Are electrons influenced by the gravitational attraction of the particles in the nucleus of an atom?

This is a question asked pretty often, especially when we
consider the fact that the distance between the nucleus and the electrons is very
small.

Now, let us take a simple case of a hydrogen atom,
which has one proton in the nucleus and one electron. The electron is negatively charged
with a charge equal to 1.6 * 10^-19 C. The proton is positively charged with a charge of
1.6* 10^-19 C.

The mass of an electron on the other hand is
approximately 9.11*10^-31 kg and that of a proton is 1.673*10^-27
kg.

If the two particles are separated by a distance r, the
force of attraction due to the electrical charges is k*Cp*Ce/ r^2, where k is a constant
equal to 9.0*10^9 N*m^2/C^2

The gravitational force of
attraction is G*Me*Mp/r^2, where G is the gravitational constant equal to
6.673*10^-11.

If we find the ratio of the electrostatic
force to the gravitational force between the particles it is equal
to:

[(1.6*10^-19) ^2* 9.0*10^9] /
[9.11*10^-31*1.673*10^-27*6.673*10^-11]

= 2.26*
10^39

So the electrostatic force is 2.26 * 10^39 times
larger than the gravitational force of
attraction.

Therefore it is evident that the
gravitational force is negligible compared to the electrostatic
force.