Ostwald's dilution law

Arrhenius concept of acids and bases was expressed quantitatively by F. W. Ostwald in the form of the dilution law in 1888.

a. Weak acids : Consider an equilibrium of weak acid HA that exists in solution partly as the undissociated species HA and partly as $\mathrm{H^{+}}$ and $\mathrm{A^{-}}$ ions. Then

The acid dissociation constant is given by Eq. (3.3),

Suppose 1 mol of acid HA is initially present in volume $V$ dm$^3$ of the solution. At equilibrium the fraction dissociated would be $\alpha$, where $\alpha$ is the degree of dissociation of the acid. The fraction of the acid that remains undissociated would be $(1-\alpha)$.

Thus, at equilibrium $[\mathrm{HA}] = \dfrac{1-\alpha}{V}\ \mathrm{mol\,dm^{-3}}$, and $[\mathrm{H^{+}}] = [\mathrm{A^{-}}] = \dfrac{\alpha}{V}\ \mathrm{mol\,dm^{-3}}$.

Substituting these in Eq. (3.3),

If $c$ is the initial concentration of the acid in mol dm$^{-3}$ and $V$ is the volume in dm$^3$ mol$^{-1}$ then $c = 1/V$. Replacing $1/V$ in Eq. (3.5) by $c$ we get

For the weak acid HA, $\alpha$ is very small, or $(1-\alpha) \cong 1$. With this, Eq. (3.5) and (3.6) reduce to

The Eq. (3.8) implies that the degree of dissociation of a weak acid is inversely proportional to the square root of its concentration, or directly proportional to the square root of the volume of the solution containing 1 mol of the weak acid.

b. Weak base : Consider 1 mol of weak base BOH dissolved in $V$ dm$^3$ of solution. The base dissociates partially as

The base dissociation constant is

Let the fraction dissociated at equilibrium be $\alpha$ and the fraction that remains undissociated be $(1-\alpha)$.

At equilibrium,

$[\mathrm{BOH}] = \frac{1-\alpha}{V}\ \mathrm{mol\,dm^{-3}}, \quad [\mathrm{B^{+}}] = [\mathrm{OH^{-}}] = \frac{\alpha}{V}\ \mathrm{mol\,dm^{-3}}$ …