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 $'x'$ $\times 10^{-1}$. The value of $x$, to th e nearest integer, is _____

Answer 1

For $1^{st}$ line
$\frac{1}{\lambda_{1}}= R z ^{2}\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)$
$\frac{1}{\lambda_{1}}= Rz ^{2} \frac{5}{36}\,\,.......(i)$
For $3^{rd}$ line
$\frac{1}{\lambda_{3}}= Rz ^{2}\left(\frac{1}{2^{2}}-\frac{1}{5^{2}}\right)$
$\frac{1}{\lambda_{3}}= Rz ^{2} \frac{21}{100} \quad \,\,....(ii)$
(ii) $+( i )$
$\frac{\lambda_{1}}{\lambda_{3}}=\frac{21}{100} \times \frac{36}{5}=1.512=15.12 \times 10^{-1}$
$x \approx 15$