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Thermodynamic modelling of concentrated aqueous electrolyte and non-aqueous systems
Fluid Phase Equilibria, 69 (1991) 155-169
155
Elsevier Science Publishers B.V., Amsterdam
Thermodynamic modelling of concentrated electrolyte and non-aqueous systems
aqueous
A.R. Kolker Institute of New Technologies, Popov str. D/17,
Build. B, Leningrad, 197022 (USSR)
(Received July 25, 1990; accepted in final form July 5, 1991)
ABSTRACT Kolker, AR., 1991. Thermodynamic modelling of concentrated non-aqueous systems. Fluid Phase Equilibria 69: 155-169.
aqueous electrolyte
and
A new approach to computer simulation of activity coefficients of components in both aqueous and non-aqueous phases is proposed. Only values of standard thermodynamic properties of components are used as initial data. These are: the difference between standard free energy of formation of the pure substance in the solid state and in solution at infinite dilution in the standard state; the heat and temperature of salt melting; the change in heat capacity on fusion; and the average numbers of nearest neighbours of pure components. To describe non-ideality of solutions at high concentrations all kinds of species interactions are taken into account, e.g. free energy of salt-salt, salt-solvent and solventsolvent interactions. The multiwmponent systems are considered to be pseudobinary. An activity coefficient for each salt is estimated by integration of the Gibbs-Duhem equation. The solubility of salts and water activity in some binary and ternary systems are calculated in accordance with experimental data. The form of the equations for activity coefficients in the organic phase is the same as that in the aqueous phase. This method is applied to the calculation of the distribution ratio of caesium nitrate and nitric acid during extraction by an extractant based on a solution of cobalt dicarbolyde in nitrobenzene.
INTRODUCTION
The problem of calculating extraction equilibria in complex multicomponent systems by theoretical considerations is far from being solved. The main task is estimating the activity coefficients of all the species, both in the aqueous and the organic phases, within a consistent framework. Among the different methods for calculating activity coefficients of salts in the aqueous phase, Pitzer’s method was found to be the most applicable (Pitzer and Kim, 1974). However, this semiempirical method requires adjustable parameters to be determined experimentally and these parameters are 0378-3812/91/$03.50
0 1991 Elsevier Science Publishers B.V. All rights reserved
156
characteristic of the solvent. Pitzer’s equation cannot be used for organic solutions and for mixed solvent systems directly because its parameters are unknown functions of solvent composition. As for activity coefficients of components in the organic phase, there is no generally acknowledged model for such solutions. Some authors use the theory of association equilibria (Sedov et al., 1988), while others use quasi-lattice (Teng et al., 1982) or local-composition theories (Prausnitz et al., 1980). In either case, a number of adjustable parameters are required. The aim of this paper is (i) to show the possibility of describing the thermodynamic properties of the aqueous phase by non-electrolyte-type equations, and (ii) to show an application of these equations to the calculation of extraction equilibria without adjustable mixture parameters in terms of only infinite dilution properties and pure component properties.
AQUEOUS SOLUTIONS
Binary aqueous solutions
The concentration dependence of chemical potentials on the mole fraction scale can be derived most conveniently by Landau’s method (Landau and Lifshitz, 1964). The Gibbs energy of solution G is described in the following form: G =N,u’ +RTn[ln
nF(P,
T, N) - 11 +RT
c k=2
dB,(P,
T, N)/k!
(1)
where N is the number of solvent moles, n is the number of solute moles and B, = (akG/hzk),. Function G must be homogeneous and of the first power. Therefore F(P, T, N) and B,(P, T, N) must be inversely proportional to the number of moles. If the concentration of solute is expressed by its mole fraction x2 = n/(N + n), functions F(P, T, N) and B,(P, T, N) must be F(P, T, N) = F(P, T)/(N + n) and B,(P, T, N) = B,(P, T)/(N + njk-‘. Differentiation of G with respect to N and n gives the following expressions for the chemical potentials of solvent (component 1) and solute (component 2): ,u~ =& +RT p2 = & + Win
In xifr
al
=G
x2.6 - ~2)
where fr and f2 are activity coefficients.
(2) (3)
157
Equation (3) differs from the commonly used expression for the chemical potential of the solute by the addition of a non-logarithmic term. At very low concentration this can be neglected, but for x2 > 0.1 it can be significant. If the concentration of the solute is expressed by the ratio n/N one can similarly obtain pu, = & + RT In f,n/N. The ratio n/N is equal to m/mw and we obtain the commonly used expression for pZ on the molal scale, where m is molality, mw is 55.51 and ,u; = & - RT In m,. In concentrated solutions all kinds of interaction should be taken into account, i.e. salt-salt, salt-solvent and solvent-solvent. In most works the excess Gibbs energy of solution g E has been considered to be the sum of two contributions, one for long-range interactions (Debye-Hiickel treatment) and one for short-range interactions (Cruz and Renon, 1978; Chen et al., 1982; Liu and Gren, 1989). However, we shall assume that the electrostatic contribution can be considered not only in the Debye-Hiickel treatment, but also taken into account indirectly with the aid of the Gibbs energy of electrolyte formation in the reference state. In this case the Debye-Hiickel term will not be present in the gE expression in the explicit form and one expects the same type of expression for the excess Gibbs energy as is found for non-electrolytes (Pitzer, 1980; Pitzer and Simonson, 1986), especially at high concentrations, when interionic forces are effectively screened from long range to short range. This approach provides the possibility of calculating the parameters in the gE expression in terms of only standard infinite dilute and pure component properties rather than adjusting them on the basis of experimental mixture data. This procedure will be demonstrated later. We shall consider a water-salt solution consisting of “salt molecules” and water molecules. The concept of “salt molecules” in solution was introduced by Hertz (Hertz and Mills, 1978) and the term means a stoichiometric composition of ions with zero total charge which has instantaneous velocity in terms of the instantaneous velocities of the component ions. In other words, the “salt molecule” is the composition of the ion which for some period of time is moving similarly to the whole molecule. In this approach the expression of gE based on the NRTL version of this local composition concept is gE =V*[&%/(+& where
+x1) +42r1&142
++)I
AZ1 = exP(r&%), & = exP(rlZ/%)Y 712 VI2 - g22, 721 = i?2, =n21 +n,, and x1 = 1 -x2. Here the values of gij g11, a12 = 1212 + 12229 a21 are the Gibbs energies of interaction between species type i and central species type j (in RT units). The values of nij are the average numbers of nearest species type i which interact with central species type j (i, j = 1,2).
158
This g E expression differs from the original expression (Renon and Prausnitz, 1968) in two respects: (i) al2 is not equal to a2i, and (ii) aij are not adjustable parameters because they can be expressed in terms of nij. In turn the latter can be calculated, as will be shown later. The activity coefficients based on this gE expression are (Kolker, 1990a) ln f1= - [ 7126412 +~2&%
+42~I~2m142
++4;,)/(~2A2,
+“212
+xJ2]6
In f2 = - [712(42 -424/h412 t-726421
-
~1~241M~2~21
+
Q,
(4)
+x212 +xJ2]x?
