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Question: The solubility product of a binary weak electrolyte is \(4\times {{10}^{-10}}\) at 298K. Its solubil...

The solubility product of a binary weak electrolyte is 4×10−104\times {{10}^{-10}} at 298K. Its solubility in mol.dm−3d{{m}^{-3}} at the same temperature is-
(A) 4×10−54\times {{10}^{-5}}
(B) 2×10−52\times {{10}^{-5}}
(C) 8×10−108\times {{10}^{-10}}
(D) 16×10−2016\times {{10}^{-20}}

Explanation

Solution

The relation between Solubility product (Ksp{{K}_{sp}}) and molar solubility(S) is Ksp=S2{{K}_{sp}}={{S}^{2}} for a binary weak electrolyte at a particular temperature. Weak electrolytes do not dissociate completely in aqueous solution.

Complete step by step solution:
For a general electrolyte solubility at equilibrium,
AxBy⇄xAy++yBx−{{A}_{x}}{{B}_{y}}\rightleftarrows x{{A}^{y+}}+y{{B}^{x-}}
The solubility product is
Ksp=[Ay+]x[Bx−]y{{K}_{sp}}={{[{{A}^{y+}}]}^{x}}{{[{{B}^{x-}}]}^{y}}
Thus the product of equilibrium concentrations of constituent ions raised to the power of their respective coefficients in the balanced equilibrium expression at a given temperature is called a Solubility product.
Molar solubility is defined as the number of moles of a compound that dissolve to give one liter of saturated solution.
If S is the molar solubility of the compound, the equilibrium concentrations of the ions in the saturated solution will be -

& [{{A}^{y+}}]=xS \\\ & [{{B}^{x-}}]=yS \\\ \end{aligned}$$ $${{K}_{sp}}={{[xS]}^{x}}{{[yS]}^{y}}$$ For a binary weak electrolyte AB at equilibrium, the chemical reaction is $$AB\rightleftarrows {{A}^{+}}+{{B}^{-}}$$ $${{K}_{sp}}={{[{{A}^{y+}}]}^{x}}{{[{{B}^{x-}}]}^{y}}$$ Here, $$\begin{aligned} & x=1 \\\ & y=1 \\\ \end{aligned}$$ Therefore, $${{K}_{sp}}=S\times S$$ $$S=\sqrt{{{K}_{sp}}}$$ $$\begin{aligned} & S=\sqrt{4\times {{10}^{-10}}} \\\ & S=2\times {{10}^{-5}}mol.d{{m}^{-3}} \\\ \end{aligned}$$ **Therefore, the correct answer is (B) $2\times {{10}^{-5}}$ $$mol.d{{m}^{-3}}$$.** **Note:** The ionic product of an electrolyte is defined in the same way as solubility product but the difference is that the ionic product consists of concentration of ions under any condition, whereas expression of solubility product consists of only equilibrium concentrations.