In which of the following systems will the radius of the first orbit `(n = 1)` be minimum ?

Options

A. doubly ionized lithium
B. singly ionized helium
C. deuterium atom
D. hydrogen atom

Explanation

To determine which system has the minimum radius of the first orbit (n = 1), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Formula for Radius of Orbits**:
The radius of the nth orbit in a hydrogen-like atom is given by the formula:
\[ R_n \propto \frac{n^2}{Z} \]
where \( R_n \) is the radius of the nth orbit, \( n \) is the principal quantum number, and \( Z \) is the atomic number.

2. **Identify the Atomic Numbers**:
We need to find the atomic numbers for the given systems:
- Lithium ion (Li²⁺): \( Z = 3 \)
- Singly ionized helium (He⁺): \( Z = 2 \)
- Deuterium (D): \( Z = 1 \)
- Hydrogen (H): \( Z = 1 \)

3. **Set \( n = 1 \)**:
For the first orbit, we set \( n = 1 \) in the formula:
\[ R_1 \propto \frac{1^2}{Z} = \frac{1}{Z} \]

4. **Analyze the Inverse Relationship**:
Since \( R_1 \) is inversely proportional to \( Z \), the radius will be minimum when \( Z \) is maximum.

5. **Compare the Atomic Numbers**:
- For Li²⁺, \( Z = 3 \)
- For He⁺, \( Z = 2 \)
- For D and H, \( Z = 1 \)

The maximum atomic number among these is for Li²⁺.

6. **Conclusion**:
Therefore, the system with the minimum radius of the first orbit (n = 1) is the doubly ionized lithium ion (Li²⁺).

### Final Answer:
The system with the minimum radius of the first orbit is **doubly ionized lithium (Li²⁺)**.

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