Solubility Equilibria of Sparingly Soluble Salts - Chemistry Class 11 -

Table of Contents
1. Solubility Product (Ksp)
2. Solubility - definition and basic ideas
3. Solubility product constant - formal derivation
4. Relation between molar solubility and Ksp
5. Factors affecting Ksp and solubility
6. The Common-ion Effect
7. Quantitative effect of a common ion - simple derivation
8. pH and solubility
9. Complex-ion formation and increased solubility
10. Ionic product (Q) and precipitation
11. Applications of Ksp and the common-ion effect
12. Worked example - effect of a common ion (symbolic)
13. Summary
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Solubility Product (K_sp)

The solubility product constant is the equilibrium constant for the dissolution of a sparingly soluble ionic solid in water. It is represented by the symbol K_sp and applies to an equilibrium between a pure solid ionic compound and its saturated aqueous solution.

K_sp is a type of equilibrium constant and its numerical value depends on temperature. For most salts, K_sp increases with rising temperature because solubility usually increases with temperature.

Solubility - definition and basic ideas

Solubility is the property of a solute to dissolve in a given solvent to form a homogeneous solution. For ionic compounds, solubility in water depends on a balance between the lattice enthalpy of the solid and the solvation (hydration) enthalpy of its ions.

• The lattice enthalpy is the energy required to separate the ions in the solid; a large lattice enthalpy tends to reduce solubility.
• The solvation enthalpy (heat released when ions are solvated by water) is always negative; a larger magnitude of solvation enthalpy favours dissolution.
• The nature of the solvent determines the magnitude of solvation enthalpy. Polar solvents such as water give large solvation enthalpies and therefore dissolve many ionic solids; non-polar solvents give small solvation enthalpies and so dissolve few ionic solids.
• For a salt to dissolve, the solvation enthalpy must compensate (in energy terms) the lattice enthalpy; otherwise the salt is sparingly soluble or insoluble.
• Solubility depends on temperature and is different for each salt.

Classification of salts on the basis of solubility

Solubility product constant - formal derivation

Consider the sparingly soluble salt barium sulphate, BaSO₄, in contact with its saturated aqueous solution. The dissolution equilibrium is:

The equilibrium constant expression for this reaction is:

Because the activity (or concentration) of a pure solid is constant, it is incorporated into the equilibrium constant. The remaining measurable equilibrium constant is called the solubility product:

Thus, at equilibrium between solid BaSO₄ and its saturated solution, the product of the molar concentrations of Ba²⁺ and SO₄^2- (each raised to the power of its stoichiometric coefficient) is constant at a given temperature. This constant is the K_sp of BaSO₄.

Relation between molar solubility and K_sp

For a salt that dissolves according to the stoichiometry:

MX(s) ⇌ M⁺(aq) + X^-(aq)

let the molar solubility of MX be s mol L^-1 (i.e., the concentration of M⁺ and X^- produced in a saturated solution). Then:

[M⁺] = s
[X^-] = s
K_sp = [M⁺][X^-] = s × s = s²

For a salt of the type M₂X (dissociating to 2 M⁺ + X^2-), if molar solubility is s then:

[M⁺] = 2s
[X^2-] = s
K_sp = (2s)²(s) = 4s³

Factors affecting K_sp and solubility

• Temperature: K_sp values change with temperature because solubility usually changes with temperature.
• Common-ion effect: The common-ion effect is the suppression of the ionization of a weak electrolyte or the reduction in solubility of a sparingly soluble salt when a strong electrolyte containing a common ion is added..
• Effect of ionic Strength: the presence of other ions (without a common ion) can alter activity coefficients and sometimes increase the apparent solubility of a sparingly soluble salt.
• Formation of ion pairs and complex ions: if ions form stable ion pairs or complex ions in solution, the apparent solubility of the solid increases.
• pH of the medium: salts containing basic anions (for example CO₃^2- or OH^-) are more soluble in acidic media because the anion can be protonated.

The Common-ion Effect

The common-ion effect is the suppression of the dissociation of a weak electrolyte when a strong electrolyte containing a common ion is added. This is a direct application of Le Châtelier's principle to equilibria involving ionization.

In general, if a salt with equilibrium M_aX_b ⇌ a M^m+ + b X^n- is in contact with its saturated solution, adding a soluble salt that provides either M^m+ or X^n- will shift the equilibrium to the left and decrease the solubility of the original salt.

Example (qualitative): When hydrogen chloride gas is passed into a saturated sodium chloride solution, the extra chloride ions push the equilibrium NaCl(s) ⇌ Na⁺ + Cl^- to the left, favouring precipitation of NaCl. This is the common-ion effect acting on a saturated solution.

