Which one of the following arrangements represents the correct order of solubilities of sparingly soluble salts $$Hg_2Cl_2$$, $$Cr_2(SO_4)_3$$, $$BaSO_4$$ and $$CrCl_3$$ respectively?

To determine the correct order of solubilities for the sparingly soluble salts $$ \text{Hg}_2\text{Cl}_2 $$, $$ \text{Cr}_2(\text{SO}_4)_3 $$, $$ \text{BaSO}_4 $$, and $$ \text{CrCl}_3 $$, we need to compare their molar solubilities. The molar solubility, denoted as $$ S $$, is the number of moles of the salt that dissolve per liter of solution. For sparingly soluble salts, the solubility product constant ($$ K_{\text{sp}} $$) relates to the solubility $$ S $$ through the dissociation reaction.

First, we write the dissociation reactions and the corresponding $$ K_{\text{sp}} $$ expressions in terms of $$ S $$:

For $$ \text{Hg}_2\text{Cl}_2 $$:
The dissociation is $$ \text{Hg}_2\text{Cl}_2 \rightarrow \text{Hg}_2^{2+} + 2\text{Cl}^- $$.
If solubility is $$ S $$, then $$ [\text{Hg}_2^{2+}] = S $$ and $$ [\text{Cl}^-] = 2S $$.
So, $$ K_{\text{sp}} = [\text{Hg}_2^{2+}][\text{Cl}^-]^2 = (S)(2S)^2 = 4S^3 $$.
Therefore, $$ S = \left( \frac{K_{\text{sp}}}{4} \right)^{\frac{1}{3}} $$.

For $$ \text{Cr}_2(\text{SO}_4)_3 $$:
The dissociation is $$ \text{Cr}_2(\text{SO}_4)_3 \rightarrow 2\text{Cr}^{3+} + 3\text{SO}_4^{2-} $$.
If solubility is $$ S $$, then $$ [\text{Cr}^{3+}] = 2S $$ and $$ [\text{SO}_4^{2-}] = 3S $$.
So, $$ K_{\text{sp}} = [\text{Cr}^{3+}]^2 [\text{SO}_4^{2-}]^3 = (2S)^2 (3S)^3 = 4S^2 \times 27S^3 = 108S^5 $$.
Therefore, $$ S = \left( \frac{K_{\text{sp}}}{108} \right)^{\frac{1}{5}} $$.

For $$ \text{BaSO}_4 $$:
The dissociation is $$ \text{BaSO}_4 \rightarrow \text{Ba}^{2+} + \text{SO}_4^{2-} $$.
If solubility is $$ S $$, then $$ [\text{Ba}^{2+}] = S $$ and $$ [\text{SO}_4^{2-}] = S $$.
So, $$ K_{\text{sp}} = [\text{Ba}^{2+}][\text{SO}_4^{2-}] = S \times S = S^2 $$.
Therefore, $$ S = \sqrt{K_{\text{sp}}} $$.

For $$ \text{CrCl}_3 $$:
The dissociation is $$ \text{CrCl}_3 \rightarrow \text{Cr}^{3+} + 3\text{Cl}^- $$.
If solubility is $$ S $$, then $$ [\text{Cr}^{3+}] = S $$ and $$ [\text{Cl}^-] = 3S $$.
So, $$ K_{\text{sp}} = [\text{Cr}^{3+}][\text{Cl}^-]^3 = (S)(3S)^3 = 27S^4 $$.
Therefore, $$ S = \left( \frac{K_{\text{sp}}}{27} \right)^{\frac{1}{4}} $$.

Now, to compare the solubilities, we assume that the $$ K_{\text{sp}} $$ values for these salts are of the same order of magnitude. This is a common approach when exact $$ K_{\text{sp}} $$ values are not provided, as the exponent in the solubility expression dominates the comparison.

Expressing $$ S $$ in terms of $$ K_{\text{sp}} $$:

• $$ S_{\text{BaSO}_4} = (K_{\text{sp}})^{\frac{1}{2}} $$
• $$ S_{\text{Hg}_2\text{Cl}_2} = \left( \frac{K_{\text{sp}}}{4} \right)^{\frac{1}{3}} = (K_{\text{sp}})^{\frac{1}{3}} \times \left( \frac{1}{4} \right)^{\frac{1}{3}} $$
• $$ S_{\text{CrCl}_3} = \left( \frac{K_{\text{sp}}}{27} \right)^{\frac{1}{4}} = (K_{\text{sp}})^{\frac{1}{4}} \times \left( \frac{1}{27} \right)^{\frac{1}{4}} $$
• $$ S_{\text{Cr}_2(\text{SO}_4)_3} = \left( \frac{K_{\text{sp}}}{108} \right)^{\frac{1}{5}} = (K_{\text{sp}})^{\frac{1}{5}} \times \left( \frac{1}{108} \right)^{\frac{1}{5}} $$

Since $$ K_{\text{sp}} $$ is assumed to be similar for all salts, the solubility $$ S $$ depends on the exponent of $$ K_{\text{sp}} $$ and the constant divisor. Higher exponents result in larger $$ S $$ for the same $$ K_{\text{sp}} $$. The exponents are:

• $$ \frac{1}{2} = 0.5 $$ for $$ \text{BaSO}_4 $$
• $$ \frac{1}{3} \approx 0.333 $$ for $$ \text{Hg}_2\text{Cl}_2 $$
• $$ \frac{1}{4} = 0.25 $$ for $$ \text{CrCl}_3 $$
• $$ \frac{1}{5} = 0.2 $$ for $$ \text{Cr}_2(\text{SO}_4)_3 $$

Ordering by the exponents: $$ 0.5 > 0.333 > 0.25 > 0.2 $$. Thus, the solubility order from highest to lowest is $$ \text{BaSO}_4 > \text{Hg}_2\text{Cl}_2 > \text{CrCl}_3 > \text{Cr}_2(\text{SO}_4)_3 $$.

Comparing with the options:

• Option A: $$ \text{BaSO}_4 > \text{Hg}_2\text{Cl}_2 > \text{Cr}_2(\text{SO}_4)_3 > \text{CrCl}_3 $$ — incorrect, as $$ \text{CrCl}_3 $$ should be greater than $$ \text{Cr}_2(\text{SO}_4)_3 $$.
• Option B: $$ \text{BaSO}_4 > \text{Hg}_2\text{Cl}_2 > \text{CrCl}_3 > \text{Cr}_2(\text{SO}_4)_3 $$ — matches our order.
• Option C: $$ \text{BaSO}_4 > \text{CrCl}_3 > \text{Hg}_2\text{Cl}_2 > \text{Cr}_2(\text{SO}_4)_3 $$ — incorrect, as $$ \text{Hg}_2\text{Cl}_2 $$ should be greater than $$ \text{CrCl}_3 $$.
• Option D: $$ \text{Hg}_2\text{Cl}_2 > \text{BaSO}_4 > \text{CrCl}_3 > \text{Cr}_2(\text{SO}_4)_3 $$ — incorrect, as $$ \text{BaSO}_4 $$ should be greater than $$ \text{Hg}_2\text{Cl}_2 $$.

Hence, the correct answer is Option B.