Question

In hydrogen atom, during the transition of electron from \( n \)th outer orbit to first Bohr orbit, a photon of wavelength \( \lambda \) is emitted. The value of \( n \) is: [\( R \) = Rydberg's constant]

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

Hint:

The Rydberg formula allows us to calculate the wavelength of emitted radiation for electron transitions in hydrogen atoms.

Answer 1

The Correct Option is A

Step 1: Using the Rydberg Formula.
The wavelength of emitted radiation when an electron transitions between two orbits in a hydrogen atom is given by the Rydberg formula: \[ \frac{1}{\lambda} = R \left( \frac{1}{n^2} - \frac{1}{1^2} \right) \] For transition from \( n \)th orbit to the first orbit, we can rewrite the equation: \[ \frac{1}{\lambda} = R \left( 1 - \frac{1}{n^2} \right) \] Thus, solving for \( n \), we get: \[ n = \frac{\lambda R}{\sqrt{\lambda R - 1}} \] Step 2: Conclusion.
Thus, the value of \( n \) is \( \frac{\lambda R}{\sqrt{\lambda R - 1}} \).