Question

Consider precipitation of ${\mathbf{X}}^{\mathbf{x}\mathbf{-}}$with role="math" localid="1655098271650" ${\mathbf{M}}^{\mathbf{m}\mathbf{+}}$:

${\mathbf{xM}}^{\mathbf{m}\mathbf{+}}\mathbf{+}{\mathbf{mX}}^{\mathbf{x}\mathbf{-}}\mathbf{\rightleftharpoons }{\mathbf{M}}_{\mathbf{x}}{\mathbf{X}}_{\mathbf{m}}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{\hspace{0.33em}}\mathbf{\hspace{0.33em}}\mathbf{\hspace{0.33em}}{\mathbf{K}}_{\mathbf{sp}}\mathbf{=}\mathbf{\left[}{\mathbf{M}}^{\mathbf{m}\mathbf{+}}{\mathbf{]}}^{\mathbf{x}}\mathbf{\right[}{\mathbf{X}}^{\mathbf{x}\mathbf{-}}{\mathbf{]}}^{\mathbf{m}}$Write mass balance equations for M and X and derive the equation

${\mathbf{V}}_{\mathbf{M}}\mathbf{=}{\mathbf{V}}_{\mathbf{x}}^{\mathbf{0}}\frac{\mathbf{(}{\mathbf{xC}}_{\mathbf{X}}^{\mathbf{0}}\mathbf{+}\mathbf{m}\mathbf{[}{\mathbf{M}}^{\mathbf{m}\mathbf{+}}\mathbf{]}\mathbf{-}\mathbf{x}\mathbf{[}{\mathbf{X}}^{\mathbf{x}\mathbf{-}}\mathbf{]}}{\mathbf{(}{\mathbf{mC}}_{\mathbf{M}}^{\mathbf{0}}\mathbf{-}\mathbf{m}\mathbf{[}{\mathbf{M}}^{\mathbf{m}\mathbf{+}}\mathbf{]}\mathbf{+}\mathbf{x}\mathbf{[}{\mathbf{X}}^{\mathbf{x}\mathbf{-}}\mathbf{]}}\mathbf{where}\mathbf{\left[}{\mathbf{X}}^{\mathbf{x}\mathbf{-}}\mathbf{\right]}\mathbf{=}{\mathbf{(}{\mathbf{K}}_{\mathbf{sp}}\mathbf{/}\mathbf{[}{\mathbf{M}}^{\mathbf{m}\mathbf{+}}{\mathbf{]}}^{\mathbf{x}}\mathbf{)}}^{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{m}$}\right.}$

Short answer

The mass balance for M and X are.

$$\begin{gathered} {\mathrm{mC}}_{\mathrm{M}}^{\mathrm{o}}\cdot {\mathrm{V}}_{\mathrm{M}}-\mathrm{m}\left[{\mathrm{M}}^{\mathrm{m}+}\right]({\mathrm{V}}_{\mathrm{M}}+{\mathrm{V}}_{\mathrm{x}}^{\mathrm{o}}) \\ ={\mathrm{xC}}_{\mathrm{x}}^{\mathrm{o}}\cdot {\mathrm{V}}_{\mathrm{x}}^{\mathrm{o}}-\mathrm{x}[{\mathrm{X}}^{(}\mathrm{x}-)\left]\right({\mathrm{V}}_{\mathrm{M}}+{\mathrm{V}}_{\mathrm{x}}^{\mathrm{o}}) \end{gathered}$$

Step-by-step solution

Step 1: Define the mass balance.

Mass balance is the sum of moles of ion and moles of precipitate. It is literally mole balance.

Step 2: derive the equation for mass balance M and X.

