[Borek]

Silver nitrate solution was added to 10 mL of sodium sulfate solution. Assuming concentrations of both solutions were high enough to produce a precipitate, at what volume of the AgNO₃ added amount of solid is at its maximum?

[SinkingTako]

this Is probably wrong, but Is it 20ml?

[Borek]

Please show how you got this result.

[SinkingTako]

Let [Ag] and [SO₄] be the initial concentration of the 2 ions in the various solutions.

Let the amount of AgNO₃ solution added be Xml

When combined, [tex][Ag]new= \frac{x[Ag]}{x+10}[/tex], [tex][SO_4]new= \frac{10[SO_4]}{x+10}[/tex]

Thus, [tex]Ksp = \frac{10x^2[Ag]^2[SO_4]}{(x+10)^3}[/tex]

[Ag]²[SO4]=1.2x10^-5 (Actually specific value doesn't matter)

Sub this into the Ksp equation, then max X is when [tex]\frac{d}{dx}  \frac{10x^2×1.2×10^{-5} }{(x+10)^3}=0[/tex]

Solving, X=20.

[Borek]

Perfect!

Funny how the result doesn't depend on the initial concentrations nor the Ksp. Also, while the differential looks rather unpleasant, finding the maximum is quite simple - it is enough to compare the numerator to zero and it is just x(x-20)=0.

I am correcting a typo in your LaTeX - 10^(-5) is not equivalent to 10^{-5}.

[SinkingTako]

I am correcting a typo in your LaTeX - 10^(-5) is not equivalent to 10^{-5}.

Thanks! Was actually trying to figure out why the ^-5 couldn't be displayed correctly, but couldn't find the problem.

[Borek]

In LaTeX {} are used for grouping of symbols, () are treated like every other character.