J. Phys. Chem. 1994,98, 654-661
654
Debye-Hiickel Theory for Hydrated Ions. 7. Thermodynamic Model for Incompletely Dissociated 1:l Electrolytes Hansjurgen Schonert Institut f"urPhysikalische Chemie, RWTH Aachen, Templergraben 59, 0-52062 Aachen, Germany Received: June 29, 1993; I n Final Form: October 14, 1993@
A recently proposed model which describes the thermodynamic properties of aqueous 1:l halides in terms of (i) Coulombic interaction via a DebyeHiickel contribution and of (ii) the thermodynamics of the hydration of the ions has been extended to include incomplete dissociation. It is applied to aqueous CsCl (273 I T/K I 523), its mixtures with NaCl and HCl, to aqueous CsI (298 I T/K I 378) and to aqueous HCl(520 IT/K I 638). The dissociation equilibrium constants derived do agree well with values estimated from other sources. The prediction of the thermodynamics of the ternary solutions of CsCl with NaCl and HCl is improved.
I. Introduction The DebyeHiickel theory,'.* which describes the thermodynamic properties of electrolyte solutions and its various extensions, for example, those by Pitzer? by Mayer's cluster expansion,4v5 and by the mean spherical approximation,6 contain-among others-the radii of the solvated ions or some equivalent thereof as adjustable or given parameters. These quantities-if fittedturn out to be greater than the crystallographic radii except for some anomalous cases: Harned and Owen express the opinion that in aqueous solutions of some cesium salts (at ambient temperatures) there may occur to some extent ion pair formation because the distances of closest approach between cations and anions as evaluated from the activity coefficients are smaller than Bjerrum distance (and the sum of the crystallographic radii of the ions).' The model theory which has been presented in this series reveals the same features.8~9Moreover, Renard and Justicelo have measured the electrical conductivityof aqueous CsCl at 298 K and have deduced therefrom a molar dissociation constant of about K 2 mol kg-I. In aqueous solutions of HCl at high temperatures we have observed the same di~crepancy,~ and it is well-known from the extrapolation of the conductivity measurements of Frantz and Marshall" and the modeling of Simonson et al.12 that HCl for T > 520 K is incompletely dissociated. This congruence between the experimentallyobserved incomplete dissociation and the anomalous behavior of the distance of closest approach, if evaluated from the thermodynamics of the solutions, suggests inclusion of the features of incomplete dissociation into the model theory. Thereby, one may remove the anomaly and find another independent way to determining the dissociation constant of weak electrolytes. With this aim in mind, we have extended in the next section the existing theory. The applications and results are reported in the last section. 11. Theory 11.1. Cibbs Energy for a Binary Solution of an Incompletely Dissociated ElectrolytewithHydratedIons. Most of the equations to be used in the following have been derived and explained in the foregoing paper^.^,^ Therefore, they are given here without further reference, if not stated otherwise. As a first step in the treatment of ternary solutions, like NaCl(A) + CsCl(B), in which the electrolyte B is assumed to be incompletely dissociated, the equations for the binary solution *Abstract published in Advance ACS Absrracts, December IS, 1993. 0022-3654/94/2098-0654604.50/0
with B are dealth with. Adhering to the former terminology,the ionic species of electrolyte A are denoted by subscripts 1 and 3 and those of electrolyte B by 2 and 3. Now, to this is added the undissociated species B with subscript 23. The Gibbs energy of the solution with B is thus
= n#f + n2c12 + n3P3 + n23/123 (1) where the subscript 1 is given to the solvent. The symbols n, and p,designate the amount of substance Q and its chemical potential. The chemical potentials of the solute species are based on the molality scale: P, = Pi + R T ln(m,y,) (2) Here m, and y, are the molality and the activity coefficient of species Q, respectively. It has been shown that
ln(y,) = (dGDH/dn,)/RT
+ N, In(( 1 + IC,)/( 1 + kbCw)) -
In N (3) if the species are considered to be hydrated and if the hydration is modeled by N, identical and independent steps with binding constant k,. The mole fraction of the "free" water is XW. The last term will be explained later. The ionic species interact by Coulombic forces and GDHis the contribution of this interaction to the Gibbs energy of the solution. For the binary solution with the 1:1 electrolyte B one has
(dGDH/dn,)= -3RTA,fi/( 1 =0 with the ionic strength
+ xB)
for 4 = 2 , 3
f o r q = 23
+
I = (m2 m,)/2
(4)
(5)
xB = a g + , d i (6) where aBis the distance of closest approach between ions 2 and 3. The Debye-Hiickel coefficients are A, and B,. For given values of the hydration parameters N , and k, the various hydration terms are calculated iteratively via the average hydration numbers
h, = N , k ~ w / ( l + k 8 w )
(7)
xw = Z/N
(8)
z = 1 - (m27;, + m3h3+ m23h23)M1
(9)
with
0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 2, I994 655
Debye-Hfickel Theory for Hydrated Ions. 7
N = Z + (m,
Likewise, there follows
+ m3 + mZ3)MI
(10)
Here MI is the molar mass of the solvent and N the argument in the last term of eq 3. The chemical potential of the solvent is pI
+ A$T13/2u(XB) + R T ln(xw)
p;
(1 1)
with U(X)
= 3(( 1 + x) - 2 In( 1 + x) - 1/( 1 + x))/x3 (12)
Up to this point we have summarized the starting equationswhich were derived in former papers. These expressions must be transformed into those for the stoichiometric, observable quantities. On this level the Gibbs energy is given by
G = nIPI + ~ B + P B ++ n d t r
(13)
where B+ and B-denote the stoichiometricquantitiesfor cations and anions of the 1:l electrolyte B. The chemical potentials on this level are (with mg+ = m b = me)
+ RT ln(mB~B+) = P L + R T In(mB7g.)
