultaneously with

354

S. P. Pinho and E. A. Macedo 50

KBr solubility/mass per cent

40

30

20

10

0 0.0

0.2

0.4

0.6

0.8

1.0

Water mass fraction (salt free basis)

•

FIGURE 7. Solubility of KBr in (water + methanol) plotted against mass fraction of water: , T = 298.15 K and
, T = 323.15 K measured in this study; ——, UNIQUAC; · · · · ··, UNIQUAC + PDH.

50

KBr solubility/mass per cent

40

30

20

10

0 0.0

0.2

0.4

0.6

0.8

1.0

Water mass fraction (salt free basis)

•

FIGURE 8. Solubility of KBr in (water + ethanol) plotted against mass fraction of water: , T = 298.15 K;
, T = 323.15 K and , T = 348.15 K measured in this study; ——, UNIQUAC; · · · · ··, UNIQUAC + PDH.

KBr solubility in several solvents

355

3.0

KBr solubility/mass per cent

2.5

2.0

1.5

1.0

0.5

0.0 0.0

0.2

0.4 0.6 0.8 Methanol mass fraction (salt free basis)

1.0

•

FIGURE 9. Solubility of KBr in (methanol + ethanol) plotted against mass fraction of methanol: , T = 298.15 K and
, T = 323.15 K measured in this study; ——, UNIQUAC; · · · · ··, UNIQUAC + PDH.

the experimental results for (methanol + ethanol), and better results were found using the UNIQUAC + PDH model as can be seen in figure 9. For the (water + ethanol + KBr), reliable experimental information was found at the same solvent compositions of the present study, but at two different temperatures, 303.15 K and 313.15 K. (12) A comparison of both models in the description of the temperature effect on the solubility is shown in figures 10 and 11, at a fixed solvent composition. The results for both models are very satisfactory. Absolute comparisons with other approaches are difficult to carry out. None of the studies known so far, have performed such a systematic work on the solubility of salts in mixed solvents. Thus, a comparison is presented in terms of the number of parameters necessary to correlate this type of equilibria, using an empirical modified Setschenow equation of the form 0 0 ln xsalt = λ1 + λ2 T + (λ3 + λ4 T )xwater + (λ5 + λ6 T )(xwater )2 .

(30)

where λ1 , λ2 , . . . , are the empirical parameters to be fitted. Using only six parameters in the Setschenow equation, it is possible to represent with the desired accuracy the solubility of KBr in mixed solvents. To compare the performances of the two proposed models against the Setschenow equation, the solubility data collected for (water + ethanol + KBr) were fitted. The resulting expression is: 0 ln xsalt = −9.4 + 8.915 · 10−3 T + (1.732 + 2.045 · 10−2 T )xwater + 0 (4.712 − 2.546 · 10−2 T )(xwater )2 .

(31)

356

S. P. Pinho and E. A. Macedo 45

KBr solubility/mass per cent

40 35 W’ (H2O) = 0.9

30 W’ (H2O) = 0.8

25 W’ (H2O) = 0.7

20 W’ (H2O) = 0.6

15 W’ (H2O) = 0.5

10 290

300

310

320

330

340

350

T/K

FIGURE 10. Solubility of KBr in (water + ethanol) plotted against temperature: correlation capabilities of the UNIQUAC and UNIQUAC + PDH models. w’(H2 O) is the water mass fraction (salt free basis) in the mixed solvent. , Stephen and Stephen; (12)
, this work; ——, UNIQUAC; · · · · ··, UNIQUAC + PDH.

•

KBr solubility/mass per cent

20

15 W’ (H2O) = 0.4

10 W’ (H2O) = 0.3

5 W’ (H2O) = 0.2

0 290

W’ (H2O) = 0.1

300

310

320

330

340

350

T/K

FIGURE 11. Solubility of KBr in (water + ethanol) plotted against temperature: correlation capabilities of the UNIQUAC and UNIQUAC + PDH models. w’(H2 O) is the water mass fraction (salt free basis) in the mixed solvent. , Stephen and Stephen; (12)
, this work; ——, UNIQUAC; · · · · ··, UNIQUAC + PDH.

•

KBr solubility in several solvents

357

50

KBr solubility/mass per cent

40

30

20

10

0 0.0

0.2

0.4

0.6

0.8

1.0

Water mass fraction (salt free basis)

•

FIGURE 12. Solubility of KBr in (water + ethanol) plotted against mass fraction of water: , T = 298.15 K;
, T = 323.15 K and , T = 348.15 K measured in this study; ——, UNIQUAC; · · · · ··, Setschenow equation.

