Application of solubility product and common ion effect

Qualitative analysis of cations is largely based on the principle of solubility product and common ion effect. Cations are separated in to six groups depending on the solubility of their salts.
Group-1 as insoluble chlorides
Only Ag⁺, Hg²⁺ and Pb²⁺ form insoluble chlorides since they have low values of K_sp.
Group-2 as insoluble sulphide in acidic medium
H₂S <========> H⁺ + HS^- ;
K₁ - first ionization constant
HS^- <========> H⁺ + S^2-;
K₂ – second ionization constant
[S^2-] = K₁K₂ [H₂S]/[H⁺]²
K_sp values of second group sulphides (PbS, CuS, SnS, HgS, As₂S₃, Bi₂S₃, Sb₂S₃) are very low. In acidic buffer, [S^2-] is decreased due to common ion effect and this results in the precipitation of Pb²⁺, Cu²⁺ etc of second group as their sulphides. Third and fourth group sulphides have high value of K_sp, hence they remain soluble.
Group- 3 as insoluble hydroxide in basic buffer of NH₄OH and NH₄Cl
The concentration of OH^- in ammoniacal solution decreases when NH₄Cl is added to it, because of the common ion effect. Thus only for least soluble hydroxide of Fe³⁺, Al³⁺, Cr³⁺ etc. ionic product exceeds the corresponding solubility products, hence only these ions are precipitated. Hydroxides of successive remain soluble due to high K_sp values.

The solubility product constant (Ksp)

The solubilityof ionic solids in water varies depending on a number of factors like lattice enthalpy of the salt and tha solvation enthalpy of the ions in a solution. As a general rule, for a salt to be able to dissolve in a particular solvent, its solvation enthalpy must be greater than its lattice enthalpy. Each salt has its characteristic solubility, which depends on temperature. We can classify salts on the basis of their solubility in three categories.
Soluble - Solubility > 0.1 M
Slightly soluble - 0.01 M < solubility < 0.1M
Sparingly soluble – solubility < 0.01M
We have now consider the equilibrium between the sparingly soluble ionic salt and its saturated aqueous solution. A solution which remains in contact with excess of the solute is said to be saturated. The amount of a solvent () in 100 ml or 1L) to form a saturated solution at a given temperature is termed as the solubility of the solute in the solvent at that temperature.
For a sparingly soluble salts like AgCl,PbI₂, BaSO₄ etc. Ionisation is very small and concentration of the salt may be considered as constant. Thus, for AgCl,
AgCl <========> Ag⁺ + Cl^-
K = [Ag⁺][Cl^-]/[AgCl]
For a pure solid substance the concentration remains constant and we can write
K_sp = K[AgCl]
=[Ag⁺][Cl^-]
Where K_sp is the solubility product constant or solubility product.
If S represents solubility of AgCl (in mol L^-1)
Then [Ag⁺]= [Cl^-] = S molL^-1
K_sp = [Ag⁺][cl^-] = S²
S = √K_sp
A solubility product constant expression is the product of the chemical equation for a solubility equilibrium, with each term raised to the power given by the coefficient in the chemical equation.
Example:-
For BaSO₄ (binary solute giving two ions)
BaSO₄ <=======> Ba²⁺+ SO₄2^-
Ksp = [Ba²⁺][SO₄2^-] = S2
For PbI₂ (Ternary solute giving three ions)
PbI₂ <=======> Pb²⁺ + 2I^-
Ksp = [Pb²⁺][I^-]2 = (s)(2s)2 = 4S³
For Al(OH)₃ (Quaternary solute giving four ions)
Al(OH)₃ <=======> Al³⁺ + 3OH^-
K_sp = [Al³⁺][OH^-]³ = (S)(3S)³ = 27Ss4
For a solute AxBy, (giving (x+y) ions)
K_sp = [A^y+]^x [B^x-]^y = (xS)^x (yS)^y = X^xY^y(S)^(x+y)
Precipitation reactions
For the sparingly soluble solute AB,
AB <=======> A⁺ +B^-
Q is the equilibrium constant with the given [A⁺] and [B^-]
Q = [A⁺][B^-]
When applied to solubility product, Q is generally called the ionic product. Precipitation occurs if Q>K_sp, then solution is said to be super saturated and precipitation cannot occur if Q_sp (a case of unsaturated solution) and a solution is just saturated if Q=K_sp.