Q ~ Bas
(14)
osmotic coefficient and 7
B
=
+
+
+
Therefore, by eq 15 the final result for the activity coefficient for an incompletely dissociated electrolyte B with hydrated species is found to be
+
ln(y,) = -3A,&/(l xB) + ln(aB) (1 k,xw)] N3 In(( 1 k 3 ) / (1
+
+
+
+ '/,[N2 ln((1 + k2)/
+ k3xw))]- In N = 1
/Z 1n(7273a,2) (24)
This expression looks very much like the corresponding one for the completely dissociated electrolyte B, except for the term with ln(ae). But it should be kept in mind that the mole fraction XW, the ionic strength and all the other terms such as (see eqs 9 and 10)
z = l-((i2+83)aB+ N =Z
~ g = + P:+
with
+
- 2RTMI@,mB
& = p;
+
ln(y3ae) = - 3 A , d / ( 1 x,) ln(a,) N3 In(( 1 k3)/(l k3xW))- In N (23b)
ln(7,)
(l-aB)&23)mBhfI
+ (1 + ae)m&[
(25)
are modified by the incompletedissociation, i.e., by the hydration A23 of the undissociated species and by a~ # 1. Another expression for the degree of dissociation is needed in order to calculate by eqs 7, 8, and 25 the various quantities of the model. This comes from the condition of equilibrium, eq 19, together with eqs 3, 4, 23, and 24:
(15)
6
as activity coefficient of electrolyte B. The mass balance between the two levels of description are
+
+
(16) ng. = n3 n23 ne+ = n, n23 or with the introduction of the degree of dissociation a g
mz = m3 = (YBmB
m23 = (1 - (YB)mB
(17)
with
In the case of vanishing hydration this expression reduces to
and with eq 5
I = (YgmB
(18)
The dissociation equilibrium is described by PZ + P3
(19) = P23 Combination of this equilibrium condition with eqs 1 and 6 gives
+ nB+ccz+ nw3
G=
(20)
A comparison of this expression with eq 13 leads to the transformation equations PI'P,
P3 = Pg.
P, = P g +
(21)
With eqs 2 4 , 14, and 17 this yields for species 2 p;
+ R T ln(m,a,y,) = pi + R T ln(m,a,) - 3RTA,&/ (1 + xB) + N,RT In(( 1 + k 2 ) / (1 + k,xw)J - R T In N = P;+
- -
+ RT WWB+)
For infinite dilution, me 0, xw = 1, N 1, YB+ a relation for the standard chemical potentials: +
P:
which is the traditional formulation for the dissociation equilibrium, except for the factor N, which accounts for the transformation of the mole fraction scale for the hydrated species to the molality scale for the stoichiometric components. For the solvent, the transformation from the model level to the component level is found by equating the chemical potentials in eqs 11 and 14: -2aBmBMI = 2AJ3"U(X,)
+ In(+)
(29)
Finally, introducing eqs 24 and 29 into the defining equation for the excess Gibbs energy per unit mass of solvent:
G""
2 m ~ R T ( 1- 9,
+ ln(7B))
(30)
one finds
(22) 1 one gets
= P;+
and hence In(?,+) = ln(y,a,)
+ xe) + ln(aB) + N , ln{(l + &,)/(l + kzxw))- In N
with
= -3A+, TK
0
Figure 3, Standard free energy of hydration, cq 73 for CsCl at (i) A, completely dissociated and (ii) a, incompletely dissociated.