+
Q2
(5)
where fi and f2 are expressed in the mole fraction scale and are appropriate to the reference state “pure substance - supercooled fused salt” (Rosen, 1979). The prime indicates the derivative with respect to the mole fraction x,, Q, =x$ Q2 = -x1x2. For calculating gij we can write (Kolker, 1990a) RT In fr(O) =,&i - & = A@ - AGF
(6)
RT in f2(0) = ,I.&- & = -AGO - AHa(1 - T/T,) + Ac,[ To - T(l + In To/T)]
(7)
where AGO = AC, - RT In mw/2, AG, = AG; - 664 and fii, &, t.~t and p! are the standard chemical potentials of components 1 and 2 in solution at infinite dilution, and in the “pure substance”, respectively, A@, AC;?:, AGY, AG,” are the standard Gibbs energies of formation at infinite dilute and in the “pure substance”, respectively, AH, and T, are the heat and temperature of salt melting, and AC, is the change in heat capacity on fusion. The term RT In m,/2 is concerned with converting values of A@ from the molal scale in which these values are given in handbooks to the mole fraction scale. If k phase transitions occur in the crystal structure of a salt with heat change Aitlr, at temperature Tk the term c AH,Jl - T/T,) should be added to the right-hand side of eqn. (7). The~~~ht-hand sides of eqns. (6) and (7) are consistent with the change in energy on removing species in a condensed phase to a distance at which they have zero interaction with each other, i.e. they are equal to -g,, and -g,. In calculating the limits, In fr as x1 + 0 and In f2 as x2 -+ 0, one can take into account that eqns. (2)~(5) are relevant to the concentration region x2 < 0.5. When x2 > 0.5 component 1 will be the solute and component 2 will be the solvent. Therefore the form of eqns. (2) and (3) will be
159
symmetrically changed. The x2 > 0.5 at Q, = -x1x2 and The functions In fr and the Gibbs-Duhem equation to limit gives
Gibbs-Duhem equation will be satisfied when Q2 =xf. In f2 have a break at x2 = 0.5. Integration of over the whole range 0 < x2 < 1 with transition
JimOln fr = In fl(0) = - [ 712 exp(r,,/n(:,)
+ rZ1] - 0.5
!lp,ln
+ 712] + 0.5
fi = In f2(0) = - [ 721 exp( T&,)
2
where ny, is the value of nil as x2 + 0 (considered to be the average number of nearest neighbours of pure component 1) and n$ is the value of nZ2 as x2 + 1 (considered to be the average number of nearest neighbours of pure component 2 in the standard state). On can utilize literature values for AG!, A@, AH,, T,, Tk and AC, (Rossini et al., 1952; Glushko, 1962-1981; Weast, 1985-1986) to calculate g,, as well as values for ny, and n$ (Petrucci, 1971). For the determination of the remaining unknown values additional equations are needed. They can be derived from the following set of equations, which is based on local composition theory:
n12/n22=x1/x* exP(r&%*) %l/%
=x2/x1
exP(72,/a,d
n12/n11 = exP(gI2/~,2 (42
+ n11Mn22
+ n21)
-
(8)
g1&21) =x1/2x2
The transition to limit as x2 + 0 results in two more equations which, together with eqns. (6) and (7), give the following set of equations for the determination of gll, gr2, g,, and n$ (as x2 + 0, n,, --f nyl, n12 + n:,, n22 + 0 and nzl + 0): 721
exp( 72lhL)
+
712 = g22
712
exp(
+
721 = cl
42/n:,
T12/4?) =
exp( &2/42
n$xP(~21/n~I)
- 4
(9)
- gd4,) + nY2[exP( -q,/nY,)
- 4 = 0
where in the final form g,, = AGO + AHo0 - T/T,) - AC&T, - T(1 + In To/T)] + EAH,J(l - T/T,) + 0.5 RT}/RT. The algorithm of calculating activity coefficients based on these relations has been given previously (Kolker, 1988). The calculation of concentration dependence of water activity for a number of salts is also presented in that work. In the present paper this model is applied to the calculation of the
160
TABLE 1 Solubility of salts rn and water activity ui in saturated solutions (25 o C) a NaF m
mexp a1
alero m m =P
al %xp
1.0 0.99 0.978 0.969
NaCl 6.0 6.14 0.758 0.753
CsCl
CF
11.3 11.37 0.655 0.658
35.2 35.6 0.054 0.040
NaI
NaNO,
KC1
12.5 12.3 0.345 0.380
11.0 10.83 0.734 0.738
4.80 4.83 0.842 0.843
5.69 5.74 0.807 0.808
8.92 8.96 0.655 0.686
3.80 3.84 0.927 0.924
CsBr
CSI
CsNO,
NaClO,
AgNO,
NH&l
1.4 1.4 0.978 0.965
10.0 9.94 0.749 0.751
15.0 15.12 0.821 0.828
7.5 7.39 0.767 0.771
5.8 5.8 0.831 0.826
3.62 3.33 0.893 0.906
a meXp and alexp are the experimental (Mikulin, 1968).
KBr
KI
KNOX
solubility and water activity in saturated
solution
solubility of a number of salts which do not form a crystal hydrate in the solid state. The equation for the solubility of such salts is -RT(ln
f2x2 -x2) = AH,(l - T/T,)
+ cAH,(l
- T/T,)
- Ac,[ T,, - T(l + 1: To/T)]
(10) The results of solubility calculations for a number of salts, m, and calculated values of water activity in saturated solutions, a,, are given in Table 1. The average error in the calculated values m and a, is determined within the framework of the model by the error in the initial data (first of all, by the error in the’ difference AG: - Ad,” and AH,). The procedure for determining the average error in the predicted a, and m values has been described earlier (Kolker, 1988) and it is in. the region l-3%. Nevertheless, it should be noted that the error in m values is much more dependent on the initial data than is the error in a,. However, in a number of data sources the values of A& AG: and AH, are noticeably different for some salts (Rossini et al., 1952; Glushko 1962-1981; Weast, 19851986). The values of AH,, and AC, and the difference AG: - A@’ which have been taken for solubility calculation are given in Kolker (1990a). In some cases the solution of the reverse problem is more expedient, i.e. determination of some thermodynamic constants in terms of experimental solubility data, because the latter are usually determined more accurately than the former. Multicomponent aqueous solutions The calculation of thermodynamic properties of multicomponent systems will be carried out, for the sake of simplicity, for ternary systems. We shall
161
consider a ternary system as a pseudobinary. This means that the system consists of a solvent (component 1) and a mixture of all dissolved salts, which are one complex component (2). Water activity is determined again using eqns. (2) and (4). Activity coefficients for each salt can be estimated by integration of the Gibbs-Duhem equation in the following manner. This equation can be written as a set of equations, each containing derivatives of solvent activity and one salt only m,(a
In a,/at72,)m.3
+ m,(a
In a,/am,)m2
+ mw(a In a,/am,),3
=0
(11)
m,(a
In a3/af722)m,
+ +(a
In a,/am,),z
+ m,(a
=0
(12)
In al/am&,
where a2, a3, m2 and m3 are the activities and molalities of salts (components 2 and 3), respectively, and m, = 1006/M, = 55.51. Introducing new independent variables namely the mole fraction of the second salt in the mixture z, and the mole fraction of the solvent x1, with z = m2/(m2 + m,), x1 = mw/(mw + m2 + m,>, and changing the old independent variables m2 and m3 in eqns. (11) and (12) for the new variables z and x1, one can obtain (Kolker, 1990b)
