ARVINS. QUISTAND WILLIAML. MARSHALL
1636
+ 1+ 1
Pl/, =
P1
(15) p1
where p12 and p1 are the depolarizations for the mixture and solvent, respectively, is the volume fraction of the solvent, and RlzO and R1° are the total Rayleigh ratios of the mixture and pure solvent, respectively.
The results are shown in Figures 7 and 8. Definite breaks are indicated in the data for dichloromethane with diiodomethane and with chloroform. Sicotte has suggested that such breaks are indicative of a change in the liquid state structuring in the solution.
Acknowledgment. We thank the Emory Biomedical Data Processing and Analysis Center for use of the computer.
The Independence of Isothermal Equilibria in Electrolyte Solutions on Changes in Dielectric Constant’ by Arvin S. Quist and William L. Marshall Reactor Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 87850
(Recebed September 18, 1967)
The importance of including the molar concentration of the solvent as a variable in the equilibrium constant (KO) is discussed. When the solvent is considered to participate actively in the equilibrium, isothermal values of KO are found to be independent of changes in dielectric constant of the solvent mixtures. Extensive examples are given in support of this general principle. The new principle contrasts with existing theory that considers log K (the conventional constant not containing solvent species as concentration variables) to be a linear function of 1/D. It is shown to apply to electrolyte behavior in water, water-organic, and organic-organic solvent systems over a wide range of temperature and pressure. For example, in water-dioxane solvent mixtures where electrolytes (ions and polar molecules) are preferentially solvated by water molecules, a plot of log K (dissociation) vs. log CH,Oin moles per liter yields a straight line of intercept KO and slope representing the net change in waters of solvation between equilibrium reactants and products. From the slope and an estimate of the hydration number of the ion pair, the “ion size” parameter can be calculated. Many examples in support of this new principle are presented both for homogeneous and heterogeneous equilibria.
Introduction In an earlier communication, we have proposed a general principle that the inclusion of solvent species of variable concentration directly into the conventional equilibrium constant ( K ) for ion-ion-pair-solvent equilibria provides a complete constant (KO) that is independent of changes in dielectric constant at constant temperature.2 It is the purpose of this paper t o present substantial evidence to support this principle, t o show that it provides an independent method for calculating hydration numbers and ion size parameters, and to present a clearer description of solution equilibria in genera1. Previously, changes in conventional, isothermal equilibrium constants involving electrolytes in aqueous-organic and organic-organic solvents have been almost universally correlated with changes in the of the solvent mixmeasurable dielectric constant (0) tures. The decrease in a conventional dissociation The Journal of Physical Chemistry
constant [expressed herein always as dissociation constants for direct comparisons and consistency] with decreasing concentration of the more polar component of a binary solvent mixture has usually been related t o the simultaneous decrease in the measurable dielectric constant of the solvent mixture. Thus the Bjerrum e q ~ a t i o nrelating ,~ ion-pair formation t o the dielectric constant of the solvent and to ionic size, was used by Fuoss and Kraus4 t o relate log K for the ionization of tetraisoamylammonium nitrate in water-dioxane mixtures t o log D. The currently accepted DenisonRamsey-Fuoss theory5a6of ion-pair formation predicts (1) Work sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corporation. (2) W. L. Marshall and A. 5. Quist, Proc. Natl. Acad. Sci. U.S., 58, 901 (1976). (3) N. J. Bjerrum, Kgl. Danske Vidensk. Selskab., 7, No. 9 (1926). (4) R. M. Fuoss and C. A. Kraus, J. Am. Chem. Soc., 5 5 , 1019 (1933).
ISOTHERMAL EQUILIBRIA IN ELECTROLYTE SOLUTIONS that log K should be alinear function of 1/D, with a slope that is proportional to the ion size. The latter relationship holds moderately well for most ionization constants when the concentration of added organic solvent is not great, but the correlation (although observed over limited ranges of very low dielectric constant) is not observed over wide ranges of solvent composition. The Denison-Ramsey-Fuoss theory of ion-pair formation is based on a sphere-in-a-continuum model and considers that! the measurable dielectric constant is the controlling factor in association. However, in recent years evidence has been a c c ~ m u l a t i n g ~to- ~indicate that ion-solvent interaction must be considered and that the dielectric constant of the solvent mixture may not be the only significant factor in ion association. If the important factors in short-range electrostatic attractions between oppositely charged ions, ion pairs, and solvent molecules in solution are the charges on the ions and the dipole moments (assumed constant at constant temperature) of the reacting species, and if ions and ion pairs in mixed solvents are selectively solvated, then a complete, isothermal constant KO involving also solvent species as variables would be expected to remain constant over a large range of solvent compositions.1° Therefore, it would appear that the Denison-RamseyFuoss theory as presently derived, using the measurable dielectric constant and considering the incomplete constant K , is probably incorrect. This theory considers also that only contact ion pairs are formed in the association p r o c e s ~ . ~Eigenll and Atkinson,I2 however, have demonstrated the existence of a three-step ionassociation process for 2-2 electrolytes, involving stepwise removal of solvent molecules from between ions. We propose that the activity (or concentration) of solvent molecules indeed be included, in molar concentration units,13 as very important variables in the complete equilibrium constant, KO. When this is done, it is found that the isothermal KO for a particular equilibrium is independent of changes in dielectric constant, whether these changes are made by varying the density of the solution system by pressurization or by adding an “inert” solvent (i.e., one that does not significantly solvate the ions or polar molecules of the electrolyte), such as dioxane, to water. Tests of this general principle have shown remarkable agreement over a wide range of conditions for a variety of aqueous-organic and some organic-organic solvents at room temperature and atmospheric pressure, and also for equilibrium behavior in water alone. ‘The agreement far surpasses that shown by existing theories that correlate the change in the incomplete equilibrium constant (not KO) with changes in the measurable dielectric constant.
