FUNDAMENTALS - Wiley

PART I

FUNDAMENTALS

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COPYRIG

HTED M

ATERIAL

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3

Ionic Interactions in Natural and Synthetic Macromolecules, First Edition. Edited by Alberto Ciferri and Angelo Perico.© 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

CHAPTER 1

ION PROPERTIES YIZHAK MARCUS

1.1 IONS AS CHARGED PARTICLES

Ions are defi ned as particles that carry electrical charges. Condensed phases, solids and liquids, are electrically neutral; that is, ions exist in them in combina-tions of positively and negatively charged particles — cations and anions, respectively — that may be bound or relatively free to migrate. An electrolyte is a neutral combination of cations and anions that can exist as a chemical substance capable of dissociating into its constituent ions in a suitable environ-ment, for example, in an aqueous solution.

Ions may be monatomic (such as Na + or Br − ), may consist of a few atoms (such as NH4

+ or SO42−) or even considerably more than a few (such as

HOC H CO6 4 2− or (C 4 H 9 ) 4 N + ), or be much larger, consisting of very many atoms.

In some of such cases, they may be referred to as polyions, constituting the dissociated part of polyelectrolytes. Some polypeptides, proteins, nucleic acids, and similar biological moieties, but also suitable synthetic molecules, are exam-ples of polyelectrolytes. Throughout this chapter, a generalized ion is desig-nated by I z ± , but when used as a subscript the charge is dropped, and I symbolizes a quantity pertaining to an ion.

Some substances that are not dissociated in solution, but are capable of donating a hydrogen ion to a basic environment, thus turn into an anion — these are weak acids, such as acetic acid, CH 3 COOH forming CH CO3 2

−. Other substances are able to add on a hydrogen ion in an acidic medium and turn into a cation — these are weak bases, such as aniline, C 6 H 5 NH 2 forming C H NH6 5 3

+. A special category consists of zwitterions: These turn into anions or cations depending on the pH of the medium (hydrogen ion defi ciency or excess), an example being glycine, + H 3 NCH 2 COO − , turning into + H 3 NCH 2 COOH in acidic media and into H NCH CO2 2 2

− in basic ones. The properties discussed

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4 ION PROPERTIES

in this chapter pertain to ions that have been formed either by strong electro-lytes directly on their dissolution or by weak electrolytes in suitable media.

1.1.1 Isolated Ions

Isolated ions may be regarded as existing in an ideal gaseous state, that is, devoid of interactions with other particles or their surroundings in general. Some quite large ions are produced in mass spectrometers, but commonly isolated ions consist of relatively few atoms. They may, however, be the centers of clusters consisting of the ion proper surrounded by a small number of solvent molecules.

The primary characteristics of isolated ions are the amount of electrical charge they carry, their mass, their shape, and their size. The amount of charge is given in terms of a multiple, z I , of the elementary units of the charge of a proton (positive) or an electron (negative), namely e = 1.60218 × 10 − 19 C. Within the scope of this book, the absolute values of z I for isolated ions range from 1 to 4 for monatomic ones and possibly somewhat larges for some complex ions. Highly ionized atoms that may be produced artifi cially or result from nuclear reactions are not considered here.

The mass of isolated ions is generally specifi ed per mole of ions; a mole consists of a very large number of individual particles, Avogadro ’ s number: N A = 6.02214 × 10 23 mol − 1 . The unit of molar mass, M I , is kg mol − 1 , but gener-ally, M I is given in g mol − 1 .

The shape of monatomic ions is, of course, strictly spherical when isolated, but they may be deformed slightly by external forces (strong electrical fi elds). Ions that consist of several atoms may have any shape, but common ones are planar ( NO3

−, CO32−), tetrahedral ( NH4

+, SO42−), octahedral ( Fe CN( )6

4−), elongated (SCN − ), or more irregular ( CH CO3 2

−, HCO3−). The sizes of ions in the isolated

state, however, are diffi cult to specify because the electrons in their periphery extend indefi nitely around the inner electronic shells and the nuclei of the atoms.

An attribute of a generalized isolated ion I with charge z + or z − (I z ± ,g) is its self - energy that is due to its charge. Per mole of isolated ions, this self - energy is

E N z e rzself A 0 II , g /4( ) ,± = 2 2 πε (1.1)

where ε 0 = 8.85419 × 10 − 12 C 2 J − 1 m − 1 is the permittivity of free space, and r I is the radius of the ion. Since, however, as stated above the size of an isolated ion is an ill - defi ned quantity, so must be its radius, hence the self - energy. Still, this concept is employed in the discussion of the hydration of ions in Section 1.3.4 .

Other thermodynamic quantities that pertain to isolated ions, on the other hand, are well defi ned. The standard molar Gibbs energy and the enthalpy of formation, Δ f G ° (I z ± ,g) and Δ f H ° (I z ± ,g), of many ions from the elements in their standard states, and the standard molar entropy and constant - pressure heat

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IONS AS CHARGED PARTICLES 5

capacity, S ° (I z ± ,g) and C P ° (I z ± ,g), all at temperature T = 298.15 K have been reported in the National Bureau of Standards (NBS) tables, 1 with values for additional ions being included in the book by Marcus. 2 The standard molar volume of an isolated ion is a trivial quantity, the same for all ions: V ° (I z ± ,g) = RT/P ° = 0.02479 m 3 mol − 1 at 298.15 K, where R = 8.31451 J K − 1 mol − 1 is the gas constant and P ° is the standard pressure of 100 kPa.

The ionization process leading from an atom, a radical, or a molecule to a cation requires the investment of energy, expressed by the ionization potential, Σ I p . The sum sign, Σ , is used for the ionization potential I p because this process of ionization can proceed in several stages up to the fi nally produced positive ion, the cation. The electron capture by an atom, a radical, or a molecule to form an anion releases energy that is expressed as the electron affi nity, EA , of such a moiety. On the other hand, the capture of an electron by a negative ion, an anion, is a very unlikely event, so that only a single electron may generally be added to a neutral species in the EA process. These energies, in electron volt units (1 eV/particle = 96.483 kJ mol − 1 ), have been reported for many ions in the aforementioned book 2 and are shown in Table 1.1 for some ions.

There are two further relevant properties of ions that depend little, if at all, on whether the ion is isolated in the ideal gas phase or is present in a con-densed phase, a solid or a liquid. These are magnetic susceptibility, χ I , and polarizability α I (or the corresponding molar refractivity, R I ∞ ). Most ions are diamagnetic; that is, they are repulsed out from a magnetic fi eld. Exceptions are ions that have an unpaired electron in their electronic shells: These are paramagnetic. The molar magnetic susceptibilities, χ Im , range from a few to several tens of − 10 − 12 m 3 mol − 1 ; that is, they have the dimensions of molar volumes. For a paramagnetic ion having n unpaired electrons, χ Im = + 1.676 n ( n + 2) × 10 − 9 m 3 mol − 1 at T = 298.15 K. Values of the diamagnetic χ Im for many ions have been reported. 2

The polarizability of an ion also has the dimension of a volume, of the order of 10 − 30 m 3 per ion. The molar refractivity is

R NI A I I4 /3)∞ = = ×( . .π α α2 5227 1024 (1.2)

It is the latter quantity that is estimated experimentally for neutral species, and is additive for the constituting ions if the ions are isolated (in a gas or infi nitely dilute solution). The reported 2 individual ionic values are based on the arbitrary but reasonable value of RD

1Na m mol∞ + − −= ×( ) .0 65 10 6 3 , where RD∞,

the infi nite dilution value obtained from the refractive index at the sodium D line (589 nm), is used in lieu of the infi nite wavelength value R ∞ .

The values of these molar ionic properties for several selected ions in the ideal gas state are shown in Table 1.1 .

1.1.2 Ions in Aqueous Environments

As a thought process, a single ion I z ± may be transferred from the ideal gas phase into water, but this process involves the passage through the water – gas

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TAB

LE

1.1

. P

rope

rtie

s of

Som

e Is

olat

ed I

ons 2

Ion

z

M I

(G m

ol − 1

)

(kJ

mol

− 1 )

(kJ

mol

− 1 )

(J K

− 1 m

ol − 1

)

(J K

− 1 m

ol − 1

)

(eV

)

χ m

( − 10

12 m

3 mol

− 1 )

(10 − 6

m 3 m

ol − 1

)

H +

1 1.

008

1536

.2

1523

.2

108.

9 20

.8

1318

− 6

.6

− 0.1

L

i + 1

6.94

68

5.8

685.

8 11

3.0

20.8

52

6 − 3

0.

08

Na +

1 22

.94

609.

4 58

0.5

148.

0 20

.8

502

2.3

0.65

K

+ 1

39.1

0 51

4.3

487.

3 15

4.6

20.8

42

5 11

.2

2.71

1 18

.04

630

681

186.

3 34

.9

458

11.5

4.

7 M

g 2 +

2 24

.31

2348

.5

2300

.3

148.

7 20

.8

2201

− 0

.7

Ca 2

2 40

.08

1925

.9

1892

.1

154.

9 20

.8

1748

− 1

1.

59

Fe 2 +

2

55.8

5 27

49.9

26

89.6

18

0.3

25.9

23

34

19.6

* 2.

1 Fe

3 +

3 55

.85

5712

.8

5669

.1

174.

0 20

.8

5296

15

.6 *

3.2

Cl −

− 1

35.4

5 − 2

33.1

− 2

41.4

15

4.4

20.8

− 3

49

28

8.63

B

r − − 1

79

.91

− 219

.1

− 245

.1

163.

6 20

.8

− 324

39

12

.24

I − − 1

12

6.91

− 1

97

− 230

.2

169.

4 20

.8

− 295

57

18

.95

SCN

− − 1

58

.08

166.

5 43

.2

− 207

35

17

− 1

62.0

1 − 3

20

− 272

.8

245.

