w. L. MASTERTON AND TEI PEI LEE
1776
Salting Coefficients from Scaled Particle Theory by W. L. Masterton and Tei Pei Lee Department of Chemistry, University of Connecticut, Storm, Connecticut
(Received October 27, 1969)
Scaled particle theory is used to derive a general expression for the salting coefficient, k,, which appears in the Setschenow equation. This expression is essentially a sum of two terms, one of which will ordinarily be positive (salting out), while the other will be negative (salting in). Calculated values of k , are in excellent agreement with experiment for systems in which the molecular and ionic diameters are relatively small. For large molecules, the agreement is poorer, as is true with classical theories of the salt effect. The magnitude of k , is very sensitive to uncertainties in the diameters of anions and cations.
Introduction
Bockris, Bowler-Reed, and I 8); culate k, are not readily available. The other theories if IC, is negative, salting in is observed (So < 8). of the salt effect suffer from this same weakness, alThe electrostatic theory of the salt effect, proposed beit to a lesser extent. originally by Debye and M.cAulay,l and refined most A few years ago, pie rot ti'^^ adapted the scaled particle recently by Consvay, Desnoyers, and Smith,2relates the theory of Reiss, et al.,Qr10to predict the solubilities of sign of k, to the effect of the nonelectrolyte on the dinonelectrolytes in water. The equation of Pierotti electric constant of xater. Species which lower the relates aqueous solubility to the sum of two free energy dielectric constant should be salted out by all electroterms: the work required to create a cavity large lytes. Electrostatic theory gives values of k, which are enough to accommodate a nonelectrolyte molecule and of about the right order of magnitude, but, contrary to the energy of interaction between the solute molecule observation, vary very little with the nature of the and the surrounding solvent. This model has been salt. Jloreover, this theory cannot explain the general reasonably successful in predicting the solubilities and phenomenon of salting in by large ion electrolytes such other thermodynamic properties of nonelectrolytes in as the tetraalkylammonium halides. water solution. The McDevit-Long theory of the salt effects atQuite recently, Shoor and Gubbins” have extended tributes the sign of k, to the influence that the salt has on the water structure. If it “compresses” the water (1) I?. Debye and J. McAulay, Phys. Z., 2 6 , 22 (1925). structure, it becomes more difficult to introduce non(2) B. E. Conway, 3. E. Desnoyers, and A. C. Smith, Phil. Trans. Roy. SOC.London, A256, 389 (1964). electrolyte molecules and salting out occurs. If, on (3) W. F. McDevit and F. A. Long, J . Amer. Chem. Soc., 74, 1773 the other hand, the water structure is “loosened” by (1952). the addition of salt, salting in is predicted. I n terms of (4) 5. O W . Bocltris, J. Bowler-Reed, and J. A. Kitchener, Trans. this picture, the AIcDevit-Long theory offers a plausible Faraday Soc., 47, 184 (1951). (5) W. L. Masterton and R. N. Schwartz, J . Phys. Chem., 69, 1546 explanation of salting in by tetraalkylammonium hal(1965). ides. It also gives relative values of k , for different (6) W. L. Masterton, T. P. Lee, and R. L. Boyington, J . Phys. salts with the same nonelectrolyte which fall in the Chem., 73, 2761 (1969). correct order (e.g., k , LiCl < IC, YaCl > k, KC1 > IC, (7) R . A . Pierotti, J . Phus. Chem., 69,281 (1965). KI). However, the absolute values of k, calculated (8) R. A. Pierotti, ibid., 67, 1840 (1963). (9) H. Reiss, H. L. Frisch, E. Helfand, and J. L. Lebowitz, J . Chem. from the equation of IIcDevit and Long are in poor Phys., 32, 119 (1960). agreement with experiment. For example, it predicts (10) H. Reiss, H. L. Frisch, and J. L. Lebowitz, ibid., 31, 369 the salting coefficient for the sodium chloride-benzene (1959). system to be 0.42; the observed value is 0.198. (11) S. K. Shoor and K. E. Gubbins, J . Phys. Chem., 73, 498 (1969) The Journal of Physical Chemistry
1777
SALTING COEFFICIENTS FROM SCALED PARTICLE THEORY Pierotti's scaled particle model to obtain an equation for the solubility of a nonelectrolyte in an aqueous salt solution. They have applied this equation to calculate the solubility of certain nonpolar gases (He, Hz, Ar, 0 2 , CH4, SFs, and C(CH3)4)in concentrated solutions of potassium hydroxide (10-50 wt % KOH). The agreement with experiment is excellent for Ar, 02, and CH4. For the other gases, agreement is not as good, particularly a t high electrolyte concentrations. This may reflect the fact that the several approximations which must be made lead to serious errors at high concentrations. Then too, one can hardly consider an alkali hydroxide to be a typical 1 : 1 electrolyte. The scaled particle approach to the calculation of salt effects has the great advantage that the required molecular parameters are readily available. For this reason among others, it is of interest to extend the derivations of Shoor and Gubbins to obtain a general expression for the salting coefficient, applicable to any salt-nonelectrolyte pair. By comparing calculated and observed values of k,, it should be possible to judge the applicability of scaled particle theory to the prediction of salt effects. This paper presents the results of such an analysis.
