Q19.77P - Chemistry: Molecular Nature Of Matter And Change

Question

Calculate the molar solubility of ${\mathbf{\text{Ag}}}_{\mathbf{\text{2}}}{\mathbf{\text{SO}}}_{\mathbf{\text{4}}}$ in 

(a)  ${\mathbf{\text{0.22 M AgNO}}}_{\mathbf{\text{3}}}$and 

(b) ${\mathbf{\text{0.22 M Na}}}_{\mathbf{\text{2}}}{\mathbf{\text{SO}}}_{\mathbf{\text{4}}}$. (See Appendix C.)

Step-by-Step Solution

Answer

(a) The molar solubility of ${\text{Ag}}_{\text{2}}{\text{SO}}_{\text{4}}$in ${\text{0.22 M AgNO}}_{\text{3}}$ is $\mathrm{S}=3.1\times {10}^{-4}\mathrm{M}$.

(b) The molar solubility of ${\text{Ag}}_{\text{2}}{\text{SO}}_{\text{4}}$ in ${\text{0.22 M Na}}_2{\text{SO}}_{\text{4}}$ is $\mathrm{S}=4.1\times {10}^{-3}\mathrm{M}$.

Concept Introduction

The solubility product is directly related to the maximum number of moles of the solute that can be dissolved in a litre of solution before the solution becomes saturated. Any additional solute precipitates out of a solution once it is saturated.

 Q_sp- ion-product expression; Q_sp  value is obtained when the concentrations of ions formed by dissolution of some compound are multiplied. When the solution is saturated Q_sp  value is called K_sp  value (solubility-product constant).

${\mathbf{\text{MX}}}_{\mathbf{\text{2}}}\mathbf{\to }{\mathbf{\text{M}}}^{\mathbf{\text{2 + }}}{\mathbf{\text{ + 2X}}}^{\mathbf{\text{ - }}}$

Solid and liquid state is not included in the K_sp  equations.

${\mathbf{\text{K}}}_{\mathbf{\text{sp}}}{\mathbf{\text{ = [M}}}^{\mathbf{\text{2 + }}}{\mathbf{\text{][X}}}^{\mathbf{\text{ - }}}{\mathbf{\text{]}}}^{\mathbf{\text{2}}}$

 S (Molar solubility) is equal to the concentration of one mol of the ion formed by dissolution of some compound.

Solubility of Ag 2 SO 4 in role="math" localid="1663350727227" AgNO 3

(a)

Write the equation of dissolution for ${\text{Ag}}_{\text{2}}{\text{SO}}_{\text{4}}$–

${\text{Ag}}_{\text{2}}{\text{SO}}_{\text{4}}\text{(s)}\iff 2{\text{Ag}}^{\text{ + }}{\text{(aq) + SO}}_4^{\text{2 - }}\text{(aq)}$

In this case, the initial concentration of ${\text{Ag}}^{\text{ + }}$ion is $\text{0.22 M + S}$, as there is ${\text{0.22 M AgNO}}_{\text{3}}$ so the equilibrium concentrations will be –

$$\begin{gathered} \left[{\mathrm{Ag}}^{+}\right]=0.22\mathrm{M}+2\mathrm{S} \\ \left[{\mathrm{SO}}_4^{2-}\right]=\mathrm{S} \end{gathered}$$

Rearranging the equation of K_sp and solving –

Square ${\text{[Ag}}^{\text{ + }}\text{]}$ as there are $2$ moles of ${\text{Ag}}^{\text{ + }}$in the equation.

$$\begin{gathered} {\mathrm{K}}_{\mathrm{sp}}=\left[{\mathrm{Ag}}^{+}\right]\left[{\mathrm{SO}}_4^{2-}\right] \\ {\mathrm{K}}_{\mathrm{sp}}=(0.22\mathrm{M}+2\mathrm{S}{)}^2·\mathrm{S} \\ {\mathrm{K}}_{\mathrm{sp}}=1.5\times {10}^{-5} \\ \mathrm{S}=3.1\times {10}^{-4}\mathrm{M} \end{gathered}$$

Therefore, the value for solubility is obtained as $\mathrm{S}=3.1\times {10}^{-4}\mathrm{M}$.

Solubility of Ag 2 SO 4 in Na 2 SO 4

(b)

In this case, the initial concentration of ${\text{SO}}_4^{\text{2 - }}$ion is $\text{0.22 M + S}$, so the equilibrium concentrations will be –

$$\begin{gathered} \left[{\mathrm{Ag}}^{+}\right]=2\mathrm{S} \\ \left[{\mathrm{SO}}_4^{2-}\right]=0.22\mathrm{M}+\mathrm{S} \end{gathered}$$

Rearranging the equation of K_sp and solving –

$$\begin{gathered} {\mathrm{K}}_{\mathrm{sp}}=\left[{\mathrm{Ag}}^{+}\right]\left[{\mathrm{SO}}_4^{2-}\right] \\ {\mathrm{K}}_{\mathrm{sp}}=(2\mathrm{S}{)}^2·(0.22\mathrm{M}+\mathrm{S}) \\ {\mathrm{K}}_{\mathrm{sp}}=1.5\times {10}^{-5} \\ \mathrm{S}=4.1\times {10}^{-3}\mathrm{M} \end{gathered}$$

Therefore, the value for solubility is obtained as $\mathrm{S}=4.1\times {10}^{-3}\mathrm{M}$.