2 44

.7

− 378

23

10

.43

− 1

99

.45

− 344

− 2

66.8

26

3.0

62.0

− 4

54

34

12.7

7

− 1

59.0

4 − 5

04.2

− 4

64.1

27

8.7

61.4

32

6 32

.4

13.8

7

− 1

61.0

2 − 7

38

− 702

25

7.9

50.6

10

.9

− 1

96

.99

− 128

0 − 1

190

286.

7 62

.5

14.6

− 2

60.0

1 − 3

21

− 300

.9

246.

1 44

.4

226

34

11.4

5

− 2

96.0

7 − 7

58

− 704

.8

363.

6 62

.4

1660

40

13

.79

− 2

95

.98

283.

0 67

.8

* P

aram

agne

tic

ion.

Δf

IH

°Δ

fI

G°

S I°

CpI°

IE

Ap/

∑R

D∞

NH

4+

NO

3−

ClO

4−

CH

CO

32−

HC

O3−

HP

O2

4−

CO

32−

SO42−

HP

O4−

6

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interface with not well - defi ned consequences. There are, therefore, very few experimental determinations that can be applied unambiguously to individual ions in aqueous solution.

The more common process that can be carried out experimentally is to dis-solve in water an entire electrolyte (consisting of a matched number of cations and anions to produce a neutral species). Conditions can be chosen for approx-imating infi nite dilution, that is, the dissolution of an infi nitesimal amount of electrolyte in a fi nite amount of water or a mole of electrolyte in a very large amount of water. It is then possible to deal with the molar quantities pertaining to the aqueous electrolyte at infi nite dilution and devise some means to deduce from the measured quantities those pertaining to the individual ions.

At infi nite dilution, the aqueous ions are remote from each other, and hence are surrounded by water only, with which each ion interacts. Therefore, the individual ionic quantities are additive, weighted by their stoichiometric coeffi cients in the electrolyte: ν + cations C z + and ν − anions A z − . For a 1:1 elec-trolyte such as NaCl ν + = ν − = z + = | z − | = 1, for a 2:1 electrolyte such as K 2 SO 4 ν + = | z − | = 2 and z + = ν − = 1. The quantitative aspects of such assignments of individual ionic properties are detailed in the appropriate sections below.

At this place, the interactions of the ions with the surrounding water are dealt with in a qualitative manner. Cations have the water molecules oriented toward them, with one of the lone pairs of electrons, carrying a fractional negative charge, pointing at them. In some cases, this results in a coordinate bond, the fractional charge penetrating unoccupied electronic orbitals of the cations. Transition metal cations, such as Fe 2 + , and also small multivalent cations such as Mg 2 + , tend to form such bonds with water. A defi nite coordina-tion geometry and number of the nearest water molecules, those in the fi rst hydration shell, result: 6 in a regular octahedron for Mg 2 + . These water mole-cules are polarized by the charge of the cation and tend to form such hydrogen bonds with water molecules in a second hydration shell that are somewhat stronger than those prevailing in pure water, so that this second shell remains with the cation as it moves in the solution.

This does not mean that the water molecules in either shell are permanently bound to the cation: On the contrary, there is a dynamic exchange of water molecules with the bulk water outside the hydration shells. For some cations (e.g., Mg 2 + ), though, this exchange is relatively slow — only 10 5.2 exchanges occur per second — compared with 10 8.5 exchanges per second for a cation such as Ca 2 + , also doubly charged, but that is not as strongly bound to the water. The latter cation, as well as the alkali metal ones (except, perhaps, Li + ), does not have a defi nite number of water molecules coordinated to them in their hydration shell. Instead, they have a distribution with a fractional average number, and there is no evidence for a stable second hydration shell. Such ions still affect the hydrogen - bonded structure of the surrounding water and in the cases of large univalent cations (K + , Rb + , and Cs + ) break this structure up to some extent. This is manifested by the faster fl owing ability of the water (lower viscosity) in solutions containing such ions (Section 1.4.3 ).

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8 ION PROPERTIES

Anions in aqueous solutions have the water molecules pointing one (or both in some cases) of their hydrogen atoms toward them, resulting in hydro-gen bonds. Anions tend to be larger and have relatively smaller electric fi elds than common cations and have no defi nite coordination number of water molecules hydrating them but rather a distribution of values with an average that may be rather small for singly charged anions ( < 2). Multivalent anions, especially oxyanions, such as CO3

2− or SO42− bind more water molecules by

accepting hydrogen bonds from them, and anions such as HSO4− or H PO2 4

− also donate hydrogen bonds to adjacent water molecules. Large univalent anions, such as Cl − and ClO4

− break the water structure around them and lower the viscosity of the solutions.

Ions with hydrophobic groups in their periphery around a buried charge, such as (C 6 H 5 ) 4 B − or (C 4 H 9 ) 4 N + , are generally only poorly hydrated, but they enhance the water structure, in the manner characteristic of pure water (tet-rahedrally oriented hydrogen bonds), as do uncharged organic molecules with similar hydrophobic groups. Ions containing hydrophobic parts, such as C H CO6 5 2

− and CH NH3 3+, arrange water molecules differently around such

parts, − C 6 H 5 and − CH 3 , compared with the arrangement near the hydrophilic part, − −CO2 or − +NH3. This behavior is even more characteristic of ionic sur-factants, such as C H SO12 25 3

− or ( )C H NH8 17 2 2+. The long organic chains tend to

associate in aqueous solutions forming micelles or other structures, the hydro-phobic “ tails ” being removed from immediate contact with the water into which the hydrophilic “ heads ” stick out.

1.1.3 Ions in Other Environments

Although water is the most important solvent, in general and within the context of this chapter, ions do exist also in other environments — in nonaque-ous and mixed solvents, and in condensed phases without any solvents — as molten salts. Some of the properties of ions in nonaqueous and mixed solvents are discussed in Section 1.5 .

Molten salts are conventionally classed as high - melting ones (examples being molten NaCl and NH 4 NO 3 ) and room temperature ionic liquid s ( RTIL s; examples being pyridinium tetrafl uoroborate and 1 - methyl - 3 - butylimidazolium bis(trifl uoromethane - sulfonyl)imide). The high - melting salts (the melting tem-peratures are 1074 K for NaCl and 443 K for NH 4 NO 3 ) are probably not relevant in the context of this chapter. RTILs, or more generally ionic liquid s ( IL s) that have melting temperatures ≤ 373 K, may have some relevance. They are characterized by negligible vapor pressures, relatively high viscosities, but adequate electrical conductivities, and are in vogue in recent years as being friendly to the environment contrary to some solvents that have been in use in industry. Their properties are not discussed further here.

1.1.4 Charged Macromolecules

This introduction to charged macromolecules is only cursory, the subject being fully treated in Chapters 3 , 4 , and 6 . Polymers that carry ionically dissociable

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groups are called polyelectrolytes, ionizing partly or completely to polyions. Synthetic polyelectrolytes generally have a linear polymeric chain, the hydro-phobic backbone, to which side groups with hydrophilic ionizable groups are attached. When these are ionized, these covalently attached ions are the “ fi xed ions ” to which correspond mobile “ counterions ” of opposite charge. An example is polystyrene sulfonic acid − CH 2 CH(C 6 H 5 SO 3 H) − , a strong, practi-cally completely ionized polyelectrolyte. On the other hand, a weak polyelec-trolyte, polymethacrylic acid − CH 2 C(CH 3 )(CO 2 H) − , is only partly ionized, depending on the pH of the solution. A polyelectrolyte that has the charged sites as parts of the polymer chain itself is polyethyleneimine ( - immonium) ( − −+C H NH C H2 4 2 2 4 ) with its counterions. Such synthetic linear polyelectro-lytes have a regularly repeating structure. They may be cross - linked to form water - insoluble materials that can exchange the counterions with other ones in the imbibed or adjacent solution: ion exchange resins, gels, and membranes. Biological examples of polyelectrolytes are proteins and nucleic acids that share many features with synthetic polyelectrolytes, but they do not have the regular repeating structures of the latter.

The special features exhibited by polyelectrolytes are due to the proximity along the polymeric chain (the backbone) of charges of the same kind that affect (repulse) one another. Even if the polyelectrolyte itself is at high dilution in the solution, the fi xed ions attached to it can never be said to be at infi nite dilution. The pH and ionic strength of the solution in which the polyelectrolyte is situated may affect its expansion or contraction, evidenced by changes in the viscosity of the solution. In the case of strong fl exible polyelectrolytes, the end - to - end distance of the chain is shortened on increasing concentration due to self - screening of the charges. Furthermore, counterions with charges opposite to those of the fi xed ions tend to associate with them to some extent (see Chapters 3 and 6 ). If ions or molecules with hydrophobic groups are present in the solution, these tend to associate with the hydrophobic backbone or hydro-phobic parts of the side groups carrying the charges. In particular, such associa-tion becomes important when surfactants are present in the solution. The solubility of polyelectrolytes in water is affected by cosolvents that have poor solvation power for the charges; this may lead to micellization or precipitation of the polyelectyrolyte. Such effects are fully discussed in Chapters 3 , 4 , and 6 .

Polyelectrolytes are characterized by being linear or branched (unless cross - linked), by their degree of polymerization, that is, their average molar mass ( MW ) and the dispersion of this average, the lengths, l , along the back-bone of the segments that carry the charged groups, and the degree of ioniza-tion, 0 ≤ α ≤ 1.

Rigid polyelectrolytes are characterized by large values of the intrinsic persistent length, q in , measuring their conformational rigidity (for instance, in DNA). This quantity should be many - fold times that of l , the distance between charges. Flexible polyelectrolytes have instead low values of q in and are extended in order to reduce the coulombic repulsion between the charges. Moreover, conformational changes occur by the screening actions of counter-ions and co - ions, if present.

