Question 614: Atoms & Nuclei

Question

The radius of the orbit of an electron in a Hydrogen-like atom is $45 a_0$, where $a_0$ is the Bohr radius. Its orbital angular momentum is $\frac{3 h}{2 \pi}$. It is given that $h$ is Planck constant and $R$ is Rydberg constant. The possible wavelength(s), when the atom de-excites, is (are)

Accepted Answer

$$ 	ext {  } L=3\left(\frac{h}{2 \pi}ight)$$
$$ 	herefore n=3 	ext {, as } L=n\left(\frac{h}{2 \pi}ight)$$
$$ \begin{array}{ll} \Rightarrow & r_n \propto \frac{n^2}{z} \ & r_3=4.5 a_0 \ \Rightarrow & z=2 \ & \frac{1}{\lambda_1}=R z^2\left(\frac{1}{2^2}-\frac{1}{3^2}ight)=4 R\left(\frac{1}{4}-\frac{1}{9}ight) \ 	herefore & \lambda_1=\frac{9}{5 R} \ & \frac{1}{\lambda_2}=R z^2\left(\frac{1}{1^2}-\frac{1}{3^2}ight)=4 R\left(1-\frac{1}{9}ight) \end{array} $$
$$ \begin{aligned} & \Rightarrow \quad \lambda_2=\frac{9}{32 R} \ &\,\,\,\,\,\,\,\,\,\,\,\, \frac{1}{\lambda_3}=R z^2\left(\frac{1}{1^2}-\frac{1}{2^2}ight)=4 R\left(1-\frac{1}{4}ight) \ & \Rightarrow \lambda_3=\frac{1}{3 R} \end{aligned} $$

Answer

$\frac{9}{32 R}$

Answer

$\frac{9}{16 R}$

Answer

$\frac{9}{15 R}$

Answer

$\frac{4}{3 R}$

Atoms & Nuclei

The radius of the orbit of an electron in a Hydrogen-like atom is 45 a_0, where a_0 is the Bohr radius. Its orbital angular momentum is \frac{3 h}{2 \pi}. It is given that h is Planck constant and R is Rydberg constant. The possible wavelength(s), when the atom de-excites, is (are)

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