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In the Bohr model of an atom, how does the mass of the nucleus affect how much the nucleus attracts its electrons?

For instance, an isotope with a heavier nucleus would attract its electrons better than a lighter isotope. That is, the energy of the electrons are more negative in the heavier isotope.

Edit: Sorry I wasn't clear. I meant to ask whether the gravitational force experienced by the nucleus towards the electron would alter the motion of the nucleus in such a way that the electrostatic attraction of the electron with the nucleus is amplified or hindered.

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Consider a proton and an electron separated by a distance $r \approx 10^{-10} m$. The electrical force between them is $-\frac{e^2}{4 \pi \epsilon_0 r^2}$ while the gravitational force is $-\frac{G m_p m_e}{r^2}$. The ratio of the two forces is $$\frac{F_g}{E_e} = \frac{4 \pi \epsilon_0 G m_p m_e}{e^2},$$ which is independent of their separation. Plugging in the numbers gives $$\frac{F_g}{F_e} \approx 10^{-41}.$$ This shows that gravitational forces are much weaker than electrical forces and so can be completely ignored.

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  • $\begingroup$ What influence would the gravitational force have, were it to be relevant? $\endgroup$ – Michael Faraday Apr 22 '20 at 12:28
  • $\begingroup$ @MichaelFaraday in atomic physics, gravity is completely negligible and I don't think has any influence at all. In, for example, neutron stars (which some people think of as a giant nucleus) gravity does have an important role to play. Perhaps that could be a question for you to pose/research? $\endgroup$ – jim Apr 22 '20 at 14:53
  • $\begingroup$ I'm sorry I meant to ask whether the gravity will change the motion of the nucleus in such a way that the electrostatic attraction between the electron and nucleus is either amplified or hindered. $\endgroup$ – Michael Faraday Apr 22 '20 at 16:36
  • $\begingroup$ @MichaelFaraday To the best iof my knowledge, no. $\endgroup$ – jim Apr 22 '20 at 20:36
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As the attraction between the nucleus and the electrons is electromagnetic,(see the numbers given by the answer of Michael Faraday) only charges and their interactions with magnetic fields can play a role in what you call "attraction". There is no mass involved.

Mass can play a role as the number of neutrons increases or decreases the mass from the average, isotopes will have a fine structure due to the different charge distributions according to the isotope, and also the different kinematics ( in Bohr model terms) .

Here is a review of how the observations are fitted using quantum mechanical models, not Bohr model.

Measurements of atomic spectra for different isotopes of the same atomic number Z show slight differences—the isotope shifts. This frequency difference of an electronic transition is usually described separately as due to the finite mass of the nucleus—the mass shift—and to the size of the nuclear charge distribution—the volume of field shift. The mass shift is dominating isotope shifts for light atoms, whereas the field shift scales as Z 5 or even up to Z 6 and thus becomes the leading contribution in the case of heavy ones.

Whether the effects can be called "attraction" is doubtful. The review is enlightening.

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