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10 ION PROPERTIES

A highly ionized ( α → 1) linear polyion in aqueous solution may be approx-imated by a charged rod with a cylindrical sheath of water of hydration. As α is decreased, the hydration water envelopes overlap only partly, down to low degrees of ionization ( α → 0) where the charged sites are hydrated individu-ally. However, the hydrophobic part of the polyion causes the arrangement of water molecules near the polymer backbone to differ from that near the ionic groups. In addition, counterions attracted electrostatically to the fi xed ions bring their own water of hydration with them, and if ion pairs are formed, the hydration shells of fi xed and counterions overlap. Thus, a complicated picture of the hydration of polyelectrolytes emerges, depending on the fl exibility of the polymeric backbone, the segment length l , the extent of ionization α , the nature of the fi xed ions and of the counterions, and on the presence or absence of additional electrolytes in the solution containing or in contact with the polyelectrolyte.

1.2 SIZES OF IONS

1.2.1 Ionic Radii in Crystals

Diffraction of X - rays and neutrons in crystals has been used for the determina-tion of the interatomic distances, or in the case of ionic crystals, the interionic distances. The interionic distances could be measured with modern technology with an accuracy of ± 0.0001 nm. They are fairly regularly additive in the sizes of the ions, so that radii may be assigned to monatomic ions and to polyatomic ones having a globular shape: tetrahedral, octahedral, or cubic. The manner of splitting the interionic distances into ionic radii is not straightforward, however.

For monatomic ions, over many years, the suggested Goldschmidt radii (based on the refractivity of ions in crystals leading to r (F − ) = 0.133 nm and r (O 2 − ) = 0.132 nm) were in substantial agreement with the Pauling radii (based on the screening of effective nuclear charges by the electronic shells, leading to r (F − ) = 0.136 nm and r (O 2 − ) = 0.140 nm). Subsequently, a completely differ-ent set was suggested by Gourary and Adrian, based on the loci of minima in the electron density between the ions, leading to r (F − ) = 0.116 nm, hence much larger cation radii than those of Goldschmidt and Pauling. Ionic radii r I depend somewhat on the temperature, 3 but this dependence is of no consequence for temperatures in aqueous solutions under ambient pressure and may play a role only in high - melting crystalline or molten salts. The dependence on the coordination number and geometry in the crystals was established in detail by Shannon and Prewitt, 4 modifying slightly the Pauling set of radii. Marcus 2 summarized the reported values and provided a “ selected ” set of ionic radii, r I , for the most commonly observed coordination numbers that correspond with the hydration numbers of ions in aqueous solutions.

Kapustinskii obtained interionic distances for crystals containing poly-atomic ions from the dependence of thermochemically derived lattice energies

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SIZES OF IONS 11

of the crystals on these distances. A set of thermochemical radii was estab-lished for polyatomic ions by Jenkins and Thakur 5 from these interionic dis-tances in crystals with monatomic counterions using the Goldschmidt radii of the latter. As mentioned above, such radii make sense for globular polyatomic ions, and a “ selected ” set was provided by Marcus. 2 For polyatomic ions deviat-ing strongly from a globular shape, the concept of a radius is ambiguous. When oblate (disklike) ions such as NO3

− or prolate (rodlike) ions such as SCN − are considered, then the half - axes a and b of the ellipsoids of revolution may be used for estimation of the volume: (4 π /3) a 2 b and (4 π /3) ab 2 with a > b , respec-tively. Caution should be used with these values when considering the distance to which other particles (solvent molecules or other ions) can approach such nonglobular ions.

1.2.2 Ionic Radii in Aqueous Solutions

Diffraction methods have been used also on aqueous salt solutions for estab-lishing the distances d I - O between the centers of ions and those of the oxygen atoms of adjacent water molecules, as summarized by Marcus 6 and by Ohtaki and Radnai. 7 It was found that if the radius of a water molecule, r W = 0.138 nm, is deducted from the d I - O values, the results correspond quite well with the set of Pauling radii of the ions. 8 The values of d I - O reported by various authors and pertaining to different concentrations have a mean uncertainty of ± 0.002 nm, much worse than the values in crystals. Still, the “ selected ” ionic radii 2 r I add up with r W ( = 0.138 nm) within this uncertainty to the d I - O values, suggesting their use for the radii of aqueous ions, as listed in Table 1.2 . The electrostatic forces acting on ions in condensed phases, crystals, and solutions are of similar magnitudes, as appear from the relevant energetics: lattice energies and ener-gies of solvation. Hence, the use of the r I derived from measurements on crystals for the radii of ions in solution is reasonable.

The concept of a constant radius for the water molecule, r W , was challenged, however, by David and Fourest. 9 They claimed that the electric fi eld of the ions polarizes the water molecules adjacent to them and, for multiply charged ions, squeezes these molecules somewhat in the direction of the ions. They suggested that r W decreases from 0.143 nm for the alkali metal cations down to 0.133 nm for the trivalent lanthanide cations. The use of the mean value, r W = 0.138 nm, increases the uncertainty of the ionic radii in solution from ± 0.002 to ± 0.005 nm according to these authors.

1.2.3 Ionic Volumes in Solution

Henceforth, the solvent (water) is designated by subscript 1 (or by subscript W as above for the radius) and the solute electrolyte by subscript 2 , as is generally the practice. The individual ionic contributions to a (molar) property of the electrolyte are designated by subscript I . Properties that pertain to infi nite

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TAB

LE

1.2

. P

rope

rtie

s of

Som

e Io

ns in

Aqu

eous

Sol

utio

ns a

t 29

8.15

K 2

Ion

r a (n

m)

(c

m 3 m

ol − 1

)

b

(cm

3 mol

− 1 )

b

h I

com

p∞

b

Δ

hydr

IH

∞

(kJ

mol

− 1 )

S I∞

(J K

− 1 m

ol − 1

)

Δ

hydr

IG

∞

(kJ

mol

− 1 )

CP

I∞

(J K

− 1 m

ol − 1

)

H +

0 − 5

.4

− 110

3 − 2

2.2

− 105

6 − 7

1 L

i + 0.

069

− 6.4

− 1

2.9

4.4

4.0

− 531

− 8

.8

− 481

− 9

N

a + 0.

102

− 6.7

− 8

.6

2.9

4.5

− 416

36

.8

− 375

− 2

8 K

+ 0.

138

3.5

− 5.9

2.

0 3.

5 − 3

34

80.3

− 3

04

− 58

0.

148

12.4

− 3

29

74.7

− 2

92

− 1

Mg 2 +

0.

072

− 32.

2 − 5

2.5

11.4

c 10

.0

− 194

9 − 1

82.5

− 1

838

− 158

C

a 2 0.

100

− 28.

9 − 3

8.5

10.0

c 9.

0 − 1

602

− 97.

5 − 1

515

− 169

Fe

2 +

0.07

8 − 3

4.4

(11.

8) c

− 197

2 − 1

82.1

− 1

848

− 188

Fe

3 +

0.06

5 − 6

0.2

(21.

1) c

− 446

2 − 3

82.5

− 4

271

− 204

C

l − 0.

181

23.3

− 4

.0

1.4

2.0

− 367

78

.7

− 347

− 5

6 B

r − 0.

196

30.2

− 3

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1.2

1.8

− 336

10

4.6

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0.22

0 41

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0 1.

5 − 2

91

133.

5 − 2

83

− 50

SCN

− (0

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0.5 c

0.1 b

− 311

16

6.5

− 287

42

(0.2

00)

34.5

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4 c 0.

7 b − 3

12

168.

8 − 3

06

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240

49.6

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20

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32)

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25

108.

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73

97

(0

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12

0.6

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00)

34.6

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22

114.

7 − 4

73

37

(0

.178

) 6.

7

(1

0.9)

c 14

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− 139

7 0.

9 − 1

315 d

− 159

0.23

0 25

.0

− 13.

8 4.

6 10

.4 b

− 113

8 d 63

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− 109

0 − 1

38

(0

.200

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.7

10.9

− 1

02

a Val

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in p

aren

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re a

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VIe

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elec

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12

c01.indd 12c01.indd 12 12/15/2011 11:16:49 AM12/15/2011 11:16:49 AM

SIZES OF IONS 13

dilution of the solute are designated by superscript ∞ and those of the pure solvent by superscript * .

The volumes that are to be assigned to ions in aqueous solutions are related to the concentration - dependent densities, ρ , of the solutions at constant tem-perature and pressure. Consider a solution made up from n 1 moles of water and n 2 moles of electrolyte. The apparent molar volume of the latter, ϕV V n V n2 1 2= −( )1

* / , is, per mole of electrolyte, that part out of V , the total volume of the solution, remaining for the electrolyte once the volume n V1 1

* (that the water would have had if there were no effect due to the ions) is subtracted. In a solution of density ρ made up from n 1 = 55.51 moles of water (1 kg) and n 2 = m 2 moles of electrolyte (i.e., at a molality m 2 ), the apparent molar volume is obtained from the densities as ϕ ρ ρ ρ ρρV M m2 2 1 21000= + −/ /1

*( ) * , M 2 being the molar mass of the solute and ρ1

* the density of pure water. This apparent molar volume of the solute is not necessarily the actual

volume that should be assigned to the electrolyte because the water near ions does not have the same molar volume as pure water has. The water near the ions is compressed, electrostricted, by the electrical fi elds of the ions. The volume to be assigned to the ions is the partial molar volume, which for a solution of molality m 2 is

V V m V m T2 2 2 2= + ∂ ∂ϕ ϕ( / )2 . (1.3)

As the molality diminishes on approaching infi nite dilution, the second term on the rhs diminishes too, so that on extrapolation to infi nite dilution φ V 2 becomes equal to the standard partial molar volume of the electrolyte: ϕV V2 2

∞ ∞= . At infi nite dilution, the contributions of the cations and anions are additive

as mentioned in Section 1.1.2 , but the way to split the measured V2∞ to the

contributions from the individual ions, V ∞ (I z ± ,aq), requires the knowledge of the value for just one ion, from which those of other ions are then derived. The value V ∞ (H + ,aq) = − 5.4 cm 3 mol − 1 at 298.15 K has been accepted as a reasonable assumption, and the derived values for some other ions are shown in Table 1.2 . In view of the steps that have led to this assumption, the ionic values have uncertainties of at least ± 0.2 z I cm 3 mol − 1 , increasing with the ionic charges. Note that for some cations, and in particular for multivalent ones, the values of V ∞ (I z ± ,aq) are negative. Such cations produce a large diminution of the volume of the water surrounding them, or in other words cause a large electrostriction.

