6566
J. Phys. Chem. B 1997, 101, 6566-6574
Solubility Parameters and the Distribution of Ions to Nonaqueous Solvents Charles F. Baes, Jr.,*,† and Bruce A. Moyer Chemical and Analytical Sciences DiVision, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831-6119 ReceiVed: March 25, 1997; In Final Form: June 4, 1997X
Provided a singly charged, spherically symmetrical ion has a radius greater than ca. 0.2 nm, its distribution to a nonaqueous solvent from water can be accounted for by assigning to it a Hildebrand solubility parameter estimated for an uncharged atom or molecule, regardless of the model assumed for the Gibbs energy of charging the ionsthat of Born, Abraham, and Liszi (J. Chem. Soc., Faraday Trans. 1 1978, 74, 1604, 2858) or Abe (J. Phys. Chem. 1986, 90, 713)sin the nonaqueous phase. For smaller ions, however, no choice of solubility parameter or ion size can alone account for their extractability. The extensive data available for the Gibbs energy of transfer of monovalent ions from water to a variety of solvents can be accounted for by assigning to the ion, in addition to a plausible solubility parameter, an effective radius rs ) (ri3 + ∆rs3)1/3, wherein ri is the radius of the ion and ∆rs is an adjustable distance characteristic of the solvent. Separate sets of ∆rs values are needed for cations and anions. While the alternative models for charging the ion all give acceptable fits to the data, the shell model of Abraham and Liszi was selected as the most appropriate. The distance ∆rs can be predicted for cations from the molar volume of the solvent and its hydrogen-bond acceptance index β. For anions, ∆rs can be predicted from the molar volume and the hydrogen-bond donation index R. On the basis of these correlations, expressions are derived for predicting the extractability of a singly charged, approximately spherical ion and its activity coefficient in the nonaqueous phase.
Introduction The distribution of ions from an aqueous phase to a nonaqueous solvent has long been under investigation and has been reviewed extensively (Persson,1 Gritzner,2 Marcus et al.,3 Markin and Volkov,4 Cox,5 Marcus,6 Moyer and Bonneson,7 Moyer and Sun8). In particular, Marcus et al.3 and, more recently, Marcus6 have collected values for the Gibbs energy of transfer ∆G°[wfs] of some 45 ions from water to 47 different solvents. The increasing availability of such data has stimulated considerable interest in the solvation effects involved in ion transfer. Although electrostatic interactions usually account for the largest share of ion solvation energy, it has been pointed out that these largely cancel in their contribution to the energetics of ion transfer, allowing “neutral” interactions such as solvent cohesion and hydrogen bonding to become decisive.3 To explore what interactions may be statistically significant, Marcus et al.3 employed a relationship of the form
∆G°[i,wfs] )
∑j ∑k AjkP(k)i[P(j)s - P(j)w]
(1)
in which ∆G°[i,wfs] is the Gibbs energy of transfer of ion i from water to solvent s; P(j) is a property of the water and the solvent such as the polarity-polarizability index9 π, the hydrogen-bond donation index9 R, the hydrogen-bond acceptance index9 β, the Hildebrand solubility parameter δ, and the molar volume V; P(k)i is a property of the ion, including z2/r or z/r (where z is the charge and r is the ion radius), and one or two other properties such as the ion volume, the softness parameter10,11 σ, or the molar refractivity RD; and Ajk are adjustable coefficients that were found to be nonzero for a few combinations of P(j) and P(k). Separate sets of Ajk values were * To whom correspondence should be addressed: 102 Berwick Dr., Oak Ridge, TN 37830; FAX, 423-482-2344; e-mail,
[email protected]. † Retired. X Abstract published in AdVance ACS Abstracts, July 15, 1997.
S1089-5647(97)01054-7 CCC: $14.00
assigned for combinations that depended on the nature of the ion. Thus, the extractability of an ion was found to depend primarily on its size and charge, and on the nonaqueous solvent properties π, R, and β for small cations, π and δ for large hydrophobic ions, and π, R, and V for anions. A total of 17 Ajk coefficients were found sufficient to account for ca. 430 ∆G° values, usually within their uncertainty of ca. 6 kJ/mol. An analogous expression was used to produce a similar fit to data on the enthalpy of transfer. The success of this statistical representation indicates the importance of neutral effects in the transfer of large, hydrophobic ions3 and supports the inclusion of such effects in models of ion transfer. Previously, Abraham and Liszi12 offered a model that included a neutral as well as an electrostatic term, a model that successfully predicted observed values of ∆G°[wfs] for selected cases. The neutral term was derived from Henry’s law constants for inert gases and hydrocarbons in the solvents of interest. Although this term is almost negligible for small ions, it becomes large and even dominates with increasing ion size.8 To generalize Abraham and Liszi’s treatment of the contribution of neutral effects to the Gibbs energy of ion transfer, one might employ solubility parameters, as hinted by Abraham and Liszi.13 We adopted this approach in modeling the extraction of alkali metal salts into organic solvents having sufficiently high relative permittivity to dissociate ion pairs. In our program SXLSQI, written to facilitate such modeling,8,14,15 the activity coefficients of neutral nonaqueous solutes are given by a solubility-parameter treatment.16 To calculate the activity coefficients of ionic species in the organic phase, the electrostatic contribution to the nonideality is represented by a Debye-Hu¨ckel term. However, since the nonaqueous ions in typical solvent extraction systems are often bulky, especially when lipophilic extractants are involved (e.g., crown ethers), a neutral contribution to the nonideality was introduced. The program SXLSQI employs for this purpose solubility-parameter assignments for all nonaqueous solute species, both neutral and charged. Although solubility © 1997 American Chemical Society
