Solvent Release upon Ion Association from Entropy Data. II - The

572

J. Phys. Chem. B 2007, 111, 572-580

Solvent Release upon Ion Association from Entropy Data. II Y. Marcus* Department of Inorganic and Analytical Chemistry, The Hebrew UniVersity, Jerusalem 91904, Israel ReceiVed: July 12, 2006; In Final Form: August 15, 2006

When a cation and an anion associate, the charge on the product is lower than that on the individual ions and solvent is released from their solvation shells to the bulk solvent. This release occurs when the associate is a solvent-shared or contact ion pair or an inner-type complex. The measurable molar entropy change involved is considered to be made up of four contributions: translational, rotational, electrostatic, and desolvation entropies. The former three can be calculated from the properties of the ions and solvents involved; hence, the fourth is obtained by difference. The release of solvent molecules from the crystalline frozen solvent to the liquid on melting is analogous to the solvent release from translational immobilization in the solvation shells of the ions. The molar entropy of melting of the solvent is used to estimate the amount of solvent released in the association process.

1. Introduction Some two decades ago, the author published a study1 analyzing entropy changes upon ion association that had been reported at the time. The analysis was in terms of the numbers of solvent molecules released from being immobilized in the solvation shells of the ion partners when they associate. The premise of this analysis was that the association of a cation and an anion to form an ion pair or a complex (the distinction is immaterial for the present purpose, if altogether2) causes a diminution of the net charge of the product relative to the sum of the charge numbers of the ion partners. The extent of solvation of ions (solvation numbers) depends strongly on their charges.3,4 Therefore, irrespective of whether solvent-shared or contact ion associates have been formed, the ion associates are solvated to a smaller extent than the separate ions. Several solvent molecules are thus released from the ionic solvation shells to join the bulk solvent. The standard molar entropy change on the association of one cation with one anion can be described in terms of several contributions. In the gas phase, when two particles unite to form a single one, there is a diminution of the translational entropy. The product of the association can be envisaged as a diatomic molecule that has rotational entropy. This differs from that of the separate ions, certainly when the latter are spherical and devoid of rotational entropy altogether. When the association takes place in solution, the electrostatic association leading to ion pairing is not expected to cause a significant change in the vibrational entropies of (polyatomic) ion partners, but the longrange electrostatic entropy effects are changed by the association of the ions. Finally, there is the entropic contribution of the desolvation that is the subject of the present analysis. It was proposed1 that, in terms of the entropy changes, the desolvation was analogous to the release of the solvent molecules from the crystal lattice of the frozen solvent to its liquid form on melting. This, then, serves as the basis for the estimation of the number of solvent molecules released from being immobilized in the solvation shells of the ions on their association. The entropy data dealt with in the former study1 were relatively few. Since then, many more data were published2 that

where R has its usual meaning. Compression of the volume at the disposal of the ion from that in its standard state in the gas (V(g) ) RT°/P°) to that in the solution (V(sol) at 1 M ≡ 1 mol dm-3) involves R ln(P°V(sol)/RT°) ) R ln[0.1 × 106 (Pa) × 10-3 (m3 mol-1)/8.3143 (J K-1 mol-1) × 298.15 (K)] ) -26.69 J K-1 mol-1 as the contribution to the translational entropy. Hence, for a particle of relative molar mass Mi at the standard temperature, pressure, and concentration (expressed on the molar scale), the translational entropy is

* Phone: +972-2-6585341. Fax: +972-2-6585319. E-mail: ymarcus@ vms.huji.ac.il.

Si(sol)°tr (J K-1 mol-1) ) 1.5R ln Mi + 82.16

merit analysis, involving ion association in several additional nonaqueous solvents as well as in water. The reported standard molar entropies of association, ∆iaS°exptl, are generally obtained from the temperature dependence of the association constant but in many cases also from calorimetric measurements, the former being expected to be less accurate, in particular in cases of weak association.2 The purpose of the present study is not to critically review these values (this was done in ref 2) but to use them as given (with the sources cited) and analyze them in terms of the solvent release on association. 2. Calculations The experimentally derived standard molar entropy change on ion association, ∆iaS°exptl, is thus assumed to consist of four terms, representing the translational, rotational, electrostatic, and desolvation contributions:

∆iaS°exptl ) ∆iaS°tr + ∆iaS°rot + ∆iaS°el + ∆iaS°desolv (1) The former three contributions can be calculated independently, so that the last, that due to the desolvation, can be obtained from the experimental quantity by difference. The translational entropy of a gaseous ion having a relative molar mass Mi at the standard temperature T° ) 298.15 K and the standard pressure P° ) 0.1 MPa is obtained from the Sackur-Tetrode equation as5

Si(g)°tr (J K-1 mol-1) ) 1.5R ln Mi + 108.85

10.1021/jp0643808 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/30/2006

(2)

(3)

Solvent Release upon Ion Association

J. Phys. Chem. B, Vol. 111, No. 3, 2007 573

The loss in translational entropy on ion association of two particles to a single one under these standard conditions is therefore1

∆iaS°tr (J K

-1

-1

mol ) ) 1.5R ln(M(/M+M-) - 82.16 (4)

The subscripts of the masses are “(” for the associate, “+” for the cation, and “-” for the anion. In this calculation, the masses of solvent molecules in the solvation shells are ignored, because their entropic effect is subsumed into that occurring on the desolvation. If the associating ions are spherical, only the product of the association has rotational entropy. In the gas phase, for a rigid rotator consisting of two (relative) masses, M+ and M-, at a distance of a apart, this is5

Si(g)°rot (J K-1 mol-1) ) R(1 - ln y)

