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Simultaneous correlation of activity coefficients for 55 aqueous electrolytes using a model with ion specific parameters
Chemical Engineering Science, Vol. 46, No. 7, pp. 181S- L821, 1991. Printed in Great Bntain.
ODE-2509/91 $3 00 + Mm 0 1991 Pergamon Press plc
SIMULTANEOUS CORRELATION OF ACTIVITY COEFFICIENTS FOR 55 AQUEOUS ELECTROLYTES USING A MODEL WITH ION SPECIFIC PARAMETERS
Department
of Chemical
YUNDA LIU and URBAN GReNf Engineering Design, Chalmers University Gothenburg,
(First received
11 February
of Technology,
S-412 96
Sweden
1990; accepted
in revised form
19 November
1990)
Abstract-The Liu-Harvey-Prausnitz model (1989a) for electrolyte solutions with ion-specific parameters is extended to simultaneously correlating the mean ionic activity coefficient data for 55 aqueous electrolytes including the H,SO,/H,O system up to the H,SO, concentration of 76 m. With only 1.6 adjustable parameters per aqueous electrolyte, an excellent agreement between the calculated values and the experimental data has been achieved. Ion-specific parameters are reported for 9 cations (H ‘, Li+, Na+, K+, Rb+,Cs+,Mg++,Ca++andZn++)andS anions (F-,Cl-, Br-, I-, OH-, NO;, CNS- and SO; -1. The results in this work are compared with those given by the Haghtalab-Vera model (1988) and the outcome is in favor of the Liu-Harvey-Prausnitz model.
INTRODUCTION
Electrolyte solutions play an important role in the chemical industry and are also relevant in other fields of science and technology. Partitioning processes in biochemical systems; precipitation and crystallization processes in geological systems such as geothermal brines or drilling muds; desalination of water and water pollution control; salting-in and salting-out effects in extraction and distillation; absorption and regeneration in heat transformers are just a few examples of cases in which the knowledge of the thermodynamic properties of electrolyte solutions is basic for a proper understanding of the phenomena. For some applications, the knowledge of the behaviors of the dilute solutions or of solutions in the intermediateconcentration range is all that is required, while for others the region close to saturation of the solution is of great importance. In the last few decades, many models have been proposed to represent and predict the thermodynamic properties of electrolyte solutions. Pitzer (1981), Maurer (1983) and Renon (1986) give excellent reviews of the subject. A typical problem of the models proposed is that their use is limited to certain range of concentrations. At higher molalities, large errors are produced by most of these models. For many industrial processes, such as crystallization processes or absorption in heat transformers, it is necessary to have models able to represent and predict the thermodynamic behavior at higher molalities. A new model meeting this requirement was proposed by Liu, Harvey and Prausnitz (1989a). The model applied a new version of the Debye-Hiickel theory for the long-range interaction contributions and a local-composition expression of the Wilson type for the short-range interaction contri-
‘Author
to whom
correspondence
should
be addressed.
butions. The new Debye-Htickel expression was specially developed taking into consideration that the Debye-Hiickel expression and the local-composition expression were not independent of each other when they were combined. Promising applicability of the model was shown by a stringent test in a simultaneous correlation of activity coefficients for 6 aqueous electrolytes, four of which were regarded as hard to handle. Ion-specific parameters were reported for 3 cations (H+ , Li+ and K+) and 2 anions (Cl- and Br-). The highest concentration of the data used in the correlation was 20 M. The objeciive of the present work is to extend the mode1 to simultaneously correlating activity coefficients for 55 aqueous electrolytes, which include the H,SO,/H,O system up to the H,SO, concentration of 76 M, and to obtaining the values of the ion-specific parameters for 9 cations (H+, Li+, Na+, K+, Rb+, Cs+, Mg++, Ca++ and Zn’+) and 8 anions (F-, Cl-, Br-, I-, OH-, NO;, CNS- and SO;-). In 1988, Haghtalab and Vera proposed a successful mode1 which was claimed to be the best two-electrolyte-specific parameter mode1 for aqueous solutions of single electrolytes. This model also consists of two contributions. The Debye-Hiickel expression of Fowler and Guggenheim is used for the long-range interaction contributions, and the local-composition expression of the NRTL type with the reference cells in the random case is used for the short-range interaction contributions. In this work, the results of the simultaneous correlation using the Liu-HarveyPrausnitz mode1 are compared with those given by the Haghtalab-Vera model. THE LIU-HARVEY-PRAUSNITZ
MODEL
The Liu-Harvey-Prausnitz model (1989a) consists of two terms, one long-range and one short-range. The long-range term is represented by a new version of the Debye-Hlckel theory, which accounts for the
1815
1816
YUNDA
Lru and URIMN GRIN
contributions due to the electrostatic forces between each central ion and the ions outside its first coordination shell. The short-range term is represented by a local-composition expression of the Wilson type, which accounts for the contributions due to the shortrange interaction forces of all kinds, including the electrostatic interaction forces between each central ion and all ions inside the first coordination shell. Although the complete dissociation of electrolytes is assumed, the mode1 is capable of describing the nonideality of the electrolyte solutions with the significant ion assqciation, which is due to the indirect consideration of the ion association in the derivation of the model (Liu et al., 1989a). Therefore, in this work, the assumption of the complete dissociation remains unchanged not only because of its simplicity but also because of the capability of the model. The long-range term for single aqueous electrolytes is given as follows: e2
WLLR = 2DkT(w, + w,)
x
1
expCK(C- r&l (1 + Icr:) W.7V‘?
tr,
-lit
vex,=,
K
v,+ xc + x,G,.,,.
X ev CK(rL - r,)l (1 + Icr:)
-
where
Tl
Gji,,i = exp C- (Sji - s,i)lR
Ic=
4n e2 n,cx,$ [- DkT
+ x.vf,
r: = rc + Lr,
II l/2 1
“’
(1)
r: = r, + Iv,. The short-range term for single aqueous lytes is given as follows:
electro-
xw(~cGcw,ww + ~,G,w,w,) xw + xcGcw, ww + x, Go,.ww
X
+(I -xw)(w, +~a) --w,G,,.,, -~,Gow,,w w
+=,
ln
c
0,
+
1
xw)
(x, + x,1
[
+
(xaG,, wc
x,
x,$-C,
w x, xc + xw
11.
(2)
The symbols in the long-range and short-range terms are explained in the Notation section. The unsymmetric mean ionic activity coefficient of a single-aqueous electrolyte is obtained as In Y:,,
= In Y:.,,~ + In ~2.~~.
