91Ó°ÊÓ

For weak acid and the weak base,\({K_b} = 5.9 \times 1{0^{ - 4}}\)

Given weak conjugated acid as\({\left( {C{H_3}} \right)_2}NH_2^ + \)

\({\left( {C{H_3}} \right)_2}NH_2^ + (aq) + {H_2}O(l) \to {\left( {C{H_3}} \right)_2}NH(aq) + {H_3}{O^ + }(aq)\)

To calculate Ionization equation

\({K_a} = \frac{{c\left( {{{\left( {C{H_3}} \right)}_2}NH} \right) \times c\left( {{H_3}{O^ + }} \right)}}{{c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right)}}\)

Ionization constant of a weak base \({\left( {C{H_3}} \right)_2}NH\)Dissociation equation is

\({\left( {C{H_3}} \right)_2}NH(aq) + {H_2}O(l) \to {\left( {C{H_3}} \right)_2}NH_2^ + (aq) + O{H^ - }(aq)\)

Ionization constant is

\({K_b} = \frac{{c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right) \times c\left( {O{H^ - }} \right)}}{{c\left( {{{\left( {C{H_3}} \right)}_2}NH} \right)}}\)

\({K_b} = 5.9 \times 1{0^{ - 4}}\)

Equation for dissociation for water is

\({H_2}O(l) + {H_2}O(l) \to {H_3}{O^ + }(aq) + O{H^ - }(aq)\)

Constant of water ionization for \({K_w}\)is

\({K_w} = c\left( {{H_3}{O^ + }} \right) \times c\left( {O{H^ - }} \right)\)

\({K_w} = 1.0 \times 1{0^{ - 14}}\)

Concentration of \({\left( {C{H_3}} \right)_2}NH\)is

\({K_b} = \frac{{c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right) \times c\left( {O{H^ - }} \right)}}{{c\left( {{{\left( {C{H_3}} \right)}_2}NH} \right)}}\)

\(\begin{aligned}{{K_b} \times c\left( {{{\left( {C{H_3}} \right)}_2}NH} \right) = c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right) \times c\left( {O{H^ - }} \right)}\\{c\left( {{{\left( {C{H_3}} \right)}_2}NH} \right) = \frac{{c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right) \times c\left( {O{H^ - }} \right)}}{{{K_b}}}}\end{aligned}\)

\({K_a} = \frac{{c\left( {{{\left( {C{H_3}} \right)}_2}NH} \right) \times c\left( {{H_3}{O^ + }} \right)}}{{c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right)}}\)

\({K_a} = \frac{{c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right) \times c\left( {O{H^ - }} \right) \times c\left( {{H_3}{O^ + }} \right)}}{{c\left( {{{\left( {C{H_3}} \right)}_2}NH_2^ + } \right) \times {K_b}}}\)

Concentration of \({\left( {C{H_3}} \right)_2}NH_2^ + \)each other, we get

\({K_a} = \frac{{c\left( {O{H^ - }} \right) \times c\left( {{H_3}{O^ + }} \right)}}{{{K_b}}}\)

Replacing \(c\left( {O{H^ - }} \right) \times c\left( {{H_3}{O^ + }} \right)\) as \({K_w}\)

\({K_a} = \frac{{{K_w}}}{{{K_b}}}\)

Ionization of a weak acid is

\(\begin{aligned}{{K_a} = \frac{{1.0 \times 1{0^{ - 14}}}}{{5.9 \times 1{0^{ - 4}}}}}\\{{K_a} = 1.7 \times 1{0^{ - 11}}}\end{aligned}\)