Question

Write the equation relating \(K_{\mathrm{a}}\) for a weak acid and \(K_{\mathrm{b}}\) for its conjugate base. Use \(\mathrm{NH}_{3}\) and its conjugate acid \(\mathrm{NH}_{4}^{+}\) to derive the relationship between \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\).

Short answer

The relationship between the acid dissociation constant (Ka) and base dissociation constant (Kb) for a weak acid and its conjugate base is given by \(Ka = K_{w} / Kb\) where \(K_{w}\) is the ion product of water. This relationship can be derived using the ionization of ammonia (NH3) and its conjugate acid ammonium (NH4+).

Step-by-step solution

Write ionization equations for the weak acid and its conjugate base

Write two ionization equations, one for NH4+ acting as a weak acid and one for NH3 as a weak base. When NH4+ ionizes, it donates a proton to become NH3, and the proton reacts with water to form hydronium ions (H3O+). This is represented as:\[ \mathrm{NH}_{4}^{+} \rightleftharpoons \mathrm{NH}_{3} + \mathrm{H}_{3}\mathrm{O}^{+} \]The ionization of NH3 acts as a weak base accepts a proton from water to become NH4+ and a hydroxide ion (OH-) is formed. This is represented as:\[ \mathrm{NH}_{3} + \mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{NH}_{4}^{+} + \mathrm{OH}^{-} \]

Write the expressions for Ka and Kb

The acidity constant Ka, is represented using the equilibrium constant expression as follows: \[ K_{a} = \frac{{[\mathrm{NH}_{3}][\mathrm{H}_{3}\mathrm{O}^{+}]}}{{[\mathrm{NH}_{4}^{+}]} \]Similarly, the base ionization constant Kb, is represented as follows:\[ K_{b} = \frac{{[\mathrm{NH}_{4}^{+}][\mathrm{OH}^{-}]}}{{[\mathrm{NH}_{3}]} \]

Derive the relationship between Ka and Kb using the ion product for water

For any conjugate acid-base pair, the product of Ka and Kb equals the ion product for water, Kw. Given the formula Kw = [H3O+][OH-] at a particular temperature usually 25°C, where Kw = 1 x 10^-14. This relationship is written as:\[ K_{w} = K_{a} \times K_{b} Or,\[ K_{a} = \frac{{K_{w}}}{{K_{b}}} \] Where Ka is the acid dissociation constant, Kb is the base dissociation constant, and Kw is the ion product of water.