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### Topic: Problem of the week - 25/02/2013  (Read 12994 times)

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#### Borek

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##### Problem of the week - 25/02/2013
« on: February 25, 2013, 08:01:07 AM »
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?
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#### SinkingTako

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##### Re: Problem of the week - 25/02/2013
« Reply #1 on: March 02, 2013, 11:21:38 AM »
this Is probably wrong, but Is it 20ml?
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#### Borek

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##### Re: Problem of the week - 25/02/2013
« Reply #2 on: March 02, 2013, 12:47:18 PM »
Please show how you got this result.
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#### SinkingTako

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##### Re: Problem of the week - 25/02/2013
« Reply #3 on: March 02, 2013, 08:33:33 PM »
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, $$[Ag]new= \frac{x[Ag]}{x+10}$$, $$[SO_4]new= \frac{10[SO_4]}{x+10}$$

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

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

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

Solving, X=20.
« Last Edit: March 03, 2013, 04:57:16 AM by Borek »
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#### Borek

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##### Re: Problem of the week - 25/02/2013
« Reply #4 on: March 03, 2013, 04:57:02 AM »
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}.
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#### SinkingTako

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##### Re: Problem of the week - 25/02/2013
« Reply #5 on: March 03, 2013, 06:13:00 AM »

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.
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#### Borek

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##### Re: Problem of the week - 25/02/2013
« Reply #6 on: March 03, 2013, 06:55:51 AM »
In LaTeX {} are used for grouping of symbols, () are treated like every other character.
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