# 1.4.1: Atomic Orbitals and Quantum Numbers (Problems)

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PROBLEM \(\PageIndex{1}\)

What are the allowed values for each of the four quantum numbers: *n*, *l*, *m _{l}*, and

*m*?

_{s}**Answer**-
n: non-zero integer

l: 0 to n-1

*m*_{l}_{ }: -l to l*m*\(\dfrac{1}{2}\) or \(\dfrac{-1}{2}\)_{s }:

PROBLEM \(\PageIndex{2}\)

Describe the properties of an electron associated with each of the following four quantum numbers: *n*, *l*, *m _{l}*, and

*m*.

_{s}**Answer**-
*n*determines the general range for the value of energy and the probable distances that the electron can be from the nucleus.*l*determines the shape of the orbital.*m*determines the orientation of the orbitals of the same_{l }*l*value with respect to one another.*m*determines the spin of an electron_{s}

PROBLEM \(\PageIndex{3}\)

Identify the subshell in which electrons with the following quantum numbers are found:

*n*= 2,*l*= 1*n*= 4,*l*= 2*n*= 6,*l*= 0

**Answer a**-
2p

**Answer b**-
4d

**Answer c**-
6s

PROBLEM \(\PageIndex{4}\)

Identify the subshell in which electrons with the following quantum numbers are found:

*n*= 3,*l*= 2*n*= 1,*l*= 0*n*= 4,*l*= 3

**Answer a**-
3d

**Answer b**-
1s

**Answer c**-
4f

PROBLEM \(\PageIndex{5}\)

Consider the orbitals shown here in outline.

- What is the maximum number of electrons contained in an orbital of type (x)? Of type (y)? Of type (z)?
- How many orbitals of type (x) are found in a shell with
*n*= 2? How many of type (y)? How many of type (z)? - Write a set of quantum numbers for an electron in an orbital of type (x) in a shell with
*n*= 4. Of an orbital of type (y) in a shell with*n*= 2. Of an orbital of type (z) in a shell with*n*= 3. - What is the smallest possible
*n*value for an orbital of type (x)? Of type (y)? Of type (z)? - What are the possible
*l*and*m*values for an orbital of type (x)? Of type (y)? Of type (z)?_{l}

**Answer a**-
x. 2

y. 2

z. 2

**Answer b**-
x. 1

y. 3

z. 0

**Answer c**-
x. 4 0 0 \(\dfrac{1}{2}\)

y. 2 1 0 \(\dfrac{1}{2}\)

z. 3 2 0 \(\dfrac{1}{2}\)

**Answer d**-
x. 1

y. 2

z. 3

**Answer e**-
x.

*l*= 0,*m*= 0_{l}y.

*l*= 1,*m*–1, 0, or_{l}=*+*1z.

*l*= 2,*m*= –2, –1, 0, +1, +2_{l}

PROBLEM \(\PageIndex{6}\)

How many electrons could be held in the second shell of an atom if the spin quantum number *m _{s}* could have three values instead of just two? (Hint: Consider the Pauli exclusion principle.)

**Answer**-
12

PROBLEM \(\PageIndex{7}\)

Write a set of quantum numbers for each of the electrons with an *n* of 4 in a Se atom.

**Answer**-
*n**l**m*_{l}*s*4 0 0 \(+\dfrac{1}{2}\) 4 0 0 \(−\dfrac{1}{2}\) 4 1 −1 \(+\dfrac{1}{2}\) 4 1 0 \(+\dfrac{1}{2}\) 4 1 +1 \(+\dfrac{1}{2}\) 4 1 −1 \(−\dfrac{1}{2}\)

PROBLEM \(\PageIndex{8}\)

Answer the following questions:

- Without using quantum numbers, describe the differences between the shells, subshells, and orbitals of an atom.
- How do the quantum numbers of the shells, subshells, and orbitals of an atom differ?

**Answer a**-
shell: set of orbitals in the same energy level

subshell: set of orbitals in the same energy level and same shape (s, p, d, or f)

orbital: can hold up to 2 electrons

**Answer b**-
shell: set of orbitals with same

*n*subshell: set of orbitals in an atom with the same values of

and*n**l*orbital: shape defined by

*l*quantum number

PROBLEM \(\PageIndex{9}\)

Sketch the boundary surface of a p_{z} and a *p _{y}* orbital. Be sure to show and label the axes.

**Answer**

PROBLEM \(\PageIndex{10}\)

Sketch the *p _{x}* and

*s*orbitals. Be sure to show and label the coordinates.

**Answer**

## Contributors

Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/85abf193-2bd...a7ac8df6@9.110).

- Adelaide Clark, Oregon Institute of Technology

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