The lines in Balmer series have their wavelengths lying between

Options

A. 1266 Å to 3647 Å
B. 642 Å to 3000 Å
C. 3647 Å to 6563 Å
D. Zero to infinity

Answer

To find the range of wavelengths in the Balmer series, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Balmer Series**:
The Balmer series corresponds to the transitions of electrons in a hydrogen atom from higher energy levels (n ≥ 3) to the second energy level (n = 2).

2. **Using the Rydberg Formula**:
The wavelength of the emitted light during these transitions can be calculated using the Rydberg formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where:
- \( R \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \) or \( 10.97 \times 10^6 \, \text{m}^{-1} \).
- \( n_1 \) is the lower energy level (for Balmer series, \( n_1 = 2 \)).
- \( n_2 \) is the higher energy level (for Balmer series, \( n_2 = 3, 4, 5, ... \) up to infinity).

3. **Calculating the Wavelength for the First Line (n2 = 3)**:
- Set \( n_1 = 2 \) and \( n_2 = 3 \):
\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \]
- Substitute the values:
\[ \frac{1}{\lambda} = 10.97 \times 10^6 \left( \frac{1}{4} - \frac{1}{9} \right) \]
- Calculate \( \frac{1}{4} - \frac{1}{9} \):
\[ \frac{1}{4} - \frac{1}{9} = \frac{9 - 4}{36} = \frac{5}{36} \]
- Now substitute back:
\[ \frac{1}{\lambda} = 10.97 \times 10^6 \times \frac{5}{36} \]
- Calculate \( \lambda \):
\[ \lambda = \frac{36}{10.97 \times 10^6 \times 5} \approx 6563 \, \text{Å} \]

4. **Calculating the Wavelength for the Last Line (n2 = ∞)**:
- Set \( n_1 = 2 \) and \( n_2 = \infty \):
\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - 0 \right) \]
- Substitute the values:
\[ \frac{1}{\lambda} = 10.97 \times 10^6 \times \frac{1}{4} \]
- Calculate \( \lambda \):
\[ \lambda = \frac{4}{10.97 \times 10^6} \approx 3647 \, \text{Å} \]

5. **Determining the Range**:
- From the calculations, we find that the wavelengths in the Balmer series range from:
\[ 3647 \, \text{Å} \text{ to } 6563 \, \text{Å} \]

6. **Selecting the Correct Option**:
- The correct option that matches this range is option 3: **3647 Å to 6563 Å**.

### Final Answer:
The lines in the Balmer series have their wavelengths lying between **3647 Å and 6563 Å**.