In the hydrogen spectrum, if the shortest wavelength in the Paschen series is denoted as $\lambda$,

Question

In the hydrogen spectrum, if the shortest wavelength in the Paschen series is denoted as $\lambda$, then the shortest wavelength in the Pfund series will be

Professor PC Thomas Classes · Answer

(a) : The shortest wavelength (series limit) occurs for a transition from infinity ($n=\infty$). For the Paschen series ($n_1 = 3$), $\frac{1}{\lambda} = R\left(\frac{1}{3^2} - 0\right) = \frac{R}{9} \implies \lambda = \frac{9}{R}$. For the Pfund series ($n_1 = 5$), $\frac{1}{\lambda'} = R\left(\frac{1}{5^2} - 0\right) = \frac{R}{25} \implies \lambda' = \frac{25}{R}$. Taking the ratio, $\frac{\lambda'}{\lambda} = \frac{25/R}{9/R} = \frac{25}{9}$. Thus, $\lambda' = \frac{25\lambda}{9}$.