Quantum Numbers Practice Questions (with Answer Key)

Quantum numbers are the “address” of an electron in an atom. They look abstract at first, but a handful of rules covers every question you’re likely to meet. These twenty-four questions are grouped into five sets that build from the basic rules to exam-style problems. Try each set, then check the answer key and explanations at the end.

The rules in one table

Quantum number	Symbol	Describes	Allowed values
principal	n	shell (size, energy)	1, 2, 3…
angular momentum	l	subshell (shape)	0 to n − 1 (0 = s, 1 = p, 2 = d, 3 = f)
magnetic	mₗ	orbital (orientation)	−l to +l, in whole steps
spin	mₛ	electron spin	+½ or −½

Useful consequences:

• Shell n contains n² orbitals and up to 2n² electrons.
• Subshell l contains 2l + 1 orbitals and up to 2(2l + 1) electrons.
• Each orbital holds at most 2 electrons, with opposite spins.

For explanations, see quantum numbers explained, shells vs subshells vs orbitals and electron spin.

Set A: Allowed values

1. What values of l are allowed when n = 3?
2. What values of mₗ are allowed when l = 2?
3. What is the smallest value of n that allows l = 3?
4. How many values of mₗ are possible for l = 1?
5. Can mₛ ever be 0? Explain.

Set B: Naming orbitals

Give the orbital name (e.g. 2p) for each combination.

6. n = 3, l = 1
7. n = 4, l = 2
8. n = 5, l = 0
9. n = 4, l = 3

Set C: Counting

10. How many orbitals are in the n = 4 shell?
11. How many electrons can the n = 3 shell hold?
12. How many orbitals are in a 5d subshell?
13. How many electrons in an atom can have n = 2 and l = 1?
14. How many electrons in an atom can have n = 3 and mₗ = 0?
15. How many electrons in an atom can have the quantum numbers n = 4, l = 2, mₗ = −1?

Set D: Possible or impossible?

Decide whether each set (n, l, mₗ, mₛ) is allowed. If not, explain why.

16. (2, 1, 0, +½)
17. (3, 3, 1, −½)
18. (2, 0, 1, +½)
19. (4, 2, −2, −½)
20. (1, 0, 0, 1)

Set E: Apply and explain

21. Give a possible set of four quantum numbers for the outermost electron of a sodium atom.
22. Give a possible set of quantum numbers for one of the 2p electrons in a carbon atom.
23. How many radial and angular nodes does a 4p orbital have?
24. Use quantum numbers to explain why the second shell holds a maximum of 8 electrons.

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Answer key

Set A

1. l = 0, 1, 2 (l runs from 0 to n − 1).
2. mₗ = −2, −1, 0, +1, +2 (five values, from −l to +l).
3. n = 4, since l can be at most n − 1. That’s why the first f subshell is 4f.
4. Three: −1, 0, +1. That’s why every p subshell has three orbitals.
5. No. Spin can only be +½ or −½. There’s no zero or intermediate value; this was shown by the Stern–Gerlach experiment.

Set B

6. 3p
7. 4d
8. 5s
9. 4f

Remember the code: l = 0, 1, 2, 3 correspond to s, p, d, f.

Set C

10. 16 orbitals (n² = 4²): one 4s, three 4p, five 4d and seven 4f.
11. 18 electrons (2n² = 2 × 3²): 3s (2) + 3p (6) + 3d (10).
12. 5 orbitals (2l + 1 with l = 2), regardless of n.
13. 6 electrons: the 2p subshell has three orbitals, each holding two.
14. 6 electrons: with n = 3, mₗ = 0 occurs once for each value of l (the 3s orbital, one of the 3p orbitals and one of the 3d orbitals). Three orbitals × 2 electrons = 6.
15. 2 electrons: n, l and mₗ together specify one orbital, which holds two electrons with opposite spins.

Set D

16. Allowed. It describes one electron in a 2p orbital.
17. Not allowed. l must be less than n; with n = 3, the maximum l is 2.
18. Not allowed. With l = 0, mₗ can only be 0.
19. Allowed. A 4d electron (mₗ = −2 is within −2 to +2).
20. Not allowed. mₛ must be +½ or −½, not 1.

Set E

21. Sodium is [Ne] 3s¹. The outer electron is in 3s: n = 3, l = 0, mₗ = 0, mₛ = +½ (or −½; either is acceptable for a single unpaired electron).
22. Carbon is 1s² 2s² 2p². A 2p electron: n = 2, l = 1, mₗ = −1, 0 or +1, mₛ = +½. By Hund’s rule, the two 2p electrons occupy different orbitals with parallel spins, for example (2, 1, −1, +½) and (2, 1, 0, +½). See Hund’s rule.
23. For 4p: n = 4, l = 1. Angular nodes = l = 1; radial nodes = n − l − 1 = 2. See radial and angular nodes.
24. For n = 2, l can be 0 or 1. l = 0 gives one orbital (mₗ = 0): 2s. l = 1 gives three orbitals (mₗ = −1, 0, +1): 2p. That’s 4 orbitals. By the Pauli exclusion principle, each orbital holds two electrons with opposite spins (mₛ = +½ and −½), so the shell holds 4 × 2 = 8 electrons.

Scoring and next steps

• 21–24 correct: strong command of quantum numbers. Try electron configuration practice questions next.
• 15–20: revisit the rules table and the counting questions in Set C, which are the most common source of errors.
• Below 15: work through quantum numbers explained and shells vs subshells vs orbitals, then try again.

The most common mistakes

1. Letting l equal n. l stops at n − 1: there’s no 1p, 2d or 3f.
2. Forgetting negative mₗ values. For l = 2, there are five values, including −1 and −2.
3. Confusing orbitals and electrons when counting. Count orbitals first, then double for electrons.
4. Giving spin as 1, 0 or −1. It’s always ±½.
5. Mixing up the letter code. s, p, d, f = 0, 1, 2, 3.

A quick way to check any set

When you’re given a set (n, l, mₗ, mₛ), check it in order:

1. Is n a positive whole number?
2. Is l between 0 and n − 1?
3. Is mₗ between −l and +l?
4. Is mₛ exactly +½ or −½?

If all four checks pass, the set is allowed. If any fails, stop: the set is impossible, and the failed check tells you why. This four-step routine takes seconds and prevents almost every error on “possible or impossible?” questions.

Key takeaways

• n ≥ 1; l = 0 to n − 1; mₗ = −l to +l; mₛ = ±½.
• Shell n has n² orbitals and 2n² electrons; subshell l has 2l + 1 orbitals.
• n, l and mₗ together identify one orbital, which holds at most two electrons.
• A set of quantum numbers is impossible if any one value breaks its rule.
• Angular nodes = l and radial nodes = n − l − 1.