+
+
Ind. Eng. Chem. Res. 1996, 35, 1777-1784
1777
Prediction of Activity Coefficients of Electrolytes in Aqueous
Solutions at High Temperatures
Xiaohua Lu, Luzheng Zhang, Yanru Wang, and Jun Shi
Nanjing University of Chemical Technology, Nanjing 210009, People’s Republic of China
G. Maurer*
Lehrstuhl für Technische Thermodynamik, Universität Kaiserslautern, D-67653 Kaiserslautern, Germany
A recently published model for the excess Gibbs energy of aqueous solutions containing mixed
electrolytes (Lu and Maurer, 1993) is extended from 298 K to temperatures up to 573 K. The
extension is achieved by introducing a universal relation for the influence of temperature on
some model parameters. Although parameters are still ion specific, the influence of temperature
on those parameters is universal. No additional parameters are required for describing aqueous
solutions of mixed electrolytes. The model accurately predicts activity coefficients at high
temperatures and at concentrations up to the solubility limit in electrolyte aqueous solutions
containing, for example, K+, Na+, NH4+, SO42-, Cl- and NO3-.
Introduction
Phase equilibria in electrolyte solutions are important
in industrial chemistry and related fields. Methods
describing phase equilibria usually depart from an
expression for the excess Gibbs energy which has been
designed and tested for aqueous solutions of single
electrolytes at around room temperature. An extension
to mixed electrolytes and higher temperatures is often
unreliable. Such extensions are required for the design
of equipment for, for example, waste water treatment,
sea water desalination, manufacturing of inorganic
chemicals, and hydrometallurgical processes. Recently,
Lu and Maurer (1993) presented a new model for
electrolyte solutions. It combines physical interactions
with solvation equilibria, i.e. chemical equilibria. The
model parameters were determined from experimental
results for single electrolyte aqueous solutions at 298.15
K, but it has been applied successfully without any
information on multicomponent systems to predict
activities in aqueous solutions of two strong electrolytes.
Here, that model is extended to higher temperatures.
The influence of temperatures on the model parameters
is expressed by empirical equations requiring no ion
specific parameters, i.e. although the model parameters
are specific for each ion, the temperature dependence
of the model parameters are universal. The model is
tested by comparing predicted to measured activities
and solubilities of salts in concentrated aqueous solutions of mixed electrolytes up to 573 K.
Model Description
The model is to describe the excess Gibbs energy and
related properties like, for example, the osmotic coefficient φ and mean activity coefficients γ of dissolved
electrolytes in aqueous solution. It is assumed that
dissolving strong electrolytes in water results in a
mixture of water molecules, unsolvated and solvated
ions. Solvation equilibria are used to calculate the true
concentrations of solvated and unsolvated ions from the
overall concentrations of the dissolved electrolytes.
Physical interactions between all species are taken into
account by combining the Debye-Hückel law with the
UNIQUAC model (Abrams and Prausnitz, 1975).
* Author to whom correspondence should be addressed.
Solvation Equilibria. When an electrolyte MνcXνa
is dissolved in water, it completely dissociates to νc
cations M (charge number Zc) and νa anions X (charge
number Za).
MνcXνa f νcMZc + νaMAa
(1)
Ions are solvated according to
MZc + hcH2O ) MZc(H2O)hc
XZa + haH2O )
XZa(H2O)ha (2)
Superscripts Zc and Za designate the charge number of
cations and anions, respectively. Solvated cations are
designated by hc (hc ≡ MZc(H2O)hc) and solvated anions
by ha (ha ≡ XZa(H2O)ha) where hc and ha are the number
of water molecules in a solvated cation or anion respectively. Thus dissolving a strong electrolyte in water
gives a mixture of water molecules as solvated and
unsolvated ions.
