5588
J . Phys. Chem. 1989, 93, 5588-5595
Thermodynamics of Ionic Hydration: Application of Scaled Particle Theory to Partial Molar Volumes and Entropies of Aqueous Cations at 25 OC Leslie Barta* and Loren G . Hepler Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6C 2G2 (Received: December 8, 1988)
Infinite dilution absolute partial molar volumes and entropies of aqueous cations at 25 “C and 1 atm have been analyzed within the context of scaled particle theory to yield the volume and entropy changes associated with ion-water interactions. The resulting values have been compared with values calculated from an electrostatic theory of solvent electrostriction that includes the effects of short-rangestructural changes in the solvent. Results of calculations for the aqueous tetraalkylammonium ions are presented and discussed in terms of structural effects of the ions on pure liquid water.
1. Introduction
Much interest has focused on accurate measurements from which the thermodynamic properties of aqueous ions at infinite dilution can be obtained. Much attention has also been given to the interpretation of these properties in terms of various theories and molecular models. Perhaps the most pervasive difficulty encountered in the interpretation of these properties, particularly the partial molar volumes, is the interplay of geometric and energetic factors. Thus, the extent of solvent ”structuraln or volumetric collapse in the presence of an electrostatic field is reflected in both the packing geometry of solvent molecules around the ion and the energy of ion-solvent interaction relative to the intermolecular potential energy of molecules of solvent. The intermolecular potential energies in turn depend on the distances between molecules and orientation with respect to field, Le., on the microscopic structure of the solution. The scaled particle theory of liquids’-3 explicitly treats the problem of the packing of solvent and solute molecules in the specific case where the solvent is a “normal” liquid (Le., lacks strong directional interactions) and where the solventsolute interaction is similarly normal. In combination with an appropriate theory of the solute-solvent interaction energy, scaled particle theory accurately accounts for the properties of pure liquids (e.g., surface tensions, compressibilities,heats of vaporization), properties of solutes (solubilities, heats of solution, entropies of solution, heat capacities of solution, partial molar volumes), and properties of solute transfer from gas phase to liquid or from one liquid to another. 1,3-5 The limitations and accuracy of scaled particle theory in application to the thermodynamics of cavity formation in water have been discussed by a few authors.>* It has been shown recently’ that scaled particle theory accurately represents the properties of transfer of nonpolar hydrophobic solutes to water, provided the solute molecules are sufficiently small (diameter
'
-500
-400
0
-300 -200
-100
0
7
100
S(int) predicted Figure I. Comparison of theoretical and experimental entropies of ionwater interaction. Units are J K-' mol-'. Perfect agreement would be indicated by the solid line.
which XM 1, U, = U M = 12.7 cm3 mol-' and 6, (eq 15) = 3.5 X 10-6 bar-'. From eq 29, pM,1,35kh = 2.6 X 10-6 bar-'. Combining this value of OM with the value at 1 atm (Table 11), we obtain DM = 0.1 and BM = 1.5 X lo9 dyn cm-*. Representative results of the calculations outlined here are presented in Table 111. Also listed in this table are values of the ratio of the mole fraction of (the less dense) component L at pressure p to the mole fraction of this component at 1 atm. As stated earlier in this section, we take this ratio to indicate the extent of structural disruption relative to the pure solvent at 1 atm. cf = 1 corresponds to no effect; f = 0 corresponds to complete disruption of the original short-range order in the solvent.) For each ion the fraction,f, to be used in eq 21 can be calculated by way of eq 26 where Ap is the effective pressure due to the electrostatic field of the ion (eq 16) minus 1 bar.
