Energy levels A, B, C of a certain atom correspond to increasing values of energy (EA < EB < EC). If λ1, λ2 and λ3 are the wavelengths of radiations corresponding to the transitions C to B, B to A and C to A respectively, the relation between these wavelengths can be written as-
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \lambda_3 = \displaystyle\frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \)
In a transition the emitted photon carries the energy difference between the levels, \( E = hc/\lambda \). The energy gap from C to A is the sum of the gaps C to B and B to A, so the energies add, not the wavelengths.
\( \displaystyle E_C - E_A = (E_C - E_B) + (E_B - E_A) \)
\( \displaystyle \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \)
\( \displaystyle \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} = \frac{\lambda_1 + \lambda_2}{\lambda_1\lambda_2} \)
\( \displaystyle \lambda_3 = \frac{\lambda_1\lambda_2}{\lambda_1 + \lambda_2} \)