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Question

Consider a M of NH3(aq) is present in a solution containing AgCl and x is the solublity of AgCl in this solution.
Given: Kf is the formation constant of [Ag(NH3)2]+,
Ksp is the solubility product of AgCl.
Which of the following relation is correct.

A
KspKf=x(a+2x)
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B
Ksp×Kf=x(a2x)
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C
KspKf=x2(a+2x)2
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D
Ksp×Kf=x2(a2x)2
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Solution

The correct option is D Ksp×Kf=x2(a2x)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 - [Ag(NH3)2]Cl,[Co(NH3)4Cl2]Cl
For [Ag(NH3)2]Cl

[Ag(NH3)2]Cl(aq)[Ag(NH3)2]+(aq)+Cl(aq)
Formation of a complex ion occurs in steps
Example - Formation of [Ag(NH3)2]+ complex ion,
Ag+(aq)+NH3(aq)K1[Ag(NH3)]+(aq)
Where K1 is formation constant or stability constant.
[Ag(NH3)]+(aq)+NH3(aq)K2[Ag(NH3)2]+(aq)
Where K2 is formation constant or stability constant.
Adding above equations we get :
Ag+(aq)+2NH3(aq)Kform[Ag(NH3)2]+(aq)
Kform=K1×K2
where, Kform is the formation constant or the stability constant for [Ag(NH3)2]+ .
Higher the value of the stability constant,more stable will be the complex.
Instability constant Kinst is numerically equal to the inverse of stability constant Kform
Kinst=1Kform

let x be the solubility of AgCl, so
AgCl(s)Ag+(aq)+Cl(aq)
xy x
Ag+(aq)+2NH3(aq)[Ag(NH3)2]+(aq)
x a 0
xy a2y y

Ksp=(xy)x

Kf=y(xy)(a2y)2
Kf is the formation constant of [Ag(NH3)2]+
If the complex is stable, then Kf will be high (in most of cases Kf is high like in this case) .
xy0xy
Adding both the equations, we get :
AgCl(s)+2NH3(aq)[Ag(NH3)2]+(aq)+Cl(aq)
a2x x x
Since the equations are added, the equilibrium constant gets multiplied and the new equilibrium constant for this reaction will be (Ksp×Kf):

Ksp×Kf=x2(a2x)2

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