(3)
DATA CONSIDERATION
Unlike the models with electrolyte-specific parameters, the models with ion-specific parameters need a simultaneous correlation of data for all selected electrolyte systems in order to obtain optimal values of ion-specific parameters, which are common to alI the electrolyte systems containing the corresponding ions. This calls for a high consistency among each set of the electrolyte data. Therefore, the selection of data sources have to be carefully considered. In this work, the focus is to find out the values of the ion-specific parameters for the 9 cations (H+, Li’, Nat, K+, Rb+, Cs+, Mg++, Ca*+ and Zn’*) and the 8 anions (F-, Cl-, Br-, I-, OH-, NO;, CNSand SO; -). There are, up to now, activity-coefficient data at 25°C for 56 aqueous electrolytes, which are out of the possible cation-anion combinations of the 9 cations and the 8 anions. The data for all these aqueous electrolytes except one, which is the aqueous hydrofluoric acid, are used in the simultaneous correlation. The exclusion of the aqueous hydrofluoric acid is due to its rather peculiar thermodynamic: behavior, which is more like those of nonuniPunivalent aqueous electrolytes. It is worth mentioning that the data for the aqueous hydrofluoric acid is also excluded in the previous works by other authors using various other models. Among the 55 aqueous electrolytes, 34 of them are uni-univalent and 21 are nonuni-univalent. We adopt the evaluated experimental data from the well-known article by Hamer and Wu (1972) for the 34 uniunivalent aqueous electrolytes, and we adopt the carefully evaluated experimental data by Goldberg et al. (1978, 1981a, b) for the 16 nonuni-univalent aqueous electrolytes. The data for the remaining 5 nonuni-univalent aqueous electrolytes, which are not included in the Goldberg’s work, come from other data sources (see Table 7). We prefer the carefully evaluated experimental data to the data from the direct experiments because the former usually possesses a higher internal and external consistency than the latter does. In the separate correlation of the data for the LiOH/H,O system, we notice that the last data point increases the deviation by almost 10 times (from 0.00146 to 0.01028, where A = 1.5). Hamer and Wu (1972) also mark the last data point to indicate that there is some inconsistency between the data point and the rest of the others. We exclude this data point in the simultaneous correlation because one inconsistent data point will influence not only the result of the corresponding electrolyte but also the results of all the other electrolytes.
Simultaneous SIMULTANEOUS
CORRELATION
correlation
of activity
AND DISCUSSION
The values of the ion-specific parameters for the 9 cations (H+, Li+, Na+, K+, Rb+, Cs+, Mg+ +, Ca++ and Zn*+) and the 8 anions (F~,Cl~,Br~,I~, OH-, NO;, CNS and SO; -) are obtained by minimizing the sum of the square deviations between the calculated and the experimental data of the mean ionic activity coefficients for the 55 aqueous electrolytes: (4)
where nj is the number of data points for the ith electrolyte. The root mean square deviation of the ith electrolyte, o~,~*, is defined as
Since the ion-specific parameter estimation is carried out based on the simultaneous correlation of all selected electrolyte data, a capable computational method for minimizing the square objective function with a large number of independent variables is very much required. Liu, Grin and Wimby (1989~) have compared seven square-function minimizing methods and concluded that the modified Marquardt method (Draper and Smith, 1966; Bard, 1974), generally, performs best. In this work, the modified Marquardt method is further compared with the Nelder-Mead method (Nelder and Mead, 1965) in minimizing the function (4). It turns out that the favor towards the modified Marquardt method remains unchanged. The ion-specific parameters are common to all electrolyte solutions containing the corresponding ions. This means, for example, that the Cl--H,0 interaction parameter should have the same value for all the aqueous chloride electrolytes. But from the results of the separate data correlation for each of the 55 aqueous electrolytes, we find that some diverse values for the same ion-specific parameter are obtained. The diversity may be explained from different aspects, such as the possible inconsistency of the data among
coefficients
for aqueous
electrolytes
the relevant electrolytes; the different degrees of the electrolyte dissociation; the semi-empirical nature of the Liu-Harvey-Prausnitz model and the calculation simplifications (like using a linear expression to describe the concentration dcpcndcncc of the dielectric constants for all electrolytes as well as using the water density instead of the real densities of the electrolyte solutions). With the increase of the number of the electrolytes in the simultaneous correlation, foreseeably the diversity problem will become more serious: meanwhile, the average number of the adjustable parameters per aqueous electrolyte in the Liu-Harvey-Prausnitz model will gradually reduce and approach one. In order to apply the Liu-Harvey-Prausnitz model to properly represent the nonideality of the aqueous electrolytes up to the very high concentration with fewer average parameter number per electrolyte, the problem of the parameter diversity has to be solved adequately. In the previous works by Liu et al. (1989a, b), it has been found that the parameter values are sensitive to the size constant, L, which is used to determine the radius of the coordination shell. Table 1 shows how the parameter values for the LiNO,/H,O system vary with different size-constant values. The results in Table 1 are from the separate fit of the mean-ionic activity coefficient data for the LiNO,/H,O system at 25°C. In this work, we try to use the size constant to reduce the parameter diversity. The applied procedure is described as follows:
(I) separately
fit the data for each of the 55 aqueous electrolytes assuming different size-constant values, (2) compare the values of the same ion-specific parameter obtained from the relevant electrolytes at different size-constant values, (3) choose a proper size-constant value for each electrolyte to make the parameter diversity become minimum. Table 2 gives the size-constant values for the 55 aqueous electrolytes, which are chosen through the above procedure.
Table 1. The parameter
values for the LiNO,/H,O system obtained assuming different values of the size constant (1) (results from fit of the mean-ionic activity coefficient data for the LiN0,/H20 system in the concentration range 0.001~20.0 M and at the temperature 298.15 K)’
1.1
1.3 1.5 1.7
0.97390 - 0.86085 - 1.73170 - 1.89981
2.36170 0.82525 0.04286 - 0.15166
- 0.63024 1.32480 2.03439 2.28983
1.12472 0.87029 0.96489 1.08237
0.0050 0.0038 0.0055 0.0062
+ Data from Hamer and Wu (1972). A value of 6 is assumed for all coordination
numbers.
parameterof anion NO;
The anionic
1817
radius,
1.79 A, is used
as the radius
1818
YUNDA
Table 2. The size constants FH+ Li+ Na+ Kf Rb+ CS’ Mg++ Cat+ zn++
1.6 1.6 1.8 2.1
0.7
for the 55 aqueous
Cl-
Br-
I-
1.2 1.7 1.0 1.1 0.9 0.7 1.0 1.3 1.4
1.3 1.6 1.9 I.1 0.9 1.7 1.4 1.2 1.5
1.1 1.2 1.5 1.0 0.8 0.8 1.2 I.2 1.6
OH-
NO;
0.4 1.5 1.8
1.2 1.3 1.4 0.9 0.6
2.4
I .o 1.2 1.3 1.2
LIU and URBAN GREN
electrolytes
CNS
1.3 0.9
SO;0.4 1.1 0.7 0.8 1.0 1.3 0.8 0.6
During the above procedure. we also found out some extraordinary parameter values for certain electrolytes. For example, no matter how the size-constant value varies, the value of the cationic radius parameter obtained from the HNO,/H,O system is always much larger than those obtained from the rest of the other electrolyte systems which contain the ion II+. To deal with such cases, we just simply let the corresponding electrolytes keep their own parameter values. The total number of the extraordinary parameter values are seven. Table 4 and Table 5 give further information about them. It is worth mentioning that the separate parameter values for some special electrolytes cause no trouble in the prediction calculation for multi-electrolyte systems, which has been confirmed in solubility prediction calculations (the results will be shown in a subsequent paper). In this work, we have not set the size constant as an adjustable parameter for each electrolyte. That is partly because through the above procedure we can obtain more information about the parameter values and partly because by using the size-constant values obtained in such a way, the results of the simultaneous correlation are aleady very satisfactory, For the sake of simplicity, the pure water density of 55,409.3 mol/m3 is used for all the 55 aqueous electrolytes. As mentioned by Liu et al. (1989a), such an assumption causes no serious deviation. Netemeyer and Glandt (1988) have examined the influence of the coordination number on the predictivity of the local-composition models and concluded that the optimal coordination number corresponds approximately to a value of 6. This conclusion is consistent with the results obtained by Liu et al. (1989b). Therefore, in this work, the coordination numbers for all the cations and anions as well as water are fixed at the value of 6 in order to enhance the predictivity of the model in the future application with the parameter values from this correlation. As in the work of Liu er al. (1989b), the dielectric constant for all aqueous electrolyte systems except the H,SO,/HI,O system is assumed to be a linear function of the water mole fraction: D = 78.3 - 46.65(1 - x,).
