I'm being asked to find the volume of a [itex]\ce{NH3}[/itex] solution ([itex]0.1\ \mathrm{M}[/itex]) to be added to [itex]10\ \mathrm{mL}[/itex] of a [itex]\ce{AgNO3}[/itex] solution ([itex]0.1\ \mathrm{M}[/itex]) to assure the 'complete formation of [itex]\ce{[Ag(NH3)2]+}[/itex]' . Both constants of formation are given: [itex]\beta_1=10^{3.3}; \ \beta_2=10^{7.2}[/itex]
As far as I know, a complete complexation would mean that [itex][\ce{Ag+}]<10^{-5}[/itex]. I tried working it out with the common method of solving those kind of complex equilibria, but the equations I get seem pretty much impossible to solve because of the volume that I have to find, which gets incorporated in them.
Their solution is really weird, but much simpler; they invoke this mathematical condition, which makes the problem easily solvable:
[itex]\frac{\ce{[Ag(NH3)2+]}}{\ce{[Ag(NH3)+]}}>10^2[/itex]
Starting from this, I can find [itex][\ce{NH3}][/itex] present in the solution, therefore the volume.
Where did it came from?