Assuming that in an atom, neutron $(m_{n})$ and electron $(m_{e})$ are bound to each other by gravitational force only and electron are moving around neutron and assuming Bohr's quantization rule to be valid.The expression for energy of electron in $n^{t h}$ level comes out to be $\frac{(a - 4) G^{2} m_{n}^{2} m_{e}^{2}}{4 n^{2} h^{2}} V$ (where $h = \frac{h}{2 π})$ . Find the value of ' $a$ '.

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Solution

Quantization of angular momentum of electron gives:
$m_{e} v r = n h$

Balancing forces we get:
$\frac{m_{e} V^{2}}{r} = \frac{G m_{e} m_{n}}{r^{2}}$

From conservation of energy:
$- E = K E + P E = \frac{1}{2} m_{e} V^{2} - \frac{G m_{e} m_{n}}{r}$

$E = - \frac{G^{2} m_{n}^{2} m_{e}^{2}}{2 n^{2} h}$

On comparing we get, $a = 2$

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Quantization of angular momentum of electron gives:mevr=nhBalancing forces we get:meV2r=Gmemnr2From conservation of energy:−E=KE+PE=12meV2−GmemnrE=−G2m2nm2e2n2hOn comparing we get, a=2