(7)
Since the dielectric constant of the pure sulfuric acid is higher than that of the pure water (at 25”C, the former has the dielectric constant value of 101 and the latter
7X.3), it is unreasonable to assume that the dielectric constant of the H,SO,/H,O system will be reduced with the increase of the H,SO, concentration. Therefore, in addition to the consideration of simplicity, we assume that the dielectric constant of the H,SO,/ H,O system is the same as that of the pure water and independent of the H,SO, concentration. The activity coefficient of electrolyte solutions in the medium- and high-concentration ranges may be largely determined by the ionic radii (Harned and Owen, 1958). In this work, we have chosen the anionic radius parameters for the 8 anions from the values of the anionic radii published in the literature, which are shown in Tahle 3. The cationic radius parameters fnr the 9 cations are treated as adjustable parameters and the values are obtained from the simultaneous correlation of the mean-ionic-activity-coefficient data for the 55 aqueous electrolytes at 25°C. Table 4 gives the correlated values for the 9 cationic radius parameters. Table 5 gives the ion-specific interaction energy parameters, gji/R T, from the simultaneous correlation. Since all interaction energy parameters in the Liu-Harvey-Prausnitz model appear as difference, i.e. gj, - gLi, an arbitrary value can be assigned to one of them. Following the work of Liu ~1 al. (1989a), we choose g,, = 0 and the values of other interaction energy parameters are obviously relative ones.
Table
3. Values
of the 8 anionic
radius
parameters from literature
F-f
1.33 1.81 1.96 2.20 1.79 2.13 1.33 2.40
cl-’ Br-7 1-t NO; t CNS-f OH-t SO;-’
i Weast and Astle (1981). r Jenkins s Marcus
and Thakur (1983).
(1979).
Table 4. Values of the 9 cationic radius parameters from simultaneous fit of mean ionic-activity coefficient data for the 55 aqueous electrolytes at 298.15 K Cation H+ Li+ Na+ K+ Rb+ CSf
r, (A) 0.46290’ 0.82S93L 0.932049; 0.47434 0.86467 0.6453 1
Cation Mg++ ca++ zn++
r, (A) 0.96615 0.89035 1.16590”
+ For the HNO,/H*O system, yH+= 1.0440. * For the Li,SO,/H,O system, yLa+ = 1.5765. § For the NaOH/H,O system, yN1+ = 0.36967. 11 For the ZnBr,/H,O system, yNn+ + = 1.3864 and for the ZnI,/H,O system. yz.+ + = 1.6347.
Simultaneous correlation of activity coefficients for aqueous Table
5. Ion-specific
FH+ Li+ Na+ K+ Rb+ cs+ Mg Ca++ zn+ + H&J
1.4330 1.4549 1.7502 1.0644
1.5040 2.2844
interaction
energy parameters, g,,/RT, from simultaneous the 55 aqueous electrolytes at 298.15 K upon
Cl-
Br-
I_
0.587850 -0.159830 1.912500 0.923250 1.781700 1.799600 1.384400 -0.063688 -0.376740 1.798900
0.309470 -0.114240 0.072562 0.786470 1.549400 -0.212590 -0.207550 0.021254 -0.232130 1.920100
0.49395 0.25694 0.33787 0.69907 1.42050 0.83251 -0.14561 -00.40421 -0.45225 1.86350
+ For the NaNO,/H,O * For the H,S0,/H20
1.39020 1.13780 1.35060 0.75664
2.23320
activity coefficient
data for
1.15050 0.23416 1.37200 0.74555 1.17980 0.42118 0.42480 -0.2308B 0.38275 1.81230+
CNS
0.62306 0.81685
1.70630
so;
-
3.1076000 -0.5236800 0.5467500 0.146oooO 0.0315380 -0.7557200 -0.3953100 0.0048656 2.1021000’
HP -1.lllOo 1.50540 -0.70880 -0.28780 -0.47121 -0.63059 -0.54461 -0.64887 - 1.46830 0.0
system, g,,;,,/RT=2.8156. -, JRT= 5.2903.
system, gso;
Tables 6 and 7 gives a comparison between the LiuHarvey-Prausnitz model (1989a) and the HaghtalabVera model (1988). The results of the Liu-HarveyPrausnitz model are from the simultaneous correlation in this work and the results of the HaghtalabVera model are from the original paper (Haghtalab and Vera, 1988). It is worth noticing that the deviation in this work is based on the logarithm of the molefraction mean ionic activity coefficient while the deviation in Haghtalab and Vera’s work is based on the logarithm of the modality mean ionic activity coefficient. For the aqueous electrolytes, the conversion between the two kinds of activity coefficients can be achieved by the relationship: lny*, ,x=lny*,.,
fit of mean-ionic
1819
setting g,,=O
NOj
OH-
electrolytes
+ In [l + 0.01g015(cU, + w,)m]. (8)
It can be shown that eq. (8) does not affect the deviation on the logarithm scale. Therefore, the deviation in this work is equivalent to that in the work of Haghtalab and Vera. To perceive the comparison between the two models more directly and clearly, it is worth noticing that the sum of the square deviations of the 55 aqueous electrolytes is I.0181for the LiuHarvey-Prausnitz model and about 7.3 for the HaghtalabVera model, which shows the great favor towards the LiuHarvey-Prausnitz model. Figures 1 and 2 show the calculated and experimental mean ionic activity coefficients at 25°C for the systems of LiBr/H,O (up to 20 M), CaCI,/H,O (up to 10 M) and H,SOJH,O (up to 76 M). The calculated values by the Liu-HarveyyPrausnitz model agree very well with the experimental data. In the simultaneous data correlation for the 55 aqueous electrolytes, the total number of adjustable parameters is 88. Among them are 9 cationic radius parameters, 17 ion-H,0 interaction energy parameters, 55 cation-anion interaction energy parameters and 7 extraordinary parameters. The average number of adjustable parameters per aqueous electrolyte is 1.6, which is much lower than those in the
Table 6. Comparison between the model of Liu er ai. (1989a) and the model of Haghtalab and Vera (1988) (results from simultaneous fit of mean-ionic activity coefficient data for the 55 aqueous electrolytes at 298.15 K); data from Hamer and Wu (1972)
Uni-univalent electrolyte
Max. molality
%“y* (Liu et al.)
