Question

List all the possible subshells and orbitals associated with the principal quantum number \(n\), if \(n=5\).

Short answer

For \(n=5\), the subshells are \(5s\), \(5p\), \(5d\), \(5f\), and \(5g\) with 1, 3, 5, 7, and 9 orbitals, respectively.

Step-by-step solution

Understanding Principal Quantum Number

The principal quantum number, denoted as \(n\), determines the main energy level of an electron in an atom. It is a positive integer. Here, \(n = 5\). This indicates that the electron can occupy the fifth energy level.

Identifying Subshells Using Angular Momentum Quantum Number

Each principal quantum number \(n\) has subshells determined by the angular momentum quantum number \(l\) which ranges from 0 to \(n-1\). For \(n = 5\), the possible \(l\) values are 0, 1, 2, 3, and 4. These correspond to the subshells: \(5s\) for \(l=0\), \(5p\) for \(l=1\), \(5d\) for \(l=2\), \(5f\) for \(l=3\), and \(5g\) for \(l=4\).

Calculating Orbital Count for Each Subshell

Each subshell can contain a specific number of orbitals, determined by the magnetic quantum number \(m_l\) which ranges from \(-l\) to \(+l\). For \(5s\) (\(l=0\)), there's 1 orbital. For \(5p\) (\(l=1\)), there are 3 orbitals. For \(5d\) (\(l=2\)), there are 5 orbitals. For \(5f\) (\(l=3\)), there are 7 orbitals. For \(5g\) (\(l=4\)), there are 9 orbitals.

Compiling the Subshells and Orbitals

The subshells for \(n=5\) are: \(5s\), \(5p\), \(5d\), \(5f\), and \(5g\). The corresponding number of orbitals are: 1 for \(5s\), 3 for \(5p\), 5 for \(5d\), 7 for \(5f\), and 9 for \(5g\). Thus, each principal quantum number \(n\) level has a set of subshells, each containing a unique number of orbitals.

Key concepts

Principal Quantum Number

The principal quantum number, represented as \(n\), is one of the key numbers essential to understanding the structure of an atom. It reveals the main energy level or shell in which an electron resides. Think of it as the floor level within a building, where electrons orbit the nucleus. Each level grows progressively further from the nucleus, increasing the electron's potential energy. For \(n = 5\), you're essentially dealing with the fifth floor of this atomic building. The principal quantum number is always a positive integer: 1, 2, 3, and so forth. As \(n\) increases, the electron is capable of being further from the nucleus, giving it more energy and a higher level capacity.

Angular Momentum Quantum Number

The angular momentum quantum number, designated as \(l\), is used to identify the shape of an electron’s orbit within a given principal quantum level. This number can have any integer value from 0 to \(n-1\).This concept is akin to the design or type of the room on a floor level. For a principal quantum number \(n = 5\):

• When \(l = 0\), it indicates a spherical or "s" subshell (\(5s\)).
• When \(l = 1\), it signifies a dumbbell-shaped "p" subshell (\(5p\)).
• When \(l = 2\), it's a "d" subshell (\(5d\)) with more complex shapes than "p".
• When \(l = 3\), it forms an "f" subshell (\(5f\)) with even more intricate shapes.
• When \(l = 4\), it denotes a "g" subshell (\(5g\)), rare in naturally occurring elements.

The angular momentum quantum number thus controls the shape of the path an electron moves through within its main energy level.

Magnetic Quantum Number

The magnetic quantum number \(m_l\) describes the orientation of an electron's orbital within a subshell. This is like pinpointing the direction in which a specific room faces in a building. For each quantum number \(l\), \(m_l\) spans from \(-l\) to \(+l\), resulting in a range of possible orientations:

• The \(5s\) subshell (\(l=0\)) includes only one orientation: \(m_l = 0\).
• The \(5p\) subshell (\(l=1\)) consists of three orientations: \(-1, 0, +1\).
• The \(5d\) subshell (\(l=2\)) has five orientations: \(-2, -1, 0, +1, +2\).
• The \(5f\) subshell (\(l=3\)) features seven orientations: \(-3, -2, -1, 0, +1, +2, +3\).
• The \(5g\) subshell (\(l=4\)) presents nine orientations: \(-4, -3, -2, -1, 0, +1, +2, +3, +4\).

The number of orbitals in a subshell corresponds with the possible values of \(m_l\). This explains why different subshells can hold a varying number of orbitals.

Subshells and Orbitals

Subshells are essentially compartments within a principal energy level designated by \(n\). Each subshell is determined by the angular momentum quantum number \(l\), and its specific orbitals are oriented as determined by the magnetic quantum number \(m_l\).For \(n = 5\):

• The subshell \(5s\) consists of 1 orbital.
• The subshell \(5p\) includes 3 orbitals.
• The subshell \(5d\) comprises 5 orbitals.
• The subshell \(5f\) contains 7 orbitals.
• The subshell \(5g\) is made up of 9 orbitals.

Each orbital within a subshell can hold a maximum of two electrons, following Pauli's exclusion principle. Thus, these orbitals are like individual beds within a room, each accommodating a specific number of electrons. The variation in room size and shape (subshells and orbitals) reflects the energy and spatial orientation of electrons within an atom.