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Question: The solubility of salt of weak acid \[\left( {{\mathbf{AB}}} \right)\] at pH 3 is \[Y \times {10^{ -...

The solubility of salt of weak acid (AB)\left( {{\mathbf{AB}}} \right) at pH 3 is Y×10−3 mol L−1Y \times {10^{ - 3}}\:{\text{mol}}\:{{\text{L}}^{ - 1}}. The value of YY is ____.
(Given that the value of solubility product of AB(Ksp)=2×10−10AB\left( {{K_{sp}}} \right) = 2 \times {10^{ - 10}} and the value of ionization constant of HB(Ka)=1×10−8HB\left( {{K_a}} \right) = 1 \times {10^{ - 8}})

Explanation

Solution

To answer this question, you should recall the concept of solubility product. The solubility product constant refers to the equilibrium constant for the dissolution of a solid substance into an aqueous solution.
The formula used:
S=Ksp([H+]Ka+1){\text{S}} = \sqrt {{{\text{K}}_{{\text{sp}}}}\left( {\dfrac{{[{{\text{H}}^ + }]}}{{{{\text{K}}_{\text{a}}}}} + 1} \right)} where S is the solubility, Ksp{{\text{K}}_{{\text{sp}}}} is the solubility product and Ka{{\text{K}}_{\text{a}}} is the ionization constant

Complete step by step answer:
We know that the solubility product is a kind of equilibrium constant and its value depends on temperature. Ksp{K_{sp}}usually increases with an increase in temperature which is explained by increased solubility.
We know the pH from which we can calculate the hydronium ion concentration: pH=3  {\text{pH}} = 3\;.
⇒[H+]=10−3M\Rightarrow [{H^ + }] = {10^{ - 3}}{\text{M}}
Now, Solubility is given by: S=Ksp([H+]Ka+1){\text{S}} = \sqrt {{{\text{K}}_{{\text{sp}}}}\left( {\dfrac{{[{{\text{H}}^ + }]}}{{{{\text{K}}_{\text{a}}}}} + 1} \right)} .
Substituting the required values and solving:
S=2×10−10([10−3]10−8+1){\text{S}} = \sqrt {2 \times {{10}^{ - 10}}\left( {\dfrac{{[{{10}^{ - 3}}]}}{{{{10}^{ - 8}}}} + 1} \right)}
⇒S=2×10−5\Rightarrow {\text{S}} = \sqrt {2 \times {{10}^{ - 5}}}
Solving this, we get:
S=4.47×10−3M{\text{S}} = 4.47 \times {10^{ - 3}}{\text{M}}
Hence, the value of Y is 4.474.47

Note:
The difference between solubility and solubility product constant. The solubility of a substance in a solvent is the total amount of the solute that can be dissolved in the solvent at equilibrium. The solubility product constant is an equilibrium constant that provides insight into the equilibrium between the solid solute and its constituent ions that are dissociated across the solution. Solubility is a property of a substance being dissolved called solute to get dissolved in a solvent to form a solution. Solubility depends on various factors such as lattice enthalpy of salt and solvation enthalpy of ions.