Question 71: Atoms & Nuclei

Question

The ratio of momentum of the photons of the $1^{	ext {st }}$ and $2^{	ext {nd }}$ line of Balmer series of Hydrogen atoms is $\alpha / \beta$. The possible values of $\alpha$ and $\beta$ are:-

Accepted Answer

According to the de Broglie relation, the momentum (p) of a photon is inversely proportional to its wavelength ( $\lambda$ ):
$$ \mathrm{p}=\frac{\mathrm{h}}{\lambda} $$
Therefore, the ratio of momenta is the inverse ratio of their wavelengths:
$$ \frac{\mathrm{p}_1}{\mathrm{p}_2}=\frac{\lambda_2}{\lambda_1} $$
The Balmer series occurs when an electron transitions from a higher energy level $\left(\mathrm{n}_2ight)$ to the second energy level ( $\mathrm{n}_1=2$ ). The wavelength is given by Rydberg formula:
$$ \frac{1}{\lambda}=\mathrm{R}\left(\frac{1}{2^2}-\frac{1}{\mathrm{n}^2}ight) $$
For $1^{	ext {st }}$ line of Balmer transition from $\mathrm{n}=3$ to $\mathrm{n}=2$.
$$ \frac{1}{\lambda_1}=\mathrm{R}\left(\frac{1}{4}-\frac{1}{9}ight)=\mathrm{R}\left(\frac{9-4}{36}ight)=\frac{5 \mathrm{R}}{36} \Rightarrow \lambda_1=\frac{36}{5 \mathrm{R}} $$
For $2^{\mathrm{nd}}$ line of Balmer transition from $\mathrm{n}=4$ to $\mathrm{n}=2$.
$$ \frac{1}{\lambda_2}=\mathrm{R}\left(\frac{1}{4}-\frac{1}{16}ight)=\mathrm{R}\left(\frac{4-1}{16}ight)=\frac{3 \mathrm{R}}{16} \Rightarrow \lambda_2=\frac{16}{3 \mathrm{R}} $$
So, the ratio $(\alpha / \beta)$,
$$ \frac{\mathrm{p}_1}{\mathrm{p}_2}=\frac{\lambda_2}{\lambda_1}=\frac{16 / 3 \mathrm{R}}{36 / 5 \mathrm{R}}=\frac{16}{3} 	imes \frac{5}{36} $$
$$\Rightarrow $$ $$ \frac{\mathrm{p}_1}{\mathrm{p}_2}=\frac{4 	imes 5}{3 	imes 9}=\frac{20}{27}=\frac{\alpha}{\beta} $$
So, $\alpha=20$ and $\beta=27$.
Therefore, the correct option is (D).

Answer

27 and 20

Answer

3 and 16

Answer

5 and 36

Answer

20 and 27

Atoms & Nuclei

The ratio of momentum of the photons of the 1^{\text {st }} and 2^{\text {nd }} line of Balmer series of Hydrogen atoms is \alpha / \beta. The possible values of \alpha and \beta are:-

Choose the correct answer: