An electron in an atom is completely described by four quantum numbers: n, l, ml, and ms. The first three quantum numbers define the orbital and the fourth quantum number describes the intrinsic electron property called spin. An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same set of all the four quantum numbers. What this means is that two electrons can share the same orbital (the same set of the quantum numbers n, l, and ml) only if their spin quantum numbers ms have different values. Since the spin quantum number can only have two values $(±\frac{1}{2})$, no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons.
The properties and meaning of the quantum numbers of electrons in atoms are briefly summarized in the table below.
| Quantum Numbers, Their Properties, and Significance | |||
|---|---|---|---|
| Name | Symbol | Allowed values | Physical meaning |
| principal quantum number | n | 1, 2, 3, 4, …. | shell, the general region for the value of energy for an electron on the orbital |
| angular momentum or azimuthal quantum number | l | 0 ≤ l ≤ n – 1 | subshell, the shape of the orbital |
| magnetic quantum number | ml | – l ≤ ml ≤ l | orientation of the orbital |
| spin quantum number | ms | $\frac{1}{2}$, $-\frac{1}{2}$ | direction of the intrinsic quantum “spinning” of the electron |
Working with Shells and Subshells
Indicate the number of subshells, the number of orbitals in each subshell, and the values of l and ml for the orbitals in the n = 4 shell of an atom.
Solution
For n = 4, l can have values of 0, 1, 2, and 3. Thus, s, p, d, and f subshells are found in the n = 4 shell of an atom. For l = 0 (the s subshell), ml can only be 0. Thus, there is only one 4s orbital. For l = 1 (p-type orbitals), m can have values of –1, 0, +1, so we find three 4p orbitals. For l = 2 (d-type orbitals), ml can have values of –2, –1, 0, +1, +2, so we have five 4d orbitals. When l = 3 (f-type orbitals), ml can have values of –3, –2, –1, 0, +1, +2, +3, and we can have seven 4f orbitals. Thus, we find a total of 16 orbitals in the n = 4 shell of an atom.
Identify the subshell in which electrons with the following quantum numbers are found: (a) n = 3, l = 1; (b) n = 5, l = 3; (c) n = 2, l = 0.
(a) 3p (b) 5f (c) 2s
Maximum Number of Electrons
Calculate the maximum number of electrons that can occupy a shell with (a) n = 2, (b) n = 5, and (c) n as a variable. Note you are only looking at the orbitals with the specified n value, not those at lower energies.
Solution
(a) When n = 2, there are four orbitals (a single 2s orbital, and three orbitals labeled 2p). These four orbitals can contain eight electrons.
(b) When n = 5, there are five subshells of orbitals that we need to sum:
| 1 orbital labeled 5s |
| 3 orbitals labeled 5p |
| 5 orbitals labeled 5d |
| 7 orbitals labeled 5f |
| +9 orbitals labeled 5g |
| 25 orbitals total |
Again, each orbital holds two electrons, so 50 electrons can fit in this shell.
(c) The number of orbitals in any shell n will equal n2. There can be up to two electrons in each orbital, so the maximum number of electrons will be $2\times n^2$
Check Your Learning
If a shell contains a maximum of 32 electrons, what is the principal quantum number, n?
n = 4
Working with Quantum Numbers
Complete the following table for atomic orbitals:
| Orbital | n | l | ml degeneracy | Radial nodes (no.) |
|---|---|---|---|---|
| 4f | ||||
| 4 | 1 | |||
| 7 | 7 | 3 | ||
| 5d |
Solution
The table can be completed using the following rules:
- The orbital designation is nl, where l = 0, 1, 2, 3, 4, 5, … is mapped to the letter sequence s, p, d, f, g, h, …,
- The ml degeneracy is the number of orbitals within an l subshell, and so is 2l + 1 (there is one s orbital, three p orbitals, five d orbitals, seven f orbitals, and so forth).
- The number of radial nodes is equal to n – l – 1.
| Orbital | n | l | ml degeneracy | Radial nodes (no.) |
|---|---|---|---|---|
| 4f | 4 | 3 | 7 | 0 |
| 4p | 4 | 1 | 3 | 2 |
| 7f | 7 | 3 | 7 | 3 |
| 5d | 5 | 2 | 5 | 2 |
Check Your Learning
How many orbitals have l = 2 and n = 3?
The five degenerate 3d orbitals