Ind. Eng. Chem. Process Des. Dev. 1902, 21, 396-400
396
Effect of the Various Parameters in the Application of Pitzer's Model to Solid-Liquid Equilibrium. Preliminary Study for Strong 1-1 Electrolytes Walter Furd and Henrl Renon* Groupe Commun R&cteurs et Processus, ENSTA-ENSMP, Equipe de Recherche Assmi& au C.N.R.S., ERA 766, 60, Bd Saint-Michel, 75272 Paris Cedex 06, France
Solubility isotherms of four ternary systems, NaCI-NaN03, NaCI-KCI, KCI-KN03, NaN0,-NaN03, are calculated using Pitzer's equation. There is good agreement between calculated curves and experimental data, but Pitzer's model applied to multicomponent systems has many parameters (eight for such systems). Effects of the various parameters of thii model have been studied in order to decrease the number of these parameters without significant precislon loss in solubilities calculations. Prediction of solubilities of such ternary systems are in good agreement with experimental data using only five of the original eight parameters.
Introduction Conception and optimization of industrial crystallization processes imply knowledge of the solubility diagrams for various operating conditions. However, obtaining such diagrams requires time-consuming experiments and therefore increases the cost of the operation. Appications of strong electrolyte models to solid-liquid equilibrium were proposed in recent publications (Correa and Vera, 1979; Lilley and Brown, 1979; Kusik and Meissner, 1978). The model should be able to predict nonideality at high ionic strength up to more than 10 M. Pitzer's model offers the best possibility of application by engineers because parameters are available for many systems. Representation is the most accurate among similar semiempirical models (at least in ita two-parameter variation) and especially prediction of multicomponent systems is among the best (Renon et al., 1981). This work shows, using, as examples of application, the systems NaC1-NaN0,-H20, KC1-NaC1-H20, KC1-KN03-H20, how solid-liquid equilibrium can be described using Pitzer's equation for the excess Gibbs energy. In the case of multicomponent systems, however, use of Pitzer's model implies knowledge of too many parameters for easy use. For instance, for a ternary system without common ion, the equation has 18 parameters, part of which can be obtained from experimental data of binary systems. The effect of the various parameters of the model will be described in order to find the minimal number of terms allowing a good representation of the solid-liquid equilibrium data. Basic Equations The equilibrium condition for generalized chemical reaction r between species x, in any system of phases (for instance solid salt = Zions in aqueous solutions)
Zv,J, m
=0
can be written as eq 2 relating the chemical potentials p m Cvmrkm = 0 (2) m
The activity coefficient in the molality scale is defined for ionic species by the equation pi(T,P,mj)= pi*(P,7') + RT In yi + RT In mi (3) where the standard chemical potential pi* is obtained from the condition 0196-4305/82/ 112 1-0396$01.25/0
yi+l as mi
-
0 with m j z i = 0
(4)
Pure water reference state is used for solvent. In the special case of a solid-liquid equilibrium reaction, the preceding equations can be written for each solid phase S in equilibrium with an aqueous solution VSrkS
+ Cvir&* + vHzOrk*H20+ CviPT 1n yimi + I
1
V H ~ O $InT U H ~ O= 0 (5) where p*H20 is the standard chemical potential defined as the chemical potential of pure water at the same temperature and pressure, VH O r differs from zero only in the case of hydrated solid s a k In the absence of solid solution, the chemical potential of the solid species ps does not depend on the composition of the solution. Combining into a single term AG,* all terms depending only on pressure and temperature, the preceding expression can be rewritten AG,*/RT =
CuirIn y; + Cvi, In mi + vH,Or In uH20 ( 6 ) i
1