(Hz&dab and Vera)
HCI HBr HI HNO,
16 11 10 28
0.029 0.021 0.029 0.012
0.024 0.040 0.062 0.018
LiCl LiBr LiI LiOH LiNO,
19.219 20 3 4+
0.027 0.017 0.018 0.002 0.011
0.052 0.095 0.020 0.021 0.016
NaNO, NaCNS
12 29 10.830 18
0.001 0.016 0.018 0.013 0.043 0.030 0.009
0.002 0.011 0.061 0.028 0.084 0.072 0.039
KF KC1 KBr KI KOH KNO, KCNS
17.5 5 5.5 4.5 20 3.5 5
0.008 0.009 0.008 0.008 0.013 0.007 0.010
0.018 0.021 0.004 0.005 0.039 0.004 0.003
RbF RbCl RbBr Rbl RbNO,
3.5 7.8 5 5 4.5
0.003 0.006 0.003 0.005 0.010
0.009 0.003 0.001 0.002 0.007
3.5 11 5 3
0.005 0.009 0.017 0.001 0.010 0.004
0.007 0.013 0.005 0.005 0.012 0.000
NaF NaCl NaBr NaI
NaOH
CsF CsCl
CsBr CSI CsOH CsNO,
20
h4 Y
1.2 1.5
+ Results from the original paper (Haghtalab 1988) + The maximum molality is 5 M in Haghtalab work.
and Vera, and Vera’s
1820
YUNDA
LIU
and
URBAN
GI&N
Table 7. Comparison between the model of Liu et al. (1989a) and the model of Haghtalab and Vera (1988) (results from simultaneous fit of mean-ionic activity coefficient data for the 55 aqueous electrolytes at 298.15 K) The model of Haghtalab and Vera
The model of Liu et al.
Max. molality
Max. Electrolyte
molality
-tny*
Reft
Bi-univalent CaCI, CaBr, Cal, Ca(N% )z MgCl, MgBr, MgI, Mg(NO,)z ZnF, ZnCI, ZnBr, ZnI, Zn(NO,l,
10 6 1.915 6 5 5 5 5 0.142 6 6 6 6
0.043 0.034 0.031 0.03 1 0.050 0.010 0.008 0.017 0.001 0.012 0.019 0.046 0.032
Uni-bivalent H,SO, Li,SO, Na,SO, K,SO, Rb$O, cs,so,
76’ 3 4.445 0.692 1.707 1.631
0.017 0.013 0.039 0.001 0.002 0.018
Bi-bivalent MgSO, ZnSO,
3 3.5
0.032 0.050
e
OL,,
6 6
0.109 0.072
2 6 5 5 5 5 _
0.007 0.046 0.018 0.025 0.046 0.022
6 6 6 6
0.029 0.058 0.149 0.02 I
f
27.5
0.232
:: b b b
5 3 0.7 1.8 1.8
0.010 0.019 0.004 0.010 0.011
g
3 3.5
0.074 0.083
c C
: c c C
e a a a a a
Ref’
f f
’ a: Goldberg(l98la); b: Goldberg(t98lb); c: Goldberg and Nuttall (1978); d: Stokes and Robison (1948); e: Pitzer and Brewer (1961); f: Robison and Stokes (1959); g: Rard and Miller (1981); h: Staples (1981). t Results from the original paper (Haghtalab and Vera, 1988). * The low-concentration range is I M.
previously published works using other ion-specific parameter models for aqueous electrolytes. The Liu-Harvey-Prausnitz model has the same problem of all other models based on the local-composition concept, i.e. the interaction energy parameters in the local-composition expressions are generally strongly correlated. But in this work, we have noticed that the problem become less serious when the average number of adjustable parameters per
aqueous electrolyte is reduced. In our case that the average parameter number is reduced to I .6, the problem become almost nonexistent. CONCLUSION
The Liu-Harvey-Prausnitz model (1989a) for electrolyte solutions with ion-specific parameters has been successfully extended to simultaneously correlating the mean-ionic activity coefficient data for 55 aqueous
100 Exp. 10
o
COlC.-
1
79
0.1
~ 0
Fig. 1. Mean ionic activity coefficients system (up to 20 M) and the CaCI,/H,O at 25°C.
for the LiBr/H,O system (up to 10 M)
0.1
0.2
0.3
0.4
0.5
Fig. 2. Mean ionic activity coefficient for the H,SO,/ system (up to 76 M) at 25°C.
H,O
Simultaneous
correlation
of activity
electrolytes. The simultaneous data correlation requires a capable computational method for minimizing the square objective function with a large number of variables. Further compared with the NelderMead method, the modified Marquardt method (Draper and Smith, 1966; Bard, 1974) has demonstrated itself to be a favorable one for this purpose. A sizeconstant selection procedure has been proposed and the results have indicated the adequateness of the procedure. At present, the parameter values obtained in this work are being applied to the solubility prediction for multi-electrolyte aqueous solutions.
Acknowledgment-The authors are very grateful to the National Swedish Board for Technical Development and the Swedish Council for Building Research for the financial support of this work. The authors are also very grateful to Dr. L. Vamling for his highly skilful and concrete help on the parameter estimation programs.
NOTATION
D
nF
dielectric constant protonic charge interaction energy parameter of j-i pairs defined quantity [see eq. (l)] Boltzmann constant molality number of data points for the ith aqueous electrolyte the total number of all ionic species per unit
r, (r,)
radius
IJ
outer
R
gas constant
Sr
sum of the square
e 9ji Gji.
ki
k m 4
volume
Si T xi
Z
Greek Y
; VC,v, cr w,, a.
parameter radius
of cations
(or anions)
of the first coordination
shell
deviations of In y * for the 55 aqueous electrolytes sum of the square deviations of In y + for the ith aqueous electrolyte temperature (K) true mole fraction of species i based on all species (molecular and ionic) coordination number letters
activity coefficient on true mole-fraction scale defined quantity [see eq. (I)] size constant algebraic valences of cations and anions respectively root mean square deviation numbers of cations and anions, respectively, produced by the complete dissociation of one electrolyte molecule
Superscripts * unsymmetric
convention
coefficients
For aqueous
electrolytes
1821
Subscripts a anion c cation ca electrolyte ca LR long-range SR short-range w water + mean ionic REFERENCES
Bard, Y., 1974, Nonlinear parameter Estimation. Academic Press, New York. Draper, N. R. and Smith, H., 1966, Applied Regression Analysis. John Wiley, New York. Goldberg, R. N. and Nuttall, R. L., 1978, Evaluated activity and osmotic coefficients for aqueous solutions: the alkaline earth metal halides, J. phys. Chem. Ref: Data 7, 263-3 IO. Goldberg, R. N., 1981a, Evaluated activity and osmotic coefficients for aqueous solutions: bi-univalent compounds of zinc. cadmium and ethylene bisltrimethvlammonium) chloride and iodide. J. piys. CL-u& Ref: data 10, l-55. Goldberg, R. N., 1981b, Evaluated activity and osmotic coefficients for aaueous solutions: thirtv-six uni-bivalent electrolytes. J. ph;s. Chem. Rex Data Ii, 671-164. Haghtalab, A. and Vera, 1. H., 1988, A nonrandom factor model for the excess Gibbs energy of electrolyte solutions. z4.I.Ch.E. J. 34, 803-813. Hamer, W. J. and Wu, Y.-C., 1972, Osmotic coefficients and activity coefficients of uni-univalent electrolytes in water at 25°C. J. phys. Ckem. Ref Data 1, 1047-1099. Harned, H. S. and Owen, B. B., 1958, The Physicul Chemistry ojE[ectrolytic Solutions, 3rd Edition. Reinhold, New York. Liu, Y., Harvey, A. H. and Prausnitz, J. M., 1989a, Thermodynamics of concentrated electrolyte solutions. Chem. Engng. Commun. 77, 43-66. Liu, Y., Wimby, M. and Or&n, U., 1989b, An activitycoefficient model for electrolyte systems. Compur. them. Engng 13,405-410. Liu. Y., Grbn, U. and Wimby, M., l98Yc, A systematic comparison of nine models formed by various combinations of three Debye-Hiickel expressions and three local-composition expressions. Fluid Phase Equilibria 53, 269-278. Maurer. G., 1983, Electrolyte solutions. Fluid Phase Equilibria 13, 269-296. Nelder, J. A. and Mead, R., 1965, A simplex method for function minimisation. Comput. J. 7, 308-313. Netemeyer, S. C. and Glandt, E. D., 1988, A systematic evaluation of local composition-based excess free-energy theories using a model binary mixture. Ind. Engng Chem. Res. 27, 1516-1524. Pitzer, K. S., f981, Characteristics of very concentrated aqueous solutions. Phys. Chem. Earth 1%$4, 249-272. Rard, J. A. and Miller, D. G., 1981, lsopiestic determination of the osmotic coefficients of aqueous Na,SO,, MgSO, and Na,SO,-MgSO, at 25°C. J. &em. Engng Data 26, 33-38. Renon, H., 1986, Electrolyte solutions. Fluid Phase Equilibria 30, 181-195. Robinson, R. A. and Stokes, R. H., 1959, Electrolyte Sofutions, 2nd Edition. Butterworths, Boston, MA. Staples, B. R., 1981, Activity and osmotic coefficients of aqueous sulfuric acid at 298. I5 K. J. phys. Chern. Ref Data 10.779-798. Stokes, R. H. and Robinson, R. A., 1948, Ionic hydration and activity in electrolyte solutions. /. Am. them. Sot. 70, 1870-1878.