Aqueous Electrolyte Behavior at S upercritical Temperatures Sodium chloride is generally considered to be completely ionized jn aqueous solutions at room temperature
1537 and atmospheric pressure. However, it becomes progressively a weaker electrolyte as the temperature increases. From electrical conductance measurements on dilute NaCl solutions at temperatures to 800” and at pressures to 4000 bars, l 3 conventional equilibrium constants ( K ) have been calculated which are related to the complete equilibrium constants as
+ kH20
1C’aC1(H20)j
Na(H20>,+
KO =
+ Cl(H20).-
‘ N ~ ( H Z O )+~ a ~ l ( ~ z ~ ) n = K/UH,OR
a ~ r t ~ i ( ~ ~ ~ ) g a ~ ~ ~ ‘
log K
=
log KO
+ k log
UH~O
(la) (1b)
(14
where KO is the complete constant assumed to be independent of U H ~ O , K is the conventional constant that varies with U H ~ O ,and the a’s are activities based on molar concentration units. In his studies on KC1 solutions at supercritical temperatures and pressures, Franckl* some 11 years ago showed that a plot of log K (at constant temperature) against log C H 2 0 (moles per liter) gave nearly a straight line. Our own recent and extensive values of log K for NaCl show a linear relationship for similar plots. The values of K (at a particular temperature and density) as obtained from the conductance measurements represent the values of this parameter at infinite dilution of the electrolyte. Under these conditions, the solvent has the same properties as the pure solvefit and its activity can be set equal to its molar concentration. Consequently, eq ICcan be written as log K = log KO
+ k log C H ~ O
(2)
where C H ~ refers O to the molar concentration of water. Thus, if water is considered a reactant of variable concentration in the equilibrium, the linear relationship beO explained. Ordinarily, in tween log K and log C H ~ is studies of equilibria in aqueous solutions under the usual conditions of 25” and atmospheric pressure, the activity of water is effectively constant and is arbitrarily assigned a value of unity, whereby the equilibrium is expressed by the conventional K . However, as seen in the above example, in studies of equilibria in solutions at supercritical temperatures and pressures where wide ranges of water concentrations are encoun(5) J. T. Denison and J. B. Ramsay, J . Am. Chem. SOC.,77, 2615 (1955). (6) R. M. Fuoss, ihid., 80, 5059 (1958). (7) A. D’Aprano and R. M. Fuoss, J . Phys. Chem., 67, 1704, 1722 (1963). (8) G. Atkinson and 5.Petrucci, J . Am. Chem. SOC., 8 6 , 7 (1964). (9) G. Atkinson and S. Petrucci, J . Phys. Chem., 70, 3122 (1966). (10) Note that the measurable dielectric constant still remains important for long-range electrostatic interactions. (11) M. Eigen and K. Tamm, 2. Elektrochem., 66, 93, 107 (1962). (12) G. Atkinson and S. K. Kor, J. Phys. Chem., 69, 128 (1965). (13) A. S. Quist and W. L. Marshall, ihid., 72, 684 (1968). (14) E. U. Franck, 2. Physik. Chem. (Frankfurt), 8 , 107 (1956).
Volume 72, Number 6 M a y 1968
ARVINS. QUISTAND WILLIAML. MARSHALL
1538 tered, it is necessary to include the activity of water as a variable in calculating the complete equilibrium constant. For the KaC1 solutions, this linear relationship was observed at temperatures from 400 to 800” and at densities from 0.30 to 0.75 g Furthermore, the values of k were essentially constant throughout this temperature range, indicating that j , nz, and n may be primary hydration numbers. A value of KO can therefore be calculated with eq 2 at each temperature. This KO is independent of density (pressure) at constant temperature, and therefore meets one of the requirements of a “true” thermodynamic equilibrium constant. This concept of a true, isothermal equilibrium constant has been experimentally verified for several different ionization equilibria over a wide range of solution densities and over wide temperature regions. The evaluations for most of the published data for aqueous solutions at high temperatures and pressures are summarized in Table I. Since the isothermal equilibrium constants KO for the equilibria summarized in Table I are independent of solvent density, it became evident to us that these constants were also independent of changes in dielectric constant of the solvent. Although at 400” the dielectric constant of water decreases from 19.5 at a density of 0.8 g t o 4.9 at a density of 0.3 g ~ m - 3 ,the ~ ~value of KO is constant. This observation is highly significant in demonstrating that the measurable dielectric constant of the solvent is not the determining factor in predicting the true equilibrium constant, in contrast to present theory on the variation of the conventional K.? In the following sections this principle is extended to include ionic equilibria in water-organic and organic-organic solvents.