(a In a,/ax,),
= -(a
1n
(a In qax,),
= -(a
In a,/ax,),x,/x,
Integration
~l/a~l),~l/~2
+
(3
In
QW$
-d/G
- (a In a,/az),,z&
(13)
(14)
of these equations by parts at z = const. gives
ln fi = ln MO)= --x1/x2 ln fi - j’ln
f&dX,
- (1 - z)j’(a
In
f3
=
In
-
f3(0L
In fJa&/x~
dx, +x,
(15)
Xl
Xl
-q/x2
iln f&i I Xl
In
fl
dx, +&a
In f,/az),,/xg
dx, +x,
Xl
(16)
The
derivative (a In fi/az>,, can be found by analytical differentiation of In fi, which is determined by eqn. (4). Further, In f2(OjZ and In f&O)= are the values of In f2 and In f3 at x2 + 0 and at z = const. # 1. They are related to In f,(O) and In f,(O) in the appropriate binary solution (z = 1) as RT In fi(0), = b2= - p;, = b2( z = 1) - [ &( z = 1) + RT In j2] = RT In f2(0) - RT In fX i.e. In fz(OjZ = In f2(_O)- In fZ. Similarly In f,(O), = In f&O) - In fs. The values In fz and In f3 are the activity coefficients in the anhydrous mixture at xi + 0 (x, + 1). These values define the interaction energy of salts with
162
each other. When In fZ and In fS are equal to zero, an ideal mixture of salts is formed. The calculation of In fi in a ternary system requires the solution of sets of eqns. (8) and (9). As initial data functions, n(2:)(z) and g,,(z) for the mixture of salts are needed. The function n’,:‘(z) for the mixture can be written as the linear combination zn’:,)(2>+ (1 -z)@(3). However, g2, cannot be expressed in that form because real mixtures of salts are not ideal. Therefore g,, in the ternary system can be written in the form g, = (Z& + (1 -z)/-&) - (z/-4 + (1 - +4)
=
%22(2)
+
(1
-
4g22P)
+ EE
where gE = z In f2 + (1 - z> In f3, and g22(2) and g22(3) are equal to the right-hand side of eqn. (7) for salts 2 and 3 respectively. To calculate In f2 and In f3 the limit xJ + 0 at z = cons& in eqns. (15) and (16) is taken. In this case In f2 + In f2 and In f3 + In f3. AS a result we obtain two more equations ln
f2(0)- 2 In f2 = /olln f&
dx, + (1 -*)/o’(d
ln
f&k),,/x~ dx, + 1 (17)
In f3(0) - 2 In f3 = /,‘ln fl/.x,2dx, -*/‘(a 0
In
fJb).,/xz dxl + 1 (18)
These form, together with eqn. (9), a set of six equations for the estimation of six unknown values: g2i, gi2, gli, ny2, In F2 and In f3. Thus, for either
t-n2
(KCI)
Fig. 1. Solubility isotherm in the system H,O (l)-KC1 (2)-NaCI calculated data, points are experimental data (Mikulin, 1968).
(3). Solid lines are
163 TABLE 2 Activities of components
in the system H,O (lbCsNO,
(2)-HNO,
1122
m3
al
%Xp
In
1.162 0.988 1.136 1.646 3.06 0.286
0.949 2.761 4.802 7.148 11.206 7.156
0.952 0.876 0.783 0.680 0.520 0.688
0.943 0.884 0.806 0.684 0.574 0.698
- 12.8 - 10.7 -8.8 - 7.7 -6.1 -7.7
aa
lexp
(3) a
a3
In a3exp
- 7.9 -6.7 -7.9
andIn a3expare experimental data (Yakimov, and Mishin, 1967).
composition of the ternary system one can solve the set of eqns. (9), (17) and (18) and then calculate In f2, In f3 and a, with the aid of eqns. (15), (16), (2) and (4). Let us consider, as an example, the application of this method to the calculation of two systems with different types of component interaction. In the system H,O (l)-KC1 (2)-NaCl (3) the component activities on the branches of solubility KC1 and NaCl have been calculated. The criterion of calculation accuracy is approximately constant values of In a2 and In a3 and their equality to the right-hand side of eqn. (10). The application of this criterion enables us to calculate the isotherm of solubility, which is shown in Fig. 1. The predicted and experimental data are in good agreement, on the whole. Thermodynamic characteristics of a more complex system of another type, with intensive interaction of the components (Yakimov and Mishin, 1967), H,O cl)-CsNO, (2)-HNO, (3), are given in Table 2, in which the results of activity calculation for both volatile components are represented. The average absolute error in predicted a, values was in this case as much as 3%. ORGANIC SOLUTIONS AND SIMULATION OF DISTRIBUTION RATIOS DURING EXTRACTION OF CsNO, AND HNO, BY EXTRACTANT BASED ON COBALT DICARBOLYDE
The designed method of calculating activity coefficients in multicomponent systems can be applied in both the aqueous and organic phase. This enables us to calculate the extraction equilibrium. If the state reference of the solute in both phases is “pure substance”, the condition of extraction equilibrium will be the equality of the activities of each component number I in phases 1 and 2 (1 = water, 2 = organic), i.e.
#)fi(l) exp( -~j’)) = ~$“)fi”) exp( -xP))
(19)
164
Besides, the condition of thermodynamic stability should be observed. The solution of the set of equations (19) should satisfy the condition of material balance m~‘)Q(‘)+ m!2)Q(2)= my&(‘), where ml’) and rni2) are the molalities at equilibrium, Q(l) and Q@ are the masses of solvents and my is the molality in the initial solution. Calculation of fl in the organic phase is usually hindered by lack of values for the standard Gibbs energy of formation in infinite dilution, and this circumstance prevents estimation of g,, directly. However, this difficulty can be overcome in the following way. The standard Gibbs energy of transfer of a component from phase 1 into phase 2, AG,,, is determined by AG,, = AG, - AG,, where AG, and AG, are the Gibbs energy of hydration and solvation, respectively. The value of AG, is usually known, and hence AG, can be calculated if AG,, is determined. Unfortunately, the theory of calculating AG,, is not sufficiently well established for complex systems. This problem is of great interest for solution theory and still remains to be solved. The value of AG,, can be determined by experiment in several ways, but probably the most convenient method in our case is determination from the distribution ratio at infinite dilution, D,(O). For each component it can be written from eqn. (19) at xl + 0 as In O,,(O) = In f/‘)(O) - In ff2)(O>. As the molar distribution ratio D,,(O) is usually determined experimentally, we obtain from consideration of eqn. (7) AG,, = AG,, - In O,,(O) - ln[p~/~~mw/(lOOO/M,,>l, where py and pi are the densities of water and organic solvent, respectively, and A4_ is the average molecular weight of the organic solvent. Let us give an example of the calculation of distribution ratios during extraction of CsNO, and HNO, by extractant based on a solution of chlorinated cobalt dicarbolyde (CCD) in nitrobenzene. This system is chosen because of its importance in the reprocessing of liquid radioactive wastes and the complexity of describing it by traditional methods based on preliminary investigation of complex formation (Rais et al., 1976).
TABLE 3 Distribution ratios of microconcentrations of CsNO, (D,) and HNO, (D,) in the system based on 0.06 M chlorinated cobalt dicarbolyde in nitrobenzene at different HNO, concentrations, C, (mol 1-l) in the aqueous phase *
CH
Dl
D l=P
D2
0.3 0.5 1.0
116 89 56
128f9 90f7 47f7
0.48 0.42 0.28
a &xp
is from experimental
data (Afonin et al. 1989).