ODE-2509/91 $3 00 + Mm 0 1991 Pergamon Press plc
SIMULTANEOUS CORRELATION OF ACTIVITY COEFFICIENTS FOR 55 AQUEOUS ELECTROLYTES USING A MODEL WITH ION SPECIFIC PARAMETERS
Department
of Chemical
YUNDA LIU and URBAN GReNf Engineering Design, Chalmers University Gothenburg,
(First received
11 February
of Technology,
S-412 96
Sweden
1990; accepted
in revised form
19 November
1990)
Abstract-The Liu-Harvey-Prausnitz model (1989a) for electrolyte solutions with ion-specific parameters is extended to simultaneously correlating the mean ionic activity coefficient data for 55 aqueous electrolytes including the H,SO,/H,O system up to the H,SO, concentration of 76 m. With only 1.6 adjustable parameters per aqueous electrolyte, an excellent agreement between the calculated values and the experimental data has been achieved. Ion-specific parameters are reported for 9 cations (H ‘, Li+, Na+, K+, Rb+,Cs+,Mg++,Ca++andZn++)andS anions (F-,Cl-, Br-, I-, OH-, NO;, CNS- and SO; -1. The results in this work are compared with those given by the Haghtalab-Vera model (1988) and the outcome is in favor of the Liu-Harvey-Prausnitz model.
INTRODUCTION
Electrolyte solutions play an important role in the chemical industry and are also relevant in other fields of science and technology. Partitioning processes in biochemical systems; precipitation and crystallization processes in geological systems such as geothermal brines or drilling muds; desalination of water and water pollution control; salting-in and salting-out effects in extraction and distillation; absorption and regeneration in heat transformers are just a few examples of cases in which the knowledge of the thermodynamic properties of electrolyte solutions is basic for a proper understanding of the phenomena. For some applications, the knowledge of the behaviors of the dilute solutions or of solutions in the intermediateconcentration range is all that is required, while for others the region close to saturation of the solution is of great importance. In the last few decades, many models have been proposed to represent and predict the thermodynamic properties of electrolyte solutions. Pitzer (1981), Maurer (1983) and Renon (1986) give excellent reviews of the subject. A typical problem of the models proposed is that their use is limited to certain range of concentrations. At higher molalities, large errors are produced by most of these models. For many industrial processes, such as crystallization processes or absorption in heat transformers, it is necessary to have models able to represent and predict the thermodynamic behavior at higher molalities. A new model meeting this requirement was proposed by Liu, Harvey and Prausnitz (1989a). The model applied a new version of the Debye-Hiickel theory for the long-range interaction contributions and a local-composition expression of the Wilson type for the short-range interaction contri-
‘Author
to whom
correspondence
should
be addressed.
butions. The new Debye-Htickel expression was specially developed taking into consideration that the Debye-Hiickel expression and the local-composition expression were not independent of each other when they were combined. Promising applicability of the model was shown by a stringent test in a simultaneous correlation of activity coefficients for 6 aqueous electrolytes, four of which were regarded as hard to handle. Ion-specific parameters were reported for 3 cations (H+ , Li+ and K+) and 2 anions (Cl- and Br-). The highest concentration of the data used in the correlation was 20 M. The objeciive of the present work is to extend the mode1 to simultaneously correlating activity coefficients for 55 aqueous electrolytes, which include the H,SO,/H,O system up to the H,SO, concentration of 76 M, and to obtaining the values of the ion-specific parameters for 9 cations (H+, Li+, Na+, K+, Rb+, Cs+, Mg++, Ca++ and Zn’+) and 8 anions (F-, Cl-, Br-, I-, OH-, NO;, CNS- and SO;-). In 1988, Haghtalab and Vera proposed a successful mode1 which was claimed to be the best two-electrolyte-specific parameter mode1 for aqueous solutions of single electrolytes. This model also consists of two contributions. The Debye-Hiickel expression of Fowler and Guggenheim is used for the long-range interaction contributions, and the local-composition expression of the NRTL type with the reference cells in the random case is used for the short-range interaction contributions. In this work, the results of the simultaneous correlation using the Liu-HarveyPrausnitz mode1 are compared with those given by the Haghtalab-Vera model. THE LIU-HARVEY-PRAUSNITZ
MODEL
The Liu-Harvey-Prausnitz model (1989a) consists of two terms, one long-range and one short-range. The long-range term is represented by a new version of the Debye-Hlckel theory, which accounts for the
1815
1816
YUNDA
Lru and URIMN GRIN
contributions due to the electrostatic forces between each central ion and the ions outside its first coordination shell. The short-range term is represented by a local-composition expression of the Wilson type, which accounts for the contributions due to the shortrange interaction forces of all kinds, including the electrostatic interaction forces between each central ion and all ions inside the first coordination shell. Although the complete dissociation of electrolytes is assumed, the mode1 is capable of describing the nonideality of the electrolyte solutions with the significant ion assqciation, which is due to the indirect consideration of the ion association in the derivation of the model (Liu et al., 1989a). Therefore, in this work, the assumption of the complete dissociation remains unchanged not only because of its simplicity but also because of the capability of the model. The long-range term for single aqueous electrolytes is given as follows: e2
WLLR = 2DkT(w, + w,)
x
1
expCK(C- r&l (1 + Icr:) W.7V‘?
tr,
-lit
vex,=,
K
v,+ xc + x,G,.,,.
