Pauli Exclusion Principle
In 1913, Niels Bohr proposed a model of the hydrogen atom that introduced the concept of quantized energy levels. Bohr assumed that the energy which an electron in an atom may have is quantized, i.e. it can possess only certain amounts of energy. Bohr assumed that electrons rotate around the nucleus in orbits with distances from the nucleus related to a quantum number n. The lowest quantized energy level has n = 1; the next highest has n = 2, etc. The properties that are predicted by Bohr's model match the experimentally observed values.
Efforts were begun immediately to extend Bohr's model of the hydrogen atom, which has only one electron, to multi-electron atoms.
The electrons of an atom will assume a configuration which has the lowest possible energy, called the ground state. In the hydrogen atom, the single electron is located in the orbit for which the quantum number n = 1. Helium has two electrons. In its ground state, both electrons are located in the lowest energy orbit with n = 1.
The lithium atom has three electrons. Again it might be assumed that in the ground state, the lowest energy configuration would require all three electrons to be in the n = 1 orbit. Experiment indicates, however, that while two electrons are in the n = 1 orbit, the third is in the orbit corresponding to n = 2. Atoms of the element beryllium have four electrons, two with n = 1 and two with n = 2.
For boron, with 5 electrons, two have n = 1, two have n = 2, and we might suppose that the fifth will have n = 3. But for some reason, unexplained by the Bohr model, the energy of this electron is much closer to the n = 2 level than to that of n = 3. A more sophisticated theory of atomic structure was needed.
In 1923, Louis de Broglie proposed that particles possess wave properties. In 1925, to explain why all electrons in an atom will not be in the lowest n = 1 energy state, Wolfgang Pauli proposed that no two electrons can simultaneously have the same wave properties. This proposal evolved into the Pauli exclusion principle.
In 1926, Erwin Schrödinger proposed a theory, known as wave mechanics, to explain the properties of particles such as electrons. The solutions of the theory's wave equation for hydrogen match exactly the predictions of the Bohr theory. Moreover, Schrödinger's wave mechanics has been successfully applied to systems with more than one electron.
Each solution of Schrödinger's wave mechanical equation for atomic systems, called an orbital, has a set of four quantum numbers associated with it. The principal quantum number n may have any positive, non-zero integer value. It is the primary determinant of the allowed energy levels of electrons. The angular momentum quantum number l may have only integer values from 0 to (n-1). The energy of an orbital depends slightly on the value of l. The magnetic quantum number m(sub)l may have any of the integer values -l to +l (including 0); and the spin quantum number s may have only the values +1/2 and -1/2. Each orbital has a different set of these four quantum numbers.
The Pauli exclusion principle may now be stated as requiring that no two electrons in an atom may occupy the same orbital, i.e. may simultaneously possess the same values of the four quantum numbers. Since the spin quantum number may have only the values +1/2 and -1/2, and since the value of spin affects neither the energy nor the rest of the wave function, we often speak of each orbital containing two electrons, one with spin +1/2 and the other with spin -1/2.
The quantum numbers corresponding to the possible energy levels of an atom are as follows (with energy increasing as the value of n and l increase):
There is only one orbital with n = 1, because when n = 1, l = 0 and m = 0. This is called the 1s orbital and two electrons may occupy it (one with spin +1/2 and one with spin -1/2).
There are five orbitals with 2: a 2s orbital with n = 2, l = 0, m = 0, containing two electrons; three 2p orbitals, of slightly higher energy, for which n = 2, l = 1, and m has the values -1, 0, and +1; each of these p orbitals may contain two electrons; six electrons may thus be accommodated in these p orbitals.
There are nine orbitals with n = 3: a 3s orbital with two electrons; three 3p orbitals of slightly higher energy that may contain a total of six electrons; and five 3d orbitals with n = 3, l = 2, and m has values of -2, -1, 0, +1, and +2. The 3d orbitals have slightly higher energy than the 3p orbitals, and may contain a total of ten electrons.
Similar orbital schemes may be constructed for larger values of the principal quantum number.
Recall that the ground state of an atom is the state of lowest energy, with the electrons of the atom occupying the orbitals of lowest energy.
The lone electron of the hydrogen atom will occupy the 1s orbital in the ground state. The two electrons of the helium atom are both in the 1s orbital, one with spin +1/2, the other with spin -1/2. The third electron of lithium must have n = 2 since there are no more unfilled orbitals with n = 1. Both the third and fourth electrons of beryllium reside in the 2s orbital, each with a different spin quantum number. The fifth electron of boron cannot be accommodated in the filled 2s orbital; the next lowest orbital is 2p.
Continuing this process, using the Pauli exclusion principle, the ground state electron configuration of the elements of the periodic table can be explained. For instance, a chlorine atom has seventeen electrons. In the ground state, they will be located in the orbitals of lowest possible energy: two electrons in the 1s orbital, two in 2s, six in 2p, two in 3s, and five in 3p. A bromine atom has thirty-five electrons, located as follows: two in 1s, two in 2s, six in 2p, two in 3s, six in 3p, ten in 3d, two in 4s, and five in 4p.
The electrons of molecular bonds can be understood in a similar way. Instead of orbitals located on a single atom, the wave theory treatment of the electrons of molecular bonds results in a set of molecular orbitals which spread over two or more atomic centers. The electrons that are shared by the atoms to form the bond belong to one of these molecular orbitals. The ground state of the molecule is that in which the bonding electrons are in the molecular orbitals of lowest energy. Again, the Pauli exclusion principle holds: only two electrons may be located in each molecular orbital, each with a different spin.
In addition to electrons, the Pauli exclusion principle applies to all sub-atomic particles with half-integral spins, known as fermions, such as neutrons and protons. The Pauli exclusion principle does not apply, however, to particles with integral spin, known as bosons, such as photons.
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