Problem 31 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

There are 4 subshells in the electron shell when \(n=4\).

Step 1: Understanding Principal Quantum Number

The principal quantum number, denoted as \(n\), determines the energy level of the shell and its distance from the nucleus. The value \(n = 4\) indicates that we are looking at the fourth shell.

Step 2: Identifying Subshells

Subshells within an electron shell are labeled as \(s\), \(p\), \(d\), and \(f\). The number of subshells in a shell is equal to \(n\). For \(n=4\), there will be 4 subshells: \(4s\), \(4p\), \(4d\), and \(4f\).

Step 3: Counting Subshells

Since we have identified the types of subshells (\(s, p, d, f\)), we can now count them. For \(n=4\), the subshells are \(4s\), \(4p\), \(4d\), and \(4f\), totaling 4 subshells.

Key Concepts

Principal Quantum Number

Quantum numbers are essential for describing the unique state of electrons in an atom. The principal quantum number, represented by the symbol \(n\), is crucial because it determines the main energy level or "shell" in which an electron is located. This number can be any positive integer (1, 2, 3, and so on).

The value of \(n\) not only indicates which energy level an electron belongs to but also its relative distance from the nucleus of the atom. A higher principal quantum number means the electron is further away from the nucleus and possesses higher energy.

• \(n = 1\), the first shell or K shell
• \(n = 2\), the second shell or L shell
• \(n = 3\), the third shell or M shell
• \(n = 4\), the fourth shell or N shell

When \(n = 4\), it corresponds to the atom's fourth shell or energy level, which can host multiple subshells.

Electron Shell

Electron shells are the layers surrounding the nucleus of an atom, formed by electrons as defined by their energy levels and their principal quantum number \(n\). Each shell corresponds to a different energy level and can contain electrons in various subshells.

The concept of electron shells helps us understand where electrons are likely to be found around the nucleus. An electron shell's capacity to host electrons is determined by the formula \(2n^2\), where \(n\) is the principal quantum number.

• For \(n = 1\), a maximum of 2 electrons can be present.
• For \(n = 2\), up to 8 electrons can be accommodated.
• For \(n = 3\), the shell can hold up to 18 electrons.
• For \(n = 4\), as many as 32 electrons can inhabit the shell.

Understanding electron shells enables chemists to predict electron configurations and chemical behavior.

Subshells

Within each electron shell, there are subshells defined by the azimuthal quantum number, usually denoted as \(l\). Each shell has a number of subshells that equals the principal quantum number \(n\).

The subshells are represented by the letters \(s\), \(p\), \(d\), and \(f\), corresponding to different shapes and energy levels:

• The \(s\) subshell (\(l = 0\)) is spherical and holds 2 electrons.
• The \(p\) subshell (\(l = 1\)) consists of three dumbbell-shaped orbitals, housing 6 electrons.
• The \(d\) subshell (\(l = 2\)) contains five orbitals, accommodating 10 electrons.
• The \(f\) subshell (\(l = 3\)) comprises seven orbitals and holds up to 14 electrons.

For a principal quantum number \(n = 4\), there are four possible subshells: 4s, 4p, 4d, and 4f. These multiple subshells allow for a more nuanced distribution of electrons, impacting an atom's chemical properties and bonding behavior.