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Question: For hydrogen atom, the figure indicates the energy level diagram and the origin of 6 spectral lines ...

For hydrogen atom, the figure indicates the energy level diagram and the origin of 6 spectral lines in emission spectrum.

Which of the following spectral line will have maximum wavelength?

A

6

A

3

B

1

D

5

Answer

6

Explanation

Solution

The energy of emitted photons is inversely proportional to their wavelength (ΔE=hc/λ\Delta E = hc/\lambda). Maximum wavelength corresponds to minimum energy difference. The energy levels are En=−13.6/n2E_n = -13.6/n^2 eV. Line 6 is n=4→n=3n=4 \to n=3. ΔE6=E4−E3=−13.6/16−(−13.6/9)=13.6(1/9−1/16)≈0.66\Delta E_6 = E_4 - E_3 = -13.6/16 - (-13.6/9) = 13.6(1/9 - 1/16) \approx 0.66 eV. Line 1 is n=4→n=2n=4 \to n=2. ΔE1=E4−E2=−13.6/16−(−13.6/4)=13.6(1/4−1/16)≈2.55\Delta E_1 = E_4 - E_2 = -13.6/16 - (-13.6/4) = 13.6(1/4 - 1/16) \approx 2.55 eV. Line 3 is n=3→n=2n=3 \to n=2. ΔE3=E3−E2=−13.6/9−(−13.6/4)=13.6(1/4−1/9)≈1.89\Delta E_3 = E_3 - E_2 = -13.6/9 - (-13.6/4) = 13.6(1/4 - 1/9) \approx 1.89 eV. Line 5 is n=2→n=1n=2 \to n=1. ΔE5=E2−E1=−13.6/4−(−13.6/1)=13.6(1−1/4)≈10.2\Delta E_5 = E_2 - E_1 = -13.6/4 - (-13.6/1) = 13.6(1 - 1/4) \approx 10.2 eV. The minimum energy difference is for line 6.