Question

The first three spectral lines of $H$ -atom in the Balmer series are given ${\lambda }_1,{\lambda }_2,{\lambda }_3$ considering the Bohr atomic model, the wave lengths of first and third spectral lines $\left(\frac{{\lambda }_1}{{\lambda }_3}\right)$ are related by a factor of approximately ${}^{\prime }{x}^{\prime }$ $\times 10^{-1}$. The value of $x$, to th e nearest integer, is _____

Answer 1

Answer: 15

For $1^{st}$ line
$\frac{1}{{\lambda }_1}=R{z}^2\left(\frac{1}{2^2}-\frac{1}{3^2}\right)$
$\frac{1}{{\lambda }_1}=R{z}^2\frac{5}{36}.......(i)$
For $3^{rd}$ line
$\frac{1}{{\lambda }_3}=R{z}^2\left(\frac{1}{2^2}-\frac{1}{5^2}\right)$
$\frac{1}{{\lambda }_3}=R{z}^2\frac{21}{100}....(ii)$
(ii) $+(i)$
$\frac{{\lambda }_1}{{\lambda }_3}=\frac{21}{100}\times \frac{36}{5}=1.512=15.12\times 10^{-1}$

$x\approx 15$