The first three spectral lines of H-atom in the Balmer series are given

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 the nearest integer, is_____________

Solution:

For $1^{\text {st }}$ line

$\frac{1}{\lambda_{1}}=R z^{2}\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)$

$\frac{1}{\lambda_{1}}=\mathrm{Rz}^{2} \frac{5}{36}$.........(1)

For $3^{\text {rd }}$ line

$\frac{1}{\lambda_{3}}=\mathrm{Rz}^{2}\left(\frac{1}{2^{2}}-\frac{1}{5^{2}}\right)$

$\frac{1}{\lambda_{3}}=\mathrm{Rz}^{2} \frac{21}{100}$          ......(2)

(ii) $+($ i $)$

$\frac{\lambda_{1}}{\lambda_{3}}=\frac{21}{100} \times \frac{36}{5}=1.512=15.12 \times 10^{-1}$

$x \approx 15$

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