secondary (angular momentum) quantum number (l)

quantum number distinguishing the different shapes of orbitals; it is also a measure of the orbital angular momentum

mathematical function that describes the behavior of an electron in an atom (also called the wavefunction)

region of space with high electron density that is either four lobed or contains a dumbbell and torus shape; describes orbitals with l = 2.

orbitals that have the same energy

a measure of the probability of locating an electron in a particular region of space, it is equal to the squared absolute value of the wave function ψ

multilobed region of space with high electron density, describes orbitals with l = 3

Heisenberg uncertainty principle

rule stating that it is impossible to exactly determine both certain conjugate dynamical properties such as the momentum and the position of a particle at the same time. The uncertainty principle is a consequence of quantum particles exhibiting wave–particle duality

magnetic quantum number (m_l)

quantum number signifying the orientation of an atomic orbital around the nucleus

dumbbell-shaped region of space with high electron density, describes orbitals with l = 1

Pauli exclusion principle

specifies that no two electrons in an atom can have the same value for all four quantum numbers

principal quantum number (n)

quantum number specifying the shell an electron occupies in an atom

field of study that includes quantization of energy, wave-particle duality, and the Heisenberg uncertainty principle to describe matter

spherical region of space with high electron density, describes orbitals with l = 0

atomic orbitals with the same principal quantum number, n

spin quantum number (m_s)

number specifying the electron spin direction, either $+ \ \frac{1}{2}$ or $- \ \frac{1}{2}$

atomic orbitals with the same values of n and l

mathematical description of an atomic orbital that describes the shape of the orbital; it can be used to calculate the probability of finding the electron at any given location in the orbital, as well as dynamical variables such as the energy and the angular momentum

6.3 Development of Quantum Theory: practice

Development of Quantum Theory: practice problems

8 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Key terms · 17

secondary (angular momentum) quantum number (l): quantum number distinguishing the different shapes of orbitals; it is also a measure of the orbital angular momentum
atomic orbital: mathematical function that describes the behavior of an electron in an atom (also called the wavefunction)
d orbital: region of space with high electron density that is either four lobed or contains a dumbbell and torus shape; describes orbitals with l = 2.
degenerate orbitals: orbitals that have the same energy
electron density: a measure of the probability of locating an electron in a particular region of space, it is equal to the squared absolute value of the wave function ψ
f orbital: multilobed region of space with high electron density, describes orbitals with l = 3
Heisenberg uncertainty principle: rule stating that it is impossible to exactly determine both certain conjugate dynamical properties such as the momentum and the position of a particle at the same time. The uncertainty principle is a consequence of quantum particles exhibiting wave–particle duality
magnetic quantum number (m_l): quantum number signifying the orientation of an atomic orbital around the nucleus
p orbital: dumbbell-shaped region of space with high electron density, describes orbitals with l = 1
Pauli exclusion principle: specifies that no two electrons in an atom can have the same value for all four quantum numbers
principal quantum number (n): quantum number specifying the shell an electron occupies in an atom
quantum mechanics: field of study that includes quantization of energy, wave-particle duality, and the Heisenberg uncertainty principle to describe matter
s orbital: spherical region of space with high electron density, describes orbitals with l = 0
shell: atomic orbitals with the same principal quantum number, n
spin quantum number (m_s): number specifying the electron spin direction, either $+ \ \frac{1}{2}$ or $- \ \frac{1}{2}$
subshell: atomic orbitals with the same values of n and l
wavefunction (ψ): mathematical description of an atomic orbital that describes the shape of the orbital; it can be used to calculate the probability of finding the electron at any given location in the orbital, as well as dynamical variables such as the energy and the angular momentum

Learn 6.3 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 6.3 in the textbook · This section on openstax.org

Development of Quantum Theory: practice problems

8 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Key terms · 17

secondary (angular momentum) quantum number (l): quantum number distinguishing the different shapes of orbitals; it is also a measure of the orbital angular momentum
atomic orbital: mathematical function that describes the behavior of an electron in an atom (also called the wavefunction)
d orbital: region of space with high electron density that is either four lobed or contains a dumbbell and torus shape; describes orbitals with l = 2.
degenerate orbitals: orbitals that have the same energy
electron density: a measure of the probability of locating an electron in a particular region of space, it is equal to the squared absolute value of the wave function ψ
f orbital: multilobed region of space with high electron density, describes orbitals with l = 3
Heisenberg uncertainty principle: rule stating that it is impossible to exactly determine both certain conjugate dynamical properties such as the momentum and the position of a particle at the same time. The uncertainty principle is a consequence of quantum particles exhibiting wave–particle duality
magnetic quantum number (m_l): quantum number signifying the orientation of an atomic orbital around the nucleus
p orbital: dumbbell-shaped region of space with high electron density, describes orbitals with l = 1
Pauli exclusion principle: specifies that no two electrons in an atom can have the same value for all four quantum numbers
principal quantum number (n): quantum number specifying the shell an electron occupies in an atom
quantum mechanics: field of study that includes quantization of energy, wave-particle duality, and the Heisenberg uncertainty principle to describe matter
s orbital: spherical region of space with high electron density, describes orbitals with l = 0
shell: atomic orbitals with the same principal quantum number, n
spin quantum number (m_s): number specifying the electron spin direction, either $+ \ \frac{1}{2}$ or $- \ \frac{1}{2}$
subshell: atomic orbitals with the same values of n and l
wavefunction (ψ): mathematical description of an atomic orbital that describes the shape of the orbital; it can be used to calculate the probability of finding the electron at any given location in the orbital, as well as dynamical variables such as the energy and the angular momentum

Learn 6.3 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 6.3 in the textbook · This section on openstax.org

Consider the orbitals shown here in outline.

This figure contains three diagrams. In x, a circle is drawn with a dot at the center. In y, two nearly ellipsoid shapes are oriented horizontally with a dot between them. In z, four shapes like those in y are oriented in an x shape with a dot at the center.

(a) What is the maximum number of electrons contained in an orbital of type (x)? Of type (y)? Of type (z)?

(b) How many orbitals of type (x) are found in a shell with n = 2? How many of type (y)? How many of type (z)?

(c) 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.

(d) What is the smallest possible n value for an orbital of type (x)? Of type (y)? Of type (z)?

(e) What are the possible l and m $_{l}$ values for an orbital of type (x)? Of type (y)? Of type (z)?