Which quantum numbers are required to determine the energy of an electron in a many-electron atom?

• A. Only the principal quantum number (n)
• B. The principal quantum number (n) and the spin quantum number (m_s)
• C. Only the magnetic quantum number (m_l)
• D. Both the principal quantum number (n) and the azimuthal quantum number (l)

Answer: (D)

Verified step by step guidance

1. Understand that in a many-electron atom, the energy of an electron depends on more than just the principal quantum number \(n\), unlike in a hydrogen atom where energy depends only on \(n\).
2. Recall the four quantum numbers: principal quantum number (\(n\)), azimuthal quantum number (\(l\)), magnetic quantum number (\(m_l\)), and spin quantum number (\(m_s\)). Each describes different properties of the electron's state.
3. Recognize that the principal quantum number \(n\) determines the main energy level or shell of the electron, which is a primary factor in energy.
4. Know that the azimuthal quantum number \(l\) defines the subshell or shape of the orbital, which affects electron-electron interactions and thus influences the energy in many-electron atoms.
5. Conclude that both \(n\) and \(l\) are required to determine the energy of an electron in many-electron atoms, while \(m_l\) and \(m_s\) do not significantly affect the energy level.