Quantum Numbers:
Only the Examples and Problems
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Fifteen Examples
Example #1: An electron cannot exist in the energy state described by which set of quantum numbers below?
(a) 3, 2, 2, −1⁄2
(b) 4, 3, 3, +1⁄2
(c) 2, 1, −3, +1⁄2
(d) 2, 0, 0, −1⁄2
(e) 1, 0, 1, −1⁄2
Example #2: Identify which sets of quantum numbers are valid for an electron. Each set is ordered (n, ℓ, mℓ, ms)
(a) 2, 2, −1, +1⁄2 (g) 2, 1, −1, +1⁄2 (b) 0, 2, 1, +1⁄2 (h) 1, 2, 0, +1⁄2 (c) 2, 0, 0, −1⁄2 (i) 1, 0, 0, ±1⁄2 (d) 3, −2, −1, −1⁄3 (j) 4, 3, 1, −1⁄2 (e) 3, 2, 1, +1⁄2 (k) 3.5, 3, 1, +1⁄2 (f) 4, 3, −5, −1⁄2 (o) 3, 2, 1, −1
Example #3: Indicate which of the following quantum states are allowed and which are disallowed under the rules governing the electronic structure of atoms.
(a) n = 2, ℓ = 1, mℓ = 0, ms = +1⁄2
(b) n = 3, ℓ = 3, mℓ = −2, ms = −1⁄2
(c) n = 4, ℓ = 3, mℓ = −2, ms = +1⁄2
(d) n = 3, ℓ = 2, mℓ = 2, ms = +1⁄3
(e) n = 2, ℓ = 1, mℓ = −2, ms = −1⁄2
(f) n = 3, ℓ = 2, mℓ = −1, ms = −1⁄2
Example #4: Explain why each of the following sets of quantum numbers would not be permissible for an electron according to the rules for quantum numbers.
(a) n = 1, ℓ = 0, mℓ = 0, ms = +1
(b) n = 1, ℓ = 3, mℓ = 3, ms = +1⁄2
(c) n = 3, ℓ = 2, mℓ = 3, ms = −1⁄2
(d) n = 0, ℓ = 1, mℓ = 0, ms = +1⁄2
(e) n = 2, ℓ = 1, mℓ = −1, ms = +3⁄2
(f) n = 4, ℓ = 3, mℓ = 5, ms = +1⁄2
Example #5: A hydrogen atom has n = 5 and mℓ = −2. What are the possible values for ℓ in this orbital?
Example #6: Which of the following is a possible set of quantum numbers in an atom?
(a) 3, 2, −1, +1
(b) 3, 3, −1, +1⁄2
(c) 3, 1, −2, −1⁄2
(d) 3, 1, 0, +1⁄2
Example #7: An orbital has n = 4 and mℓ = −1. What are the possible values of ℓ for this orbital?
Example #8: In potassium how many electrons will have ℓ = 0 as one of its quantum numbers.
Example #9: In a single atom, what is the maximum number of electrons that can have the quantum numbers n = 4 and mℓ = 2
Example #10: Determine which set(s) of quantum numbers is NOT allowed:
(a) n = 5, ℓ = 3, mℓ = −1, ms = +1⁄2
(b) n = 1, ℓ = 0, mℓ = 0, ms = −1⁄2
(c) n = 2, ℓ = 2, mℓ = 2, ms = +1⁄2
(d) n = 4, ℓ = 1, mℓ = 0, ms = −1⁄2
(e) n = 6, ℓ = 4, mℓ = −3, ms = +1⁄2
Example #11: All the following sets of quantum numbers describe nonexistent orbitals. Find the mistake in each one.
(a) n = 0, ℓ = 3, mℓ = −3, ms = +1⁄2
(b) n = 3, ℓ = −1, mℓ = 0, ms = +1⁄2
(c) n = 3, ℓ = 2, mℓ = −3, ms = −1⁄2
(d) n = 5, ℓ = 3, mℓ = −2, ms = −1
Example #12: An electron in an atom is in the n = 3 and ℓ = 1 quantum state. Identify the possible values of mℓ that it can have.
