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Q.
What will be the ratio of the wavelength of the first line to that of the second line of Paschen series of $H$ atom?
Structure of Atom
Solution:
First time of paschen series $n_{1}=3, n_{2}=4$
$\frac{1}{\lambda_{1}}=R\left[\frac{1}{9}-\frac{1}{16}\right]$
$\frac{1}{\lambda_{1}}=\frac{7 R}{144}$
$\lambda_{1}=\frac{144}{7 R }$
Second line of paschen series $n_{1}=3,\,\, n_{2}=5$
$\frac{1}{\lambda_{2}}=R\left[\frac{1}{9}-\frac{1}{25}\right]$
$\frac{1}{\lambda_{2}}=\frac{16 R}{225} \quad \lambda_{2}=\frac{225}{16 R}$
$=\frac{\lambda_{1}}{\lambda_{2}}=\frac{144}{7 R} \times \frac{16 R}{225}$
$=\frac{2304}{1575}=\frac{256}{175}$