Q.vi. Derive the relationship between degree of dissociation and dissociation constant in weak electrolytes.

[!NOTE]
The textbook numbers this item vi. -- its own printed numbering skips v. (iv. jumps straight to vi.), so from here on the printed numbers sit one ahead of the question positions. Nothing is missing from the book.

Step 1. General setup. Consider a general weak electrolyte AB that ionizes as $AB(aq)\rightleftharpoons A^+(aq)+B^-(aq)$ (this applies identically whether AB is a weak acid HA, giving Ka, or a weak base BOH, giving Kb -- sections 3.4.1/3.4.2 develop both in parallel). Start with 1 mol of AB dissolved in V dm3 of solution.

Step 2. Equilibrium amounts. If the degree of dissociation at equilibrium is $\alpha$, then the amount of AB remaining un-ionized is $(1-\alpha)$ mol, and the amounts of A+ and B- formed are each $\alpha$ mol (one of each ion per molecule of AB that dissociates).

Step 3. Equilibrium concentrations. Dividing by the volume V: $[AB]=\dfrac{1-\alpha}{V}$, $[A^+]=[B^-]=\dfrac{\alpha}{V}$ mol dm-3.

Step 4. Substitute into the equilibrium constant. $K=\dfrac{[A^+][B^-]}{[AB]}=\dfrac{(\alpha/V)(\alpha/V)}{(1-\alpha)/V}=\dfrac{\alpha^2}{(1-\alpha)V}$.

Step 5. Introduce c = 1/V. Writing $c=1/V$ as the initial molar concentration, $K=\dfrac{\alpha^2c}{1-\alpha}$ -- this is the exact relationship between degree of dissociation and the dissociation constant for any weak electrolyte (matches Eq. 3.6 for Ka and Eq. 3.10 for Kb). …

Repeat the general Ostwald's-dilution-law derivation (equilibrium amounts -> concentrations -> substitute into K -> introduce c -> simplify for small alpha) in the generi …

• Deriving only the acid-specific (Ka) version when the question asks for the GENERAL relationship applicable to any weak electrolyte (acid or base). …