Distribution of Ions to Nonaqueous Solvents
J. Phys. Chem. B, Vol. 101, No. 33, 1997 6567
parameters have previously been applied to ionic systems in isolated cases (Barton,17 Chapter 12), previous workers apparently have not described the activity coefficients of ions in this manner. Accordingly, we were led to explore further how one may associate solubility parameters with ions. Our first objective was to provide a basis for assignment of values to such parameters. The second objective, which developed naturally from the first, was to estimate the extractability of these ions from an aqueous phase. If filled, the array of ∆G°[wfs] values from Marcus6 could contain over 2000 values, but it is only partially filled. In the present treatment, 19 singly charged ions with roughly spherical symmetry were selected. For these there were 415 values of ∆G°[wfs], involving some 40 solvents. The approach was first to develop expressions for the activity coefficient of the nonaqueous ion and its partition, expressions that included, in addition to solubility parameters, alternative functions for the electrostatic term. These expressions were then fitted to the ∆G°[wfs] values in order to select the most appropriate such function and to fix adjustable parameters. In accord with previous attempts to compare theory with experiment,7,8 we found it beneficial to include hydrogen-bonding effects in our treatment. These were again introduced in the form of the Kamlet-Taft R and β indices. Solubility Parameters and Activity Coefficients Hildebrand solubility parameters (δ), hereinafter referred to simply as solubility parameters, are a measure of the energy of interaction of an uncharged molecule with its like neighbors in the pure liquid state. When different such liquids are mixed, the activity coefficient of component i in the mixture is often given quite well by the expression18
ln γi )
Vi (δ - δ h )2 + ln(Vi /Vm) + 1 - Vi /Vm RT i
(2)
∑φiδi
(2a)
and Vm is the volume containing a total of 1 mol of components,
Vm )
∑xiVi
(2b)
The activity coefficient γi thus defined is on the mole fraction scale. At infinite dilution of component i in a pure solvent s, it is given by
ln γi,0 )
Vi (δ - δs)2 + ln(Vi /Vs) + 1 - Vi /Vs RT i
(3)
Activity Coefficients for Ions In order to arrive at an analogous definition for the activity coefficient if the solute is an ion of charge zi instead of a neutral atom or molecule, imagine a spherical ion of radius ri relieved of its charge and introduced into the solvent s at infinite dilution. Its activity coefficient will be given by eq 3, the molar volume Vi and the solubility parameter δi being those of a hypothetical pure liquid of neutral atoms or molecules. Then return the charge zi to the ion. The energy associated with this may be represented (from Slater and Frank19) by
2
∞ 4πr dr ∫ r 4 32π e r D(r) 2
(4)
i
0
Here, e is the elementary charge, 0 is the permittivity of a vacuum, and D(r) is the relative permittivity or dielectric “constant” at distance r from the center of the ion. Thus, for a mole of ions of radius ri,
1 dr ∫r∞r2D(r)
∆Gc ) Bzi2
(5)
i
wherein
B ) e2Nav/8π0 ) 69.4677 kJ nm mol-1
(5a)
(Nav is Avogodro’s number). If the dielectric constant is that of the solvent Ds, independent of r, and the radius of the ion is replaced by an effective radius rs that can be treated as an adjustable parameter, then
zi2B ∆Gc ) Dsrs
(6)
Except for the replacement of ri by rs, this expression is implicit in the Born equation for the Gibbs energy of transfer of an ion from a vacuum to a medium of dielectric constant D.
zi2B 1 1∆G ) ri D
(
)
(7)
An alternative expression for the energy of the charging process, suggested by Abraham and Liszi,12 is obtained by assuming that the ion is surrounded by a shell of solvent of thickness rs - ri in which the dielectric constant (D0) is lower than that (Ds) of the bulk solvent. The resulting expression for the Gibbs energy of charging is
in which Vi is the molar volume (cm3/mol) of component i and δh (J1/2/cm3/2) is the average of the solubility parameters of all the components, weighted by their volume fractions (φi),
δh )
zi2e2
∆G )
∆Gc ) zi2B
[
∫rr r21D s
dr +
i
∫r∞r21D
dr
s
0
s
]
(8)
giving
[( )
]
1 1 1 1 + ri rs D0 Dsrs
∆Gc ) zi2B
(8a)
An additional representation of the charging process, suggested by Abe,20 is obtained by assuming that the ion resides in a cavity of radius rs within which the dielectric constant is that of a vacuum (unity), and that beyond this spherical region, the dielectric constant is given by the function
D(r) ) Ds(1-rs/r)
(9)
It assigns a value of unity to D(r) at r ) rs and a value of Ds at r ) ∞. The expression for the Gibbs energy of charging produced by this model is
[
∆Gc ) zi2B
1 dr ∫rr r12 dr + ∫r∞r2D(r) s
i
s
]
(10)
and when eq 9 is introduced, one obtains
[ (
∆Gc ) zi2B
)]
Ds - 1 1 1 - 1ri rs Ds ln Ds
(10a)
6568 J. Phys. Chem. B, Vol. 101, No. 33, 1997
Baes and Moyer
Adding the charging term to eq 3, we obtain for the activity coefficient of ion i at infinite dilution
ln γi,0 )
Vi ∆Gc,i,s (δi - δs)2 + ln(Vi/Vs) + 1 - Vi/Vs + RT RT (11)