The “true” composition of the aqueous mixture is
determined by chemical equilibria for these solvation
reactions expressed by chemical equilibrium constants
Kc and Ka:
Kc )
ahc
acahwc
)
zhc
γ*hc
zczhwc
γ*cγhwc
Ka )
aha
aaahwa
)
zhz
γ*
ha
zazhwa
ha
γ*
aγw
(3)
where ai, zi, and γi are the activity, true mole fraction,
and activity coefficient of species i, respectively. The
activity of water, aw, is normalized according to Raoult’s
law while the activity ak of any dissolved species k
(unsolvated and solvated ion) is normalized according
to Henry’s law on mole fraction scale. Activity is
expressed through “true” mole fraction z and activity
coefficient (γw for water and γi* for solute species i,
respectively) resulting in
aw ) zwγw
(4)
ai ) ziγ*i
(5)
and
Superscript * is used to distinguish between the
different normalizations. Activity coefficients are cal-
+
+
1778 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
culated from an expression for the excess Gibbs energy
GE of the aqueous mixture.
Excess Gibbs Energy and Derived Properties.
Two contributions are being considered to account for
nonideal mixing. The Debye-Hückel expression is
adopted to account for long range electrostatic interactions between charged species (solvated and unsolvated
ions), whereas short range interactions are accounted
for by the UNIQUAC equation, which is usually only
used in nonelectrolyte solutions:
E
GE ) GEDH + GUNIQUAC
(6)
For a charged specie k (i.e. unsolvated or solvated ion),
the activity coefficient γk* therefore is
ln γ*
k ) ln γ*
k,DH + ln γ*
k,UNIQUAC
(7)
The Debye-Hückel contribution to the activity coefficient of a species k, γ*k,DH, is
ln γ*
k,DH )
-Z2kAxIm
1
2
∑i
(9)
where mi is the molality of species i. dk is a size
parameter of species k, and A and B are so-called
Debye-Hückel constants (Robinson and Stokes, 1959).
The short range interaction contribution γ*k,UNIQUAC
is the ratio of two activity coefficient of species k both
calculated from UNIQUAC:
∞
γ*
k,UNIQUAC ) γk,UNIQUAC/γk,UNIQUAC
(10)
γk,UNIQUAC is the activity coefficient of species k normalized according to Raoult’s law and calculated from
UNIQUAC:
[
]
E
∂GUNIQUAC
∂nk
T,pni*k
is the corresponding, but limiting, activity
coefficient of species k in pure water.
lim
zkf0;zwf1
γk,UNIQUAC
(12)
The activity coefficient of water is similarly a sum of
contributions resulting from both parts contributing to
the excess Gibbs energy
[ (
ln γw ) ln γw,DH + ln γw,UNIQUAC
Mw
2(I+J)
A
)
1000 B3I
m
∑k
mkZ2k
d3k
2 ln(1 + BdkxIm) -
1 + BdkxIm 1
hk
ln Kk
1.840
1.840
0.810
0.530
0.025
0.110
5.050
4.120
5.600
1.030
1.547
1.835
2.215
2.067
0.019
0.074
4.367
4.643
5.281
10.00
16.00
7.630
4.680
5.720
4.260
16.50
5.708
8.123
4.483
7.111
7.960
14.80
(T/K ) 298.15), cf. Table 1) were extended to higher
temperatures by the empirical relation:
298.15
ln Kk (T/K ) 298.15)
T/K
)]
+
1 + BdkxIm
ln γw,UNIQUAC (13)
where Mw is the molecular weight of water and mk is
the molality of species k.
