D3.2 Multi-electron Atoms
The ideas already developed about quantum numbers, orbitals, and sizes and shapes of electron-density distributions apply to all atoms. However, when there are two or more electrons in an atom, there are repulsive forces between the electrons in addition to the attractive forces between electrons and the nucleus. These repulsions affect electron energies.
For example, the energy levels in a He+ ion (which, like H, has a single electron) are significantly lower than those in a H atom because of the stronger Coulomb’s law attraction between the one electron and the 2+ charge of the He nucleus. However, in a neutral He atom, which has two electrons, electron-electron repulsions between the electrons raise energy levels significantly compared to He+, and a He atom is not as stable as we might have expected.
For atoms with many electrons, the effect of electron-electron repulsions differs for different subshells. Therefore orbital energy depends on both n and ℓ quantum numbers. For the same value of n (the same shell), as ℓ increases the energy also increases. Thus s-subshell electrons have lower energy than p-subshell electrons, which are lower than d-subshell electrons, and so forth. Orbitals within the same subshell (for example 2px, 2py, and 2pz) all have the same energy; orbitals that have the same energy are said to be degenerate.
The Austrian physicist Wolfgang Pauli formulated what is now called the Pauli exclusion principle:
- Each electron in an atom must have a different set of values for the four quantum numbers.
- If two electrons share the same orbital (have the same n, ℓ, and mℓ), then their spin quantum numbers ms must have different values; we say the two electrons have opposite spin.
- Because ms can only have two values, +½ or -½, no more than two electrons can occupy the same orbital.
By applying the Pauli exclusion principle, the arrangement of electrons in any multi-electron atom can be determined by recognizing that the ground state of an atom has all of its electrons in orbitals with the lowest energies possible.
Activity: Arrangement of Electrons in Li
Describe the arrangement of electrons in a ground state Li atom (which contains three electrons) by specifying the set of four quantum numbers for each electron. Which subshells are occupied in a Li atom? Write your answers in your notebook and then compare with the answers below.
Quantum numbers for each electron
The ground state of Li has electrons in the lowest-energy orbitals. Orbital energy depends on n and ℓ, so begin with the lowest n value and assign quantum numbers based on the rules for quantum numbers and the Pauli exclusion principle. When all combinations for n = 1 are written, go to n = 2. Continue until all electrons have been assigned different sets of the four quantum numbers.
- electron 1: n = 1, ℓ = 0, mℓ = 0, ms = +½
- electron 2: n = 1, ℓ = 0, mℓ = 0, ms = -½
- electron 3: n = 2, ℓ = 0, mℓ = 0, ms = +½ (or -½)
Subshells occupied
The subshells occupied in a ground state Li atom are 1s and 2s subshells. The 2s subshell is half filled because it has only a single electron.
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