in which ∆Gc,i,s is the Gibbs energy of charging of the ion in solvent s. In this expression, δi and Vi, as already noted, are properties of a (hypothetical) pure liquid of atoms or molecules without charge that are otherwise identical to the ion. The activity coefficient refers to this hypothetical liquid. The last term will contain alternative forms of ∆Gc (from eqs 6, 8a, or 10a), but all involve as variables only the dielectric constant of the solvent Ds, the ionic radius ri, and/or an effective radius rs. The Partition of an Ion between Two Solvents Now consider an ion Iz distributed between two solvents, s1 and s2,
Iz(s1) a Iz(s2)
(12)
If the activity of the ion in each phase is referred to the same standard state, the hypothetical pure liquid just considered, then at equilibrium these activities will be the same. Hence
xi,1γi,1 ) xi,2γi,2
(13)
and the ratio Rx of mole fractions at equilibrium is
Rx )
xi,2 γi,1 ) xi,1 γi,2
(14)
The corresponding ratio RM of molar concentrations for ions at infinite dilution is given by
V1 RM,0 ) Rx,0 V2
(15)
Combining eqs 11, 14, and 15, we obtain for the equilibrium ratio of molar concentrations at infinite dilution
ln RM,0 )
(
)
Vi 1 1 [2δ (δ - δi) - δ22 + δ12] + Vi + RT i 2 V2 V1 (∆Gc,i,1 - ∆Gc,i,2) (16) RT
The last term takes various forms, depending on the choice of the expression for ∆Gc, from eq 6, 8a, or 10a. In these equations, the effective radius rs will be evaluated in various ways from the crystal radius and the approximate radius of a solvent molecule. In eq 8a, the dielectric constant of the solvation shell (D0) will be fixed at 2.12 Thus, one might expect the partition of a spherical, singly charged ion between two mutually insoluble solvents to be given by properties that may be assigned to the ion (δi, Vi, and ri), by properties that may be assigned to the solvent (δs, Vs, and Ds), and by an effective distance (rs). Of these quantities, there is little difficulty in assigning values to ri, Vi, δs, Vs, and Ds (Tables 1 and 2). It remains necessary to assign values to δi and rs, the latter a quantity that may depend on both the solvent and the ion and, of course, on the expression used for the Gibbs energy
of charging. These matters will be explored by fitting data on the Gibbs energy of transfer of ions from water to a variety of solvents. Fitting Data on the Gibbs Energy of Transfer of Ions Marcus et al.3 and Marcus6 list ∆G°[wfs] for the transfer of ions from water to various solvents (Table 1) at infinite dilution on the molarity scale. They adopted the convention that ∆G°[wfs] is the same for the tetraphenylarsonium cation and the tetraphenylborate anion (Table 2). In fitting eq 16 to these data, we employed the following relationship:
∆G°[wfs] ) ∆G°[wfr] - RT ln RM,0
(17)
By choosing one solvent as the reference solvent (r) and the corresponding Gibbs energy of transfer ∆G°[wfr] as an adjustable quantity (Table 2), it was possible to restrict the calculated values of RM,0 to the transfer of ions between this and another nonaqueous solvent (s). In this way, we avoided applying this model to the aqueous phase, where an alternative (e.g., Pitzer21) treatment of ionic activity coefficients is usually employed. In all tests of eqs 16 and 17, the molar volume Vi of the ion was calculated from the crystal radius ri (Table 2),
Vi /cm3 mol-1 ) 3406.6(ri /nm)3
(18)
assuming a void fraction of 0.26 for the cubic closest packing of spheres in the hypothetical pure liquid. In the first attempts to fit eqs 16 and 17 to the ∆G°[wfs] data, the solubility parameter was adjusted, along with ∆G°[wfr], for each ion, after the distance rs was fixed in the alternative expressions for ∆Gc (eqs 6, 8a, and 10a). In eqs 6 and 10a, rs was placed equal to ri; in eq 8a, rs was placed equal to the sum of ri and an effective solvent thickness rsv calculated from its molar volume by the inverse of the relationship in eq 18,
rsv /nm )
(Vs /cm3 mol-1)1/3 15.0467
(19)
These assumptions produce fairly good fits to the data for ions with crystal radii larger than about 0.2 nm (Table 3) and solubility parameters that are plausible in magnitude, with uncertainties generally less than 2 J1/2cm-3/2. As the ionic radius becomes smaller than 0.2 nm, however, the solubility parameter becomes increasingly uncertain, partly because ∆Gc becomes large relative to the solubility parameter term in eq 16 and partly because the fit to the data becomes poorer. The result was about the same regardless of the choice of expression for ∆Gc. The recommended values of δi listed for ions in Table 2 were obtained by comparison with δ values that are known or can be estimated for neutral atoms and molecules of similar size and structure. One or more choices of such reference atoms or molecules are listed for each ion in Table 4, and the estimated values of δi for the ions were obtained by averaging selected reference values. It may be seen that the fitted values of δi for the larger ions in Table 3 have a satisfactory similarity to the estimated values in Table 2. Since this similarity has little dependence on the expression used for ∆Gc, it appears that solubility parameters should be useful in developing a model for the extractability and nonaqueous activity coefficients of ions. We therefore adopted the values of δi estimated in Table 4 (and listed in Table 2) and held these fixed in subsequent refinements of the model. This seemed especially desirable both because the contribution of the solubility parameter term to ∆G°-
Distribution of Ions to Nonaqueous Solvents
J. Phys. Chem. B, Vol. 101, No. 33, 1997 6569
TABLE 1: Solvent Parameters solventa
δsb (J1/2/cm3/2)
Vsb (cm3/mol)
Dsb
βd
rsve (nm)
∆rs(cations)f (nm)
∆rs(anions)f (nm)
MeOH EtOH PrOH BuOH HxOH TFE HOENSH EG
29.66 26.08 24.49 23.45 21.90 23.87 27.77 32.40
41.00 59.00 75.00 92.00 125.32 72.00 70.50 56.00
32.26 24.39 20.41 17.54 13.30 26.32 31.69 37.74
Alcohols 0.98 0.86 0.84 0.84 0.80 1.51
0.66 0.75 0.90 0.84 0.84 0.00 0.52
0.228 (0.009) 0.242 (0.011) 0.236 (0.012) 0.238 (0.012) 0.210 (0.012) 0.281 (0.021) 0.246 (0.025) 0.212 (0.010)
