Find out the number of cation (s) which form(s) black ppt. (soluble in hot and dilute `HNO_(3)`) on passing `H_(2)S` gas into their salt solution?
`Mg^(2+)(aq.)Cu^(2+)(aq.),Ba^(2+)(aq.),Fe^(3+)(aq.),Ag^(+)(aq.),Al^(3+)(aq.),Hg^(2+)(aq.),Pb^(2+)(aq.),Mn^(2+)(aq.)`

Answer

To determine the number of cations that form a black precipitate soluble in hot and dilute HNO₃ when H₂S gas is passed into their salt solutions, we will analyze each given cation step by step.

### Step-by-Step Solution:

1. **Identify the given cations:**
The cations provided are:
- Mg²⁺
- Cu²⁺
- Ba²⁺
- Fe³⁺
- Ag⁺
- Al³⁺
- Hg²⁺
- Pb²⁺
- Mn²⁺

2. **Analyze each cation for black precipitate formation with H₂S:**
- **Mg²⁺:** Does not form a precipitate with H₂S.
- **Cu²⁺:** Forms a black precipitate of CuS (copper sulfide).
- **Ba²⁺:** Does not form a precipitate with H₂S.
- **Fe³⁺:** Does not form a precipitate with H₂S.
- **Ag⁺:** Forms a black precipitate of Ag₂S (silver sulfide).
- **Al³⁺:** Forms a white precipitate of Al(OH)₃, not black.
- **Hg²⁺:** Forms a red precipitate of HgS (mercury sulfide), not black.
- **Pb²⁺:** Forms a black precipitate of PbS (lead sulfide).
- **Mn²⁺:** Forms a pink precipitate of MnS (manganese sulfide), not black.

3. **List the cations that form black precipitates:**
From the analysis:
- Cu²⁺ → CuS (black)
- Ag⁺ → Ag₂S (black)
- Pb²⁺ → PbS (black)

4. **Check solubility in hot and dilute HNO₃:**
- **CuS:** Soluble in hot and dilute HNO₃.
- **Ag₂S:** Soluble in hot and dilute HNO₃.
- **PbS:** Soluble in hot and dilute HNO₃.

5. **Count the number of cations that meet the criteria:**
The cations that form a black precipitate and are soluble in hot and dilute HNO₃ are:
- Cu²⁺
- Ag⁺
- Pb²⁺

Thus, the total number of cations that form a black precipitate soluble in hot and dilute HNO₃ is **3**.

### Final Answer:
The number of cations that form a black precipitate soluble in hot and dilute HNO₃ is **3** (Cu²⁺, Ag⁺, Pb²⁺).