In which of the following transitions will the wavelength of light emitted be minimum?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → n = 2 to n = 1
In a hydrogen-like atom the photon emitted in a transition has the energy equal to the difference of the two levels, and the wavelength is \( \lambda = hc/\Delta E \). The largest energy gap therefore gives the shortest wavelength. For hydrogen, \( \Delta E = 13.6\left(\frac{1}{n_f^{2}} - \frac{1}{n_i^{2}}\right)\mathrm{eV} \).
\( 5\rightarrow4:\ 13.6\left(\frac{1}{16} - \frac{1}{25}\right) \approx 0.31\,\mathrm{eV} \)
\( 4\rightarrow3:\ 13.6\left(\frac{1}{9} - \frac{1}{16}\right) \approx 0.66\,\mathrm{eV} \)
\( 3\rightarrow2:\ 13.6\left(\frac{1}{4} - \frac{1}{9}\right) \approx 1.89\,\mathrm{eV} \)
\( 2\rightarrow1:\ 13.6\left(1 - \frac{1}{4}\right) = 10.2\,\mathrm{eV} \)
The gap is largest for \( n = 2\rightarrow1 \), so the wavelength is minimum for this transition.
Note: The same question was also asked in CUET (UG) 2023 on 08-06-2023 (Slot 3) and on 27-05-2023 (Slot 3).