Question

How many subshells occur in the electron shell with the principal quantum number \(n=5 ?\)

Short answer

There are 5 subshells in the electron shell with \(n=5\).

Step-by-step solution

Understanding the Principal Quantum Number

The principal quantum number, denoted by \(n\), determines the energy level of an electron in an atom. Each energy level (or shell) can contain one or more subshells. The value of \(n=5\) indicates that we are working within the fifth energy level.

Determining Subshell Types

In any given shell, the number of subshells is equal to the value of \(n\). Therefore, if \(n=5\), the shell can have 5 subshells. These subshells correspond to the azimuthal quantum number \(l\) values from 0 to (\(n-1\)).

Listing Possible Subshells

Enumerate the values of \(l\) for \(n=5\). The possible values are \(l=0,1,2,3,4\). These correspond to the subshells s, p, d, f, and g, respectively.

Key concepts

Principal Quantum Number

The principal quantum number, commonly represented by the letter \( n \), is a fundamental concept in quantum mechanics used to describe the unique quantum state of an electron in an atom. This number indicates the primary energy level—or shell—of an electron in an atom. The principal quantum number can take any positive integer value, such as 1, 2, 3, and so on. As the value of \( n \) increases, the energy level, as well as the average distance of the electron from the nucleus, increases. Typically, these energy levels are depicted as concentric circles around the nucleus.

With each increment in \( n \), a new energy level is introduced. These energy levels can contain multiple electron subshells.
For instance:

• \( n = 1 \): Contains one subshell (1s)
• \( n = 2 \): Contains two subshells (2s and 2p)
• \( n = 3 \): Contains three subshells (3s, 3p, and 3d)

This structure is vital in understanding the arrangement of electrons in atoms, influencing chemical behavior and bonding.

Electron Subshells

Electron subshells are subdivisions within a main energy level and are specified by both the principal and azimuthal quantum numbers. For a given principal quantum number \( n \), there are \( n \) possible subshells. Each subshell is defined by its azimuthal quantum number, \( l \), ranging from 0 to \( n-1 \).

The subshell types, based on \( l \) values, are named as follows:

• \( l = 0 \): s subshell
• \( l = 1 \): p subshell
• \( l = 2 \): d subshell
• \( l = 3 \): f subshell
• \( l = 4 \): g subshell

The number of orbitals available in each subshell type can be calculated using the formula \( 2l+1 \), meaning each subshell can accommodate a unique number of electrons. The understanding of these subshells helps explain why elements display different chemical characteristics based on their electronic configuration.

Energy Levels

Energy levels in an atom are defined by the principal quantum number, \( n \), and can be visualized as the different floors of a building where electrons can reside. As \( n \) increases, the energy of the electron and its distance from the nucleus increase, resulting in electrons occupying higher energy shells. The energy levels are integral in determining the permissible energy and position states for electrons.

Furthermore, within each main energy level, subshells appear, which can hold different numbers of electrons. Generally, the shell closest to the nucleus has the lowest energy. As electrons fill higher energy levels:

• More complexity in the atom's structure is seen.
• New chemical properties and behaviors emerge.

A key takeaway is that the energy levels and structures dictate the electron configuration of elements, which is essential in chemistry for predicting how atoms will interact and bond with each other.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \( l \), is directly tied to the shape and number of subshells within a given energy level. It ranges from 0 to \( n-1 \) and determines distinct subshells in an electron shell.

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

• \( l = 0 \) represents the s subshell.
• \( l = 1 \) denotes the p subshell.
• \( l = 2 \) indicates the d subshell.
• \( l = 3 \) illustrates the f subshell.
• \( l = 4 \) symbolizes the g subshell.

The azimuthal quantum number not only determines the shape but also impacts the energy of the orbitals within the same main energy level. Therefore, understanding \( l \) values is vital for predicting the spatial distribution of electrons around an atom, influencing its magnetic and spectral properties.