(G S Manku, email: manku.gs@gmail.com)

[10.2].1 BORN’S EQUATION

Assuming ions to be spherically charged spheres, solubility of ionic electrolyte in a polar solvent of dielectric constant ε can be calculated by equating electrostatic work done for charging a sphere in a dielectric medium with the free energy and solute concentration (Born’s equation). The work done W in charging a sphere of radius r to charge z is given as

W = z²e² / 8πεr

Here e is the electronic charge. Assuming similar expressions to hold for cation of radius r_{+ ,} charge ν₊ and anion of radius r_– , charge ν _– charge ν _–, total work done for N molecules ionizing to will be

W = Nν₊z₊²e² / 8πεr₊  +  Nν_–z_–²e² / 8πεr_–
= (Ne² / 8πεr)  (ν₊z₊²  + ν_–z_–² )

Here r is the average ionic radius of the two types of ions. Work done in charging the ions of a given electrolyte in solvent 1 with respect to work necessary to charge in another solvent 2 will be (if similar equations hold for both the solvents)

W = {(Ne² / 8πε₁r)  (ν₊z₊²  + ν_–z_–² )}₁
–  {(Ne² / 8πε₂ r)  (ν₊z₊²  + ν_–z_–² )}₂
= (Ne² / 8πr)  (ν₊z₊²  + ν_–z_–² ) (1/ ε₁  -1/ ε₂ )

Here, r  is now the average ionic radius of the electrolyte ions in the two solvents.

The ideal relation between the free energy G and the solute concentration c is:

ΔG = ΔG⁰ + RT ln c

Equating this with energy required for transferring ions from solvent 1 to solvent 2, (ionization of electrolyte giving ν₊ cations and ν_– anions)

ΔG₁ –  ΔG₂ = RT ln c₁ – RT ln c₂
 = RT ln {( ν₊c₁) ^ν1 –  (ν_–c ₁ ) ^ν2) / ( ν₊c₂) ^ν1 –  (ν_–c ₂ ) ^ν2 ) }

From these relations,

(Ne² / 8πr)  (ν₊z₊²  + ν_–z_–² ) (1/ ε₁  -1/ ε₂ )
= RT ln {( ν₊c₁) ^ν1 –  (ν_–c ₁ ) ^ν2) / ( ν₊c₂) ^ν1 –  (ν_–c ₂ ) ^ν2 ) }

Because of charge symmetry of z with respect to ν,

ν₊z₊²  + ν_–z_–² =  z₊z_– (ν₊ + ν_– )

And the above equation becomes

(Ne² / 8πr) z₊z_– (ν₊ + ν_– ) (1/ ε₁  -1/ ε₂ ) = RT (In c₁ – ln c­₂) (ν₊ + ν_– )

Or, ln RT ln c₁ =   ln c₂ +  (Ne² z₊z_– / 8πrRT) (1/ ε₁  -1/ ε₂ )

This is Born equation and can be used to find the solubility of a solute in any solvent 1 by knowing

1. Its solubility in any solvent 2
2. Dielectric constant of the solvents 1 and 2
3. Ionic radii of the ions formed.

The Born equation is too ideal and has many obvious limitations:

1. Interionic interactions are not considered so that it applies only to dilute solutions.
2. The ionization of solute in a solvent may be not complete, especially if solvent has a low dielectric constant.
3. In saturated solutions, the concentrations and the molarities are not same due to reduced activity coefficients.

The agreement between solubility of ionic solutes and Born equation is rather poor, even after considering the activity coefficients.

Ricci and Davis proposed an empirical equation for solubility of ionic solutes in nonaqueous solvents. They assumed that the activity coefficients of the ions (γ) in saturated solutions of sparingly soluble salts is almost constant regardless of the solvent (for silver acetate, γ = 0.79 ±  0.01 in 10 – 30 % aqueous acetone and dioxane), and applied Debye – Hückel theory to saturated but dilute solutions to get

A ε₁^{– 3/2}c₁^1/2 =  A ε₂^{– 3/2}c₂^1/2
Or, log c₁  = log c₂ + 3(log ε₁ – log ε₂)

This equation is better for solubility of sparingly soluble salts.

[10.2],2 ION ASSOCIATIONS

In concentrated solutions, electrostatic interactions between ions become significant, If ions come close enough, electrostatic interactions become more than the thermal agitations, which leads to disorder the ions. If two oppositely charged ions come close together, they may form an ion pair due purely electrostatic attractions. This is different from the covalent bond formation and is shown as A⁺B^–. In solvents of low dielectric medium, ion pair formation is, however, rather extensive.

