Consider the energy levels of atom as shown in the figure. The ratio of wavelengths \( \lambda_{1}/\lambda_{2} \) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{1}{3} \)
In a transition the energy of the emitted photon equals the difference between the two levels, \( \Delta E = hc/\lambda \), so the wavelength is inversely proportional to the energy gap.
Readings from the figure: \( \lambda_1 \) is emitted from \( -E \) to \( -2E \), and \( \lambda_2 \) from \( -E \) to \( -\frac{4}{3}E \).
\( \Delta E_1 = -E - (-2E) = E \)
\( \displaystyle \Delta E_2 = -E - \left(-\frac{4}{3}E\right) = \frac{E}{3} \)
\( \displaystyle \frac{\lambda_1}{\lambda_2} = \frac{\Delta E_2}{\Delta E_1} = \frac{E/3}{E} = \frac{1}{3} \)