Chapter 19, Problem 110A

Interpretation Introduction

Interpretation : The conjugate acid-base pair ${\text{K}}_{\text{a}}{\text{×K}}_{\text{b}}{\text{=K}}_{\text{w}}$ has to be generalized.

Concept Introduction : The adjective conjugate is used to differentiate the reactants from the products of the forward reaction. This word's root is a Latin verb that means "to connect together."

Answer to Problem 110A

The derivation of conjugate acid-base pair ${\text{K}}_{\text{a}}{\text{×K}}_{\text{b}}{\text{=K}}_{\text{w}}$ has been generalized.

Explanation of Solution

When a base pick up a hydrogen ion, a conjugate acid is created as an ion or molecule.

An ion or molecule that is left over after an acid loses a hydrogen ion is known as a conjugate base.

Two ions or molecules that are connected by the loss or gain of one hydrogen ion are referred to as a conjugate acid-base pair.

Consider a weak acid and its conjugate base.

  $\text{HA}\rightleftharpoons {\text{H}}^{+}+{\text{A}}^{-}$

The acid dissociation constant is given as

  ${\text{K}}_{\text{a}}=\frac{\left[{\text{H}}^{+}\right]\left[{\text{A}}^{-}\right]}{\left[\text{HA}\right]}$

Consider a weak base and its conjugate acid.

  ${\text{A}}^{-}+{\text{H}}_{\text{2}}\text{O}\rightleftharpoons {\text{OH}}^{-}+\text{HA}$

The base dissociation constant is given as

  ${\text{K}}_{\text{b}}=\frac{\left[\text{HA}\right]\left[{\text{OH}}^{-}\right]}{\left[{\text{A}}^{-}\right]}$

The formula of ion product constant is given as

  ${\text{K}}_{\text{w}}=\left[{\text{H}}^{+}\right]\left[{\text{OH}}^{-}\right]$

The product of acid dissociation constant and base dissociation constant is given as

  $\begin{array}{c}{\text{K}}_{\text{a}}{\text{×K}}_{\text{b}}\text{=}\frac{\left[{\text{H}}^{+}\right]\cancel{\left[{\text{A}}^{-}\right]}}{\cancel{\left[\text{HA}\right]}}\times \frac{\cancel{\left[\text{HA}\right]}\left[{\text{OH}}^{-}\right]}{\cancel{\left[{\text{A}}^{-}\right]}}\\ =\left[{\text{H}}^{+}\right]\left[{\text{OH}}^{-}\right]\\ ={\text{K}}_{\text{w}}\end{array}$

The product of acid dissociation constant and base dissociation constant gives the ion product constant.