The shortest wavelength of the Brackett series of a hydrogen-like atom (atomic number of `Z` ) is the same as the shortest wavelength of the Balmer series of hydrogen atom. The value of `z` is

Options

A. 3
B. 4
C. 5
D. 2

Explanation

To solve the problem, we need to find the atomic number \( Z \) of a hydrogen-like atom, given that the shortest wavelength of the Brackett series of this atom is the same as the shortest wavelength of the Balmer series of a hydrogen atom.

### Step-by-Step Solution:

1. **Understanding the Series**:
- The Brackett series transitions occur when an electron falls to the \( n = 4 \) energy level.
- The Balmer series transitions occur when an electron falls to the \( n = 2 \) energy level.

2. **Formula for Wavelength**:
- The formula for the wavelength in terms of wave number is given by:
\[ \frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
- Where \( R \) is the Rydberg constant, \( Z \) is the atomic number, \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

3. **Brackett Series for Hydrogen-like Atom**:
- For the Brackett series, \( n_1 = 4 \) and \( n_2 \) can be \( 5, 6, \ldots \). The shortest wavelength corresponds to \( n_2 \to \infty \):
\[ \frac{1}{\lambda_{Brackett}} = R Z^2 \left( \frac{1}{4^2} - 0 \right) = R Z^2 \left( \frac{1}{16} \right) \]

4. **Balmer Series for Hydrogen**:
- For the Balmer series, \( n_1 = 2 \) and \( n_2 \) can be \( 3, 4, \ldots \). The shortest wavelength corresponds to \( n_2 \to \infty \):
\[ \frac{1}{\lambda_{Balmer}} = R \left( \frac{1}{2^2} - 0 \right) = R \left( \frac{1}{4} \right) \]

5. **Setting the Wavelengths Equal**:
- Since the shortest wavelengths are equal:
\[ R Z^2 \left( \frac{1}{16} \right) = R \left( \frac{1}{4} \right) \]

6. **Canceling \( R \)**:
- We can cancel \( R \) from both sides:
\[ Z^2 \left( \frac{1}{16} \right) = \frac{1}{4} \]

7. **Solving for \( Z^2 \)**:
- Rearranging gives:
\[ Z^2 = 4 \]

8. **Finding \( Z \)**:
- Taking the square root:
\[ Z = 2 \]

### Final Answer:
The value of \( Z \) is \( 2 \).