If is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength of the electron present in the second orbit of hydrogen atom? [ : any integer] [2025]

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Solution

Q.
If $a_{0}$ is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength $(λ)$ of the electron present in the second orbit of hydrogen atom [ $n$ : any integer] [2025]

1 $\frac{2 a_{0}}{n π}$

2 $\frac{4 π a_{0}}{n}$

3 $\frac{8 π a_{0}}{n}$

4 $\frac{4 n}{π a_{0}}$

Ans.
(3)

By de Broglie's equation:

$λ = \frac{h}{m v}$

$\Rightarrow m v = \frac{h}{λ} -I$

By quantization of angular momentum:

$m v r = n \frac{h}{2 π} -II$

$II / I$ gives:

$\frac{m v r}{m v} = \frac{n λ}{2 π}$

$2 π r = n λ$

$2 π \times a_{0} \frac{n^{2}}{Z} = n λ$

$2 π \times a_{0} \frac{n}{Z} = λ$

For second Bohr orbit, $n$ = 2 and for H atom Z = 1

$2 π \times a_{0} \frac{2}{1} = λ$

$λ = \frac{4 π a_{0}}{1}$

Putting $n$ = 2 in option (3) gives the correct answer.

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