Only the Examples and Problems

**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, m_{ℓ}, m_{s})

(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, m_{ℓ}= 0, m_{s}= +^{1}⁄_{2}

(b) n = 3, ℓ = 3, m_{ℓ}= −2, m_{s}= −^{1}⁄_{2}

(c) n = 4, ℓ = 3, m_{ℓ}= −2, m_{s}= +^{1}⁄_{2}

(d) n = 3, ℓ = 2, m_{ℓ}= 2, m_{s}= +^{1}⁄_{3}

(e) n = 2, ℓ = 1, m_{ℓ}= −2, m_{s}= −^{1}⁄_{2}

(f) n = 3, ℓ = 2, m_{ℓ}= −1, m_{s}= −^{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, m_{ℓ}= 0, m_{s}= +1

(b) n = 1, ℓ = 3, m_{ℓ}= 3, m_{s}= +^{1}⁄_{2}

(c) n = 3, ℓ = 2, m_{ℓ}= 3, m_{s}= −^{1}⁄_{2}

(d) n = 0, ℓ = 1, m_{ℓ}= 0, m_{s}= +^{1}⁄_{2}

(e) n = 2, ℓ = 1, m_{ℓ}= −1, m_{s}= +^{3}⁄_{2}

(f) n = 4, ℓ = 3, m_{ℓ}= 5, m_{s}= +^{1}⁄_{2}

**Example #5:** A hydrogen atom has n = 5 and m_{ℓ} = −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 m_{ℓ} = −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 m_{ℓ} = 2

**Example #10:** Determine which set(s) of quantum numbers is NOT allowed:

(a) n = 5, ℓ = 3, m_{ℓ}= −1, m_{s}= +^{1}⁄_{2}

(b) n = 1, ℓ = 0, m_{ℓ}= 0, m_{s}= −^{1}⁄_{2}

(c) n = 2, ℓ = 2, m_{ℓ}= 2, m_{s}= +^{1}⁄_{2}

(d) n = 4, ℓ = 1, m_{ℓ}= 0, m_{s}= −^{1}⁄_{2}

(e) n = 6, ℓ = 4, m_{ℓ}= −3, m_{s}= +^{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, m_{ℓ}= −3, m_{s}= +^{1}⁄_{2}

(b) n = 3, ℓ = −1, m_{ℓ}= 0, m_{s}= +^{1}⁄_{2}

(c) n = 3, ℓ = 2, m_{ℓ}= −3, m_{s}= −^{1}⁄_{2}

(d) n = 5, ℓ = 3, m_{ℓ}= −2, m_{s}= −1

**Example #12:** An electron in an atom is in the n = 3 and ℓ = 1 quantum state. Identify the possible values of m_{ℓ} 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, m_{ℓ}= 0, m_{s}= +^{1}⁄_{2}

(b) n = 1, ℓ = 3, m_{ℓ}= 3, m_{s}= +^{1}⁄_{2}

(c) n = 3, ℓ = 2, m_{ℓ}= −2, m_{s}= −^{1}⁄_{2}

(d) n = 2, ℓ = 1, m_{ℓ}= −1, m_{s}= +^{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, m_{ℓ}= −1, m_{s}= −^{1}⁄_{2}

(b) n = 6, ℓ = 2, m_{ℓ}= 1, m_{s}= +^{1}⁄_{2}

(c) n = 1, ℓ = −1, m_{ℓ}= −2, m_{s}= +^{1}⁄_{2}

(d) n = 6, ℓ = 0, m_{ℓ}= 1, m_{s}= +^{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 m_{ℓ} for an electron in a 5d orbital? Write the n, ℓ, m_{ℓ} for each of the orbitals in the 5d subshell.