here, describe the activity/osmotic coefficient in the complete temperature range with the same standard deviation. On this basis no preference can be given to one of the models, and KcScl seems to be more like an adjustable parameter than a dissociation constant. But the fact that the unrealistic distance of closest approach in the model with complete dissociationis now removed clearly favors the second model. This conclusion is strengthened by the agreement of the dissociationconstant evaluated from the model with the experimentallydetermined value.lO Another test will be the prediction of the thermodynamic properties of ternary solutions as explained several time^.^.^ The extensive measurement~'~ of the emf of galvanic cells with HzO HCl(A) CsCl(B) are apt for this test. With the help of eqs 48,49,and 52 and the parameters for HCl(A) from ref 9 and CsCl(B) from Table 1, one can calculate the ternary properties for the model with incomplete dissociation, whereas the foregoing equations and tabulations9are sufficient to do the same for the other model. In this way the Harned coefficient aA in Harned's rule
+
+
m = mA + mB= constant
log(7,) = log(y:) - aAmB,
(74) was calculated and compared with the experimental value, Table 2. It can be seen that the inclusion of the incompletedissociation for CsCl improves the agreement with the experimental value. Up to m = 0.5 mol k g l the differences between experiment and model are shifted near to the error limit; and although at m = 1 mol kg-1 the differences are greater the model with incomplete dissociation allows for a fair estimation. In this connection it seems worthwhile to point out that the short-rangeinteractionparameters of Pitzer's theory in the ternary HCl CsCl show a molality dependencewhich is quite different from those for HCl + LiCl, HCl + NaCI, and HCl KCl; Figure 4 in ref 3. This may also signalize the incomplete dissociation. The system H2O + NaCl(A) CsCI(B) has been studied by the isopiestic vapor pressure technique.15 Because all measurements are outside the range of applicability of the model m > 1 mol k g l , only an indirect comparison is possible. The experimental value of the osmotic coefficient @(exp)was calculated at constant total molality m = 1 mol kg-1 and at the mole fractions
+
+
+
where the interaction parameter 0 1 2 (case I1 in ref 15) and the binaries @A and %were taken from the literature.13J6 The model value was calculated by eq 50,the parameters of Table 1, and for NaCl from the parameters of Tables 3 and 4.9 In Table 3 the standard deviation u(@) is listed, together with its value for CsCl considered to be completely dissociated. This comparison also supports the model of an incompletely dissociated CsCl. The thermodynamic data for aqueous solutions of CsI from 298 to 373 K are from the work of Saluga et al.1' and those for CsBr at 298.15 K from Robinson and Stokes.' As is the case with CsC1, the distances of closest approach ucsBr and act[ evaluated from the smoothed experimental data with the assumption of complete dissociationturn out to be smaller than the sums of the crystallographic radii.8 This criterion again was taken to apply the model with incomplete dissociationwhere the distances of closest approach were fixed as listed in Table 1. The binding constants for the anions were those determined for the completely dissociated sodium salts? and the binding constant for the cation is from Table 1. Only one parameter, the dissociation constant, remains to be evaluated. The result is added to Table 1. Figure 2 shows that Kcst increases with increasing temperature-in the temperature range studied-as also found for KCCI. A word of caution must be added. In the model presented, the parameters describing the hydration and the dissociation are correlated; see Figure 1. The range of fit for the parameters thus establishedcan be made smaller only if reciprocal salt pairs, MX, NX, MY,and NY, are treated as was done* at T = 298.15 K. But in the case discussed here no data for such a pair are available in a wide temperature range. Hence, the dissociation constants as given in Table I are uncertain-within the frame of the model-to approximately 30%. III.2. Aqueous HCI. As outlined in the Introduction, HCl is incompletely dissociated at high temperatures. For moderately concentrated solutions, m I 1 mol k g l , this is detectable by the thermodynamic properties for T > 520 K. It is therefore interesting to find out if the dissociation constant evaluated from the present model does sufficiently agree with the extrapolated values of the conductometric determination.' The smoothed experimental data are from model I11 of the work of Holmes et a1.18 The Debye-Hiickel coefficients A, and B, are calculated from the results of Uematsu and Franck19and DAns Lax20and smoothed along the saturation pressure with eq 72;the parameters q1 thus found are given in Table 4. The convention, the relation of the binding constants for the chloride ion and the potassium ion is fixed (independent of temperature) which has been used so far for T I 523 K to regulate the relationship between the hydration properties of the ions and of the electroneutral components, cannot be extended beyond T = 523 K because of lack of the experimental data for aqueous KCl in this high-temperature range. It is therefore replaced by a new convention:
kcl- = 0.060,T L 520 K
(76)
The distance of closest approach is assumed to be
aHCl= 0.300 nm (77) This value-with an uncertainty of severalpercent-is suggested by theradiiofthechlorideion(=0.181 nm)andofHsO+ (10.138 nm). With these assumptions the model parameters to be fitted are the binding constant of the cation, kH+, and the dissociation constant Thus, the procedure is the same as used for CsCI, visualized in Figure 1, including the assumption of eq 70. The
660 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994
Schhert
TABLE 2 Hamed Coefficient in the System HCI((A) + CsCI(B); Calculated for Incomplete Dissociation a A ( d c 1) and for Complete Dissociation a ~ ( c a l c2) and the Standard Deviation ~ Y A for ) the Model with Incomplete Dissociation T/OC m/mol.kg-l ~~
5 0.1599 0.1440 0.1405