Expression for the Salting Coefficient In practice, the Setschenow eq 1 is found to be valid only at low salt concentrations, i.e., as c -t 0. To obtain an expression for k,, it is convenient to differentiate eq 1 with respect to c
of eq 4. The problem now becomes one of deriving general expressions for ha, k,, and k , in terms of parameters characteristic of the nonelectrolyte and the ions of the salt. For simplicity, the derivation will be limited to 1: 1 salts and slightly soluble nonelectrolytes a t 25"; the extension to other systems is an obvious one. We start with the expression for k,, since this is the easiest to evaluate. Expression for k,. For slightly soluble nonelectrolytes, p1 can be dropped from the summation12 and we can write ZP, = P2
-1ogs =
4
+ -+ log kT c P5 2.3kT 2.3kT Blh
BIS
__
j=1
L,
=
[d'"lV.;kT']
[d(y;;pkT)]c+O k,
+ IC, + k,
(7)
where N is Avogadro's number. Applying the defining equation for the apparent molal volume, 4, to 1 cc of solution, it is readily shown that the variation of the number density of water, pz, with concentration is given by the equation
where dz is the density of pure water (0.997 g/ml at 25"), M2 is the molecular weight of water, 18.02 g/mol, and 4 is the apparent molal volume of the salt in solution. Substituting for pz, p3, and p4 in eq 6, we obtain, after rearranging 2CMZ l+--Nd2 Mz lOOOdz
[
At low concentrations, i.e., as c Zp, =
(3)
In
Ndz M2 ~
--f
1000
0
2cM2 C+ +-lOO0dz 1000
(10)
To obtain k,, we neglect the (small) variation of 4 with c and write
"= =
d log Zpj dc
[
0.016
IC+,
- 4.34 X
2M2
=
23ood2
- -90
2300
10-44~
(11) where ,$t is the apparent molal volume of the salt at infinite dilution. Expression for k,. Shoor and Gubbins derive the following expression for the interaction energy between
+
C-CO
=
(6)
P4
Nc/1000
p3 = p4
In
where glh is the free energy change when a cavity large enough to hold the nonelectrolyte molecule is formed in the solution, is the free energy change when the nonelectrolyte is introduced into the cavity, and p5 is the number density (particles/cc) of a solution species. Following Shoor and Gubbins, we use the subscript to represent the nonelectrolyte, 2 the solvent (water), 3 the cation (Na+, K+, etc.), and4 the anion (Cl-, I-, etc.). Combining eq 2 and 3
P3
For a 1 : 1 electrolyte
Zp5=--
Shoor and Gubbins" derived from scaled particle theory the following expression for log S
+ +
+ [d
';7c,"q c-0
(4)
(5)
where IC, k,, and k, represent the contributions to the salting coefficient of each of the three terms on the right
(12) To justify dropping P I , we must show that its concentration dependence is negligibly small compared to that of p2, P S , or p4. The Setschenow equation can be written in the form In p l / p i O = - 2.3k,c, or p l = p10e-2.