**Problem #1:** Which of the following is a possible set of quantum numbers that describes an electron?

(a) n = 3, ℓ = 2, m _{ℓ}= −3, m_{s}= −^{1}⁄_{2}(d) n = 3, ℓ = 1, m _{ℓ}= −1, m_{s}= +^{1}⁄_{2}(b) n = 0, ℓ = 0, m

_{ℓ}= 0, m_{s}= +^{1}⁄_{2}(e) n = 4, ℓ = −3, m _{ℓ}= −1, m_{s}= +1(c) n = 4, ℓ = 2, m

_{ℓ}= −1, m_{s}= 0(f) none

**Problem #2:** Each electron orbital is characterized by 3 quantum numbers: n, ℓ, and m_{ℓ}.

n is known as the ____ quantum number.

ℓ is known as the ____ quantum number.

m_{ℓ}is known as the ____ quantum number.

**Problem #3:** Each electron orbital is characterized by 3 quantum numbers: n, ℓ, and m_{ℓ}.

n specifies ___.

ℓ specifies ___.

m_{ℓ}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 m_{ℓ}?

(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, m_{ℓ}= 1, m_{s}= +^{1}⁄_{2}

(b) n = 3, ℓ = 2, m_{ℓ}= 2, m_{s}= +^{1}⁄_{2}

(c) n = 4, ℓ = 1, m_{ℓ}= −1, m_{s}= −^{1}⁄_{2}

(d) n = 4, ℓ = 3, m_{ℓ}= 3, m_{s}= −^{1}⁄_{2}

(e) n = 5, ℓ = 0, m_{ℓ}= 0, m_{s}= +^{1}⁄_{2}

**Problem #8:** Which of the following set of quantum numbers (ordered n, ℓ, m_{ℓ}, m_{s}) 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, m_{ℓ}= +1

(c) n = 5, m_{s}= +^{1}⁄_{2}

(d) n = 3, ℓ = 2

(e) n = 1, ℓ = 0, m_{ℓ}= 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?

**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, m_{ℓ}= 0, m_{s}= +^{1}⁄_{2}

(b) n = 3, m_{ℓ}= +2, m_{s}= +^{1}⁄_{2}

(c) n = 3, m_{ℓ}= 0, m_{s}= +^{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, m_{ℓ}= 3, m_{s}= −^{1}⁄_{2}

(b) n = 4, ℓ = 3, m_{ℓ}= 4, m_{s}= −^{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, m_{ℓ}= −1

**Problem #14:** Which of the following combinations of quantum numbers are allowed?

(a) n = 1, ℓ = 1, m_{ℓ}= 0

(b) n = 3, ℓ = 0, m_{ℓ}= 0

(c) n = 1, ℓ = 0, m_{ℓ}= −1

(d) n = 2, ℓ = 1, m_{ℓ}= 2

**Problem #15:** The following sets of quantum numbers, listed in the order n, ℓ, m_{ℓ}, and m_{s} were written for the last electron added to an atom. Identify which sets are valid:

n ℓ m _{ℓ}m _{s}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. m_{ℓ}

IV. m_{s}

(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, m_{ℓ}, m_{s}) 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, m_{ℓ}= 0, m_{s}= +^{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, m_{ℓ}, m_{s}) 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, m_{ℓ}= 3

(b) n = 2, ℓ =1, m_{ℓ}= −1

(c) n = 3, ℓ = 1, m_{ℓ}= −1

(d) n = 4, ℓ = 3, m_{ℓ}= 3

**Problem #24:** Identify which sets of quantum numbers are valid for an electron. Each set is ordered (n, ℓ, m_{ℓ}, m_{s})

(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, m_{ℓ}, m_{s}) 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, m_{ℓ}= +1

(c) n = 4

(d) n = 4, ℓ = 3, m_{ℓ}= +1, m_{s}= +^{1}⁄_{2}

(e) n = 2, ℓ = 2, m_{ℓ}= 2, m_{s}= −^{1}⁄_{2}

**Bonus Problem #1:** All of the following sets of quantum numbers (ordered n, ℓ, m_{ℓ}, m_{s}) 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, m_{ℓ}= +2

(b) n = 2, ℓ = 1, m_{ℓ}= −2

(c) n = 1, ℓ = 1, m_{ℓ}= 0