The results, in terms of AAD, for the UNIQUAC model, the UNIQUAC + PDH model, and the Setschenow equation are, respectively, 3.6, 4.0, and 4.7 per cent. Indeed, the UNIQUAC model represents more accurately the salt solubility diagram for the (water + ethanol + KBr). Figure 12 presents a comparison between the performances of the UNIQUAC model and the Setschenow equation in the calculation of the solubility of KBr in (water + ethanol) mixed solvent: the UNIQUAC model gives a more rigorous description of both the influence of the temperature and of the solvent composition on the KBr solubility. A stringent test of the proposed models and methodology suggested in this work to correlate and predict the salt solubility in mixed solvents, is the study of their capabilities for the description of more complex mixtures. In our experimental programme, the measurement of the solubility of KBr in (water + methanol + ethanol) solvent mixtures at T = 313.15 K was carried out. Table 7 summarizes the results obtained for the predictions. The deviations between the experimental solubilities and the calculated values with the UNIQUAC and the UNIQUAC + PDH models are also indicated. It is possible to conclude that the average deviation, for both models, is around 3.1 per cent and, therefore, the models can be used with good accuracy for prediction purposes. The good quality of the predictions are given in figure 13. The UNIQUAC + PHD model gives a better agreement over the whole solvent composition range, as indicated by the average deviations given in table 7.

358

S. P. Pinho and E. A. Macedo 50

KBr solubility/mass per cent

40

30

20

10

0 0.0

0.2

0.4

0.6

0.8

1.0

Water mass fraction (salt free basis)

FIGURE 13. Solubility of KBr in (water + methanol + ethanol) plotted against mass fraction of water at wethanol /wmethanol = 1.00: , T = 313.15 K measured in this study; ——, UNIQUAC; · · · · ··, UNIQUAC + PDH predictions.

•

TABLE 7. Experimental and predicted solubility of KBr (grams of salt per 100 grams of saturated solution) at T = 313.15 K in (water + methanol + ethanol) solvents. The mixed solvent composition is expressed as water mass 0 fraction in salt free basis (wwater ), and in the ratio between the ethanol and methanol mass fractions System composition 0 wwater

0 wethanol 0 wmethanol

UNIQUAC

UNIQUAC + PDH

Experimental

Calculated

Calculated

solubility

solubility

Errora

solubility

Errora

0.790

0.772

2.28

0.759

3.92

0.0000

1.000

0.2000

1.001

4.553

4.382

3.76

4.686

2.92

0.4000

1.001

13.277

12.236

7.84

13.828

4.15

0.5993

1.000

23.492

22.711

3.32

24.204

3.03

0.7998

1.001

33.631

33.673

0.12

33.586

0.13

Average

3.46

a error (percentage) = |(experimental − calculated)/experimental|*100.

2.83

KBr solubility in several solvents

359

7. Conclusions The simple apparatus designed for the measurement of salt solubilities in mixed solvents using an analytical method proved to be very successful. High precision and accurate results were obtained and generally the solubilities are reproducible when compared with the values reported in the literature. A new formulation for the calculation of the solubility of salts in pure and mixed solvents is presented. It involves the symmetric convention of normalization of the activity coefficients, for all species, and the mole fraction on an ionized basis. In this way, a considerable simplification is introduced since the salt solubility product can be evaluated in terms of its thermodynamic properties such as the melting temperature, the enthalpy of fusion, and the change of heat capacity, between the liquid and solid state. Two models are proposed to correlate and/or predict the salt solubilities, the UNIQUAC + PDH model and the UNIQUAC model with linear temperature dependent parameters between the salt and the solvents. Both models correlate satisfactorily and with similar accuracy the solubility of salts in mixed solvents, but it is clearly shown that temperature and electrostatic effects are both very important in this type of equilibrium calculations. The average deviation obtained is around 3.6 per cent. When correlation of salt solubility diagram of a specific system is required, the UNIQUAC model is preferred due to the better results obtained and to its simplicity. The Setschenow empirical equation, using the same number of parameters as for the two proposed models, gives poorer representation of the solubilities of salts in mixed solvents. These models and this methodology are a valuable tool for the correlation and prediction of salt solubility in mixed solvents. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Furter, W. F. Chem. Eng. Comm. 1992, 116, 35–40. Korin, E.; Soifer, L. J. Chem. Eng. Data 1997, 42, 1251–1253. Emons, H.-H.; Janneck, E.; Pollmer, K. Z. Anorg. Allg. Chem. 1984, 511, 135–147. Labban, A. K. S.; Marcus, Y. J. Solution Chem. 1991, 20, 221–232. Lorimer, J. W. Pure Appl. Chem. 1993, 65, 183–191. Perkyns, J.; Pettitt, B. M. J. Phys. Chem. 1994, 98, 5147–5151. Liu, Y.; Watanasiri, S. Chem. Eng. Progress 1999, 95, 25–42. Chiavone-Filho, O. Phase Behavior of Aqueous Glycol Ether Mixtures: (1) Vapor–Liquid Equilibria (2) Salt Solubility. Ph.D. Thesis, Department of Chemical Engineering, Technical University of Denmark, Lyngby. 1993. Kolker, A.; de Pablo, J. J. Ind. Eng. Chem. Res. 1996, 35, 228–233. Kolker, A.; de Pablo, J. J. Ind. Eng. Chem. Res. 1996, 35, 234–240. Stephen, H.; Stephen, T. Solubilities of Inorganic and Organic Compounds: Binary Systems. Pergamon Press: Oxford. 1963. Stephen, H.; Stephen, T. Solubilities of Inorganic and Organic Compounds: Ternary Systems. Pergamon Press: Oxford. 1964. Linke, W. F.; Seidell, A. Solubilities of Inorganic and Metal-organic Compounds: 4th edition. Vol. I. American Chemical Society: Washington. 1958. Linke, W. F.; Seidell, A. Solubilities of Inorganic and Metal-organic Compounds: 4th edition. Vol. II. American Chemical Society: Washington. 1965. Lloyd, E.; Brown, C. B.; Glynwyn, D.; Bonnell, R.; Jones, W. J. J. Chem. Soc. 1928, 658. Germuth, F. G. J. Franklin Inst. 1931, 212, 343–349.

