How do you determine the \[{{\text{K}}_{\text{a}}}\] of weak acid and \[{{\text{K}}_{\text{b}}}\] of weak base?

Hint: To answer this question we should know what is \[{{\text{K}}_{\text{a}}}\] ,\[{{\text{K}}_{\text{b}}}\] .These are used to determine the strength of acids and bases. \[{{\text{K}}_{\text{a}}}\] is ionization constant for weak acids and \[{{\text{K}}_{\text{b}}}\] is ionization constant for weak base. So depending on these acids and bases can be weak and strong. If an acid or base is completely ionized in aqueous solution it is a strong acid or strong base. But if it is slightly ionized then it is a weak acid or weak base.

Complete step by step answer:
As we know that the reaction of weak acid with water produces \[{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{ + }}}\] or \[{{\text{H}}^{\text{ + }}}\]
To determine acid ionization constant \[{{\text{K}}_{\text{a}}}\] , following is the equation for ionization of weak acid $\left[ {HA} \right]$
 in water and [\[{{\text{A}}^{\text{ - }}}\]] is the conjugate base.
\[{\text{HA(aq) + }}{{\text{H}}_{\text{2}}}{\text{O(l)}} \rightleftarrows {{\text{H}}_{\text{3}}}{{\text{O}}^{\text{ + }}}{\text{(aq) + }}{{\text{A}}^{\text{ - }}}{\text{(aq)}}\]
The equilibrium constant for the equation can be written as,
\[{\text{K = }}\dfrac{{{\text{[}}{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{ + }}}{\text{][}}{{\text{A}}^{\text{ - }}}{\text{]}}}}{{{\text{[}}{{\text{H}}_{\text{2}}}{\text{O][HA]}}}}\]
Since concentration of water is constant so \[{\text{K}}\] can be written as \[{\text{K[}}{{\text{H}}_{\text{2}}}{\text{O]}}\] and new constant form is \[{{\text{K}}_{\text{a}}}\]and also known as acid dissociation constant.
\[{{\text{K}}_{\text{a}}}{\text{ = K[}}{{\text{H}}_{\text{2}}}{\text{O] = }}\dfrac{{{\text{[}}{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{ + }}}{\text{][}}{{\text{A}}^{\text{ - }}}{\text{]}}}}{{{\text{[HA]}}}}\]
We can write \[{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{ + }}}\] as \[{{\text{H}}^{\text{ + }}}\] but no \[{{\text{H}}^{\text{ + }}}\] ion exist in aqueous medium so we use \[{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{ + }}}\].
Now weak bases on reaction with water produces \[{\text{O}}{{\text{H}}^{\text{ - }}}\] (hydroxide) ions.
To determine base ionization constant \[{{\text{K}}_{\text{b}}}\] ,following is the equation for ionization of weak base $\left[ B \right]$ in water and [\[{\text{B}}{{\text{H}}^{\text{ + }}}\]] is conjugate acid.
\[{\text{B(aq) + }}{{\text{H}}_{\text{2}}}{\text{O(l)}} \rightleftarrows {\text{B}}{{\text{H}}^{\text{ + }}}{\text{(aq) + O}}{{\text{H}}^{\text{ - }}}{\text{(aq)}}\]
The equilibrium constant for the equation can be written as,
\[{{\text{K}}_b}{\text{ = K[}}{{\text{H}}_{\text{2}}}{\text{O] = }}\dfrac{{{\text{[B}}{{\text{H}}^{\text{ + }}}{\text{][O}}{{\text{H}}^{\text{ - }}}{\text{]}}}}{{{\text{[B]}}}}\]
If we know the value of \[{{\text{K}}_{\text{a}}}\] and \[{{\text{K}}_{\text{b}}}\] we can calculate \[{\text{p}}{{\text{K}}_{\text{a}}}\] and \[{\text{p}}{{\text{K}}_{\text{b}}}\]

Note:
 We should be noted that acid- base ionization constants are measured in terms of \[{{\text{H}}^{\text{ + }}}\] and \[{\text{O}}{{\text{H}}^{\text{ - }}}\] and therefore they don’t have any unit. Example of strong acid is \[{\text{HCl}}\] and \[{\text{C}}{{\text{l}}^{\text{ - }}}\] (chloride ion) is its weak conjugate base. Similarly \[{\text{C}}{{\text{H}}_{\text{3}}}{\text{COOH}}\] is weak acid and its strong base is \[{\text{C}}{{\text{H}}_{\text{3}}}{\text{CO}}{{\text{O}}^{\text{ - }}}\] (acetate ion).It should be noted that the larger the \[{{\text{K}}_{\text{a}}}\] ,the stronger is the acid(\[{\text{p}}{{\text{K}}_{\text{a}}}\]). Similarly the larger the \[{{\text{K}}_{\text{b}}}\] , the stronger is the base (\[{\text{p}}{{\text{K}}_{\text{b}}}\]) .