Quantum Numbers

There are 4 quantum numbers: n, l, m_l and m_s. Used together they describe the locations of electrons in orbitals within an atom. No two electrons in an atom have the same set of four quantum numbers.

n = The Principle Quantum Number. It is sometimes called the SHELL number in which electrons having similar energies are contained. n can be any whole number : 1,2,3,4,5.... If n = 1, there is one orbital in that shell (an s orbital). If n = 2, there are two orbital types in that shell (an s and p set of orbitals). If n = 3, there are three orbital types in that shell (an s, a p set, and d set of orbitals). If n = 4, there are four orbital types in that shell (an s, a p set, a d set, and an f set of orbitals).

l = Azimuthal Quantum Number. It is sometimes called the ORBITAL TYPE quantum number. The possibilities for l range from 0 to n-l. Each possibility represents an orbital type.

When l = 0 an s orbital is described.

When l = 1 a set of three p orbitals is described.

When l = 2 a set of five d orbitals is described.

When l = 3 a set of seven f orbitals is described.

When l = 4 a set of nine g orbitals is described.

After this, all orbitals are in alphabetical order -- h,i,j,k .....

m_l = Magnetic Quantum Number is associated with the DIRECTION of an orbital. The possibilities for m range from + l to - l in whole number steps. Each m_l set (from + l to - l) represents a particular orbital set. For instance when l = 1, (a set of p orbitals), m_l = 1,0,-1. Each m_l number represents one of the three orbitals. We will not be concerned with which m_l number corresponds to a particular direction of an orbital. Be aware however that in a p orbital set there is a p_x, p_y, and p_z . Also be aware that there is a d_z2, d_x2__y2 d_xy, d_xz and d_yz, orbitals in a d orbital set.

m_s = Spin Quantum Number. This describes the SPIN of an electron in a particular orbital. There

can only be two electrons in any one orbital. These two electrons can only have opposite spins. The two possibilities are +1/2 and -1/2. The electrons must have opposite spin otherwise two electrons would be able to have the same set of four quantum numbers, and that is impossible.

Consider the following example: There are 15 electrons in phosphorus.

Electron # n l m_l m_s

1 1 0 0 +1/2 Filled ls orbital

2 1 0 0 -1/2

3 2 0 0 +1/2

4 2 0 0 -1/2 Filled 2s orbital

5 2 1 1 +1/2

6 2 1 1 -1/2

7 2 1 0 +1/2

8 2 1 0 -1/2 Filled 2p orbitals

9 2 1 -1 +1/2

10 2 1 -1 -1/2

11 3 0 0 +1/2 Filled 3s orbital

12 3 0 0 -1/2

13 3 1 1 +1/2 Half filled p orbitals with

14 3 1 0 +1/2 one electron in each

15 3 1 -1 +1/2