X ev CK(rL - r,)l (1 + Icr:)
-
where
Tl
Gji,,i = exp C- (Sji - s,i)lR
Ic=
4n e2 n,cx,$ [- DkT
+ x.vf,
r: = rc + Lr,
II l/2 1
“’
(1)
r: = r, + Iv,. The short-range term for single aqueous lytes is given as follows:
electro-
xw(~cGcw,ww + ~,G,w,w,) xw + xcGcw, ww + x, Go,.ww
X
+(I -xw)(w, +~a) --w,G,,.,, -~,Gow,,w w
+=,
ln
c
0,
+
1
xw)
(x, + x,1
[
+
(xaG,, wc
x,
x,$-C,
w x, xc + xw
11.
(2)
The symbols in the long-range and short-range terms are explained in the Notation section. The unsymmetric mean ionic activity coefficient of a single-aqueous electrolyte is obtained as In Y:,,
= In Y:.,,~ + In ~2.~~.
(3)
DATA CONSIDERATION
Unlike the models with electrolyte-specific parameters, the models with ion-specific parameters need a simultaneous correlation of data for all selected electrolyte systems in order to obtain optimal values of ion-specific parameters, which are common to alI the electrolyte systems containing the corresponding ions. This calls for a high consistency among each set of the electrolyte data. Therefore, the selection of data sources have to be carefully considered. In this work, the focus is to find out the values of the ion-specific parameters for the 9 cations (H+, Li’, Nat, K+, Rb+, Cs+, Mg++, Ca*+ and Zn’*) and the 8 anions (F-, Cl-, Br-, I-, OH-, NO;, CNSand SO; -). There are, up to now, activity-coefficient data at 25°C for 56 aqueous electrolytes, which are out of the possible cation-anion combinations of the 9 cations and the 8 anions. The data for all these aqueous electrolytes except one, which is the aqueous hydrofluoric acid, are used in the simultaneous correlation. The exclusion of the aqueous hydrofluoric acid is due to its rather peculiar thermodynamic: behavior, which is more like those of nonuniPunivalent aqueous electrolytes. It is worth mentioning that the data for the aqueous hydrofluoric acid is also excluded in the previous works by other authors using various other models. Among the 55 aqueous electrolytes, 34 of them are uni-univalent and 21 are nonuni-univalent. We adopt the evaluated experimental data from the well-known article by Hamer and Wu (1972) for the 34 uniunivalent aqueous electrolytes, and we adopt the carefully evaluated experimental data by Goldberg et al. (1978, 1981a, b) for the 16 nonuni-univalent aqueous electrolytes. The data for the remaining 5 nonuni-univalent aqueous electrolytes, which are not included in the Goldberg’s work, come from other data sources (see Table 7). We prefer the carefully evaluated experimental data to the data from the direct experiments because the former usually possesses a higher internal and external consistency than the latter does. In the separate correlation of the data for the LiOH/H,O system, we notice that the last data point increases the deviation by almost 10 times (from 0.00146 to 0.01028, where A = 1.5). Hamer and Wu (1972) also mark the last data point to indicate that there is some inconsistency between the data point and the rest of the others. We exclude this data point in the simultaneous correlation because one inconsistent data point will influence not only the result of the corresponding electrolyte but also the results of all the other electrolytes.
Simultaneous SIMULTANEOUS
CORRELATION
correlation
of activity
AND DISCUSSION
The values of the ion-specific parameters for the 9 cations (H+, Li+, Na+, K+, Rb+, Cs+, Mg+ +, Ca++ and Zn*+) and the 8 anions (F~,Cl~,Br~,I~, OH-, NO;, CNS and SO; -) are obtained by minimizing the sum of the square deviations between the calculated and the experimental data of the mean ionic activity coefficients for the 55 aqueous electrolytes: (4)
where nj is the number of data points for the ith electrolyte. The root mean square deviation of the ith electrolyte, o~,~*, is defined as
Since the ion-specific parameter estimation is carried out based on the simultaneous correlation of all selected electrolyte data, a capable computational method for minimizing the square objective function with a large number of independent variables is very much required. Liu, Grin and Wimby (1989~) have compared seven square-function minimizing methods and concluded that the modified Marquardt method (Draper and Smith, 1966; Bard, 1974), generally, performs best. In this work, the modified Marquardt method is further compared with the Nelder-Mead method (Nelder and Mead, 1965) in minimizing the function (4). It turns out that the favor towards the modified Marquardt method remains unchanged. The ion-specific parameters are common to all electrolyte solutions containing the corresponding ions. This means, for example, that the Cl--H,0 interaction parameter should have the same value for all the aqueous chloride electrolytes. But from the results of the separate data correlation for each of the 55 aqueous electrolytes, we find that some diverse values for the same ion-specific parameter are obtained. The diversity may be explained from different aspects, such as the possible inconsistency of the data among
coefficients
for aqueous
electrolytes
the relevant electrolytes; the different degrees of the electrolyte dissociation; the semi-empirical nature of the Liu-Harvey-Prausnitz model and the calculation simplifications (like using a linear expression to describe the concentration dcpcndcncc of the dielectric constants for all electrolytes as well as using the water density instead of the real densities of the electrolyte solutions). With the increase of the number of the electrolytes in the simultaneous correlation, foreseeably the diversity problem will become more serious: meanwhile, the average number of the adjustable parameters per aqueous electrolyte in the Liu-Harvey-Prausnitz model will gradually reduce and approach one. In order to apply the Liu-Harvey-Prausnitz model to properly represent the nonideality of the aqueous electrolytes up to the very high concentration with fewer average parameter number per electrolyte, the problem of the parameter diversity has to be solved adequately. In the previous works by Liu et al. (1989a, b), it has been found that the parameter values are sensitive to the size constant, L, which is used to determine the radius of the coordination shell. Table 1 shows how the parameter values for the LiNO,/H,O system vary with different size-constant values. The results in Table 1 are from the separate fit of the mean-ionic activity coefficient data for the LiNO,/H,O system at 25°C. In this work, we try to use the size constant to reduce the parameter diversity. The applied procedure is described as follows:
(I) separately
fit the data for each of the 55 aqueous electrolytes assuming different size-constant values, (2) compare the values of the same ion-specific parameter obtained from the relevant electrolytes at different size-constant values, (3) choose a proper size-constant value for each electrolyte to make the parameter diversity become minimum. Table 2 gives the size-constant values for the 55 aqueous electrolytes, which are chosen through the above procedure.
Table 1. The parameter
values for the LiNO,/H,O system obtained assuming different values of the size constant (1) (results from fit of the mean-ionic activity coefficient data for the LiN0,/H20 system in the concentration range 0.001~20.0 M and at the temperature 298.15 K)’
1.1
1.3 1.5 1.7
0.97390 - 0.86085 - 1.73170 - 1.89981
2.36170 0.82525 0.04286 - 0.15166
- 0.63024 1.32480 2.03439 2.28983
1.12472 0.87029 0.96489 1.08237
0.0050 0.0038 0.0055 0.0062
+ Data from Hamer and Wu (1972). A value of 6 is assumed for all coordination
numbers.