Bjerrum’s model for ion pair formation assumes that (1) ions are hard non-polarizable spheres, (2) only coulombic forces exist between the ions, and (3) dielectric constant of the medium between ion pair is same as that in the bulk of the solvent. Number of ions (dn) of the type i in spherical shell of thickness dr at a distance r from the selected j–ion by given by Beckmann’s distribution law as

dn = n_i exp (U/kT)(4πr²dr)

Where k = R/N is the Boltzman’s constant, n₁ = number of i-ions per unit volume (cm³ or ml) solution, U = work required to separate the i-ions at a distance r to infinity from a particular j-ion, 4πr²⁺dr is the volume of the spherical shell of thicken dr at a distance r from the center. As U is the coulombic potential energy of the charge in the presence of dielectric constant ε,

U = – z₁z₂e²/4πεr.

Hence, Beckman’s distribution equation becomes

dn_i = 4πr²n_i exp (– z₁z₂e²/4πεkTr) dr

The probability of finding i-ion in the shell of thickness dr (may be as small as 1 pm) from a selected j-ion as a function of r shows a minimum at r = r₀. The value of r₀ can easily be calculated by differentiating the above equation and putting d/dr (dn_i/dr) = 0:

d/dr (dn_i/dr) =  2r.4πrn_i exp (– z₁z₂e²/4πεkTr) 4πr²⁺dr  –
(z₁z₂e²/4πεkTr² ) 4πr²n_i exp (– z₁z₂e²/4πεkTr)
= [ 8πr  – z₁z₂e²/εkT ] n_i exp (– z₁z₂e²/4πεkTr)

Hence d/dr (dn_i/dr) = 0 at r = r₀ gives

r₀ =  z₁z₂e²/4πεkT

The Bjerrum’s equation applies only to electrostatic interactions and breaks down in case of covalent bonding, e.g. for weak electrolytes.

The distance r₀ for ion pair formation cannot be determined directly, but can be evaluated from the equilibrium between the ion pair formation and dissociation of ions. Degree of association α will be proportional to the number of i-ions that are at a distance ≤ r₀ and can be calculated by integrating the expression for dn­_i/dr from a distance a (point of contact of the two ions) to the distance r_0..

[10.2].3 CLASSIFICATION OF SOLVENTS

 The different solvents can be classified in many ways depending upon their physical and chemical properties.

Protonic and Aprotic solvents: Protonic solvents autoionize to form a solvated hydron (proton), e.g. water, ammonia, hydrogen fluoride, sulphuric acid, acetic acid, etc. Aprotic solvent do not ionize to give a hydronium ion, e.g. SO₂, BrF₃, CS₂, CHCl₃, etc.

The protonic solvents can be further classified as

• acidic (strong tendency to ionize into hydronium ions, like sulphuric acid, HF, or acetic acid),
• basic (strong tendency to accept hydronium ions, like ammonia, and
• amphoteric (can both accept as well as donate hydronium ions like water and alcohols.

However, this classification over-emphasizes the acid – base behavior as all the protonic solvents can accept as well donate a hydronium ion depending upon the nature of the solute.

The aprotic solvents can be further classified as

1. nonpolar, non-ionizing and non-solvating like benzene, carbon tetrachloride or dichloromethane;
2. highly polar and good coordinating ligands like acetonitrile, dimethylacetamide, dimethyl sulphoxide, etc. and (3) highly polar auto-ionizing like , OPCl₃, BrF₃, which are highly reactive and difficult to keep pure.

2. Ionizing and Non-ionizing Solvents based on their polar and nonpolar nature. Non-ionizing solvents have low dipole moments and low dielectric constants, whereas polar nature. Ionizing solvents have polarity, high dipole moments and high dielectric constants

However, such classifications are of no use as they are based on the studies of only a limited number of solvents and do not give any basic information of the solvent properties

Solubility of ionic compounds, especially in aprotic solvents, involves electrostatic as well as specific solute–solvent interactions Gutman has proposed a donor number (DN) as a measure of basicity of a solvent. It is defined as the negative enthalpy of the reaction of aa solvent with antimony trichloride:

Solvent + SbCl₃ →  [(Solvent)_nSbCl₃]                          ΔH = – DN

However, there seems to no relation between DN and the dielectric constant (ε) or polarity of solvents. Solvents with high ε may be poor donors (nitrobenzene, ε = 35.9, DN = 2.7), diethyl carbonate (ε = 89.1, DN = 16.4) and best donor solvents may have a low ε (pyridine, ε = 12.3, DN = 33 or tributyl phosphate ε = 12.3, DN = 33).