The equation of mass balance of M and X

${\mathrm{V}}_{\mathrm{M}}={{\mathrm{V}}_{\mathrm{x}}}^{\mathrm{o}}\left(\frac{{{\mathrm{C}}_{\mathrm{x}}}^{°}+\left[{\mathrm{M}}^{+}\right]-\left[{\mathrm{X}}^{-}\right]}{{{\mathrm{C}}_{\mathrm{M}}}^{°}-\left[{\mathrm{M}}^{+}\right]-\left[{\mathrm{X}}^{-}\right]}\right)$

The titration reaction of ${\mathrm{M}}^{\mathrm{m}+}$with role="math" localid="1655099209443" ${\mathrm{X}}^{\mathrm{x}-}$

$$\begin{gathered} {\mathrm{xM}}^{\mathrm{m}+}+{\mathrm{mX}}^{\mathrm{x}-}\rightleftharpoons {\mathrm{M}}_{\mathrm{x}}{\mathrm{X}}_{\mathrm{m}\left(\mathrm{s}\right)} \\ {{\mathrm{mC}}_{\mathrm{M}}}^{\mathrm{o}},{\mathrm{V}}_{\mathrm{M}}\hspace{0.33em}{{\mathrm{xC}}_{\mathrm{x}}}^{°},{\mathrm{V}}_{\mathrm{x}}^{\mathrm{o}} \end{gathered}$$

${\mathrm{mC}}_{\mathrm{M}}^0$is initial concentration of ${\mathrm{M}}^{\mathrm{m}+}$

${\mathrm{V}}_{\mathrm{M}}$is volume of added${\mathrm{M}}^{\mathrm{m}+}$

${\mathrm{xC}}_{\mathrm{x}}^0$is concentration of ${\mathrm{X}}^{\mathrm{x}-}$

${\mathrm{V}}_{\mathrm{x}}^0$is volume of${\mathrm{X}}^{\mathrm{x}-}$

The mass balance of M and x .

$$\begin{gathered} {\mathrm{mC}}_{\mathrm{M}}^{\mathrm{o}}\cdot {\mathrm{V}}_{\mathrm{M}}=\mathrm{m}\left[{\mathrm{M}}^{\mathrm{m}+}\right]({\mathrm{V}}_{\mathrm{M}}+.{\mathrm{V}}_{\mathrm{x}})+\mathrm{mol}{\mathrm{M}}_{\mathrm{X}}{\mathrm{X}}_{\mathrm{m}\left(\mathrm{s}\right)} \\ {\mathrm{mC}}_{\mathrm{M}}^{\mathrm{o}}\cdot {\mathrm{V}}_{\mathrm{M}}=\mathrm{m}\left[{\mathrm{M}}^{\mathrm{m}+}\right]({\mathrm{V}}_{\mathrm{M}}+{\mathrm{V}}_{\mathrm{x}})+{\mathrm{molM}}_{\mathrm{X}}{\mathrm{X}}_{\mathrm{m}\left(\mathrm{s}\right)} \end{gathered}$$

On rearranging the above equation

we get $$\begin{gathered} {\mathrm{mC}}_{\mathrm{M}}^{\mathrm{o}}\cdot {\mathrm{V}}_{\mathrm{M}}-\mathrm{m}\left[{\mathrm{M}}^{\mathrm{m}+}\right]({\mathrm{V}}_{\mathrm{M}}+{\mathrm{V}}_{\mathrm{x}}^{\mathrm{o}}) \\ ={\mathrm{xC}}_{\mathrm{x}}^{\mathrm{o}}\cdot {\mathrm{V}}_{\mathrm{x}}^{\mathrm{o}}-\mathrm{x}\left[{\mathrm{X}}^{\mathrm{x}-}\right]({\mathrm{V}}_{\mathrm{M}}+{\mathrm{V}}_{\mathrm{x}}^{\mathrm{o}}) \end{gathered}$$

Thus the solution is ${\mathrm{V}}_{\mathrm{M}}={{\mathrm{V}}_{\mathrm{x}}}^{\mathrm{o}}\left(\frac{{{\mathrm{C}}_{\mathrm{x}}}^{°}+\left[{\mathrm{M}}^{+}\right]-\left[{\mathrm{X}}^{-}\right]}{{{\mathrm{C}}_{\mathrm{M}}}^{°}-\left[{\mathrm{M}}^{+}\right]-\left[{\mathrm{X}}^{-}\right]}\right)$