If the series limit wavelength of the Lyman series of hydrogen atom is `912Å`, then the series limit wavelength of the Balmer series of the hydrogen atom is

Options

A. `912Å`
B. `1824Å`
C. `3648Å`
D. `456Å`

Answer

To find the series limit wavelength of the Balmer series of the hydrogen atom given that the series limit wavelength of the Lyman series is 912 Å, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Series Limits**:
The series limit wavelength for any spectral series in hydrogen can be calculated using the formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, \( n_1 \) is the principal quantum number of the lower energy level, and \( n_2 \) is the principal quantum number of the higher energy level.

2. **Lyman Series Calculation**:
For the Lyman series, the transitions are from \( n_2 \) (higher level) to \( n_1 = 1 \) (lower level). The series limit occurs when \( n_2 \) approaches infinity. Thus:
\[ \frac{1}{\lambda_L} = R \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \left( 1 - 0 \right) = R \]
Given that \( \lambda_L = 912 \, \text{Å} \), we can express this as:
\[ R = \frac{1}{912 \, \text{Å}} \]

3. **Balmer Series Calculation**:
For the Balmer series, the transitions are from \( n_2 \) (higher level) to \( n_1 = 2 \) (lower level). Again, the series limit occurs when \( n_2 \) approaches infinity:
\[ \frac{1}{\lambda_B} = R \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = R \left( \frac{1}{4} - 0 \right) = \frac{R}{4} \]

4. **Relating the Two Series**:
Since we have \( R = \frac{1}{912 \, \text{Å}} \), we can substitute this into the equation for the Balmer series:
\[ \frac{1}{\lambda_B} = \frac{1}{912 \, \text{Å}} \cdot \frac{1}{4} \]
This gives:
\[ \frac{1}{\lambda_B} = \frac{1}{3648 \, \text{Å}} \]

5. **Final Calculation**:
Therefore, the wavelength for the Balmer series limit is:
\[ \lambda_B = 3648 \, \text{Å} \]

### Conclusion:
The series limit wavelength of the Balmer series of the hydrogen atom is **3648 Å**.