It is possible to derive from Pitzer's model the equation for In yi and In U H 0 as functions of the molalities of all species in solution. kor each system, the value of AG,* may be taken from the literature or evaluated using equilibrium data. In this work, we use solid-liquid equilibrium data of binary systems to obtain the corresponding AG,* values. Finally, solution of eq 6 using Newton's method gives the molalities at equilibrium taking into account all reactions when enough independent variables are given. Pitzer's Model: Expression for In ri. Using interaction parameters of second or third order, the expression of excess Gibbs energy given by Pitzer is
GE/nwRT= f
+ CZBijmimj + CZOijmimj + i l i l
'/ZC(CmklZkl)Cijmimj + '/sCZC+ijkmimjmk (7) i
l
k
i l k
In this equation, second-order parameters are written B , if i and j are two ions of different sign, but contrary to the original notation of Pitzer, the summation is made over all ions i and j without ordering the cation before the anion; therefore B , = Bji. They are written 0, if i and j are ions of same signs (ei = 0). Likewise, third-order parameters C, are written only for two ions of different sign with Cij = Cji and J / i j k is written for three different ions 0 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 397 12
-
h,, 2t 1
*t
2 -
,
o ! 0
1
2
3
4
5
5
0
7
mLl-
(-a+,
with $ijk = $jki = $kij and $ikk = 0. Only Bij and Cij may be obtained from data on binary systems: electrolyte-water. Determination of Bi, and $ijk can be obtained only from data on systems made of water and two or more electrolytes. Long range interactions of electrostatic type are taken into account into the term f . We retain Pitzer's simplificative hypothesis that only parameter Bij depends on the ionic strength Z and $ijk = 0 if species i , j , and k have charges of the same sign. Partial derivation of the expression for GE yields expressions for ionic activity coefficient In yi =
ani
7
!
2
1
3
TIC
Figure 1. Solubility curves for the system NaNOS-NaCl-H,O using parameters given by Pitzer (Pitzer and Mayorga, 1973;Pitzer and Kim, 1974): experimental points (A),calculated curve using parameters $O), j3('), and C (-), using parameters fl(O), $I), and $ given in Tables I1 and I11 and using all original parameters ( - - -).
GGE/RT
0
Figure 2. Comparison of experimental and calculated solubility curves for the system KNOS-KC1-H20 using parameters given by Pitzer (Pitzer and Mayorga, 1973;Pitzer and Kim, 1974): experimental points (A),calculated curve using parameters $O), $'), and C (-), using parameters $O), $I), and given in Tables I1 and I11 (---), and using all original parameters (---).
+
1 - - 40
-
1
2
3
1
m
2: -f'+ 2CBijmj + ,ECB;kmjmk 2 1 I k
+ 2CBijmj + I
1
---i
0
2:
5
4
-
5
5
7
no.
Figure 3. Comparison of experimental and calculated solubility curves for the system NaC1-KCl-H20 using parameters given by Pitzer (Pitzer and Mayorga, 1973;Pitzer and Kim, 1974): experimental points (A),calculated curve using parameters @(O), b('),and C (-), using parameters $O), @(I), and given in Tables I1 and I11 (---), and using all original parameters (---).
+
(8)
where B>k= d B j k ! U , f ' = df/dl, and where the depen= 0). f and dence of Bij on ionic strength is neglected parameters Bij are functions of ionic strength. The expressions given by Pitzer (Pitzer and Kim, 1974; Pitzer and Mayorga, 1974) are
ADHis the Debye-Huckel constant
p.0)
Bij =
11 + -[1 21
PijcO)
For the 1-1,l-2,l-3,l-4, 2-2 electrolytes
- (1 + 2W2) e ~ p ( - 2 P / ~ )(10) ] and 1-5 electrolytes and for the
Zpij(1)
Bij = @..@I + -[ l - (1 + 1.4plz) e~p(-1.4p/~)] + ' (1.4)'Z
Application of Pitzer's Model to Solid-Liquid Equilibria of Multicomponent Systems Using Original Parameters. Diagrams at 25 OC for systems (I)
NaC1-NaNO3-H20, (11) NaCl-KCl-H,O, and (111) KC1KNO3-H20 were calculated using eq 2 and 8 and parameters given in Pitzer's articles (Pitzer and Kim, 1974; Pitzer and Mayorga, 1973). In each case, calculations have been repeated with and without mixing parameters Bi, and $ijk (see Figures 1 to 3) to test the possibility to predict equilibrium diagrams using data for binary systems only. Calculated curves are in good agreement with experimental points if we retain all terms in eq 8. Eight parameters are needed for a ternary system with a common ion, six parameters (four second-order B , and two thirdorder C,) obtained by fitting osmotic coefficients of binary