Quantum Numbers: Angular Momentum Quantum Number: Videos & Practice Problems

The angular momentum quantum number (l), a crucial aspect of atomic structure, defines the shape of an atom's orbitals and is always less than the principal quantum number (n), following the rule:

$l=0ton-1$

Each subshell, denoted by the letters s, p, d, and f, corresponds to l values of 0, 1, 2, and 3, respectively. The s subshell is spherical, p subshells have dumbbell-shaped orbitals, d subshells have clover-shaped orbitals with one unique dumbbell with a ring, and f subshells have complex shapes like pacifiers and double clovers. Understanding these subshell shapes is essential for grasping electron configuration and orbital geometry within an atom.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as l, plays a crucial role in defining the shape of an atom's electron cloud. This quantum number is intrinsically linked to the principal quantum number, n, which indicates the energy level of an electron in an atom. The relationship between these two quantum numbers is such that for a given principal quantum number n, the angular momentum quantum number l can take on values ranging from 0 to n - 1. This means that l is always less than n, establishing a clear limitation in their relationship.

Understanding this relationship is essential for predicting the possible shapes of atomic orbitals, which are influenced by the value of l. For instance, when l equals 0, the orbital is spherical (s orbital), while higher values of l correspond to more complex shapes, such as p, d, and f orbitals. This foundational knowledge is vital for grasping the behavior of electrons in various elements and their corresponding chemical properties.

Angular Momentum Quantum Number Example

In quantum mechanics, the principal quantum number, denoted as n, indicates the energy level of an electron in an atom. The azimuthal quantum number, l, describes the shape of the orbital and can take on integer values from 0 to n - 1. When n is equal to 4, the possible values for l can be calculated as follows:

Since l ranges from 0 to n - 1, we substitute n with 4:

l = 0, 1, 2, 3

This means that the possible values for l when n equals 4 are 0, 1, 2, and 3. Therefore, the correct answer is that l can take on the values 0, 1, 2, or 3.

Angular Momentum Quantum Number and Subshell

The quantum number l, known as the angular momentum quantum number, plays a crucial role in defining the subshells of an atom. Each shell can be divided into subshells, also referred to as sublevels, which are designated by specific letters based on the value of l. The relationship between l and the subshell letters is as follows:

• When l = 0, the subshell is designated as s.
• When l = 1, the subshell is designated as p.
• When l = 2, the subshell is designated as d.
• When l = 3, the subshell is designated as f.

This means that knowing the value of l allows you to identify the corresponding subshell letter, and vice versa. Understanding this relationship is essential for grasping the structure of electron configurations in atoms.

Quantum Numbers and Subshell Example

In quantum mechanics, the principal quantum number (n) indicates the energy level or shell of an electron in an atom. For an electron located in the 3rd principal level, the value of n is 3. This means the electron is in the third shell of the atom.

The azimuthal quantum number (l) defines the shape of the orbital and is associated with the sublevels within a principal energy level. The values of l correspond to specific sublevel designations: for s, l = 0; for p, l = 1; for d, l = 2; and for f, l = 3. Since the electron is found in a d sublevel, the value of l is 2.

Therefore, for an electron in the 3rd principal level and d sublevel, the possible values are:

n = 3

l = 2

This understanding of quantum numbers is essential for predicting the electron configuration and behavior of elements in the periodic table.

Provide all the possible values of l for a 2 energy level.

• A. 0
• B. 0, 1
• C. 0, 1, 2
• D. 0, 1, 2, 3
• E. 1

Answer: (B)

How many sublevels are contained in the third shell (n = 3) for a given atom?

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Answer: (C)

Angular Momentum Quantum Number and Orbital Shape

The angular momentum quantum number, denoted as l, provides crucial insights into the shapes of the orbitals that electrons occupy within an atom. Each value of l corresponds to a specific subshell and its characteristic shape:

When l = 0, the subshell is designated as s, which has a spherical shape. This means that the electron density is evenly distributed around the nucleus, resembling a sphere.

For l = 1, the subshell is p, which features three distinct shapes often described as dumbbells or ellipses. These shapes are oriented in three-dimensional space, allowing for a more complex distribution of electron density.

When l = 2, the subshell is referred to as d. This subshell has five unique shapes, with the first four resembling four-leaf clovers, while the fifth shape can be visualized as a dumbbell encircled by a ring.

