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 Quı́mica, 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–Hückel
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 Gestão, Instituto
Politécnico de Bragança, Portugal (E-mail: spinho@ipb.pt).
b E-mail: eamacedo@fe.up.pt
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–Hückel (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.
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(Received 11 July 2000; in final form 24 April 2001)
WE-249