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Question: The \[{{\text{K}}_{{\text{SP}}}}\] for \[{\text{AgCl}}\] is \[2.8 \times {10^{ - 10}}\] at a given t...

The KSP{{\text{K}}_{{\text{SP}}}} for AgCl{\text{AgCl}} is 2.8×10−102.8 \times {10^{ - 10}} at a given temperature. The solubility of AgCl{\text{AgCl}} in 0.010.01 molar HCl{\text{HCl}} solution at this temperature will be:
A.2.8×10−12 mol L−1{\text{2}}{\text{.8}} \times {\text{1}}{{\text{0}}^{ - 12}}{\text{ mol }}{{\text{L}}^{ - 1}}
B.2.8×10−8 mol L−1{\text{2}}{\text{.8}} \times {\text{1}}{{\text{0}}^{ - 8}}{\text{ mol }}{{\text{L}}^{ - 1}}
C.5.6×10−8 mol L−1{\text{5}}{\text{.6}} \times {\text{1}}{{\text{0}}^{ - 8}}{\text{ mol }}{{\text{L}}^{ - 1}}
D.2.8×10−4 mol L−1{\text{2}}{\text{.8}} \times {\text{1}}{{\text{0}}^{ - 4}}{\text{ mol }}{{\text{L}}^{ - 1}}

Explanation

Solution

KSP{{\text{K}}_{{\text{SP}}}} depends only on temperature. Due to the common ion effect, the solubility of common ions decreases but other ions increase in order to keep KSP{{\text{K}}_{{\text{SP}}}} constant.

Complete step by step answer:
The salts which are completely soluble in water are called highly soluble salt. Sparingly soluble salts are not completely soluble in water as they are only partial soluble. For example salts like AgCl, AgBr, ZnS{\text{AgCl, AgBr, ZnS}} etc.
KSP{{\text{K}}_{{\text{SP}}}} or solubility product constant is define for solution containing sparingly soluble salt and it is the product of concentration of cation and anion in aqueous solution of sparingly soluble salt with each concentration term raised to the power of their stoichiometric coefficients.
The concentration of ions is to be taken at saturated conditions. The saturated conditions are one where salt gets maximum soluble in water. On adding sparingly soluble salt say AgCl{\text{AgCl}} in a solution of its common ion say HCl{\text{HCl}} . The solubility of common ion that is chloride ion decrease and solubility of other ion that is silver ion will increase in order to keep KSP{{\text{K}}_{{\text{SP}}}} constant.
According to the question, KSP{{\text{K}}_{{\text{SP}}}} of AgCl{\text{AgCl}} is 2.8×10−102.8 \times {10^{ - 10}} which is very low.
So if AgCl{\text{AgCl}} is mixed with 0.01{\text{0}}{\text{.01}} molar solution of HCl{\text{HCl}} which contains 0.01 moles of chloride ions, the chloride ion coming from AgCl{\text{AgCl}} will be negligible as compared to chloride concentration coming from HCl{\text{HCl}} in mixture. Thus KSP{{\text{K}}_{{\text{SP}}}} for this will be as follow:
Ksp=[Ag+][Cl−]{{\text{K}}_{{\text{sp}}}} = \left[ {{\text{A}}{{\text{g}}^ + }} \right]\left[ {{\text{C}}{{\text{l}}^ - }} \right]
2.8×10−10=[Ag+]0.012.8 \times {10^{ - 10}} = \left[ {{\text{A}}{{\text{g}}^ + }} \right]0.01
Rearranging this:
2.8×10−100.01=[Ag+]\dfrac{{2.8 \times {{10}^{ - 10}}}}{{0.01}} = \left[ {{\text{A}}{{\text{g}}^ + }} \right]
⇒[Ag+]=2.8×10−8 mol L−1\Rightarrow \left[ {{\text{A}}{{\text{g}}^ + }} \right] = {\text{2}}{\text{.8}} \times {\text{1}}{{\text{0}}^{ - 8}}{\text{ mol }}{{\text{L}}^{ - 1}}

Thus, the correct option is B.

Note:
Equilibrium constant has concentration in terms of reactant and product present at equilibrium conditions but KSP{{\text{K}}_{{\text{SP}}}} have concentration terms of ions at saturated conditions. When we heat a saturated solution, more amount of solute can be added because of the increase in volume. The solution thus formed is known as a supersaturated solution.