0.110 (0.020) 0.120 (0.019) 0.127 (0.020) 0.140 (0.019)
0.90
0.229 0.259 0.280 0.300 0.333 0.276 0.304 0.254
Amides and Amines 111.11 0.71 0.48 181.82 0.62 0.80 37.04 0.00 0.69 28.40 0.00 0.79 37.74 0.00 0.76 32.15 0.00 0.78 29.41 0.00 1.05 23.60 0.00 0.80 6.80 0.26 0.50
0.227 0.259 0.283 0.320 0.301 0.334 0.372 0.328 0.300
0.207 (0.008) 0.207 (0.009) 0.199 (0.008) 0.199 (0.009) 0.197 (0.007) 0.197 (0.010) 0.192 (0.010) 0.185 (0.008) 0.225 (0.016)
0.113 (0.018)
0.250 0.275 0.295 0.298 0.312 0.324
0.241 (0.010) 0.246 (0.013) 0.239 (0.019) 0.242 (0.012) 0.260 (0.013) 0.257 (0.016)
0.180 (0.016)
Rc
0.099 (0.022)
FA NMF DMF DEF DMA DEA HMPT TMU PhNH2
39.37 32.86 24.90 21.70 22.14 20.25 18.44 21.70 22.90
40.00 59.00 77.00 111.40 93.00 127.00 176.00 120.76 91.53
MeCN EtCN PrCN iPrCN PhCN BzCN
24.49 22.10 21.50 20.10 17.20 23.40
53.00 70.90 87.87 90.26 103.06 115.70
Py 2CNPy PYRR NMPy
21.68 27.20 13.00 23.24
81.00 97.00 69.48 96.00
Pyridines and Pyrroles 12.20 0.00 0.64 95.24 0.00 0.29 8.13 32.26 0.00 0.77
0.288 0.305 0.273 0.304
0.216 (0.011) 0.260 (0.016) 0.305 (0.021) 0.185 (0.008)
Nitro Compouds 37.04 0.22 0.06 35.09 0.00 0.30
0.251 0.312
0.255 (0.011) 0.279 (0.014)
0.180 (0.014) 0.171 (0.018)
0.220 (0.011) 0.228 (0.011) 0.164 (0.009)
0.184 (0.018)
36.36 28.60 24.83 20.40 25.20 18.70
Nitriles 0.19 0.40 0.00 0.39 0.00 0.40 0.00 0.00
0.37 0.41
0.197 (0.015) 0.217 (0.015) 0.205 (0.020)
0.189 (0.020)
0.173 (0.021) 0.186 (0.020)
MeNO2 PhNO2
26.08 22.14
54.00 103.00
Me2CO γBulac THF
20.25 25.80 20.30
74.00 76.49 81.84
20.83 39.00 7.58
0.55
0.279 0.282 0.289 0.267 0.292 0.285
0.253 (0.017) 0.245 (0.016) 0.301 (0.014)
0.125 (0.034) 0.160 (0.024) 0.127 (0.018)
Sulfur Compouds 0.00 0.76 0.00 0.81
0.275 0.303
0.199 (0.007) 0.209 (0.010)
0.179 (0.015) 0.198 (0.020)
0.324 0.292
0.178 (0.008) 0.237 (0.009)
0.179 (0.015)
Ketones and Ethers 0.08 0.43 0.00
CH2Cl2 11DClE 12DClE
19.00 20.00 20.20
64.50 85.00 79.00
Chloro Compouds 8.93 0.13 0.10 10.00 0.10 0.10 10.31 0.00 0.10
DMSO TMS
24.70 27.39
71.00 95.00
46.51 43.48
TMP PC
25.30 27.20
116.23 85.00
Phosphates and Carbonates 16.39 0.00 0.77 64.52 0.00 0.40
a
MeOH ) methanol, EtOH ) ethanol, PrOH ) 1-propanol, BuOH ) 1-butanol, HxOH ) 1-hexanol, EG ) 1,2-ethanediol, HOENSH ) 2-mercaptoethanol, TFE ) 2,2,2-trifluoroethanol, FA ) formamide, NMF ) N-methylformamide, DMF ) N,N-dimethylformamide, DEF ) N,Ndiethylformamide, DMA ) N,N-dimethylacetamide, DEA ) N,N-diethylacetamide, HMPT ) hexamethyl phosphoric triamide, TMU ) N,N,N′,N′tetramethylurea, PhNH2 ) aniline, MeCN ) acetonitrile, EtCN ) propionitrile, PrCN ) butyronitrile, iPrCN ) isobutyronitrile, PhCN ) benzonitrile, BzCN ) benzyl cyanide, Py ) pyridine, 2CNPy ) 2-cyanopyridine, PYRR ) pyrrole, NMPy ) 1-methylpyrrolidin-2-one, MeNO2 ) nitromethane, PhNO2 ) nitrobenzene, Me2CO ) acetone, γBulac ) γ-butyrolactone, THF ) tetrahydrofuran, CH2Cl2 ) dichloromethane, 11DClE ) 1,1dichloroethane, 12DClE ) 1,2-dichloroethane, DMSO ) dimethylsulfoxide, TMS ) tetramethylene sulfone (sulfolane), TMP ) trimethyl phosphate, PC ) 1,2-propylene carbonate. b From Marcus et al.,3 Riddick et al.,25 and Barton.17 c H-bond donation index, from Marcus.26 d H-bond acceptance index, from Marcus.26 e Calculated from the molar volume by means of eq 19. f Solvent distances in eq 20; they are adjusted values from case 2, Table 6 (with the standard deviation in parentheses).
[wfr] is small for small ions (and therefore could not be established from the data) and because a successful model should employ parameters that can be established independently. A second series of tests, with fixed solubility parameters, was carried out in which the distances in eqs 6, 8a, and 10a were adjusted to obtain the best fit to the data. This seems an appropriate procedure since ionic radii are often adjusted in application of the Born equation22 and the shell model of Abraham and Liszi, which requires some basis for choosing the shell thickness.12,23
As in the previous tests (Table 3), good fits were usually obtained for the larger ions (ri > 0.2 nm), regardless of the form chosen for ∆Gc (Table 5). The rs values given by eq 6sto be viewed as the effective ionic radiusswere reasonably well determined for all the cations, but they were too small for several of the larger ones and too large for the smallest ones. For the anions, the ion sizes thus obtained were more realistic for the larger ones and generally more uncertain for all, especially for the smaller ones. The ri values given by eq 8asto be viewed as the radius of the ion within the shell of solventswere too
6570 J. Phys. Chem. B, Vol. 101, No. 33, 1997
Baes and Moyer
TABLE 2: Ion Parameters
TABLE 4: Estimated Solubility Parameters of Ions (J1/2/cm3/2)a
∆G°[wfr]d b
c
calcf (kJ/mol)
ion
solventsa
Li+ Na+ K+ Rb+ Cs+ NH4+ Me4N+ Et4N+ Pr4N+ Bu4N+ Ph4As+
30 37 34 30 33 7 19 16 15 12 27
(10.0) (10.0) 10.9 13.1 14.0 9.6 13.3 15.0 15.7 16.1 19.8
0.069 0.102 0.139 0.149 0.169 0.147 0.280 0.337 0.379 0.413 0.426
16.0 5.0 -9.0 -22.0 -33.0g
59.0 54.5 48.6 46.7 43.0 47.1 17.6 1.5 -11.9 -23.7 -28.1
FClBrINO3ClO4TcO4Ph4B-
13 23 24 24 8 22 14 27
17.2 16.9 20.8 26.5 25.8 25.8 25.8 19.8
0.133 0.182 0.196 0.222 0.189 0.238 0.252 0.420
65.0 52.0 38.0 25.0 32.0 17.0 15.0 -33.0g
66.4 42.9 36.6 25.5 39.8 19.1 13.8 -33.1
δi
(J1/2
obse (kJ/mol)
cm-3/2)
ri (nm)
56.0 52.0 46.0 37.0
a The number of solvents for which ∆G°[wfs] values are reported.6 From Table 4. c The value for TcO4- is from Moyer and Bonneson;7 the others are from Marcus et al.3 d The Gibbs energy of transfer of the ion from water to the reference solvent, 1,2-dichloroethane. e From Marcus.6 f Values calculated from eq 21 (set 1, Table 7). g These values are assumed to be equal.