systems, and two parameters (one of second-order Bij and one of third-order $ijk) obtained by fitting osmotic coefficients for ternary systems. These results are surprisingly good, in particular for system I, where ionic strength is about 11M at the point where two solid phases coexist. The parameters we used are indeed those given by Pitzer (Pitzer and Kim, 1974; Pitzer and Mayorga, 1973) and they are obtained from data of osmotic coefficients of solutions, the ionic strength of which is less or equal to 6 M (Robinson and Stokes, 1959). Figures 1,2, and 3 illustrate the need of mixing parameters 0 and 9 to predict the experimental solid-liquid equilibria in multicomponent systems and therefore the need of treating ternary systems data to obtain them. The results for the quinary system IV, NaC1-NaNO3KC1-KN03-Hz0, are given in Table I. Twelve parameters (p(O),p(l), and C type) obtained from values of osmotic
398 Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
$
s a 5 3 .d
0
c
f
Figure 4. Comparison of experimental and calculated solubility curves for the system NaNO,-NaCl-&O using parameters given in Tables I1 and 111: experimental points (A),calculated curve using $O), and C (-), using Po)and (- - -), and using and
o(’),
a(”’,o(”,
e (-.-).
z +
L I
E“ +
5
E
E‘ +
5
E‘ +
i
E
C
F +
F i3
E
2
E
3
6 “
C
5
m Figure 5. Comparison of Experimental and calculated solubility curves for the system KN03-KC1-H20 using parameters given in Tables I1 and I11 experimental points (A),calculated curve using $O), /3(’), and C (-), using pCo) and 8‘’) (- - -), and using $l), and
e (-.-I.
coefficients for binary systems electrolyte-water and six mixing parameters obtained from experimental osmotic coefficients for ternary systems were needed in the calculation. In this case, the prediction of the experimental data (Reinders, 1915) is not so good, but it should be remembered that ionic strength is very high (up to more than 16 M)and that parameters are optimized using experimental data for solutions, the ionic strength of which is lgss than or equal \o 6 M. Application of Pitzer’s Equation with a Lower Number of Interaction Parameters. The comparison of the influence of various parameteEs was established using parameters obtained from the treatment of identical sets of experimental activity coefficient data (Robinson and Stokes, 1959),by the least-square method. The treatment yields the set of parameters p(O) and p(l) or the other set @O), $l), and C. Our values which are given in Table I1 differ from those given by Pitzer because we did not use weighting factors. However, this does not affect significantly the difference between calculated and ekperimental activity coefficients for binary systems. Solid-liquid equilibrium curves for systems (I) NaCl-NaNO,-H,O, (11) NaC1-KC1-H20, (111) KC1-KN03-H20, and (IV) NaN02-NaN03-H20, have been calculated using the two sets of parameters. Comparison of the results is shown in Figures 4 to 7. Calculations of solid-liquid equilibria using only the B terms give a significant departure between calculated and experimental curves. Adding C terms does not improve the results, even at high ionic strength. Besides, for those systems, there is no significant influence on the calculated
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
Table 11. ParametersP(O), P ( l ) , and C Calculated Using a Least-Square Method from Experimental Values of (Robinson and Stokes, 1969)
399
'yt
electrolytes MX NaCl
KCl
KNO,
NaNO,
NaNO,
0.003
0.051
0.2037
0.1625
0.00013
-0.0023
a
0.002
a 0.01
0.0041 0.1968
0.0282 0.3505
a
a
Optimization of 3 Parameters
"'MAX
0.0757 (0.0765) 0.2795 (0.2664) 0.0014 (0.00127) 6 M
std dev
0.001
fl(O)MX P M X
0.0825 0.2492 6 M 0.004
P(o)MX P(1)MX
CMX
0.0464 (0.04835) 0.2274 (0.2122) -0.0004 (-0.0008) 4.5 M 0.001
-0.0802 (-0.0816) 0.0722 (0.0494) 0.0065 (0.0066) 3.5 M 0.002
Optimization of 2 Parameters
MAX
0.0448 0.2336 4.5 M 0.001
-0.0598 0.0022 3.5 M 0.006
0.04 a In t h e case of NaNO, and NaNO, electrolytes, we used, for optimization, experimental values of T~ up to 6 M and one s t d dev
value deduced from experimental data for saturated binary systems. (Pitzer and Mayorga, 1973).
