Which of the following statement(s) is (are) correct about the spectrum of hydrogen atom?

Question

Aicharya · Accepted Answer

For Balmer Series:

Shortest wavelength, $\frac{E_{o}}{4} = \frac{h c}{λ_{s}}$

Longest wavelength, $(\frac{E_{o}}{4} - \frac{E_{0}}{9}) = \frac{h c}{λ_{l}} = \frac{5 E_{o}}{36}$

$∴ \frac{λ_{l}}{λ_{s}} = \frac{9}{5}$

Range of energy level in Balmer series,

$(\frac{5 E_{0}}{36}, \frac{E_{0}}{4})$

Range of energy level in Paschen series,

$(\frac{7 E_{0}}{144}, \frac{E_{0}}{9})$

Hence, no overlap.

For Lyman series:

For shortest wavelength,

$E_{0} = \frac{h c}{λ_{0}}$

For general wavelength,

$(E_{0} - \frac{E_{0}}{n^{2}}) = \frac{h c}{λ}$

$\Rightarrow \frac{1}{(1 - \frac{1}{n^{2}})} = \frac{λ}{λ_{0}}$

$∴ λ = (\frac{n^{2}}{n^{2} - 1}) λ_{0}$

or $λ = (1 + \frac{1}{n^{2} - 1}) λ_{0}$ $= (1 + \frac{1}{m^{2}}) λ_{0}$

as $m$ should be an integer.

Range of energy level in Lyman series, $(\frac{3 E_{0}}{4}, E_{0})$

$\Rightarrow$ Lyman and Balmer series wavelengths do not overlap.

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Solution