Example #13: What are all the possible values of ℓ when n = 3?
(a) ℓ = 0, 1, 2, 3(b) ℓ = −2, −1, 0, 1, 2
(c) ℓ = −3, −2, −1, 0, 1, 2, 3
(d) ℓ = 0, 1, 2
Example #14: Which of the following combination of quantum numbers is/are allowed?
(a) n = 1, ℓ = 0, mℓ = 0, ms = +1⁄2
(b) n = 1, ℓ = 3, mℓ = 3, ms = +1⁄2
(c) n = 3, ℓ = 2, mℓ = −2, ms = −1⁄2
(d) n = 2, ℓ = 1, mℓ = −1, ms = +3⁄2
Example #15: Which of the following combinations of quantum numbers are allowed for an electron in a one-electron atom?
(a) n = 4, ℓ = 2, mℓ = −1, ms = −1⁄2
(b) n = 6, ℓ = 2, mℓ = 1, ms = +1⁄2
(c) n = 1, ℓ = −1, mℓ = −2, ms = +1⁄2
(d) n = 6, ℓ = 0, mℓ = 1, ms = +1⁄2
Bonus Example #1: Assign a correct set of four quantum numbers for the valence electron in a sodium atom.
Bonus Example #2: What are the possible values of n and mℓ for an electron in a 5d orbital? Write the n, ℓ, mℓ for each of the orbitals in the 5d subshell.
Probs 1-10
Problem #1: Which of the following is a possible set of quantum numbers that describes an electron?
(a) n = 3, ℓ = 2, mℓ = −3, ms = −1⁄2 (d) n = 3, ℓ = 1, mℓ = −1, ms = +1⁄2 (b) n = 0, ℓ = 0, mℓ = 0, ms = +1⁄2
(e) n = 4, ℓ = −3, mℓ = −1, ms = +1 (c) n = 4, ℓ = 2, mℓ = −1, ms = 0
(f) none
Problem #2: Each electron orbital is characterized by 3 quantum numbers: n, ℓ, and mℓ.
n is known as the ____ quantum number.
ℓ is known as the ____ quantum number.
mℓ is known as the ____ quantum number.
Problem #3: Each electron orbital is characterized by 3 quantum numbers: n, ℓ, and mℓ.
n specifies ___.
ℓ specifies ___.
mℓ specifies ___.(a) The subshell or orbital shape.
(b) The energy and average distance from the nucleus.
(c) The orbital orientation.
Problem #4: Give the orbital designation (1s, 2p, 3d, etc.) of electrons with the following combination of principal and azimuthal quantum numbers.
(a) n = 1, ℓ = 0
(b) n = 2, ℓ = 1
(c) n = 3, ℓ = 2
(d) n = 5, ℓ = 3
(e) n = 6, ℓ = 0
(f) n = 4, ℓ = 2
Problem #5: For the quantum number ℓ values below, how many possible values are there for the quantum number mℓ?
(a) 5; (b) 3; (c) 2; (d) 1
Problem #6: What does a set of four quantum numbers tell you about an electron? Compare and contrast the locations and properties of two electrons with quantum number sets (4, 3, 1, +1⁄2) and (4, 3, −1, +1⁄2).
Problem #7: Identify the shell/subshell that each of the following sets of quantum numbers refers to.
(a) n = 2, ℓ = 1, mℓ = 1, ms = +1⁄2
(b) n = 3, ℓ = 2, mℓ = 2, ms = +1⁄2
(c) n = 4, ℓ = 1, mℓ = −1, ms = −1⁄2
(d) n = 4, ℓ = 3, mℓ = 3, ms = −1⁄2
(e) n = 5, ℓ = 0, mℓ = 0, ms = +1⁄2
Problem #8: Which of the following set of quantum numbers (ordered n, ℓ, mℓ, ms) are possible for an electron in an atom?