b
TABLE 3: Test of Eqs 6, 8a, and 10a, with Adjustment of δia,b ric ion Li+ Na+ K+ NH4+ Rb+ Cs+ Me4N+ Et4N+ Pr4N+ Bu4N+ Ph4As+ FClNO3BrIClO4TcO4Ph4B-
(nm)
eq
6d
δi
0.069 -387 (410) 0.102 -147 (87) 0.139 -4 (23) 0.147 173 (134) 0.149 2 (19) 0.169 21 (9) 0.280 22 (3) 0.337 21 (2) 0.379 22 (1) 0.413 21 (1) 0.426 21 (1) 0.133 162 (90) 0.182 58 (16) 0.189 27 (18) 0.196 46 (10) 0.222 30 (4) 0.238 24 (5) 0.252 29 (2) 0.420 21 (1)
eq σg
8ae
δi
4.16 -431 (447) 3.21 -128 (20) 2.70 -15 (31) 1.90 -224 (223) 2.10 -8 (29) 1.98 10 (14) 1.13 24 (5) 1.11 21 (2) 1.04 27 (1) 1.08 21 (2) 1.69 21 (1) 4.35 180 (74) 3.79 55 (12) 1.65 22 (7) 2.92 41 (7) 1.73 29 (3) 1.88 25 (5) 1.24 28 (2) 1.66 21 (1)
eq σg
δi
10af
ion
est. δ
ref species
ref δ
sourcesb
Li Na+ K+ Rb+ Cs+ NH4+ Me4N+ Ete4N+ Pr4N+ Bu4N+ Ph4As+ F-
(10.0) (10.0) 10.9 13.1 14.0 9.6 13.3 15.0 15.7 16.1 19.8 17.2
He Ne Ar Kr Xe CH4 Me4C Et4C Pr4C Bu4C Ph4C MoF6 WF6 UF6 CCl4 SiCl4 GeCl4 SnCl4 Ar SiBr4 Br2 Kr SnI4 I2 Xe OsO4 OsO4 OsO4 -NO3 -SO3 Ph4C
10.9 13.1 (14.0) 9.6 13.3 15.0 15.7 16.1 19.8 17.0 16.4 18.2 17.6 15.5 16.6 17.8 10.9 18.0 23.5 13.1 23.9 28.8 (14.0) 25.8 25.8 25.8 25.0 26.1 19.8
(1) (1) (2) (3) (4) (4) (4) (4) (4) (5) (5) (5) (5) (5) (5) (5) (1) (5) (5) (1) (5) (5) (2) (5) (5) (5) (6) (6) (4)
+
Cl-
16.9
Br-
20.8
I-
26.5
ClO4TcO4NO3-
25.8 25.8 25.8
Ph4B-
19.8
a
σg
4.17 -1816 (573) 5.82 4.17 -495 (130) 4.78 3.52 -95 (27) 3.20 2.76 180 (204) 2.92 2.98 -76 (23) 2.54 2.86 -22 (10) 2.19 1.63 14 (3) 1.06 1.42 17 (1) 0.85 1.90 19 (1) 0.85 1.28 20 (1) 0.86 1.96 20 (1) 1.58 3.61 56 (116) 5.65 2.95 30 (21) 5.00 0.61 5 (35) 3.12 2.16 25 (14) 4.07 1.22 17 (6) 2.64 1.88 13 (6) 2.24 1.34 22 (3) 1.72 1.93 20 (1) 1.56
a In these tests, the Gibbs energy of transfer ∆G°[wfs] was calculated from eqs 16 and 17, with ∆Gc evaluated alternatively by eqs 6, 8a, and 10a. The quantities adjusted were the solubility parameter (with the standard deviation in parentheses) and ∆G°[wfr]. b For some ions one or more solvents were excluded from the fits listed because the deviations of calculated from observed values of ∆G°[wfs] seemed excessive. c From Table 2. d In tests of eq 6, the ion size rs was fixed at the crystal radius ri. e In tests of eq 8a, the ion radius ri was fixed at the crystal radius, and rs was taken to be the sum of ri and rsv from eq 19. f In tests of eq 10a, the cavity size rs was placed equal to the crystal radius ri. g Standard deviation of calculated values of ∆G°[wfs] from the values reported by Marcus,6 in multiples of the assigned uncertainty (3 or 6 kJ/mol).
small for the larger cations (except Ph4As+) and too uncertain for the smaller ones to permit a useful comparison. For the largest anions, the ion sizes were plausible within their uncertainties, while for the smaller ones (except NO3-) they were less uncertain, but some were too small. The rs values
An estimate of δ for each ion was made by choosing a neutral reference atom or molecule of approximately the same size and structure and assigning a δ value to it. If more than one reference species suggested itself, an average of selected values was taken. b Sources: (1) Barton,17 p 323; (2) ibid., p 38, gives 19 J1/2 mol-1/2 cm-3/2 at 166 K, which is corrected approximately to 298 K; (3) ibid., p 84; (4) ibid, p 160; (5) ibid., p 34; (6) ibid., p 161.
given by eq 10asto be viewed as the radius of the cavity in which the ion residesswere comparable to or greater than ionic radii ri for both cations and anions, but were very uncertain for the small anions. The tests in Tables 3 and 5 show that, while plausible solubility parameters can produce good fits to the data for the larger ions, the fits given for the smaller ones by adjusting either rs or the solubility parameter are too poor to permit a confident selection of either. Clearly, if plausible solubility parameters are to yield good fits for all these ions, a better representation of the effective distance rs in the expressions for ∆Gc is needed. The representation finally chosen is
rs ) (rin + ∆rsn)1/n
(20)
in which the distance ∆rs is treated as an adjustable parameter that can be assigned to each solvent rather than each ion. Thus, if the ionic radius ri is larger than this distance, then the effective radius rs approaches ri; if ri is smaller than ∆rs, then rs approaches ∆rs. In fitting this function to the data, it soon became clear that the value of ∆rs for a given solvent was not the same for cations and anions. Accordingly, the data for each were treated separately. While tests showed that fits to the data were not sensitive to the values chosen for the integer n in the range 1-4, odd values are preferable in case ∆rs should in some cases be negative. More plausible values of ∆rs were obtained for n ) 3 than for n ) 1. The former value was also preferred because it it produces values of rs that rapidly approach one of
Distribution of Ions to Nonaqueous Solvents
J. Phys. Chem. B, Vol. 101, No. 33, 1997 6571
TABLE 5: Test of Eqs 6, 8a, and 10a, with Adjustment of the Ion Sizea,b eq 8ad
eq 6
eq 10ae
ric (nm)
rs
σf
ri
σf
rs
σf
Na+ K+ NH4+ Rb+ Cs+ Me4N+ Et4N+ Pr4N+ Bu4N+ Ph4As+
0.069 0.102 0.139 0.147 0.149 0.169 0.280 0.337 0.379 0.413 0.426
0.13 (0.05) 0.16 (0.03) 0.15 (0.03) 0.15 (0.02) 0.16 (0.03) 0.15 (0.03) 0.37 (0.12) 0.19 (0.03) 0.18 (0.03) 0.17 (0.04) 0.33 (0.09)
3.74 2.97 2.71 1.97 2.76 2.11 2.27 1.09 1.33 1.42 1.77
0.44 (0.58) 0.56 (0.60) 0.53 (0.60) 0.35 (0.52) 0.58 (0.65) 0.39 (0.30) 0.18 (0.14) 0.13 (0.07) 0.09 (0.03) 0.05 (0.04) 0.59 (0.44)
4.12 3.78 3.62 1.49 3.30 3.07 2.28 1.65 0.89 1.00 2.00
0.45 (0.21) 0.39 (0.12) 0.34 (0.08) 0.22 (0.04) 0.34 (0.07) 0.46 (0.10) 0.31 (0.04) 0.31 (0.03) 0.30 (0.04) 0.64 (0.28) 0.50 (0.11)
4.02 3.54 2.90 2.36 2.34 2.52 1.04 0.91 1.10 1.91 1.57
FClNO3BrIClO4TcO4Ph4B-
0.133 0.182 0.189 0.196 0.222 0.238 0.252 0.420
0.44 (0.98) 0.33 (1.26) 0.48 (0.34) 0.46 (0.44) 0.43 (0.33) 0.42 (0.12) 0.30 (0.09) 0.34 (0.10)
4.61 4.27 1.19 3.18 1.46 1.59 1.34 1.76
0.04 (0.11) 0.03 (0.03) 0.71 (0.41) 0.02 (0.04) 0.37 (0.13) 0.08 (0.20) 0.31 (0.11) 0.62 (0.47)
4.26 3.15 1.09 2.72 1.44 1.56 1.36 1.95
0.72 (1.30) 0.56 (0.39) 2.14 (1.96) 1.27 (1.22) 1.19 (0.52) 0.56 (0.12) 0.50 (0.09) 0.51 (0.11)
4.59 4.23 1.21 3.11 1.56 1.42 1.01 1.53
ion Li+
a In these tests, the Gibbs energy of transfer ∆G°[wfs] was calculated from eqs 16 and 17, with ∆Gc calculated alternatively by eqs 6, 8a, or 10a. The quantities adjusted were the distances rs or ri (with the standard deviation in parentheses) and ∆G°[wfr]. The solubility parameters of the ions were fixed at the values estimated in Table 4. b For some ions one or more solvents were excluded from the fits listed because the deviations of calculated values from observed values of ∆G°[wfs] seemed excessive. c From Table 2. d In tests of eq 8a, the ion size ri was adjusted, and rs was taken to be the sum of ri and rsv from eq 19. e In tests of eq 10a, the radius of the cavity in which the ion resides was adjusted. f Standard deviation of calculated values of ∆G°[wfs] from the values reported by Marcus,6 in multiples of the assigned uncertainty (3 or 6 kJ/mol).