(5)

where y ) 0.242 54/IT°, I ) (M+M-/M()a2 is the moment of inertia, and the numerical coefficient is for a in nanometers. Expression 5 is valid for an unsymmetrical linear rotator (symmetry number ) 1) and provided that the quantity y2/90 can be neglected relative to unity. Then, for the associate under the standard conditions,

∆iaS°rot (J K-1 mol-1) ) -R ln(M(/M+M-) + 2R ln[a (nm)] + 67.47 (6) (Note the misprint in eq 2 of ref 1, where the sign before the mass-dependent term is + instead of -, but the calculations there were made with the correct sign.) If the associating ions are not spherical, for example, tetraalklammonium or sulfate ions, then they have rotational entropy of their own that is partly lost on the association. Such ions can still rotate around the axis between the cation and the anion but do not tumble at all random directions as they would if free. It is difficult to determine the actual change in rotational entropy in the general case, but an estimate of the lost rotational entropy as one-third of that of the free nonspherical ion was made;1 see also ref 6. Again, the effects of the solvent molecules in the solvation shells of the ions and of the association product upon the change in the rotational entropy is neglected, with the rotation pertaining to gaseous, bare ions. The contributions of the translational and rotational entropies expressed in eqs 4 and 6 to the total entropy change are thus independent of the solvent. The effect of the solvent is related to the electrostatic contribution. Within the solvation shells of the ions, the electrostatic effect is essentially independent of the solvent because of dielectric saturation, as argued by Abraham and Liszy7 and by Marcus et al.8 Therefore, the entropy change on ion association, calculable according to the Born equation, takes place only beyond a distance d from the surface of the bare ions or of the associate, where d nm is a characteristic linear dimension of the solvent molecules. For an ion with a radius ri and charge zi in a solvent of relative permittivity r at T°, the electrostatic entropy is

Si(s)°el (J K-1 mol-1) ) 6.945 × 104[r-2(dr/dT)]zi2(ri + d)-1 (7) where the numerical coefficient is 109NAe2/8π0, for ri and d in nanometers. The radius of the ion associate is taken as that of a sphere of the same volume as that of its ellipsoid of rotation, that is, r( ) (r+r-a)1/3, and its charge is z( ) z+ - z-. The

electrostatic entropy change on ion association under the standard conditions is therefore

∆iaSi°el (J K-1 mol-1) ) -6.945 × 104[r-2(dr/dT)][z+2/(r+ + d) + z-2/(r- + d) - z(2/(r( + d)] (8) Note that for all the solvents of interest dr/dT < 0 and that for association of a symmetrical electrolyte (1:1, 2:2, etc.) z( ) z+ - z- ) 0, so that the last term in the third factor of eq 8 vanishes. The entropic contribution of the desolvation of the ion partners on ion association is obtained by difference from the experimentally determined standard molar entropy of association from eq 1. It represents the losses of the standard molar solvation entropies of the ion partners and its gain by the ion associate in a cycle between the gas and solution phases. It should be related to the number of solvent molecules released to the bulk solvent. This is done on the assumption that the entropy change involved is akin to that occurring when solvent molecules are released from the crystal lattice of the frozen solvent on melting, albeit not at the melting temperature, Tm, but at the standard temperature T° ) 298.15 K. It is, therefore, necessary to extrapolate the molar entropy of melting, ∆mS°, from Tm to T°. For some solvents, the molar entropy of the frozen, crystalline solvent is known as a function of the temperature, S(c)°(T), as is that of the liquid solvent, S(l)°(T). If S(c)°(T) is linear, as it is for some solvents, it can be extrapolated to form the difference:

∆mS°(T°) ) S(l)°(T°) - S(c)(T°, extrapolated)

(9)

Otherwise, if the molar heat capacity, Cp°(T), of the solvent, both below Tm and above it up to T°, is known, the change in the molar enthalpy of melting, ∆mH°, from its value at Tm to that of the superheated solid at T° can be evaluated:

∆mH°(T°) ) ∆mH°(Tm) +

∫TT°[Cp°(l) - Cp°(c)] dT

(10)

m

In other cases, standard molar enthalpies, H° - H°0, for the crystalline and liquid states are reported as a function of the temperature. Then,

∆mS°(T°) ) ∆mH°(T°)/T° ) [H°(l)(T°) - H°(c)(T°, extrapolated)]/T° (11) assuming equilibrium between the superheated frozen solvent and its liquid. Finally, the number of solvent molecules released on ion association is taken as

∆iandesolv ) ∆iaS°desolv/∆mS°(T°)

(12)

3. Results The extrapolated molar entropies of melting, ∆mS°(T°), of several solvents, for which standard molar entropies of ion association, ∆iaS°exptl, have been reported, are shown in Table 1. Also shown there are the characteristic solvent measures d and the values at T° of r-2(dr/dT). The solvent linear sizes d from ref 9 are deemed “diameters”, but this is an approximation, since the solvent molecules are not necessarily globular. For a discussion of the actual sizes of the molecules of solvents, see ref 13; the set of d values shown in Table 1 are in general agreement with the values given there.

574 J. Phys. Chem. B, Vol. 111, No. 3, 2007

Marcus

TABLE 1: The Relevant Solvent Properties

solventa

Tmb (K)

water methanol ethanol 1-propanol 2-propanol acetone acetonitrile NMF DMF DMA DMSO

273.2 175.5 158.7 147.0 185.2 178.5 229.3 270.6 212.9e 254.2e 291.7

∆mS° (J K-1 mol-1) at Tmc at T° d

db (nm)

-105 × r-2 (dr/dT)b (K-1)

21.9 18.3 31.7 36.5 29.2 32.3 35.6 38.6 42.7e 40.1e 49.3

0.276 0.408 0.469 0.515 0.516 0.482 0.436 0.485 0.521 0.557 0.513

5.78 18.62 25.34 21.35 35.84 22.96 11.57 4.87 13.95 16.12 5.88g

23.0 21.2 28.2 34.7 31.4 27.2 34.6 ∼39 52.7f 43.9f 48.3

a NMF ) N-methylformamide, DMF ) N,N-dimethylformamide, DMA ) N,N-dimethylacetamide, DMSO ) dimethylsulfoxide. b From ref 9: Tm, p 70 ff; r and dr/dT, p 95 ff; d, p 88 ff. c ∆mS°(Tm) ) ∆mH°(Tm)/Tm with ∆mH° from ref 10. d See the text. e Reference 11. f From S°(cr) data provided by N. N. Smirnova11 extrapolated to T°. g The value of dr/dT from ref 12.

