The Rydberg constant for hydrogen is 10967700 m^–1. Calculate the short and long wavelength limits of Lyman series.

Solution

For Lyman series, the wave number is given by

v = \frac{1}{λ} = R_{H} (\frac{1}{1^{2}} - \frac{1}{n^{2}})

For the short wavelength limit (λ = λ_s), n = ∞

\Rightarrow

\bar{v_{s}} = \frac{1}{λ_{s}} = R_{H} (\frac{1}{1^{2}} - \frac{1}{\infty^{2}}) = R_{H}

∴

λ_{s} = \frac{1}{R_{H}} = \frac{1}{10967700} m = 9.116 \times 10^{- 8} m = 911.6 Å

For long wavelength limit (λ = λ_L.), n = 2

∴

{\bar{v}}_{L} = \frac{1}{λ_{L}} = R_{H} (\frac{1}{1^{2}} - \frac{1}{2^{2}}) = \frac{3}{4} R_{H}

So,

λ_{L} = \frac{4}{3 R_{H}}

= \frac{4}{3} \times 911.6 Å = 1215 \overset{o}{A}

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