Problem 31 - Chemistry and Chemical Reactivity

Question

How many subshells occur in the electron shell with the principal quantum number \(n=4 ?\)

Step-by-Step Solution

Answer

There are 4 subshells in the electron shell with \(n=4\).

Step 1: Understand Principal Quantum Number

The principal quantum number, denoted as \(n\), determines the electron shell or energy level of an atom. It must be a positive integer (e.g., 1, 2, 3, ...). In this exercise, \(n = 4\) represents the fourth energy level.

Step 2: Determine Available Subshells

The number of subshells within a shell is equal to the value of the principal quantum number \(n\). This means if \(n = 4\), there are 4 subshells at this energy level.

Step 3: List Subshells by Angular Momentum Quantum Number

The angular momentum quantum number \(l\) ranges from 0 to \(n-1\). For \(n = 4\), the possible values of \(l\) are 0, 1, 2, and 3. These correspond to the subshells denoted as \(s\), \(p\), \(d\), and \(f\), respectively.

Step 4: Confirm Subshell Count

We have identified four subshells: 4s (\(l = 0\)), 4p (\(l = 1\)), 4d (\(l = 2\)), and 4f (\(l = 3\)). Thus, for \(n = 4\), there are indeed 4 subshells.

Key Concepts

Principal Quantum Number

The principal quantum number, often symbolized as \(n\), plays a crucial role in understanding the electronic structure of atoms. It determines the main electron shell or the energy level within an atom. The principal quantum number is always a positive integer, such as 1, 2, 3, and so on.
It serves as an indicator of the size and energy of the electron cloud in an atom. Specifically, as \(n\) increases, the electron cloud becomes larger, and the electrons within that shell have more energy. This is because electrons tend to be further away from the nucleus. For example, an electron in the \(n=4\) shell has more energy than an electron in the \(n=3\) shell. The principal quantum number is a fundamental concept that helps categorize an atom's electrons into different shells.
In practical terms:

• \(n=1\) represents the first shell (closest to the nucleus).
• \(n=2\) represents the second shell, and so forth.

Subshells

Subshells are subdivisions of electron shells based on the angular momentum quantum number. The maximum number of subshells that can occur within a particular electron shell is determined by the principal quantum number \(n\). Specifically, there are \(n\) subshells within an electron shell corresponding to \(n\).
Each subshell is associated with a distinct angular momentum quantum number \(l\) that specifies its shape and can be expressed in terms that we commonly encounter as \(s\), \(p\), \(d\), and \(f\). The range of values \(l\) can take is from 0 to \(n-1\). For example, when \(n=4\), the possible subshells are:

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

Totaling four subshells (4s, 4p, 4d, and 4f) for this energy level.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted typically as \(l\), is an integral quantum number that describes the subshells or orbitals of electron shells. It indicates the shape of the electron cloud within a particular subshell.
Its value must range from 0 up to \(n-1\), where \(n\) is the principal quantum number of the shell. The value of \(l\) not only correlates with the shape but also associates with the subshell designation as \(s, p, d,\) or \(f\). Here’s how the values correspond:

• \(l = 0\): Corresponds to the \(s\) subshell (spherical shape).
• \(l = 1\): Corresponds to the \(p\) subshell (dumbbell shape).
• \(l = 2\): Corresponds to the \(d\) subshell (cloverleaf shape).
• \(l = 3\): Corresponds to the \(f\) subshell (complex shape).

Understanding \(l\) is essential because it helps explain the different shapes and orientations of electron clouds, thereby influencing chemical bonding and molecular geometry significantly.