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Which one of the following arrangements represents the correct order of solubilities of sparingly soluble salts $\mathrm{Hg}_2 \mathrm{Cl}_2, \mathrm{Cr}_2\left(\mathrm{SO}_4\right)_3$, $\mathrm{BaSO}_4$ and $\mathrm{CrCl}_3$ respectively?
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Verified Answer
The correct answer is:
$\mathrm{BaSO}_4>\mathrm{Hg}_2 \mathrm{Cl}_2>\mathrm{CrCl}_3>\mathrm{Cr}_2\left(\mathrm{SO}_4\right)_3$
$\mathrm{BaSO}_4>\mathrm{Hg}_2 \mathrm{Cl}_2>\mathrm{CrCl}_3>\mathrm{Cr}_2\left(\mathrm{SO}_4\right)_3$
$$
\text { } \begin{aligned}
& \mathrm{Cr}_2\left(\mathrm{SO}_4\right)_3 \leftrightharpoons 2 \mathrm{Cr}^{+++}+3 \mathrm{SO}_4^{--} \\
& \mathrm{K}_{\mathrm{sp}}=(2 s)^2(3 s)^3=4 \mathrm{~s}^2 \times 27 s^3=108 s^5 \\
& \mathrm{~s}=\left(\frac{\mathrm{K}_{s p}}{108}\right)^{1 / 5} \\
& \mathrm{Hg}_2 \mathrm{Cl}_2 \leftrightharpoons 2 \mathrm{Hg}^{2+}+2 \mathrm{Cl}^{-} \\
& \mathrm{K}_{\mathrm{sp}}=(2 s)^2 \times(2 s)^2=16 s^4 \\
& \mathrm{~s}=\left(\frac{\mathrm{K}_{s p}}{16}\right)^{1 / 4} \\
& \mathrm{BaSO}_4 \leftrightharpoons \mathrm{Ba}^{++}+\mathrm{SO}_4^{-} \\
& \mathrm{K}_{\mathrm{sp}}=s^2 \\
& \mathrm{~s}=\sqrt{\mathrm{K}_{\mathrm{sp}}} \\
& \mathrm{CrCl}_3 \leftrightharpoons \mathrm{Cr}^{3+}+3 \mathrm{Cl}^{-} \\
& s
\end{aligned}
$$
$$
\begin{aligned}
& \mathrm{K}_{\mathrm{sp}}=s \times(3 s)^3=27 s^4 \\
& s=\left(\frac{\mathrm{K}_{s p}}{27}\right)^{1 / 4}
\end{aligned}
$$
Hence the correct order of solubilities of salts is
$$
\sqrt{\mathrm{K}_{\mathrm{sp}}}>\left(\frac{\mathrm{K}_{\mathrm{sp}}}{16}\right)^{1 / 4}>\left(\frac{\mathrm{K}_{\mathrm{sp}}}{27}\right)^{1 / 4}>\left(\frac{\mathrm{K}_{\mathrm{sp}}}{108}\right)^{1 / 5}
$$
\text { } \begin{aligned}
& \mathrm{Cr}_2\left(\mathrm{SO}_4\right)_3 \leftrightharpoons 2 \mathrm{Cr}^{+++}+3 \mathrm{SO}_4^{--} \\
& \mathrm{K}_{\mathrm{sp}}=(2 s)^2(3 s)^3=4 \mathrm{~s}^2 \times 27 s^3=108 s^5 \\
& \mathrm{~s}=\left(\frac{\mathrm{K}_{s p}}{108}\right)^{1 / 5} \\
& \mathrm{Hg}_2 \mathrm{Cl}_2 \leftrightharpoons 2 \mathrm{Hg}^{2+}+2 \mathrm{Cl}^{-} \\
& \mathrm{K}_{\mathrm{sp}}=(2 s)^2 \times(2 s)^2=16 s^4 \\
& \mathrm{~s}=\left(\frac{\mathrm{K}_{s p}}{16}\right)^{1 / 4} \\
& \mathrm{BaSO}_4 \leftrightharpoons \mathrm{Ba}^{++}+\mathrm{SO}_4^{-} \\
& \mathrm{K}_{\mathrm{sp}}=s^2 \\
& \mathrm{~s}=\sqrt{\mathrm{K}_{\mathrm{sp}}} \\
& \mathrm{CrCl}_3 \leftrightharpoons \mathrm{Cr}^{3+}+3 \mathrm{Cl}^{-} \\
& s
\end{aligned}
$$
$$
\begin{aligned}
& \mathrm{K}_{\mathrm{sp}}=s \times(3 s)^3=27 s^4 \\
& s=\left(\frac{\mathrm{K}_{s p}}{27}\right)^{1 / 4}
\end{aligned}
$$
Hence the correct order of solubilities of salts is
$$
\sqrt{\mathrm{K}_{\mathrm{sp}}}>\left(\frac{\mathrm{K}_{\mathrm{sp}}}{16}\right)^{1 / 4}>\left(\frac{\mathrm{K}_{\mathrm{sp}}}{27}\right)^{1 / 4}>\left(\frac{\mathrm{K}_{\mathrm{sp}}}{108}\right)^{1 / 5}
$$
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