Determine the solubilities of silver chromate,barium chromate,ferric hydroxide,lead chloride,and mercurous iodide at $298 \, K$ from their solubility product constants. Determine also the molarities of individual ions. (Given $K_{sp}$ values: $Ag_{2}CrO_{4} = 1.1 \times 10^{-12}$,$BaCrO_{4} = 1.2 \times 10^{-10}$,$Fe(OH)_{3} = 1.0 \times 10^{-38}$,$PbCl_{2} = 1.6 \times 10^{-5}$,$Hg_{2}I_{2} = 4.5 \times 10^{-29}$)

$(1)$ Silver chromate $(Ag_{2}CrO_{4})$:
$Ag_{2}CrO_{4} \rightleftharpoons 2Ag^{+} + CrO_{4}^{2-}$
$K_{sp} = [Ag^{+}]^{2} [CrO_{4}^{2-}] = (2s)^{2}(s) = 4s^{3}$
$1.1 \times 10^{-12} = 4s^{3} \Rightarrow s = 0.65 \times 10^{-4} \, M$
$[Ag^{+}] = 1.30 \times 10^{-4} \, M, [CrO_{4}^{2-}] = 0.65 \times 10^{-4} \, M$
$(2)$ Barium chromate $(BaCrO_{4})$:
$BaCrO_{4} \rightleftharpoons Ba^{2+} + CrO_{4}^{2-}$
$K_{sp} = [Ba^{2+}][CrO_{4}^{2-}] = s^{2}$
$1.2 \times 10^{-10} = s^{2} \Rightarrow s = 1.09 \times 10^{-5} \, M$
$[Ba^{2+}] = 1.09 \times 10^{-5} \, M, [CrO_{4}^{2-}] = 1.09 \times 10^{-5} \, M$
$(3)$ Ferric hydroxide $(Fe(OH)_{3})$:
$Fe(OH)_{3} \rightleftharpoons Fe^{3+} + 3OH^{-}$
$K_{sp} = [Fe^{3+}][OH^{-}]^{3} = (s)(3s)^{3} = 27s^{4}$
$1.0 \times 10^{-38} = 27s^{4} \Rightarrow s = 1.39 \times 10^{-10} \, M$
$[Fe^{3+}] = 1.39 \times 10^{-10} \, M, [OH^{-}] = 4.17 \times 10^{-10} \, M$
$(4)$ Lead chloride $(PbCl_{2})$:
$PbCl_{2} \rightleftharpoons Pb^{2+} + 2Cl^{-}$
$K_{sp} = [Pb^{2+}][Cl^{-}]^{2} = (s)(2s)^{2} = 4s^{3}$
$1.6 \times 10^{-5} = 4s^{3} \Rightarrow s = 1.58 \times 10^{-2} \, M$
$[Pb^{2+}] = 1.58 \times 10^{-2} \, M, [Cl^{-}] = 3.16 \times 10^{-2} \, M$
$(5)$ Mercurous iodide $(Hg_{2}I_{2})$:
$Hg_{2}I_{2} \rightleftharpoons Hg_{2}^{2+} + 2I^{-}$
$K_{sp} = [Hg_{2}^{2+}][I^{-}]^{2} = (s)(2s)^{2} = 4s^{3}$
$4.5 \times 10^{-29} = 4s^{3} \Rightarrow s = 2.24 \times 10^{-10} \, M$
$[Hg_{2}^{2+}] = 2.24 \times 10^{-10} \, M, [I^{-}] = 4.48 \times 10^{-10} \, M$