Question

Suppose an electron is attracted towards the originby a force $\frac{k}{r}$ 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 the $n^{th}$ orbital of the electrons is found to be '$r_n$' and the kinetic energy of the electron to be '$T_n$'. Then which of the following is true?

• A. $T_n\propto \frac{1}{n^2},r_n\propto n^2$
• B. $T_n$ independent of n, $r_n\propto n$
• C. $T_n\propto \frac{1}{n},r_n\propto n$
• D. $T_n\propto \frac{1}{n},r_n\propto n^2$

Answer 1

The correct option is B $T_n$ independent of n, $r_n\propto n$

$\frac{k}{r}=\frac{m{v}^2}{r}$
$\Rightarrow m{v}^2=k$ (Independent of n)
Hence, $T=\frac{1}{2}m{v}^2$ is also independent of n.
Now, $mvr=\frac{nh}{2\pi }$
So, $r\propto n$