The ratio of wavelength values of series limit lines 2 ($\mathrm{n}=\mathrm{\alpha }$) of Balmer series and Paschen series are

Solution:

For limiting line in Balmer series,

${\mathrm{n}}_1=2 \mathrm{and} {\mathrm{n}}_2=\mathrm{\alpha }$

For limiting line Paschen series

${\mathrm{n}}_1=3 \mathrm{and} {\mathrm{n}}_2=\mathrm{\alpha }$

Wavelength of limiting line in Balmer series,

$\frac{1}{{\mathrm{\lambda }}_{\mathbf{B}}}=\mathrm{R}\left[\frac{1}{{\mathrm{n}}_1^2}-\frac{1}{{\mathrm{n}}_2^2}\right]$

$Then, \frac{1}{{\mathrm{\lambda }}_{\mathbf{B}}}=\mathrm{R}\left[\frac{1}{{\left(2\right)}_1^2}-\frac{1}{{\left(\mathrm{\alpha }\right)}_2^2}\right]=\mathrm{R}\left[\frac{1}{4}-0\right]$

$\frac{1}{{\mathrm{\lambda }}_{\mathrm{B}}}=\frac{\mathrm{R}}{4} \mathrm{or} {\mathrm{\lambda }}_{\mathrm{B}}=\frac{4}{\mathrm{R}}$

Wavelength of limiting line in Paschen series,

$\frac{1}{{\mathrm{\lambda }}_{\mathrm{P}}}=\mathrm{R}\left[\frac{1}{{\left(3\right)}^2}-\frac{1}{{\left(\infty \right)}^2}\right]=\mathrm{R}\left[\frac{1}{9}-0\right]$

$\frac{1}{{\mathrm{\lambda }}_{\mathrm{P}}}=\mathrm{R}\left[\frac{1}{9}\right] \mathrm{or} {\mathrm{\lambda }}_{\mathrm{P}}=\frac{9}{\mathrm{R}}$

The ratio of wavelengths values of limiting lines for Balmer and Paschen series is given in the below.

$\frac{{\mathrm{\lambda }}_{\mathrm{B}}}{{\mathrm{\lambda }}_{\mathrm{P}}}=\frac{\left({\displaystyle \frac{4}{\mathrm{R}}}\right)}{\left({\displaystyle \frac{9}{\mathrm{R}}}\right)}\Rightarrow \frac{{\mathrm{\lambda }}_{\mathrm{B}}}{{\mathrm{\lambda }}_{\mathrm{P}}}=\frac{4}{9}$