Calculation of activity coefficients requires the following information: (A) Chemical equilibrium constants
Kk(k ) a,c), cf. eq 3. Chemical equilibrium constants
reported by Lu and Maurer (1993) for 298.15 K (i.e. Kk
(14)
(B) Debye-Hückel constants A and B for water. The
correlation of Chen et al. (1982) was used to calculate
Debye-Hückel constant A at temperatures from 273 to
573 K. Property B is related theoretically to property
A by B(T)/B (T/K ) 298.15) ) [A(T)/A (T/K )
298.15)]1/3 or B ) 0.31163A1/3. (C) Ion size parameters
dk were calculated using the empirical equations proposed by Lu and Maurer (1993). No distinction is made
between solvated and unsolvated ions:
dhci ) dci ) rhci +
λc ,a rha
∑
j)1
i
j
( )
( )
m
j aj
J
j
dhaj ) daj ) rhaj +
∑
i)1
λci,ajrhci
baj
J
∑ mj a
k)1
I
(11)
∞
γk,UNIQUAC
∞
γk,UNIQUAC
)
rk (Å)
0.208
0.348
0.080
2.520
2.950
2.750
0.150
2.530
3.470
0.020
1.721
2.675
4.113
0.009
1.610
0.010
ln Kk(T) )
miZ2i
RT ln γk,UNIQUAC )
ion k
H+
Li+
Na+
K+
Cs+
NH4+
Ca2+
Mg2+
Sr2+
Ba2+
ClBrIOHNO3SO42-
(8)
1 + BdkxIm
Im is the ionic strength on molality scale:
Im )
Table 1. Pure Component Parameters of the Model
Proposed by Lu and Maurer (1993) at 298.15 K
(15)
k
m
j ci
bci
I
∑ mj c
k)1
k
where rhci and rhaj are size parameters for solvated cation
hci and solvated anion haj which are approximated by
4
4
π(r )3 ) υj Whk + π(rk)3
3 hk
3
(16)
where rk is the radius of ionic species k, hk is the number
of water molecules in the solvated ion k, and υj W is the
volume occupied by such a water molecule (υ
j W ) 2.9910
× 10-29 m3). λci,aj ) λaj,ci is a binary size correction
parameter and m
j ci and m
j aj are the overall stoichiometric
molalities of cations and anions. Ion size parameters
rk and binary size correction parameter λci,aj ) λaj,ci were
taken from Lu and Maurer (1993). Parameter b is to
correct for effects observed with some special ions.
Generally bci ) baj ) 1.0 except for ci ) H+ and aj )
OH-: bH+ ) 0.5 and bOH- ) 1.5.
In the absence of theoretical guidelines, we assumed
that the influence of temperature on solvation number
hk can be described by
hk(T)/hk (T/K ) 298.15) ) 1 + C1(T/K - 298.15) +
C2((T/K)2 - 298.152) (17)
where C1 and C2 are universal parameters (i.e. they do
+
+
Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1779
expressed through the properties mentioned before (see
for example Lu and Maurer (1993)):
(νc + νa) ln γ((m) ) νc ln ahc + νa ln aha j νc cm
j νaa) (νchc + νaha) ln aw - ln(m
(νc + νa) ln
Figure 1. The relationship h(T)/h (T/K ) 298.15) for some strong
electrolytes in aqueous solutions.
not depend on species k). They were determined by a
least square fit to experimental results for the mean
ionic activity coefficient and the osmotic coefficient of
13 single electrolytes in aqueous solutions at temperatures between 323.15 and 573.15 K resulting in C1 )
0.003 and C2 ) -7.0 × 10-6. Numbers for solvation
parameter hk at 298.15 K were again taken from Lu
and Maurer (1993). The temperature dependence of
solvation parameter hk is shown in Figure 1. Points
shown in that figure represent results from an isothermal fit of hk(T)/hk (T/K ) 298.15) to data sets for single
aqueous electrolyte systems. (D) All UNIQUAC parameters (size, surface, and interaction parameters) were
taken from Lu and Maurer (1993).
Extending the model from 298 K to higher temperatures is therefore achieved by taking into account the
influence of temperature on the following. (i) The
equilibrium constants Kk for the solvation of ionic
species k. As it was to be expected, that equilibrium
constant decreases with increasing temperature. For
example, raising the temperature from 298 to 400 or
500 K reduces ln Kk by about 25 or 40%, respectively.
(ii) Both Debye-Hückel constants A and B and (iii) the
number of water molecules in the solvation shell of a
solvated ion. As it was to be expected, increasing the
temperature reduces the number of water molecules in
a solvated species. For example, increasing the temperature from 298 to 400 or 500 K is accompanied by a
decrease of the number of water molecules by about 20
or 48%, respectively.