360

S. P. Pinho and E. A. Macedo

17. 18. 19. 20. 21. 22.

Stenger, V. A. J. Chem. Eng. Data 1996, 41, 1111–1113. Chiavone-Filho, O.; Rasmussen, P. J. Chem. Eng. Data 1993, 38, 367–369. Delesalle, G.; Heubel, J. J. Bull. Soc. Chim. Fr. 1972, 7, 2626–2631. Rosen, A. M. Doklady Akademii Nauk SSSR 1979, 249, 135–139. Pitzer, K. S. J. Am. Chem. Soc. 1980, 102, 2902–2906. Prausnitz, J. M.; Lichenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria : 3rd edition. Prentice-Hall, Englewood Cliffs: New Jersey. 1998. Lohmann, J.; Joh, R. J.; Gmehling, J. J. Chem. Eng. Data 1997, 42, 1176–1180. Zhu, J.-Q.; Yu, Y.-S.; He, C.-H. Fluid Phase Equilib. 1999, 155, 85–94. Peres, A. M.; Macedo, E. A. Fluid Phase Equilib. 1996, 123, 71–95. Peres, A. M.; Macedo, E. A. Carbohydr. Res. 1997, 303, 135–151. Peres, A. M.; Macedo, E. A. Fluid Phase Equilib. 1997, 139, 47–74. Peres, A. M.; Macedo, E. A. Ind. Eng. Chem. Res. 1997, 36, 2816–2820. Kolker, A. R. Fluid Phase Equilib. 1992, 74, 109–125. Kolker, A. R. Fluid Phase Equilib. 1991, 69, 155–169. Abrams, D. S.; Prausnitz, J. M. AIChE J. 1975, 21, 116–128. Sander, B.; Fredenslund, Aa.; Rasmussen, P. Chem. Eng. Sci. 1986, 41, 1171–1183. Macedo, E. A.; Skovborg, P.; Rasmussen, P. Chem. Eng. Sci. 1990, 45, 875–882. Li, J.; Polka, H.-M.; Gmehling, J. Fluid Phase Equilib. 1994, 94, 89–114. Pitzer, K. S.; Simonson, J. M. J. Phys. Chem. 1986, 90, 3005–3009. Simonson, J. M.; Pitzer, K. S. J. Phys. Chem. 1986, 90, 3009–3013. Koh, D. S. P.; Khoo, K. H.; Chan, C.-Y. J. Sol. Chem. 1985, 14, 635–651. Chen, C.-C.; Evans, L. B. AIChE J. 1986, 32, 444–454. Achard, C.; Dussap, C. G.; Gros, J. B. Fluid Phase Equilib. 1994, 98, 71–89. DIPR Tables of Physical and Thermodynamic Properties of Pure Compounds. AIChE: New York. 1984. Maryott, A.; Smith, E. R. Tables of Dielectric Constants of Pure Liquids. National Bureau of Standards, Circular 514. 1951. Franks, F. Water, A Comprehensive Treatise. Plenum Press: New York. 1973. Barata, P. A.; Serrano, M. L. Fluid Phase Equilib. 1997, 141, 247–263. Larsen, B. L.; Rasmussen, P.; Fredenslund, Aa. Ind. Eng. Chem. Res. 1987, 26, 2274–2286. Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Glasses. John Wiley & Sons: New York. 1968. Kikic, I.; Fermeglia, M.; Rasmussen, P. Chem. Eng. Sci. 1991, 46, 2775–2780. Dahl, S.; Macedo, E. A. Ind. Eng. Chem. Res. 1992, 31, 1195–1201. Levenberg, K. Q. Appl. Math. 1944, 2, 164–168. Marquardt, D. W. J. Soc. Ind. Appl. Math. 1963, 11, 431–435. Chase, M. W., Jr; Davies, C. A.; Downey, J. R., Jr; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. J. Phys. Chem. Ref. Data 1985, 14, Suppl. 1. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions: 2nd edition. Butterworths: London. 1970. Hamer, W. J.; Wu, Y.-C. J. Phys. Chem. Ref. Data 1972, 1, 1047–1099.

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

(Received 11 July 2000; in final form 24 April 2001)

WE-249