Question

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

Short answer

There are 4 subshells: 4s, 4p, 4d, and 4f.

Step-by-step solution

Understanding Principal Quantum Number

The principal quantum number, often represented as \(n\), indicates the energy level of an electron in an atom. For \(n=4\), it means we are considering the fourth electron shell.

Identifying Possible Subshells

The number of subshells in a given electron shell is equal to the value of the principal quantum number \(n\). Hence, for \(n=4\), there are 4 subshells possible. These are denoted by the azimuthal (angular momentum) quantum number \(l\) that can take on values from \(0\) to \(n-1\).

Listing Subshells for \(n=4\)

For \(n=4\), the possible subshells are:- When \(l=0\), the subshell is "s".- When \(l=1\), the subshell is "p".- When \(l=2\), the subshell is "d".- When \(l=3\), the subshell is "f".Thus, the subshells available are 4s, 4p, 4d, and 4f.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a key concept in quantum mechanics. It is like a label that tells us about the main energy level or shell where an electron resides within an atom. Think of it as the address number of a building, indicating which floor you need to go to. In this analogy, 'floor' represents the energy level.
For electrons, the value of \( n \) not only dictates their energy but also tells us about their average distance from the nucleus. A higher \( n \) value means the electron is found further from the nucleus and possesses more energy.
In the exercise given, when \( n=4 \), it implies we are considering the fourth shell. This principal quantum number is significant because it helps define the possible subshells in that energy level. Remember, the higher the principal quantum number, the more subshells and various electron arrangements an atom can hold.

Subshells

Subshells are subdivisions of the principal quantum number and help define the shape and energy of an electron's orbital. Each shell has multiple subshells, and the number of these subshells in any given shell is equal to the principal quantum number \( n \).
For instance, with \( n=4 \), there are four possible subshells.
The subshells are characterized by the azimuthal quantum number \( l \) and are labeled by different letters:

• When \( l=0 \), it's called the "s" subshell.
• When \( l=1 \), it's the "p" subshell.
• When \( l=2 \), it's the "d" subshell.
• When \( l=3 \), we have the "f" subshell.

Therefore, for the fourth shell \( n=4 \), we have the subshells 4s, 4p, 4d, and 4f. These subshells hold specific numbers of electrons, and knowing them helps predict the chemical and physical behavior of elements.

Azimuthal Quantum Number

The azimuthal quantum number, or the angular momentum quantum number, is denoted by \( l \). This quantum number describes the shape of an electron's orbital and consequently influences the subshells within a principal energy level.
It defines the possible shapes (spherical, dumbbell, etc.) that an electron's orbit can take, giving us further insight into the nature of chemical bonds and molecular geometry.
The azimuthal quantum number \( l \) can have integer values ranging from 0 to \( n-1 \) for any principal quantum number \( n \). 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.

Thus, for \( n=4 \), the azimuthal quantum numbers can range from 0 to 3, giving us four subshells with their distinct orbitals. The combination of the principal and azimuthal quantum numbers determines the complexity and variety of the chemical behavior of atoms.