the limiting distances ri or ∆rs, whichever is larger, and because it implies an intuitively satisfying additivity of effective volumes. The choice of the function for ∆Gc also had little effect on the fit to the data. Use of eq 10a for ∆Gc is based on perhaps the most satisfying model, one involving a dielectric constant that increases continuously with distance from the cavity of radius rs containing the ion. This condition, however, is not consistent with the Debye-Hu¨ckel and Bjerrum24 models often used respectively to estimate the activity coefficients and the association of ions at finite concentrations in a nonaqueous solvent. There is a difficulty as well with the simple Born model (eq 6), in that the effective radius rs should be viewed here as the radius of the ion. The values of rs obtained from fits to the data, however, are in the range 0.13-0.44 nm, too large to be realistic radii for the smaller ions. In the remaining model (eq
Figure 1. Correlation of ∆G°[wfr] with the ion radius. The filled circles and solid curve indicate cations; the open circles and dashed curve indicate anions. The curves are calculated from eq 21, with parameters from set 1 in Table 7.
8a), rs is the radius of the ion plus the effective thickness of the solvation shell, wherein the dielectric constant (D0) is taken to be 2. We prefer this model for the representation of the data. The fits to the data obtained by use of eq 20 are summarized in Table 6. Case 1 results from adjusting ∆rs for each solvent while fixing the values of ∆G[wfr]sthe Gibbs energy of transfer of each ion from water to the reference solvent (1,2dichloroethane)sat the observed values (Table 2). Since some of the latter values were unavailable, a method was sought for predicting them. Predicting ∆G°[wfr] Values. This quantity correlates strongly with the crystal radii of the cations and the anions. The curves in Figure 1 are produced by
∆G°[wfr] ) C1(1 + C2ri)[1 + C3(1 + C2ri)]
(21)
with the C1, C2, and C3 values listed in set 1 of Table 7. (The awkward form of this expression has the virtue that it reduces the covariance of C1, C2, and C3 with one another.) Similar correlations are produced by the other solvents for which there is a comparable number of data points, but 1,2-dichloroethane appears to give the best one. The resulting smoothed and interpolated values of ∆G°[wfr] are listed in Table 2. Use of these values and eq 20 produced the fits listed as case 2 in Table 6 and the values of ∆rs listed in Table 1. The quality of the fits obtained in cases 1 and 2 (Table 6) is perhaps not surprising in view of the large number of adjustable parameters employed, the ∆rs values for each solvent. It would be far more satisfying if these effective distances could be related to independently known properties of the solvent.
TABLE 6: Fits Obtained by Use of Eq 20a cations
anions
case
source of ∆G°[wfr]b
source of ∆rsb
paramsc
pointsd
σe
paramsc
pointsd
σe
1 2 3 4
Table 2, obsf eq 21 [1] eq 21 [1] eq 21 (2)
adjustment adjustment eqs 22 (4) eqs 22 (5)
39 39 [3] 3 [3] 6
252 252 229 229
1.45 1.42 2.21 2.00
22 22 [3] 3 [3] 6
136 136 149 149
1.70 1.58 2.26 1.89
a
All fits were made assuming eq 8a for ∆Gc and n ) 3 in eq 20. b Numbers in brackets refer to parameter sets listed in Table 7 that were fixed before the data were fit; numbers in parentheses refer to parameter sets listed in Table 7 that were adjusted. c The number of parameters adjusted to obtain the indicated fit; the numbers in brackets refer to additional parameters in eq 21 fitted to the observed values of ∆G°[wfr]. d The number of ∆G°[wfs] values from Marcus6 fitted. In cases 1 and 2: for cations, eight points were excluded (Me4N+, Pr4N+, and Ph4B+ in FA; Cs+, Me4N+, Et4N+, Pr4N+, and Bu4N+ in CHCl3); for anions, seven solvents were excluded (totaling 18 points, 4 in TFE, 1 in NMF, 2 in DEA, 1 in iPrCN, 5 in Py, 1 in THF, and 4 in CHCl3), and one point for Ph4B- in FA. In cases 3 and 4: the solvents HOENSH, iPrCN, PYRR, and γBulac were excluded because R and β values were unavailable (17 cations points and 1 anion point); for cations, in addition to the 8 points noted above, the 6 points in THF were excluded; for anions, in addition to the 1 point in iPrCN, the 4 points in CHCl3 and the 1 in FA were again excluded. e Standard deviation of calculated values from observed values of ∆G°[wfs] in multiples of the assigned uncertainty (3 or 6 kJ/mol). f Since no observed values have been reported for Li+ and NH4+, these were calculated from eq 21, with the parameters of set 1, Table 7.
6572 J. Phys. Chem. B, Vol. 101, No. 33, 1997
Baes and Moyer
TABLE 7: Parameter Values in Eqs 21 and 22 set 1a 2c
3d 4f 5c
ion type
C1
cations anions cations anions
1.0487751E+02g 9.6134965E+01 8.0080896E+01 1.1339028E+02
cations anions cations anions cations anions
2.1103699E-01 2.4515807E-01 2.5947986E-01 2.5705753E-01 1.8661384E-01 2.1320844E-01
C2 Eq 21 -2.9280578E+00 -3.4350118E+00 -3.1245225E+00 -3.2377659E+00 Eqs 22a and 22b -9.9277831E-02 2.6806288E-01 -8.4765133E-02 2.6456219E-01 -8.8837223E-02 3.2871323E-02
C3 -3.7039987E-01 5.0475758E-01 -2.0897244E-01 7.2562293E-01 5.0692925E+00 2.6102835E-01 5.6751187E+00 2.4896952E-01 1.7958478E-01 2.3365640E-01
σ 0.68b 0.92b
1.763e 0.943e
a This parameter set was obtained by fitting eq 21 to the observed values of ∆G°[wfr] in Table 2. b These standard deviations are in multiples of the uncertainties in the observed values of ∆G°[wfr] (6 kJ/mol). c These parameter sets were obtained by fitting the data as indicated in case 4, Table 6. d This parameter set was obtained by fitting the ∆rs values listed in Table 1. e These standard deviations are in multiples of the uncertainties in ∆rs (Table 1). f This parameter set was obtained by fitting the data as indicated in case 3, Table 6. g Read as 1.048775 × 102.
