Question

Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

• A. The ratio of the longest wavelength to the shortest wavelength in Balmer series is $\frac{9}5$
• B. There is an overlap between the wavelength ranges of Balmer and Paschen series
• C. The wavelength of Lyman series are given by $\left(1+\frac{1}{ m ^{2}}\right) \lambda_{0}$, where $\lambda_{0}$ is the shortest wavelength of Lyman series and $m$ is an integer
• D. The wavelength ranges of Lyman and Balmer series do not overlap

Answer 1

The Correct Option is D

Concept: The spectral lines of hydrogen atom arise due to electron transitions between energy levels. The wavelength of emitted radiation is given by:

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \) is the Rydberg constant, \( n_1 \) is the lower energy level and \( n_2 \) is the higher energy level.

Lyman Series: \( n_1 = 1 \), \( n_2 = 2, 3, 4, \ldots \)

• Belongs to the ultraviolet (UV) region
• Shortest wavelength (limit): \( n_2 = \infty \):
  \[ \frac{1}{\lambda_{\min}^{\text{Lyman}}} = R_H \Rightarrow \lambda_{\min}^{\text{Lyman}} = \frac{1}{R_H} \approx 91.2 \, \text{nm} \]
• Longest wavelength: \( n_2 = 2 \):
  \[ \frac{1}{\lambda_{\max}^{\text{Lyman}}} = R_H \left(1 - \frac{1}{4}\right) = \frac{3R_H}{4} \Rightarrow \lambda_{\max}^{\text{Lyman}} \approx 121.6 \, \text{nm} \]
• Range: 91.2 nm to 121.6 nm

Balmer Series: \( n_1 = 2 \), \( n_2 = 3, 4, 5, \ldots \)

• Belongs to the visible region
• Shortest wavelength (limit): \( n_2 = \infty \):
  \[ \frac{1}{\lambda_{\min}^{\text{Balmer}}} = R_H \left( \frac{1}{4} \right) \Rightarrow \lambda_{\min}^{\text{Balmer}} \approx 364.6 \, \text{nm} \]
• Longest wavelength: \( n_2 = 3 \):
  \[ \frac{1}{\lambda_{\max}^{\text{Balmer}}} = R_H \left( \frac{1}{4} - \frac{1}{9} \right) = \frac{5R_H}{36} \Rightarrow \lambda_{\max}^{\text{Balmer}} \approx 656.3 \, \text{nm} \]
• Range: 364.6 nm to 656.3 nm

Conclusion:

The Lyman series lies entirely in the UV region (91–122 nm) while the Balmer series lies in the visible region (365–656 nm), so there is no overlap between the two.

Correct Answer: Option (D): The wavelength ranges of Lyman and Balmer series do not overlap