Question

Which one of the relations is correct between time period and number of orbits while an electron is revolving in an orbit?

• A. $ {{n}^{2}} $
• B. $ \frac{1}{{{n}^{2}}} $
• C. $ {{n}^{3}} $
• D. $ \frac{1}{n} $

Answer 1

Answer: (C)

Time required for electron to go around once is
$T=\frac{2 \pi r_{n}}{v_{n}} ?$ (i)
where $r_{n}$ is radius of $n^{th}$ orbit and $v_{n}$ is velocity.
Multiply and divide Eq. (i) by $mv ^2_n$, we get
$T=\frac{2 \pi m r_{n} v_{n}}{m v_{n}^{2}}=\pi \frac{n h}{\frac{1}{2} m v_{n}^{2}}=\pi \frac{n h}{K_{n}}$
where $K_{n}$ is kinetic energy in $n^{th}$ orbit,
$T=-\pi \frac{n h}{E}$
$T=\frac{h}{2} n \frac{n^{2}}{13.6 \,eV }\left(\because h=\frac{h}{2 \pi}\right.$ and $\left.E=-\frac{13.6}{n^{2}}\right) $
$\Rightarrow T \propto n^{3} .$