Select all that apply:
(a) 3, 2, 2, −1⁄2 (f) 5, 3, −3, +1⁄2 (b) 2, 1, 3, +1⁄2 (g) 3, 1, −2, −1⁄2 (c) −3, 2, 2, −1⁄2 (h) 5, 3, 0, +1⁄2 (d) 3, 3, 1, −1⁄2 (i) 3, 2, −1, ±1⁄2 (e) 3, 2, 1, −1 (j) 3, 2, −1, 0
Problem #9: For principal quantum number n = 4, the total number of orbitals having ℓ = 3 is?
Problem #10: The maximum number of electrons that can have principal quantum number n = 3 and spin −1⁄2 is?
Bonus Problem #1: Give the maximum number of electrons in an atom that can have these quantum numbers:
(a) n = 4
(b) n = 5, mℓ = +1
(c) n = 5, ms = +1⁄2
(d) n = 3, ℓ = 2
(e) n = 1, ℓ = 0, mℓ = 0
Bonus Problem #2: What is the Principal Quantum Number of the first shell to have:
(a) s orbitals?
(b) p orbitals?
(c) d orbitals?
(d) f orbitals?
Probs 11-25
Problem #11: What is the maximum number of electrons that can be identified with the following set of quantum numbers?
(a) n = 4, ℓ = 0, mℓ = 0, ms = +1⁄2
(b) n = 3, mℓ = +2, ms = +1⁄2
(c) n = 3, mℓ = 0, ms = +1⁄2
Problem #12: What is the maximum number of electrons that can have the following sets of quantum numbers?
(a) n = 4, ℓ = 3, mℓ = 3, ms = −1⁄2
(b) n = 4, ℓ = 3, mℓ = 4, ms = −1⁄2
Problem #13: In an atom, what is the maximum number of electrons that can have the following quantum numbers?
(a) n = 6, ℓ = 4
(b) n = 6, ℓ = 4, mℓ = −1
Problem #14: Which of the following combinations of quantum numbers are allowed?
(a) n = 1, ℓ = 1, mℓ = 0
(b) n = 3, ℓ = 0, mℓ = 0
(c) n = 1, ℓ = 0, mℓ = −1
(d) n = 2, ℓ = 1, mℓ = 2
Problem #15: The following sets of quantum numbers, listed in the order n, ℓ, mℓ, and ms were written for the last electron added to an atom. Identify which sets are valid:
n ℓ mℓ ms I. 2 1 0 +1⁄2 II. 2 2 -1 +1⁄2 III. 2 0 1 −1⁄2 IV. 4 2 2 +1⁄2
Which of the following sets of quantum numbers is/are allowed?
(a) I and III
(b) I and IV
(c) I, II, and III
(d) II, III, and IV
(e) They are all allowed.
Problem #16: Which of the following quantum number cannot be the same for an electron in the 2p orbital and one in the 3d orbital?
I. n
II. ℓ
III. mℓ
IV. ms
(a) I only
(b) I and II only
(c) I, II, III
(d) I, II, IV
(e) I, II, III, IV
Problem #17: Which of the following is not a valid set of four quantum numbers? (Order ---> n, ℓ, mℓ, ms) Why?
(a) 2, 1, 0, −1⁄2
(b) 3, 1, −1, −1⁄2
(c) 1, 0, 0, +1⁄2
(d) 2, 0, 0, +1⁄2
(e) 1, 1, 0, +1⁄2
Problem #18: Determine which sets of quantum numbers are correct and which are incorrect.
(a) 14, 9, −3, −1⁄2
(b) 9, 5, −1, 0
(c) 15, 2, -6, +1⁄2
(d) 7, 10, 0, +1⁄2
(e) 10, 9, 1, +3⁄4
Problem #19: Classify each set of quantum numbers as possible or not possible for an electron in an atom.