of the solvent species as reactants of variable concentration might provide equilibrium constants that were truly independent of changes in solvent composition. I n the examples described above, the concentration of water was varied by changing the pressure on the solutions. The molarity of water can also be changed by adding an “inert” solvent, such as dioxane. The addition of this inert solvent does not appear to affect significantly the hydration of ions since the water molecule is much more polar than the dioxane molecule and consequently water molecules selectively solvate the Therefore, the relative hydration of ions should be nearly independent of total water concentration, and the hydration numbers in an equation like eq l a should remain constant over a wide range of water-dioxane compositions. Kunze and Fuossl9 have calculated conventional ionization constants for NaCl for the equilibrium represented by K in eq ICfrom conductance measurements in several water-dioxane mixtures at 25”. When the ionization constants reported
040
-0.5
0.08
I/(d!electric constunt) 0.06
0.04
0.02
I ~
1
,
Equilibrium Behavior in Water-Organic Solvents at Room Temperature The above expectations and observations indicated that for ionic equilibria in mixed solvents the inclusion Table I
Electrolyte
Solvent system
HSO4HzS04
KHSOa NaCl KC1 HCl KOH
HzO to HzO to HsO t o HzO to HzO to H20 to HzO to
4000 bars 4000 bars 4000 bars 4000 bars 2700 bars 2700 bars 2700 bars
Temp, OC
100-300 400-800 400-800 400-800 400-750 400-700 400-700
Net ohange in waters of hydration on ionization
20 11
8.5 10 9 9
8
Lit. ref
a b a c
d
e e
a A. S. Quist and W. L. hfarshall, J. Phys. Chem., 70, 3714 A. S. Quist, et al., ibid., 69, 2726 (1965). ’ Ref(1966). erence 13. Reference 14. ‘ E. U. Franck, 2. Physilc. Chem. (Frankfurt), 8 , 192 (1956).
’
The Journal of Physical Chemistry
Figure 1. Log K (dissociation) of sodium chloride us. log CH~O (moles per liter) i n dioxane-water mixtures; also compared us. l/(dielectric constant, D); K values of Kunze and Fuoss (1963). (15) W. J. Moore, “Physical Chemistry,” 3rd ed, Prentice-Hall, Ino., Englewood Cliffs, N. J., 1982,p 174. (16)A. S. Quist and W. L. Marshall, J . Phys. Chem., 69, 3185 (1965). (17) T.W. Davis, J. E. Ricci, and C. G. Sauter, J. A m . Chem. Soc., 61,3274 (1939). (18)A. Fratiello and D. C. Douglass, J. Chem. Phys., 39, 2017 (1983). (19) R.W.Kunze and R. M. Fuoss, J. Phys. Chem., 67, 911 (1963).
ISOTHERMAL :EQUILIBRIA IN ELECTROLYTE SOLUTIONS
1539
Table I1 0
c i-
-2
,$ I
t
-g -3
lop Cnzo knolrr/llter)
Figure 2. Logarithm of the conventional equilibrium constant K for several salts us. log C H ~ O (moles per liter) in dioxane-water mixtures at 25'.
by Kunze and Fuoss (Kin eq IC)are plotted against log CHI0,a straight line is obtained as shown in Figure 1with a random deviation of It0.005 pK unit ( = k l % in K ) . This linear relationship indicates that the complete equilibrium is represented by an equation like eq la, as was the case with the NaCl data at supercritical temperatures and pressures. It also indicates the constancy of the hydration numbers with varying water concentration. Figure 1 also includes a plot of log K for NaCl in water-dioxane mixtures as a function of 1/D and shows a maximum deviation of 0.25 pK unit (a factor of 1.8 in K ) , which is typical of the behavior of most alkali metal halides in these mixtures. Equilibrium constants for many ionization processes have been measured in water-dioxane mixtures a t 25". We have plotted log K (for the ionization process that does not consider hydration) as a function of log CH~O over as much as 0.5 log unit for many types of equilibria in water-dioxane mixtures, and in nearly all of these cases we have obtained straight lines, yielding constant values for KO according to eq 2. Several examples of these plots are shown in Figure 2, and Table I1 summarizes our evaluations of some of the literature data according to our proposal for the equilibrium including hydration. Of the examples shown in Table 11, only those involving RbI and LiCl showed significant deviations from linearity. Table I1 also contains the results of our evaluations of literature data for some acetone-water and methanol-water systems. Of the examples cited, appreciable deviations from linearity were observed for the equilibria involving MnSO1, Mn m-benzenedisulfonate,
Eleotrolyte
Solvent system
LiCl NaC1 KCl RbCl CSCl RbBr RbI LiI MgSO4 MgS04 MnSO4 MnSO4 MnS04 MnSO4 Mn mBDSaa Mn mBDS Mn mBDS Ca mBDS Ca mBDS LaFe(CN)G LaFe(CN)e (Bu)4NBr (CH3)dN picrate (CBH11)4NN03 ( C5Hl&NNOa Acetic acid Acetic acid Acetic acid HzO (Kw) HzO(Kw) NHaOH CH3NH30H
Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water to 2000 bars Water-dioxane Water-acetone Water-methanol Water to 2000 bars Water-dioxane Water-acetone Water-methanol Water-acetone Water-methanol Water-dioxane Water to 2000 bars Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water to 2000 bars Water-methanol Water-dioxane Water to 2000 bars Water to 2000 bars Water to 2000 bars
Net change in waters of hydration on ionization
Lit. ref
3.6-7.2 6.4 6.5 6.3 6.2 6.3 3-8 5.1 8.2 8.4 9 6.6 4.6 9 9 6 4.5 4.4 2.6 10.8 8.4 5.8 8.4
U
b C
d e
f 9
h i j k 1
m n k 1 m 0
0
P
Q r
10 (ion pairs) 2 . 4 (triple ions) 7.6 12 2.2-2.8 6.2-6.6 20 28 25
r s t U 2)
W
2
Y z
z
T. L. Fabry and R. M. FUOSS, J . Phys. Chem., 68, 971 Reference 19. J. E. Lind, Jr., and R. M. FUOSS, J . Phys. Chem., 65,999 (1961). R. W. Kunze and R. M. Fuoss, ibid., 67, 914 (1963). e J. C. Justice and R. M.FUOSS, ibid., 67, J. E. Lind, Jr., and R. M. FUOSS, ibid., 66, 1722 1707 (1963). (1962). T. L. Fabry and R. M. FUOSS, ibid., 68, 974 (1964). G. Atkinson and Y. Mori, J . Chem. Phys., 45, 4716 (1966). H. S. Dunsmore and J. C. James, J . Chem. SOC.,2925 (1951). F. H. Fisher, J . Phys. Chem., 66, 1607 (1962). G. Atkinson and C. J. Hallada, J . Am. Chem. SOC.,84, 721 (1962). Reference 8. C. J. Hallada and G. Atkinson, J . Am. Chem. Soc., 83,3759 (1961). " F. H. Fisher and D. F. Davis, J . Phys. Chem., 69, 2595 (1965). ' H. Tsubota and G. Atkinson, J . Am. Chem. Soc., 87, 164 (1965). J. C. James, J . Chem. SOC.,1094 (1950). ' S. D. Hamann, P. J. Pearce, and W. Strauss, J. Phys. Chem., 68, 375 (1964). P. L. Mercier and C. A. Kraus, Proc. Natl. Acad. Sci., 41, 1033 (1955). Reference 4. R. M. Fuoss and C. A. Kraus, J . Am. Chem. SOC.,55, 2387 (1933). H. S. Harned, et al., ibid., 58, 1912 (1936); ref 33. ' A. J. Ellis and D. W. Anderson, J . Chem. SOC.,1765 (1961). T. Shedlovsky and R. L. Kay, J . Phys. Chem., 60, 151 (1956). H. S. Harned and L. D. Fallon, J . Am. Chem. SOC.,61, 2374 (1939). S. D. Hamann, J. Phys. Chem., 67,2233 (1963). * S. D. Hamann and W. Strauss, Trans. Faraday SOC.,51, 1684 (1955). m-Benzenedisulfonate. a
(1964).
'
'
'
'
(20) S. Lua and S. Meiboom, J. Chem. Phya., 40, 1058 (1964).
Volume 72#Number 6 May 1968
1540 and acetic acid in methanol-water solutions. Also, the values of the slopes of the lines (log K vs. log CHIO) were generally lower in acetone-water and methanolwater mixtures as compared to the values obtained from the dioxane-water mixtures, which indicates that fewer waters of hydration were involved in the change. These comparisons indicate that there is some solvation by acetone and methanol, which might be expected after considering that these two molecules are more polar than the dioxane molecule. Mixed methanolwater complexes of cobalt have previously been reported in methanol-water mixtures at low temperatures.20 The literature also contains the results of several investigations on the change in a conventional equilibrium constant with pressure at 25”. We have evaluated these results, considering hydration as part of the equilibria and using literature values for the density (concentration) of water at the various pressures, and again we observe the linear relationship between log K and log CH~O.The values obtained for the net changes in waters of hydration are included in Table 11. Several features of the hydration changes as reported in Table I1 should be mentioned. For the alkali metal halides (with the exception of the lithium salts) the net change in waters of hydration for the ionization of the neutral molecule (or ion pair) is usually near 6. Hydration changes near 9 are observed with JSgSO, and MnSOzIin both water-dioxane mixtures and in aqueous solutions under pressure. However, the net change in hydration for the dissociation of weak acids in aqueous solutions under pressure at room temperature is large compared to the value obtained in water-dioxane mixtures. Similarly large changes are observed for the dissociation constants of weak bases under pressure at room temperature. These large hydration changes may possibly be related to the “extra” mobility of the hydrogen and hydroxide ions, the structure of water at room temperature, and their changes when solvent concentrations are varied by pressure alone. From the above results, it is clear that the use of the molar concentration of water in the complete equilibrium constant expression provides an explanation of the change in the conventional equilibrium “constant” with water concentration over wide ranges of waterorganic solvent compositions. By comparison, the deviation from linearity of plots of log K against 1/D is usually quite large, and therefore in disagreement with presently accepted theory.
