An electron in the ground state of the hydrogen atom has the orbital radius of while that for the electron in third excited state is . The ratio of the de-Broglie wavelengths of electron in the ground state to that in excited state is [2025]

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Solution

Q.
An electron in the ground state of the hydrogen atom has the orbital radius of $5.3 \times 10^{- 11} m$ while that for the electron in third excited state is $8.48 \times 10^{- 10} m$ . The ratio of the de-Broglie wavelengths of electron in the ground state to that in excited state is [2025]

1 4

2 9

3 3

4 16

Ans.
(1)

According to de-Broglie hypothesis, $λ = \frac{h}{m v}$

$m v r = \frac{n h}{2 π} \Rightarrow m v = \frac{n h}{2 π r}$

$λ = \frac{2 π r h}{n h} \Rightarrow λ \propto \frac{r}{n}$

$\frac{λ_{1}}{λ_{4}} = \frac{r_{1} n_{4}}{n_{1} r_{4}} = \frac{5.3 \times 10^{- 11} \times 4}{1 \times 84.8 \times 10^{- 11}}$

$\Rightarrow \frac{λ_{1}}{λ_{4}} = \frac{1}{4}$

Note: Most appropriate answer will be option (1).

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