Question

Suppose an electron is attracted towards the origin by a force $k/r$, where $k$ is a constant and $r$ is the distance of the electron from the origin. By applying Bohr model to this system, the radius of $n^{th}$ orbit of the electron is found to be $r_n$ and the kinetic energy of the electron is found to be $T_n$. Then which of the following is true?

• A. $T_{n}\propto\frac{1}{n^{2}}$
• B. $T_{n}$ is independent of $n ; r_n \propto n$
• C. $T_{n}\propto\frac{1}{n}$ and $r_{n}$
• D. $T_{n}\propto\frac{1}{n}$ and $r_{n}\propto n^{2}$

Answer 1

Answer: (B)

Applying Bohr model to the given system,
$\frac{mv^{2}}{r_{n}} = \frac{k}{r_{n}}\quad...\left(i\right) $
and $mvr_{n} = \frac{nh}{2 \pi}$ or $v = \frac{nh}{2\pi mr_{n}}$
Put in $\left(i\right), \frac{m}{r_{n}}\times \frac{n^{2}h^{2}}{4\pi^{2}m^{2}r_{n}^{2}} = \frac{k}{r_{n} } $
$\Rightarrow r_{n}^{2} = \frac{n^{2}h^{2}}{4\pi^{2}mk}\quad...\left(ii\right) $
$ \therefore r_{n}^{2} \propto n^{2}$ or $r_{n} \propto n $
$ K.E$. of the electron,
$ T_{n} = \frac{1}{2}mv^{2} $
$= \frac{1}{2}m\frac{ n^{2}h^{2}}{4\pi^{2}m^{2} r_{n}^{2}}$
$ =\frac{n^{2}h^{2}}{8\pi^{2} m r_{n}^{2}} $
Using $\left(ii\right)$, we get,
$T_{n} = \frac{n^{2}h^{2}4\pi^{2}mk}{8\pi^{2}mn^{2}h^{2}} = \frac{k}{2}$
$ \therefore T_{n}$ is independent of $n$