Predicting ∆rs Values. That such a correlation of ∆rs with solvent properties might be possible can be judged by inspection of the ∆rs values listed for cations and anions in Table 1. It is obvious that they correlate poorly with the rsv radii derived from the molar volumes of the solvents. This is especially true for cations. Rather, these ∆rs values occur in relatively narrow ranges for the alcohols, the amides, and the nitriles. Moreover, there is a general decrease in ∆rs as the electron-pair donor ability of the solvent (and its H-bond acceptance index β) increases. For the anions, there is greater variability of ∆rs with solvent type, but strong hydrogen-bond donor ability (a high R value) yields the smallest values (alcohols and the one amide, FA, with no N-substituents). Indeed, there is a general decrease in ∆rs for anions in all solvents as R increases. While for cations and for anions, the absolute values of ∆rs were found to be less well determined than were their differences from solvent to solvent, these differences make some chemical sense. The correlations of ∆rs for cations with β and for anions with R were strong enough to permit the prediction of ∆rs values with useful accuracy. The functions chosen for this purpose were
∆rs ) C1rsvC2/(1 + C3β)2
for cations
(22a)
∆rs ) C1rsvC2/(1 + C3R)2
for anions
(22b)
With these expressions it was possible to fit the ∆rs values listed in Table 1 with standard deviations for anions comparable to their uncertainties and deviations for cations that were somewhat larger (Table 7, set 3). But use of Eqs 22 to generate ∆rs values, along with the calculated values of ∆G°[wfr] in Table 2, was less successful in fitting most of the data (Table 6, case 3), 378 points, excluding solvents with unknown values of R and β (Table 1). With adjustment of the parameters in eq 21 giving ∆G°[wfr], the fit to these data was improved (Table 6, case 4), but at the cost of poorer agreement of calculated with observed values of ∆G°[wfr]. Discussion and Conclusions Combinations of ∆rs and ∆G°[wfr] values derived in various ways were used to fit the ∆G°[wfs] data. The results (Table 6) show, not surprisingly, that the best fits (case 2) are given by using the largest number of adjusted parameters, i.e., values of ∆rs adjusted for each solvent (Table 1) and parameters generating values of ∆G°[wfr] from eq 21 (Table 2). The deviation plots in Figure 2 are obtained from this fit. Replacing
Figure 2. Deviation of calculated from observed values of ∆G°[wfs] given by case 2, Table 6. The solvents are ordered from left to right as they appear from top to bottom in Table 1.
the individual ∆rs values with ones generated by eqs 22 gives poorer fits (cases 3 and 4), but with far fewer adjustable parameters. The conclusion to be drawn from these tests is that if one wishes to predict ∆G°[wfs] for an ion not included in Table 2, one should estimate ∆G°[wfr] from eq 21, using the parameters of set 1 (Table 7). If the ∆rs value for the solvent is among those listed in Table 1, it should be used in eq 20. If none is listed, then eqs 22 may be used to estimate it, with parameters of set 3 or 4 (Table 7). In the former case, the accuracy of the estimate of ∆G°[wfs] should be about (9 kJ/ mol; in the latter case about (12 kJ/mol. The first objective of this study was to estimate the solubility parameters of ions. It was found (Table 3) that solubility
Distribution of Ions to Nonaqueous Solvents
J. Phys. Chem. B, Vol. 101, No. 33, 1997 6573
parameters obtained for larger ions by fitting available data for the Gibbs energy of transfer were comparable to those one might estimate for ions if they had no charge (Table 4). Moreover, similarly estimated solubility parameters for smaller ions were found to be consistent with the ∆G°[wfs] data if the effective distance rs is assigned by use of eq 20. These solubility parameters for ions may be used as follows in estimating their activity coefficients in a nonaqueous phase. At finite concentrations, the activity coefficient on the mole fraction scale should be given by
ln γi )
Vi (δ - δ h )2 + ln(Vi/Vm) + RT i 1 - Vi/Vm +
∆Gc,i,s + F(I) (23) RT
wherein the volume-fraction weighted average of the solubility parameters of all the components (δh , eq 2a) and the volume of one mole of solution (Vm, eq 2b) appear, as well as a function of the ionic strength F(I), to represent the effects of long-range Coulombic interactions. The expression for the activity coefficient on the molarity scale (gi) is obtained from
() ()
xi ci gi ) γi ci xi
xsf1
gi γi
( )( )
Vm 1000 1 ) γi 1000 Vs γi xsf1
(24) xsf1
wherein the ratios in parentheses are evaluated in the limit as the mole fraction of the solvent goes to unity. From eqs 23 and 24,
ln gi )
(
)
Vi 1 1 [(δ - δh )2 - (δi - δs)2] + Vi + F(I) RT i Vs Vm (25)
The function of the ionic strength remains to be evaluated in order to obtain the desired activity coefficient. At the low ionic strengths in the nonaqueous phases usually involved in the extraction of ions, this may be represented by the DebyeHu¨ckel function
zi2AxI F(I) ) 1 + BaxI
(26)
wherein
2πNave6/1000k3 x A)
(26a)
8πNave2/1000k x B)
(26b)
(DsT)3/2
DsT
where k is Boltzmann’s constant and a is the distance of closest approach. If it is further assumed that this distance is the Bjerrum critical distance,24
e2/2k aB ) |z+z-| DsT
(26c)
(equal to 28.01/Ds nm for singly charged ions at 25 °C), then
zj2AxI
F(I) ) 1 + |z+z-|AxI
(27)
Bjerrum assumed that at distances smaller than aB, ions would be associated because their electrostatic energy given by a constant Ds would be greater than twice the thermal energy. The electrostatic energy given at this distance by the shell model adopted in the present treatment is larger than twice the thermal energy because D is lower than that of the solvent over a portion of this distance; hence, this model suggests a stronger tendency of ions to associate. Still, the shell model can be considered roughly consistent with the Bjerrum model provided the effective distance rs remains smaller than aB. This will be true for singly charged ions as long as Ds is smaller than 28/rs. The second objective of this study, prediction of the distribution of an ion from water to an immiscible solvent, has led to the following relationship (from eqs 8a, 16, 17, and 20):