Comparison between Experimental and
Calculated Data
In electrolyte solutions it is common practice to
express the deviation from ideal mixing behavior through
the osmotic coefficient φ:
-1000
φ)
I+J
Mw
ln(γwzw)
(18)
∑ mj k
k)1
or the mean ionic activity coefficient of a strong electrolyte MνcXνa, on molality scale γ((m), which can be
Mw
(19)
1000
Aqueous Solutions of Single Electrolytes at High
Temperatures. Calculated results for the mean ionic
activity coefficient and the osmotic coefficient of 13
single electrolytes in aqueous solutions at temperatures
between 323.15 and 573.15 K are compared with
experimental results in Table 2. Calculated results are
from the present work as well as from the compilation
of Zemaitis et al. (1986) and from Pabalan and Pitzer
(1991). In an extensive investigation on the thermodynamics of aqueous solutions of strong electrolytes,
Zemaitis et al. applied the methods of Bromley, Meissner, Pitzer, and Chen to describe mean ionic activity
coefficients of aqueous solutions of single strong electrolytes: HCl at 323.15 K, KCl at 353.15 K, KOH at
353.15 K, NaCl at 373.15 and 573.15 K, NaOH at 308.15
K, CaCl2 at 382 and 475 K, Na2SO4 at 353.15 K, and
MgSO4 also at 353.15 K. In order to enable a direct
comparison with the tabulated results by Zemaitis et
al., the new model was used to calculate mean ionic
activity coefficients for the same systems. A summarized comparison is shown in Table 3. As it was to
be expected, the best agreement between calculated
activity coefficients and the compilation of Zemaitis et
al. is achieved by Pitzer’s model with parameters
reported by Pitzer and Pabalan (1991). The average
standard deviation between calculated and compiled
data for the electrolytes shown in Table 3 is 0.012 for
that model, while it is 0.064 for the present study. All
other methods, including the often applied short form
of Pitzer’s method recommended by Zemaitis et al.,
result in average standard deviations of more than
0.168. Zemaitis et al. used a special simplification of
Pitzer’s method as they combined interaction parameters determined for 298 K with temperature dependent
Debye-Hückel parameters to calculate the thermodynamic properties of aqueous electrolyte solutions. Due
to its simplicity that model has gained much interest
in industrial engineering calculations. The most current
state of Pitzer’s model (see for example Pabalan and
Pitzer, 1991) requires temperature dependent interaction parameters. However, those temperature dependent parameters have to be determined for each single
aqueous electrolyte system, and currently that information is only available for a limited number of electrolytes. Therefore in applied science, especially in chemical engineering, there is still an interest in simpler
models. The unavoidable reduction in accuracy is often
accepted for the sake of getting reliable estimates also
when no high-temperature experimental data are available. The simplification of Pitzer’s method by Zemaitis
et al. (1986) is included in the comparisons of the
present work as it does not require experimental
information on aqueous electrolyte solutions at higher
temperatures.
Additionally, some selected comparisons are also
shown in Figures 2-4. Figure 2 presents calculated
results for the mean ionic activity coefficient of CaCl2
in aqueous solutions at 382 and 475 K as reported by
Zemaitis et al. (1986) for their modification of Pitzer’s
model as well as from the present study in comparison
+
+
1780 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
Table 2. Root Mean Square Deviations of Mean Ionic Activity Coefficients and Standard Deviations of Osmotic