-80 -80
-70
-60
-50
-40
-30
-20
-10
V(int) predicted Figure 2. Comparison of theoretical and experimental volumes of ion-
water interaction. Units are cm3 mol-'. Perfect agreement would be indicated by the solid line. 4. Results and Discussion
For each ion the fraction of residual structure, the maximum potential energy of ion-water interaction, and the corresponding molar entropy of water in the first layer of water molecules are listed in columns 2, 3, and 4 of Table IV. Column 5 lists the total entropy change of water per mole of ion. On the basis of calculations of the electrostatic field in the second layer of water molecules, for monovalent and divalent ions we have found that explicit consideration of only one hydration sphere is needed in addition to the Born continuum model. For 3+ ions, we have estimated an additional contribution of --70 J K-'mol-' due to water in the second layer (f 0.25, n2 = 8, AS, -35.8 J K-' mol-'). The long-range polarization contribution, SBorn, is
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5594 The Journal of Physical Chemistry, Vol. 93, No. 14, 1989
given in the sixth column. The total entropy of interaction is listed in the last column and compared in Figure 1 with the values calculated from the experimental data as described in section 2. There is good agreement between the values of Sin,derived from theory and from experimental data. Table V and Figure 2 summarize the results of our calculation of the volumes of ion-water interaction. The values obtained for are in good agreement with the value of 15.5 cm3 mol-' found by Millero et al.45from correlations of electrostriction partial molar volumes and compressibilitiesof aqueous electrolytes. As discussed earlier, in order to calculate the total electrostriction, we need to consider the number of water molecules near the ion. Our calculated values of the quantityf(co1umn 2, Table IV) suggest that complete disappearance of the local short-range order of the solvent occurs only for ions of high ionic potential (Be2+,3+ ions). When complete disruption occurs, the number of water molecules in the first layer is limited primarily by the size of the ion relative to the size of the water molecule. As noted by both Swaddle and MakZ2and Matteoli,13a hexagonal-close-packed assembl of water while a molecules encloses an interstitial void of radius 0.57 cubic-close-packed assembly encloses a void of radius 1.006 A. In the first case, the void has six nearest neighbors with a molar volume of 1 1.8 cm3 mol-'; in the second case, the void has eight nearest neighbors with a molar volume of 12.5 cm3 mol-'. On this basis we might expect the volume of water near BeZ+to be 11.8 cm3 mol-' and that of water near Ca2+,Cd2+,and Yb3+ to be 12.5 cm3 mol-'. Our calculated values (column 3, Table V) are somewhat larger, significantly larger in the case of Ca2+ and Cd2+. In our treatment of ionic entropies, we found that for divalent cations there is a small remaining structural correlation of water molecules in the primary shell with those in the second shell (see values off, column 1, Table V). This result is consistent with finding that the molar volume of water in the first shells of Ca2+ and Cd2+is significantly larger than the volume anticipated on the basis of close packing. Matteoli13 has pointed out that if the radius of the ion is larger than the radius of the interstitial void in a close-packed assembly of water molecules, then the packing efficiency is reduced, the volume of void space increases, and the local molar volume of water increases. For ions of high field this increase is compensated to some extent since water molecules beyond the first layer can be drawn closer to the ion. With this in mind we have assigned values of nl as shown in Table V. For the monovalent ions, where the extent of structural disruption is not so large, we consider that the number of water molecules in the first layer is determined primarily by the original tetracoordinate structure of the solvent. For the 3+ ions we have explicitly considered a second layer of water molecules contributing --20 cm3 mol-' to the total volume change (n2 8, molar volume 15.6 cm3 mol-'). Our calculated result for VD(Be2+)is considerably more negative than the experimental value. The dataz1-&for this ion, however, are suspect, due to the presence of hydrolysis products. Discrepancies of 11 and 8 cm3 mol-I are observed for the partial molar volumes of Ni2+ and Zn2+ (predicted values less negative than the experimental values). Only a small part of this discrepancy can be attributed to the choice of ionic radii. Calculations based on the Shannon radii of these ions recommended by Swaddle 1 cm3 mol-l smaller than our and MakZ2lead to values of V,, values of V,,, based on Goldschmidt's radii. Similarly, use of Shannon's radii leads to values of qntless than 1.O cm3 mol-' more negative than those listed in Table V. Thus, the choice of ionic radii can account for only -2 cm3 mol-' of the observed differences in experimental and theoretical values of the partial molar volumes of these ions. In view of the good agreement of our theoretical values with experimental values for other divalent cations (excluding Be2+,discussed above) and in view of the good agreement that we obtained for the entropies of transfer, a source of error more likely than the specific choices for our model pa-
8:
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(45) Millero, F. J.; Ward, G. K.; Lepple, F. K.; Hoff, E. V. J . Phys. Chem. 1974, 78, 1636. (46) Sidgwick, N. V.; Lewis, N. B. J . Chem. SOC.1926, 1287.
Barta and Hepler TABLE VI: Cavity and Interaction Terms for Limiting Partial Molar Volumes of Tetraalkylammonium Ions, Calculated from Scaled Particle Theow" ion
Me4N+ Et4N+ Pr4N+ Bu4N+
radiusb 2.51 (3.47) 3.08 (4.00) 3.49 (4.52) 3.81 (4.94)
V(abs) 84.17 143.72 209.04 270.26
V,: 73.20 (165.0) 122.02 (238.2) 168.06 (330.8) 211.13 (420.1)
Vn,C 9.84 (-80.8) 20.58 (-95.5) 39.86 (-121.8) 58.01 (-149.8)
Units are as follows: radius, A; volumes, cm3 mol-'. Values in parentheses are from ref 47; other values are from ref 11. CValue~ in parentheses are from ref 12.
