Which of the following series fall in the visible range of electromagnetic spectrum of a hydrogen atom ?

Options

A. Brackett series
B. Lyman series
C. Balmer series
D. Paschan series

Answer

To determine which series of the hydrogen atom falls within the visible range of the electromagnetic spectrum, we need to analyze the different series and their corresponding wavelengths.

### Step-by-Step Solution:

1. **Understand the Visible Range**:
The visible range of the electromagnetic spectrum is approximately from 400 nm to 700 nm.

2. **Identify the Series**:
The series of transitions in a hydrogen atom include:
- Lyman series (n1 = 1)
- Balmer series (n1 = 2)
- Paschen series (n1 = 3)
- Brackett series (n1 = 4)
- Pfund series (n1 = 5)

3. **Calculate Wavelengths Using Rydberg Formula**:
The Rydberg formula for the wavelength of emitted light during electron transitions in hydrogen is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant (approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \)), \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

4. **Calculate for Balmer Series**:
For the Balmer series, \( n_1 = 2 \). The transitions can be to \( n_2 = 3, 4, 5, \ldots \).
- For \( n_2 = 3 \):
\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{9 - 4}{36} \right) = \frac{5R}{36} \]
\[ \lambda = \frac{36}{5R} \]
Substituting \( R \):
\[ \lambda = \frac{36}{5 \times 1.097 \times 10^7} \approx 656 \, \text{nm} \]
- This wavelength (656 nm) falls within the visible range.

5. **Check Other Series**:
- **Lyman Series**: Wavelengths are shorter than 400 nm (ultraviolet).
- **Paschen Series**: Wavelengths are longer than 700 nm (infrared).
- **Brackett Series**: Wavelengths are also longer than 700 nm (infrared).
- **Pfund Series**: Wavelengths are even longer (far infrared).

6. **Conclusion**:
The only series that falls within the visible range of the electromagnetic spectrum is the **Balmer series**.

### Final Answer:
The Balmer series falls in the visible range of the electromagnetic spectrum of a hydrogen atom.