Dual Nature Of Radiation And Matter

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

where is the Rydberg constant, is the lower energy level and is the higher energy level.

Lyman Series: ,

• Belongs to the ultraviolet (UV) region
• Shortest wavelength (limit): :
  
• Longest wavelength: :
  
• Range: 91.2 nm to 121.6 nm

Balmer Series: ,

• Belongs to the visible region
• Shortest wavelength (limit): :
  
• Longest wavelength: :
  
• 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

Step 1: Understanding the given spectrum of the hydrogen atom
The spectrum of the hydrogen atom is characterized by distinct series, such as the Lyman series, Balmer series, etc., that correspond to different electronic transitions within the atom.
In the case of the hydrogen atom, the wavelengths of the emitted radiation in these series can be described using the Rydberg formula:
where:
- is the Rydberg constant for hydrogen,
- and are the principal quantum numbers of the initial and final states, respectively.
The Balmer series corresponds to transitions where and takes values from 3 to infinity.
The Lyman series corresponds to transitions where and takes values from 2 to infinity.
Step 2: Analyzing the statements
Statement (A): The ratio of the longest wavelength to the shortest wavelength in the Balmer series is
In the Balmer series, the longest wavelength corresponds to the transition from to , and the shortest wavelength corresponds to the transition from to .
Using the Rydberg formula, we can calculate the wavelengths for these transitions and find that the ratio of the longest wavelength to the shortest wavelength is indeed . Hence, statement (A) is correct.
Statement (D): The wavelength ranges of the Lyman and Balmer series do not overlap
The Lyman series lies in the ultraviolet range, and the Balmer series lies in the visible range of the spectrum.
Since the wavelength for the Lyman series is shorter than the wavelength for the Balmer series, their wavelength ranges do not overlap. Hence, statement (D) is also correct.
Step 3: Conclusion
Therefore, the correct options are:
(A): The ratio of the longest wavelength to the shortest wavelength in the Balmer series is
(D): The wavelength ranges of Lyman and Balmer series do not overlap