Question

The ratio of the wave number corresponding to the first line of Lyman series of $H$-atom and third line of Paschen series of a hydrogen-like sample is $9: 16$. Then find the third excitation potential in terms of volt for this $H$-like samples.

Answer 1

Wave number
$N=\frac{1}{\lambda}=R z^{2}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)$
$\frac{N_{1}}{N_{2}}=\frac{z_{1}^{2}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)}{z_{2}^{2}\left(\frac{1}{n_{1}^{' 2}}-\frac{1}{n_{2}^{' 2}}\right)}$
$=\frac{(1-1 / 4)}{z^{2}\left(\frac{1}{9}-\frac{1}{36}\right)}$
$\Rightarrow \frac{N_{1}}{N_{2}}=\frac{9}{z^{2}}=\frac{9}{16}$
$\Rightarrow z=4$
Hence third excitation potential $=12.75 \times 16=204\, V$.