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Step-by-Step Solution
1. Determine the Moles of Each Complex
• The first solution is 1 L of 0.02 M
$[\mathrm{Co}(\mathrm{NH}_{3})_{5}\,\mathrm{SO}_4]\,\mathrm{Br}$.
• Moles of this complex in the first solution
= (0.02 mol/L) × (1 L) = 0.02 moles.
• The second solution is 1 L of 0.02 M
$[\mathrm{Co}(\mathrm{NH}_{3})_{5}\,\mathrm{Br}]\,\mathrm{SO}_4$.
• Moles of this complex in the second solution
= (0.02 mol/L) × (1 L) = 0.02 moles.
2. Identify the Free Ions in Each Solution
• In $[\mathrm{Co}(\mathrm{NH}_{3})_{5}\,\mathrm{SO}_4]\,\mathrm{Br}$, the $\mathrm{SO}_4^{2-}$ group is within the coordination sphere, so the free ion in solution is $\mathrm{Br}^{-}$.
→ Hence, the total free $\mathrm{Br}^{-}$ moles in that 1 L solution is 0.02 moles.
• In $[\mathrm{Co}(\mathrm{NH}_{3})_{5}\,\mathrm{Br}]\,\mathrm{SO}_4$, the $\mathrm{Br}^{-}$ is within the coordination sphere, so the free ion in solution is $\mathrm{SO}_4^{2-}$.
→ Hence, the total free $\mathrm{SO}_4^{2-}$ moles in that 1 L solution is 0.02 moles.
3. Mix the Two Solutions
• When these two 1 L solutions are mixed, the total volume becomes 2 L.
• Free $\mathrm{Br}^{-}$ in the combined solution = 0.02 moles.
• Free $\mathrm{SO}_4^{2-}$ in the combined solution = 0.02 moles.
4. Divide the Mixture into Two Equal Parts
• The resulting 2 L mixture is split into two equal parts, each of volume 1 L.
• Therefore, each 1 L part (labeled as X) contains half the total moles of free ions:
→ $\mathrm{Br}^{-}$ = 0.01 moles
→ $\mathrm{SO}_4^{2-}$ = 0.01 moles
5. Precipitation with AgNO₃ (Formation of Y)
• The first 1 L portion (X) is treated with excess $\mathrm{AgNO}_3$.
• $\mathrm{Ag}^{+}$ ions precipitate all $\mathrm{Br}^{-}$ ions as $\mathrm{AgBr}$.
• Since there are 0.01 moles of $\mathrm{Br}^{-}$, the moles of $\mathrm{AgBr}$ formed (Y) is also 0.01 moles.
6. Precipitation with BaCl₂ (Formation of Z)
• The second 1 L portion (X) is treated with excess $\mathrm{BaCl}_2$.
• $\mathrm{Ba}^{2+}$ ions precipitate all $\mathrm{SO}_4^{2-}$ ions as $\mathrm{BaSO}_4$.
• Since there are 0.01 moles of $\mathrm{SO}_4^{2-}$, the moles of $\mathrm{BaSO}_4$ formed (Z) is also 0.01 moles.
7. Final Answer
The number of moles of Y and Z are 0.01 and 0.01, respectively.