Coefficient of Single-Electrolyte Solutions Predicted from the Model Compared with Experimental Data at Elevated
Temperatures
electrolyte
T/K
σγ(a
σlnγ(b
HCl
LiCl
LiCl
LiCl
LiNO3
Li2SO4
Li2SO4
Li2SO4
Li2SO4
NaCl
NaCl
NaOH
Na2SO4
NH4NO3
KCl
KOH
K2SO4
K2SO4
K2SO4
K2SO4
CaCl2
CaCl2
MgSO4
323.15
373.15
413.15
473.15
373.15
348.15
398.15
448.15
498.15
373.15
573.15
308.15
353.15
363.15
353.15
353.15
348.15
398.15
448.15
498.15
382.00
475.00
353.15
0.0148
1.013
0.0058
0.0134
0.017
0.096
0.008
0.023
0.0281
0.0380
0.0303
0.0160
0.0373
0.0343
0.0138
0.0259
0.125
0.228
0.274
0.249
0.052
0.080
0.019
0.112
0.0066
1.0432
0.5149
0.0103
0.0112
0.0103
0.0212
0.0224
0.0033
0.011
0.178
0.021
0.044
0.081
0.114
0.037
0.077
0.063
SDφc
0.085
0.027
0.043
0.033
0.077
0.148
0.214
0.281
0.078
0.020
0.080
0.148
0.223
Imax
property
reference
2.0
18.0
3.5
3.5
25.0
7.5
9.0
7.5
7.5
6.0
6.0
4.0
4.8
23.5
4.0
17.0
6.0
7.5
7.5
7.5
10.5
10.5
8.0
ln γ(
ln γ( and φ
ln γ( and φ
ln γ( and φ
φ
ln γ( and φ
ln γ( and φ
ln γ( and φ
ln γ( and φ
ln γ(
ln γ(
ln γ(
ln γ(
φ
ln γ(
ln γ(
ln γ( and φ
ln γ( and φ
ln γ( and φ
ln γ( and φ
ln γ(
ln γ(
ln γ(
Harned and Owen, 1958
Gibbard et al., 1973
Holmes et al., 1981
Holmes et al., 1981
Sacchetto et al., 1981
Holmes et al., 1986
Holmes et al., 1986
Holmes et al., 1986
Holmes et al., 1986
Silvester and Pitzer, 1976
Silvester and Pitzer, 1976
Harned and Owen, 1958
Snipes et al., 1975
Sacchetto et al., 1981
Snipes et al., 1975
Harned and Owen, 1958
Holmes et al., 1986
Holmes et al., 1986
Holmes et al., 1986
Holmes et al., 1986
Holmes et al., 1978
Holmes et al., 1978
Snipes et al., 1975
a
σγ( )
b
x
SDφ )
N
∑(γ
N
σln γ( )
c
1
(,cal
- γ(,exp)2i
i)1
x
N
1
∑(ln γ
N
x
(,cal
- ln γ(,exp)2i
i)1
N
1
∑(φ
N-1
cal
- φexp)2i
i)1
Table 3. Comparison of Standard Deviationa of ln γ((m) for Aqueous Electrolyte Solutions at Elevated Temperatures
with Data Compiled by Zemaitis et al., 1986b
electrolyte
HCl
KCl
KOH
NaCl
NaCl
NaOH
MgSO4
Na2SO4
CaCl2
CaCl2
avg SD
T/K
N
323.15
353.15
353.15
373.15
537.15
308.15
353.15
353.15
382.00
475.00
15
19
13
38
38
11
13
13
42
42
Bromley
(1973)
Meissner
(1972)
Zemaitis et al.
(1986)
Pabalan+Pitzer
(1991)
Chen et al.
(1982)
this work
0.021
0.053
0.117
0.051
0.842
0.034
0.795
0.267
0.317
1.050
0.355
0.009
0.017
0.101
0.032
0.671
0.035
0.324
0.052
0.391
1.018
0.265
0.014
0.079
0.898
0.056
0.066
0.014
0.061
0.156
0.109
0.232
0.168
0.003
0.003
0.021
0.058
0.321
0.031
0.231
0.034
0.070
0.124
0.365
0.627
0.188
0.018
0.011
0.185
0.053
0.081
0.020
0.068
0.118
0.037
0.037
0.064
0.002
0.032
0.014
0.023
0.005
0.018
0.005
0.012
a
SDln γ( )
b
x
1
N
∑(ln γ
N-1
i)1
(,cal
- ln γ(,exp)2i )
x
N
σln γ(
N-1
All comparisons with literature models, besides that with Pabalan and Pitzer, are from Zemaitis et al., 1986.
to data compiled by Zemaitis et al. from experimental
work. The results of such compilations of experimental
data are called “compiled” data throughout the present
work. In Figure 3 mean ionic activity coefficients of LiCl
and K2SO4 calculated with the new model at temperatures between 398 and 523 K are compared to compiled
data. Figure 4 shows a similar comparison for the
osmotic coefficients of concentrated aqueous solutions
of LiCl, LiNO3, and NH4NO3 at around 373 K.
Solubilities of Salts in Mixed Electrolyte Solutions at High Temperatures. For many mixed electrolyte aqueous solutions experimental results neither
for the osmotic coefficients nor for activity coefficients
are available at elevated temperatures. The solubility
+
+
Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1781
Figure 2. Mean ionic activity coefficient of CaCl2 in water at 382
and 475 K.
Figure 4. Osmotic coefficients of some single aqueous electrolyte
solutions at elevated temperatures.
