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,
$\frac{m{v}^2}{r_n}=\frac{k}{r_n}...\left(i\right)$
and $mv{r}_n=\frac{nh}{2\pi }$ or $v=\frac{nh}{2\pi m{r}_n}$
Put in $\left(i\right),\frac{m}{r_n}\times \frac{n^2{h}^2}{4{\pi }^2{m}^2{r}_n^2}=\frac{k}{r_n}$
$\Rightarrow r_n^2=\frac{n^2{h}^2}{4{\pi }^{2}mk}...\left(ii\right)$
$\therefore r_n^2\propto n^2$ or $r_n\propto n$
$K.E$. of the electron,
$T_n=\frac{1}{2}m{v}^2$
$=\frac{1}{2}m\frac{n^2{h}^2}{4{\pi }^2{m}^2{r}_n^2}$
$=\frac{n^2{h}^2}{8{\pi }^{2}m{r}_n^2}$
Using $\left(ii\right)$, we get,
$T_n=\frac{n^2{h}^{2}4{\pi }^{2}mk}{8{\pi }^{2}m{n}^2{h}^2}=\frac{k}{2}$
$\therefore T_n$ is independent of $n$