Deriving the Periodic Chart of Atoms

James Richard Fromm

The structure of the periodic chart of the elements follows directly from the possible values of the four quantum numbers and one additional principle. That principle is the Pauli exclusion principle proposed by the German physicist Wolfgang Pauli in 1929:

No two electrons in any atom can have the same set of four quantum numbers.

The Pauli exclusion principle allows the determination of the maximum number of electrons possible in any given orbital, subshell, or shell. Since the spin quantum number can have only two different values, the number of electrons per orbital depends upon the type of orbital. The number of electrons an orbital can contain will be twice the number of different orbitals, or energy levels, available within the subshells of that type. These numbers are two electrons for an s orbital, six electrons for a p orbital, ten electrons for a d orbital, fourteen electrons for a f orbital, and eighteen electrons for a g orbital.

The same principle allows the calculation of the number of electrons which can be accommodated within any given shell. These are:

 Shell 1: s: 2 electrons Shell 2: s + p: 2 + 6 = 8 electrons Shell 3: s + p + d: 2 + 6 + 10 = 18 electrons Shell 4: s + p + d + f: 2 + 6 + 10 + 14 = 32 electrons Shell 5: s + p + d + f + g: 2 + 6 + 10 + 14 + 18 = 50 electrons

Building Up the Periodic Chart

An electron, which is negative, approaching a positive nucleus is attracted to the nucleus and will assume the innermost orbital available to it since that will be the orbital of lowest energy. The electronic structure of an atom therefore builds up outward from the innermost to the outermost orbitals; an electron will always attempt to occupy the lowest unoccupied orbital, which is called its ground state. To move an electron to a higher unoccupied orbital (an excited state) is possible, but requires energy. The energy difference between the orbitals must be provided by absorption of a photon of light of the appropriate wavelength. The electron will decay from the excited state back to the ground state, emitting a photon, after some period of time.

Hydrogen has but one electron, so that electron can occupy the orbital with n = 1 (and therefore l = 0 and m = 0) and spin +1/2. This structure is often indicated as 1s1. Helium, with two electrons, can have one of them with n = 1, l =0, m = 0, s = +1/2 and one with n = 1, l = 0, m = 0, and s = -1/2, giving an overall structure indicated as 1s2.

Lithium has three electrons, so the quantum numbers for two of them are the same as for the two electrons of helium. The third electron cannot fit in the shell with n = 1 at all, since no possible unused quantum numbers are left unused; given that n = 1, both l and m must be zero, and both of the possible values of s have been used. The third electron must then have the quantum numbers n = 2, l = 0, m = 0, s = +1/2. This electron configuration is usually indicated as 1s22s1. Beryllium, with four electrons, follows the same pattern with the electronic configuration 1s22s2.

In the shell with n = 1, only two electrons were possible because l and m could only be zero. With n = 2, however, l can not only be zero but can also be one, and when the value of l is one then the value of m can be -1, 0, or +1. The fifth electron of boron then is n = 2, l = 1, m = -1, s = +1/2 which is indicated by 1s22s22p1. The remaining elements in the first row of the periodic chart are:

 C:1s22s22p2 N:1s22s22p3 O:1s22s22p4 F:1s22s22p5 Ne:1s22s22p6

With the ten electrons of neon the first two shells are now completed and can hold no more electrons. This building-up procedure (called aufbau, from the German word for building up) continues with sodium, whose outermost electron is 3s1:

 Na: 1s22s22p63s1 or [Ne]3s1 Mg: 1s22s22p63s2 or [Ne]3s2 Al: [Ne]3s23p1 Si: [Ne]3s23p2 P: [Ne]3s23p3 S: [Ne]3s23p4 Cl: [Ne]3s23p5 Ar: [Ne]3s23p6 or [Ar]

Up to this point the orbitals have simply been filled in order of expected energy as if it were in order of quantum numbers: 1s, 2s, 2p, 3s, 3p, ... Beyond this point the order of filling changes because the order of quantum numbers and the order of actual energies are not the same; the 4s orbital and the 3d orbital are comparable in energy. Building up the orbital structure in one-proton, one-electron steps gives the 4s as the orbital lower in energy, and the electronic structures are:

 K: [Ar]4s1 Ca: [Ar]4s2 Sc: [Ar]4s23d1 Ti: [Ar]4s23d2 V: [Ar]4s23d3 --- Kr: [Ar]4s23d104p6 = [Kr] Rb: [Ar]4s23d104p65s1 = [Kr]5s1

The approximate relative energies of the orbitals of electrons are shown on the left-hand side of the Figure below. A useful mnemonic device for the formally expected order of filling of these orbitals using the aufbau principle is shown on the right-hand side of the Figure.