Selective Solvation in Nonaqueous Solvent Mixtures Selective solvation of electrolytes might also be expected to occur in nonaqueous solvent mixtures. The logarithms of the conventional ionization constants of electrolytes in these systems would therefore be expected to show the same linear dependence on the The Journal of Physical Chemistry
ARVINS. QUIST AND WILLIAM L. MARSHALL t/idieleclric constant1
0.20
o , ,
0
0.16 I
,
0.2
0.(2 I
,
008 ,
04
004
,
06
08
40
139 CNITROBEN~ENE lmoles/l~terl
Figure 3. Log K (dissociation) of several tetrabutylammonium salts us. log nitrobenzene molarity in carbon tetrachloridenitrobenzene mixtures; also compared os. l/(dielectric constant, D);K values of Hirsch and Fuoss (1960); T = 25’.
logarithm of the molarity of the polar component where the other component is essentially nonpolar as was observed for aqueous systems. Fuoss and Hirsch21 have reported ionization constants for several quaternary ammonium salts in nitrobenzene-carbon tetrachloride mixtures. Since nitrobenzene is much more polar (dipole moment of 4.3 D) than carbon tetrachloride (dipole moment of 0), selective solvation by nitrobenzene would be expected. Figure 3 shows a linear relationship from pure nitrobenzene to moderately low concentrations when the logarithm of these ionization constants is plotted against the logarithm of the nitrobenzene molarity. Included also in Figure 3 are the comparative plots of log K against 1/D. The initial slopes of the log K VS. log Cnitrobenzene lines appear to have nearly the same value (6.4) for all the quaternary ammonium salts given in Figure 3. Although it seems surprising that a deviation from linearity at low concentrations of nitrobenzene occurs in carbon tetrachloride-nitrobenzene solutions (Figure 3, lower segment) and not in corresponding dioxane-water solutions (Figure 1) , this behavior might be explained by considering the smaller polarity characteristic of nitrobenzene (D = 35) as compared to water (D = 78). Thus, the comparative energy of solvation with nitrobenzene could be less than that with water and might allow a small but detectable solvation by CCL. If minor solvation by CC14 does occur, then an expression as follows might apply (21) R.M.Fuoss and E. Hirsch, J . Am. Chem. SOC.,82, 1013, 1018 (1960).
1541
ISOTHERMAL EQUILIBRIA IN ELECTROLYTE SOLUTIONS Kdo
+
Bu4N picrate(eolv) Ic(CeHsNO2) Bu4N+(,,1,,
KdO
=:
4- p(Cc14)
+ picrate-(solv)
+apiomte(soiv)
(34
K / (C’C~H~NO~C~CC~J
MA ( ~ ) U ‘ H ~ O
+ jH20 = MA(H20),(aq) K,O
=
aMA(HzO)g(sq) aMA(s)a’HzO
(4a) (4b)
(54 (5b)
Equations 4 and 5 are related by the complete equilibrium constant for the dissociation of the neutral species in solution MA(H20)’bq)
+ (r - j)HzO
=
M(HzO)m+ Kdo
=
+ A(H20),-
UM ( H ~ O ) , + ~ A ( H ~ O ) , ~MA(H~O)+JH~O(’-’)
1
(64 (6b)
Dickson, Blount, and Tune1122have measured the solubility of anhydrous CaS04in water from 100 to 275” and a t pressures to 1000 bars. When the logarithm of the molar solubility of CaS04 is plotted against log CH~O, at 250°, a straight line is obtained. If it is considered that Cas04 is nearly completely ionized under these conditions,28a hydration change of 28 is calculated according to the equilibrium of eq 4b. Since the solid phase is anhydrous, this number, although large, corresponds to the sum of the hydration numbers of the calcium and sulfate ions and not to the net change of
I
I
1
i i
.i t
1
(3c)
where the activity of the solid phase is a constant and may be set equal to unity a t all temperatures and pressures, and where the solute is completely ionized in solution. When the solute remains undissociated in solution, the solubility equilibrium may be written as MA(s)
I
T-600.C
g -2 >
-1 F
-3
r
-4
-I
-6
U M (H’o), + ~ A ( H * O ) , -
‘
-4
Solubility Equilibria in Aqueous Solution In accordance with the concept of water as a necessary concentration variable, a complete solubility product equilibrium for the dissolution of an anhydrous salt may be written as
KspO =
I
(3b)
*
+ r H 2 0 = RI(H20),+ + A(H20),-
1
0
where Kdo is now the complete constant and k and p represent the net (selective) changes in solvation in the reaction equilibrium of solvents CsH6N02and CCh, respectively. By a method of least squares, values of k of 7.7 and p of 0.61, respectively, were obtained that provided a fit to the conventional K’s of 15%, within the experimental precision of the K’s, over the entire range of compositions (but not including pure nitrobenzene, where eq 3 cannot apply).