parameterof anion NO;
The anionic
1817
radius,
1.79 A, is used
as the radius
1818
YUNDA
Table 2. The size constants FH+ Li+ Na+ Kf Rb+ CS’ Mg++ Cat+ zn++
1.6 1.6 1.8 2.1
0.7
for the 55 aqueous
Cl-
Br-
I-
1.2 1.7 1.0 1.1 0.9 0.7 1.0 1.3 1.4
1.3 1.6 1.9 I.1 0.9 1.7 1.4 1.2 1.5
1.1 1.2 1.5 1.0 0.8 0.8 1.2 I.2 1.6
OH-
NO;
0.4 1.5 1.8
1.2 1.3 1.4 0.9 0.6
2.4
I .o 1.2 1.3 1.2
LIU and URBAN GREN
electrolytes
CNS
1.3 0.9
SO;0.4 1.1 0.7 0.8 1.0 1.3 0.8 0.6
During the above procedure. we also found out some extraordinary parameter values for certain electrolytes. For example, no matter how the size-constant value varies, the value of the cationic radius parameter obtained from the HNO,/H,O system is always much larger than those obtained from the rest of the other electrolyte systems which contain the ion II+. To deal with such cases, we just simply let the corresponding electrolytes keep their own parameter values. The total number of the extraordinary parameter values are seven. Table 4 and Table 5 give further information about them. It is worth mentioning that the separate parameter values for some special electrolytes cause no trouble in the prediction calculation for multi-electrolyte systems, which has been confirmed in solubility prediction calculations (the results will be shown in a subsequent paper). In this work, we have not set the size constant as an adjustable parameter for each electrolyte. That is partly because through the above procedure we can obtain more information about the parameter values and partly because by using the size-constant values obtained in such a way, the results of the simultaneous correlation are aleady very satisfactory, For the sake of simplicity, the pure water density of 55,409.3 mol/m3 is used for all the 55 aqueous electrolytes. As mentioned by Liu et al. (1989a), such an assumption causes no serious deviation. Netemeyer and Glandt (1988) have examined the influence of the coordination number on the predictivity of the local-composition models and concluded that the optimal coordination number corresponds approximately to a value of 6. This conclusion is consistent with the results obtained by Liu et al. (1989b). Therefore, in this work, the coordination numbers for all the cations and anions as well as water are fixed at the value of 6 in order to enhance the predictivity of the model in the future application with the parameter values from this correlation. As in the work of Liu er al. (1989b), the dielectric constant for all aqueous electrolyte systems except the H,SO,/HI,O system is assumed to be a linear function of the water mole fraction: D = 78.3 - 46.65(1 - x,).
(7)
Since the dielectric constant of the pure sulfuric acid is higher than that of the pure water (at 25”C, the former has the dielectric constant value of 101 and the latter
7X.3), it is unreasonable to assume that the dielectric constant of the H,SO,/H,O system will be reduced with the increase of the H,SO, concentration. Therefore, in addition to the consideration of simplicity, we assume that the dielectric constant of the H,SO,/ H,O system is the same as that of the pure water and independent of the H,SO, concentration. The activity coefficient of electrolyte solutions in the medium- and high-concentration ranges may be largely determined by the ionic radii (Harned and Owen, 1958). In this work, we have chosen the anionic radius parameters for the 8 anions from the values of the anionic radii published in the literature, which are shown in Tahle 3. The cationic radius parameters fnr the 9 cations are treated as adjustable parameters and the values are obtained from the simultaneous correlation of the mean-ionic-activity-coefficient data for the 55 aqueous electrolytes at 25°C. Table 4 gives the correlated values for the 9 cationic radius parameters. Table 5 gives the ion-specific interaction energy parameters, gji/R T, from the simultaneous correlation. Since all interaction energy parameters in the Liu-Harvey-Prausnitz model appear as difference, i.e. gj, - gLi, an arbitrary value can be assigned to one of them. Following the work of Liu ~1 al. (1989a), we choose g,, = 0 and the values of other interaction energy parameters are obviously relative ones.
Table
3. Values
of the 8 anionic
radius
parameters from literature
F-f
1.33 1.81 1.96 2.20 1.79 2.13 1.33 2.40
cl-’ Br-7 1-t NO; t CNS-f OH-t SO;-’
i Weast and Astle (1981). r Jenkins s Marcus
and Thakur (1983).
(1979).
Table 4. Values of the 9 cationic radius parameters from simultaneous fit of mean ionic-activity coefficient data for the 55 aqueous electrolytes at 298.15 K Cation H+ Li+ Na+ K+ Rb+ CSf
r, (A) 0.46290’ 0.82S93L 0.932049; 0.47434 0.86467 0.6453 1
Cation Mg++ ca++ zn++
r, (A) 0.96615 0.89035 1.16590”
+ For the HNO,/H*O system, yH+= 1.0440. * For the Li,SO,/H,O system, yLa+ = 1.5765. § For the NaOH/H,O system, yN1+ = 0.36967. 11 For the ZnBr,/H,O system, yNn+ + = 1.3864 and for the ZnI,/H,O system. yz.+ + = 1.6347.
Simultaneous correlation of activity coefficients for aqueous Table
5. Ion-specific
FH+ Li+ Na+ K+ Rb+ cs+ Mg Ca++ zn+ + H&J
1.4330 1.4549 1.7502 1.0644
1.5040 2.2844
interaction
energy parameters, g,,/RT, from simultaneous the 55 aqueous electrolytes at 298.15 K upon
Cl-
Br-
I_
0.587850 -0.159830 1.912500 0.923250 1.781700 1.799600 1.384400 -0.063688 -0.376740 1.798900
0.309470 -0.114240 0.072562 0.786470 1.549400 -0.212590 -0.207550 0.021254 -0.232130 1.920100
0.49395 0.25694 0.33787 0.69907 1.42050 0.83251 -0.14561 -00.40421 -0.45225 1.86350
+ For the NaNO,/H,O * For the H,S0,/H20
1.39020 1.13780 1.35060 0.75664
2.23320
activity coefficient
data for
1.15050 0.23416 1.37200 0.74555 1.17980 0.42118 0.42480 -0.2308B 0.38275 1.81230+
CNS
0.62306 0.81685
1.70630
so;
-
3.1076000 -0.5236800 0.5467500 0.146oooO 0.0315380 -0.7557200 -0.3953100 0.0048656 2.1021000’
HP -1.lllOo 1.50540 -0.70880 -0.28780 -0.47121 -0.63059 -0.54461 -0.64887 - 1.46830 0.0
system, g,,;,,/RT=2.8156. -, JRT= 5.2903.
system, gso;
Tables 6 and 7 gives a comparison between the LiuHarvey-Prausnitz model (1989a) and the HaghtalabVera model (1988). The results of the Liu-HarveyPrausnitz model are from the simultaneous correlation in this work and the results of the HaghtalabVera model are from the original paper (Haghtalab and Vera, 1988). It is worth noticing that the deviation in this work is based on the logarithm of the molefraction mean ionic activity coefficient while the deviation in Haghtalab and Vera’s work is based on the logarithm of the modality mean ionic activity coefficient. For the aqueous electrolytes, the conversion between the two kinds of activity coefficients can be achieved by the relationship: lny*, ,x=lny*,.,
fit of mean-ionic
1819
setting g,,=O
NOj
OH-
electrolytes
+ In [l + 0.01g015(cU, + w,)m]. (8)
It can be shown that eq. (8) does not affect the deviation on the logarithm scale. Therefore, the deviation in this work is equivalent to that in the work of Haghtalab and Vera. To perceive the comparison between the two models more directly and clearly, it is worth noticing that the sum of the square deviations of the 55 aqueous electrolytes is I.0181for the LiuHarvey-Prausnitz model and about 7.3 for the HaghtalabVera model, which shows the great favor towards the LiuHarvey-Prausnitz model. Figures 1 and 2 show the calculated and experimental mean ionic activity coefficients at 25°C for the systems of LiBr/H,O (up to 20 M), CaCI,/H,O (up to 10 M) and H,SOJH,O (up to 76 M). The calculated values by the Liu-HarveyyPrausnitz model agree very well with the experimental data. In the simultaneous data correlation for the 55 aqueous electrolytes, the total number of adjustable parameters is 88. Among them are 9 cationic radius parameters, 17 ion-H,0 interaction energy parameters, 55 cation-anion interaction energy parameters and 7 extraordinary parameters. The average number of adjustable parameters per aqueous electrolyte is 1.6, which is much lower than those in the
Table 6. Comparison between the model of Liu er ai. (1989a) and the model of Haghtalab and Vera (1988) (results from simultaneous fit of mean-ionic activity coefficient data for the 55 aqueous electrolytes at 298.15 K); data from Hamer and Wu (1972)
Uni-univalent electrolyte
Max. molality
%“y* (Liu et al.)