(a) 3, 2, −3, +1⁄2 (e) 3, 2, 0, −2 (b) 4, 3, −2, +1⁄2 (f) 4, 3, 4, −1⁄2 (c) −2, 1, 0, −1⁄2 (g) 2, 1, 0, +1⁄2 (d) 2, 2, 2, +1⁄2 (h) 4, 2, −2, +1⁄2
Problem #20: What is wrong with the following set of quantum numbers?
n = 2, ℓ = 2, mℓ = 0, ms = +1⁄2
Problem #21: Give the quantum numbers for all orbitals in the 5f subshell.
Problem #22: Which of the following set of quantum numbers (ordered n, ℓ, mℓ, ms) are possible for an electron in an atom?
(a) 3, 2, 0, −2
(b) 3, 4, 0, +1⁄2
(c) 3, 1, 0, −1⁄2
(d) 4, 2, −1, −3⁄2
(e) 2, 1, −2, +1⁄2
(f) −1, 0, 0, −1⁄2
(g) 4, 2, 1, −1⁄2
(h) 2, 1, 3, +1⁄2
Problem #23: Which set of quantum numbers cannot occur together to specify an orbital?
(a) n = 3, ℓ = 2, mℓ = 3
(b) n = 2, ℓ =1, mℓ = −1
(c) n = 3, ℓ = 1, mℓ = −1
(d) n = 4, ℓ = 3, mℓ = 3
Problem #24: Identify which sets of quantum numbers are valid for an electron. Each set is ordered (n, ℓ, mℓ, ms)
(a) 3, 1, −1, +1⁄2 (g) 4, 3, 4, −1⁄2 (b) 3, 2, −1, 0 (h) 2, 1, 1, +1⁄2 (c) 3, 2, 1, +1⁄2 (i) 4, 3, 1, −1⁄2 (d) 3, −2, −2, −1⁄2 (j) 1, 0, 0, −1⁄2 (e) 2, 3, 1, +1⁄2 (k) 2, −1, 1, −1⁄2 (f) 1, 3, 0, +1⁄2 (ℓ) 0, 1, 1, −1⁄2
Problem #25: Which of the following set of quantum numbers (ordered n, ℓ, mℓ, ms) are possible for an electron in an atom?
(a) 4, 3, 4, −1⁄2 (e) 3, 2, −3, +1⁄2 (b) 2, 1, 0, +1⁄2 (f) 2, 2, 2, +1⁄2 (c) −2, 1, 0, −1⁄2 (g) 3, 2, 1, −1 (d) 5, 3, −3, +1⁄2 (h) 4, −2, 1, −1⁄2
Problem #26: Determine the number of electrons that can be described by each of the given quantum numbers:
(a) n = 3, ℓ = 2
(b) n = 3, ℓ = 3, mℓ = +1
(c) n = 4
(d) n = 4, ℓ = 3, mℓ = +1, ms = +1⁄2
(e) n = 2, ℓ = 2, mℓ = 2, ms = −1⁄2
Bonus Problem #1: All of the following sets of quantum numbers (ordered n, ℓ, mℓ, ms) are not possible for an electron in an atom. Identify the error in each one.
(a) 0, 0, 0, 1⁄2 (e) −4, 3, 1, 1⁄2 (b) 2, 1, 0, −1 (f) 3, 2, 2, 1⁄3 (c) 5, 3, 4, 1⁄2 (g) 3, 1, 2, −1⁄2 (d) 3.5, 2, 1, −1⁄2 (h) 3, 2, −3, 1⁄2
Bonus Problem #2: One QN in each set is not allowed. Identify the mistake and replace it with one that is allowed.
(a) n = 3, ℓ = 3, mℓ = +2
(b) n = 2, ℓ = 1, mℓ = −2
(c) n = 1, ℓ = 1, mℓ = 0
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