0.003
The values in parentheses are those given by Pitzer
Table 111. Results of Optimization of Parameters e or $ from Solubility Diagrams system para- NaClmeter NaNO,
m
e IL
I-
't
ll 9-1 0
1
2
3
4
&
5
5
-0.0279 -0.0035
T
10
+,.\
NaClKCl
NaN0,NaNO,
-0.000815 -0.0203 -0.00023 -0.004
l2
-0.0144 -0.0012
T
,
7
m IC Figure 6. Comparison of experimental and calculated solubility curves for the system NaC1-KCl-H20 using parameters given in Tables and I11 experimental points (A),calcualted curve using Po), fl(l),and C (-), using $O) and $') (---), and using $O), $'), and 0 (-4. 12
KC1KNO,
? +
0
2
1
5
E
10
1 2 1 4
"0;
Figure 8. Comparison of experimental and calculated solubility curves for the system NaN0,-NaN02-H20: experimental points (A),calculated curve using parameters $O), $I), and J. given in Tables I1 and I11 (-) and using parameters @(O) and 8") given in Table I1 (- - -1.
I
of-9
t
'
2
I
6
m
8
1
1 0 1 2 1 4
NGt
Figure 7. Comparison of experimental and calculated solubility curves for the system NaN03-NaN02-H20 using parameters given in Tables II and IIk experimental points (A),calculated curve using parameters Po), $l), and C (-), using $O) and $l) (- - -), and using j3(O), fl(l),and 6 (---).
values of In yi of binary systems whether C parameters are used or not. On the contrary,good representation of experimental data is obtained using parameters $O), pl) and mixing parameters 8, assuming, as Pitzer does, that 8 does not depend on ionic strength. p(O) and @(') parameters are the same as before and 8 parameters were obtained from a least-square fit of solid-liquid equilibrium data (Seidell and Linke, 1965) for ternary systems. Results of the calculations are summarized in Table I11 and calculated curves appear in Figures 4 to 7.
Solubility curves can be calculated using only parameters of second order for solutions of ionic strength up to more than 10 M. But the parameters 8Cl-N obtained for systems NaCl-NaN0, and KCl-KNOB are%ifferent, against the fact that Pitzer's hypothesis implies that Bu parameters do not depend on the other ions present in solution. Indeed, BZjparameters given by Pitzer have the same value whatever the system where ions i or j are present. If the two systems NaC1-NaN0, and KC1-KNO, are compared, it appears that maximum ionic strength is 11.03 M in the first system and 6.98 M for the second. The different values of &).NOs obtained from data in the two systems can be explained by the breakdown of Pitzer's hypothesis of nondependence of 8 on ionic strength if no is added. Therefore, we tried to use terms neglecting 8 terms. A good representation of experimental points is obtained parameters are obtained from solid-liquid again. equilibrium data. Results of this optimization are given in Table I11 and calculated diagrams are shown in Figures 1 to 3 and 8.
