Step 1: Understanding the Concept:
In the systematic qualitative analysis of inorganic cations, different metal ions are precipitated sequentially using specific group reagents. We must identify which of the given metals specifically belongs to the analytical group whose reagent is $H_2S$ gas passed in an alkaline medium ($NH_4OH$). After identifying the metal, we state its highest known oxidation state.
Step 2: Key Formula or Approach:
Group II reagent: $H_2S$ in acidic medium ($HCl$). Precipitates $Pb^{2+}, Cu^{2+}$.
Group III reagent: $NH_4OH$ in presence of $NH_4Cl$. Precipitates $Fe^{3+}$ as $Fe(OH)_3$.
Group IV reagent: $H_2S$ in basic medium ($NH_4OH$). Precipitates $Mn^{2+}, Zn^{2+}, Ni^{2+}, Co^{2+}$.
Step 3: Detailed Explanation:
Let's classify the given ions into their analytical groups:
1. $Pb^{2+}$: Belongs to Group I (precipitates as chloride) and Group II (precipitates as sulfide in acidic medium).
2. $Cu^{2+}$: Belongs to Group II. It precipitates as $CuS$ when $H_2S$ is passed in an acidic medium.
3. $Fe^{3+}$: Belongs to Group III. Its specific group reagent is $NH_4OH$ (with $NH_4Cl$), precipitating it as $Fe(OH)_3$.
4. $Mn^{2+}$: Belongs to Group IV. The specific group reagent for Group IV is $H_2S$ passed in an alkaline medium provided by $NH_4OH$. This precipitates manganese as Manganese(II) sulfide ($MnS$).
Thus, the metal that is specifically precipitated by $H_2S$ in the presence of $NH_4OH$ as its group reagent is Manganese (Mn).
The question asks for the highest possible oxidation state of this metal.
Manganese has the electronic configuration $[Ar] 3d^5 4s^2$. By losing all of its 4s and 3d valence electrons, it can achieve a maximum oxidation state of +7 (commonly seen in the permanganate ion, $MnO_4^-$).
Step 4: Final Answer:
The highest possible oxidation state is +7.