J. Phys. Chem. 1984,88, 4358-4361
4358
in which VxA and VxBare the potential well depths, respectively, for XA and XB in eq 1, and g(s) is a monotonically decreasing function in the normalized reaction coordinate, s. AGO and AG* are
= a and "(1)
= (VXB - VXA)(g(o)- d 1 ) ) AG* = VXB(g(o) - g(s')) + VXAkdl) - d1-s*))
a = 1 - (1 - ao)f2(x) Since a = 1 - g('l2) and AGO*= VXA(1 - 2g('/2)), then
in which s* is the transition-state value for s. If VxA is fixed in the reaction series and VxBis allowed to vary, then a can be shown to be a=
- g(s*))/(g(O) -
Since N ( a ) = aAGo - AG* and s* = g-'(g(O)-a(g(O)-g( l))), then N ( a ) is (note: g-' is the inverse function of g; that is, if g(s) = z, then s = g-'(z)) N(a) = -VXA(g(l) + ( d o ) - g(l)la - g(l - g-'(g(O)-a(g(O)-g(l))))) In most models in which g(s) is some specific function, it is usually true that g( 1) = 0 and g(0) = 1. Therefore, in general N ( a ) can be written N ( a ) = -VXA((Y - g(l-g-'(l-a)))
Moreover, AGO can be written as VxB - VxA. Since VxBvaries as 0 iVxB i+m as a varies between 0 and 1, and since " ( a ) = AGO, then N'( 1) = +a and "(0) = -Vu (remember that "(0)
= b). Therefore, from eq 20 and 21, one can write
fi(4 = g( 1-g-W-4)
/g(!h>
h ( x ) = (1 - a)/&) Since it is always true that N ( a ) = N(ti), then one can write a - g(1-g-yl-a))
= a - g(1-g-yl-a))
If 1 - a is replaced with g(l-g-*(l-a)) on the right-hand side of the equation above, an identity results. Therefore, it must be true that 1 - a = g(1-g-yl-a))
Since 1 - a = g(l-g-'(l-a))
then
h ( x ) =fz(x) Q.E.D.
Determination of Equilibrium Constants for Complex Species Formed from Slightly Soluble Salts J. J. Fritz Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: October 13, 1981; In Final Form: March 16, 1984)
Errors introduced by neglecting the variation of activity coefficients with solution composition in the determinationof equilibrium constants for complex species are discussed, with numerical examples. The techniques needed for proper inclusion of activity coefficients are described, as well as the requirements on the experimental data necessary to make them effective. The utility and validity of less rigorous determinations are discussed.
Introduction Many salts whose solubility in pure water is severely limited acquire substantial solubility in the presence of excess amounts of their anion due to the formation of (usually charged) complexes of the sort M,L, between the cation (M) and anion (L). Prime examples of this sort of behavior are the halides of Ag(I), Cu(I), and Hg(1). These and other systems have been the subject of many investigations using a considerable variety of experimental techniquesi designed to determine the equilibrium constants for formation of the complex species either from its parent ions (stability constants) or from the solid salt and ligand anion (hereafter referred to as formation constants). Despite the fact that the better methods have experimental precision of the other of 1% in the data obtained2 and despite use of sophisticated g r a p h i ~ a land ~ , ~c o m p ~ t a t i o n a l ~methods -~ for extracting equi(1) See, for example, A. E. Martell and L. G. SillCn, Eds., "Stability Constants of Metal-Ion Complexes", Chemical Society, London, 1964; Supplement No. 1, Chemical Society, London, 1971. (2) L. Johannson, Coord. Chem. Rev., 3, 293 (1968). (3) E. Berne and I. Leden, Z . Naturforsch., A , 8A, 719 (1953).
