Suppose an electron is attracted towards the origin by a force $k / r$ where ' $k$ ' is a constant and ' $r$…
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 the $n^{\text {th }}$ orbital of the electron is found to be ' $r_n$ ' and the kinetic energy of the electron to be $T_n$. Then which of the following is true?
$T_n \propto 1 / n^2, \quad r_n \propto n^2$
$T_n$ independent of $n, r_n \propto n$
$T_n \propto 1 / n, \quad r_n \propto n$
$T_n \propto 1 / n, \quad r_n \propto n^2$
Solution
$
\begin{aligned}
& \frac{\mathrm{k}}{\mathrm{r}}=\frac{m v^2}{\mathrm{r}} \\
& \mathrm{mv}=\mathrm{k} \quad \text { (independent or } r \text { ) } \\
& \mathrm{n}\left(\frac{\mathrm{h}}{2 \pi}\right)=m v r \Rightarrow r \propto \mathrm{n} \text { and } \mathrm{T}=\frac{1}{2} m v^2 \text { is independent of } \mathrm{n} .
\end{aligned}
$