D 2=P
165
The distribution ratio of microconcentration of CsNO, when the HNO, concentration vanishes is approximately 11,1<0>= 180 at a CCD concentration of 0.06 M (Afonin et al., 1989). This gives AG,, = 2800 cal mol-‘. The distribution ratio of HNO,, D,,(O), is not precisely known, but DJO) must be less than unity and a value of about 0.5 at infinite dilution leads to AGSz= 7500 cal mol-r. Distribution ratios calculated with these values of AG,, at different HNO, concentrations in the aqueous phase, C,, are given in Table 3.
CONCLUSIONS
It has been shown in this paper that a description of the thermodynamic properties of both aqueous and organic phases at high concentrations by equations of a similar type, without mixture adjustable parameters, in terms of infinite dilution properties and pure component properties is possible. These equations can be used for the calculation of extraction equilibria and the solubility of substances. In a number of cases, solution of the reverse problem can, however, be more expedient, i.e. the determination of some thermodynamic characteristics of components in terms of experimental solubility or extraction data.
ACKNOWLEDGEMENT
The referee’s comments are much appreciated
by the author.
LIST OF SYMBOLS
A% CFl CCD Dl Q,(O) D,,(O> fi
fi(O)
activity of component number i function in eqn. (1) change in heat capacity on fusion molar concentration of HNO, in water chlorinated cobalt dicarbolyde distribution ratio of component number I in extraction system distribution ratio of component number 1 at infinite dilution on the mole fraction scale distribution ratio of component number I at infinite dilution on the molar scale activity coefficient of component number i mean of fi at infinite dilution
166
gij
G
w-l AC0 A@
AGj’
A% A% AHk AH0
ml m$‘) (2) mz
Krg Ml n nij 42 4,
N P Ql Q2
mean of fi at infinite dilution and z = const. # 1 (i = 2, 3) mean of fi in anhydrous salt mixtures at x1 + 0 (i = 2, 3) function in eqn. (1) molar excess Gibbs energy molar excess Gibbs energy of anhydrous salt mixtures Gibbs energy of interaction between species type i and central species type j in RT units (i, j = 1, 2) Gibbs energy of solution in eqn. (1) mean of the difference AG: - AGi mean of the difference AG, - RT In m,/2 standard Gibbs energy of formation of component number i in an infinitely dilute solution standard Gibbs energy of formation of component number i in the state “pure substance” standard Gibbs energy of solvation of component number 1 standard Gibbs energy of the component transfer from phase 1 to phase 2 heat of phase transition number k in a crystal structure of a salt heat of salt fusion molality of component number i (i = 2, 3) molality of component number I at equilibrium in phase 1 in extraction system molality of component number I at equilibrium in phase 2 in extraction system molality of component number I in initial aqueous solution in extraction system average molecular weight of organic phase in extraction system molecular weight of solvent (water) number of solute moles in eqn. (1) average number of nearest neighbours of type i which interact with central species type j (i, j = 1, 2) mean of n12 as x2 + 0 the mean of n,, as x2 --, 0 (considered as the average number of nearest neighbours of pure solvent) mean of n22 as x2 -+ 1 (considered as the average number of nearest neighbours of pure solute) number of solvent moles in eqn. (1) pressure in eqn. (1) x22at x2 < 0.5; -x1x2 at x2 > 0.5 -x1x2 at x2 < 0.5; xt at x2 > 0.5
167 Q(l) Q(2)
R T TIC To ‘i 4 (2)
Xl
z
solvent mass of phase 1 in extraction system solvent mass of phase 2 in extraction system universal gas constant temperature temperature of phase transition in a crystal structure of a salt temperature of salt melting mole fraction of component number i mole fraction of component number 1 in phase 1 in extraction system mole fraction of component number I in phase 2 in extraction system fraction of second salt in salts mixture (= m2/(wz2 + m,))
Greek letters
mean of the sum n,j + njj(i, j = 1, 2) standard chemical potential of component number i in an infinitely dilute solution standard chemical potential of component number i in the state “pure substance” &-RT In wzv,, density of the aqueous phase in the extraction system density of the organic phase in the extraction system mean of the difference gij - gjj Subscripts c
exp
H i
i, j k
1
erg S
tr x
molar scale experimental hydration number of component (1 = solvent; 2, 3 = solutes) species type in binary system (i, j = 1, 2) number of phase transitions in the crystal structure of a salt, summation index in eqn. (1). number of the component in the extraction system (1 = CsNO,; 2 = HNO,) organic solvation transfer mole fraction scale
168
Superscripts
1, Cl), (2) ,
E
infinite dilution (for lzll and lt12 as x2 --) 0) anhydrous (for nz2 as x2 + 1) number of phases in eqn. (19) (15 aqueous, 2 = organic) infinite dilute solution (for pi and AGF) anhydrous state (for fi and gE) derivative with respect to the mole fraction x1 (in eqns. (4) and (5)) excess
REFERENCES Afonin, M.A., Kopyrin, A.A., Korolev, V.V., Lazarev, L.N., Pochitonov, Yu.A. and Romanovskii, V.N., 1989. Optimization of extraction of cesium, strontium, rare earth and transplutonium elements from liquid highly radioactive wastes by extractant based on cobalt dicarbolyde. In: A.A. Kopyrin (Ed.), Chemistry and Technology of Rare Earth Elements. Lensoviet Instititute of Technology, Leningrad, pp. 17-24 (in Russian). Chen, C.-C., Britt, H.I., Boston, J.F. and Evans, L.B., 1982. Local composition model for excess Gibbs energy of electrolyte systems. AIChE J., 28: 588-596. Cruz, J.-L. and Renon, H., 1978. A new thermodynamic representation of binary electrolyte solution nonideality in the whole range of concentrations. AIChE J., 24: 817-825. Glushko, V.P. (Ed.), 1962-1981. Thermal Constants of Compounds, ~01s. 1-15. VINITI, Moscow (in Russian). Hertz, H. and Mills, R., 1978. Velocities correlations in aqueous electrolyte solutions from diffusion, conductance and transference data. J. Phys. Chem., 82: 952-960. Kolker, A.R., 1988. Calculation of concentration dependence of water activity in water-salts systems. Zh. Fiz. Khim., 62: 30-36 (in Russian). Kolker, A.R., 1990a. Calculation of salts solubility in binary systems. Dokl. Akad. Nauk. SSSR., 310: 634-639 (in Russian). Kolker, A.R., 199Ob. Simulation of thermodynamic properties of binary and ternary watersalts systems. In: A.A. Kopyrin (Ed.), Chemistry and Technology of Rare Earth Elements. Lensoviet Institute of Technology, Leningrad, pp. 9-19 (in Russian). Landau, L.D. and Lifshitz, E.M., 1964. Statistical Physics. Nauka, Moscow, 567 pp. (in Russian). Liu, Yunda and Gren, U., 1989. A systematic comparison of nine models formed by various combinations of three Debye-Hiickel expressions and three local-composition expressions for representation of non-ideality in electrolyte solutions. Fluid Phase Equilibria, 53: 269-277. Mikulin, G.M. (Ed.), 1968 Physical Chemistry of Electrolyte Solutions. Chimia, Leningrad, 419 pp. (in Russian). Petrucci, S., 1971. Ionic Interactions from Dilute Solutions to Fused Salts, Vol. 1. New York 410 pp. Pitzer, K.S., 1980. Electrolytes. From dilute solution to fused salts. J. Am. Chem. Sot., 102: 2902-2906.
