Question 105: Atoms & Nuclei

Question

The radius of the orbit of an electron in a hydrogen-like atom is 4.5a0, where a0 is the Bohr radius. Its orbital angular momentum is $${{3h} \over {2\pi }}$$. 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

The orbital angular momentum is
$$L = {{nh} \over {2\pi }}$$
$${{3h} \over {2\pi }} = {{nh} \over {2\pi }} \Rightarrow n = 3$$
The radius of the orbit is
$$4.5({a_0}) = {a_0}\left( {{{{n^2}} \over Z}} ight)$$
$$Z = {{{n^2}} \over {4.5}} = {{{3^2}} \over {4.5}} = {9 \over {4.5}} = 2$$
Thus, the possible transitions are 3$$	o$$2, 3$$	o$$1 and 2$$	o$$1. For the transition 3$$	o$$2, the wavelength is
$${1 \over \lambda } = R{(2)^2}\left[ {{1 \over 4} - {1 \over 9}} ight] = 4R\left[ {{{9 - 4} \over {36}}} ight] = {{5R} \over 9}$$
$$ \Rightarrow \lambda  = {9 \over {5R}}$$
For the transition 3$$	o$$1, the wavelength is
$${1 \over \lambda } = R{(2)^2}\left[ {1 - {1 \over 9}} ight] = 4R\left( {{8 \over 9}} ight) = {{32R} \over 9}$$
$$ \Rightarrow \lambda  = {9 \over {32R}}$$
For the transition 2$$	o$$1, the wavelength is
$${1 \over \lambda } = R{(2)^2}\left[ {1 - {1 \over 4}} ight] = 4R\left( {{3 \over 4}} ight) = 3R$$
$$ \Rightarrow \lambda  = {1 \over {3R}}$$

Answer

$${9 \over {32R}}$$

Answer

$${9 \over {16R}}$$

Answer

$${9 \over {5R}}$$

Answer

$${4 \over {3R}}$$

Atoms & Nuclei

The radius of the orbit of an electron in a hydrogen-like atom is 4.5a0, where a0 is the Bohr radius. Its orbital angular momentum is ${{3h} \over {2\pi }}$. 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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