TABLE VII: Cavity and Interaction Terms for Standard Molar Entropies (J K-' mol-') of Transfer from Cas Phase to Water for Tetraakylammonium Ions," Calculated from Scaled Particle Theory ion A.S0(g+w) S," Sint
Me4N+ Et4N+ Pr4N+
-112.0 -193.8 -298.5
-98.4 -140.0 -174.7
45.3 5.6 -64.5
Data not available for tetrabutylammonium ion rameters is the physical simplicity of our electrostatic model for partial molar volumes of ions.
5. Analysis of the Standard-State Partial Molar Volumes and Entropies of Transfer of Aqueous Tetraalkylammonium Ions Using Scaled Particle Theory Scaled particle theory was first applied to the analysis of the standard-state properties of the aqueous tetraalkylammonium (TAM) ions by Hirata and Arakawa.I2 By equating the volume of cavity formation (V,J with the intrinsic volume of an ion and taking values of the ionic radii from Robinson and these authors calculated the volumes of interaction (qnt) from the relationship VD(X,abs) = V,, Vint, with limiting partial molar volumes from Millero's2' compilation and VD(H+,abs) = -5.0 cm3 mol-'. The values of Vht were found to be large and negative and to become more negative with increasing radius of the ion. These results are summarized in Table VI. Masterson, Bolocofsky, and Lee" have reevaluated the radii of the aqueous TAM ions using the scaled particle theory of the salt effect on solubility of nonpolar gases in aqueous media. The values that they obtained are also listed in Table VI. These authors noted that Robinson and Stokes'47 radii are maximum estimates and subject to considerable uncertainty when applied to the aqueous ions. It was also observed that the value of the radius of (CH3)4N+obtained from the scaled particle theory of gas solubility is in good agreement with the value obtained by Hepler et a*!I from analysis of internuclear distances in the chloride and bromide salts of this ion. For these and other reasons we have preferred to use the scaled particle theory radii of the TAM ions in our calculations. Using eq 1, 2, and 4, Millero's recommended values of the limiting partial molar volumes, and VD(H+,abs) = -5.4 cm3 mol-', we have calculated cavity and solute-solvent interaction terms as listed in Table VI. As shown there, the cavity terms based on radii from scaled particle theory lead to values of the interaction terms that are positive and increase with increasing ionic radius. We have also calculated entropies of ion-water interaction, using eq 3-5 and entropies of transfer from Johnson and Martir1.4~ The results are listed in Table VII. As for the monatomic cations, the values of both V,, and Sint are determined predominantly by short-range interactions. (For
+
(47) Robinson, R. A.; Stokes, R.H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959; pp 124-125. (48) Hepler, L. G.; Stokes, J. M.; Stokes, R. H. Trans. Faraday Soc. 1965, 61, 20.
(49) Johnson, D. A,; Martin, J . F. J . Chem. SOC.,Dalton Trans. 1973, 1585.
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 5595
Thermodynamics of Ionic Hydration the bare TAM ions the long-range polarization contributions are V,,, < 2 cm3 mol-' and Sbrn< -16 J K-'mol-'). Many authors have interpreted the properties of the TAM ions in aqueous solution in terms of enhanced structure of the solvent resulting from interaction of the solvent with the alkyl groups. With reference to the two-state model for liquid water presented in section 3, enhanced structuring ( M L) is characterized by a positive change in volume and a negative change in entropy. On this basis our calculations suggest that only (C3H7)4N+and presumably also (C4H9)4N+increase the degree of local solvent structure. Values of Frit > 0 and Si,,, 0 for (C2H5)4N+are consistent with a local expansion of the solvent without substantial structural changes. Positive values of both V,,, and Sintfor (CH3)4N+suggest that this ion disrupts the local solvent structure but maintains a high local molar volume of solvent due to geometric factors. A similar interpretation has been given by BunzlSO on the basis of a careful infrared study of aqueous solutions of the TAM ,bromides at several temperatures (see also ref 51). We stated in the Introduction that, for aqueous ions, scaled particle theory is most appropriately applied to the calculation of cavity terms for small monatomic ions. There may be an objection that scaled particle theory should not be used to calculate cavity terms for large polyatomic species such as the TAM ions. We note first that scaled particle theory overcomes one of the important difficulties in treating the partial molar volumes by including in its formulation the void space related to the packing efficiency of a system of hard spheres. Second, the ionic radii of these species are not well-defined physically; scaled particle theory yields operational values of these radii that are consistent with gas solubility data and with theoretical calculations as described above. Finally, scaled particle theory leads to values of the volumetric and entropic contributions of ion-water interaction that suggest an interpretation of solvent structural perturbations that is supported by spectroscopic information. It is therefore suggested that, as for monatomic ions, scaled particle theory offers the best available method for calculating the intrinsic thermodynamic properties of the TAM ions in water and thus for separating hard-sphere effects from effects due to ion-water interactions.