Most authors have dealt with the standard partial molar volumes of ions, VI

∞, that is, the infi nite dilution values. The values of the ionic V I at fi nite con-centrations are not known accurately because the additivity of the individual volumes breaks down due to interionic interactions. The apparent molar volumes of electrolytes can be expressed as ϕ ϕV V S c bc2

21 2= + +∞

V/ , where c is

the molar concentration (in mol dm − 3 ), S V is the theoretical slope of the square root term (Debye – H ü ckel theory; 1.85 dm 3/2 · mol − 1/2 at 298.15 K), and b is an empirical parameter specifi c for each electrolyte.

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14 ION PROPERTIES

The standard partial molar volume of an ion in aqueous solution, VI∞, can

be construed as being the sum of its intrinsic volume, VIintr∞ , and the electrostric-

tion that the ion has caused in the water around it, VIelec∞ , the latter being a

negative quantity. Attempts have been made to estimate each of these quanti-ties independently, but so far the results are not very conclusive.

The intrinsic volume of an ion in a solution is a rather nebulous concept, but it should be independent of the concentration and, substantially, also of the temperature. If the molar volume of the “ bare ” unhydrated ion, ( )4 3 3πN rA I/ , is considered to represent VIintr

∞ , then a much too small negative or even a posi-tive quantity must be assigned to VIelec

∞ in order to make up the measured VI∞.

Thus, the radius of the “ bare ” ion should be modifi ed (enlarged) for it to be useful in this respect. This enlargement has been supposed to represent the void spaces between the water molecules and the ion and among themselves. A factor of k = 1.23 was proposed 10 for the alkali metal and the halide ions, producing

V N krIintr A I/∞ = ( )( ) ,4 3 3π (1.4)

a more reasonable estimate, in fair agreement with other ones. These, for V 2intr , are the sums of the cation and anion values, such as the extrapolated molar volumes of molten alkali metal halides down to room temperature or the limits of V 2 at very high concentrations, where all the water present is already com-pletely compressed, so that V 2elec → 0. 11

The electrostriction caused by an ion has been estimated on the basis of the electrostatic effects the very high electric fi eld of an ion has on the dielec-tric medium, the water with which the ion is surrounded. One consequence of this fi eld is a large pressure exerted on the water; the other is the sharp decrease of the dielectric permittivity of the latter, down to dielectric satura-tion. Such calculations are complicated due to the mutual dependences of the pressure and the compressibility on the one hand, and of the fi eld strength and the permittivity on the other. Values of VIelec

∞ have recently 12 been calculated for the alkali metal and alkaline earth metal cations and the halide, perchlo-rate, and sulfate anions for aqueous solutions at fi ve temperatures between 273.15 and 373.15 K, with those for 298.15 K being shown in Table 1.2 . The corresponding intrinsic volumes are V V VIintr I Ielec

∞ ∞ ∞= − , and of course, all are positive.

1.3 THERMODYNAMICS OF AQUEOUS IONS

Ions in aqueous solutions are characterized by several thermodynamic quanti-ties, in addition to the volumes discussed above. Some of these quantities are the molar heat capacities (at constant pressure) and entropies; others are the molar changes of enthalpy or Gibbs energy on the transfer of an ion from its isolated state in the ideal gas to the aqueous solution. The latter quantities

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THERMODYNAMICS OF AQUEOUS IONS 15

pertain also to the dissolution of an electrolyte in water, being parts in a ther-modynamic cycle describing the transfer of the electrolyte itself to the gas phase, its dissociation there into its constituent ions that are then transferred into the solution. In all these measures and processes, it is not possible to deal experimentally (contrary to thought processes) with individual ions but always only with entire electrolytes or with such combinations (sums or dif-ferences) of ions that are neutral. The assignment of values to individual ions requires the splitting of the electrolyte values by some extrathermodynamic assumption that cannot be proved or disproved within the framework of thermodynamics. 2

1.3.1 Molar Heat Capacities of Aqueous Ions

When the heat of solution of an electrolyte in water to form a dilute solution is measured calorimetrically at several temperatures, the standard partial molar (constant pressure) heat capacity of the electrolyte, CP2

∞ , is obtained from the temperature coeffi cient of heats of dilution, extrapolated to infi nite dilution. Alternatively, the difference between the specifi c heat of a dilute solution of the electrolyte and that of water is obtained by fl ow microcalorimetry to yield the same quantity. Such determinations are accurate to ± 1 to ± 3 J K − 1 mol − 1 .

As mentioned above, it is necessary to assume a value for one ion in order to obtain the so - called “ absolute ” standard molar ionic heat capacities, CPI

∞ , these values being additive at infi nite dilution. The value 1 114 J K mol− − at 298.15 K has been suggested 13 on equating the standard molar heat capacities of aqueous tetraphenylphosphonium and tetraphenylborate ions (the TPTB assumption). These ions should have nearly the same value due to their chemical similarity and similar sizes and the charges of opposite sign being buried well inside the tetraphenyl structure. 14 Unfortunately, the CPI

∞ of these bulky ions are large, hence the uncertainty involved in equating them, due to slight differences in the sizes and induced partial charges in the phenyl rings, is also large. However, no more satisfactory method for splitting CP2

∞ into the CPI

∞ of the constituent ions was found. Values of the latter are shown in Table 1.2 . The values of multicharged ions are seen to be large and negative, and those of polyatomic ions are more positive (or less negative) than those of monatomic ions of the same charge class.

1.3.2 Molar Entropies of Aqueous Ions and Their Entropies of Hydration

The standard molar entropies of aqueous electrolytes, S2∞, are obtained from

the temperature coeffi cients of the electromotive forces of galvanic cells or of the solubilities of sparingly soluble salts. The values for individual ions need an assumption concerning the value of one ion, as in the cases of the standard molar volumes and heat capacities. The value chosen is S ∞ (H + ,aq) = − 22.2 ± 1.4 J K − 1 mol − 1 at 298.15 K, based on data for thermocells. 15 The derived values

CP H aq 71∞ + = − ±( , )

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16 ION PROPERTIES

for other ions, so - called “ absolute ” standard molar ionic entropies, are shown in Table 1.2 . The values are seen to increase with the masses of the ions but to be small or negative for multicharged ions.

The standard molar entropy of hydration of an ion is Δhydr I I IS S S∞ ∞= − °, the difference between its standard molar entropy in the aqueous solution (Table 1.2 ) and the standard molar entropy of the isolated ion in the ideal gas phase (Table 1.1 ). These standard molar entropies of hydration of ions are related to the effect that ions have on the structure of water. Various schemes of expressing this relationship have been proposed, 16 the main point being the subtraction from Δhydr IS∞ of the entropic effect of a neutral molecule of a similar size as the ion has and of the electrostatic effects of the electric fi eld of the ion in terms of the entropy. The latter are obtained from the temperature coeffi cient of the Born expression for the Gibbs energy of hydration (Section 1.3.4 ). The remainder then expresses the effect of the ion on the water struc-ture: If positive, the ion is said to be structure - breaking; if negative, it is structure - making. To the former category belong large univalent ions (e.g., K + , I − , ClO4

− ) and to the latter small or multivalent ions (e.g., Li + , Mg 2 + , CO32−),

while others are borderline in this respect (Section 1.4.3 ).

1.3.3 Enthalpies of Hydration of Ions

When an ion is transferred, in a thought process, from its isolated state in the ideal gas phase into water at infi nite dilution, a large amount of energy is released due to the interaction of the ion with the surrounding water. Its molar volume is compressed in this process from RT/P ° to the standard partial molar volume in the solution. The net relevant energetic amount is the change in the enthalpy, Δhydr IH ∞, but this cannot be determined experimentally for individual ions. The values for complete electrolytes, ΔhydrH2

∞, are obtained from their heats of solution and lattice energies, yielding the differences between the standard molar enthalpy of formation of the infi nitely dilute aqueous solute 1 and the sum of the standard molar enthalpies of formation of the ideal gaseous ions (Table 1.1 ).

A value must be estimated for one ion in order to split the experimentally available values dealing with entire electrolytes into the ionic contribution. The value Δ hydr H ∞ (H + ,aq) = − 1103 ± 7 kJ mol − 1 results from equating the stan-dard enthalpies of hydration of the tetraphenyphosphonium and tetraphenyl-borate ions (the TPTB assumption). The choice of these ions has been briefl y discussed in Section 1.3.1 , but contrary to the case of the heat capacities, the values of Δhydr IH ∞ for these bulky ions are small compared with those of small ions, so that the uncertainty involved in equating the values for these reference ions is also small. This estimate for the hydrogen ion is compatible with several other reliable values suggested on the basis of other considerations, ranging from − 1091 ± 10 to − 1104 ± 17 kJ mol − 1 . The values for a number of ions are shown in Table 1.2 and are expected to be accurate to within ± 7 z I kJ mol − 1 . 2 The values are all negative, as expected (heat is released), of similar magnitude

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THERMODYNAMICS OF AQUEOUS IONS 17

for singly charged ions, whether cations or anions (though becoming less nega-tive with increasing sizes) but becoming considerably more negative for mul-ticharged ions, by a factor of the order of z 2 .

On departing from infi nite dilution, molar enthalpies of hydration of elec-trolytes may be estimated by adding to the sum of the cation and anion values of Δhydr IH ∞ the relative partial molar heat content of the solute, L 2 , equal and of opposite sign to the experimentally measurable 17,18 enthalpy of dilution of the electrolyte, Δ dil H 2 . At fi nite concentrations, the heat content and the enthalpy of hydration may therefore be smaller or larger than at infi nite dilu-tion, depending on the enthalpies involved in the interactions between neigh-boring ions. These are obtainable from the temperature derivatives of the activity coeffi cients

L vRT T P m22= − ∂ ∂±( ln ) ,,γ / (1.5)

where v is the number of ions in a formula of the solute electrolyte.