In some cases, the extrapolation of the molar entropy of melting from Tm to T° is over a short temperature span only. For dimethylsulfoxide (DMSO), the H° - H°0 data given by Clever and Westrum14 are readily extrapolated to yield ∆mS° from eq 11. For water, the linear Cp°(c)(T) data of Giauque and Stout15 permit the estimate of ∆mH°(T°) from eq 10, and hence of ∆mS°(T°) from eq 11. The changes of ∆mS° for these two cases are (1 J K-1 mol-1, so that for N-methylformamide (NMF), for which no relevant thermodynamic quantities were found, the approximation ∆mS°(T°) ≈ ∆mS°(Tm) can be made. The error in ∆mS°(T°) is expected to be no larger than (1 J K-1 mol-1, since both a nonstructured solvent, DMSO, and a structured one, water, have close values of ∆mS°(T°) and ∆mS°(Tm) for the small difference in temperature as for NMF. More care has to be taken in the cases of solvents for which the extrapolation from Tm to T° is long. The S° data for the crystalline state of methanol (above the solid-solid transition temperature) reported by Carlson and Westrum,16 of ethanol reported by Parks,17 of 1-propanol reported by Counsell, Lees, and Martin,18 and of 2-propanol reported by Andon, Counsell, and Martin19 can be linearly extrapolated to T° to yield ∆mS°(T°) from eq 9. For acetone, the Cp°(T) data of Kelley20 above the anomaly are used with eq 10 to give ∆mH°(T°), hence ∆mS°(T°). The Cp°(T) data of Putnam, McEachern, and Kilpatrick21 are similarly used for acetonitrile. No relevant thermodynamic data were found for the amides, N,N-dimethylformamide (DMF) and -acetamide (DMA); hence, these were measured in relation to the present study by Smirnova and co-workers.11 It should be noted in passing that most of the required thermodynamic data mentioned above are fairly old and for many solvents that are widely used currently such data are absent altogether. The standard molar entropies of ion association, ∆iaS°, at T° ) 298.15 K on the molar scale obtained from the literature for the systems studied here are shown in Table 2 for aqueous solutions and in Table 3 for nonaqueous ones. Also shown in these tables are the calculated contributions of the translational, ∆iaStr, rotational, ∆iaSrot, and electrostatic, ∆iaSel, entropies. The required properties of the ions (M and r) and of the solvents (d, r, and dr/dT) are from standard compilations.3,9 Table 4 lists the rotational entropies of most of the gaseous (i.e., isolated) polyatomic ions included in Tables 2 and 3, calculated from the data provided by Loewenschuss and Marcus.5 These values are between 50 and 110 J K-1 mol-1, and if, as estimated previously,1 one-third of this is lost by the

restricted rotation in the associated ions, the amounts of ∆iandesolv would be augmented by 1-4 units. The ∆iaSel values depend on the validity of the Born equation for the solvent effects. For symmetrical ion pairs z( ) 0 and the last term in the third factor of eq 8 vanishes. This means that the ion associate has no long-range electrostatic effects expressible as a contribution to the entropy of association. Nonsymmetrical ion pairs or complexes, however, do make such a contribution, and this depends on the assumed size of the associate, r(. In all cases, the distance from the center of the charged particle beyond which the electrostatic effect takes place depends on the size of d, and this, again, is a cause of some uncertainty. The last two columns of Tables 2 and 3 contain the values of ∆iaSdesolv and ∆iandesolv obtained from eqs 1 and 12. 4. Discussion A positive standard molar entropy of ion association, ∆iaSexptl, is often associated with a positive ∆iaHexptl value, still resulting in a negative ∆iaGexptl value. This involves an association constant of >1, denoting significant association, and in these cases, the association is entropy driven. Several authors1,22,29,32,34,35,46,62 have expressed the view that a positive ∆iaSexptl value reflects the release, consequent to the association, of solvent molecules to the bulk solvent from the solvation shells of the ion partners forming the associate. Except for the author’s previous publication,1 none has tried to estimate the number of solvent molecules thus released. Yokoyama and co-workers,34,35 however, did analyze ∆iaSexptl (of outer-sphere associates of aqueous hexammine- and tris(ethylenediamine)cobalt(III) with anions) essentially as done here. They did so in terms of the sum of the entropy change for the association in the gaseous phase, ∆iaS(g), and the changes involved in the entropy of solvation (hydration) of the participants in the association. The ∆iaS(g) value was taken as in the present study as the sum ∆iaS°tr + ∆iaS°rot (plus a term representing interionic vibration, ignored in the present work). Furthermore, in the analysis of Yokoyama and co-workers,34,35 the rotation of polyatomic ions (nitrate and perchlorate) was considered as hindered in the associate, leading to a loss of 3141% of their rotational entropy, similar to the present estimate of one-third. These authors, however, did not consider longrange electrostatic effects, ∆iaS°el, that pertain to those effects of the solvent on the entropy of association beside those due to the change in the solvation of the participants in the association, ∆iaS°desolv. It is now necessary to assess the reality of the assumptions made in deriving the ∆iaSdesolv values and their effect on these values. Of the calculated quantities, the ∆iaStr values are straightforward, obtained from the net masses of the associating ions (i.e., without their solvation shells). The ∆iaSrot values were calculated on the basis that the interionic distance is a ) r+ + r-, that is, for a contact ion pair, and disregarding the residual rotational entropy that polyatomic ions have. If a large proportion of the associates formed in a given system are of the solvent-shared types, then a larger contribution of ∆iaSrot to the experimental value would result. This is because solvent-shared ion pairs have larger a values (r+ + r- + d), resulting in a larger ∆iaSrot value. As a consequence, the ∆iaSdesolv value obtained from eq 1 by difference is smaller and the resulting ∆iandesolv values obtained from eq 12 are also smaller. On the contrary, rotation of polyatomic ions around the interionic axis decreases ∆iaSrot, hence allowing for larger ∆iaSdesolv and ∆iandesolv values, as already discussed above.