*ksc, where plo is the number density of the nonelectrolyte in pure water. Expanding the exponential and retaining only the first two terms: P I = plo - 2.3k.pl0c. Realizing that p l o = NC10/1000, where Clo is the molar concentration of nonelectrolyte in pure water, we have API/AC= (-2.8ksNC1o)/1000. From eq 7 Apa/Ac = Apa/Ac = N / 1 0 0 0 . For slightly soluble nonelectrolytes, C10 is very small, ks is ordinarily less than unity, and 2.3ksClo Na+; I- > Cl-) on the leading term in eq 19. Combining these two trends, it is clear that scaled particle theory predicts that IC, will become algebraically smaller (less salting out or more salting in) as the sizes and polarizabilities of the cation and anion increase. This general trend is observed experimentally. An exception is LiC1, which ordinarily salts out less than NaCl. Interestingly enough, scaled particle theory predicts that LiCl should be out of line, primarily because its apparent molal volume is abnormally large (cf Table 11). I n many respects, the expression for k, derived from scaled particle theory resembles that from the theory of Bockris, et aL4 In both cases, k, is essentially a sum of two terms, one of which leads to salting out (k,), the other to salting in (k,). In the Bockris approach, the first term takes into account electrostatic forces and is always positive; the second term, which is usually negative, arises from dispersion forces, as does k,. Both theories predict that the second term will The Journal of Physical Chemistry
Table 111: Effect of Uncertainties in Parameters on the Calculated Value of k, System
H2-NaCl Hz-KI CH4-NaCl CHrKI SF8-NaCI SFa-KI
Aa/a = 0.05
Aoi/ui
Aualsa
= 0.05
= 0.05
Audlu4 = 0.06
-0.001 -0.001 -0.002 -0.003 -0,005 -0.006
$0.008 +0.006 +O.OlO $0.006 +0.015 +0.009
+0,005
+o. 022
+O.Oll
+0.035 + O . 039 $0,062 $0.078 +O. 123
$0.010 $0.020 +0.022 +0.042
become more important as t,he size and polarizability of the ions increase. The advantage of scaled particle theory is that the parameters necessary to evaluate k, are more readily available. Estimated Uncertainties in Calculated Salting Coe$cients. Of the several parameters required to evaluate k, for a particular salt-nonelectrolyte pair, 23 and 2 4 , the numbers of electrons in the ions, are known exactly. The apparent molal volumes, +o, of the salts are known quite accurately. The polarizabilities, all as, and (24, are somewhat less certain, but literature values generally agree with each other within a few per cent. The parameters most subject to error are the energy parameter of the nonelectrolyte, € I l k , its molecular diameter, ul, and the ionic diameters u3 and u4, probably in that order. Hirschfelder, Bird, and Curtiss16 list values of el/k and ~1 which, for a given nonelectrolyte, may vary by as much as 10%. The problem of estimating ionic diameters is even more serious. We have chosen to use for u3 and u4 values calculated by simply doubling the crystallographic radii. Alternatively, one could use hydration radii, calculated by (15) J. 0. Hirschfelder. C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” John Wiley and Sons, New York, N . Y., 1964.
SALTING COEFFICIENTS FROM SCALED PARTICLE THEORY
1781
Table IV : Comparison of Observed and Calculated Salting Coefficients at 25"
.
LiCl
Obsd
He Ne Ar Kr Hz 0 2
Nz CHI C2Hd CzHe SFs
d
0.050a 0.081 0.059" 0.079 0.096" 0.087 0.116" 0.082 0.076b 0.088 O.lOOb 0.097 0.095' 0.102 0.097c 0.095 0.089C 0.090 0.124C 0.089 0.1450 0.131
e
0.112 0.060
0.098 0.119
0.091 0.108 0.140 0.136 0.175 0.192 0.266
,----
Obsd
0.0810 0.097" 0.133" 0.146a 0.114b 0.14lb 0.12lc 0.127' 0.127' 0.162' 0.1950
NaC1d
0.102
0.100
---
----KI--
-------KCl---------,
8
0.151 0.080
Obsd
d
0.06@ ...
