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The $K_{sp}$ of $Ag_2CrO_4, AgCl, AgBr$ and $AgI$ are respectively, $1.1 \times 10^{-12}, 1.8 \times 10^{-10},$$ 5.0 \times 10^{-13}, 8.3 \times 10^{-17}.$ Which one of the following salts will precipitate last if $AgNO_3$ solution is added to the solution containing equal moles of $NaCl, NaBr, NaI$ and $Na_2CrO_4$ ?
$AgBr$
$Ag_2CrO_4$
$AgI$
$AgCl$
Solution
$\mathrm{Ag}_{2} \mathrm{CrO}_{4} \rightleftharpoons 2 \mathrm{Ag}^{+}+\mathrm{CrO}_{4}^{2-}$
solubility product $\mathrm{K}_{\mathrm{sp}}=(2 \mathrm{s})^{2} \times \mathrm{S}=4 \mathrm{s}^{3}$
$\mathrm{K}_{\mathrm{sp}}=\left(1.1 \times 10^{-12}\right)$
$s=3 \sqrt{\frac{\mathrm{K}_{\mathrm{sp}}}{4}}=0.65 \times 10^{-4}$
$\mathrm{AgCl} \rightleftharpoons \mathrm{Ag}^{+}+\mathrm{Cl}^{-}$
$\mathrm{K}_{\mathrm{sp}}=5 \times 5 \quad\left(\mathrm{K}_{\mathrm{sp}}=1.8 \times 10^{-10}\right)$
$s=\sqrt{\mathrm{K}_{\mathrm{sp}}}=1.34 \times 10^{-5}$
AgBr $\rightleftharpoons \mathrm{Ag}^{+}+\mathrm{Br}^{-}$
$\mathrm{K}_{\mathrm{sp}}=5 \times 5 \quad\left(\mathrm{K}_{\mathrm{sp}}=5 \times 10^{-13}\right.$
$S=\sqrt{K_{s p}}=0.71 \times 10^{-6}$
$\mathrm{Agl} \rightleftharpoons \mathrm{Ag}^{+}+\mathrm{I}^{-}$
$\left.\mathrm{Ksp}=\mathrm{S} \times \mathrm{S} \quad \mathrm{K}_{\mathrm{sp}}=8.3 \times 10^{-17}\right)$
$S=\sqrt{K_{\mathrm{sp}}}=0.9 \times 10^{-8}$
therefore, solubility of $\mathrm{Ag}_{2} \mathrm{CrO}_{4}$ is highest so it will precipitate last.