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 the $n$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?

• A. $T_n\propto 1/n^2,r_n\propto n^2$
• B. $n,r_n\propto n$
• C. $T_n\propto 1/n,r_n\propto n$
• D. $T_n\propto 1/n,r_n\propto n^2$

Answer 1

Answer: (B)

$n,r_n\propto n$

$F=\frac{F}{r}$-(1)

according to bohr model

$\mathrm{mvr}=\frac{nh}{2\pi }$-(2)

Frim (1) $F=\frac{m{v}^2}{r}=\frac{k}{r}$

$\Rightarrow \begin{array}{c}k=m{v}^2-3\\ v=\sqrt{\frac{k}{m}}\end{array}$

from

(11) and (3)

$m\sqrt{\frac{k}{m}}r=\frac{nh}{2x}$

$\sqrt{mk}r=\frac{nh}{2x}$

$r=\frac{nh}{2\pi \sqrt{mk}}\Rightarrow r\propto n$

$kE=\frac{1}{2}m{v}^2=T_n$

From $\left(3\right)$

$\frac{1}{2}m\frac{k}{m}=T_n$

$\begin{array}{cc}\frac{k}{2}=T_n& \\ & =\frac{T_n\text{constant}}{}\\ \text{option}B& \end{array}$