0.091
...
0.117 0.114 0.111 0.129
0.132 0.160
0.137 0.131 0.132 0.134 0.202
0.188 0.184 0.236 0.260 0.358
0.1245 0.102b
0.123 0.146
... ,. . .. . .. . .. .
0.165a
0.088 0.100 0.096 0.099 0.113 0.119 0.113 0.111 0.111 0.171
e
0,123 0.066 0.108 0.131
0.100 0.119 0.154 0.151 0.193 0.212 0.292
Obsd
0.083a 0.0800. 0.108a
d
0.076 0.067
0.067 0.1200 0,060 0 . 0 8 1 ~ 0.080 , , . 0.082 0 . 1 0 0 ~ 0.085 0.0970 0.076 O.06lc 0.065 0.101C 0.062 0.145" 0.113
e
0,070 0.037 0.061 0.074 0.057 0.068 0.087 0.086 0.109 0.120 0.166
5 T.J. Morrison and N . B. B. Johnstone, J. Chem. SOC., 3655 (1955). F. A. Long and W. F. McDevit, Chem. Reo., 51, 119 (1952). T. J. Morrison and F. Billett, J. Chem. SOC.,3819 (1952). Calculated from scaled particle theory, eq 11, 19,and 32. e Calculated from McDevit-Long theoryS3 Molar volumes of nonelectrolytes taken from J. H. Hildebrand and R. L. Scott, "Solubility of Nonelectrolytes," 3rd ed, Van Nostrand-Reinhold Co. Inc., Princeton, N. J., 1950.
c
various means,2 which are generally much larger than the crystal radii. In Table I11 we have listed the effect upon the calculated value of ks of increasing E&, (rl, m, and 6 4 by 5%. Clearly, the most critical parameters are u3 and ~ 4 precisely , the ones that are least accurately known. Shoor and Gubbins" suggest evaluating these parameters by empirically fitting salt effect data for nonelectrolytes of low polarizability. The values of ~3 and u4 they obtain for K + and OH- fall within a few per cent of those calculated from crystal radii. It is clear from Table I11 that the uncertainties in the various parameters become more critical with large, highly polarizable molecules (SFO > CH, > Hz) or ions (K+ > N a f ; I- > C1-). One can expect the k, values for systems involving such species to show relatively poor agreement with experiment. This conclusion is reinforced by the fact that in deriving the expression for k, from scaled particle theory, it is assumed that the entropy of interaction between solute and solvent species is negligible and that the molecular distribution is uniform." These assumptions are likely to break down with large molecules and/or ions. They would almost certainly lead to serious errors with polar nonelectrolytes. Comparison with Experiment. In view of the various approximations implicit in scaled particle theory and the uncertainties in the several parameters required to calculate k,, the values calculated using eq 11, 19, and 32 and reported in Table IV are in remarkably good agreement with experiment. In about two thirds of the systems, the salting coefficient predicted from scaled particle theory is closer to the observed value than that calculated from the McDevit-Long theory. This effect is most pronounced with the aliphatic hydrocarbons and sulfur hexafluoride, where, for LiC1, NaC1, and KC1, the McDevit-Long theory predicts salting
coefficients which are nearly twice as large as those observed. The poorest agreement with experiment is found with the krypton systems; here the McDevit-Long theory clearly gives better results. Scaled particle theory also gives relatively poor results for systems involving potassium iodide; the calculated values of k, are almost always significantly smaller than those observed. This suggests that the diameter we have chosen for the iodide ion should be modified; an increase of 0.1 A in u4 would bring most of the salting coefficients for K I to within a few per cent of the experimental values. The nonelectrolyte for which the greatest amount of salt effect data is available is probably benzene. In Table V, the observed salting coefficients for benzene with various alkali halides are compared to the predictions of scaled particle theory (column a ) , the McDevitLong theory (column b ) , and the electrostatic theory (column c ) . Clearly, the agreement is rather poor in all three cases. The scaled particle and RIcDevit-Long Table V : Salting Coefficients for Benzene at 25"
NaCl KCl NaBr LiCl RbCl KBr NaI CSCl CSI
a
a
C
d
Obsd
0,105 0.077 0.058 0.048 -0.024 0,030 0,036
0.42 0.34 0.35 0.31 0.31 0.34 0.27
