Energy levels A, B, C of a certain atom corresponds to increasing value of energy \( E_A \lt E_B \lt E_C \). If \( \lambda_1, \lambda_2, \lambda_3 \) are the wavelengths of radiation corresponding to the transitions C → B, B → A and C → A respectively then :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \lambda_3 = \displaystyle\frac{\lambda_1\cdot\lambda_2}{\lambda_1 + \lambda_2} \)
By Bohr's frequency condition, the photon emitted in a transition carries the energy difference of the two levels, \( E = hc/\lambda \). Energies add, not wavelengths.
\( \displaystyle E_C - E_A = (E_C - E_B) + (E_B - E_A) \)
\( \displaystyle \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \;\Rightarrow\; \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} = \frac{\lambda_1 + \lambda_2}{\lambda_1\lambda_2} \)
\( \displaystyle \lambda_3 = \frac{\lambda_1\lambda_2}{\lambda_1 + \lambda_2} \)