How to determine the minimum pH at which iron(III) hydroxide dissolves completely?


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> Knowing $K\_\mathrm{s} = 6\cdot10^{-38}$ for $\ce{Fe(OH)3}$ in neutral solutions, calculate the minimum pH of an acidic solution in order to completely dissolve $\pu{10 mg}$ of $\ce{Fe(OH)3}$. Data: $V = \pu{0.1 L}$.
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My idea was to calculate the amount of $\ce{Fe^3+}$ and $\ce{OH-}$ ions, and then add enough acid containing $\ce{H+}$ ions in order to neutralise the $\ce{OH-}$ ions corresponding to the amount of $\ce{Fe^3+}$. To do this I used the equilibrium constant leaving the hydroxide ions as the unknown. Then I calculated the difference between the initial hydroxide concentration (corresponding to $\pu{10 mg}$ of salt) and the needed concentration to reach $K\_\mathrm{s}$. This corresponds to the concentration of hydronium ions in the acidic solution, and I can calculate pH.


Doing this I get $\mathrm{pH} = 3.03$. Is this how it should be done?




The OP first stated the problem as:



> 
> Knowing $K\_\mathrm{s} = 6\cdot10^{-38}$ for $\ce{Fe(OH)3}$ in neutral solutions, calculate the minimum pH of an acidic solution in order to completely dissolve $\pu{10 mg}$ of $\ce{Fe(OH)3}$. Data: $V = \pu{0.1 L}$.
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I took that to mean that acid was added to a solution containing 10 mg of iron (iii) hydroxide and the pH of the solution when the iron (iii) hydroxide was totally dissolved was desired.\*


Knowing that $K\_\mathrm{sp} = 6\cdot10^{-38}$ for $\ce{Fe(OH)3}$ then
\begin{align}
 K\_\mathrm{sp} &= \ce{[Fe^{3+}][OH^-]^3} = 6\cdot10^{-38},&
\text{ so }&&
\ce{[OH^-]} &=\sqrt[3]{\frac{6\cdot10^{-38}}{\ce{[Fe^{3+}]}}}
\end{align}


You need to solve for $\ce{[Fe^{3+}]}$ which you can calculating knowing that there is $\pu{10 mg}$ of $\ce{Fe(OH)\_3}$ in 0.1 liters of solution. It has a molecular weight of $\pu{107 g/mol}$, so there is 
$$0.010/107 = \pu{9.35\* 10^-5 mol//L}\text{ of }\ce{Fe(OH)\_3}.$$ 


Going back to the equation for the $K\_\mathrm{sp}$:
$$\ce{[OH^-]} =\sqrt[3]{\frac{6\cdot10^{-38}}{\ce{[Fe^{3+}]}}} = \sqrt[3]{\frac{6\cdot10^{-38}}{9.35\cdot10^{-5}}} = 8.63\cdot10^{-12}$$


Knowing $\ce{[OH^-]}$ you can calculate $\ce{[H^+]}$ via


$$\ce{[H^+]} = \frac{1\cdot10^{-14}}{\ce{[OH^-]}} = \frac{1\cdot10^{-14}}{8.63\cdot10^{-12}} = 1.16\cdot10^{-3}$$


and then pH: 


$$\mathrm{pH} = -\log\left(1.16\cdot10^{-3}\right) = 2.94.$$




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Now the OP has modified the problem statement to be:



> 
> Knowing $K\_\mathrm{s} = 6\cdot10^{-38}$ for $\ce{Fe(OH)3}$ in neutral solutions, calculate the minimum pH of an acidic solution in order to completely dissolve $10\ \mathrm{mg}$ of $\ce{Fe(OH)3}$. Data: $V = \pu{0.1 L}$.
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We again end up with the final pH being just acidic enough to dissolve $\pu{10 mg}$ of iron (iii) hydroxide in $\pu{100 ml}$ of water, but obviously the solution must start out more acidic. The final solution will again have a pH of $2.94$ or a $\ce{[H^+]} = 1.16\cdot10^{-3}$.


In $\pu{100 mL}$ of a solution of pH $2.94$ there are $\pu{0.116 mmol}$ of acid. 


We calculated before that there were $\pu{9.35\*10^{-5} mol}$ of $\ce{Fe^{3+}}$, but there are three $\ce{OH^-}$ anions for every cation of $\ce{Fe^{3+}}$, so there are $3 \times 9.35\cdot10^{-5} = \pu{2.81\*10^{-4} mol}$ of $\ce{OH^-}$. 


The total amount of substance (in millimoles) of acid needed is 
$$0.116 + 0.281 = 0.397.$$


For $\pu{100 ml}$ to contain $\pu{0.397 mmol}$ the solution would need to be 3.97 millimolar in $\ce{[H^+]}$, or the pH = $2.40$. 


If you start with $\pu{100 ml}$ of solution with pH $2.40$ and dump in $\pu{10 mg}$ of $\ce{Fe(OH)3}$, then the final pH will be $2.94$ and all of the $\ce{Fe(OH)3}$ will be just dissolved. 


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