Some compounds of transition metals do not show simple common-ion behaviour because transition metal ions often form complex ions with ligands such as Cl^-. For example, cuprous chloride, CuCl, is sparingly soluble in water, but on addition of excess chloride ions it dissolves due to formation of a soluble complex ion (for example [CuCl₂]^-), increasing the apparent solubility.

Quantitative effect of a common ion - simple derivation

Consider MX(s) ⇌ M⁺ + X^- with K_sp = [M⁺][X^-] and molar solubility in pure water equal to s.

[M⁺] = s
[X^-] = s
K_sp = s²

Now suppose a soluble salt that supplies X^- is present and its concentration of X^- from that source is c (c ≫ s). In the new equilibrium:

[M⁺] = s'
[X^-] = c + s'
K_sp = s'(c + s')
If c ≫ s', then s' ≈ K_sp/c.

Thus the solubility of MX in the presence of a large concentration of the common ion X^- is approximately inversely proportional to the common-ion concentration.

pH and solubility

Salts containing anions that are the conjugate bases of weak acids (for example CO₃^2-, PO₄^3-, F^-, OH^-) exhibit solubility that depends strongly on pH.

• Lowering the pH (adding H⁺) converts basic anions into their protonated forms (for example CO₃^2- → HCO₃^- → H₂CO₃), removing free anion from the equilibrium and shifting the dissolution equilibrium to the right, thereby increasing solubility.
• For metal hydroxides, increasing the acidity increases their solubility because OH^- is consumed by H⁺, so M(OH)_n(s) ⇌ Mⁿ⁺ + n OH^- shifts to the right.

These pH effects are exploited in processes such as removing carbonate hardness from water by precipitation or dissolving carbonate deposits by acid treatment.

Complex-ion formation and increased solubility

When a metal ion forms a stable complex with a ligand present in solution, the free metal ion concentration decreases and the dissolution equilibrium of the sparingly soluble salt shifts to the right, increasing solubility.

For example, for AgCl(s):
AgCl(s) ⇌ Ag⁺ + Cl^-, K_sp = [Ag⁺][Cl^-]

In the presence of excess Cl^-, complex ions such as [AgCl₂]^- can form:
Ag⁺ + 2 Cl^- ⇌ [AgCl₂]^-
Formation constant of the complex reduces free [Ag⁺], so more AgCl dissolves to re-establish K_sp. Thus complex formation increases apparent solubility.

Ionic product (Q) and precipitation

The ionic product or reaction quotient Q for a dissolution is defined exactly like K_sp but for the instantaneous concentrations (not necessarily at equilibrium):

Q = [M⁺]^a[X^-]^b (where a and b are stoichiometric numbers)

• If Q < K_sp, the solution is unsaturated and no precipitation occurs; the solid can dissolve further.
• If Q = K_sp, the solution is saturated and is at equilibrium.
• If Q > K_sp, the solution is supersaturated and precipitation will occur until Q is reduced to K_sp.

Applications of K_sp and the common-ion effect

• Water treatment and purification: In removing hardness, addition of soluble carbonate (Na₂CO₃) precipitates calcium as CaCO₃ because the common-ion/solubility equilibria drive CaCO₃ out of solution; the precipitate can be filtered off.
• Obtaining pure precipitates: Controlled precipitation using common ions gives fine, pure precipitates (for example, producing fine CaCO₃ for industrial uses such as toothpaste).
• Salting-out of soaps: Soaps (sodium salts of long-chain fatty acids) are precipitated by adding NaCl; the common Na⁺ ion reduces the solubility of the soap, aiding its separation and purification.
• Gravimetric analysis and qualitative inorganic analysis: Controlled precipitation using K_sp principles allows selective separation of ions as insoluble salts.
• Analytical chemistry: Predicting whether a precipitate forms in a mixture of ions depends on comparing Q and K_sp.

Worked example - effect of a common ion (symbolic)

Consider AgCl(s) ⇌ Ag⁺ + Cl^- with K_sp = [Ag⁺][Cl^-] and molar solubility in pure water equal to s.

[Ag⁺] = s
[Cl^-] = s
K_sp = s²

If the solution already contains chloride ions at concentration c (for example from NaCl), then at the new equilibrium:

[Ag⁺] = s'
[Cl^-] = c + s'
K_sp = s'(c + s')

If c ≫ s', then s' ≈ K_sp / c, showing that the solubility of AgCl is reduced when a common ion Cl^- is present.

Summary

The solubility product constant (K_sp) is a central concept for understanding the solubility of sparingly soluble salts. Solubility depends on lattice and solvation enthalpies, temperature, presence of common ions, pH, complex formation and ionic interactions. The common-ion effect reduces solubility and is routinely used in water treatment, precipitation techniques and analytical chemistry. Comparing the instantaneous ionic product (Q) with K_sp allows prediction of precipitation, while formation of complexes or changes in pH can greatly alter solubility in ways that can be exploited in laboratory and industrial processes.