Problem 63 - 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 ?

Step-by-Step Solution

Answer

The maximum number of electrons for each subshell is: (a) 3p: 6 electrons, (b) 5d: 10 electrons, (c) 2s: 2 electrons, (d) 4f: 14 electrons.

Step 1: Identify the quantum numbers for 3p subshell

The main quantum number (n) is 3, and since it is a p-subshell, the angular momentum quantum number (l) must be 1. The possible values of m_l are -1, 0 and 1, and each electron can have a spin quantum number of either +1/2 or -1/2.

Step 2: Calculate the maximum number of electrons

For each value of m_l (-1, 0, and 1), there can be two electrons – one with a spin quantum number of +1/2 and one with -1/2. That gives us a maximum of 2 electrons for each m_l value, thus the 3p subshell can hold a maximum of 2 * 3 = 6 electrons. #b) 5d subshell#

Step 1: Identify the quantum numbers for 5d subshell

The main quantum number (n) is 5, and since it is a d-subshell, the angular momentum quantum number (l) must be 2. The possible values of m_l are -2, -1, 0, 1, and 2, and each electron can have a spin quantum number of either +1/2 or -1/2.

Step 2: Calculate the maximum number of electrons

For each value of m_l (-2, -1, 0, 1, and 2), there can be two electrons – one with a spin quantum number of +1/2 and one with -1/2. That gives us a maximum of 2 electrons for each m_l value, thus the 5d subshell can hold a maximum of 2 * 5 = 10 electrons. #c) 2s subshell#

Step 1: Identify the quantum numbers for 2s subshell

The main quantum number (n) is 2, and since it is an s-subshell, the angular momentum quantum number (l) must be 0. The only possible value of m_l is 0, and each electron can have a spin quantum number of either +1/2 or -1/2.

Step 2: Calculate the maximum number of electrons

For the only value of m_l (0), there can be two electrons – one with a spin quantum number of +1/2 and one with -1/2. Thus, the 2s subshell can hold a maximum of 2 * 1 = 2 electrons. #d) 4f subshell#

Step 1: Identify the quantum numbers for 4f subshell

The main quantum number (n) is 4, and since it is an f-subshell, the angular momentum quantum number (l) must be 3. The possible values of m_l are -3, -2, -1, 0, 1, 2, and 3, and each electron can have a spin quantum number of either +1/2 or -1/2.

Step 2: Calculate the maximum number of electrons

For each value of m_l (-3, -2, -1, 0, 1, 2, and 3), there can be two electrons – one with a spin quantum number of +1/2 and one with -1/2. That gives us a maximum of 2 electrons for each m_l value, thus the 4f subshell can hold a maximum of 2 * 7 = 14 electrons.

Key Concepts

Quantum Numbers

Understanding quantum numbers is key to grasping the arrangement of electrons in an atom. Each electron in an atom is described by a set of four quantum numbers. These numbers provide insights into the energy level (or shell), the shape of the orbital, the orientation of the orbital, and the electron's spin.

• Principal Quantum Number (n): This number, typically a positive integer, specifies the energy level or shell. For instance, in the 3p subshell, the principal quantum number is 3.
• Angular Momentum Quantum Number (l): This number can range from 0 to n-1 and corresponds to the type of subshell. The values of l indicate s, p, d, and f subshells, respectively. The 3p subshell, therefore, has an l value of 1.
• Magnetic Quantum Number (m_l): Based on the value of l, this number specifies the orientation of the subshell's orbital. For p, where l is 1, m_l can be -1, 0, or +1.
• Spin Quantum Number (m_s): This is either +1/2 or -1/2, representing the two possible spin states of an electron within an orbital.

Combining these numbers provides a comprehensive way to determine the position and behavior of electrons in an atom.

Subshells

Subshells are divisions within electron shells and are determined by the angular momentum quantum number, denoted by 'l.' Each shell (n) can have multiple subshells, identified by s, p, d, and f:

• s-subshell: Here, l = 0. The simplest subshells which contain a single orientation, allowing for two electrons. For example, the 2s subshell can have 2 electrons.
• p-subshell: With l = 1, these subshells can hold a maximum of 6 electrons across its three orientations. Hence, the 3p subshell can accommodate 6 electrons.
• d-subshell: When l = 2, these subshells offer five orientations, with a capacity of 10 electrons collectively, illustrated by the 5d subshell containing 10 electrons.
• f-subshell: With l = 3, offering seven orientations and able to hold up to 14 electrons. Therefore, the 4f subshell can hold a maximum of 14 electrons.

Each orientation can accommodate two electrons, thanks to the two possible spin quantum numbers. Subshells explain why electron configurations are diverse and precisely structured.

Spin Quantum Number

The spin quantum number, symbolized as m_s, is fascinating because it adds an additional degree of uniqueness to each electron within an atom. Electrons spin on their own axes, much like how Earth rotates, but in quantum terms.

• There are only two possible spin quantum numbers: +1/2 and -1/2. These values represent two possible spin directions.
• Despite its name, the "spin" does not describe a literal spinning action; instead, it's a quantum property without a classical equivalent.
• Spin quantum numbers are crucial because they allow electrons to coexist in the same orbital, giving each a distinct identity. This aligns perfectly with Pauli's Exclusion Principle, which states that no two electrons in an atom can have the same four quantum numbers.

The introduction of spin means that each orbital, regardless of its type, can hold two electrons—each "spinning" in opposite directions.