169 Pitzer, K.S., and Kim, J.J., 1974. Thermodynamics of electrolytes. Activity and osmotic coefficients for mixed electrolytes. J. Am. Chem. Sot., 96: 5701-5707. Pitzer, K.S. and Simonson, J.M., 1986. Thermodynamics of multicomponent, miscible, ionic Systems: Theory and equations. J. Phys. Chem., 90: 3005-3009. Prausnitz, J.M., Andersen, T., Grens, E., Eckert, C., Hsieh, R. and O’Connel, J.P., 1980. Computer Calculations for Multicomponent Vapour-Liquid and Liquid-Liquid Equilibria. Prentice Hall, Englewood Cliffs, NJ. Rais, J., Selucky, P. and Kyrs, M., 1976. Extraction of alkali metals into nitrobenzene in the presence of univalent polyhedral borate anions. J. Inorg. Nucl. Chem., 38: 1376-1378. Renon, H. and Prausnitz, J.M., 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J., 14: 135-144. Rosen, A.M., 1979. Reference state “pure substance” as a tool for investigation of chemical interactions in electrolyte solutions. Dokl. Akad. Nauk SSSR, 249: 135-139 (in Russian). Rossini, F., Wagman, D., Evans, W., Levine, S. and Jaffe, I., 1952. Selected Values of Chemical Thermodynamic Properties. Circular NBS 500, Washington, 1268 pp. Sedov, V.M., Kopyrin, A.A., Proyayev, V.V. and Burtcev, I.A., 1988. The determination of thermodynamic characteristics of extraction equilibria in the systems containing alkylammonium salts and organic phosphorus compounds. In: Int. Solv. Extr. Conf., ISEC ‘88, Moscow, Conf. Papers, 1: 135-137. Teng, Teng, Li, Yi-Gui and Zhang, Liang-Ping, 1982. An investigation of the thermodynamics of solvent extraction of metals. Hydrometallurgy, 8: 261-272. Weast, R.C. (Ed.), 1985-1986. Handbook of Chemistry and Physics, 66th Edn. CRC Press, Boca Raton, FL. Yakimov, M.A. and Mishin, V.Ya., 1967. An investigation of heterogeneous equilibria in ternary metal nitrate-nitric acid-water systems. Zh. Neorg. Khim., 12: 1054-1063 (in Russian).
155
Elsevier Science Publishers B.V., Amsterdam
Thermodynamic modelling of concentrated electrolyte and non-aqueous systems
aqueous
A.R. Kolker Institute of New Technologies, Popov str. D/17,
Build. B, Leningrad, 197022 (USSR)
(Received July 25, 1990; accepted in final form July 5, 1991)
ABSTRACT Kolker, AR., 1991. Thermodynamic modelling of concentrated non-aqueous systems. Fluid Phase Equilibria 69: 155-169.
aqueous electrolyte
and
A new approach to computer simulation of activity coefficients of components in both aqueous and non-aqueous phases is proposed. Only values of standard thermodynamic properties of components are used as initial data. These are: the difference between standard free energy of formation of the pure substance in the solid state and in solution at infinite dilution in the standard state; the heat and temperature of salt melting; the change in heat capacity on fusion; and the average numbers of nearest neighbours of pure components. To describe non-ideality of solutions at high concentrations all kinds of species interactions are taken into account, e.g. free energy of salt-salt, salt-solvent and solventsolvent interactions. The multiwmponent systems are considered to be pseudobinary. An activity coefficient for each salt is estimated by integration of the Gibbs-Duhem equation. The solubility of salts and water activity in some binary and ternary systems are calculated in accordance with experimental data. The form of the equations for activity coefficients in the organic phase is the same as that in the aqueous phase. This method is applied to the calculation of the distribution ratio of caesium nitrate and nitric acid during extraction by an extractant based on a solution of cobalt dicarbolyde in nitrobenzene.
INTRODUCTION
The problem of calculating extraction equilibria in complex multicomponent systems by theoretical considerations is far from being solved. The main task is estimating the activity coefficients of all the species, both in the aqueous and the organic phases, within a consistent framework. Among the different methods for calculating activity coefficients of salts in the aqueous phase, Pitzer’s method was found to be the most applicable (Pitzer and Kim, 1974). However, this semiempirical method requires adjustable parameters to be determined experimentally and these parameters are 0378-3812/91/$03.50
0 1991 Elsevier Science Publishers B.V. All rights reserved
156
characteristic of the solvent. Pitzer’s equation cannot be used for organic solutions and for mixed solvent systems directly because its parameters are unknown functions of solvent composition. As for activity coefficients of components in the organic phase, there is no generally acknowledged model for such solutions. Some authors use the theory of association equilibria (Sedov et al., 1988), while others use quasi-lattice (Teng et al., 1982) or local-composition theories (Prausnitz et al., 1980). In either case, a number of adjustable parameters are required. The aim of this paper is (i) to show the possibility of describing the thermodynamic properties of the aqueous phase by non-electrolyte-type equations, and (ii) to show an application of these equations to the calculation of extraction equilibria without adjustable mixture parameters in terms of only infinite dilution properties and pure component properties.
AQUEOUS SOLUTIONS
Binary aqueous solutions
The concentration dependence of chemical potentials on the mole fraction scale can be derived most conveniently by Landau’s method (Landau and Lifshitz, 1964). The Gibbs energy of solution G is described in the following form: G =N,u’ +RTn[ln
nF(P,
T, N) - 11 +RT
c k=2
dB,(P,
T, N)/k!
(1)
where N is the number of solvent moles, n is the number of solute moles and B, = (akG/hzk),. Function G must be homogeneous and of the first power. Therefore F(P, T, N) and B,(P, T, N) must be inversely proportional to the number of moles. If the concentration of solute is expressed by its mole fraction x2 = n/(N + n), functions F(P, T, N) and B,(P, T, N) must be F(P, T, N) = F(P, T)/(N + n) and B,(P, T, N) = B,(P, T)/(N + njk-‘. Differentiation of G with respect to N and n gives the following expressions for the chemical potentials of solvent (component 1) and solute (component 2): ,u~ =& +RT p2 = & + Win
In xifr
al
=G
x2.6 - ~2)
where fr and f2 are activity coefficients.
