Find number of basic radicals among the following cations, which can form soluble complex on adding excess of `NH_(4) ` solution.
`Cd^(2+)(aq.),Pb^(2+)(aq.),Ni^(2+)(aq.),Mn^(2+)(aq.),Zn^(2+)(aq.),Ag^(+)(aq.),Hg^(2+)(aq.),Fe^(3+)(aq.),Mg^(2+)(aq.)`

Explanation

To solve the problem of finding the number of basic radicals among the given cations that can form soluble complexes upon adding excess of \( NH_4 \) solution, we will analyze each cation one by one.

### Step-by-Step Solution:

1. **Identify the Cations**: The cations provided are:
- \( Cd^{2+} \)
- \( Pb^{2+} \)
- \( Ni^{2+} \)
- \( Mn^{2+} \)
- \( Zn^{2+} \)
- \( Ag^{+} \)
- \( Hg^{2+} \)
- \( Fe^{3+} \)
- \( Mg^{2+} \)

2. **Check Each Cation for Soluble Complex Formation**:
- **\( Cd^{2+} \)**: Forms \( Cd(OH)_2 \) and then \( [Cd(NH_3)_4]^{2+} \) in excess \( NH_4 \) solution. **(Forms complex)**
- **\( Pb^{2+} \)**: Forms \( Pb(OH)_2 \) but does not form a soluble complex with excess \( NH_4 \). **(Does not form complex)**
- **\( Ni^{2+} \)**: Forms \( Ni(OH)_2 \) and then \( [Ni(NH_3)_6]^{2+} \) in excess \( NH_4 \) solution. **(Forms complex)**
- **\( Mn^{2+} \)**: Forms \( Mn(OH)_2 \) but does not form a soluble complex with excess \( NH_4 \). **(Does not form complex)**
- **\( Zn^{2+} \)**: Forms \( Zn(OH)_2 \) and then \( [Zn(NH_3)_4(OH)_2] \) in excess \( NH_4 \) solution. **(Forms complex)**
- **\( Ag^{+} \)**: Forms \( Ag(NH_3)_2^{+} \) in excess \( NH_4 \) solution. **(Forms complex)**
- **\( Hg^{2+} \)**: Forms \( Hg(OH)_2 \) but does not form a soluble complex with excess \( NH_4 \). **(Does not form complex)**
- **\( Fe^{3+} \)**: Forms \( Fe(OH)_3 \) but does not form a soluble complex with excess \( NH_4 \). **(Does not form complex)**
- **\( Mg^{2+} \)**: Forms \( Mg(OH)_2 \) but does not form a soluble complex with excess \( NH_4 \). **(Does not form complex)**

3. **Count the Cations that Form Soluble Complexes**:
- The cations that form soluble complexes are:
- \( Cd^{2+} \)
- \( Ni^{2+} \)
- \( Zn^{2+} \)
- \( Ag^{+} \)

Thus, there are **4 cations** that can form soluble complexes.

### Final Answer:
The number of basic radicals among the given cations that can form soluble complexes upon adding excess \( NH_4 \) solution is **4**.

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