Question

List all the possible subshells and orbitals associated with the principal quantum number \(n\), if \(n=4\).

Short answer

For \(n=4\), the subshells are 4s, 4p, 4d, and 4f, with 1, 3, 5, and 7 orbitals respectively.

Step-by-step solution

Understanding the Principal Quantum Number

The principal quantum number \(n\) determines the energy level or shell of an electron in an atom. When \(n=4\), the energy level is the fourth shell of the atom.

Identifying Possible Subshells with n=4

For a given principal quantum number \(n\), the azimuthal quantum number \(l\) can take values from 0 to \(n-1\). Each value of \(l\) corresponds to a different type of subshell: - \(l = 0\) corresponds to the \(s\) subshell. - \(l = 1\) corresponds to the \(p\) subshell. - \(l = 2\) corresponds to the \(d\) subshell. - \(l = 3\) corresponds to the \(f\) subshell.Therefore, for \(n = 4\), the possible subshells are: 4s, 4p, 4d, and 4f.

Determining the Number of Orbitals in Each Subshell

Each subshell can have a certain number of orbitals, determined by the magnetic quantum number \(m_l\). The values of \(m_l\) range from \(-l\) to \(+l\), including zero:- The 4s subshell (\(l=0\)) has 1 orbital (\(m_l = 0\)).- The 4p subshell (\(l=1\)) has 3 orbitals (\(m_l = -1, 0, +1\)).- The 4d subshell (\(l=2\)) has 5 orbitals (\(m_l = -2, -1, 0, +1, +2\)).- The 4f subshell (\(l=3\)) has 7 orbitals (\(m_l = -3, -2, -1, 0, +1, +2, +3\)).

Key concepts

principal quantum number

The principal quantum number, symbolized as \(n\), is a fundamental concept in quantum mechanics. It helps determine many characteristics of electrons in an atom. At its core, \(n\) defines the main energy level or shell where an electron is located, which is critical because it relates directly to the electron's energy. As the value of \(n\) increases, the electron is located further away from the nucleus, indicating higher energy levels.

For example, when \(n = 1\), the electron is in the closest shell to the nucleus, known as the first energy level. For \(n = 4\), which is our specific interest here, the electrons reside in the fourth energy level. This defines an extended space around the atom due to increased energy and distance from the nucleus. By understanding the value of the principal quantum number, we can systematically examine what other types of subshells and orbitals come into play.

azimuthal quantum number

The azimuthal quantum number, denoted as \(l\), plays a pivotal role in deeper characterizing the electron's behavior within a principal energy level. This number signifies the shape of the electron's orbital, which essentially gives the subshell type. The azimuthal quantum number can vary from 0 up to \(n-1\), providing distinct subshells based on its specific value.

Each \(l\) corresponds to a particular type of subshell:

• \(l = 0\) represents the \(s\) subshell.
• \(l = 1\) stands for the \(p\) subshell.
• \(l = 2\) designates the \(d\) subshell.
• \(l = 3\) correlates with the \(f\) subshell.

Thus, for \(n = 4\), the possible values of \(l\) range from 0 to 3, leading to the presence of four subshells: 4s, 4p, 4d, and 4f. Understanding \(l\) is essential, as it provides insights into the geometry and complexity of the orbitals available at any given energy level.

magnetic quantum number

The magnetic quantum number, which is indicated by \(m_l\), further splits the subshells into their component orbitals. This quantum number defines the orientation of the orbital within a given subshell. The range of \(m_l\) values spans from \(-l\) to \(+l\).

Each specific subshell type defined by \(l\) will have a different number of orbitals associated with it:

• For \(l = 0\), the 4s subshell has only 1 orbital, corresponding to \(m_l = 0\).
• For \(l = 1\), the 4p subshell includes 3 orbitals, with \(m_l = -1, 0, +1\).
• For \(l = 2\), the 4d subshell contains 5 orbitals, ranging from \(m_l = -2\) to \(+2\).
• For \(l = 3\), the 4f subshell holds 7 orbitals, spanning \(m_l = -3\) to \(+3\).

The magnetic quantum number brings the overall picture of electron configuration into focus, helping us understand how these electrons distribute in three-dimensional space around the nucleus.

subshells and orbitals

Subshells and orbitals form the core of an atom's electron configuration. A subshell signifies a group of orbitals within a principal energy level and is characterized by specific \(l\) values. Each type of subshell hosts a different number of orbitals, which influences how electrons are arranged in an atom. In atomic structure, conducting electrons fill subshells in a particular order dictated by their energy requirements.

The primary subshells for \(n = 4\) include:

• 4s, with 1 orbital.
• 4p, having 3 orbitals.
• 4d, encompassing 5 orbitals.
• 4f, including the maximum 7 orbitals.

In each orbital, electrons pair up or remain unpaired based on the atom's electron configuration and their quantum mechanical properties. By understanding subshells and orbitals, we can predict electron arrangements and chemical properties of elements.