(2) (3)
157
Equation (3) differs from the commonly used expression for the chemical potential of the solute by the addition of a non-logarithmic term. At very low concentration this can be neglected, but for x2 > 0.1 it can be significant. If the concentration of the solute is expressed by the ratio n/N one can similarly obtain pu, = & + RT In f,n/N. The ratio n/N is equal to m/mw and we obtain the commonly used expression for pZ on the molal scale, where m is molality, mw is 55.51 and ,u; = & - RT In m,. In concentrated solutions all kinds of interaction should be taken into account, i.e. salt-salt, salt-solvent and solvent-solvent. In most works the excess Gibbs energy of solution g E has been considered to be the sum of two contributions, one for long-range interactions (Debye-Hiickel treatment) and one for short-range interactions (Cruz and Renon, 1978; Chen et al., 1982; Liu and Gren, 1989). However, we shall assume that the electrostatic contribution can be considered not only in the Debye-Hiickel treatment, but also taken into account indirectly with the aid of the Gibbs energy of electrolyte formation in the reference state. In this case the Debye-Hiickel term will not be present in the gE expression in the explicit form and one expects the same type of expression for the excess Gibbs energy as is found for non-electrolytes (Pitzer, 1980; Pitzer and Simonson, 1986), especially at high concentrations, when interionic forces are effectively screened from long range to short range. This approach provides the possibility of calculating the parameters in the gE expression in terms of only standard infinite dilute and pure component properties rather than adjusting them on the basis of experimental mixture data. This procedure will be demonstrated later. We shall consider a water-salt solution consisting of “salt molecules” and water molecules. The concept of “salt molecules” in solution was introduced by Hertz (Hertz and Mills, 1978) and the term means a stoichiometric composition of ions with zero total charge which has instantaneous velocity in terms of the instantaneous velocities of the component ions. In other words, the “salt molecule” is the composition of the ion which for some period of time is moving similarly to the whole molecule. In this approach the expression of gE based on the NRTL version of this local composition concept is gE =V*[&%/(+& where
+x1) +42r1&142
++)I
AZ1 = exP(r&%), & = exP(rlZ/%)Y 712 VI2 - g22, 721 = i?2, =n21 +n,, and x1 = 1 -x2. Here the values of gij g11, a12 = 1212 + 12229 a21 are the Gibbs energies of interaction between species type i and central species type j (in RT units). The values of nij are the average numbers of nearest species type i which interact with central species type j (i, j = 1,2).
158
This g E expression differs from the original expression (Renon and Prausnitz, 1968) in two respects: (i) al2 is not equal to a2i, and (ii) aij are not adjustable parameters because they can be expressed in terms of nij. In turn the latter can be calculated, as will be shown later. The activity coefficients based on this gE expression are (Kolker, 1990a) ln f1= - [ 7126412 +~2&%
+42~I~2m142
++4;,)/(~2A2,
+“212
+xJ2]6
In f2 = - [712(42 -424/h412 t-726421
-
~1~241M~2~21
+
Q,
(4)
+x212 +xJ2]x?
+
Q2
(5)
where fi and f2 are expressed in the mole fraction scale and are appropriate to the reference state “pure substance - supercooled fused salt” (Rosen, 1979). The prime indicates the derivative with respect to the mole fraction x,, Q, =x$ Q2 = -x1x2. For calculating gij we can write (Kolker, 1990a) RT In fr(O) =,&i - & = A@ - AGF
(6)
RT in f2(0) = ,I.&- & = -AGO - AHa(1 - T/T,) + Ac,[ To - T(l + In To/T)]
(7)
where AGO = AC, - RT In mw/2, AG, = AG; - 664 and fii, &, t.~t and p! are the standard chemical potentials of components 1 and 2 in solution at infinite dilution, and in the “pure substance”, respectively, A@, AC;?:, AGY, AG,” are the standard Gibbs energies of formation at infinite dilute and in the “pure substance”, respectively, AH, and T, are the heat and temperature of salt melting, and AC, is the change in heat capacity on fusion. The term RT In m,/2 is concerned with converting values of A@ from the molal scale in which these values are given in handbooks to the mole fraction scale. If k phase transitions occur in the crystal structure of a salt with heat change Aitlr, at temperature Tk the term c AH,Jl - T/T,) should be added to the right-hand side of eqn. (7). The~~~ht-hand sides of eqns. (6) and (7) are consistent with the change in energy on removing species in a condensed phase to a distance at which they have zero interaction with each other, i.e. they are equal to -g,, and -g,. In calculating the limits, In fr as x1 + 0 and In f2 as x2 -+ 0, one can take into account that eqns. (2)~(5) are relevant to the concentration region x2 < 0.5. When x2 > 0.5 component 1 will be the solute and component 2 will be the solvent. Therefore the form of eqns. (2) and (3) will be
159
symmetrically changed. The x2 > 0.5 at Q, = -x1x2 and The functions In fr and the Gibbs-Duhem equation to limit gives
Gibbs-Duhem equation will be satisfied when Q2 =xf. In f2 have a break at x2 = 0.5. Integration of over the whole range 0 < x2 < 1 with transition
JimOln fr = In fl(0) = - [ 712 exp(r,,/n(:,)
+ rZ1] - 0.5
!lp,ln
+ 712] + 0.5
fi = In f2(0) = - [ 721 exp( T&,)
2
where ny, is the value of nil as x2 + 0 (considered to be the average number of nearest neighbours of pure component 1) and n$ is the value of nZ2 as x2 + 1 (considered to be the average number of nearest neighbours of pure component 2 in the standard state). On can utilize literature values for AG!, A@, AH,, T,, Tk and AC, (Rossini et al., 1952; Glushko, 1962-1981; Weast, 1985-1986) to calculate g,, as well as values for ny, and n$ (Petrucci, 1971). For the determination of the remaining unknown values additional equations are needed. They can be derived from the following set of equations, which is based on local composition theory:
n12/n22=x1/x* exP(r&%*) %l/%
=x2/x1
exP(72,/a,d
n12/n11 = exP(gI2/~,2 (42
+ n11Mn22
+ n21)
-
(8)
g1&21) =x1/2x2
The transition to limit as x2 + 0 results in two more equations which, together with eqns. (6) and (7), give the following set of equations for the determination of gll, gr2, g,, and n$ (as x2 + 0, n,, --f nyl, n12 + n:,, n22 + 0 and nzl + 0): 721
exp( 72lhL)
+
712 = g22
712
exp(
+
721 = cl
42/n:,
T12/4?) =
exp( &2/42
n$xP(~21/n~I)
- 4
(9)
- gd4,) + nY2[exP( -q,/nY,)
- 4 = 0
where in the final form g,, = AGO + AHo0 - T/T,) - AC&T, - T(1 + In To/T)] + EAH,J(l - T/T,) + 0.5 RT}/RT. The algorithm of calculating activity coefficients based on these relations has been given previously (Kolker, 1988). The calculation of concentration dependence of water activity for a number of salts is also presented in that work. In the present paper this model is applied to the calculation of the
160
TABLE 1 Solubility of salts rn and water activity ui in saturated solutions (25 o C) a NaF m
mexp a1
alero m m =P
al %xp
1.0 0.99 0.978 0.969
NaCl 6.0 6.14 0.758 0.753
CsCl
CF
11.3 11.37 0.655 0.658
35.2 35.6 0.054 0.040
NaI
NaNO,
KC1
12.5 12.3 0.345 0.380
11.0 10.83 0.734 0.738
4.80 4.83 0.842 0.843
5.69 5.74 0.807 0.808
8.92 8.96 0.655 0.686
3.80 3.84 0.927 0.924
CsBr
CSI
CsNO,
NaClO,
AgNO,
NH&l
1.4 1.4 0.978 0.965
10.0 9.94 0.749 0.751
15.0 15.12 0.821 0.828
7.5 7.39 0.767 0.771
5.8 5.8 0.831 0.826
3.62 3.33 0.893 0.906
a meXp and alexp are the experimental (Mikulin, 1968).
