Suppose the potential energy between an electron and a proton at a distance $r$ is given by $- k e^{2} / 3 r^{3}$ . Use Bohr's theory to obtain energy level of such a hypothetical atom.

Open in App

Solution

Given, potential energy, $U = - \frac{k e^{2}}{3 r^{3}}$
The electrostatic force between electron and proton at a distance r is given by, $F = \frac{d U}{d r} = \frac{k e^{2}}{r^{4}}$
According to Bohr's first postulate, $\frac{m v^{2}}{r} = F = \frac{k e^{2}}{r^{4}} . . . (1)$
According to Bohr's second postulate, $m v r = \frac{n h}{2 π} . . . . . . . . (2)$
From (1) and (2), $r = \frac{4 π^{2} e^{2} k m}{n^{2} h^{2}}$
Total energy of the electron, $E = K + U = \frac{1}{2} m v^{2} - \frac{k e^{2}}{3 r^{3}} = \frac{k e^{2}}{2 r^{3}} - \frac{k e^{2}}{3 r^{3}} = \frac{k e^{2}}{6 r^{3}}$ using (1)
Substituting the value of $r$ , we get $E = \frac{n^{6} h^{6}}{6 (2 π)^{6} m^{3} k^{2} e^{4}}$

Suggest Corrections

Similar questions

\frac{k e^{2}}{3 r^{3}}

r

\frac{k e^{2}}{3 r^{3}},

k

U = K e^{2} / 3 r^{3}

r

\frac{- K e^{2}}{3 r^{3}}

r

U = - k e

k

n^{t h}

Join BYJU'S Learning Program