Problem 71 - Chemistry: The Central Science

Question

What is the maximum number of electrons that can occupy each of the following subshells? (a) \(3 p\), (b) \(5 d\), (c) \(2 s\), (d) \(4 f\).

Step-by-Step Solution

Answer

The maximum number of electrons that can occupy each subshell is as follows: (a) 3p subshell can hold 6 electrons, (b) 5d subshell can hold 10 electrons, (c) 2s subshell can hold 2 electrons, and (d) 4f subshell can hold 14 electrons.

Step 1: (a) Maximum number of electrons in the 3p subshell

In the 3p subshell, the principal quantum number \(n\) is 3 and the angular momentum quantum number \(l\) is equal to 1 (for the p subshell). Using the formula for the maximum number of electrons in a subshell: \[ 2(2l+1) = 2(2\times1+1) = 2(3) = 6\] So, the maximum number of electrons in the 3p subshell is 6.

Step 2: (b) Maximum number of electrons in the 5d subshell

In the 5d subshell, the principal quantum number \(n\) is 5 and the angular momentum quantum number \(l\) is equal to 2 (for the d subshell). Using the formula for the maximum number of electrons in a subshell: \[ 2(2l+1) = 2(2\times2+1) = 2(5) = 10\] So, the maximum number of electrons in the 5d subshell is 10.

Step 3: (c) Maximum number of electrons in the 2s subshell

In the 2s subshell, the principal quantum number \(n\) is 2 and the angular momentum quantum number \(l\) is equal to 0 (for the s subshell). Using the formula for the maximum number of electrons in a subshell: \[ 2(2l+1) = 2(2\times0+1) = 2(1) = 2\] So, the maximum number of electrons in the 2s subshell is 2.

Step 4: (d) Maximum number of electrons in the 4f subshell

In the 4f subshell, the principal quantum number \(n\) is 4 and the angular momentum quantum number \(l\) is equal to 3 (for the f subshell). Using the formula for the maximum number of electrons in a subshell: \[ 2(2l+1) = 2(2\times3+1) = 2(7) = 14\] So, the maximum number of electrons in the 4f subshell is 14.

Key Concepts

Understanding Quantum Numbers

Quantum numbers are like an address for electrons in an atom, helping us understand where they are likely to be found. These numbers describe the unique quantum state of an electron in an atom and are essential for determining electron configurations. There are four quantum numbers needed:

• The Principal Quantum Number) \(n\): Indicates the energy level and size of the electron's orbit or shell. It's a positive integer (1, 2, 3,...).
• The Angular Momentum Quantum Number \(l\): Defines the shape of the orbital, and its values range from 0 to \(n-1\). For example, \(l = 0\) for s orbitals, \(l = 1\) for p orbitals.
• The Magnetic Quantum Number \(m_l\): Describes the orientation in space of a particular orbital, and its values range from \(-l\) to \(+l\).
• The Spin Quantum Number \(m_s\): Represents the spin direction of the electron, which can be either +1/2 or -1/2.

Understanding these numbers helps predict electron distribution in atoms and is crucial for chemistry and physics studies.

Exploring Subshells

Subshells are subdivisions of electron shells based on the shape and energy of orbitals. The shell corresponding to a principal quantum number \(n\) consists of various subshells characterized by different angular momentum quantum numbers \(l\). The main types of subshells include:

• s subshell: \(l = 0\), spherical shape, found once in every shell.
• p subshell: \(l = 1\), dumbbell shape, appears from the second shell onward.
• d subshell: \(l = 2\), more complex shapes, appears from the third shell onward.
• f subshell: \(l = 3\), even more complex shapes, appears from the fourth shell onward.

Each subshell contains one or more orbitals, and their energy levels also increase as \(l\) gets larger. Understanding subshells is crucial for determining how electrons are arranged in an atom.

Determining Maximum Electrons in a Subshell

To find out the maximum number of electrons in a subshell, we use a straightforward formula: \(2(2l+1)\). Here, \(l\) is the angular momentum quantum number, which identifies the type of subshell:

• For an s subshell with \(l = 0\), the formula gives: \[2(2\times0+1) = 2(1) = 2\] Maximum electrons: 2
• For a p subshell with \(l = 1\), it calculates to: \[2(2\times1+1) = 2(3) = 6\] Maximum electrons: 6
• For a d subshell with \(l = 2\), the formula becomes: \[2(2\times2+1) = 2(5) = 10\] Maximum electrons: 10
• For an f subshell with \(l = 3\), it results in: \[2(2\times3+1) = 2(7) = 14\] Maximum electrons: 14

This formula relies on understanding the shape and number of orbitals within each subshell. It helps predict electron arrangements and balances charge within atoms. Knowing the electron capacity of each subshell is vital for constructing electron configurations accurately.