Calculate the concentrations of ALL solute species in 0.25 M H3PO4(aq). Ka1 = 7.3 x 10-3, Ka2 = 6.2 x10-8, Ka3 = 2.1 x 10-13?

As Kₐ₁ ≫ Kₐ₂ ≫ Kₐ₃, the effect of lower degree dissociation is negligible to the higher degree dissociation.


First dissociation :
H₃PO₄(aq) + H₂O(l) ⇌ H₂PO₄⁻(aq) + H₃O⁺(aq) …. Kₐ₁ = 7.3 × 10⁻³

Initial concentrations :
[H₃PO₄]ₒ = 0.25 M
[H₂PO₄⁻]ₒ = [H₃O⁺]ₒ = 0 M

Equilibrium concentrations :
Assume that x M of H₃PO₄ is consumed on establishment of the equilibrium.
[H₃PO₄] = (0.25 - x) M
[H₂PO₄⁻] = [H₃O⁺] = x M

Kₐ₁ = [H₂PO₄⁻] [H₃O⁺] / [H₃PO₄]
7.3 × 10⁻³ = x² / (0.25 - x)
x² + (7.3 × 10⁻³)x - (1.825 × 10⁻³) = 0
Solving the quadratic equation, and rejected the negative answer.
x = 0.039
[H₃PO₄] = (0.25 - 0.039) M = 0.21 M
[H₂PO₄⁻] = [H₃O⁺] = 0.039 M


Second dissociation :
H₂PO₄⁻ (aq) + H₂O(l) ⇌ HPO₄²⁻(aq) + H₃O⁺(aq) …. Kₐ₂ = 6.2 × 10⁻⁸

Initial concentrations :
[H₂PO₄⁻]ₒ = [H₃O⁺]ₒ = 0.039 M
[HPO₄²⁻]ₒ = 0 M

Equilibrium concentrations :
Assume that y M of H₂PO₄⁻ is consumed on establishment of the equilibrium :
[H₂PO₄⁻] = (0.039 - y) M ≈ 0.039 M, by assuming that 0.039 ≫ y
[H₃O⁺] = (0.039 + y) ≈ 0.039 M, by assuming that 0.039 ≫ y
[HPO₄²⁻] = y M

Kₐ₂ = [HPO₄²⁻] [H₃O⁺] / [H₂PO₄⁻]
6.2 × 10⁻⁸ = 0.039y / 0.039
y = 6.2 × 10⁻⁸ (This value agrees with the assumption that 0.039 ≫ y)
[HPO₄²⁻] = 6.2 × 10⁻⁸ M


Third dissociation :
HPO₄²⁻ (aq) + H₂O(l) ⇌ PO₄³⁻(aq) + H₃O⁺(aq) …. Kₐ₃ = 2.1 × 10⁻¹³

Initial concentrations :
[HPO₄²⁻]ₒ = 6.2 × 10⁻⁸ M
[H₃O⁺]ₒ = 0.039 M
[PO₄³⁻]ₒ = 0 M

Equilibrium concentrations :
Assume that z M of HPO₄²⁻ is consumed on establishment of the equilibrium :
[HPO₄²⁻] = (6.2 × 10⁻⁸ - z) M ≈ 6.2 × 10⁻⁸ M, by assuming that 6.2 × 10⁻⁸ ≫ z
[H₃O⁺] = (0.039 + z) ≈ 0.039 M, by assuming that 6.2 × 10⁻⁸ ≫ z
[PO₄³⁻] = z M

Kₐ₂ = [PO₄³⁻] [H₃O⁺] / [HPO₄²⁻]
2.1 × 10⁻¹³ = 0.039z / (6.2 × 10⁻⁸)
z = 3.3 × 10⁻¹⁹ (This value agrees with the assumption that 6.2 × 10⁻⁸ ≫ z)
[PO₄³⁻] = 3.3 × 10⁻¹⁹ M


Answer :
[H₃PO₄] = 0.21 M
[H₂PO₄⁻] = [H₃O⁺] = 0.039 M
[HPO₄²⁻] = 6.2 × 10⁻⁸ M
[PO₄³⁻] = 3.3 × 10⁻¹⁹ M