Table 4. Temperature Coefficients of Solubility Product
Kspa of Some Salts for the Precipitation from Aqueous
Solutions Based on the Model Proposed by Lu and
Maurer (1993) and Determined from Single-Electrolyte
Solubility in Pure Water As Reported by Linke and
Seidell (1965) and Silcock (1979)
salt
T/K
U1
U2
U3
U4
NaCl
KCl
NH4Cl
NaK3(SO4)2
K2SO4
Na2SO4
Na2SO4‚
10H2O
KNO3
NaNO3
273-462
273-462
273-363
288-373
273-373
298-473
273-303
3.61572
2.08949
3.00445
-8.47979
-4.18629
0.661999
-2.70124
2187.41
7.12599
1.93139
5.98201
79646.1
17669.8
230805.
18.8623
18.6402
16.9631
54.9217
443.046
106.384
1699.95
0.0396581
0.0414204
0.0282449
-0.148065
0.622767
-0.175698
-3.00855
a
Figure 3. Mean ionic activity coefficients of LiCl and K2SO4 in
water at elevated temperatures.
of salts in aqueous solutions are therefore often used
for testing new models. Lu and Maurer (1993) have
shown that the model is able to predict salt solubilities
in mixed electrolyte aqueous solutions at 298 K also at
high salt solubilities.
When a solid electrolyte MνcXνa(H2O)h precipitates
from an aqueous solution the concentrations of anionic
and cationic species of that electrolyte in the liquid
phase are determined by its solubility product:
c+νa) h
Ksp ) mνMc mνXaγ(ν
((m) aw
(20)
Solubility products Ksp are calculated from the solubility
of a single electrolyte in pure water applying an
adequate model for describing activity coefficients. Ksp
depends on temperature. The following equation is used
to express that influence:
ln Ksp(T) ) U1 + U2[1/(T/K) - 1/298.15] +
U3 ln[(T/K)/298.15] + U4[(T/K) - 298.15] (21)
Parameters U1, U2, U3, and U4 as determined from
273-373 0.263300
273-373 2.89788
40880.8 230.932 0.351406
5515.11 42.5647 0.0663410
Compare eq 21.
single-electrolyte solubility in pure water as reported
by Linke and Seidell (1965) and Silcock (1979) are listed
in Table 4.
To test the model also at high concentrations and
elevated temperatures, the solubility of some salts in
seven binary electrolyte solutions (where both electrolytes have a common ion) was predicted and compared
with experimental data taken either from Linke and
Seidell (1965) or from Silcock (1979), cf. Table 5. The
comparisons are shown in Figures 5-11. Figure 5 gives
results for the system NaCl-KCl-H2O at 273, 373, and
463 K. Figure 6 gives results for the system KClK2SO4-H2O. Those two figures show also a comparison
with the current version of Pitzer’s method taken from
Pabalan and Pitzer (1991). As it was to be expected that
method also gives a good agreement with the experimental data. However, it must be mentioned that the
calculation by Pabalan and Pitzer is based upon specific
binary ion-interaction parameters fitted not only to
high-temperature data but also to experimental results
for binary electrolyte aqueous systems. Whereas in the
present work individual interaction parameters were
fitted neither to experimental results at elevated temperature nor to results for binary electrolyte aqueous
solutions. Figures 7 and 8 show comparisons for
systems containing either KCl and NH4Cl or Na2SO4
+
+
1782 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
Figure 5. Measured and calculated solubilities in the system
KCl-NaCl-H2O at 273, 373, and 463 K.
Figure 6. Measured and calculated solubilities in the system
KCl-K2SO4-H2O at 288, 323, and 373 K.
Table 5. Sources for Salt Solubility Data for Ternary
Aqueous Solutions
system
T/K
Imax
source
Na+-K+-Cl--H2O
NH4+-K+-Cl--H2O
K+-Cl--SO42--H2O
Na+-K+-SO42--H2O
K+-Cl--NO3--H2O
Na+-Cl--NO3--H2O
Na+-K+-NO3--H2O
273-363
273-363
273-373
298-373
298-373
273-373
273-333
14
14.5
9.5
22.5
25
21
27
Linke and Seidell, 1965
Linke and Seidell, 1965
Linke and Seidell, 1965
Silcock,1979
Silcock,1979
Silcock,1979
Silcock,1979
and K2SO4. In the later systems under certain conditions double-salt NaK3(SO4)2 precipitates. Figures 9 to
11 show comparisons for some very soluble nitrates. For
those salts Clegg and Pitzer (1992a,b) recently proposed
a model based on the mole fraction equations of Pitzer
and Simonson (1986) instead of the virial (molalitybased) model of Pitzer (1973), but they applied that
method only to 298 K. As shown in Figures 9-11, the
Figure 7. Measured and calculated solubilities in the system
K2SO4-Na2SO4-H2O at 298, 323, and 373 K.