TABLE 2: Standard Molar Entropies of Ion Association in Aqueous Solutions, the Derived Molar Entropy of Desolvation, and the Number of Solvent Molecules Released on the Association associate

ref

∆iaS°exptl (J K-1 mol-1)

∆iaS°tr (J K-1 mol-1)

∆iaS°rot (J K-1 mol-1)

∆iaS°el (J K-1 mol-1)

∆iaS°desolv (J K-1 mol-1)

∆iandesolv

LiB(OH)4 NaCl NaSO4NaS2O3KCl KS2O3MgCl+ MgSO4 MgS2O3 MgFe(CN)6MgFe(CN)62CaCl+ CaSO4 CaS2O3 CaFe(CN)6CaFe(CN)62SrFe(CN)62SrFe(CN)62BaFe(CN)6BaFe(CN)62MnF+ MnNO3+ MnClO3+ MnSO4 CoF+ CoNO3+ CoClO3+ CoSO4 NiF+ NiNO3+ NiClO3+ NiSO4 CuF+ CuNO3+ CuClO3+ CuSO4 ZnF+ ZnNO3+ ZnClO3+ ZnSO4 CdSO4 UO2Cl+ UO2SO4 FeSO4+ Co(NH3)6Cl2+ Co(NH3)6Br2+ Co(NH3)6I2+ Co(NH3)6NO32+ Co(NH3)6ClO42+ Coen3Cl2+ Coen3Br2+ Coen3I2+ Coen3NO32+ Coen3ClO42+ LaSO4+ PrSO4+ NdSO4+ EuSO4+ GdSO4+ TbSO4+ DySO4+ HoSO4+ ErSO4+ TmSO4+ YbSO4+ LuSO4+

22 23 24 25 23 25 23 26 25 27 28 23 26 25 27 28 27 28 27 28 29 30 30 31 29 30 30 31 29 30 30 26 29 30 30 26 29 30 30 26 26 32 33 31 34 34 34 34 34 35 35 35 35 35 36 36 36 36 36 36 36 36 36 36 36 36

22.9 -33 52 15 -19 32 28 62 16 38.5 133 31 65 22 36.8 115 32.6 96 35.2 131 56.9 -18.8 -22.0 71 41.4 -19.2 -5.1 63 33.1 -10.3 -24.5 63 65.3 -18.3 -28.4 69 51.9a -18.0 -16.9 65 74 26.0 132 163 41.6 34.9 29.9 25.2 18.4 38.5 34.2 29.9 27.4 22.9 221.7b 226.4 227.4 230.3 228.3 226.3 227.6 226.4 226.1 225.1 224.4 223.4

-105.3 -115.1 -118.6 -119.0 -118.7 -124.2 -115.5 -119.2 -119.5 -120.6 -120.6 -118.8 -123.9 -124.4 -126.1 -126.1 -133.7 -133.7 -137.4 -137.4 -115.2 -124.3 -125.9 -126.5 -115.4 -124.7 -126.4 -127.1 -115.4 -124.7 -126.3 -127.0 -115.7 -125.2 -126.9 -127.6 -115.7 -125.4 -127.1 -127.9 -131.4 -125.2 -135.3 -126.7 -124.2 -131.8 -135.4 -129.6 -133.6 -125.0 -133.2 -137.3 -130.8 -135.2 -132.6 -132.6 -132.8 -133.0 -133.2 -133.2 -133.3 -133.4 -133.5 -133.5 -133.6 -133.7

62.8 68.4 73.8 74.7 72.8 78.8 66.8 72.2 73.6 82.3 82.3 70.8 76.9 78.2 86.8 86.8 92.3 92.3 95.4 95.4 64.0 74.5 75.6 72.7 63.5 74.4 75.5 77.7 63.1 74.0 75.1 77.3 63.5 74.6 75.7 77.9 63.7 74.8 76.0 78.2 81.6 83.3 89.6 76.8 87.0 92.5 95.6 91.2 94.8 88.4 94.3 97.6 92.7 96.7 82.9 82.9 82.8 82.8 82.7 82.7 82.6 82.6 82.6 82.7 82.7 82.7

19.6 19.4 77.5 32.9 18.5 32.4 45.5 77.9 76.7 89.0 104.5 42.5 74.5 73.2 86.1 102.9 84.8 102.2 82.8 101.0 44.8 44.1 44.1 76.5 45.6 450 45.0 77.5 46.3 45.7 45.7 78.3 45.9 45.2 45.2 77.8 45.6 45.0 45.0 77.5 75.0 30.5 64.1 128.6 35.6 35.7 36.0 35.7 36.2 33.8 34.0 34.3 34.0 34.5 118.2 119.4 119.6 120.6 120.8 121.0 121.6 121.8 122.0 122.3 122.5 122.8

