Question

A hydrogen atom is in an excited state of principal quantum number \( n \), it emits a photon of wavelength \( \lambda \), when it returns to the ground state. The value of \( n \) is

• A. \( \sqrt{\frac{R}{\lambda}} \)
• B. \( \sqrt{\frac{R - 1}{\lambda}} \)
• C. \( \frac{\lambda R}{1} \)
• D. \( \frac{\lambda R}{\lambda - 1} \)

Hint:

The wavelength of emitted photons in hydrogen atoms depends on the transition between energy levels, and can be calculated using the Rydberg formula.

Answer 1

The Correct Option is A

Step 1: Using the Rydberg Formula.
The Rydberg formula for the wavelength of the emitted photon is given by: \[ \frac{1}{\lambda} = R \left( \frac{1}{n^2} - \frac{1}{1^2} \right) \] where \( R \) is the Rydberg constant and \( n \) is the principal quantum number. Solving for \( n \) using this equation, we get \( n = \sqrt{\frac{R}{\lambda}} \). Step 2: Conclusion.
The correct answer is (A), \( \sqrt{\frac{R}{\lambda}} \).