Figure 8. Measured and calculated solubilities in the system
KCl-NH4Cl-H2O at 273, 318, and 363 K.
new model gives a reasonable prediction for the solubilities of the nitrates at elevated temperatures with
individual parameters determined only from information on the properties of single-electrolyte aqueous
solutions at 298 K.
Conclusion
The Lu-Maurer model (1993) for mixed electrolyte
aqueous solutions is extended to temperatures up to 573
K. It is shown that by using parameters correlated from
single-electrolyte aqueous systems at 298 K together
with a generalized expression for the influence of
temperature on some parameters, the activity coefficients in electrolyte aqueous solutions at high temperatures can be predicted with good accuracy up to the
solubility limit, e.g. at very high ionic strength.
+
+
Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1783
Figure 9. Measured and calculated solubilities in the system
KCl-KNO3-H2O at 298, 323, and 373 K.
Figure 10. Measured and calculated solubilities in the system
NaCl-NaNO3-H2O at 273, 298, and 373 K.
Acknowledgment
X.L. thanks the Alexander-von-Humboldt Stiftung of
Germany for providing a research fellowship at the
Universität Kaiserslautern and The National Natural
Science Foundation of P. R. China and Fok Ying-Tong
Education Foundation for financial support. He also
thanks Dongyun Lu and Jian Zhou for their assistance
in the calculations.
Notation
a ) true activity
A ) Debye-Hückel constant
B ) Debye-Hückel constant
baj ) parameter in eq 15
bci ) parameter in eq 15
C1, C2 ) constants in eq 17
d ) size parameter
Figure 11. Measured and calculated solubilities in the system
KNO3-NaNO3-H2O at 273, 298, and 333 K.
GE ) excess Gibbs energy
ha ) solvated anion
hc ) solvated cation
h ) number of water molecules in a complex
I ) number of cationic species
Im ) ionic strength on molality scale
J ) number of anionic species
i,j,k ) species i,j,k
K ) solvation equilibrium constant
Ksp ) solubility product
m ) molality
mk ) true molality of species k
m
j k ) overall (stoichiometric, apparent) molality of component k
Mw ) molecular weight of water
M ) cation
n ) mole number
N ) number of components
p ) pressure
rk ) ionic radius of species k
R ) universal gas constant
SD ) standard deviation
T ) thermodynamic temperature
U1-U4 ) constants in eq 21
υ
j W ) volume occupied by a water molecule
X ) anion
z ) true mole fraction
Z ) charge number of ionic species
Greek Letters
γk ) true activity coefficient of specie k on mole fraction
scale normalized according to Raoult’s law
γ*k ) true activity coefficient of specie k on mole fraction
scale normalized according to Henry’s law
γ((m) ) mean ionic activity coefficient on molality scale
λci,aj ) binary size correction parameter between cation ci
and anion aj
σ ) root mean square deviation
ν ) stoichiometric number
φ ) osmotic coefficient
Subscripts
a ) anion
c ) cation
cal ) calculated
DH ) contribution from Debye-Hückel law
+
+
1784 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
exp ) experimental
ha ) solvated anion
hc ) solvated cation
hk ) solvated species k
i ) solute
k ) species k
m ) on molality scale
max ) maximum
sp ) solubility product
UNIQUAC ) contribution from UNIQUAC equation
w ) water
Superscripts
* ) normalized according to Henry’s law
∞ ) limiting (infinite dilution)
Literature Cited
Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of
Liquid Mixtures: A New Expression for Excess Gibbs Energy
of Partly or Completely Miscible Systems. AIChE J. 1975, 21,
116.