45.8 -5.8 19.3 26.5 8.4 44.0 31.2 31.0 -14.7 -11.6 66.8 36.5 37.6 -5.0 -9.3 51.4 -10.1 35.2 -5.0 72.0 63.3 -13.2 -15.9 43.3 47.7 -13.9 0.8 34.9 39.2 -5.3 -18.9 34.4 71.6 -12.9 -22.4 40.9 58.3 -12.4 -10.0 37.2 48.8 37.4 113.6 84.2 43.4 38.6 33.9 28.0 21.0 41.3 39.2 35.3 31.4 26.9 153.2 156.4 157.3 159.9 157.9 155.8 156.7 155.1 155.1 153.8 152.8 151.6

2.0 0 0.8 (3.5) 1.2 (4.0) 0.4 1.9 (4.7) 1.4 1.3 (4.0) 0 (2.3) 0 (2.6) 2.9 (5.9) 1.6 1.6 (4.3) 0 (2.5) 0 (2.3) 2.2 (5.2) 0 (2.3 1.5 (4.5) 0 (2.5) 3.1 (6.1) 2.8 0 (1.8) 0 (1.9 1.9 (4.6) 2.1 0 (1.8) 0 (2.6) 1.5 (4.3) 1.7 0 (2.1) 0 (1.7) 1.5 (4.0) 3.1 0 (1.8) 0 (1.6) 1.8 (4.5) 2.5 0 (1.8) 0 (2.1) 1.6 (4.3) 2.1 (4.8) 1.6 4.9 (7.6) 3.7 (6.4) 1.9 1.7 1.5 1.2 0.9 1.8 1.7 1.5 1.4 1.2 6.7 (9.4) 6.8 (9.5) 6.9 (9.6) 7.0 (9.7) 6.9 (9.6) 6.8 (9.5) 6.9 (9.6) 6.7 (9.4) 6.7 (9.4) 6.7 (9.4) 6.6 (9.3) 6.6 (9.3)

a Reference 37 reports 75.3 J K-1 mol-1, increasing ∆ n b -1 mol-1 calculated from literature ia desolv by 1.0. Reference 33 reports ∆iaSexptl ) 116.6 J K data for LaSO4+ formation, decreasing ∆iandesolv by 4.6. Similarly, values of ∆iaSexptl lower by ∼100 J K-1 mol-1 were reported there for other lanthanide sulfate associates up to HoSO4+, leading to ∆iandesolv values lower by ∼4.35 units.


Marcus

TABLE 3: Standard Molar Entropies of Ion Association in Nonaqueous Solutions, the Derived Molar Entropy of Desolvation, and the Number of Solvent Molecules Released on the Association ref



CaClO4+ SrClO4+ BaClO4+ CoClO4+ NiClO4+ CuClO4+ ZnClO4+ CdClO4+

38 38 38 38 38 38 38 38

97 101 101 79 83 103 87 90

-124.0 -130.1 -132.8 -127.2 -127.2 -127.8 -128.0 -131.7

NaI KI RbI CsI Me4NI Et4NI Pr4NBr Pr4NI Pr4NI Pr4NClO4 BuNI

39 39 39 39 39 39 40 39 40 40 39

77 82 76.5 71 34.5 49 42.8 57 41.5 40.2 69.5

-119.2 -124.6 -131.3 -134.2 -130.2 -134.1 -132.4 -136.2 -136.2 -134.2 -137.4

LiCl NaBr NaI NaSCN NaClO4 KI KI RbI

41 42 42 43 42 42 41 42

98.5 122.1 107.7 103 105.0 113.4 75.5 110.7

NaI NaSCN NaBPh4

44 43 44

LiI NaI NaI NaI NaBPh4 Me4NI Et4NI

associate

∆iaS°rot (J K-1 mol-1) Methanol 77.4 82.1 84.9 78.3 78.0 78.6 78.8 828.3



∆iandesolv

100.6 98.4 94.8 105.0 106.1 105.4 105.0 101.4

43.0 50.6 54.0 22.9 26.1 46.9 31.2 38.0

2.0 2.4 2.5 1.1 1.2 2.1 1.5 1.8

73.3 78.7 83.5 86.6 88.0 92.4 91.8 94.9 94.9 94.2 96.7

56.4 54.5 54.1 53.0 49.0 47.4 47.2 46.3 46.3 45.6 45.5

66.5 73.4 70.1 65.6 27.7 43.4 36.2 51.9 36.5 34.6 64.7

2.4 2.6 2.5 2.3 1.0 1.5 1.3 1.8 1.3 1.2 2.3

-104.1 -118.1 -119.2 -117.1 -118.7 -124.6 -124.6 -131.3

1-Propanol 59.1 71.3 73.3 71.6 74.0 78.7 78.7 83.5

46.7 44.9 44.2 44.4 43.7 42.9 42.9 42.6

96.9 124.0 109.4 104.1 106.0 116.4 78.5 118.5

2.8 3.6 3.2 3.0 3.1 3.4 2.3 3.4

141.3 130 77.4

-119.2 -117.1 -120.4

2-Propanol 73.3 71.6 82.2

47.5 47.7 42.9

139.7 127.9 72.4

4.4 4.1 2.3

45 45 46 44 44 47 47

-13.1 4.5 48.1 79.2 51.2 54 63

-105.6 -119.2 -119.2 -119.2 -120.4 -130.2 -134.2

51.7 50.0 50.0 50.0 45.0 43.6 42.2

-21.6 0.4 44.0 75.1 44.5 52.6 53.6

0 0 1.6 2.8 1.6 1.9 2.0

LiI LiClO4 LiClO4 NaI NaClO4 KI KI KClO4 KClO4 Me4NI Me4NClO4 Et4NClO4 MgClO4+

48 49 50 48 49 41 50 41 50 50 50 50 49

-25.1 -10.5 44.6 -10.5 -1.3 38.2 46.9 52.9 44.2 36.9 36.2 33.9 -46.4

-105.7 -105.5 -105.5 -119.2 -118.7 -124.6 -124.6 -123.8 -123.8 -130.2 -129.0 -132.6 -119.3

