Question

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

Answer 1

Since H atom is a bounded system U cannot be positive

U = - $k{e}^2/3r^3$

Electrostatic force of attraction at a distance r will be

F = -dU/dr

$=-k{e}^2/r^4$

Assuming a circular radius of the orbit

$mv2/r=k{e}^2/r^4$

$=m{v}^2=k{e}^2/r^3$

$=1/2m{v}^2=k{e}^2/2r^3$

$=K.E=k{e}^2/2r^3$

E = K.E + U

$=k{e}^2/2r^3+(-k{e}^2/3r^3)$

$=k{e}^2/6r^3$

Gives the energy level of the atom