MA(s)
I
DATA OF SOURIRAJAN AND KENNEDY (49621
-
~ B u ~picratc(solv)CkCeHsNOzcpCClr N
=:
I -
-R2
0
0.2 0.4 0.6 109 C H ~ holer/liierl,BASEO Q
0.8 4.0 4.2 IA’ 4.6 ON DENSITY OF SOLUTION+DENSITYOF WATER
4.8
Figure 4. Logarithm of the molar solubility of NaC1 us. log molar concentration of water a t 600”
hydration numbers between the solvated ion pair and the ions as listed in Tables I and 11. When the solubility data for NaCl at 6OOoz4are treated in a similar manner, the graph shown in Figure 4 is obtained. At low solution densities where NaCl exists nearly completely in the form of neutral speciesl3and eq 5b would be expected to apply predominantly, the slope of the line gives a hydration number of approximately 2 for this neutral species. At higher solution densities, NaC1 has been observed to ionize appreciably, and so the slope of the line approaches that number equal to ‘/z the sum of the hydration numbers of the sodium and chloride ions. In Figure 4 the densities of the solutions have been assumed to be those of pure water, a reasonable assumption at low solubilities but invalid for the concentrated solutions. This causes the graph to become nonlinear at high concentrations of water where the solubility of NaCl is quite large. If a t high solubilities the mean activity coefficient of NaCl would change markedly with changing water concentration, nonlinearity could occur also from this effect. However, at solution saturation, this coefficient would be expected to remain constant as discussed in the next section. The solubilities of inorganic salts in water-organic solvent mixtures also exhibit the two types of behavior shown by eq 4a and 5a. Ricci and coworkers have measured the solubilities of several salts in water-dioxane mixtures; among these are AgAc,” Ag2S04,17 Ba(22) F. W. Dickson, C. W. Blount, and G. Tunell, Am. J . Sci., 261, 61 (1963). (23) W. L. Marshall and R. Slusher, J . Phys. Chem., 70, 4015 (1966). (24) S. Sourirajan and G. C. Kennedy, Am. J . Sei., 260, 115 (1962). Volume 72, Number 6 May 1968
1542
ARVINS. QUISTAND WILLIAML. MARSHALL I
Table I11 Temp,
From solubilities
Cas04 KIOa Zn(IO& AgAc AgzSOd NaNOa Ba(IOs)z. HzO
OC
Water to 1000 bars Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane Water-dioxane
Total waters of hydration of ions in solution
250 25 25 25 25 25 25
28 9.4 14.1 6.4 17.1 3.2 18.1
Lit. ref
a b b C C
d C
a Reference 22. Reference 25. Reference 17. B. Selikson and J. E. Ricci, J. Am. Chem. Soc., 64,2474 (1942).
-5
I 0.4
0.6
0.8
I 4.0
IS^ c H
silver acetate in saturated solution a t 25" was found to remain essentially constant in water and in several compositions of ethanol-water, acetone-water, and dioxanewater. Since all of these solutions contain water, on the basis of the concepts in this paper we may write for the complete solubility product equilibrium
I
I2 44 irnoler/l~lerl
1.6
4.8
2.0
Figure 5. Log solubility (moles per liter) for several salts us. log C H ~ O in dioxane-water mixtures a t 25'; data of Ricci, el al. (1939, 1942). (Added in proof: add $0.4 and -0.2 to log solubilities for AgzSO4 and Zn(I03)z, respectively; corrected upper slopes = 5.7 for both AgzS04 and Ba(IOs)z.HzO, and 3.2 for AgCZH802.)
(IO&.H20,17 KIOs,26and Z n ( 1 0 ~ ) ~ .For ~ 6 most of these salts, &hen the logarithm of molar solubility is plotted against the logarithm of water molarity, the straight line that is observed a t high water concentrations in all cases undergoes a change in slope a t moderately low water concentrations, as shown in Figure 5 . I n this figure, the slope a t very low concentrations of water approaches that corresponding to the hydration number of the neutral molecules (eq 5b). At high water molarities the slope approaches a value related to the sum of the hydration numbers of the ions. For silver sulfate, this sum is equal to three times the slope since the product of the ion concentrations is proportional to the cube of the solubility of silver sulfate. Table I11 summarizes the results of our interpretation of some of the solubility measurements contained in the literature, where HzO in Ba(IOa)2.H20is included in the total. Constancy of Electrolyte Activity Coefficients in Saturated Water-Organic Mixtures Ricci and Daviszehave mentioned several systems in which the activity coefficients of an electrolyte in its saturated solution, based on the use of the DebyeHuckel limiting law with the measured dielectric constants of the solvent mixtures, are nearly constant and independent of the nature of the water-organic solvents. For example, the mean ionic activity coefficient of The Journal of Physical Chemistry
Kspo= aAg(HzO)m+aAo(HzO)n-aAgAc(s)a7Ha0
Ksp
aTH20
(7b)
If the activity of solid silver acetate (a constant) by convention is taken to be unity, if the activity of water is set equal to its molar concentration (a reasonable assumption for these solutions of low concentrations), and if the conventional Kspis expressed by
KBP= aAg(HIO)m+aAo(HxO)n=
Qspfh2
(sa)
then eq 7b may be written as 1%
&sp
log &Po
+ r log CH*O- 2 logf*
(9)
For saturated soIutions of silver acetate in acetone-water, ethanol-water, and dioxane-water at relatively high water concentrations we have obtained the following relationships from the experimental data
log Qsp = r = 6.0,5.4, 6.4 (respectively) 3 log CHaO
(10)
Since
(the true, isothermal equilibrium constant is invariant), then
for these systems, which is what Ricci and others have (25) J. E. Ricci and G. J. Nesse, J . Am. Chem. SOC.,64, 2305 (1942). (26) J. E. Ricci and T. W. Davis, ibid., 62,407 (1940).