28.2 27.8 27.8 27.2 26.8 26.2 26.2 25.9 25.9 23.5 23.1 22.3 62.0

-10.1 3.7 58.8 8.2 16.6 57.9 66.6 71.8 63.1 55.6 54.3 52.3 -62.0

0 0.1 1.7 0.2 0.5 1.7 1.9 2.1 1.8 1.6 1.6 1.5 0

CoCl+ NiCl+ ZnCl+

51 51 51

33 18 79.4

23.9 24.1 23.9

59.3 44.5 105.9

1.5 1.2 2.7

LiSCN LiSCN TlCl TlBr TlI MgSCN+ MnCl+ MnCl+

52 53 54 54 54 53 55 56

34 10 62 53 41 64 75.1 74

29.6 29.6 28.2 28.0 27.5 64.4 63.7 63.7

47.7 23.7 81.0 74.2 62.9 47.0 61.0 59.9

0.9 0.4 1.5 1.4 1.2 0.9 1.2 1.1

Ethanol

Acetone 62.5 73.3 73.3 73.3 82.2 88.0 92.4

Acetonitrile 62.5 63.5 63.5 73.3 74.0 78.7 78.7 79.0 79.0 88.0 87.8 91.9 72.8 N-Methylformamide -120.8 70.6 -120.8 70.2 -121.3 70.9 N,N-Dimethylformamide -105.0 61.8 -105.0 61.8 -124.7 77.5 -132.7 83.5 -136.8 87.3 -117.6 70.2 -120.5 70.9 -120.5 70.9



TABLE 3. (Continued) associate

ref




MnBr+ CoCl+ CoCl+ CoBr+ NiCl+ NiBr+ CuCl+ CuCl+ CuBr+ ZnCl+ ZnBr+ ZnSCN+ YCl2+ YBr2+ LaCl2+ LaBr2+ CeCl2+ PrCl2+ NdCl2+ NdBr2+ SmCl2+ EuCl2+ GdCl2+ TbCl2+ TbBr2+ DyCl2+ HoCl2+ ErCl2+ TmCl2+ TmBr2+ YbCl2+ LuCl2+

56 55 56 56 56 56 57 58 58 59 60 60 61 61 62 61 62 62 62 61 62 62 62 62 61 62 62 61 62 61 62 62

84 100.5 87 95 84 82 190 165 132 142 170 80 108 38 129 42.9 118 109 107 40 103 104 104 102 35 109 110 122 141 42 138 144

N,N-Dimethylformamide (Continued) -125.6 75.2 -120.8 70.6 -120.8 70.6 -126.1 75.1 -120.8 70.2 -126.1 74.7 -121.2 70.7 -121.2 70.7 -126.7 75.3 -121.3 70.9 -126.9 75.6 -124.9 75.3 -122.5 72.5 -128.9 77.8 -123.9 74.5 -131.2 80.2 -123.9 74.2 -123.9 74.2 -124.0 74.1 -131.3 80.0 -124.1 74.1 -124.1 74.0 -124.2 74.0 -124.2 74.0 -131.8 79.9 -124.2 73.9 -124.3 73.8 -124.3 73.8 -124.3 73.7 -132.0 79.8 -124.4 73.7 -124.4 73.7

LiCl MnCl+ MnBr+ MnSCN+ CoCl+ CoBr+ CoSCN+ NiCl+ NiBr+ ZnSCN+ YBr2+ LaBr2+ CeBr2+ PrBr2+ NdBr2+ SmBr2+ EuBr2+ GdBr2+ TbBr2+ DyBr2+ HoBr2+ ErBr2+ TmBr2+ YbBr2+ LuBr2+

63 64 64 65 64 64 65 64 64 65 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66

38.5 140 140 63 184.0 178 100 176.5 150 100 89 122 136 135 126 100 94 96 89 97 98 103 107 120 118

N,N-Dimethylacetamide -104.1 59.1 -120.5 70.9 -125.6 75.2 -128.9 75.0 -120.8 70.6 -126.1 75.1 -124.3 74.9 -120.8 70.2 -126.1 74.7 -124.9 75.3 -128.8 77.8 -131.2 80.2 -131.2 80.0 -131.2 80.0 -131.3 80.0 -131.5 79.9 -131.6 79.9 -131.7 79.9 -131.8 79.9 -131.9 79.8 -131.9 79.8 -132.0 79.8 -132.0 79.8 -132.1 79.8 -132.1 79.7

LiN3 LiNCO NaNCO KNCO MnCl+ CoCl+ NiCl+ CuCl+ CuCl+ CuBr+ CuI+

67 68 68 68 67 67 67 67 69 69 69

25 43 43 25 62.9 66 62 109 62 49 45

-104.5 -104.5 -115.9 -119.7 -120.5 -120.8 -120.8 -121.2 -121.2 -126.7 -129.0

Dimethylsulfoxide 60.2 60.7 70.2 74.6 70.9 70.6 70.2 70.7 70.7 75.3 78.3



∆iandesolv

63.6 64.4 64.4 64.3 65.0 64.8 64.5 64.5 64.5 64.4 64.3 64.1 99.1 100.3 97.5 97.8 98.1 98.3 98.5 98.8 99.0 99.1 99.3 99.5 99.8 99.8 100.0 100.1 100.3 100.6 100.4 100.6