KBr
KI
KNOX
solubility and water activity in saturated
solution
solubility of a number of salts which do not form a crystal hydrate in the solid state. The equation for the solubility of such salts is -RT(ln
f2x2 -x2) = AH,(l - T/T,)
+ cAH,(l
- T/T,)
- Ac,[ T,, - T(l + 1: To/T)]
(10) The results of solubility calculations for a number of salts, m, and calculated values of water activity in saturated solutions, a,, are given in Table 1. The average error in the calculated values m and a, is determined within the framework of the model by the error in the initial data (first of all, by the error in the’ difference AG: - Ad,” and AH,). The procedure for determining the average error in the predicted a, and m values has been described earlier (Kolker, 1988) and it is in. the region l-3%. Nevertheless, it should be noted that the error in m values is much more dependent on the initial data than is the error in a,. However, in a number of data sources the values of A& AG: and AH, are noticeably different for some salts (Rossini et al., 1952; Glushko 1962-1981; Weast, 19851986). The values of AH,, and AC, and the difference AG: - A@’ which have been taken for solubility calculation are given in Kolker (1990a). In some cases the solution of the reverse problem is more expedient, i.e. determination of some thermodynamic constants in terms of experimental solubility data, because the latter are usually determined more accurately than the former. Multicomponent aqueous solutions The calculation of thermodynamic properties of multicomponent systems will be carried out, for the sake of simplicity, for ternary systems. We shall
161
consider a ternary system as a pseudobinary. This means that the system consists of a solvent (component 1) and a mixture of all dissolved salts, which are one complex component (2). Water activity is determined again using eqns. (2) and (4). Activity coefficients for each salt can be estimated by integration of the Gibbs-Duhem equation in the following manner. This equation can be written as a set of equations, each containing derivatives of solvent activity and one salt only m,(a
In a,/at72,)m.3
+ m,(a
In a,/am,)m2
+ mw(a In a,/am,),3
=0
(11)
m,(a
In a3/af722)m,
+ +(a
In a,/am,),z
+ m,(a
=0
(12)
In al/am&,
where a2, a3, m2 and m3 are the activities and molalities of salts (components 2 and 3), respectively, and m, = 1006/M, = 55.51. Introducing new independent variables namely the mole fraction of the second salt in the mixture z, and the mole fraction of the solvent x1, with z = m2/(m2 + m,), x1 = mw/(mw + m2 + m,>, and changing the old independent variables m2 and m3 in eqns. (11) and (12) for the new variables z and x1, one can obtain (Kolker, 1990b)
(a In a,/ax,),
= -(a
1n
(a In qax,),
= -(a
In a,/ax,),x,/x,
Integration
~l/a~l),~l/~2
+
(3
In
QW$
-d/G
- (a In a,/az),,z&
(13)
(14)
of these equations by parts at z = const. gives
ln fi = ln MO)= --x1/x2 ln fi - j’ln
f&dX,
- (1 - z)j’(a
In
f3
=
In
-
f3(0L
In fJa&/x~
dx, +x,
(15)
Xl
Xl
-q/x2
iln f&i I Xl
In
fl
dx, +&a
In f,/az),,/xg
dx, +x,
Xl
(16)
The
derivative (a In fi/az>,, can be found by analytical differentiation of In fi, which is determined by eqn. (4). Further, In f2(OjZ and In f&O)= are the values of In f2 and In f3 at x2 + 0 and at z = const. # 1. They are related to In f,(O) and In f,(O) in the appropriate binary solution (z = 1) as RT In fi(0), = b2= - p;, = b2( z = 1) - [ &( z = 1) + RT In j2] = RT In f2(0) - RT In fX i.e. In fz(OjZ = In f2(_O)- In fZ. Similarly In f,(O), = In f&O) - In fs. The values In fz and In f3 are the activity coefficients in the anhydrous mixture at xi + 0 (x, + 1). These values define the interaction energy of salts with
162
each other. When In fZ and In fS are equal to zero, an ideal mixture of salts is formed. The calculation of In fi in a ternary system requires the solution of sets of eqns. (8) and (9). As initial data functions, n(2:)(z) and g,,(z) for the mixture of salts are needed. The function n’,:‘(z) for the mixture can be written as the linear combination zn’:,)(2>+ (1 -z)@(3). However, g2, cannot be expressed in that form because real mixtures of salts are not ideal. Therefore g,, in the ternary system can be written in the form g, = (Z& + (1 -z)/-&) - (z/-4 + (1 - +4)
=
%22(2)
+
(1
-
4g22P)
+ EE
where gE = z In f2 + (1 - z> In f3, and g22(2) and g22(3) are equal to the right-hand side of eqn. (7) for salts 2 and 3 respectively. To calculate In f2 and In f3 the limit xJ + 0 at z = cons& in eqns. (15) and (16) is taken. In this case In f2 + In f2 and In f3 + In f3. AS a result we obtain two more equations ln
f2(0)- 2 In f2 = /olln f&
dx, + (1 -*)/o’(d
ln
f&k),,/x~ dx, + 1 (17)
In f3(0) - 2 In f3 = /,‘ln fl/.x,2dx, -*/‘(a 0
In
fJb).,/xz dxl + 1 (18)
These form, together with eqn. (9), a set of six equations for the estimation of six unknown values: g2i, gi2, gli, ny2, In F2 and In f3. Thus, for either
t-n2
(KCI)
Fig. 1. Solubility isotherm in the system H,O (l)-KC1 (2)-NaCI calculated data, points are experimental data (Mikulin, 1968).
(3). Solid lines are
163 TABLE 2 Activities of components
in the system H,O (lbCsNO,
(2)-HNO,
1122
m3
al
%Xp
In
1.162 0.988 1.136 1.646 3.06 0.286
0.949 2.761 4.802 7.148 11.206 7.156
0.952 0.876 0.783 0.680 0.520 0.688
0.943 0.884 0.806 0.684 0.574 0.698
- 12.8 - 10.7 -8.8 - 7.7 -6.1 -7.7
aa
lexp
(3) a
a3
In a3exp
- 7.9 -6.7 -7.9
andIn a3expare experimental data (Yakimov, and Mishin, 1967).
composition of the ternary system one can solve the set of eqns. (9), (17) and (18) and then calculate In f2, In f3 and a, with the aid of eqns. (15), (16), (2) and (4). Let us consider, as an example, the application of this method to the calculation of two systems with different types of component interaction. In the system H,O (l)-KC1 (2)-NaCl (3) the component activities on the branches of solubility KC1 and NaCl have been calculated. The criterion of calculation accuracy is approximately constant values of In a2 and In a3 and their equality to the right-hand side of eqn. (10). The application of this criterion enables us to calculate the isotherm of solubility, which is shown in Fig. 1. The predicted and experimental data are in good agreement, on the whole. Thermodynamic characteristics of a more complex system of another type, with intensive interaction of the components (Yakimov and Mishin, 1967), H,O cl)-CsNO, (2)-HNO, (3), are given in Table 2, in which the results of activity calculation for both volatile components are represented. The average absolute error in predicted a, values was in this case as much as 3%. ORGANIC SOLUTIONS AND SIMULATION OF DISTRIBUTION RATIOS DURING EXTRACTION OF CsNO, AND HNO, BY EXTRACTANT BASED ON COBALT DICARBOLYDE
The designed method of calculating activity coefficients in multicomponent systems can be applied in both the aqueous and organic phase. This enables us to calculate the extraction equilibrium. If the state reference of the solute in both phases is “pure substance”, the condition of extraction equilibrium will be the equality of the activities of each component number I in phases 1 and 2 (1 = water, 2 = organic), i.e.
