## Quantum Numbers

Quantum Mechanics

The theory or subject of quantum mechanics is based on advanced physics, chemistry and high level mathematics. For nearly a century, a large number of individuals have devoted their entire careers to the theory of quantum mechanics. Quantum equations produce numbers that are used to describe subatomic particles. Chemists use them to describe the atomic orbital shapes electrons produce for individual atoms, as well as how elements and molecules will react when brought together. Four distinct quantum numbers describe each electron. No two electrons in the same atom will have the same four quantum numbers. Chemistry teachers often liken quantum numbers to apartment addresses in that no two apartments in a building will have the same address. Four quantum numbers (n, , m, ms) can be assigned to any electron. The four quantum numbers can be predicted using orbital-filling diagrams, electron-configuration diagrams or the position of the element on the periodic table. Even though chemists are able to predict the four quantum numbers for any electron,  a few elements have an electron that is not found in its anticipated or predicted orbital. However, the explanations for these cases are better left to theoretical physicists and chemists.

Principal Quantum Number (n)

The first quantum number, also known as the principal quantum number (n), tells how far the electron is from the nucleus of the atom. The larger the number, the greater the distance the electron is from the nucleus. The values for n are n = 1, 2, 3

Azimuthal Quantum Number ()

An azimuthal quantum number () or second quantum number, identifies the energy sublevel (s, p, d, f, g) in which the electron is found. The numbers used to identify the sublevel are s = 0, p = 1, d = 2 and f = 3. The second quantum number is also known as the angular quantum number or orbital quantum number. The g energy sublevel will be used starting with element 121 and larger atomic number elements.

Magnetic Quantum Number (m

Orbitals are assigned a magnetic quantum number (m) that tells the direction of the orbital with respect to the other orbitals in the atom's shell. The  values used for identifying the orbital depends on the sublevel: s has 0; p has -1, 0, +1; d has -2, -1, 0, +1, +2;  and, f has -3, -2, -1, 0, +1, +2, +3.

Spin Quantum Number (ms)

A spin quantum number, (ms), is assigned to distinguish between two electrons occuping the same orbital. The spin quantum number has two values: ms=+½ for an arrow up and ms=-½ for an arrow down.

Quantum Numbers Example One: Helium's 1s² Electrons

Helium has the electron-configuration diagram 1s². The 1 in the  electron-configuration diagram identifies the principal quantum number: n=1. The s in the electron-configuration diagram identifies the azimuthal quantum number (): =0. The azimuthal quantum number for any s sublevel has just m = 0. See figure 1. Since helium has one s  orbital, the two electrons are in that one orbital. However, they must have opposite spin to occupy the same orbital: ms=+½ for an arrow up and ms=-½ for an arrow down. The quantum numbers (n, , m, ms) for the two electrons are: 1, 0, 0, +½ and 1, 0, 0, -½.

Quantum Numbers Example Two: Neon's 2p⁶ Electrons

[He]2s22p6 is the electron-configuration diagram for neon. The 2 in the 2p6 in electron-configuration diagram identifies the principal quantum number: n = 2. The p in the electron-configuration diagram identifies the azimuthal quantum number (): =1. The choices are s = 0, p = 1, d = 2 and f = 3. The magnetic quantum numbers (m) for a p orbital are m = -1, 0, +1. Since neon has three 2p⁶ orbitals, the six electrons are in the three orbitals. Each orbital will have two electrons with opposite spin: ms = +½ for an arrow up and ms = -½ for an arrow down. The quantum numbers (n, , m, ms) for the six electrons in the 2p⁶ are: 2, 1,-1, +½;  2, 1, 0, +½;  2, 1, +1, +½; 2, 1, -1, -½;  2, 1, 0, -½; and,  2, 1, +1, -½. See figure 3.

Quantum Numbers Example Three: Zinc's 3d10 Electrons

The electron-configuration diagram for zinc is [Ar]4s23d10. The 3 in the 3d10 from the electron-configuration diagram identifies the principal quantum number: n=3. The d in the electron-configuration diagram identifies the azimuthal quantum number (): =2. The values used for identifying each orbital for the d are  m=  -2, -1, 0, +1, +2. Since zinc has five 3d orbitals, containing 10 electrons, each orbital will have two electrons with opposite spin: ms = +½ for an arrow up and ms = -½ for an arrow down. The quantum numbers (n, , m, ms) for the ten electronsorbital-filling are: 3, 2, -2, +½;  3, 2, -1, +½;  3, 2, 0, +½; 3, 2, +1, +½;  3, 2, +2, +½;  3, 2, -2, -½;  3, 2, -1, -½;  3, 2, 0, -½;  3, 2, +1, -½; and,  3, 2, +2, -½. See figure 4.