70.8 86.3 72.8 81.8 69.7 68.6 175.9 150.9 118.9 128.0 157.0 65.5 57.9 -11.2 80.9 -3.9 69.5 60.4 58.3 -7.4 54.1 55.0 54.9 52.8 -12.9 59.6 60.5 72.4 91.3 -6.4 88.2 94.1

1.3 1.6 1.4 1.6 1.3 1.3 3.3 2.9 2.3 2.4 3.0 1.2 1.1 0 1.5 0 1.3 1.1 1.1 0 1.0 1.0 1.0 1.0 0 1.1 1.1 1.4 1.7 0 1.7 1.8

33.0 69.5 69.3 69.2 70.2 70.0 69.9 70.8 70.6 69.9 109.1 106.6 107.2 107.4 107.6 108.1 108.2 108.4 108.6 108.9 109.1 109.2 109.4 109.6 109.8

50.5 120.2 121.1 42.7 164.0 159.0 79.5 156.4 130.8 79.8 31.0 66.4 80.0 78.9 69.9 43.5 37.5 39.4 32.3 40.1 41.0 45.9 49.8 62.7 60.6

1.2 2.7 2.8 1.0 3.7 3.6 1.8 3.6 3.0 1.8 0.7 1.5 1.8 1.8 1.6 1.0 0.9 0.9 0.7 0.9 0.9 1.0 1.1 1.4 1.4

12.8 12.7 12.3 12.0 27.2 27.5 27.6 27.6 27.6 27.5 27.4

56.5 74.0 76.3 76.1 85.3 88.7 84.9 131.9 84.9 84.9 68.3

1.2 1.5 1.6 1.6 1.8 1.8 1.8 2.7 1.8 1.5 1.4


Marcus

TABLE 3. (Continued) associate ZnCl+ ZnCl+ ZnBr+ ZnI+ CdCl+ CdBr+ CdI+ HgCl+ HgBr+ HgI+

ref 69 67 69 69 69 69 69 69 69 69



Dimethylsulfoxide (Continued) -121.3 70.9 -121.3 70.9 -126.9 75.6 -129.3 78.6 -123.3 73.5 -130.1 78.9 -123.2 82.3 -124.7 74.8 -132.7 81.0 -136.5 84.9

112 122 110 50 41 43 50 140 149 147

TABLE 4: Rotational Entropies of Isolated Polyatomic Ions ion

Si(g)°rot (J K-1 mol-1)

Me4N+ N3NCOSCNNO3ClO3ClO4-

105.9 53.3 60.0 65.9 81.3 88.2 91.4


ion

Si(g)°rot (J K-1 mol-1)

SO42B(OH)4S2O32Fe(CN)63Fe(CN)64Co(NH3)63+

92.2 81.9 95.0 108.8 103.6 181.1

The electrostatic contribution to the entropy, ∆iaSel, depends on the validity of the Born equation beyond the solvation shell. The solvent permittivity near an ion is illustrated for several systems in the author’s publications,71,72 indicating that only beyond a certain distance from the edge of the bare ion the permittivity, , regains rather abruptly its bulk value. Within the solvation sphere, the relative permittivity is nearly constant and its value, ∼2, is assumed to be independent of the temperature, thus having an insignificant entropic contribution. However, the spatial extent of the solvation sphere, here expressed as the parameter d for a given solvent, is assumed to be the same for all ions (and charge asymmetrical associates) in this solvent. The applied “radius”, r( ) (r+r-a)1/3, of the ion associate, for lack of a better estimate, is another possible cause for uncertainty in the electrostatic contribution (for charge asymmetrical associates only). It remains to discuss the analogy, in terms of the entropy effects, between solvent release from solvation shells and the melting of crystalline solvents, employed for obtaining ∆iandesolv from eq 12. Solvent molecules in the frozen crystalline form have internal vibrational degrees of freedom and limited oscillations around their equilibrium positions but are devoid of rotational and translational degrees of freedom. These are liberated on melting, and each solvent molecule moves throughout the volume of the liquid at random and rotates randomly as well. Still, the molecules are confined and restricted somewhat by their neighbors with which they interact. Solvent molecules in the solvation shells of ions are immobilized there in the sense that they move in the company of other solvent molecules and the ion and not individually and their rotation is appreciably hindered. Exchange with solvent molecules from the bulk solvent does not change the total entropy ascribable to the solvent molecules in the solvation shell, but the release of these molecules to the bulk solvent without return does affect this entropy. Once released to the bulk, they move and rotate freely but interact with their neighboring molecules as do solvent molecules on melting. Thus, the molar entropy of fusion is the nearest analog to the entropy change per mole of solvent molecules released from their immobilization in ionic solvation shells.