Finally, for l = 3, the subshell is known as f, which contains seven shapes. The first three shapes can be likened to pairs of pacifiers, while the remaining four resemble double four-leaf clovers.

Understanding these shapes is essential, as they influence the arrangement of electrons in an atom and ultimately affect the chemical properties and reactivity of elements. The angular momentum quantum number thus plays a vital role in quantum mechanics and atomic theory.

Quantum Numbers and Orbital Shape Example

In quantum mechanics, understanding atomic orbitals and their corresponding quantum numbers is crucial for describing the behavior of electrons in atoms. Each electron in an atom is characterized by a set of four quantum numbers: n, l, m_l, and m_s. The principal quantum number n indicates the energy level and can take any positive integer value (1, 2, 3, ...). The azimuthal quantum number l defines the subshell and can take values from 0 to n - 1.

For the d subshell, the azimuthal quantum number l is equal to 2. This means that the possible values for n must be greater than 2, as l must always be less than n. Therefore, if l is 2, n must be at least 3. The relationship between n and l can be summarized as follows: l can take values from 0 up to n - 1, ensuring that l is always less than n.

When evaluating the options for quantum numbers, it is essential to check that the values adhere to these rules. For example, if an option states that n is equal to 2, it is invalid because l cannot equal n. Similarly, if n is 3, l can only be 0, 1, or 2, making it impossible for l to be 2 if n is less than 3.

In conclusion, the correct set of quantum numbers for a d subshell must have n equal to 3 or greater, with l equal to 2. Therefore, an example of valid quantum numbers could be n = 5, l = 2, which satisfies all the necessary conditions, making it the best answer in this context.

Which of the following orbitals possesses the most orbital shapes?

• A. 2p
• B. 7s
• C. 4d
• D. 5p
• E. 5f

Answer: (E)

Here’s what students ask on this topic:

Which characteristic is given by the angular momentum quantum number?

The angular momentum quantum number, often represented by the letter 'l', determines the shape of an electron's orbital within an atom. It is one of four quantum numbers used to describe the unique quantum state of an electron. The angular momentum quantum number can take on any integer value from 0 to $n-1$, where 'n' is the principal quantum number associated with the electron's energy level.

For a given energy level ($n$), 'l' defines the subshells or sublevels that are present, such as s (sharp), p (principal), d (diffuse), and f (fundamental), corresponding to 'l' values of 0, 1, 2, and 3, respectively. Each of these subshells has a distinct shape, which in turn affects the distribution of electron density around the nucleus. For instance, s orbitals are spherical, while p orbitals are dumbbell-shaped. The angular momentum quantum number is crucial for determining the allowed shapes for an electron's probability cloud within an atom and plays a key role in the chemical bonding and properties of elements.

How can I find the angular momentum quantum number?

The angular momentum quantum number, often represented by the letter 'l', is a quantum number that determines the shape of an electron's orbital and contributes to the angular momentum of an electron within an atom. It can be found by considering the principal quantum number 'n' of the electron's orbital. The angular momentum quantum number 'l' can take on any integer value from 0 to (n-1).

For example, if an electron is in the second energy level (n=2), the possible values of 'l' can be 0 or 1. These values correspond to different types of orbitals: 'l' = 0 is an s-orbital (spherical), 'l' = 1 is a p-orbital (dumbbell-shaped), and so on for higher values of 'l' (d-orbitals for 'l' = 2, f-orbitals for 'l' = 3, etc.).

To find the angular momentum quantum number, you need to know the principal quantum number of the electron's orbital and then list out the possible 'l' values within the range 0 to (n-1). Each of these values of 'l' will correspond to a different orbital shape within that energy level.

What is the angular momentum quantum number?

The angular momentum quantum number, often symbolized as l, is a quantum number that determines the shape of an electron's orbital within an atom. It is one of four quantum numbers used to describe the unique quantum state of an electron. The angular momentum quantum number can take on any integer value from 0 to n-1, where n is the principal quantum number associated with the electron's energy level.

For a given energy level (n), l defines the subshell or sublevel in which the electron resides, with each subshell corresponding to a different shape of the orbital. For example, if l = 0, the orbital is spherical and is called an s orbital. If l = 1, the orbital is dumbbell-shaped and is known as a p orbital. Higher values of l correspond to d and f orbitals, which have more complex shapes.

The angular momentum quantum number is crucial for determining the allowed orbits for electrons in atoms and thus plays a key role in the structure and behavior of atoms in chemical reactions and physical processes.