Question

Suppose the potential energy between electron and proton at a distance $r$ is given by $-\frac{K e^{2}}{2 r^{3}}$. Using Bohr's theory choose the correct statements :

• A. Energy in the nth orbit is proportional to $n^{3}$
• B. Energy in the nth orbit is proportional to $n^{6}$
• C. Energy is proportional to $m^{2}(m$ : mass of electron $)$
• D. Energy is proportional to $m^{-3}(m:$ mass of electron)

Answer 1

Answer: (B), (D)

We use the given potential energy for calculation of force electron as
$F=\frac{d U}{d r}=\frac{3}{2} \frac{K e^{2}}{r^{4}}$
$\Rightarrow K E \frac{1}{2} m v^{2}=\frac{3}{4} \frac{K e^{2}}{r^{3}}$
Using Bohr's II Postulate $m v r=\frac{n h}{2 \pi}$ we get
$m\left(\frac{n h}{2 \pi m r}\right)^{2}=\frac{3}{2} \frac{K e^{2}}{2 \pi}$
$\Rightarrow r \propto \frac{1}{n^{2}}$
and $r \propto m$
hence option (B) and (D) are correct as energy is inverssely proportional to $r^{3}$.