ISOTHERMAL EQUILIBRIA IN ELECTROLYTE SOLUTIONS
1543
observed using the Debye-Huckel limiting law to calculate f* for these systems. If the extended DebyeHuckel equation involving the ion-size parameter is used instead of the limiting law, the calculated activity coefficient should still remain constant with change in water concentration since it appears that the hydration number (and therefore ionic size)Z7 does not change with water concentration. Consequently, the constancy of the mean ionic activity coefficients for saturated silver acetate solutions in several solvent mixtures appears t o be related to the apparent selective hydration of ions over a wide range of solvent composition and the related constancy of the complete equilibrium constant. On the basis of their observed constancies of activity coefficients for electrolytes in various solvents, and with the use of the Debye-Huckel theory, Ricci and Davisz6 proposed that a plot of log solubility us. log D would give a straight line of slope 3. They showed an order of magnitude agreement for many solvent systems, but considerably better agreement with dioxane-water mixtures. With this latter solvent system where we O consider dioxane to be an “inert” diluent, log C H ~ is nearly a linear function of log D over a very wide range of C K ~ Oand , therefore if linearity is observed for log it will also solubility (or log K ) plotted against log CH~O, be approximately observed for log solubility (or log K ) vs. log D . The dielectric constant is, therefore, a function chiefly of the molar concentrations of polar species in the solvent system.
value of 2 molecules of HzO for the number involved in the hydration of the neutral NaC1 ion pair. The average ion volume then can be expressed as
Liquid-Liquid Equilibria in Aqueous Inorganic Systems Uranyl sulfate-water systems are unique in that they are among the very few examples of electrolyte-water systems (that do not contain organic solvents) that exhibit liquid-liquid immiscibility. Secoy2*was the first to report this characteristic behavior exhibited by uranyl sulfate solutions at high temperatures and pressures. In later studies at this l a b o r a t ~ r ythe , ~ ~effect of pressure on the temperature at which the second liquid phase appears in these solutions was determined. This system was studied to 400” and 1700 bars, and it was found that temperature of liquid-liquid immiscibility was strongly dependent on solvent density. Since the molar concentration of water varies directly with its density, another direct link between a phenomenon that probably involves ion-solvent interactionsao and the molar concentration of water is shown. The Ion=Size Parameter and Related Comments on Debye-Huckel Theory and Structure of Water The principle can be used to calculate values for it, the ion-size parameter of the extended Debye-Huckel theory. For example, from plots such as Figure 1, the value of IC can be estimated. For NaCl, this value was 6.4. From the available solubility information on NaC1 and related salts (Figures 4 and 5), we have estimated a
rav
=
[VNa+
+
VCl-
+ (k + 2)(2vH+ + vOZ-)]/2 (13)
where VNa+,Vc1-, VH-, and Voz-are ionic volumes obtained from a tabulation of crystallographic radii31 ( T ~ ~=+ 0.95 A, r c l - = 1.81 A, T O Z = 1.40 A), and ~ V His+considered negligible compared with V O X - . Since V8v = ~(&)3/6,it follows that & for sodium chloride solutions is 4.9 8. This value is in very good agreement with that, of 5.2 given by Robinson and Stokes and obtained from the evaluation of transport numbers by extended Debye-Huckel theory.3z The constancy of the slope, n, of Figure 1 produces an ion size parameter that is independent of the added “inert” solvent. This constancy of n and of KO also implies that any isothermal changes in the structure of water may be unimportant to the electrolyte-solvent equilibrium. Although for the example given above, a value of 2 was roughly estimated for the hydration number of NaClO, the solubility data in mixed solvents were not available for a more exacting estimate. When a value of 4 was used for the hydration number of the neutral NaCl species, an ion size of 5.2 was obtained. The attainment of rather exact ion-size parameters for a variety of electrolytes by this method may have to await additional solubility measurements of electrolytes in dioxane-water mixtures, or the availability of reliable hydration numbers of neutral species by other methods. The use of a complete KO does not contradict DebyeHuckel theory and its use of the dielectric constant of the solvent or solvent mixture in calculating mean ionic activity coefficients. Long-range interionic effects on ions are certainly still dependent on the solvent dielectrics. The concept indeed appears to support the extended Debye-Huckel theory by providing comparable hydrated ion-size parameters. The actual value of KO will depend upon the dipole moments (and ionic charges) or the various reactants and products in the equilibrium. These values will change with temperature. Knowledge of these quantities might provide a means for an a priori calculation of KO.