[10.2].4 GENERAL BEHAVIOR OF SOLVENTS

[10.2].4.1 General

Water has long been recognized as a good solvent because of its ease of availability and its purification. No other solvent is so versatile to handle. High availability, purity, wide liquid range, dissolution tendencies for both covalent and ionic compounds, non-corrosive nature, high dielectric constant that reduces the interionic interactions, dipolar nature and coordination tendencies that stabilizes many ions and molecules giving high hydration energies, amphoteric behavior and other physico-chemical properties make water almost a unique solvent. However, many other liquids possess these properties to different extents, but have similar behavior.

[10.2].4.2 Properties of Ionizing Solvents

The most important properties of the ionizing solvents are given below.

1. Ionizing solvents in pure state dissociate to some extent into ions and are weak conductors of electricity:

NH₃  + NH₃ ⇌ NH₄⁺ + NH₂^–
H₂O + H₂O ⇌ H₃O⁺ + OH^–
HF  +  HF  ⇌ H₂F⁺ + F^–
SO₂ + SO₂ ⇌ SO²⁺ + SO₃^{2 –}

2. Ionizing solvents are usually polar molecules that can solvate the ions through ion – dipole interactions and weaken the interionic attraction forces between the ions in the solid crystals.
3. The ionizing solvents have high dielectric constant. Dipole moment is product of charge separated and the distance between the charged ends, but the dielectric constant depends on the extent to which the molecules orient themselves in an electric field to destroy the field.
4. Polar molecules tend to associate due to dipole–dipole interactions. Association is more in protonic solvents due to hydrogen bonding and leads to a higher boiling points and increase the liquid range of the solvent.
5. Solvent should be easily available, equally liquefiable and have a good liquid range.

An examination of commonly used solvents (Table [10.2].1) shows that these properties may not be available in many solvents. HF and HCN are dangerous to health; NH₃, SO₂ and H₂S have low boiling points;acetic acid and dioxane have low dipole moments. Many organic solvents (alcohols, benzene, chloroform, carbon tetrachloride and dichloromethane have favorable handling properties but are not good solvents for ionic solutes.

Table [10.2].1 Physical Characteristics of Some Solvents

olvent                           M.P.          B.P.    Dipole Moment Dielectric constant
Water                         273 K      373 K     1.84 D          81.3 at 288 K
Ammonia                  195.5 K    240 K    1.46 D           22 at 240 K
Hydrogen fluoride  190 K       293.1 K  1.90 D          83.6 at 273 K
Sulphur dioxide      198 K      290 K      1.61 D           17.3 at 298 K
Hydrogen cyanide                  299 K      2.93 D          106.8 at 298 K
Hydrogen sulphide  187.5     212.9 K   1.10 D           10.2 at 213 K
Acetic acid                289.8 K   391 K      0                     7.1 at 298 K
Benzene                    278.8 K   353 K      0                     13.1
Sulphuric Acid        283.7 K   611 K (d)  1.83 D           110
Ethanol                     161 K     353.5 K     1.7 D             25
Chloroform              210 K     334 K        1.1 D             5.1
Carbon tetrachloride 251 K 353 K        0                    13.1
Carbon disulphide  160 K    319 K        1.1 D              2.7
Diethyl ether            155 K    308 K        1.1 D              4
Nitrobenzene           279 K    484 K        4.18 D            35.5
———————————————————————————
d = decomposes

[10.2].4.3 Type Reactions in Solvents

Though in solvents, specific reactions may be different, same types of reactions are defined for solvents as for water. Activity coefficients of solutes can be determined from measured half-cell potentials using using Nernst equation and Debye-Hückel theory in aqueous as well as nonaqueous solutions. Some typical reactions are given below.

[10.2].4.3.1 Acid–Base Behavior

By the Cady–Elsey solvent system definition of acids and bases (Article 1.3, Lewis and Other Definitions), acids are those substances that by direct interaction with solvent, increase the concentration of its characteristic cation formed by autoionization, whereas bases are those substances that by direct interaction with solvent, increase the concentration of its characteristic anion. The acid–base behavior depends upon (i) inherent acid–base nature of the solvent, (ii) inherent character of the solute and (iii) solute–solvent interactions. The solvent with high dielectric constant ε only can support ionization of solutes.