ln
( ) ( )]
ci,s,0 Vi 1 1 ) [2δi(δs - δr) - δs2 + δr2] + Vi + ci,w,0 RT Vs Vr
[x
zi2B RT
3
1 3
ri + ∆rs
(
3
)
1 1 D0 Ds
1
xri + ∆rr 3
3
3
1 1 D0 Dr
-
∆G°[wfr] (28) RT
To employ this equation (1) the solubility parameter of the ion (δi) can be estimated from a similar structure without charge; (2) the molar volume of the ion (Vi) is derived from the ion radius (ri) by eq 18, or ri may be derived from Vi by its inverse (eq 19); (3) the solvent distances ∆rr and ∆rs may be obtained from Table 1, or the latter may be estimated from eqs 22 and parameters listed in Table 7, if R and β values are available for the solvent; (4) D0 is assigned a value of 2; (5) the Gibbs energy of transfer of the ion from water to the reference solvent (∆G°[wfr]) is calculated from ri by eq 21, with the parameters listed in set 1 of Table 7. The subscript r, of course, refers to the reference solvent chosen (here 1,2-dichloroethane). This treatment should be useful, provided the ion is monovalent and reasonably spherical, as are those chosen here. The deviations of calculated from observed values of ∆G°[i,wfs]sor of the equivalent -RT ln(cs,0/cw,0)stend to be larger for solvents of low dielectric constant, although the correlation is by no means a strong one. Nonetheless, eq 28 probably should not be applied to solvents with D values less than 7. Similarly, it should not be applied to the smallest ion of all, H+. Acknowledgment. This research was funded in part by the Division of Chemical Sciences, Office of Basic Energy Sciences, U.S. Department of Energy, under Contract No. DE-AC05960R22424 with Oak Ridge National Laboratory, managed by Lockheed Martin Energy Research Corp. The authors are indebted to Dr. Y. Marcus for communicating the data upon which this work is based prior to its publication and for his helpful comments on the manuscript. Notation hydrogen-bond donation index hydrogen-bond acceptance index solubility parameter (J1/2 cm-3/2) volume-averaged solubility parameter (J1/2 cm-3/2, eq 2a) dielectric constant Gibbs energy of charging an ion at infinite dilution in a solvent (kJ/mol) ∆G°[wfs] Gibbs energy of transfer of an ion from water to the solvent of interest, hypothetical 1 M solutions (kJ/mol)
R β δ δ h D ∆Gc
6574 J. Phys. Chem. B, Vol. 101, No. 33, 1997 ∆G°[wfr] Gibbs energy of transfer of an ion from water to the reference solvent 1,2-dichloroethane, hypothetical 1 M solutions (kJ/mol) γ activity coefficient, mole fraction scale g activity coefficient, molarity scale I ionic strength (molarity) φ volume fraction ri radius of ion i (nm) a distance associated with the ion (nm) in the expression rs for the Gibbs energy of charging; in eq 6, it is the effective radius of the ion; in eq 8a, it is the distance from the center of the ion to the outer limit of the solvent shell; in eq 10a, it is the radius of the spherical cavity containing the ion ∆rs a distance associated with the solvent (nm) which, with ri, yields rs (eq 20) a solvent radius derived from its molar volume (eq 19) rsV equilibrium constant of the reaction in eq 12, molarity RM scale (eq 15) Rx equilibrium constant of the reaction in eq 12, mole fraction scale (eq 14) V molar volume (cm3/mol) the volume of a solution containing a total of 1 M of Vm components x mole fraction
References and Notes (1) (2) (3) 3613. (4)
Persson, I. Pure Appl. Chem. 1986, 58, 1153. Gritzner, G. Pure Appl. Chem. 1988, 60, 1743. Marcus, Y.; Kamlet, M. J.; Taft, R. W. J. Phys. Chem. 1988, 92, Markin, V. S.; Volkov, A. G. Electrochim. Acta 1989, 34, 93.
Baes and Moyer (5) Cox, B. G.; Schneider, H. Coordination and Transport Properties of Macrocyclic Compounds in Solution; Elsevier: New York, 1992; pp 2526. (6) Marcus, Y. Ion Properties; Marcel Dekker: New York, 1997. (7) Moyer, B. A.; Bonneson, P. V. In Supramolecular Chemistry of Anions; Bianchi, A., Bowman-James, K., Espana, E., Eds.; VCH: New York, 1997. (8) Moyer, B. A.; Sun, Y. In Ion Exchange and SolVent Extraction; Marinsky, J. A., Marcus, Y., Eds.; Marcel Dekker: New York, 1997. (9) Kamlet, M. J.; Abboud, J.-L. M.; Abraham, M. H.; Taft, R. W. J. Org. Chem. 1983, 48, 2877. (10) Marcus, Y. Isr. J. Chem. 1972, 10, 659. (11) Marcus, Y. Thermochim. Acta 1986, 104, 389 (12) Abraham, M. H.; Liszi, J. J. Inorg. Nucl. Chem. 1981, 43, 143. (13) Abraham, M. H.; Liszi, J. J. Chem. Soc., Faraday Trans. 1 1978, 74, 1604, 2858. (14) Deng, Y.; Sachleben, R. A.; Moyer, B. A. J. Chem. Soc., Faraday Trans. 1995, 91, 4215. (15) Sun, Y.; Chen, Z.; Cavenaugh, K. L.; Sachleben, R. A.; Moyer, B. A. J. Phys. Chem. 1996, 100, 9500. (16) Baes, C. F., Jr.; Moyer, B. A.; Case, G. N.; Case, Faith I. Sep. Sci. Technol. 1990, 25, 1675. (17) Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters, 2nd ed.; CRC Press: Boca Raton, FL, 1991. (18) Henley, E. J.; Seader, J. D. Equilibrium-Stage Separation Operations in Chemical Engineering; John Wiley & Sons: New York, 1981; pp 183-192. (19) Slater, J. C.; Frank, N. H. Electromagnetism; McGraw-Hill Book Co.: New York, 1947; pp 90-104. (20) Abe, T. J. Phys. Chem. 1986, 90, 713. (21) Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill, Inc.: New York, 1995; pp 292-309. (22) Johnson, D. A. Some Thermodynamic Aspects of Inorganic Chemistry, 2nd ed.; Cambridge University Press: Cambridge, 1982; p 124. (23) Marcus, Y. J. Chem. Soc., Faraday Trans. 1991, 87, 2995. (24) Bjerrum, N. Kgl. Danske Vidensk, Selskab. 1926, 7. (25) Riddick, J. A.; Bunger, W. B.; Sakano, T. Techniques of Organic Chemistry: Vol. II; Organic SolVents, 4th ed.; John Wiley & Sons, Inc.: New York, 1986. (26) Marcus, Y. Chem. Soc. ReV. 1993, 22, 408.