1.3.4 Gibbs Energies of Hydration of Ions

The standard molar ionic Gibbs energy of hydration Δhydr IG∞ has traditionally been estimated from the Born expression, resulting from the following ideal-ized process. Consider an isolated ion in the gaseous phase, I z ± (g), that is dis-charged, producing a neutral particle. The electric self energy E self (I z ± ,g) (Section 1.1.1 ) must be provided for this process. The neutral particle is then transferred into the bulk of liquid water, there being no electric energetic component for crossing the gas/liquid boundary. The neutral particle is then charged up to the original value, producing the infi nitely dilute aqueous ion, I z ± ,aq ∞ . The energy of interaction with the surrounding water is thereby released. This depends on the permittivity of the water ε πε ε*

0 r4= *, where εr* is

the temperature - and pressure - dependent relative permittivity ( εr* .= 78 4 at

298.15 K and ambient pressure). The net effect of this idealized process rep-resenting the hydration of the ion is

Δhydr I A I I r*/4 1/G N e z r∞ −= −( ) ( ).2

02 1 1πε ε (1.6)

The problem with this mode of calculation is the use of the same value of the radius r I for the ion in the aqueous solution and the isolated state and the use of the relative permittivity of pure water for the description of the interaction of the ion with its immediate surroundings, where dielectric saturation, due to the high electric fi eld of the ion, occurs. Various schemes have been proposed to counter this problem, such as adding a quantity Δ r to the ionic radius and/or splitting the process into two spatial regions: one adjacent to the ion, where dielectric saturation occurs and εr D

* ≈ n2 (the square of the refractive index), and the other beyond this, where the bulk value εr

* prevails. The use of such devices permits the estimation of reasonably correct Δhydr IG∞ values. 19

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18 ION PROPERTIES

Such values of the standard molar Gibbs energy of hydration, Δhydr IG∞, must be thermodynamically consistent with the combination of the standard molar enthalpy and entropy of hydration:

Δ Δ Δhydr I hydr I hydr IG H T S∞ ∞ ∞= − . (1.7)

The value for the hydrogen ion, Δ hydr G ∞ (H + ,aq) = − 1056 ± 6 kJ mol − 1 is thus consistent 19 with the values of Δ hydr H ∞ (H + ,aq) and S ∞ (H + ,aq) given above and S ° (H + ,g) in Table 1.1 as required. It is also compatible with the estimate 15 − 1066 ± 17 kJ mol − 1 but not with − 1113 ± 8 kJ mol − 1 obtained from the cluster pair approximation used by Kelly et al. 20 As discussed by Marcus, 21 the assump-tions involved in the latter value lead to a surface potential of water of Δ χ = 0.34 ± 0.08 V, which in turn, is not consistent with the recent estimate 22 of Δ χ = 0.1 V deemed to be the most nearly correct one.

So - called “ real ” standard molar Gibbs energies of hydration are obtained from the electromotive force of cells consisting of a downward fl owing jet of aqueous solution and a solution along the surface of a tube concentric with the jet, with a small air (vapor) gap between them. The “ real ” Δhydr I

RG∞ differ from the thermodynamic ones Δhydr IG∞ by z I F Δ χ , where the algebraic value of the ionic charge z I is to be used and where F = 96485.3 C mol − 1 is Faraday ’ s constant. The uncertainties connected with the value of Δ χ make the use of the measurable “ real ” standard molar Gibbs energies of hydration unattract-ive for obtaining individual ionic values for the desired quantity, Δhydr IG∞.

1.4 ION TRANSPORT

Ions in solution move around spontaneously due to their thermal energy. The speed of their movements is a quantity that can be determined experimentally for individual ions, contrary to the thermodynamic quantities dealt with in Section 1.3 . The ions may carry with them some of their hydration shells, depending on how strong the bonding between the ion and the water of hydra-tion is. The movement of ions depends on the presence or absence of fi elds, that is, gradients in the forces that cause the ions to migrate. An external fi eld could be a pressure gradient, causing the fl ow of the solution as a whole. It could be an electrical fi eld, causing ions of opposite charges to move in oppo-site directions. A directional concentration gradient at fi nite concentration causes directional diffusion of ions. The inherent movement of ions in the absence of a fi eld is their self - diffusion and can occur at infi nite dilution or at fi nite ones in a homogeneous solution.

1.4.1 Self - Diffusion of Ions

The rate of self - diffusion of ions is commonly obtained from other transport quantities, such as the conductivity, but can as well be determined by the use of isotopic labeling of the ions. For this purpose, a diaphragm cell with equal

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ION TRANSPORT 19

concentrations of the electrolyte in the two stirred compartments is employed, in one of which ions of one kind are labeled by a radioactive tracer. Other methods of measuring diffusion of ions, for example, by NMR with nonradio-active isotopes, have also been used. The rates of migration of the labeled ion truly measures self - diffusion at the nominal concentrations employed. When these are extrapolated to infi nite dilution, the corresponding value of the limit-ing diffusion coeffi cient, DI

∞, is obtained. The values are of the order of 10 − 9 m 2 s − 1 .

Values of the limiting self - diffusion coeffi cients of the ions dealt with in this chapter at 298.15 K are shown in Table 1.3 . It is seen that the more strongly hydrated an ion is, the lower is its rate of self - diffusion, an exception being the hydrogen ion. This ion does not diffuse in water by massive movement of the ion carrying its hydration shell but by the Grotthuss mechanism of the positive charge hopping from one water molecule to the next, and hence is much faster.

1.4.2 Ionic Conductivities

The most characteristic properties of ions are their abilities to move in the direction of an electrical fi eld gradient imposed externally, cations toward the

TABLE 1.3. Transport Properties of Some Ions at 298.15 K 2

Ion DI∞

(10 − 9 m 2 s − 1 ) λI

∞ (cm 2 Ω − 1 mol − 1 )

r ISt (nm)

B I (dm 3 mol − 1 )

H + 9.311 349.8 0.026 0.068 Li + 1.029 38.7 0.238 0.146 Na + 1.334 50.1 0.184 0.085 K + 1.957 73.5 0.125 − 0.009 NH4

+ 1.958 a 73.6 0.125 − 0.008 Mg 2 + 0.706 106.1 0.174 0.385 Ca 2 0.792 119.0 0.155 0.298 Fe 2 + 0.719 107 0.172 0.42 Fe 3 + 0.604 204 0.136 0.69 Cl − 2.032 76.4 0.121 − 0.005 Br − 2.08 78.1 0.118 − 0.033 I − 2.045 76.8 0.120 − 0.073 SCN − 1.758 66 0.142 − 0.022 NO3

− 1.902 71.5 0.129 − 0.045 ClO4

− 1.792 67.4 0.137 − 0.058 CH CO3 2

− 1.089 40.9 0.225 0.246 HCO3

− 1.185 44.5 0.207 0.13 H PO2 4

− 0.879 33 0.279 0.34 0.923 138.6 0.133 0.294 1.065 160 0.115 0.206 HPO4

− 0.439 66 0.279 0.382

a Calculated from the molar conductivity.

CO32−

SO42−

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20 ION PROPERTIES

negatively charged cathode, anions in the opposite direction. The rate of move-ment of ions in an electric fi eld is governed by their mobilities uI

∞, measuring their speed in m s − 1 at unit fi eld, 1 V m − 1 .

When an external electric fi eld is imposed on an electrolyte solution by electrodes dipped into the solution, the solution conducts a current. The inten-sity of the current is proportional to the potential difference between the electrodes, the proportionality coeffi cient being the resistance of the solution. This resistance, or its reciprocal, the conductivity of an electrolyte solution, is readily measured accurately with an alternating potential at a rate of ∼ 1 kHz in a virtually open circuit in order to avoid electrolysis at the electrodes. The molar conductivity Λ 2 can be extrapolated to infi nite dilution by an appropri-ate theoretical expression (see below), yielding Λ2

∞. This quantity can be split into the ionic contributions, the limiting molar ionic conductivities λI

∞, by using experimentally measured (and extrapolated to infi nite dilution) transport numbers, t+

∞ and t t−∞

+∞= −1 . For a binary electrolyte, Λ2

∞+∞

−∞= +λ λ and λ+

∞+∞ ∞= ⋅t Λ2 .

Values of the limiting ionic molar conductivities in water at 298.15 K are shown in Table 1.3 . Molar ionic conductivities, λI

∞, are | z | time the formerly widely used equivalent conductivities, and their commonly used units are S cm 2 mol − 1 (S = Ω − 1 ). For many ions, the λI

∞ are accurate to ± 0.01 S cm 2 mol − 1 . The mobilities, uI

∞, of ions at unit electric fi eld gradient are directly propor-tional to the limiting ionic molar conductivities:

u z FI I /∞ ∞= λ , (1.8)

as are also the self - diffusion coeffi cients:

D RT z FI I /∞ ∞= λ 2 2. (1.9)

In fact, the latter have been obtained for most ions from the conductivities rather than from isotope labeling. Ion mobilities (hence molar conductivities and self - diffusion coeffi cients) increase with increasing temperatures and a fi vefold increase in Λ2

∞, the limiting molar conductivity of many electrolytes, has been reported between 273 and 373 K. This is mainly due of the corre-sponding decrease of the viscosity of the solvent. However, the transference numbers are also temperature - sensitive, though only mildly.

