Chapter 16, Problem 77QP

Interpretation Introduction

Interpretation:

The relation between $K_{\text{a}}$ and $K_{\text{b}}$ of a weak acid and its conjugate base is to be written and proved.

Concept introduction:

Relationship between $\left[{\text{H}}_3{\text{O}}^{+}\right]$ and $\left[{\text{OH}}^{-}\right]$ is:

$K_{\text{w}}=\left[{\text{H}}_3{\text{O}}^{+}\right]\left[{\text{OH}}^{-}\right]$ …… (1)

Value of $K_{\text{w}}$ is $1.0\times {10}^{-14}$.

The ionization of a weak base takes place as:

${\text{B}}_{\left(\text{aq}\right)}+{\text{H}}_2{\text{O}}_{\left(\text{l}\right)}\rightleftharpoons {\text{BH}}^{\text{+}}{}_{\left(\text{aq}\right)}+{\text{OH}}^{-}{}_{\left(\text{aq}\right)}$

$K_{\text{b}}$ is the measure of ionization of a base and is known as the base-ionization constant, which is specific at a particular temperature.

$K_{\text{b}}=\frac{\left[{\text{BH}}^{+}\right]\left[{\text{OH}}^{-}\right]}{\left[\text{B}\right]}$ …… (2)

The ionization of the weak acid takes place as:

${\text{HA}}_{\left(\text{aq}\right)}+{\text{H}}_2{\text{O}}_{\left(\text{l}\right)}\rightleftharpoons {\text{H}}_{\text{3}}{\text{O}}^{\text{+}}{}_{\left(\text{aq}\right)}+{\text{A}}^{-}{}_{\left(\text{aq}\right)}$

$K_{\text{a}}$ is the measure of dissociation of an acid and is known as the acid-ionization constant, which is specific at a particular temperature.

$K_{\text{a}}=\frac{\left[{\text{H}}_3{\text{O}}^{+}\right]\left[{\text{A}}^{-}\right]}{\left[\text{HA}\right]}$ …… (3)