a. Show that the solubility of \(\mathrm{Al}(\mathrm{OH})_{3}\), as a function of \(\left[\mathrm{H}^{+}ight]\), obeys the equation

\[S=\left[\mathrm{H}^{+}ight]^{3} K_{\mathrm{sp}} / K_{\mathrm{w}}{ }^{3}+K K_{\mathrm{w}} /\left[\mathrm{H}^{+}ight]\]

where \(S=\) solubility \(=\left[\mathrm{Al}^{3+}ight]+\left[\mathrm{Al}(\mathrm{OH})_{4}{ }^{-}ight]\)and \(K\) is the equilibrium constant for

\[\mathrm{Al}(\mathrm{OH})_{3}(s)+\mathrm{OH}^{-}(a q) ightleftharpoons \mathrm{Al}(\mathrm{OH})_{4}^{-}(a q)\]

b. The value of \(K\) is 40.0 and \(K_{\text {sp }}\) for \(\mathrm{Al}(\mathrm{OH})_{3}\) is \(2 \times\) \(10^{-32}\). Plot the solubility of \(\mathrm{Al}(\mathrm{OH})_{3}\) in the \(\mathrm{pH}\) range 4-12.