Which two of the four quantum numbers determine the energy level of an orbital in a multielectron atom?

Verified step by step guidance

1. Identify the four quantum numbers: principal quantum number \( n \), azimuthal (angular momentum) quantum number \( l \), magnetic quantum number \( m_l \), and spin quantum number \( m_s \).
2. Understand that the principal quantum number \( n \) primarily determines the energy level of an electron in an atom, as it indicates the shell or energy level in which the electron resides.
3. Recognize that the azimuthal quantum number \( l \) also influences the energy level in multielectron atoms by defining the shape of the orbital and contributing to the energy through electron-electron interactions.
4. Note that the magnetic quantum number \( m_l \) and the spin quantum number \( m_s \) do not directly affect the energy level of the orbital in a multielectron atom.
5. Conclude that the two quantum numbers that determine the energy level of an orbital in a multielectron atom are the principal quantum number \( n \) and the azimuthal quantum number \( l \).

Key Concepts

Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. There are four quantum numbers: principal (n), azimuthal (l), magnetic (m_l), and spin (m_s). Each quantum number provides specific information about the electron's energy, shape, orientation, and spin.

Principal Quantum Number (n)

The principal quantum number (n) indicates the main energy level or shell of an electron in an atom. It can take positive integer values (1, 2, 3, ...), with higher values corresponding to higher energy levels and greater distances from the nucleus. In multielectron atoms, n significantly influences the energy of the orbitals.

Azimuthal Quantum Number (l)

The azimuthal quantum number (l) defines the shape of the orbital and is associated with the angular momentum of the electron. It can take integer values from 0 to (n-1) for each principal quantum number. The value of l determines the type of orbital (s, p, d, f) and contributes to the energy of the electron in a multielectron atom.