Question

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

Short answer

There are 4 subshells in the electron shell with \(n=4\): \(4s\), \(4p\), \(4d\), and \(4f\).

Step-by-step solution

Understanding Principal Quantum Number

The principal quantum number, denoted as \(n\), defines the main energy level or shell of an electron in an atom. It determines the number of subshells within that shell. Each shell with a principal quantum number \(n\) has \(n\) subshells.

Identifying Number of Subshells for n=4

Given that \(n=4\), the number of subshells in this shell is equal to the value of \(n\). Therefore, the shell with \(n=4\) will contain 4 subshells. These subshells can be designated as \(s\), \(p\), \(d\), and \(f\).

Confirming Subshells

For \(n=4\), the subshells are labeled as follows: the \(4s\), \(4p\), \(4d\), and \(4f\) subshells. This confirms the presence of 4 subshells, each corresponding to a different orbital shape and capacity for electrons.

Key concepts

Principal Quantum Number

The principal quantum number, often represented by the symbol \( n \), plays a crucial role in understanding the structure of atomic orbitals. It indicates the main energy level or shell that an electron occupies within an atom.
The principal quantum number is always a positive integer, starting from 1 and can increase infinitely. However, for most practical purposes, the values typically range from 1 to 7.

• The energy level associated with a higher \(n\) value implies it is further from the atomic nucleus.
• As \( n \) increases, the size of the orbitals, the amount of energy, and the possible number of electrons all increase.
• Each energy level determined by \(n\) has a maximum number of electrons, calculated using the formula \( 2n^2 \).

Given that \(n = 4\) implies we're looking at the fourth energy level or shell, it further affects the number and type of subshells present in this energy level.

Electron Subshells

Electron subshells are subdivisions within each main electron shell, described by the principal quantum number. As noted, the number of subshells within a given shell is determined by the shell's principal quantum number \( n \).
There are several types of subshells, namely, \(s\), \(p\), \(d\), and \(f\). Each type of subshell has a unique shape and can hold a fixed number of electrons.

• The \(s\) subshell is spherical and can hold up to 2 electrons.
• The \(p\) subshell is dumbbell-shaped and can hold up to 6 electrons.
• The \(d\) subshell can hold up to 10 electrons and has a more complex shape.
• The \(f\) subshell can hold up to 14 electrons.

When \(n = 4\), there are 4 subshells designated as \(4s\), \(4p\), \(4d\), and \(4f\), each capable of holding different numbers of electrons, totaling up to 32 electrons.

Electron Configuration

Electron configuration is a method used to describe the distribution of electrons in an atom’s orbitals. It's a key concept that helps us understand chemical properties and the periodic table’s organization.
The notation involves listing the occupied subshells in order of increasing energy, filling each subshell with electrons up to its maximum capacity.
For example, if we consider a full electron shell with \(n = 4\), its electron configuration would look something like:

• \(4s^2\)
• \(4p^6\)
• \(4d^{10}\)
• \(4f^{14}\)

The superscript numbers indicate the number of electrons in each subshell, totaling the maximum of 32 electrons in the fourth shell. This notation helps visualize the arrangement of electrons, which is essential for predicting an atom's behavior and how it bonds with other atoms.

Atomic Orbitals

Atomic orbitals are regions within an atom where electrons are likely to be found. They form the basis of understanding electron configuration, subshells, and atomic interactions.
Each type of subshell consists of a specific number of orbitals:

• The \(s\) subshell contains 1 orbital.
• The \(p\) subshell contains 3 orbitals.
• The \(d\) subshell contains 5 orbitals.
• The \(f\) subshell contains 7 orbitals.

Each orbital can hold a maximum of 2 electrons. Thus, for the \(n = 4\) shell, we have:

• 1 \(s\) orbital, accommodating 2 electrons.
• 3 \(p\) orbitals, accommodating 6 electrons.
• 5 \(d\) orbitals, accommodating 10 electrons.
• 7 \(f\) orbitals, accommodating 14 electrons.

These orbitals represent spatial areas and allow chemists to predict how atoms interact, react, and bond to form molecules, offering a deep insight into chemical behavior and properties.