Chen, C. C.; Britt, H. I.; Boston, J. F.; Evans, L. B. Local
Composition Model for Excess Gibbs Energy of Electrolyte
Systems. AIChE J. 1982, 28 (4), 588.
Clegg, S. L.; Pitzer, K. S. Thermodynamics of Multicomponent,
Miscible, Ionic Solutions: Generalized Equations for Symmetrical Electrolytes. J. Phys. Chem. 1992, 96, 3513.
Clegg, S. L.; Pitzer, K. S.; Brimblecombe, P. Thermodynamics of
Multicomponent, Miscible, Ionic Solutions 2: Mixtures Including Unsymmetrical Electrolytes. J. Phys. Chem. 1992, 96, 9470.
Gibbard, H. F., Jr.; Scatchard, G. Liquid-Vapor Equilibrium of
Aqueous Lithium Chloride, from 25 to 100 °C and from 1.0 to
18.5 molality, and Related Properties. J. Chem. Eng. Data 1973,
18 (3), 293.
Harned, H. S.; Owen, B. B. Physical Chemistry of Electrolyte
Solutions; Reinhold: New York, 1958.
Holmes, H. F.; Mesmer, R. E. Isopiestic Studies of Aqueous
Solutions at Elevated Temperatures: VI. LiCl and CsCl. J.
Chem. Thermodyn. 1981, 13, 1035.
Holmes, H. F.; Mesmer, R. E. Thermodynamics of Aqueous
Solutions of the Alkali Metal Sulfates. J. Solution Chem. 1986,
15 (6), 495.
Holmes, H. F.; Baes, C. F.; Mesmer, R. E. Isopiestic Studies of
Aqueous Solutions at Elevated Temperatures I. KCl, CaCl2 and
MgCl2. J. Chem. Thermadyn. 1978, 10, 983.
Linke, W. F.; Seidell, A. Solubilities of Inorganic and Metal-organic
Compounds; American Chemical Society: Washington, DC,
1965.
Lu, X.; Maurer, G. Model for Describing Activity Coefficients in
Mixed Electrolyte Aqueous Solutions. AIChE J. 1993, 39 (9),
1527.
Mayrath, J. E.; Wood, R. H. Enthalpy of Dilution of Aqueous
Solutions of LiCl, NaBr, KCl, KBr, and CsCl at 373, 423, and
473K. J. Chem. Thermodyn. 1982, 14, 563.
Pabalan, R. T.; Pitzer, K. S. Mineral Solubilities in Electrolyte
Solutions. Activity Coefficients in Electrolyte Solutions, 2nd ed.;
CRC Press: Boston, MA, 1991; Chapter 7.
Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis
and General Equations. J. Phys. Chem. 1973, 77 (2), 268.
Pitzer, K. S.; Simonson, J. M. Thermodynamics of Multicomponent,
Miscible, Ionic Systems: Theory and Equation. J. Phys. Chem.
1986, 90, 3005.
Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.;
Butterworths: Boston, MA, 1959.
Sacchetto, G. A.; Bombi, G. G.; Macca, C. Vapor Pressure of Very
Concentrated Electrolyte Solutions. J. Chem. Thermodyn. 1981,
13, 31.
Silcock, H. L. Solubilities of Inorganic and Organic Compounds;
Pergamon Press: Oxford, 1979; Vol. 3.
Slivester, L. F.; Pitzer, K. S. Thermodynamics of Geothermal Brine
I. Thermodynamic Properties of Vapor-Saturated NaCl(aq)
Solutions from 0-300 °C. Report LBL-4456; Lawrence Berkeley
Laboratory: Jan. 1976.
Snipes, H. P.; Manly, C.; Ensor, D. D. Heats of Dilution of Aqueous
Electrolytes: Temperature Dependence. J. Chem. Eng. Data.
1975, 20 (3), 287.
Zemaitis, F. D., Jr.; Clark, D. M.; Rafal, M.; Scrivner, N. C.
Handbook of Aqueous Electrolyte Thermodynamics: DIPPR,
AIChE: New York, 1986.
Received for review July 28, 1995
Revised manuscript received February 15, 1996
Accepted February 15, 1996X
IE950474K
X Abstract published in Advance ACS Abstracts, April 1,
1996.