Question

If S and \[{{K}_{sp}}\] are respectively solubility and solubility product of the sparingly soluble binary electrolyte, then
(A) \[S\text{ }=\text{ }{{K}_{sp}}\]
(B) \[S\text{ }=\text{ }{{K}_{sp}}^{2}\]
(C) \[S\text{ }=\text{ }\surd {{K}_{sp}}\]
(D) \[S\text{ }=\text{ }1/2\text{ }{{K}_{sp}}\]

Answer 1

Hint: Solubility (S) and solubility product (\[{{K}_{sp}}\]) are determined for sparingly soluble salts (saturated or equilibrium point attain earlier between precipitate(s) of salt and dissolved part of salt (aq) in solution) and which is a good electrolyte. Almost all the salts are good electrolytes, which means all the salts easily dissociate into ions in an aqueous solution.

Complete Step by Step Solution:
Let us first consider a salt that is slightly soluble in the solvent (say water), \[AgCl\left( s \right)\]. If we add silver chloride to water, it starts to dissolve in water slightly but at any instant, silver chloride stops to get dissolve in water and settles down to the base in the form of precipitates (insoluble silver chloride) which is in solid form (generally salts are solid).

In other words, at any instant, the solution attains a saturated point and equilibrium set up between the insoluble part of silver chloride which is a solid form (precipitates), and the soluble part of silver chloride which is in an aqueous form such as
\[AgCl\left( s \right)\rightleftarrows AgCl(aq)\]

Now, being a strong electrolyte, silver chloride does not exist in the molecular form in solution but dissociates in ions. In short, we can say equilibrium exists between an insoluble part of silver chloride which is in solid form (ppt.) and ions of the soluble part of silver chloride such as
\[AgCl(s)\rightleftarrows A{{g}^{+}}(aq)+C{{l}^{-}}(aq)\] \[AgCl(s)\rightleftarrows A{{g}^{+}}(aq)+C{{l}^{-}}(aq)\]

Solubility product is the mathematical expression which is defined as the product of concentrations of both ions that present in saturated solution both raised to the power of their stoichiometry coefficient if any such as,
\[~{{K}_{sp}}\left( solubility\text{ }product \right)\text{ }=\text{ }\left[ A{{g}^{+}} \right]\left[ C{{l}^{-}} \right]\]

To find solubility (S), the solubility of Ag+ is S, and the solubility of Cl- is also S such as
\[{{K}_{sp}}=\text{ }S\text{ }\times \text{ }S\]
\[{{K}_{sp}}=\text{ }{{S}^{2}}\]
\[S\text{ }=\text{ }\surd {{K}_{sp}}\]
Thus, the correct option is C.

Note: Given equation is
\[AgCl(s)\rightleftarrows A{{g}^{+}}(aq)+C{{l}^{-}}(aq)\] \[{{K}_{c}}=[A{{g}^{+}}][C{{l}^{-}}]/[AgCl]\]
The equilibrium constant of a given reaction is given as
\[{{K}_{c}}=[A{{g}^{+}}][C{{l}^{-}}]/[AgCl]\]
As in denomination, the concentration of solid is needed to calculate equilibrium constant but the concentration of product which is in solid form is always 1 so equilibrium constant is the same as solubility product constant such as
\[{{K}_{c}}=[A{{g}^{+}}][C{{l}^{-}}]={{K}_{sp}}\]