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Solubility Product

Consider a M ...

Question

Consider $a M$ of $N H_{3} (a q)$ is present in a solution containing $A g C l$ and $x$ is the solublity of $A g C l$ in this solution.
Given: $K_{f}$ is the formation constant of $[A g (N H_{3})_{2}]^{+}$ ,
$K_{s p}$ is the solubility product of $A g C l$ .
Which of the following relation is correct.

\frac{K_{s p}}{K_{f}} = \frac{x}{(a + 2 x)}

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K_{s p} \times K_{f} = \frac{x}{(a - 2 x)}

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\frac{K_{s p}}{K_{f}} = \frac{x^{2}}{(a + 2 x)^{2}}

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K_{s p} \times K_{f} = \frac{x^{2}}{(a - 2 x)^{2}}

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Solution

The correct option is D $K_{s p} \times K_{f} = \frac{x^{2}}{(a - 2 x)^{2}}$
Theory:
Complex salt :
Formed due to the combination of simple salts or molecular compounds and they retain their identity in the solution as well as the solid state.
Example - $[A g (N H_{3})_{2}] C l, [C o (N H_{3})_{4} C l_{2}] C l$
For $[A g (N H_{3})_{2}] C l$

$[A g (N H_{3})_{2}] C l (a q) ⇌ [A g (N H_{3})_{2}]^{+} (a q) + C l^{-} (a q)$
Formation of a complex ion occurs in steps
Example - Formation of $[A g (N H_{3})_{2}]^{+}$ complex ion,
$A g^{+} (a q) + N H_{3} (a q)_{1} [A g (N H_{3})]^{+} (a q)$
Where $K_{1}$ is formation constant or stability constant.
$[A g (N H_{3})]^{+} (a q) + N H_{3} (a q)_{2} [A g (N H_{3})_{2}]^{+} (a q)$
Where $K_{2}$ is formation constant or stability constant.
Adding above equations we get :
$A g^{+} (a q) + 2 N H_{3} (a q)_{f o r m} [A g (N H_{3})_{2}]^{+} (a q)$
$K_{f o r m} = K_{1} \times K_{2}$
where, $K_{f o r m}$ is the formation constant or the stability constant for $[A g (N H_{3})_{2}]^{+}$ .
Higher the value of the stability constant,more stable will be the complex.
Instability constant $K_{i n s t}$ is numerically equal to the inverse of stability constant $K_{f o r m}$
$K_{i n s t} = \frac{1}{K_{f o r m}}$

let $x$ be the solubility of $A g C l$ , so
$A g C l (s) ⇌ A g^{+} (a q) + C l^{-} (a q)$
$x - y x$
$A g^{+} (a q) + 2 N H_{3} (a q) ⇌ [A g (N H_{3})_{2}]^{+} (a q)$
$x a 0$
$x - y a - 2 y y$

$K_{s p} = (x - y) x$

$K_{f} = \frac{y}{(x - y) (a - 2 y)^{2}}$
$K_{f}$ is the formation constant of $[A g (N H_{3})_{2}]^{+}$
If the complex is stable, then $K_{f}$ will be high (in most of cases $K_{f}$ is high like in this case) .
$∴ x - y \approx 0 \to x \approx y$
Adding both the equations, we get :
$A g C l (s) + 2 N H_{3} (a q) ⇌ [A g (N H_{3})_{2}]^{+} (a q) + C l - (a q)$
$a - 2 x x x$
Since the equations are added, the equilibrium constant gets multiplied and the new equilibrium constant for this reaction will be $(K_{s p} \times K_{f})$ :

$K_{s p} \times K_{f} = \frac{x^{2}}{(a - 2 x)^{2}}$

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