Problem 53 - Chemistry The Molecular Science

Question

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

Step-by-Step Solution

Answer

There are 5 subshells in the shell with principal quantum number \(n=5\).

Step 1: Understanding Principal Quantum Number

The principal quantum number (\(n\)) indicates the shell or energy level of an atom where electrons are found. It tells us about the size of the electron cloud. For \(n=5\), this refers to the 5th electron shell.

Step 2: Identifying Possible Azimuthal Quantum Numbers

The azimuthal quantum number (\(l\)), also known as the angular momentum quantum number, defines the shape of the orbital. For any principal quantum number \(n\), \(l\) can take on integer values from 0 to \(n-1\). So, for \(n=5\), \(l\) can be 0, 1, 2, 3, or 4.

Step 3: Relating Azimuthal Quantum Number to Subshells

Each unique value of \(l\) corresponds to a different subshell with the following designations: \(l = 0\) is an s subshell, \(l = 1\) is a p subshell, \(l = 2\) is a d subshell, \(l = 3\) is an f subshell, and \(l = 4\) is a g subshell.

Step 4: Counting the Number of Subshells

Since for \(n=5\), \(l\) can be 0, 1, 2, 3, or 4, we have five different subshells: 5s, 5p, 5d, 5f, and 5g.

Key Concepts

Principal Quantum Number

The principal quantum number, often denoted as \( n \), plays a crucial role in atomic physics. It indicates the main energy level or shell where an electron resides in an atom. Think of it as a measure of how far an electron can be from the nucleus rather than its specific location. As the value of \( n \) increases, the electron shell becomes larger, allowing electrons within that shell to have higher energy levels and be farther from the nucleus.

For instance, when \( n = 1 \), we're talking about the smallest and innermost shell closest to the nucleus. By the time \( n = 5 \), which is our focus here, we're discussing a quite expansive shell, the 5th energy level. This means electrons in this shell have a much higher energy and occupy a larger region in space within the atom.

To sum up, \( n \) not only gives us information about energy levels but also helps determine the electron’s distance from the nucleus, influencing properties like the atom's size and how it interacts with other atoms.

Azimuthal Quantum Number

The azimuthal quantum number, represented as \( l \), is an essential component for describing electron orbitals in an atom. While the principal quantum number \( n \) tells us about the energy level, \( l \) provides insight into the shape of the electron's path, or orbital, within that level.

For any given principal quantum number \( n \), \( l \) can have integer values ranging from 0 up to \( n-1 \). These values determine the specific shapes of orbitals - each associated with different subshell designations. Let’s look at these:

• \( l = 0 \) corresponds to an 's' subshell, characterized by a spherical shape.
• \( l = 1 \) corresponds to a 'p' subshell, recognized by its dumbbell shape.
• \( l = 2 \) relates to a 'd' subshell, which is more complex with clover-like shapes.
• \( l = 3 \) refers to an 'f' subshell, known for even more complex patterns.
• \( l = 4 \) leads to a 'g' subshell, theoretical at \( n = 5 \).

Understanding these shapes is crucial, as they influence how electrons fill orbitals and subshells and determine the chemical bonding and properties of elements. So, \( n = 5 \) allows \( l \) to take values from 0 to 4, showcasing a variety of subshells and accompanying electron configurations.

Electron Subshells

Electron subshells are the divisions within each principal energy level of an atom. They are determined by the azimuthal quantum number \( l \) and help define the energies and locations of electrons within an atom.

For any main energy level defined by the principal quantum number \( n \), there are \( n \) possible subshells. This gives rise to different subshells, each with a specific designation and shape related to \( l \). For \( n = 5 \), the possible values for \( l \) are 0 through 4, creating five distinct subshells: 5s, 5p, 5d, 5f, and 5g.

Here's a quick rundown of these subshells:

• The 5s subshell has the simplest spherical shape, housing up to 2 electrons.
• The 5p subshell is known for its three dumbbell-shaped orientations, containing up to 6 electrons.
• 5d subshells can fit up to 10 electrons in their complex clover-like orbitals.
• 5f subshells, with seven different orientations, hold up to 14 electrons.
• The 5g subshell, though not encountered in any naturally occurring elements, theoretically allows 18 electrons in even more intricate orbital patterns.

Understanding the distribution of electrons among these subshells is vital for predicting an element's chemical properties and behaviors. It explains how atoms bond, as electrons located in different subshells interact in unique ways to form chemical compounds.