Atomic Physics

The hydrogen atomic spectrum in the Lyman series corresponds to electron transitions from higher energy levels ( ) to the first energy level ( ). The wavelength of the emission can be calculated using the Rydberg formula for hydrogen:

where:

• is the wavelength of the emitted light.
• is the Rydberg constant ( ).
• is the lower energy level.
• is the higher energy level.

For the first emission in the Lyman series, and . Therefore, the formula becomes:

So,

The wavelength expressed in terms of the Rydberg constant is , which matches the correct answer:

To determine the atomic number Z of an H-like ion where the wavelength matches between two different transitions, we analyze the energy transitions in hydrogen-like atoms.

1. Lyman Series Transition (Hydrogen Atom):
For the Lyman series (transition to n=1):


For the first line (n=2 → n=1):

2. Balmer Series Transition (H-like Ion):
For the second line of Balmer series (n=4 → n=2) in an H-like ion:

3. Equating the Transitions:
Since both transitions produce the same wavelength:


Canceling common terms:

4. Solving for Z:

Final Answer:
The atomic number of the H-like ion is .