Using Bohr's model, the orbital period of electron in hydrogen atom in $\mathrm{n}^{\text {th }}$ orbit is (…

Using Bohr's model, the orbital period of electron in hydrogen atom in $\mathrm{n}^{\text {th }}$ orbit is ( $\mathrm{m}=$ mass of electron, $\mathrm{h}=$ Planck's constant, $\mathrm{e}=$ electronic charge, $\varepsilon_0=$ permittivity of free space)
  1. $\frac{2 \varepsilon_0^2 \mathrm{n}^2 \mathrm{~h}^2}{\mathrm{me}^4}$
  2. $\frac{4 \varepsilon_0^2 \mathrm{n}^2 \mathrm{~h}^2}{\mathrm{me}^2}$
  3. $\frac{4 \varepsilon_0^2 \mathrm{n}^3 \mathrm{~h}^3}{m \mathrm{e}^4}$
  4. $\frac{4 \varepsilon_0 \mathrm{n}^2 \mathrm{~h}^2}{\pi \mathrm{me}^2}$

Solution

$\mathrm{r}=$ radius of $\mathrm{n}^{\text {th }}$ orbit $=\frac{\mathrm{n}^2 \mathrm{~h}^2}{\pi \mathrm{mZe}^2}$ $\mathrm{v}=$ speed of $\mathrm{e}^{-}$in $\mathrm{n}^{\text {th }}$ orbit $=\frac{\mathrm{Ze}^2}{2 \varepsilon_0 n h}$ $\mathrm{T}=\frac{2 \pi \mathrm{r}}{\mathrm{v}}$
Substituting values of v and r , $\therefore \quad \mathrm{T}=\frac{4 \varepsilon_0^2 \mathrm{n}^3 \mathrm{~h}^3}{m \mathrm{Z}^2 \mathrm{e}^4}$
For hydrogen, $\mathrm{Z}=1$ $\therefore \quad T=\frac{4 \varepsilon_0^2 \mathrm{n}^3 \mathrm{~h}^3}{\mathrm{me}^4}$

Asked in: MHT CET 2024 (10 May Shift 1)

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