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Hydrogen spectrum – series limits
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The ratio of wavelengths of the last line of Balmer series and the last line of Lyman series is
Series limits: Balmer $4/R$, Lyman $1/R$.
For the last line of the Balmer series: $\frac{1}{\lambda_b} = R\left[\frac{1}{2^2} - \frac{1}{\infty^2}\right] \Rightarrow \lambda_b = \frac{4}{R}$ For the last line of the Lyman series: $\frac{1}{\lambda_l} = R\left[\frac{1}{1^2} - \frac{1}{\infty^2}\right] \Rightarrow \lambda_l = \frac{1}{R}$ $\frac{\lambda_b}{\lambda_l} = \frac{4/R}{1/R} = 4$