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What are the heights of the electron orbits of an atom? (How far apart are the energy levels of the electron relative to the center of the atomic nucleus?) How fast do electrons move in their orbits?

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I'm afraid that the whole question is predicated on a misunderstanding (albeit one that many popular sources rather encourage). Electrons don't "orbit" in the sense that planets orbits the sun; they occupy "orbitals" which is a very different, and fundamentally quantum mechanical thing. Your questions do have reasonable analogues in this view ("What is the RMS radial position?" is a reasonable way to answer the height question, and so on), but the answer won't mean quite what you expect them to mean. It might help if you told us what your level of preparation is. – dmckee Feb 14 '13 at 17:18
The second part is fully answered in How fast do electrons travel in an atomic orbital? – dmckee Feb 14 '13 at 17:20
Two related questions that don't answer the height query but might help understand the distinction I made in my first comment: Why do electrons occupy the space around nuclei, and not collide with them? and Why don't electrons crash into the nuclei they “orbit”?. – dmckee Feb 14 '13 at 18:20
Is it confirmed by experiment or just a calculated theoretically? Can you describe this experiment? (about orbitals) – Robotex Feb 15 '13 at 12:48
Have a look at this link for orbitals . It was the data that drove physics theories to Quantum Mechanics rather than primitive models like the Bohr model. – anna v Mar 7 '14 at 4:39

In the (incorrect) Bohr model of the hydrogen atom, for the ground state (n=1), the distance of the electron from the nucleus is 52.9 picometers (pm). The constant $a_0$ is defined to be this distance. The distance of the electron in the nth energy level is $a_0n^2$.

According to the exact solution of the Schrodinger equation for the hydrogen atom, in the ground state:

  1. $a_0$ (52.9pm) is the most probable distance of the electron from the nucleus.

  2. There is a 0.323 probability that the electron is closer than $a_0$, 0.677 probabilty that it is further away.

  3. The average distance of the electron from the nucleus is $\frac{3}{2}a_0$

  4. The probability that the electron is between r and r + dr from the nucleus is:

$$ \frac{4}{a_0}r^2e^{-\frac{2r}{a_0}}dr$$

Concerning velocity, the Heisenberg uncertain principle implies that given the knowledge that the electron is confined in its position to the degree explained above, the uncertainty in speed is on the order of $10^7 m/s$.

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