∆iandesolv

27.5 27.5 27.5 27.4 26.8 26.7 26.6 26.5 26.4 26.4

134.9 144.9 133.8 73.3 64.0 67.5 74.3 163.3 174.2 172.3

2.8 3.0 2.8 1.5 1.3 1.4 1.5 3.4 3.6 3.6

On considering the results of Tables 2 and 3, a few of the entries in the ∆iaSdesolv column are seen to be negative, so that 0 was entered in the ∆iandesolv column. For those cases involving monoatomic ions (NaCl in water23 and LiI and NaI in acetone45 and in acetonitrile48), the reported ∆iaSexptl values are themselves negative (or very small). For NaCl in water, there is good indication that ion association may not take place at all, an insignificant association constant having been deduced by Hanna and Pethybridge,70 contrary to the estimates by De Robertis et al.23 The ∆iaSexptl values reported by the latter have anyway an error limit of (15 J K-1 mol-1, within which positive values of ∆iaSdesolv are obtained. The reliability of the infrared spectroscopic method employed by Perelygin and Klimchuk45,48 for obtaining the entropies of association is questionable. It involves changes in the solvent vibration intensities in the presence of the salts but no direct observation of cation-anion bond vibrations, although contact ion pairs are assumed. Therefore, no great weight should be placed on the entropy values so obtained, as also for the perchlorate salts in acetonitrile.49 For lanthanide bromides in DMF,61 although the reported ∆iaSexptl value is positive, it is rather small when allowance was made also for higher association, being much smaller than that for the lanthanide chlorides62 and for the bromides in the analogous DMA.66 When only the first associate LaBr2+ was assumed to be formed,61 then ∆iaSexptl increased from 43 to 196 J K-1 mol-1 according to the data shown, leading to a positive ∆iaSdesolv value. This would be the case also for the other lanthanide bromides studied in DMF. For cases where one of the associating ions is polyatomic, the reason for the negative calculated ∆iaSdesolv values can very well be that residual rotation of such ions in the ion associate was completely disregarded in the present calculation. Such may be the cases of the thiosulfates and the hexacyanoferrate(III) salts in water. This certainly is the reason for the different results noted for the aqueous divalent metal sulfate associates and the methanolic divalent metal perchlorate associates in the present study and the former one,1 where the entropy of rotation of the anions was allowed for, albeit reduced by one-third from the value of the gaseous ion. If this were done in the present calculations, that is, allowing for partial rotation of the anions in the associate, then 54, 58, 62, 63, 70, and 71 J K-1 mol-1 should be added to the ∆iaSdesolv value of the aqueous nitrates, chlorates, sulfates, thiosulfates, and hexacyanoferrates (II) and (III), respectively, yielding positive values of this quantity in all cases. The resulting ∆iandesolv values are shown in Table 2 in parentheses. For the sulfates, the latter values are in substantial agreement with values calculated recently from the volume changes on ion pairing4 and with the values reported in the previous study of the entropies.1

Solvent Release upon Ion Association The ∆iaSdesolv values of the aqueous hexammine- and tris(ethylenediamine)cobalt(III) outer-sphere complexes in Table 2 appear to indicate that no rotational entropy of the multiatomic cations is lost on the association. For the ethylenediamine complexes (for which no rotational entropy of the isolated cation was provided in ref 5), the present values of ∆iandesolv are in good agreement with those obtained from the volume changes,4 so presumably, the values for the hexammine complexes need no correction for rotation. What is evident from these values is that it is the anions that lose much of their hydration shell when they associate electrostatically with the complex cations, whereas water molecules in the second shell around the central cobalt atom are not appreciably affected by the association. For the divalent metal perchlorate ion pairs in methanol, again, it appears from the data in Table 3 that it is the perchlorate anion that sheds its solvation (1.5 methanol molecules on average4) on association. The small nickel and cobalt cations retain their solvation, but the larger calcium, strontium, barium, and cadmium ions do release some of their solvating methanol molecules. Little can be said about the values of ∆iandesolv shown in Table 3 for ion pairs in the other alcoholic solvents, except that the values appear to increase somewhat as the relative permittivity of the solvent decreases in the series of alcohols. This may mean that a relatively larger proportion of the ion pair formed is of the contact rather than of the solvent-shared type as decreases. 5. Conclusions It may be tempting to interpret the values of ∆iandesolv deduced in the present study as being able to distinguish between innerand outer-sphere complexation. However, even with outersphere association, several solvent molecules may be released from the solvation shells of the participating ions, but more would be released when inner-sphere association takes place. As found on the whole for the nonaqueous solvents, the lower their permittivity, the stronger the electrostatic association and the larger the number of solvent molecules released to the bulk. Positive ∆iaHexptl (g10 kJ mol-1) values combined with positive ∆iaSexptl values (with T∆iaSexptl > ∆iaHexptl, ensuring that association does take place) may be construed as definitely indicating that the association is of the outer-sphere type. Innersphere association would cause a compensation of the desolvation enthalpy by that of the association and hence lead to negative ∆iaHexptl values or only small positive ones. References and Notes (1) Marcus, Y. J. Solution Chem. 1987, 16, 735. (2) Marcus, Y.; Hefter, G. Chem. ReV. 2006, 106, 4585. (3) Marcus, Y. Ion Properties; Dekker: New York, 1997; p 177 ff. (4) Marcus, Y. J. Phys. Chem. B 2005, 109, 18541. (5) Loewenschuss, A.; Marcus, Y. Chem. ReV. 1984, 84, 89. (6) Marcus, Y.; Loewenschuss, A. Annu. Rep. C, R. Soc. Chem. 1984, 1985, 130, 131. (7) Abraham, M. H.; Liszi, J. J. Chem. Soc., Faraday Trans. 1 1978, 74, 1604, 2858. (8) Marcus, Y.; Kamlet, M. J.; Taft, R. W. J. Phys. Chem. 1988, 92, 3613. (9) Marcus, Y. The Properties of SolVents; Wiley: Chichester, U.K., 1998. (10) Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic SolVents, 4th ed.; Wiley: New York, 1986. (11) Smirnova, N. N. Private communication, 2006, Nizhny Novgorod, to be published. (12) Casteel, J. F.; Sears, P. G. J. Chem. Eng. Data 1974, 19, 196. (13) Marcus, Y. J. Phys. Org. Chem. 2003, 16, 398. (14) Clever, H. L.; Westrum, E. F., Jr. J. Phys. Chem. 1970, 74, 1309. (15) Giauque, W. F.; Stout, J. W. J. Am. Chem. Soc. 1936, 58, 1144. (16) Carlson, H. G.; Westrum, E. F., Jr. J. Chem. Phys. 1971, 54, 1464. (17) Parks, G. S. J. Am. Chem. Soc. 1925, 47, 338.

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Solvent Release upon Ion Association from Entropy Data. II - The

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