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Acknowledgment. We thank the Natural Sciences and Engineering Research Council of Canada for support of this and related research.
List of Symbols B, D E e
f i
K L M
empirical constants electrostatic field electronic charge fraction of original solvent structure remaining in the presence of the electrostatic field of an ion integer subscript denoting the ith shell of water molecules around an aqueous ion equilibrium constant for interconversion of states in the two-state model the less dense of the components in the two-state model the more dense of the components in the two-state model
(50) Bunzl, K. W. J . Phys. Chem. 1967, 71, 1358. (51) Verrall, R. E. In Water: A Comprehenriue Treatise; Franks, F., Ed.; Plenum: New York, 1973; Vol. 3, pp 252-256.
N
Avogadro's constant number of water molecules in the ith shell of solvent around an ion, or moles of water per mole of ion for the ith shell P effective pressure due to the electrostatic field of an ion AP effective pressure minus 1 atm (1 bar 1 atm) r distance from the center of an ion to the center of the ith shell of water molecules ria radius of the interstitial void in a close-packed assembly of water molecules approximated as hard spheres Tion hard-sphere radius of an ion 'err effective electrostatic radius of an ion sa" molar entropy of cavity formation Sin, molar entropy of ion-water interaction AS,,, structural molar entropy of pure liquid water at 25 "C and 1 atm aSo(g- w) standard molar entropy of transfer of an ion from gas phase (ideal gas standard state) to water (infinitely dilute solute standard state) at 25 " C and 1 atm change in molar entropy of water due to the mean ionic field ASw(i) in the ith shell entropy of an isolated system of water molecules in an '(h) electrostatic (ionic) field entropy of an isolated system of water molecules in the field S(W) of a central water molecule T temperature, K t temperature, OC UW molar volume of bulk water ui molar volume of water in the ith shell V ( X ,abs) infinitelv dilute standard-state absolute oartial molar volume of ionic species X molar volume of cavity formation molar volume of ion-solvent interaction contribution of nonelectrostatic interactions to the molar volume of ion-solvent interactions molar volumes of components L and M in the two-state model molar volume of water at 1 atm molar volume of water at pressure p intrinsic volume per mole of ions volume change of solvent due to electrostatic field (electrostriction), per mole of ions change in molar volume of water due to the mean ionic field in the ith shell mole fractions of components L and M in the two-state model of liquid water generalized thermodynamic molar property long-range polarization contribution to ion-solvent interaction, per mole of ions reduced number density formal charge ni
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Greek Symbols isobaric expansivity = (l/u)(au/aT), isothermal compressibility = -( 1/ ~ ) ( B u / t + p ) ~ c dielectric constant P chemical potential P number density U hard-sphere molecular diameter Registry No. Lit, 17341-24-1; Na', 17341-25-2; , ' K 24203-36-9; Rb', 22537-38-8; CS', 18459-37-5; Ag', 14701-21-4; TI', 22537-56-0; Be2', 22537-20-8; Mg2+, 22537-22-0; CaZ+, 14127-61-8; Sr2', 2253739-9; Ba2+, 22541-12-4; Ni2', 14701-22-5; Cu2+, 15158-11-9; Co2+, 22541-53-3; Fe2', 15438-31-0; Zn2', 23713-49-7; Mnzt, 16397-91-4; Cd2', 22537-48-0; Hg2', 14302-87-5; Pb2', 14280-50-3; AI'+, 2253723-1; Cr3', 16065-83-1; Fe3+, 20074-52-6; Yb3+, 18923-27-8; La3', 16096-89-2; Me4N+, 51-92-3; Et4N+, 66-40-0; Pr4Nt, 13010-31-6; Bu~N', 10549-76-5. ff
P