#)fi(l) exp( -~j’)) = ~$“)fi”) exp( -xP))
(19)
164
Besides, the condition of thermodynamic stability should be observed. The solution of the set of equations (19) should satisfy the condition of material balance m~‘)Q(‘)+ m!2)Q(2)= my&(‘), where ml’) and rni2) are the molalities at equilibrium, Q(l) and Q@ are the masses of solvents and my is the molality in the initial solution. Calculation of fl in the organic phase is usually hindered by lack of values for the standard Gibbs energy of formation in infinite dilution, and this circumstance prevents estimation of g,, directly. However, this difficulty can be overcome in the following way. The standard Gibbs energy of transfer of a component from phase 1 into phase 2, AG,,, is determined by AG,, = AG, - AG,, where AG, and AG, are the Gibbs energy of hydration and solvation, respectively. The value of AG, is usually known, and hence AG, can be calculated if AG,, is determined. Unfortunately, the theory of calculating AG,, is not sufficiently well established for complex systems. This problem is of great interest for solution theory and still remains to be solved. The value of AG,, can be determined by experiment in several ways, but probably the most convenient method in our case is determination from the distribution ratio at infinite dilution, D,(O). For each component it can be written from eqn. (19) at xl + 0 as In O,,(O) = In f/‘)(O) - In ff2)(O>. As the molar distribution ratio D,,(O) is usually determined experimentally, we obtain from consideration of eqn. (7) AG,, = AG,, - In O,,(O) - ln[p~/~~mw/(lOOO/M,,>l, where py and pi are the densities of water and organic solvent, respectively, and A4_ is the average molecular weight of the organic solvent. Let us give an example of the calculation of distribution ratios during extraction of CsNO, and HNO, by extractant based on a solution of chlorinated cobalt dicarbolyde (CCD) in nitrobenzene. This system is chosen because of its importance in the reprocessing of liquid radioactive wastes and the complexity of describing it by traditional methods based on preliminary investigation of complex formation (Rais et al., 1976).
TABLE 3 Distribution ratios of microconcentrations of CsNO, (D,) and HNO, (D,) in the system based on 0.06 M chlorinated cobalt dicarbolyde in nitrobenzene at different HNO, concentrations, C, (mol 1-l) in the aqueous phase *
CH
Dl
D l=P
D2
0.3 0.5 1.0
116 89 56
128f9 90f7 47f7
0.48 0.42 0.28
a &xp
is from experimental
data (Afonin et al. 1989).
D 2=P
165
The distribution ratio of microconcentration of CsNO, when the HNO, concentration vanishes is approximately 11,1<0>= 180 at a CCD concentration of 0.06 M (Afonin et al., 1989). This gives AG,, = 2800 cal mol-‘. The distribution ratio of HNO,, D,,(O), is not precisely known, but DJO) must be less than unity and a value of about 0.5 at infinite dilution leads to AGSz= 7500 cal mol-r. Distribution ratios calculated with these values of AG,, at different HNO, concentrations in the aqueous phase, C,, are given in Table 3.
CONCLUSIONS
It has been shown in this paper that a description of the thermodynamic properties of both aqueous and organic phases at high concentrations by equations of a similar type, without mixture adjustable parameters, in terms of infinite dilution properties and pure component properties is possible. These equations can be used for the calculation of extraction equilibria and the solubility of substances. In a number of cases, solution of the reverse problem can, however, be more expedient, i.e. the determination of some thermodynamic characteristics of components in terms of experimental solubility or extraction data.
ACKNOWLEDGEMENT
The referee’s comments are much appreciated
by the author.
LIST OF SYMBOLS
A% CFl CCD Dl Q,(O) D,,(O> fi
fi(O)
activity of component number i function in eqn. (1) change in heat capacity on fusion molar concentration of HNO, in water chlorinated cobalt dicarbolyde distribution ratio of component number I in extraction system distribution ratio of component number 1 at infinite dilution on the mole fraction scale distribution ratio of component number I at infinite dilution on the molar scale activity coefficient of component number i mean of fi at infinite dilution
166
gij
G
w-l AC0 A@
AGj’
A% A% AHk AH0
ml m$‘) (2) mz
Krg Ml n nij 42 4,
N P Ql Q2
mean of fi at infinite dilution and z = const. # 1 (i = 2, 3) mean of fi in anhydrous salt mixtures at x1 + 0 (i = 2, 3) function in eqn. (1) molar excess Gibbs energy molar excess Gibbs energy of anhydrous salt mixtures Gibbs energy of interaction between species type i and central species type j in RT units (i, j = 1, 2) Gibbs energy of solution in eqn. (1) mean of the difference AG: - AGi mean of the difference AG, - RT In m,/2 standard Gibbs energy of formation of component number i in an infinitely dilute solution standard Gibbs energy of formation of component number i in the state “pure substance” standard Gibbs energy of solvation of component number 1 standard Gibbs energy of the component transfer from phase 1 to phase 2 heat of phase transition number k in a crystal structure of a salt heat of salt fusion molality of component number i (i = 2, 3) molality of component number I at equilibrium in phase 1 in extraction system molality of component number I at equilibrium in phase 2 in extraction system molality of component number I in initial aqueous solution in extraction system average molecular weight of organic phase in extraction system molecular weight of solvent (water) number of solute moles in eqn. (1) average number of nearest neighbours of type i which interact with central species type j (i, j = 1, 2) mean of n12 as x2 + 0 the mean of n,, as x2 --, 0 (considered as the average number of nearest neighbours of pure solvent) mean of n22 as x2 -+ 1 (considered as the average number of nearest neighbours of pure solute) number of solvent moles in eqn. (1) pressure in eqn. (1) x22at x2 < 0.5; -x1x2 at x2 > 0.5 -x1x2 at x2 < 0.5; xt at x2 > 0.5
167 Q(l) Q(2)
R T TIC To ‘i 4 (2)
Xl
z
solvent mass of phase 1 in extraction system solvent mass of phase 2 in extraction system universal gas constant temperature temperature of phase transition in a crystal structure of a salt temperature of salt melting mole fraction of component number i mole fraction of component number 1 in phase 1 in extraction system mole fraction of component number I in phase 2 in extraction system fraction of second salt in salts mixture (= m2/(wz2 + m,))
Greek letters
mean of the sum n,j + njj(i, j = 1, 2) standard chemical potential of component number i in an infinitely dilute solution standard chemical potential of component number i in the state “pure substance” &-RT In wzv,, density of the aqueous phase in the extraction system density of the organic phase in the extraction system mean of the difference gij - gjj Subscripts c
exp
H i
i, j k
1
erg S
tr x
molar scale experimental hydration number of component (1 = solvent; 2, 3 = solutes) species type in binary system (i, j = 1, 2) number of phase transitions in the crystal structure of a salt, summation index in eqn. (1). number of the component in the extraction system (1 = CsNO,; 2 = HNO,) organic solvation transfer mole fraction scale
168
Superscripts
1, Cl), (2) ,
E
infinite dilution (for lzll and lt12 as x2 --) 0) anhydrous (for nz2 as x2 + 1) number of phases in eqn. (19) (15 aqueous, 2 = organic) infinite dilute solution (for pi and AGF) anhydrous state (for fi and gE) derivative with respect to the mole fraction x1 (in eqns. (4) and (5)) excess
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