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\[M|{{M}^{2+}}\](saturated solution of a sparingly soluble salt, \[M{{X}_{2}}\])|| \[{{M}^{2+}}\](0.001 mol \[d{{m}^{-3}}\])|M
The emf of the cell depends on the difference in concentrations of \[{{M}^{2+}}\]ions at the two electrodes.
The emf of the cell at 298 is 0.059 V.
The solubility product (\[{{K}_{sp}};mo{{l}^{3}}d{{m}^{-9}}\]) of \[M{{X}_{2}}\]at 298 based on the information available for the given concentration cell is:
[take \[2.303\times R\times 298/F=0.059V\]]
A. \[1\times {{10}^{-15}}\]
B. \[4\times {{10}^{-15}}\]
C. \[1\times {{10}^{-12}}\]
D. \[4\times {{10}^{-12}}\]

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Last updated date: 18th Jun 2024
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Answer
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Hint: The solubility of a substance in a solvent is defined as the total amount of the solute dissolved in the solvent at equilibrium and the solubility product constant is defined as an equilibrium constant which provides insight into the equilibrium between the solid solute and its constituent ions dissociated in the solution.

Complete Solution :
In electromagnetism and electronics, electromotive force which is denoted by emf is measured in volts. It is the electrical action produced by a non-electrical source. Devices which are known as transducers provide an emf by converting other forms of energy into electrical energy, such as batteries which convert chemical energy to electrical energy. The solubility product constant is the equilibrium constant for the dissolution of a solid substance into an aqueous solution. It is denoted by the symbol \[{{K}_{sp}}\]. It generally depends on the temperature; Higher the temperature higher is solubility product.

There are two electrodes which are showing the reactions :
\[M\to {{M}^{2+}}+2{{e}^{-}}\]
\[{{M}^{2+}}+2{{e}^{-}}\to M\]
\[{{E}_{cell}}=E{{{}^\circ }_{cell}}-\dfrac{0.059}{2}\log \dfrac{{{C}_{1}}}{{{C}_{2}}}\]
\[0.059=-\dfrac{0.059}{2}\log \dfrac{{{C}_{1}}}{{{C}_{2}}}\]
\[\log \dfrac{0.001}{[{{M}^{2+}}]}=2\]
\[\dfrac{0.001}{[{{M}^{2+}}]}=2\]
\[[{{M}^{2+}}]={{10}^{-5}}\]
\[M{{X}_{2}}\rightleftarrows {{M}^{2+}}+2{{X}^{-}}\]

\[{{K}_{sp}}\]= \[4{{S}^{3}}\]; Where S = \[{{10}^{-5}}\]
\[\therefore {{K}_{sp}}=4\times {{({{10}^{-5}})}^{3}}\]=\[4\times {{10}^{-15}}\]
So, the correct answer is “Option B”.

Note: Solubility depends on a number of parameters like lattice energy of salt and solvation enthalpy of ions in the solution are of most importance. When a salt is dissolved in a solvent the strong forces of attraction of solute must be overcome by the interactions between ions and the solvent. The solvation enthalpy of ions is always negative which means that energy is released during this process. The nature of the solvent determines the amount of energy released during solvation that is solvation enthalpy.