Law of dilution

${\displaystyle K_d=\frac{[{\text{A}}^{+}][{\text{B}}^{-}]}{[\text{AB}]}=\frac{{\alpha }^2}{1-\alpha }\cdot c_0}$

Where the square brackets denote concentration, and c₀ is the total concentration of electrolyte.

Using ${\displaystyle \alpha ={\mathrm{\Lambda }}_c/{\mathrm{\Lambda }}_0}$, where ${\displaystyle {\mathrm{\Lambda }}_c}$ is the molar conductivity at concentration c and ${\displaystyle {\mathrm{\Lambda }}_0}$ is the limiting value of molar conductivity extrapolated to zero concentration or infinite dilution, this results in the following relation:

${\displaystyle K_d=\frac{{\mathrm{\Lambda }}_c^2}{({\mathrm{\Lambda }}_0-{\mathrm{\Lambda }}_c){\mathrm{\Lambda }}_0}\cdot c_0}$

Derivation

Consider a binary electrolyte AB which dissociates reversibly into A⁺ and B⁻ ions. Ostwald noted that the law of mass action can be applied to such systems as dissociating electrolytes. The equilibrium state is represented by the equation:

${\displaystyle \text{AB}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{A}}^{+}+{\text{B}}^{-}}$

If α is the fraction of dissociated electrolyte, then αc₀ is the concentration of each ionic species. (1 - α) must, therefore be the fraction of undissociated electrolyte, and (1 - α)c₀ the concentration of same. The dissociation constant may therefore be given as

${\displaystyle K_d=\frac{[{\text{A}}^{+}][{\text{B}}^{-}]}{[\text{AB}]}=\frac{(\alpha c_0)(\alpha c_0)}{(1-\alpha )c_0}=\frac{{\alpha }^2}{1-\alpha }\cdot c_0}$

For very weak electrolytes ⁠${\displaystyle \alpha \ll 1}$⁠, implying that (1 - α) ≈ 1.

${\displaystyle K_d=\frac{{\alpha }^2}{1-\alpha }\cdot c_0\approx {\alpha }^2{c}_0}$

This gives the following results;

${\displaystyle \alpha =\sqrt{\frac{K_d}{c_0}}}$

Thus, the degree of dissociation of a weak electrolyte is proportional to the inverse square root of the concentration, or the square root of the dilution. The concentration of any one ionic species is given by the root of the product of the dissociation constant and the concentration of the electrolyte.

${\displaystyle [{\text{A}}^{+}]=[{\text{B}}^{-}]=\alpha c_0=\sqrt{K_d{c}_0}}$