0,166 0.156 0.163 0.172 0.153 0,153 0.158
0.188 0.162 0,153 0.124 0.089 0.128 0.137
0.198 0.166 0.155 0.141 0.140 0.119 0.095
-0.032
0.26
0.150
0.085
0.088
-0.101
0.11
0.141
0.034 -0.006
Calculated from scaled particle theory, using o ( ~= 9.89 X UI = 5.27 X 10-8 cm, e l / k = 440. Calculated from McDevit-Long theory.8 c Calculated from electrostatic theorya2 d Calculated as in (a), but taking el/k = 214. Volume 74, Number 8 April 16, 1970
1782 theories predict the correct salt order, but the absolute values of k, are too small in the former case and too large in the latter. Electrostatic theory gives salting coefficients of the right order of magnitude but does not yield the correct salt order. Indeed, it predicts essentially the same value of k , for each of the alkali halides, contrary to what is observed. The salting coefficients for benzene calculated from scaled particle theory using the parameters of ref 7 are uniformly low by about 0.08-0.10 unit. This suggests that we may be overestimating the effect of the interaction between the benzene molecule and the surrounding ions. If the free energy of interaction were smaller, IC, would be less negative and the salting coefficient, IC,, would be larger. One way to obtain better agreement between scaled particle theory and experiment for benzene would be to adjust the values of el/k and ul,which determine the magnitude of IC,. In this connection, it is interesting to see what would happen if me were to use Pierotti's value for u1 but calculate ~ l / kfrom the MavroyannisStephen theory (eq 18), which was used to obtain E ~ / I C and q / k . The values of k, obtained in this manner are given in column d of Table V. These numbers are in quite good agreement with the experimentally determined salting coefficients. Only with RbCl and, to a lesser extent, NaI, are there large discrepancies when values of ~ l / calculated k from eq 18 are used to evaluate 16, and hence k,. Lest it be supposed that this treatment will always give salting coefficients in better agreement with experiment, it should be pointed out that we have tested it with all of the systems listed in Table IV and find little overall improvement. Values of cl/k calculated from eq 18, assuming the nonelectrolyte diameters given in Table 11, give somewhat better agreement for the
The Journal of Physical Chemistry
W. L. MASTERTON AND TEI PEI LEE ethane systems, but poorer results for sulfur hexafluoride. It may well be that the discrepancies found for benzene are of a more fundamental origin related to the approximations which are made in scaled particle theory. In conclusion, we can say that: (1) For nonpolar solutes of relatively small size, scaled particle theory leads to values of k, which are in good agreement with experiment. On the whole, it would appear to be an improvement over the McDevit-Long theory in these cases. (2) With large solute molecules, agreement with experiment is relatively poor, as is the case with all present theories of the salt effect. It is possible, however, to select solute parameters which give good agreement with experiment for those salts for which salting coefficients are available. These parameters can then be used to predict quite accurately the salting coefficients for other electrolytes. (3) Values of k, calculated from scaled particle theory are quite sensitive to ionic size (u3, u4). This effect is particularly serious with large ions. In order to get good agreement with experiment for systems involving Rb+, Csf, or tetraalkylammonium ions, it may be necessary to use a semiempirical method similar to that employed by Shoor and Gubbins" to obtain u3. With smaller ions, crystal radii can be used to obtain satisfactory values of u3 and u4.
Acknowledgment. This work was supported by the National Science Foundation under Grant GP-6163 and by funds provided by the United States Department of the Interior as authorized under the Water Resources Research Act of 1964, Public Law 88-379. Computer time was donated by the University of Connecticut Computer Center under funds provided by the Kational Science Foundation.
