C. Structure and Properties of Matter · Structure and Properties of Matter

Quantum Numbers and the Shapes of Orbitals

C3.2 C2About 65 minutes

Learning goals

I can describe the meaning and allowed values of the four quantum numbers.
I can decide whether a set of quantum numbers is allowed and name the orbital it describes.
I can describe the shapes of s and p orbitals and count orbitals in any sublevel.
I can state the Pauli exclusion principle and use it to explain why an orbital holds two electrons.

Words to know

principal quantum number (n)
Gives an orbital's main energy level and size; n = 1, 2, 3 ...

secondary quantum number (l)
Gives the sublevel and shape; l = 0 to n − 1 (s, p, d, f).

magnetic quantum number (mₗ)
Gives an orbital's orientation; mₗ = −l to +l.

spin quantum number (mₛ)
Gives the electron's spin, +½ or −½.

Pauli exclusion principle
No two electrons in an atom can have the same four quantum numbers.

degenerate orbitals
Orbitals with the same energy, such as the three 2p orbitals.

An address for every electron

In a big city, a full mailing address has a province, a city, a street and a house number. In an atom, every electron has a similar four-part address: its four quantum numbers. They come straight out of Schrödinger's equation and tell you the electron's energy level, the shape of its orbital, the orbital's orientation and the electron's spin.

Remember from the last lesson: an orbital is a region where an electron is likely to be found. Quantum numbers tell you which orbital.

The four quantum numbers

1. Principal quantum number, n (n = 1, 2, 3, ...)

Gives the main energy level (shell) and the orbital's size.
Larger n means a larger orbital, an electron farther from the nucleus on average, and higher energy.

2. Secondary (angular momentum) quantum number, l (l = 0, 1, 2, ..., n − 1)

Gives the sublevel and the orbital's shape.
Each value has a letter: l = 0 → s, l = 1 → p, l = 2 → d, l = 3 → f.
Level n has n sublevels: n = 1 has only 1s; n = 2 has 2s and 2p; n = 3 has 3s, 3p and 3d; n = 4 has 4s, 4p, 4d and 4f.

3. Magnetic quantum number, mₗ (mₗ = −l, ..., 0, ..., +l)

Gives the orientation of the orbital in space.
The number of values = the number of orbitals in the sublevel: s has 1, p has 3, d has 5, f has 7.

4. Spin quantum number, mₛ (mₛ = +½ or −½)

Electrons behave as if they spin, making each a tiny magnet with one of two orientations, often drawn as ↑ and ↓.
An orbital holds at most two electrons, and they must have opposite spins.

n	Allowed l (sublevel)	Orbitals in each sublevel	Orbitals in level (n²)	Max electrons (2n²)
1	0 (1s)	1	1	2
2	0 (2s), 1 (2p)	1, 3	4	8
3	0 (3s), 1 (3p), 2 (3d)	1, 3, 5	9	18
4	0, 1, 2, 3 (4s, 4p, 4d, 4f)	1, 3, 5, 7	16	32

How the quantum numbers build the first four energy levels.

The quantum-number tree for n = 3. Each mₗ box is one orbital, and each orbital holds at most two electrons (mₛ = +½ and −½).

Orbital shapes

s orbitals (l = 0) are spherical. A 2s orbital is a larger sphere than 1s, with a spherical node (a surface of zero probability) inside it.
p orbitals (l = 1) have two lobes on opposite sides of the nucleus, like a dumbbell. The three p orbitals in a sublevel point along the x, y and z axes: px, py and pz. They have equal energy.
d orbitals (l = 2) mostly have four lobes (one has two lobes and a ring). There are five of them.
f orbitals (l = 3) have more complex shapes; there are seven.

Orbitals in the same sublevel (for example, the three 2p orbitals) have the same energy in an isolated atom. Orbitals with the same energy are called degenerate.

Shapes of s and p orbitals. An s orbital is a sphere. The three p orbitals are dumbbells at right angles to each other, along the x, y and z axes, with a node (zero probability) at the nucleus.

Electron spin and the Pauli exclusion principle

In 1925 Wolfgang Pauli stated the Pauli exclusion principle: no two electrons in an atom can have the same four quantum numbers. Since two electrons in the same orbital share n, l and mₗ, they must differ in mₛ. That is why an orbital holds at most two electrons, and why they have opposite spins.

Spin has a real, measurable effect. Atoms with unpaired electrons are attracted into a magnetic field (paramagnetic). Atoms with all electrons paired are weakly pushed out (diamagnetic). You will use this in the next lesson.

Worked example

Example 1: Is this set allowed?

Which of these sets of quantum numbers (n, l, mₗ, mₛ) are allowed? (a) (2, 1, −1, +½) (b) (3, 3, 0, −½) (c) (4, 2, −3, +½) (d) (1, 0, 0, −½)

(a) n = 2 allows l = 0 or 1; l = 1 allows mₗ = −1, 0, +1; mₛ = +½ is fine. Allowed (a 2p electron).
(b) l must be at most n − 1 = 2. l = 3 is not allowed.
(c) l = 2 allows mₗ from −2 to +2. mₗ = −3 is not allowed.
(d) n = 1, l = 0, mₗ = 0, mₛ = −½. Allowed (a 1s electron).

