On which quantum numbers does the energy of an electron depend in each of the following? a. a one-electron atom or ion b. an atom or ion with more than one electron

For a one-electron atom or ion, the energy of the electron only depends on the principal quantum number (n). The principal quantum number essentially determines the energy level the electron occupies in the atom. It is represented by a whole positive number (n=1, 2, 3, ...). The energy of the electron increases as n increases, meaning electrons farther from the nucleus have higher energy. In a one-electron atom or ion, angular, magnetic, and electron spin quantum numbers do not affect the energy level.

In atoms or ions with more than one electron, the energy of an electron depends on both the principal quantum number (n) and the azimuthal quantum number (l). While the principal quantum number determines the energy level, the azimuthal quantum number (also known as the angular momentum or orbital quantum number) determines the shape of the electron cloud. l can have values from 0 to (n-1). The value of l affects the energy of the electron due to the fact that electrons experience both the electrostatic attraction from the nucleus and repulsive forces from other electrons occupying the atom or ion. In a multi-electron atom or ion, the energy increases with increasing n and l values, so the electron's energy is dependent on both of these quantum numbers. It is important to note that magnetic (m_l) and spin (m_s) quantum numbers still do not impact the energy of the electron in an atom with more than one electron.