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: Understanding Principal Quantum Number

The principal quantum number, represented by \(n\), determines the electron shell. For any electron shell defined by a principal quantum number \(n\), the number of subshells is equal to \(n\).

Step 2: Applying the Formula

Since the principal quantum number given is \(n=5\), the number of subshells is simply equal to 5 because according to quantum mechanics, each electron shell \(n\) contains exactly \(n\) subshells.

Key Concepts

Understanding the Principal Quantum Number

The principal quantum number, symbolized by \(n\), is essential in quantum mechanics. It defines the energy level or shell of an electron in an atom. The value of \(n\) is a positive integer (1, 2, 3, etc.), and as \(n\) increases, the electron is found on average further from the nucleus.

Key points about the principal quantum number include:

• Determines the size of the electron cloud; a higher \(n\) means a larger cloud.
• Sets the stage for how many subshells are within the electron shell.
• Influences the energy level of the electron; higher \(n\) means higher energy.

Understanding \(n\) helps us determine the structure and properties of atoms, which form the basis for the chemical behavior of materials.

Exploring the Electron Shell

The electron shell refers to the path where electrons orbit around the nucleus. It's layered based on the principal quantum number \(n\). Each shell holds a certain number of electrons determined by the formula \(2n^2\).

These shells are crucial because:

• They define the atom's size and energy by holding varying numbers of electrons.
• The shell number \(n\) correlates directly to the number of subshells in that shell.
• Each shell is a step further from the nucleus as the electron energy increases with \(n\).

Thus, the electron shell not only indicates position but also contributes to the atom's chemical identity.

Insight into Subshells

Subshells are the divisions within each electron shell and are determined by the azimuthal quantum number, \(l\), which can range from 0 to \(n-1\). These are represented by letters \(s, p, d,\) and \(f\).

For any shell with a principal quantum number \(n\), there are exactly \(n\) subshells:

• For \(n=1\), there is only one subshell: \(s\).
• For \(n=2\), there are two subshells: \(s\) and \(p\).
• For \(n=5\), as in the exercise, there are five subshells: \(s, p, d, f,\) and \(g\).

Each subshell has orbitals that can hold a specific number of electrons, guiding electron configuration and chemical bonds.

This structured arrangement helps define the complex behavior and properties of elements.