Quantum Numbers: Magnetic Quantum Number: Videos & Practice Problems

Quantum Numbers and Orbitals Example

In the context of quantum mechanics and atomic structure, the principal quantum number (n) indicates the energy level of an electron in an atom. For the 4th principal level, n is equal to 4. The sublevel designation provides further detail about the shape and type of orbitals present. In this case, the f sublevel is specified, which corresponds to a set of orbitals characterized by their complex shapes and the ability to hold a maximum of 14 electrons.

Thus, when identifying the orbitals in the 4th principal level and f sublevel, we refer to the 4f orbitals. The notation indicates that these orbitals are part of the fourth energy level and belong to the f sublevel, confirming that the correct identification is indeed the 4f set of orbitals.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_l \), describes the orientation of atomic orbitals and provides insight into the location of electrons within these orbitals. Each quantum number has specific limitations, and for \( m_l \), its values are determined by the angular momentum quantum number \( l \). The range of \( m_l \) extends from \(-l\) to \(+l\), including all integer values in between.

For instance, if \( l = 2 \), then \( m_l \) can take on the values of \(-2, -1, 0, +1, +2\). This relationship can be summarized as follows: when \( l = 0 \), \( m_l = 0 \); when \( l = 1 \), \( m_l \) ranges from \(-1\) to \(+1\); when \( l = 2\), it ranges from \(-2\) to \(+2\); and when \( l = 3\), it ranges from \(-3\) to \(+3\).

The subshell letters—s, p, d, and f—correspond to specific values of \( l \): \( l = 0 \) for s, \( l = 1 \) for p, \( l = 2 \) for d, and \( l = 3 \) for f. Knowing the subshell letter allows us to determine the associated \( l \) value, and consequently, the possible \( m_l \) values. For example, the s subshell has one spherical orbital (with \( m_l = 0 \)), while the p subshell has three orbitals corresponding to \( m_l = -1, 0, +1\).

In terms of electron capacity, each subshell can hold a maximum number of electrons: the s subshell can accommodate 2 electrons, the p subshell can hold 6, the d subshell can contain 10, and the f subshell can accommodate 14 electrons. This capacity is crucial for understanding electron configurations and will be further explored in relation to the spin quantum number.

Overall, the magnetic quantum number \( m_l \) is essential for determining the spatial orientation of orbitals and the distribution of electrons in an atom, which is foundational for understanding atomic structure and behavior.

Magnetic Quantum Example

In quantum mechanics, the magnetic quantum number, denoted as \( m_l \), is crucial for understanding the orientation of orbitals in an atom. The values of \( m_l \) are determined by the azimuthal quantum number \( l \), which corresponds to the type of subshell. For the f subshell, \( l \) is equal to 3. This means that the possible values for \( m_l \) range from \(-l\) to \(+l\), which translates to the values \(-3, -2, -1, 0, +1, +2, +3\) for the 7f set of orbitals.

Therefore, the valid magnetic quantum numbers for the 7f orbitals are \(-3, -2, -1, 0, +1, +2, +3\). Any value outside this range, such as \( +4 \), is not a valid magnetic quantum number for the f subshell. Thus, when identifying which number is not valid for the 7f orbitals, \( +4 \) is the correct answer, as it exceeds the allowable range for \( m_l \).

How many different values of m_l are possible for a 4d set of orbitals?

• A. 1
• B. 3
• C. 7
• D. 5
• E. 2

Answer: (D)

Select a correct set of values for an electron found within the designated 5d orbital.

• A. n = 5, l = 2, m_l = 0
• B. n = 5, l = 3, m_l = +1
• C. n = 5, l = 3, m_l = 0
• D. n = 5, l = 5, m_l = – 2
• E. n = 5, l = 2, m_l = +5

Answer: (A)

Which of the following statements is false?
a) A set of d orbitals contains 5 orbitals.
b) A set of 4s orbitals would have more energy than a set of 3p orbitals.
c) The second shell of an atom possesses d orbitals.
d) A set of f orbitals contains 3 orbitals.
e) The first energy level contains only s orbitals.

• A. a, b
• B. b, d
• C. c, d
• D. d, e
• E. b, e

Answer: (C)

Here's what students ask on this topic:

What is the magnetic quantum number and what does it represent in an atom?

The magnetic quantum number, denoted as $m_l$, represents the orientation of an orbital within a subshell in an atom. It arises from the angular momentum quantum number $l$ and indicates the specific orbital where electrons are likely to be found. The values of $m_l$ range from $-l$ to $+l$, including all integers in between. For example, if $l=\; 2$, then $m_l$ can be $-2,\; -1,\; 0,\; +1,\; +2$. Each value corresponds to a different orbital orientation within that subshell, helping to define the spatial distribution of electrons.

How are the magnetic quantum number values related to the subshell types (s, p, d, f)?

The magnetic quantum number values correspond directly to the number of orbitals in each subshell type. For the $s$ subshell, where $l=\; 0$, $m_l$ is 0, indicating one orbital. For the $p$ subshell ($l=\; 1$), $m_l$ ranges from -1 to +1, giving three orbitals. The $d$ subshell ($l=\; 2$) has $m_l$ values from -2 to +2, totaling five orbitals. Lastly, the $f$ subshell ($l=\; 3$) has seven orbitals with $m_l$ values from -3 to +3. These values define the orientation and number of orbitals available for electrons in each subshell.

Why does the magnetic quantum number range from -l to +l, and what does this range signify?

The magnetic quantum number $m_l$ ranges from $-l$ to $+l$ because it represents all possible orientations of an orbital within a given subshell. The angular momentum quantum number $l$ defines the shape of the orbital, and $m_l$ specifies its spatial orientation relative to an external magnetic field or coordinate axis. This range includes all integer values between $-l$ and $+l$, ensuring that every possible orbital orientation is accounted for. For example, if $l=\; 1$, the values $-1,\; 0,\; +1$ correspond to the three different p orbitals oriented along the x, y, and z axes.

How does the magnetic quantum number relate to the maximum number of electrons in each subshell?

The magnetic quantum number $m_l$ determines the number of orbitals in a subshell, and since each orbital can hold a maximum of two electrons (due to the spin quantum number), it indirectly sets the maximum electron capacity of that subshell. For example, the $s$ subshell has one orbital ($m_l=\; 0$), so it can hold 2 electrons. The $p$ subshell has three orbitals ($m_l=\; -1,\; 0,\; +1$), allowing 6 electrons. The $d$ subshell has five orbitals, holding up to 10 electrons, and the $f$ subshell has seven orbitals, holding up to 14 electrons. This relationship is fundamental for understanding electron configurations.

What is the significance of the magnetic quantum number in determining electron location within an atom?

The magnetic quantum number $m_l$ is significant because it specifies the exact orbital within a subshell where an electron is likely to be found. While the principal quantum number $n$ indicates the energy level and the angular momentum quantum number $l$ defines the shape of the orbital, $m_l$ determines the orbital's orientation in space. This helps in understanding the spatial distribution of electrons around the nucleus. For example, in the p subshell, the three orbitals (with $m_l$ values -1, 0, +1) correspond to orbitals oriented along the x, y, and z axes, respectively. This orientation is crucial for predicting chemical bonding and molecular geometry.