0022-3654/84/2088-4358$01.50/0
librium constants from the data, one still finds substantial disagreements in the numerical values of the constants and even in the identity of the complexes formed.' (Note: ref 3-7 contain only prominent examples of the techniques used; for critiques of these and other methods see reviews by Vacca et al.7 and by Bond? No more recent reviews of techniques were found in the literature.) The primary difficulty in extracting equilibrium constants from experimental data arises from the need to relate concentrations and activities in the course of the calculation. Typically,l measurements are made of a property dependent on concentration (solubility, optical density) or on activity (cell measurements) for solutions with a variety of ratios of dissolved salt to complexforming ligand. The desired equilibrium constant is expressed in terms of activities with respect to a selected reference state. (4) B. Hedstrom, Acta Chem. Scand., 9, 613 (1955). (5) J. Rydberg, Acta Chem. Scand., 15, 1723 (1961). (6) L. G. SillBn, Acta Chem. Scand., 16, 159 (1962). (7) A. Vacca, A. Sabatini, and M. A. Gristina, Coord. Chem. Rev., 8, 45 (1972). (8) A. M. Bond, Coord. Chem. Rev., 6, 377 (1971).
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 19, 1984 4359
Equilibrium Constants for Complex Species In the course of the calculation, it is necessary to make a mass balance in terms of concentrations, whether the original measurements are concentration or activity dependent. Thus, the ability to relate concentrations to activities can be crucial to successful interpretation of the experimental data. In an effort to circumvent this problem, many investigations have been carried out at one or more constant nominal ionic strengths maintained by use of an "inert" salt along with that providing the ligand anion. The data have then been analyzed on the assumption that all activity coefficients, or at least the essential ratios of activity coefficients, were constant at a given ionic nominal strength, so that concentrations and activities could be used interchangeably. (This assumption will hereafter be referred to as the "constant activity coefficient approximation".) The analysis then gives a set of formation constants (Km,,')or stability constants supposedly characteristic of the specified ionic strength maintained by the chosen inert salt. (See, for example, the equilibrium constants tabulated in ref 1.) The assumption that activity coefficients (or their ratios) are constant a t a given nominal ionic strength is at best an approximation and at worst can lead to serious misinterpretation of the experimental data. A major purpose of this paper is to examine the difficulties which it can produce, some obvious and some more subtle. Fortunately, if sufficient data of high quality are available, all necessary activity coefficients can be evaluated along with the desired equilibrium constants. This has been done9J0for solutions of cuprous chloride in aqueous HCl and in a number of soluble chlorides. Some of the results of this more exact treatment will be used below to illustrate the difficulties inherent in the constant activity coefficient approximation. Thereafter, the final part of this paper will discuss the type of experimental data required for the evaluation of the necessary equilibrium constants without the assumptions of constant activity coefficients. Thermodynamics
The formation of a complex species MmXncan be described either by the reaction mM+ with a stability constant
+ nX- = M,X,("") P,,
(1)
given by
or by the reaction mMX(s)
-+ (n - m)X- = MmXim-")
(3)
with a formation constant Kmngiven by
(4) All quantities on the right-hand sides of eq 2 and 4 are activities with respect to some specified reference state. In order to make a mass balance on the material present, one requires the concentrations of the various species present. The procedure is described below for solubility measurements; similar equations can be written for other types of measurement. The molar solubility of a salt MX in a solution of another salt N X is given by S = Cm(Mmxn)
(5)
m,n
where the quantities in parentheses are concentrations. Similarly, the initial concentration of anion X - is given by (X-),, = ( X - ) +
E(n - m ) ( M m X n )
m.n
From eq 4 we have
(9) J. J. Fritz, J . Phys. Chem., 84, 2241 (1980). (IO) J. J. Fritz, J . Phys. Chem., 85, 890 (1981).
