JEE Main 2025 — Structure Of Atom Question with Solution

Bohr radius of hychogon atom $\to a_0$

According to Bohr, the equation used to calculate the angular momentum of an election in a hydrogen atom is

$\mathrm{mvr}=\frac{\mathrm{nh}}{2\pi }$ ..... (1)

$m\to$ mass of electron

$v\to$ velocity of electron

$r\to$ radius of the orbit

$n\to$ orbit. number. in. which electron is present.

Given that; election is present in second orbit, $n=2$

The radius of the second orbit $r_2=a_0\times 2^2=4a_0$

General formula for radius of $n^{th}$ orbit,

$r_n=a_0\times n^2$

From (1)

$$\begin{array}{rl}& mvr=n\frac{h}{2\pi }\\ & 2\pi r=n\frac{h}{mv}\end{array}$$

$\frac{h}{mv}=\lambda$ (de Broglie relation ship, $\lambda \to$ de Broglie Wavelength

So, $2\pi r=n\lambda$

For the electron in the second orbit, $2\pi r_2=n\lambda$

Substitute for $r_2$,

$$\begin{array}{rl}& 2\pi \times 4a_0=n\lambda \\ & 8\pi a_0=n\lambda \\ & \therefore \lambda =\frac{8\pi a_0}{n}\end{array}$$

Correct answer: Option 1) $\frac{8\pi a_0}{n}$