Solubility of AgCl is much higher in NH_4OH than water due to complex ion format

Question

Solubility of AgCl is much higher in NH_4OH than water due to complex ion formation. A solution was made by shaking a certain mass of silver chloride with 6.0 M NH_4OH until the solution is saturated with silver chloride and very little solid silver chloride remains. The total volume of that solution is 500. mL.(Ksp for AgCl = 1.8 times 10^-10 and K_f for [Ag(NH_3)_2]^+ = 1.6 times 10^7} Calculate the mass of silver chloride that did dissolve in that solution. Mass of AgCl = Calculate the concentration of silver ion in that solution.

Explanation / Answer

From the given data,

Ag+ + 2NH3 <==> Ag(NH3)2^2+ ------------Kf

AgCl(s) <==> Ag+ + Cl- -----------------------Ksp

Adding,

AgCl(s) + 2NH3 <====> Ag(NH3)2^2+ + Cl- --------------Keq

Keq = Kf x Ksp = 1.8 x 10^-10 x 1.6 x 10^7 = 2.88 x 10^-3

a) ICE chart

                    AgCl(s) + 2NH3 <====> Ag(NH3)2^2+ + Cl-

I                      -              6 M                         -                 -

C                    -               -2x                        +x              +x

E                    -              6-2x                        x                x

So,

Keq = [Ag(NH3)2^2+][Cl-]/[NH3]^2

2.88 x10^-3 = x^2/(6 - 2x)^2

2.88 x10^-3 = x^2/36 - 24x + 4x^2  

x^2 = 0.01152x^2 - 0.07x + 0.104

0.99x^2 + 0.07x - 0.104 = 0

x = 0.29 M

So,

moles of AgCl dissolved = 0.29 M x 0.5 L = 0.145 mol

mass of AgCl dissolved = 0.145 x 143.32 g/mol = 20.78 g

b) concentration of silver ion in solution,

= Ksp/[Cl-]

= 1.8 x 10^-10/0.29

= 6.21 x 10^-10 M

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