\(Al(OH)_{3}\rightleftharpoons Al^{3+}_{(aq)}+3OH^{-}_{(aq)}\)
\(K_{sp}=[Al^{3+}][OH^{-}]^{3}\)
Al(OH)₃ precipitates when ionic product > K_sp
K_s = 1.0 \(\times\) 10^-15m, [Al³⁺] = 1.0 \(\times\) 10^-3m
1.0 \(\times\) 10^-15 = [1.0 \(\times\) 10^-3][OH^-]³
\(\left[\mathrm{OH}^{-}\right]^3=\frac{1.0 \times 10^{-15}}{1.0 \times 10^{-3}}\)
[OH^-]³ = 1.0 \(\times\) 10^-12
(or)
[OH^-]³ = 10^-4M
[H⁺][OH^-] = 10^-4M
[H⁺] = \(\frac{10^{14}}{10^{-4}}\) =10^-10
Here,
pH = 10
ie., At pH = 10, Al(OH)₃ gets precipitated on the addition of NH₄Cl & NH₄OH solution.

K_sp =1.1 \(\times\)10^-2, [K₂, CrO₄]= 0.1M
\(\underset {s}{Ag_{2}CrO_{4}}\rightleftharpoons \underset{2s}{2Ag^{+}}+\underset{s}{CrO^{2-}_{4}}\)
\(\underset{0.1M}{K_{2}CrO_{4}}\rightleftharpoons \underset{0.2M}{2K^{+}}+\underset{2 \times 0.1M}{CrO^{2-}_{4}}\)
\({\left[\mathrm{Ag}^{+}\right] } =2 \mathrm{~s} ;\left[\mathrm{CrO}_{4}^{2-}\right]=(\mathrm{S}+0.1) \simeq 0.1  \)
\(\mathrm{~K}_{\mathrm{sp}} =\left[\mathrm{Ag}^{+}\right]^{2}\left[\mathrm{CrO}_{4}^{2-}\right]  \)
\(1.1 \times 10^{-12} =(2 \mathrm{~s})^{2}(0.1)  \)
\(1.1 \times 10^{-12} =0.4 \mathrm{~s}^{2}  \)
\(0.4 \mathrm{~s}^{2} =1.1 \times 10^{-12}  \)
\(\mathrm{~s}^{2} =\frac{1.1 \times 10^{-12}}{0.4}=2.75 \times 10^{-12}  \)
\(\mathrm{~s} =\sqrt{2.75 \times 10^{-12}}  \)
\(\mathrm{~s} =1.658 \times 10^{-6} \mathrm{M}\)

\(Ag_{2}CrO_{4}(s)\rightleftharpoons 2Ag^{+}_{aq}+CrO^{2-}_{4}(aq)\)
\(K_{sp}=[Ag^{+}]^{2}[CrO_{4}^{2-}]\)
\(=(5\times10^{-5})^2(4.4\times10^{-4})\)
=\((1.1\times10^{-12})\)

\(\mathrm{CaF}_{2(\mathrm{~s})} \rightleftharpoons \mathrm{Ca}_{\mathrm{(aq)}}^{2+}+2 \mathrm{~F}^{-}_  {(\mathrm{aq}) }\)
                            s           2s 
\({\left[\mathrm{Ca}^{2+}\right] } =3.3 \times 10^{-4} \mathrm{M}  \)
\({\left[\mathrm{F}^{-}\right] } =2 \times 3.3 \times 10^{-4}=6.6 \times 10^{-4} \mathrm{M}  \)
\(\mathrm{K}_{\mathrm{sp}} =\left[\mathrm{Ca}^{2+}\right]\left[\mathrm{F}^{-}\right]^{2}  \)
\(=\left(3.3 \times 10^{-4}\right)\left(6.6 \times 10^{-4}\right)^{2}=1.44 \times 10^{-10}\)

\(\mathrm{Ag}_{2} \mathrm{CrO}_{4(\mathrm{~s})} \rightleftharpoons 2 \mathrm{Ag}_{(\mathrm{aq})}^{+}+\mathrm{CrO}_{4{(\mathrm{aq})}}^{2-}\\ \quad s \quad \quad \quad \quad \quad 2s \quad \quad \quad \quad s\)
\(\left[\mathrm{Ag}^{+}\right]=2 \mathrm{~s}+0.01 \)
\(\simeq 0.01 \)
\((\because 2 s<<0.01) \)
\(\left[\mathrm{CrO}_{4}^{2-}\right]=\mathrm{S} \)
\(\mathrm{AgNO}_{3(\mathrm{~s})} \rightleftharpoons \mathrm{Ag}_{(\mathrm{aq})}^{+}+\mathrm{NO}_{3_{(\mathrm{aq})}}^{-}\\ 0.01M \quad \quad 0.01M \quad \quad 0.01M \quad\)
\(\mathrm{K}_{\mathrm{sp}}=\left[\mathrm{Ag}^{+}\right]^{2}\left[\mathrm{CrO}_{4}^{2-}\right]  \)
\(1 \times 10^{-12}=(0.01)^{2}(\mathrm{~s})  \)
\(\mathrm{S}=\frac{1 \times 10^{-12}}{10^{-4}}=1 \times 10^{-8} \mathrm{M}\)