(6)
in which the required ratio of activity coefficients can be obtained from a suitable set of mean ion activity coefficients. One can then write S = CKm,'(X-)n-m
(8)
m,n
and a similar expression for (X-),,. In eq 7 and 8, the Km,,'are apparent formation constants suitable for use with concentrations for the particular solution composition present. The assumption of constant activity coefficients implies that at a given ionic strength the Km,,'are constants which can then be determined by application of eq 6 and 8 to a set of measurements of S a ~ f [ ( X - ) ~ ] . In fact, the Km,,' depend on solution composition as well as on ionic strength, through the activity coefficient ratios of eq 7. These ratios can be expressed in terms of both ionic strength and solution composition by a suitable virial model of activity coefficient behavior. The model developed by Pitzer and co-workersl1.l2has proved adequate to interpret the behavior of complexes formed by cuprous c h l ~ r i d e ~and . ' ~is used in the examples described below. Use of the model requires determination of three parameters for each ion pair present in solution. Parameters are available in the literature" for many common ion pairs. In order to apply the model to systems forming complexes, it is necessary to evaluate the parameters for ion pairs involving each complex species along with the desired equilibrium constants. This requirement increases the complexity of the optimization procedure but does not render it especially formidable. The Nature of the Activity Coefficient Problem
The assumption that activity coefficients or their ratios can be considered constant so long as the nominal ionic strength (before complex formation) is fixed is erroneous for two reasons: 1. Substitution of one ion for another (for example, an "inert" anion for ligand anion) will alter the ionic environment in such a way as to change the activity coefficients of all ions. 2. When complexes are formed whose charges are different from that of the ligand anion, the ionic strength must change, again altering the activity coefficients of all species present. Examples of both effects are illustrated in Table I11 of the Appendix, which presents data calculated with the aid of formation constants and virial coefficients capable of representing the solubility of CuCl in two aqueous systems within 1-2%. The table contains calculated mean ion activity coefficients for representative solutions of CuCl in aqueous HC1-HClO, mixtures (2.00 and 5.00 M ) and in NaC1-NaC1O4 mixtures (5.00 M only). Examination of Table I11 will indicate that the effects cited above are significant and may be quite large, especially at high ionic strengths. The effects of changing the ionic environment are less serious for the ratios of activity coefficients, as is evident in the last three columns of Table 111. The effects are almost negligible for the complex having the same charge as the ligand anion (Cl-) but can be quite large for ratios involving complexes of higher charge. Problems Created by Variation in Activity Coefficients Representation of Data. The assumption that activity coef-
ficients (or their ratios) depend only on the nominal ionic strength can lead to serious difficulties in representing data. For example, Hikita et al.I3 were able to fit their extensive data on the solubility of CuCl in aqueous HCl-HC10, to only about 10% using the assumption of constant activity coefficients (Le., that the K,,,' of eq 7 and 8 depended on the nominal ionic strength alone), although the precision and consistency of the observations were much better than this. A more detailed analysis? which included evaluation of the virial parameters needed to evaluate activity coefficients, represented their data (56 points at five ionic strengths), to an average (root mean square) deviation of 1%. Similarly, Ahrland and Rawsthornel, made both potentiometric and solubility (11) K. S . Pitzer and G. Mayorga, J . Phys. Chem., 77, 2300 (1973). (12) K. S . Pitzer and J. J. Kim, J . Am. Chem. Soc., 96, 5701 (1974). (13) H. Hikita, H. Ishikawa, and E. N. Esaka, Nippon Kagaku Kaishi, 1, 13 (1978). (14) S.Ahrland and J. Rawsthorne, Acta Chem. Scund., 24, 157 (1974).
4360
Fritz
The Journal of Physical Chemistry, Vol. 88, No. 19, 1984
TABLE I: Apparent Formation Constants of Chloro-Cupro Complexes at 25 OC and Variation with Ionic Strength and Solution Composition"se
system
nominal ionic strength
any HCI-HC104
0
HCI-HC104
5.00
NaCI-NaC104
5.00
method
Kl2/
constant y constant y variable y constant y constant y variable y constant y b constant y variable y
2.00
source
K3s/
K13'
x
10-5
0.0604
0.0144
3.4
9
0.052 0.052 0.049
0.036 0.040 0.045-0.053
0.0053 0.0027 5.6 X 10-4-8.8 X
11 this work 9
0.031 0.032 0.027
0.03 1 0.028 0.044-0.045
0.0067 0.0055 0.001 5-0.0042
11 this work 9
0.042 0.043 0.041-0.040
0.083 0.062 0.082-0.100