Previous Proposals A relationship between an ionization constant in a mixed solvent system and the amount of one of the (27) See ref 4. (28) C. H. Secoy, J . Am. Chem. Sac., 72, 3343 (1950). (29) W. L. Marshall and J. S. Gill, J . Inorg. NucZ. Chem., 2 5 , 1033 (1963). (30) H. L. Friedman, J. Phys. Chem., 6 6 , 1595 (1962). (31) L. Pauling, “The Nature of the Chemical Bond,” Cornel1 Cniversity Press, Ithaca, N. Y., 1945. (32) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed revised, Butterworth and Co. Ltd., London, 1965, p 158. Volume 73, Number 6 M a y 1968
ARVINS. QUISTAND WILLIAML. MARSHALL
1544 solvents present has been reported previously. and Fal10n~~ used an empirical equation log K = - C - C'N
Harned (14)
for the interpolation of ionization constants of acetic acid in several water-dioxane mixtures (to 82% dioxane). In this equation, K is the ionization constant (not considering hydration) and N is the mole fraction of water (or dioxane). Feakins and French34used an approach to the problem of ion solvation in waterorganic solvent mixtures as developed by Hudson and S a ~ i l l eto~ show ~ that a plot of the standard potential (Ec0,molar scale) of the cell H2(Pt)IHC11AgC1-Ag in these mixtures was linearly related to the logarithm of the volume fraction of water in the solvent medium. Aksnes3'j considered electrostatic effects in the ionization of weak acids in water-dioxane mixtures and concluded that the main reason for the decrease in the ionization constant of the weak acid with increasing dioxane concentration was due to a statistical effect; that is, water must be regarded as a reactant because it hydrates the proton. He concluded that the dielectric constant to be used in the Born equation (for calculating the attraction between oppositely charged ions in solution) should be the dielectric constant of the water only in the solvent mixture. The relationship between the solubility of substances in supercritical water and the density of the water has been recognized for some time. Kennedy3?mentioned the linearity between the solubility of quartz and the logarithm of the specific volume of supercritical water. R l ~ r e yalso ~ ~commented on the relation between solubility and solvent density in connection with his measurements on the solubility of quartz in steam. Jasmund39 showed that the concentration of KC1 in steam could be represented by the equation where A and n are constants. This equation can also be written as
Styrikovich40has used equations similar to eq 15 to obtain hydration numbers for several inorganic substances in high temperature water. Franck41 derived by statistical methods a relationship between the logarithm of the mole fraction of solid in a gas phase and the logarithm of the density of the gas phase. More recently, T ~ r ' y a nhas ~ ~ calculated hydration numbers
The Journal of Physical Chemistry
from the solubilitites of several slightly soluble salts in isodielectric ethanol-water and methanol-water mixtures of high water content.
Conclusions It has been apparent for some times that theories of ion-pair formation based on the Born equation are not adequate to describe completely the experimentally observed behavior. By considering solvent species as reactants of variable concentration in the equilibrium process, the isothermal equilibrium constant is found to be invariant with changes in molar solvent concentrations (changed by means of pressure or composition changes) which, when molar activity units are used, is one of the requirements of a true equilibrium constant. Although linear relationships between conventional equilibrium constants (not including waters of hydration) and some function of the concentration of water have been reported in the literature (usually for equilibria involving the ionization of weak acids), this is believed to be the first time that such a relationship has been applied to the appropriate concentration unit (molarity) l 3 for complete equilibrium constants, and it is the first time that it has been shown to apply over a wide range of temperature, pressure, and solvent composition. This exacting linear relationship over wide ranges of solvent composition is not observed when conventional ionization constants are related to the reciprocal of the dielectric constant, which is the relationship predicted by the usual electrostatic theories of ionic association. We expect the present approach to lead to a better understanding of the details of ion association and ion-solvent interaction and to furnish a powerful tool for extrapolations into regions where data do not exist. (33) H. S. Harned and L. D. Fallon, J . Am, Chem. Soc., 61, 2377 (1939). (34) D. Feakins and C. M. French, J . Chem. SOC.,2581 (1957). (35) R. F. Hudson and B. Saville, ibid., 4114 (1955). (36) G. Aksnes, Acta Chem. Scand., 16, 1967 (1962). (37) G. C. Kennedy, Econ. Geol., 45, 629 (1950). (38) G. W. Morey and J. M. Hesselgesser, Trans. A S M E , 73, 865 (1951). (39) K. Jasmund, Heidelberger Beitr. Mineral. Petrogr., 3 , 380 (1953); Chem. Abstr., 47, 11899 (1953). (40) M. A. Styrikovich, I. Kh. Khaibullin, and D. G. Tskhvirashvili, Dokl. Akad. Nauk SSSR, 100, 1123 (1955); Chem. Abstr., 49, 1707c (1955). (41) E. U.Franck, 2. Physik. Chem. (Frankfurt), 6 , 345 (1956). (423 Ya. I. Tur'yan, Rum. J . Inorg. Chem., 10, 369 (1965).