(Hz&dab and Vera)
HCI HBr HI HNO,
16 11 10 28
0.029 0.021 0.029 0.012
0.024 0.040 0.062 0.018
LiCl LiBr LiI LiOH LiNO,
19.219 20 3 4+
0.027 0.017 0.018 0.002 0.011
0.052 0.095 0.020 0.021 0.016
NaNO, NaCNS
12 29 10.830 18
0.001 0.016 0.018 0.013 0.043 0.030 0.009
0.002 0.011 0.061 0.028 0.084 0.072 0.039
KF KC1 KBr KI KOH KNO, KCNS
17.5 5 5.5 4.5 20 3.5 5
0.008 0.009 0.008 0.008 0.013 0.007 0.010
0.018 0.021 0.004 0.005 0.039 0.004 0.003
RbF RbCl RbBr Rbl RbNO,
3.5 7.8 5 5 4.5
0.003 0.006 0.003 0.005 0.010
0.009 0.003 0.001 0.002 0.007
3.5 11 5 3
0.005 0.009 0.017 0.001 0.010 0.004
0.007 0.013 0.005 0.005 0.012 0.000
NaF NaCl NaBr NaI
NaOH
CsF CsCl
CsBr CSI CsOH CsNO,
20
h4 Y
1.2 1.5
+ Results from the original paper (Haghtalab 1988) + The maximum molality is 5 M in Haghtalab work.
and Vera, and Vera’s
1820
YUNDA
LIU
and
URBAN
GI&N
Table 7. Comparison between the model of Liu et al. (1989a) and the model of Haghtalab and Vera (1988) (results from simultaneous fit of mean-ionic activity coefficient data for the 55 aqueous electrolytes at 298.15 K) The model of Haghtalab and Vera
The model of Liu et al.
Max. molality
Max. Electrolyte
molality
-tny*
Reft
Bi-univalent CaCI, CaBr, Cal, Ca(N% )z MgCl, MgBr, MgI, Mg(NO,)z ZnF, ZnCI, ZnBr, ZnI, Zn(NO,l,
10 6 1.915 6 5 5 5 5 0.142 6 6 6 6
0.043 0.034 0.031 0.03 1 0.050 0.010 0.008 0.017 0.001 0.012 0.019 0.046 0.032
Uni-bivalent H,SO, Li,SO, Na,SO, K,SO, Rb$O, cs,so,
76’ 3 4.445 0.692 1.707 1.631
0.017 0.013 0.039 0.001 0.002 0.018
Bi-bivalent MgSO, ZnSO,
3 3.5
0.032 0.050
e
OL,,
6 6
0.109 0.072
2 6 5 5 5 5 _
0.007 0.046 0.018 0.025 0.046 0.022
6 6 6 6
0.029 0.058 0.149 0.02 I
f
27.5
0.232
:: b b b
5 3 0.7 1.8 1.8
0.010 0.019 0.004 0.010 0.011
g
3 3.5
0.074 0.083
c C
: c c C
e a a a a a
Ref’
f f
’ a: Goldberg(l98la); b: Goldberg(t98lb); c: Goldberg and Nuttall (1978); d: Stokes and Robison (1948); e: Pitzer and Brewer (1961); f: Robison and Stokes (1959); g: Rard and Miller (1981); h: Staples (1981). t Results from the original paper (Haghtalab and Vera, 1988). * The low-concentration range is I M.
previously published works using other ion-specific parameter models for aqueous electrolytes. The Liu-Harvey-Prausnitz model has the same problem of all other models based on the local-composition concept, i.e. the interaction energy parameters in the local-composition expressions are generally strongly correlated. But in this work, we have noticed that the problem become less serious when the average number of adjustable parameters per
aqueous electrolyte is reduced. In our case that the average parameter number is reduced to I .6, the problem become almost nonexistent. CONCLUSION
The Liu-Harvey-Prausnitz model (1989a) for electrolyte solutions with ion-specific parameters has been successfully extended to simultaneously correlating the mean-ionic activity coefficient data for 55 aqueous
100 Exp. 10
o
COlC.-
1
79
0.1
~ 0
Fig. 1. Mean ionic activity coefficients system (up to 20 M) and the CaCI,/H,O at 25°C.
for the LiBr/H,O system (up to 10 M)
0.1
0.2
0.3
0.4
0.5
Fig. 2. Mean ionic activity coefficient for the H,SO,/ system (up to 76 M) at 25°C.
H,O
Simultaneous
correlation
of activity
electrolytes. The simultaneous data correlation requires a capable computational method for minimizing the square objective function with a large number of variables. Further compared with the NelderMead method, the modified Marquardt method (Draper and Smith, 1966; Bard, 1974) has demonstrated itself to be a favorable one for this purpose. A sizeconstant selection procedure has been proposed and the results have indicated the adequateness of the procedure. At present, the parameter values obtained in this work are being applied to the solubility prediction for multi-electrolyte aqueous solutions.
Acknowledgment-The authors are very grateful to the National Swedish Board for Technical Development and the Swedish Council for Building Research for the financial support of this work. The authors are also very grateful to Dr. L. Vamling for his highly skilful and concrete help on the parameter estimation programs.
NOTATION
D
nF
dielectric constant protonic charge interaction energy parameter of j-i pairs defined quantity [see eq. (l)] Boltzmann constant molality number of data points for the ith aqueous electrolyte the total number of all ionic species per unit
r, (r,)
radius
IJ
outer
R
gas constant
Sr
sum of the square
e 9ji Gji.
ki
k m 4
volume
Si T xi
Z
Greek Y
; VC,v, cr w,, a.
parameter radius
of cations
(or anions)
of the first coordination
shell
deviations of In y * for the 55 aqueous electrolytes sum of the square deviations of In y + for the ith aqueous electrolyte temperature (K) true mole fraction of species i based on all species (molecular and ionic) coordination number letters
activity coefficient on true mole-fraction scale defined quantity [see eq. (I)] size constant algebraic valences of cations and anions respectively root mean square deviation numbers of cations and anions, respectively, produced by the complete dissociation of one electrolyte molecule
Superscripts * unsymmetric
convention
coefficients
For aqueous
electrolytes
1821
Subscripts a anion c cation ca electrolyte ca LR long-range SR short-range w water + mean ionic REFERENCES
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