The mobility of an ion and hence its electric conductivity depends on its size and on the viscosity of the solvent, ηW

* , for aqueous solutions. According to Nernst, Stokes, and Einstein, a quantity called the Stokes radius may be assigned to an ion:

r F NISt A I W*

I/ z /= ∞( ) .2 6π η λ (1.10)

The parameter 6 in the denominator arises from the assumption of perfect slipping of the hydrated ion in the aqueous environment; otherwise, for perfect sticking, the parameter would be 4. Ionic Stokes radii, shown in Table 1.3 , are

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ION TRANSPORT 21

of the same order of magnitude as the ionic radii r I measuring the sizes of ions in crystals and solutions, shown in Table 1.2 , but are not directly related to them, except for large tetraalkylammonium ions. In fact, in many cases, the Stokes radii are smaller than the crystal ionic radii, although they are supposed to pertain to the hydrated ions and ought to be larger than the latter. It is interesting to note that the Stokes radii of ions are not sensitive to the solvents in which they are dissolved, as the products η λ1

*I∞ of the viscosities of the sol-

vents and the ionic molar conductivities (the Walden products) are approxi-mately constant. 23

At fi nite concentrations, ion – ion interactions cause the conductivities of electrolytes to decrease, not only if ion pairs are formed (Section 1.6.2 ) but also due to indirect causes. The modern theory, for example, that of Fernandez - Prini, 24 takes into account the electrophoretic and ionic atmosphere relaxation effects. The molar conductance of a completely dissociated electrolyte is

Λ Λ2 2 2 2= − + + ′ ′ − ′′ ′′∞2 2

1/223/2Sc Ec c J R c J R cln ( ) ( ) . (1.11)

Here S , E , J ′ , and J ″ are explicit expressions, containing contributions from relaxation and electrophoretic effects, the latter two depending also on ion - distance parameters R .

1.4.3 Ionic Effects on the Viscosity

The dynamic viscosity of water, ηW* , although rather small compared with other

liquids, is caused by the extensive network of hydrogen bonds existing in it that must be partly broken for the water to fl ow. 16 Ions affect the dynamic viscosity of the solution, η , some electrolytes enhancing it, whereas others diminishing it. The effect is described up to fairly concentrated solutions by the Jones – Dole expression:

[( ) ] .η η/ W*

21/2− = +1 2Ac Bc (1.12)

The A coeffi cients can be calculated theoretically from the conductivities, but the B coeffi cients are empirical and are obtained as the limiting slopes of plots of [( ) ]η η/ W

*2

1/2− −1 c versus c21/2. In order to have the individual ionic effects on

the viscosity, the B coeffi cients must be split according to some reasonable assumption. The generally accepted one relates to the mobilities of the ions: B + / B − ≈ u + / u − . Over a fairly wide temperature range in water, this leads to B (Rb + ,aq) = B (Br − ,aq), although B (K + ,aq) = B (Cl − ,aq), valid over a narrow temperature range, has often been used, differing by ± 0.002 dm 3 mol − 1 . 25 Selected viscosity B coeffi cients of ions at 298.15 K are listed in Table 1.3 . They are seen to be positive for small and multivalent ions but negative for univa-lent large ions.

These algebraic signs have led to the classifi cation of ions into water struc-ture makers ( B > 0, increasing the viscosity) and water structure breakers

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22 ION PROPERTIES

( B < 0, diminishing it and increasing the fl uidity of the solution). The absolute magnitudes of the B coeffi cients indicate the extents of the effects of the ions on the structure of water in the solution. They agree with other measures of such effects, for example, those derived from the relaxation of NMR signals and from the entropies of hydration of ions (Section 1.3.2 ). 16 As the tempera-ture is increased, the negative B values become less negative and may change sign at a characteristic temperature. This is explained by the diminishing extent of hydrogen - bonded structure in the water as the temperature is raised, so that structure - breaking ions have less structure to break.

1.5 ION – SOLVENT INTERACTIONS

Water is the most important solvent for electrolytes dissociating into ions. Therefore, the properties of ions in aqueous solutions have been described above in detail. There are some aspects of the interaction of ions with water that have not been touched upon in the previous discussion. One of them is the number of water molecules in the hydration shells. This aspect requires, of course, departure from the strict primitive model described below since it recognizes the molecular nature of the aqueous solvent. This number must be defi ned operationally since diverse methods are sensitive in different ways to this number.

However, ions can be found also in nonaqueous solvents and in mixed solvent consisting of water and another solvent. A brief account of the relevant ion properties in such environments is presented here. The number of solvent molecules surrounding an ion, its solvation number, is of interest, as mentioned in Section 1.2.3 concerning hydration numbers. More defi nite are the thermo-dynamic quantities related to the transfer of ions from a source solvent, say water, to a target solvent or solvent mixture. As for the thermodynamic quanti-ties characterizing ions in aqueous solutions dealt with above, also the transfer quantities can be measured for entire electrolytes only, and an extrathermo-dynamic assumption is required in order to describe the transfer properties of individual ions.

If an ion exists in a mixture of solvents, its solvation by the components of the mixture depends not only on its affi nity to each of these but also on the mutual interactions of the solvents. The composition of the solvation shell of the ion will generally differ from that of the bulk solvent. This can be deter-mined from thermodynamic data, including the standard molar Gibbs energy of transfer of the ion from the source solvent (say water) to the target aqueous solvent mixture.

1.5.1 Solvation Numbers

What is said in the following regarding hydration numbers generally applies also to solvation numbers. Water molecules around ions are arranged in con-

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ION–SOLVENT INTERACTIONS 23

centric shells: The nearest ones, in the fi rst hydration shell, are relatively strongly bound to the ion and move together with it, as do at least some of the water molecules in a second hydration shell, if present. Large ions with a single charge (e.g., (C 2 H 5 ) 4 N + and larger tetraalkylammonium ions) may not have a hydration shell altogether, but have the water in enhanced icelike tet-rahedral structures around them as for nonionic hydrophobic solutes, but in any case large ions (e.g., K + and Cl − ) lack a well - formed second hydration shell. A second shell characterizes multiply charged small ions (e.g., Mg 2 + ), the water molecules in it being hydrogen - bonded to those in the fi rst shell more strongly than the hydrogen bonds in pure water. Beyond the hydration shells, the water molecules are still affected by the electric fi eld of the ionic charge and the possibility of being hydrogen - bonded to the inner water molecules, but the hydrogen - bonded structure of this region is less ordered than in pure water (see above). Only further out from the ion does the water become bulk water, having the properties of pure water, unless, of course, at fi nite concentrations of the electrolyte, counterions show the effects of their presence.

Hydration numbers are the time - average numbers of water molecules residing in the fi rst (and second, if formed) hydration shell of ions. Coordinate bonds may be formed between the water molecules in the fi rst hydration shell and the ion, the bonds being then strongly directional, and a defi nite hydration number results, equaling the coordination number (e.g., 4 for Be 2 + , 6 for Mg 2 + and Fe 2 + ). If only nondirectional electrostatic association takes place, then geometric constraints may occur, smaller ions having smaller hydration numbers than larger ions, although the water molecules are bonded more energetically to the former. Over time, water molecules depart from the hydra-tion shells and others come in, so that the time - average is a dispersion of numbers with a noninteger average value.

There are several methods for the determination of the hydration numbers, but there is unfortunately no strong agreement between the results. If a primi-tive model of the ions (hard spheres) and the solvent (a compressible dielec-tric) is employed, it may be assumed that the electric fi eld of the ion causes a certain molar compression of the electrostricted solvent in its solvation shell independent of the nature of the ion. 26 The molar compression of electrostricted water is Δ V Welec = − 2.9 cm 3 mol − 1 at 298.15 K 12,26 obtained from the compress-ibility and the pressure derivative of the relative permittivity of pure water. The ratio

h V VIelec Ielec Welec/∞ ∞= Δ (1.13)

between the molar ionic electrostriction, VIelec∞ , and Δ V Welec can be construed to

represent the time - average hydration number of the ion. Alternatively, the ion and the water in its fi rst hydration shell may be con-

sidered to be uncompressible by an external pressure, the electric fi eld having produced the maximal possible compression. Then the hydration number is defi ned by the compressibility as

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24 ION PROPERTIES

h V P VTIcomp I W W/ /∞ ∞= − ∂ ∂1 ( ) .* *κ (1.14)

Here ( )∂ ∂∞V P TI / is a negative quantity, the standard partial molar compress-ibility of the ion, and κW

* is the isothermal compressibility of water at ambient pressures. To obtain individual ionic values of ( )∂ ∂∞V P TI / from experimental values for electrolytes requires the assumption of a value for one ion. That for chloride, ( / Cl aq cm GPa molI∂ ∂ = − ±∞ − − −V P T) ( , ) . .16 5 1 5 3 1 1 at 298.15 K, was suggested. 27 The hydration numbers from these two methods are shown in Table 1.2 and are in only fair mutual agreement. Other values were obtained by other methods and for ions for which no other value of the hydration number is available, the approximation h z rI I/ /nm∞ = 0 360. ( ) can be used. 2

Hydration numbers are expected to diminish as the concentration of the electrolyte increases, mildly at low concentrations but strongly when the hydration shells of oppositely charged ions start to overlap. The average dis-tance between ions in a solution is inversely proportional to the cube root of the concentration:

d N c cavA

1/3I I

1/3

I I

1/3nm= ⋅( ) = ⋅( )−

− −∑ ∑ν ν1 1844. , (1.15)

the summation extending over all the ions present with their stoichiometric coeffi cients ν I at concentrations c I in mol dm − 3 . 28 The size of a hydrated ion may be taken as d I - O + r W = r I + 2 r W (Section 1.2.2 ), so that it is possible to estimate the concentration at which the hydration shells start to overlap. For aqueous NaCl with ionic radii r I of 0.102 and 0.181 nm for the cation and anion and with r W = 0.138 nm, the sum of the radii of the hydrated ions is 0.835 nm, so that at a concentration of 1.43 mol dm − 3 the hydration shells start to overlap. Below the overlap limit, experimental values of ( ∂ V I / ∂ P ) T may be used for the estimation of the hydration numbers at fi nite concentrations from the expres-sion for h Icomp given above.