+
+
+
400
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
For the systems under study with a common ion, it is possible to represent the whole solid-liquid equilibrium curve using four parameters derived from data on binary systems electrolytewater and only one mixing parameter obtained from the solid-liquid data of the ternary system. Osmotic coefficients for the various ternary systems were also calculated using the same equation with the mixing parameter 1c/ optimized from liquid-solid equilibrium experimental data. Experimental osmotic coefficients are well predicted. With 1c/ parameters from Table 111, differences between experimental and calculated osmotic coefficients are less than 2.5% considering various molalities for the systems KN03-NaN03, NaNO3-NaC1, KN0,-NaC1, KN03-KC1, NaN03-KC1, and KC1-NaCl (Bezboruah et al., 1970; Robinson, 1961). Parameters given by Pitzer are obtained by fitting the experimental osmotic coefficients and were used to predict solubilities; here it shows that the reverse process can be used successfully. Conclusion This work shows that, in the case of four strong 1-1 multicomponent systems, solid-liquid equilibrium can be predicted using Pitzer's approach of excess properties for electrolyte solutions. We have studied mixtures of two strong 1-1 electrolytes with a common ion. For such systems, the original Pitzer's model requires six parameters obtained from experimental activity or osmotic coefficienh for systems MX-H20 and MY-H20 and two parameters from such data for the MX-MY-H20 system. In this case, application of Pitzer's model requires eight parameters: fiMX(o), BMX"), BMY'O), PMY'", OXY, CMX, CMY, and. 1 c / q Y : We have shown, in the second part, that a simplified model could be used. Liquid-solid equilibrium diagrams can be represented using only four interaction parameters of second order deduced from experimental activity or osmotic coefficients for binary systems, PMX(O), &#, and only one "mixing" parameter + M x y obtained from experimental solubility data for the ternary systems. Nomenclature a = activity of water AM = Debye-Htickel constant Bij = interaction parameter of second order relating to i and j ions Cij = interaction parameter of third order relating to i and j ions d, =. density of water D = dielectric constant of water
e = charge of electron f = function defined in eq 9 GE = excess Gibbs energy relating to the solution AG,* = variation of standard GIBBS energy in reaction r I = ionic strength k = Boltzmann constant mi = molality of ionic species i n, = number of kg of solvent N = Avogadro's number P = pressure R = gas constant T = temperature X, = component m Zi = charge on the ion i pi,('), &i(') = interaction parameters of second order defined in eq 10 and 11 yi = activity coefficient on the molality scale p,,, = chemical potential of component m pi*. = standard chemical potential on a molality scale for ion 1
p*H20
= standard chemical potential for water
vmr = stoichiometric number of moles of component m in
reaction r Oij = mixing parameter of second order relating to i $ijb = mixing parameter of third order relating to
ions
and j ions
i, j , and k
Subscripts i = ionic species j = ionic species k = ionic species r = reaction S = solid species m = ionic, solid species or the solvent in case of hydrated salt
Literature Cited Bezboruah, C. P.; Covington, A. K.; Robinson, R. A. J. Chem. Thefmodyn. 1970, 2, 431. Correa, H. A.; Vera, J. H. Can. J . Chem. Eng. 1979, 53, 204. Kusik, C. L.; Meissner, H. P. AICMSymp. Ser. 1978, 7 4 , 14. Liiiey, T. H.; Brown, D. J. "Fluids and Fluid Mixtures"; Science and Technoiogy Press: Guildford, Surrey, England, 1979; p 206. Piker, K. S.; Maywa, G. J. Phys. Chem. 1973. 77, 2300. Piker, K. S.;Mayorga. G. J. SoluNon Ct".1974. 3, 539. Pltzer. K. S.; Kim, J. J. J. Am. Chem. Soc. 1974, 96, 5701. Relnders, W. Z. Ancfg. Chem. 1815, 93, 202. Renon, H.; Bell, F-X.; Plananche, H.; FOrst, W. Proceedings 2nd World Congress of Chemical Engineering, 1981, Vol. V, p 542. Robinson, R. A. J. Phys. Chem. 1961, 65, 662. Robinson, R. A.; Stokes, R. M. "Electrolytes Solutions"; Butterworths: London, 1959. Seidell, A.; Linke, W. F. "Solubilttiis of Inorganic and Organic Compounds"; Van Nostrand: New York, 1965; Vol. 11.
Receiued for reuiew September 10, 1980 Accepted January 11, 1982