(a) and (d) are allowed; (b) and (c) are not.

Worked example

Example 2: Counting orbitals and electrons

How many orbitals and how many electrons can the n = 4 level hold? How many are in the 4f sublevel?

Orbitals in level n = n² = 4² = 16 (1 + 3 + 5 + 7).
Electrons = 2n² = 2(16) = 32.
4f: l = 3, so mₗ = −3, −2, −1, 0, +1, +2, +3, which is 7 orbitals and 14 electrons.

n = 4: 16 orbitals, 32 electrons. 4f: 7 orbitals, 14 electrons.

Worked example

Example 3: Naming an orbital

Name the orbital described by n = 5, l = 2, and state how many electrons it can hold.

n = 5 gives the 5th energy level.
l = 2 is the d sublevel.
So the orbital is a 5d orbital. Any single orbital holds at most 2 electrons; the whole 5d sublevel (5 orbitals) holds 10.

A 5d orbital; 2 electrons in that orbital (10 in the 5d sublevel).

Allowed or not?

A quick check list for a quantum-number set:

n must be a positive whole number.
l runs from 0 up to n − 1.
mₗ runs from −l to +l (including 0).
mₛ is +½ or −½ only.

Just for fun

The letters s, p, d and f come from old descriptions of spectral lines: sharp, principal, diffuse and fundamental. After that, chemists gave up on adjectives and just carried on alphabetically with g, h and i.

Hands-on activity

Build the orbital address book

You need: an egg carton (12 cups) or a muffin tin, a second egg carton or a sheet of paper with circles drawn on it, and about 20 small objects such as dried beans or buttons. Grown-up note: keep small objects away from young children.

Label one cup 1s, the next 2s, the next three cups 2p, then one 3s and three 3p. That uses 9 cups. Draw 5 more circles on paper labelled 3d.
Each cup is one orbital. Put beans in, using the rule that each cup holds at most 2 beans, and if two beans share a cup, place one bean on its side and one standing up to show opposite spins.
Count how many beans fit in level 1 (cups 1s), level 2 (2s and 2p) and level 3 (3s, 3p, 3d).
Write the four quantum numbers for one bean in a 2p cup and one bean in a 3d circle.

What to look for: level 1 holds 2, level 2 holds 8, level 3 holds 18: the 2n² rule. A correct address for a 2p bean is, for example, n = 2, l = 1, mₗ = 0, mₛ = +½. For 3d: n = 3, l = 2, mₗ any value from −2 to +2, mₛ = ±½.

Your turn

What does each quantum number describe? (n, l, mₗ, mₛ)

List the possible values of l and mₗ for n = 2.

Is (3, 2, +2, −½) allowed? If so, which sublevel?

Is (2, 2, 0, +½) allowed?

How many electrons can the 3p sublevel hold?

State the Pauli exclusion principle.

Describe the shape of an s orbital and of a p orbital.

More practice: warm-up

Which letter goes with l = 2?

How many orbitals are in an f sublevel?

What are the two possible values of mₛ?

How many sublevels does n = 3 have?

Maximum number of electrons in one orbital?

More practice: on your own

Give all four quantum numbers for each electron in a helium atom (1s²).

What is the lowest n value that has a d sublevel? Explain.

How many orbitals in an atom can have n = 3 and mₗ = 0?

Find the error in (4, 1, −2, +½) and fix it in two different ways.

How many electrons in one atom can have n = 2 and mₛ = +½?

Challenge

Show that the number of orbitals in level n is always n². Check it for n = 5.

Suppose (in an imaginary universe) mₛ could have three values instead of two. How many electrons could the n = 2 level hold, and what would the first noble gas be?

A d orbital has two angular nodes (planes or cones of zero probability through the nucleus). Using the pattern s = 0 and p = 1, predict the number of angular nodes for an f orbital and state the rule.

Check your understanding

Quiz

What does the secondary quantum number, l, determine?

Which set of quantum numbers (n, l, mₗ, mₛ) is NOT allowed?

How many orbitals are in the 4d sublevel?

What is the maximum number of electrons in the n = 3 level?

Which principle says that two electrons in the same orbital must have opposite spins?

Which orbital is described by n = 2, l = 1?

For parents and teachers

Big idea (C3.2): four quantum numbers describe every electron in an atom. Their rules (l up to n − 1; mₗ from −l to +l; mₛ = ±½) produce the 2, 8, 18, 32 pattern of shell capacities that shapes the periodic table.

How to help: quiz your teen with quick sets like "(3, 1, 0, +½): allowed? which orbital?" until the four rules are automatic. The egg-carton model helps students who like hands-on work.

Common difficulties:

Letting l equal n (it must be at most n − 1).
Forgetting that mₗ includes 0 and negative values.
Confusing orbitals with electrons: the 3p sublevel has 3 orbitals but holds 6 electrons.