The molar compression of solvents other than water in the electric fi eld of an ion, Δ V 1elec , has been determined from their compressibility and the pres-sure derivative of their relative permittivity as it was for water. Typical values are considerably larger than for water, but relative to the molar volume of the pure solvent, V1

*, some are of the same magnitude as for water (0.161). 26 Solvation numbers have been deduced from the ratios of the ionic electrostric-tion volume, VIelec

∞ (when available for various ion/solvent systems), and the molar compression of the solvent, Δ V 1elec . They are commensurate with the hydration numbers of univalent ions but smaller than the hydration numbers (about one - half) for multivalent ions, probably because of geometric con-straints for the bulkier solvents.

1.5.2 Salting Out and Salting In

As the ions are solvated, there remains less “ free ” solvent to accommodate other solutes, so that a general result of the presence of ions in a solution is a

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diminished solubility of nonelectrolyte solutes, marked by subscript 3 . (The mutual interactions of ions are treated in Section 1.6 .) This phenomenon is called “ salting out ” and is expressed up to fairly high concentrations, c 2 , of the electrolyte by log( )*s s kc3 3 2/ = , where s3

* is the solubility of the nonelectrolyte in the absence and s 3 that in the presence of the electrolyte. The coeffi cient k is called the Setchenov salting - out constant. There are some systems (see below) where k < 0, and salting in then occurs, the solubility being enhanced by the presence of the electrolyte, but as a rule k > 0 and it is diminished. The mag-nitude of k increases with the molar volume of the nonelectrolyte, V 3 , and with the intensity of solvation of the electrolyte. The latter can be described by the electrostriction that the ion causes, and this leads to the McDevit and Long 29 formulation for the Setchenov constant:

k V V RT= − 3 2 10elec T/(ln ) .κ (1.16)

Here V 2elec is the molar electrostriction by the electrolyte, the (negative) dif-ference between the partial molar volume of the electrolyte and the sum of the intrinsic volumes of its ions and κ T is the isothermal compressibility of the solvent.

This discussion pertains to electrolytes consisting of “ small ” ions, such as Na + and Cl − , to solvents such as water, and to nonpolar solutes, such as benzene. In systems like these, the coulombic forces between the ions and the solvent leading to their solvation cause the salting out. This does not exclude the pos-sibility of direct interactions between the solute and the ions by hydrogen bonding or donor – acceptor interactions, which may overcompensate the cou-lombic forces and cause salting in ( s s3 3> *) (see Chapter 6 ). Other cases of salting in occur with very poorly solvated ions (tetraalkylammonium ones in water) and nonelectrolytes that are more polar that the solvent (have a higher permittivity).

1.5.3 Ion Transfer Thermodynamics

The transfer of an ion from water, as the source solvent, to a target solvent or solvent mixture has thermodynamic consequences in terms of energies, entro-pies, volumes, and so on. It is, in principle, possible to evaluate these from a comparison of the solvation energetics of the ions, that is, transfer from the ideal gas into the source solvent on the one hand and into the target solvent or mixture on the other. However, this procedure would involve rather small differences between large numbers and be inaccurate. It is, therefore, prefer-able to determine the transfer energetics directly, and derive from them, if desirable, the solvation quantities. This preference arises from the fact that the properties of hydrated ions and the hydration properties of ions have been most extensively and accurately determined.

The sign of the standard molar Gibbs energy of transfer of an ion I ± from water to the target solvent (or mixture) S, Δ t I I aq SG∞ ± →( , ), determines

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26 ION PROPERTIES

whether this ion prefers to be in an aqueous environment ( Δ t IG∞ > 0) or have S as its near environment ( Δ t IG∞ < 0). Values of the standard molar Gibbs energy of transfer electrolytes, Δ tG2

∞, can be determined for sparingly soluble electrolytes from their relative solubilities in water and in S, or for well - soluble electrolytes electrochemically from the electromotive force of cells, the two back - to - back half - cells of which have water and S as the solvent but being otherwise identical, or from polarographic half - wave potentials in water and in S. The standard molar enthalpy of transfer of electrolytes, Δ tH2

∞, can be determined from their relative heats of solution in water and in S and the standard molar entropy of transfer electrolytes, Δ tS2

∞, can be determined from the temperature coeffi cients of the solubilities, electromotive forces, or half - wave potentials establishing the Δ tG2

∞ values. The standard molar volumes of transfer Δ tV2

∞ are obtained from the apparent molar volumes, and these, in turn, from the densities of the solutions in water and in S.

In attempts to split the electrolyte values into the individual ionic contribu-tion, it is expedient to employ the same extrathermodynamic assumption for all these three thermodynamic quantities in order to arrive at consistent values, obeying

Δ Δ Δt t tG H T S2 2 2∞ ∞ ∞= − . (1.17)

A variant of the TPTB assumption (Section 1.3.1 ), namely that the transfer quantities of tetraphenylarsonium cations and tetraphenylborate anions are equal, has been widely used (the TATB assumption). Although Ph 4 As + is slightly larger than Ph 4 P + , which in turn, is slightly larger than BPh4

−, these small differences are deemed to lead to results within experimental errors of the transfer data and be of little consequence for the individual ionic transfer energetics. 30 The results for transfer of ions into many pure organic solvents have been summarized by Marcus 2 and those for transfer into mixed aqueous - organic solvents by him with coworkers. 31 – 33 Some results for the standard molar ionic volumes of transfer, Δ t IV ∞, and heat capacity of transfer, Δ t PIC∞ , into pure organic solvents have also been reported. 34

The transfer of many ions from water into most organic solvents is an unfa-vorable process that would not occur spontaneously: Δ t I I aq SG∞ ± → >( , ) 0. This is generally the case for transfer of cations into alcohols, ketones, esters, nitriles, nitro compounds, and halogenated alkanes. Exceptions are the transfer of Ag + and Cu + into nitriles and of “ soft cations ” (such as Ag + , Cd 2 + , and Hg 2 + ) into solvents such as pyridine and those in which the oxygen donor atom is thio-substituted. Furthermore, solvents that are strong electron pair donors, such as ethane - 1,2 - diol, propylene carbonate, amides, sulfolane, dimethyl sulfoxide, and trimethyl phosphate, have Δ t I I aq SG z∞ + → <( , ) 0, and cations are preferen-tially solvated by them. Anions, in general, prefer the aqueous environment that provides hydrogen bonds with them, an exception being the strong hydro-gen bond donor solvent 2,2,2 - trifl uoroethanol. Bulky ions, whether cations or anions, do prefer the less structured (less hydrogen - bonded) environment of

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most organic solvents. This pertains to tetraalkylammonium cations, the tetra-phenyl moieties mentioned above, triiodide, picrate, and so on.

1.5.4 Preferential Solvation of Ions in Mixed Solvents

When an ion is located in a solvent mixture, S A + S B , its environment generally has a composition differing from that of the bulk mixture due to preferential solvation of the ion by one of the components of the solvent mixture. It stands to reason that if an ion I ± has a favorable Gibbs energy of transfer into a pure component of the mixture, say S A , Δ t I AI aq SG∞ ± → <( , ) 0, but an unfavorable one into the other component, Δ t I BI aq SG∞ ± → >( , ) 0, then it will be preferen-tially surrounded by molecules of S A . This statement must be modifi ed in view of the mutual interactions of the solvent components of the mixture. In some cases, the preferential solvation in the mixture is practically complete, and then selective solvation takes place. The quantitative aspect of these preferences is of interest, that is, what fraction of the solvation shell of the ion is occupied by each solvent component, and there are two approaches that have been used in order to deal with it. 35

In the quasi - lattice quasi - chemical ( QLQC ) approach, 36 the ion I ± and the molecules of the two solvents, S A and S B , are distributed on sites of a quasi - lattice characterized by a lattice parameter Z . It specifi es the number of neighbors each particle has, independently of the nature of the particles. The pair - wise interaction energies e IA , e IB , e AA , e AB , and e BB , weighted according to the numbers of the corresponding nearest neighbors, determine the total con-fi gurational energy of the system. Ideal entropy of mixing of the particles on the quasi - lattice sites is assumed. The quasi - chemical aspect relates to the rela-tive strength of the mutual interactions of the solvent molecules and those with the ion. A set of equations is provided by this approach to determine the local (i.e., around the ion) mole fraction of component S A : x xIA

LIBL( )= −1 , hence

the solvation number of the ion by S A : ZxIAL , and eventually the equilibrium

constant for the replacement of S B by S A : K x x x xBA

IAL

B A IBL/= . 37 These expressions

require the excess Gibbs energy of mixing of the equimolar solvent mixture ( x A = 0.5) in the absence of the solute and the standard molar Gibbs energy of transfer of the ion I ± from solvent S A to the mixture as a function of the composition up to x B = 1. The latter data establish the lattice parameter Z within ± 2 units.

The inverse Kirkwood – Buff integral method ( IKBI ) does not involve a model such as the QLQC method does, hence is rigorous, but requires deriva-tives of the Gibbs energy of transfer and the excess Gibbs energy of mixing 38 with respect to the solvent composition. These functions, however, may often not be known suffi ciently accurately for obtaining meaningful derivatives. The Kirkwood – Buff integrals need for their evaluation, in addition to these deriva-tives, also the isothermal compressibility of the mixture and the partial molar volumes of the ion and the two solvent components in it as a function of the solvent composition. The expressions yielding the local mole fraction xIA

L

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28 ION PROPERTIES

require, furthermore, an estimate of the correlation volume, that is, the volume around the ion in which it affects the composition of the local solvent mixture. Finally, there is the problem in obtaining the required information from exper-imental data, in that the latter pertain to entire electrolytes, and their applica-tion to single ions has some bearing on the meaning of the Kirkwood – Buff integrals. Application of the TATB assumption (Section 1.5.3 ) concerning the splitting of the standard molar Gibbs energy of transfer of an electrolyte into the ionic contributions has been employed to circumvent this theoretical problem. 35

Both methods have been applied to several ions in aqueous and completely nonaqueous solvent mixtures, provided that ionic dissociation is complete, with fair agreement when they have been applied to a given system. 35

1.6 ION – ION INTERACTIONS

At practical concentration, that is, beyond infi nite dilution, ions in solution interact with each other electrostatically due to their charges: Ions of like charge sign repel each other and those of unlike charge sign attract each other. These interactions compete with the thermal movement of all the particles in the solution and are screened by the dielectric medium of the solvent in which the ions are located. Water, having a high dielectric permittivity, is very effective in this screening. The overall interactions, involving ion solva-tion in addition to ion – ion interactions are quite complicated, and approxima-tions have to be applied in order to handle the resulting behavior of the ions theoretically.

