A hypothetical sparingly soluble compound ( $\left.\mathrm{PO}_4\right)_5$, dissociates in water to

Question

A hypothetical sparingly soluble compound ( $\left.\mathrm{PO}_4\right)_5$, dissociates in water to form $\mathrm{Ga}^{5+}$ and $\mathrm{PO}_4^{3-}$ ions. If ' s ' denotes the molar solubility of this compound, determine the correct relationship between ' s ' and its solubility product constant  ${K }_{\mathrm{sp}}$.

Professor PC Thomas Classes · Answer

The given sparingly soluble compound is $\mathrm{Ga}_3\left(\mathrm{PO}_4ight)_5$.
When this compound dissociates in water, it forms $\mathrm{Ga}^{5+}$ and $\mathrm{PO}_4^{3-}$ ions.
The dissociation equilibrium can be written as:
$
\mathrm{Ga}_3\left(\mathrm{PO}_4ight)_5(s) ightleftharpoons 3 \mathrm{Ga}^{5+}(a q)+5 \mathrm{PO}_4^{3-}(a q)
$
Let 's' be the molar solubility of $\mathrm{Ga}_3\left(\mathrm{PO}_4ight)_5$.
According to the stoichiometry of the dissociation:
The concentration of $\mathbf{G a}^{\mathbf{5 +}}$ ions at equilibrium will be $\mathbf{3 s}$.
The concentration of $\mathrm{PO}_4^{3-}$ ions at equilibrium will be $5 s$.
The solubility product constant, $\mathbf{K}_{	ext {sp }}$, is defined as the product of the concentrations of the ions, each raised to the power of its stoichiometric coefficient in the balanced dissociation equation.
So, for $\mathrm{Ga}_3\left(\mathrm{PO}_4ight)_5: \mathrm{K}_{	ext {sp }}=\left[\mathrm{Ga}^{5+}ight]^3\left[\mathrm{PO}_4^{3-}ight]^5$
$
\begin{aligned}
& \mathrm{K}_{	ext {sp }}=(3 s)^3(5 s)^5 \
& \mathrm{~K}_{	ext {sp }}=\left(3^3 \cdot s^3ight) \cdot\left(5^5 \cdot s^5ight) \
& \mathrm{K}_{	ext {sp }}=\left(27 \cdot s^3ight) \cdot\left(3125 \cdot s^5ight) \
& \mathrm{K}_{	ext {sp }}=(27 	imes 3125) \cdot s^{(3+5)} \
& \mathrm{K}_{	ext {sp }}=84375 \cdot s^8 \
& s=\left(\frac{\mathrm{K}_{	ext {sp }}}{84375}ight)^{\frac{1}{8}}
\end{aligned}
$