In a protonic solvent, the cation is just a solvated hydronium ion (proton), so that the Brönsted and the solvent system definitions are the same. Thus, in ammonia as solvent, NH₄⁺ is acid whereas NH­₂^–, NH^2– and  N^3– ions (amide, imide and nitride respectively) are bases. Typical neutralization reactions in ammonia (solvent) are:

NH₄Cl + NaNH₂ → NaCl + 2NH₃
2NH₄Cl + PbNH → PbCl₂ + 3NH₃
3NH₄Cl + Na₃N → NaCl + 4NH₃
NH₄Cl + NaOH → NaCl + NH₃ + H₂O.

In non-protonic solvents (benzene, carbon tetrachloride), Lewis type of neutralization reactions can take place.  The solvent system definition of acid and base is useless here. However, for self-ionizing solvents (SO₂, BrF₃, POCl₃), the acid–base reaction will be

2SO₂ ⇌ SO²⁺ + SO₃^2–
SOCl₂ + Na₂SO₃ ⇌ 2NaCl + 2SO₂

A solute may act acidic or basic depending on the solvent. For example, acetic acid is a strong acid in ammonia, a weak acid in water and a base in sulphuric acid solutions:

CH₃COOH + NH₃ ⇌ CH₃COO^– + NH₄⁺          (strong acid)
CH₃COOH + H₂O ⇌ CH₃COO^– + H₃O⁺          (weak acid)
CH₃COOH + H₂SO₄ ⇌ CH₃COOH₂⁺ + HSO₄ ^–  (weak base)

The differentiating range of a solvent is controlled by the autoionization constant for the solvent. It is 0–14 for aqueous solutions, 0–27 for ammonia and 0–3 for sulphuric acid. All acids stronger than the characteristic cation of the solvent behave as strong acids, and all bases stronger than the characteristic anion of the solvent behave as strong bases. These get leveled off to the solvent cation or the anion.

[10.2].4.3.2 Metathetical Reactions

A metathetical reaction can be predicted only we know the solubility product of the compound in the solvent. The solubility product of a salt changes so much from solvent to solvent, that it can be reversed by a change of solvent. For example, in aqueous solutions, addition of KI to AgNO₃ solution forms a precipitate of AgI, but in ammonia solutions, addition of KNO₃ solution to a solution of AgI precipitates KI:

AgI + KNO₃ → KI ↓ + AgNO₃     (in ammonia solutions)

Metathetical reactions cannot be generalized due to specific solute–solvent interactions.

[10.2].4.3.3 Solvolytic Reactions

In a solvolytic reaction, the solvent reacts with the solute changing the normal concentration of the solvent cations or anions, and the phenomenon is called solvolysis. As an example, solvolysis of POCl₃ in some solvents is as:

POCl₃ + 3HOH → OP(OH)₃ + 3HCl
POCl₃ + 3ROH → OP(OR)₃ + 3HCl
POCl₃ + 6HNH₂ → OP(NH₂)₃ + 3NH₄Cl

In nonprotonic solvent, an example is

SnCl₂ + 2SOCl₂ → 2SOCl⁺ + SnCl₃ ^–

[10.2].4.3.4  Redox Reactions

If the solvent does not undergo redox reaction, the redox reactions in aqueous and in nonaqueous solvents are similar. In ammonia solutions, the reference hydrogen electrode is the half-cell for the reaction

(1/2)H₂(g) + NH₃ ⇌ NH₄⁺ + e ^–        E⁰ = 0.000 V

However, the E⁰ values for half reactions vary considerably from solvent to solvent due to the solvation energy and the chemical reactivity of solvents. In many cases, the order of reactivity of the substances may get altered in nonaqueous solutions. In many cases, the reactions appear very unique. For example, sodium can be electrolyzed from aqueous solutions containing pyridine using mercury cathode. Sodium and beryllium can be electro-deposited from ammonia solutions. The dimercury(2+) cation (Hg₂²⁺) ion disproportionates to Hg + Hg²⁺ in ammonia solutions.

The order of – E⁰ (in volts) for alkali and alkaline earth metals in aqueous solutions is

Li (3.05) > Cs (3.04) > K (2.93) ) Rb (2.92) > Ba (2.90) >
Sr (2.89) > Ca (2.87) > Na (2.71) > Mg (2.37)

In ammonia solutions, it changes to

Li (2.34) > Sr (2.30) > Ba (2.20) > Ca (2.17) > Cs (2.08) >
Rb (2.06) > K (2.04) > Na (1.85) > Mg (1.74)