Ostwald's Dilution Law

Ostwald's dilution law relates a weak acid's dissociation constant $K_a$ to its degree of dissociation $\alpha$ (the fraction of the total moles that dissociates at equilibrium, $\alpha = \dfrac{\text{moles dissociated}}{\text{total moles}}$) and its concentration $C$. Working through acetic acid, $CH_3COOH \rightleftharpoons H^+ + CH_3COO^-$, with initial concentration $C$ and degree of dissociation $\alpha$, the equilibrium concentrations are $(1-\alpha)C$, $\alpha C$ and $\alpha C$ respectively, so $K_a=\dfrac{(\alpha C)(\alpha C)}{(1-\alpha)C}=\dfrac{\alpha^2 C}{1-\alpha}$. Because a weak acid dissociates only to a small extent, $\alpha$ is small enough that $(1-\alpha)\approx1$, simplifying the law to $K_a \approx \alpha^2 C$, so $\alpha = \sqrt{K_a/C}$. This is Ostwald's dilution law: as dilution increases (C decreases), the degree of dissociation of a weak electrolyte increases -- for example, for an acid of $K_a=4\times10^{-4}$, $\alpha$ is 0.2 at $C=1\times10^{-2}$ M but rises to $\alpha=2$ ... i.e. the book's own worked figures show $\alpha=0.2$ at $10^{-2}$ M and a ten-times-larger dissociation at $10^{-4}$ M, a hundred-fold dilution. From $\alpha$, the hydrogen ion concentration follows as $[H^+]=\alpha C = \sqrt{K_a C}\cdot\sqrt{C}/\sqrt{C}$, which simplifies cleanly to $[H^+]=\sqrt{K_a C}$; the analogous result for a weak base is $[OH^-]=\sqrt{K_b C}$.

Dissociation of acetic acid in terms of degree of dissociation. The bookkeeping table used to derive Ostwald's dilution law for one mole of acetic acid dissociating to degree $\alpha$ in a solution of total concentration C.

Example 8.4 – Ka from percentage dissociation. A 0.10M solution of a weak electrolyte is 1.20% dissociated at $25^\circ C$; find its dissociation constant. $\alpha = 1.20\% = 1.2\times10^{-2}$. Using $K_a = \alpha^2 C = (1.2\times10^{-2})^2 \times 0.1 = 1.44\times10^{-4}\times10^{-1} = 1.44\times10^{-5}$. …