Question

Energy levels A, B, and C of a certain atom correspond to increasing values of energy, i.e., \( E_A<E_B<E_C \). If \( \lambda_1, \lambda_2, \) and \( \lambda_3 \) are the wavelengths of a photon corresponding to the transitions shown, then:

• A. \( \lambda_3 = \lambda_1 + \lambda_2 \)
• B. \( \lambda_3 = \frac{(\lambda_1 + \lambda_2)}{\lambda_1 \lambda_2} \)
• C. \( \lambda_3^2 = \lambda_1^2 + \lambda_2^2 \)
• D. \( \lambda_3 = \frac{\lambda_1 \lambda_2}{(\lambda_1 + \lambda_2)} \)

Hint:

When an electron transitions between energy levels, the wavelength follows the reciprocal relation in photon emission.

Answer 1

The Correct Option is D

Step 1: Use the Energy Relation for Wavelengths From energy conservation: \[ E_3 = E_1 + E_2 \] Since energy and wavelength are related by: \[ E = \frac{hc}{\lambda} \] we get: \[ \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \] Dividing by \( hc \): \[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]
Step 2: Solve for \( \lambda_3 \) \[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \]