Minimum ammonia to dissolve 0.1 mol AgCl with Ksp and Kf

[zorro]

Homework Statement

What is the minimum quantity of ammonia which must be added to 1L of a solution in order to dissolve 0.1 mole silver chloride by forming [Ag(NH3)2]+.
Given Ksp- 10^-10 and Kf- 10^8

Homework Equations

The Attempt at a Solution

I used the equation (which I derived) s=c^2/(Ksp)(Kf) (assuming almost all amount of AgCl converts to complex and some other approximations)
where s is the solubility of AgCl and c is the concentration of NH3 (both in mol/litre)

I solved and got c as 1 mole.
But answer given is 1.2 mole.

Are my assumptions wrong, or there is some other thing?

 

[Borek]

Show your derivation.

 

[zorro]

Let s be the solubility of AgCl in cM NH3. Let x mole/L be the amount of salt forming complex

At equilibrium.
Ag+ = s-x
ksp=(s-x)(s)
kf= x/(s-x)(c-2x)^2
Since kf is very high, x approaches s. Thus s-x would be very small. Let this value be y. Since x is small, 2x can be ignored w.r.t c.

i.e. c-2x = c
ksp=ys
kf=s^2/ksp(c)^2
i.e.
s=c x (ksp x kf)^1/2

Sorry, I wrote the earlier expression wrong.

 

[Borek]

You want x to be around 0.1M. It is not small.

 

[zorro]

Oops. Then that will make the calculations very lengthy.
Is there any method by which we solve this way and then add 0.2 mole by giving some reason?

 

[Borek]

What is [Cl^-]?

What is maximum possible [Ag⁺]?

What is complex concentration?

Calculate ammonia concentration that fits.

--

 

[zorro]

max [CL-] is 0.1
max [Ag+] is 10^-9

How to calculate ammonia concentratrion that fits?

 

[Borek]

What is complex concentration?

Look at Kf.

 

[zorro]

Complex concentration will be practically 0.1.
From this
[NH3]^2 = 0.1/(10^-9)(10^8)
I got [NH3] as 1M
not 1.2.