Law of dilution

Wilhelm Ostwald’s dilution law is a relationship between the dissociation constant K_d and the degree of dissociation α of a weak electrolyte. The law takes the form

${\displaystyle K_{d}={\cfrac {\ce {[A+][B^{-}]}}{\ce {[AB]}}}={\frac {\alpha ^{2}}{1-\alpha }}\cdot c_{0}}$

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

Concerning conductivity, this results in the following relation:

$K_{c}={\cfrac {\Lambda _{c}^{2}}{(\Lambda _{0}-\Lambda _{c})\Lambda _{0}}}\cdot c$

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:

${\displaystyle {\ce {AB<=>{A+}+B^{-}}}}$

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

${\displaystyle K_{d}={\cfrac {\ce {[A+][B^{-}]}}{\ce {[AB]}}}={\cfrac {(\alpha c_{0})(\alpha c_{0})}{(1-\alpha )c_{0}}}={\cfrac {\alpha ^{2}}{1-\alpha }}\cdot c_{0}}$

For very weak electrolytes (however, neglecting 'α' for most weak electrolytes yields counterproductive result) $\alpha \ll 1$, implying that (1 - α) ≈ 1.

$K_{d}={\frac {\alpha ^{2}}{1-\alpha }}\cdot c_{0}\approx \alpha ^{2}c_{0}$

This gives the following results;

$\alpha ={\sqrt {{\cfrac {K_{d}}{c_{0}}}}}$

Thus, 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.

${\displaystyle {\ce {[A+]}}={\ce {[B^{-}]}}=\alpha c_{0}={\sqrt {K_{d}c_{0}}}}$

Limitations

The law holds good only for weak electrolytes and fails completely in the case of strong electrolytes. The value of 'α' is determined by conductivity measurements by applying the formula Λ/Λ∞. The value of 'α' determined at various dilutions of an electrolyte when substituted in Eq. (i) gives a constant value of K only in the case of weak electrolytes like CH3COOH, NH4OH, etc. the cause of failure of Ostwald's dilution law in the case of strong electrolytes is due to the following factors"

(i) The law is based on the fact that only a portion of the electrolyte is dissociated into ions at ordinary dilution and completely at infinite dilution. Strong electrolytes are almost completely ionized at all dilutions and Λ/Λ∞ does not give accurate value of 'α'.

(ii) When concentration of the ions is very high, the presence of charges on the ions appreciably effects the equilibrium. Hence, law of mass action in its simple form cannot be strictly applied in the case of strong electrolytes.

• K_c: constant of dissociation
• $\Lambda _{c}$: equivalent conductivity
• $\Lambda _{0}$: boundary conductivity
• c: concentration of electrolyte.

See also

Autosolvolysis