Problem 32 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the formula

The number of subshells in an electron shell is determined by the principal quantum number, denoted as \(n\). The number of subshells is equal to \(n\).

Step 2: Substitute the value of n

For this problem, the principal quantum number \(n\) is given as 5. Therefore, substitute \(n = 5\) into the formula which states that the number of subshells is equal to \(n\).

Step 3: Calculate the number of subshells

By substituting \(n = 5\), we determine that there are 5 subshells in the electron shell with principal quantum number 5. These subshells are s, p, d, f, and g.

Key Concepts

Subshells

Subshells are specific regions within an electron shell where electrons are likely to be found. They define the different shapes and energy levels that electrons can occupy. Each subshell is characterized by the azimuthal quantum number, represented by the letter \(l\). The value of \(l\) determines the shape of the orbital within a subshell.
For instance:

• The \(^l=0\) subshell corresponds to the \(s\) orbital, which is spherical.
• The \(^l=1\) subshell corresponds to the \(p\) orbitals, which are dumbbell-shaped.
• Higher \(l\) values correspond to more complex shapes like \(d\) and \(f\) subshells.

Understanding subshells is crucial because it allows scientists to predict electron configurations and chemical behaviors of atoms. Each principal energy level contains multiple subshells, and each subshell can hold a specific maximum number of electrons. For example, the \(s\) subshell can hold up to 2 electrons, the \(p\) can hold up to 6, and so forth. This knowledge aids in detailing how electrons are arranged around the nucleus.

Principal Quantum Number

The principal quantum number, denoted by \(n\), is a key player in quantum mechanics. It determines the overall size and energy of an electron shell. Essentially, \(n\) tells you how far away the electron is likely to be from the nucleus. The principal quantum number is a positive integer \((n=1, 2, 3, \ldots)\), and higher values correspond to larger and more energetic shells.
The principal quantum number directly relates to the number of subshells within an electron shell. For any given \(n\), there are \(n\) subshells. For example:

• If \(n=1\), there is only one subshell (the \(s\) subshell).
• If \(n=2\), there are two subshells (\(s\) and \(p\)).
• If \(n=3\), there are three subshells (\(s\), \(p\), and \(d\)).

This pattern continues as you increase \(n\), illustrating why the quantum number is so essential for understanding atomic structure. It provides a systematic approach to organizing electrons in multi-electron atoms, which further impacts chemical bonding and periodic trends.

Electron Shell

An electron shell is a grouping of electrons surrounding the nucleus of an atom. Think of it as a layer that holds electrons in specific energy levels around the nucleus. Shells are determined by their principal quantum number \(n\), which reflects their energy level and capacity for electrons. As \(n\) increases, the shell's size and energy level increase as well.
Each shell is made up of subshells, which are distinguished by their shapes and energy. These subshells include \(s\), \(p\), \(d\), and \(f\), among others. Each of these has a defined number of orbitals:

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

Each orbital can hold two electrons, explaining why different shells can accommodate different numbers of electrons. Understanding electron shells is fundamental because chemical reactions often occur when electrons move between these shells. Therefore, recognizing the structure and layout of electron shells helps in comprehending how atoms interact with one another.