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Spin Quantum Number (s):

The spin quantum number tells whether a given electron is spin up (+1/2) or spin down (-1/2). An orbital contains two electrons, and each of those electrons must have different spins.

It is often convenient to depict orbitals in an orbital energy diagram, as seen below in . Such diagrams show the orbitals and their electron occupancies, as well as any orbital interactions that exist. In this case we have the orbitals of the hydrogen atom with electrons omitted. The first electron shell (n = 1) contains just the 1s orbital. The second shell (n = 2) holds a 2s orbital and three 2p orbitals. The third shell (n = 3) holds one 3s orbital, three 3p orbitals, and five 3d orbitals, and so forth. Note that the relative spacing between orbitals becomes smaller for larger n. In fact, as n gets large the spacing becomes infinitesimally small.

Figure %: Energy diagram of the unoccupied atomic orbitals of hydrogen. Potential energy is on the y-axis.

You will see such energy diagrams quite often in your continuing study of chemistry. Notice that all orbitals with the same n have the same energy. Orbitals with identical energies are said to be degenerate (not in the moral sense!). Electrons in higher-level orbitals have more potential energy and are more reactive, i.e. more likely to undergo chemical reactions.

Multi-electron atoms

When an atom only contains a single electron, its orbital energies depend only on the principle quantum numbers: a 2s orbital would be degenerate with a 2p orbital. However, this degeneracy is broken when an atom has more than one electron. This is due to the fact that the attractive nuclear force any electron feels is shielded by the other electrons. s-orbitals tend to be closer to the nucleus than p-orbitals and don't get as much shielding, and hence become lower in energy. This process of breaking degeneracies within a shell is known as splitting. In general s orbitals are lowest in energy, followed by p orbitals, d orbitals, and so forth.

Figure %: Splitting of orbital energies in multi-electron systems

The energy diagram of imply a further fact about the energy of electrons. Note that the energy levels in these diagrams do not follow a continuous line: an atom is either in one energy subshell or it is in another. There is no in between. In this way, the diagram perfectly represents the quantized nature of electrons, meaning that electrons can only exist at specific and defined energy levels. The energy level of an electron in a particular energy shell can be determined according to the following equation:

En = /frac-2.178x10-18joulesn2    

where n is the principal quantum number and En is the energy level at that quantum number. When an electron absorbs a specific quanta of energy it can jump to a higher energy level. It can also give off a specific quanta and fall back to a lower energy level. An atom whose electrons are at their lowest energy levels is said to be in the ground state. The discovery of the quantum nature of energy and electrons, first formulated by Max Planck in 1900, led to the creation of an entirely new field, quantum mechanics.

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