The simplest approximation is called the “ restricted primitive model ” and considers the ions as charged conducting spheres dispersed uniformly in a continuum made up of a compressible dielectric. The ions are characterized by their charges (sign and magnitude) and sizes (radii), the solvent, whether single or a mixture, by its permittivity, compressibility, and thermal expansibil-ity. Within this model and in dilute solutions of electrolytes, the well - known Debye – H ü ckel theory describes the chemical potentials of the electrolyte, μ 2 , and that of the solvent, μ 1 , suffi ciently well. The former is directly related to the activity of the electrolyte and to its mean ionic activity coeffi cient, γ 2 ± (on the molal scale) or y 2 ± (on the molar scale). The latter, μ 1 , is directly related to the activity of the solvent, a 1 , and to its osmotic coeffi cient,

ϕ ν1 2 2 1 11000= −( )ln ,/ m M a (1.18)

where ν 2 is the number of ions per formula of the electrolyte of molality m 2 , and M 1 is the molar mass of the solvent (in g mol − 1 ; for water 1000/ M 1 = 55.51). The solvent activity is roughly equal to the ratio of the vapor pressure of the solvent in the solution to that of the pure solvent: p p1 1/ * (when the vapor pres-

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ION–ION INTERACTIONS 29

sures are small and the vapors can be considered to approximate ideal gases). It is also related to the depression of the freezing point and elevation of the boiling point of the solvent.

1.6.1 Activity and Osmotic Coeffi cients

The limiting (Debye – H ü ckel) expression for the mean molal activity coeffi -cient of an electrolyte is log( γ 2 ± ) = − Az + z − I 1/2 , where A depends on the solvent and the temperature ( A = 0.511 in water at 298.15 K). The logarithm of γ 2 ± is thus proportional to the square root of the ionic strength I m zI= ∑0 5. I I

2, the summation extending over all the ions in the solution, which may contain a mixture of electrolytes. This limiting expression is valid only up to, say, m = 0.01 m ( ∼ 0.01 mol dm − 3 ) for uni - univalent electrolytes.

Beyond this concentration, the extended Debye – H ü ckel expression

log( ) ( )γ 2 1± + −= − +Az z I BaI1/2 1/2/ (1.19)

should be used, valid, in turn, up to, say, m = 0.2 m ( ∼ 0.15 mol dm − 3 ). Beyond this concentration again, or for higher valent electrolytes, a linear term in the ionic strength, bI , has to be added, up to quite high concentrations. In these extended expressions, B is a solvent - and temperature - dependent constant and a is the mean distance of closest approach of the ions (their diameters, if considered the same for cations and anions), whereas the coeffi cient b is a completely empirical fi tting parameter. For many purposes the product Ba = 1.5 kg 1/2 mol − 1/2 at any temperature and for any solvent may be used, the onus of fi tting the experimental values of log( γ 2 ± ) is then placed on the value of the parameter b .

A key quantity in the Debye – H ü ckel theory is the screening length, κ − 1 , the average reciprocal of the radius of the “ ionic atmosphere ” surrounding an ion in the solution. The square of screening length is proportional to the ionic strength of the solution:

κ ε ε2A 0 r/ ) ,= (2 10002 2N e RT I (1.20)

this being the source of the dependence of log( γ 2 ± ) on I 1/2 . The numerical value of this screening length is thus κ ε= − − − −502 9. ( ) ( )r

1/2 1/2 3 1/2 1/K /mol dm nmT I and depends on the reciprocal square roots of the relative permittivity and of the temperature, becoming κ = 3.26( I/ mol dm − 3 ) 1/2 nm − 1 for water at 298.15 K. The dimensionless product κ a features in the denominator of the extended Debye – H ü ckel expression for log( γ 2 ± ) and in the expression for the osmotic coeffi -cient. The logarithm of the solvent activity is given by the Debye – H ü ckel theory as

ln ln( ) ( ( ).a m M V N a1 2 2 1 131 1000 24= − + +ν π κ σ κ/ / )A (1.21)

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30 ION PROPERTIES

The σ ( κ a ) function is

σ κ κ κ κ κ κ κ( ) ( ) [( ) ( ) ln( )] ( ) . ( )a a a a a a a= + − + − + ≈ − +− −3 1 1 2 1 1 1 83 1 1.5 22, (1.22)

where the approximation is valid for κ a ≤ 1. An alternative formulation for the activity and osmotic coeffi cients is that

of Pitzer 39 that includes the Debye – H ü ckel limiting law but treats differently its extension to practical concentrations for fully dissociated electrolytes, in fact up to several moles per kilogram or per cubic decimeter. For 1:1 electro-lytes, the resulting expressions are ln( γ 2 ± ) = f γ + B γ m + C γ m 2 and

ϕ ϕ ϕ ϕ121= + + +f B m C m , (1.23)

the functions f are the electrostatic Debye – H ü ckel terms and the B and C coeffi cients are electrolyte - specifi c fi tting parameters. For electrolyte types other than 1:1, factors in the charge numbers and stoichiometric coeffi cients have to be included. Two universal constants, b = 1.2 and α = 2.0, are employed in the full expressions as well as the solvent - and temperature - dependent A φ arising from the Debye – H ü ckel theory. For details, the series of papers by Pitzer and coworkers should be consulted. 40

1.6.2 Ion Pairing

In electrolyte solutions consisting of relatively poorly solvated ions and of solvents of relatively low permittivity, the screening of the charges by the solvent is inadequate to prevent ionic association at suffi ciently high concen-trations. In most cases, the association stops at ion pairing: one cation with one anion, but in a solvent of low permittivity, triple ions may be formed or even larger associates. The treatment of ion pairing offered by Bjerrum, using the restricted primitive model (Section 1.5 ), has been validated over time, and can be formulated as follows. 41 Ions that are nearer each other than a certain distance

q z z N e RT z z l= =+ − + −A 0 r B/8 / )2 2πε ε ( , (1.24)

where l B is the Bjerrum length (but necessarily more remote than the distance of closest approach a ), are considered to be paired, those at larger distances from each other are free. Due to the electrostatic forces, only ions of opposite charge signs are likely to approach each other to a distance ≤ q . An equilib-rium constant, K A , may be formulated for the equilibrium C z + + A z − � C z + A z − . If the fraction α of the c 2 molar electrolyte is dissociated and 1 − α is paired, then

K cA /= −( ) .1 22α α (1.25)

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ION–ION INTERACTIONS 31

A dimensionless parameter b = q / a is employed and the resulting association constant is

K N b Q bA A/= ( ) ( )4 1000 3π (1.26)

using the function Q b x x xb( ) exp( )= ∫ −2

4 d , with x as an auxiliary variable, and the integral is to be solved numerically. At 298.15 K, the value of the key vari-able b for any solvent is given by

log log log /nmi j r( ) . ( ) log( ),b z z a= + − −1 448 ε (1.27)

log(298.15 K/ T ) should be added to the numerical constant at other tempera-tures. The distance of closest approach a is taken as the mean diameter of the ions that should not be smaller than the sum of their crystal ionic radii: a ≥ r + + r − . The values of log Q ( b ) have been tabulated 42 and range from − 1.358 at b = 2.1 (the lowest practical value) through zero for b = 5.9 to positive values at large b for which Q ( b ) ≈ exp( b )/ b 4 . The parameters b and Q ( b ) depend on the temperature and the solvent (through its relative permittivity ε r ) and on the electrolyte through the mean diameter of its ions, a , and are readily calcu-lated. From them, the association constant K A is obtained and the fraction associated, 1 − α , as a function of the concentration c 2 .

Experimentally, the dissociated fraction of the electrolyte, α , is commonly obtained from conductivity data, although other techniques have also been widely employed. In the case of the conductivities, the expression

Λ Λ2 2 2 2 2 2 2= − + + ′ ′ − ′′ ′′∞ S c E c c J R c J R c( ) ( ) ( ) ( )( ) ( )( )α α α α α1/2 3/ln 22 (1.28)

has to be used (Section 1.4.2 ), recognizing that the actual concentration of the ions is α c 2 instead of c 2 .

In aqueous solutions, with its high relative permittivity, ion pairing between univalent ions is rare, unless they are only poorly hydrated and can approach each other to within q = 0.357 nm (at 298.15 K). However, for more highly charged ions, appreciable ion pairing does occur in aqueous solutions at con-centrations of the order of 1 mol dm − 3 for 1:2 or 2:1 type salts and even at concentrations of 0.1 mol dm − 3 for 2:2 or higher types of salts. The equilibrium constants K A for the latter (e.g., MgSO 4 ) are of the order of 100 to 200 dm 3 mol − 1 .

In this formal theory, based on the restricted primitive model of the elec-trolyte solution, there is nothing that indicates how intimately the cation and anion are bound together in the ion pair. In fact, for suffi ciently low ε r values or large charges of the ions, the cutoff distance q beyond which ions are con-sidered to be free is manifold larger than the mean ionic diameters a . It is clear that solvent molecules penetrate the intervening distance, and if the “ restricted ” is removed from the model, it is possible to specify how many solvent mole-cules separate the cation and the anion. If this number is zero, this means that

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32 ION PROPERTIES

a “ contact ion pair ” ( CIP ) is formed, if it is 1 then a “ solvent - shared ion pair ” ( SIP ) is formed, if it is 2, then a “ solvent - separated ion pair ” ( SSIP or S2IP) is formed. Several forms of such ion pairs can exist at equilibrium with each other, depending on the solvent permittivity and the electrolyte concentration. Methods for ascertaining the situation with regard to these forms of ion pairs have been reviewed. 41

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