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 ratio of wavelengths 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 the nearest integer, is .

Answer 1

For $1^{st}$ line
$\frac{1}{{\lambda }_1}=R{Z}^2(\frac{1}{2^2}-\frac{1}{3^2})$
$\frac{1}{{\lambda }_1}=R{Z}^2\frac{5}{36}...\left(i\right)$
For $3^{rd}$ line
$\frac{1}{{\lambda }_3}=R{Z}^2(\frac{1}{2^2}-\frac{1}{5^2})$
$\frac{1}{{\lambda }_3}=R{Z}^2\frac{21}{100}...\left(ii\right)$
Dividing the equations $\left(ii\right)\&\left(i\right)$
$\frac{{\lambda }_1}{{\lambda }_3}=\frac{21}{100}\times \frac{36}{5}=1.512=15.12\times {10}^{-1}$
$x\approx 15$
$\therefore 15$ is the required value.