Problem 65 - 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 of the given subshells are: (a) 3p subshell: \(6\), (b) 5d subshell: \(10\), (c) 2s subshell: \(2\), and (d) 4f subshell: \(14\).

Step 1: Identify the quantum numbers

For a 3p subshell, the principle quantum number (n) is 3 and the angular momentum quantum number (l) is 1 (since p corresponds to l=1).

Step 2: Calculate the maximum number of electrons

Using the formula 2(2l+1), we can calculate the number of electrons in the 3p subshell: \(2(2(1)+1)=2(3)=6\) Ultimately, the 3p subshell can hold a maximum of 6 electrons. #b) 5d Subshell Maximum Electrons#

Step 1: Identify the quantum numbers

For a 5d subshell, the principle quantum number (n) is 5 and the angular momentum quantum number (l) is 2 (since d corresponds to l=2).

Step 2: Calculate the maximum number of electrons

Using the formula 2(2l+1), we can calculate the number of electrons in the 5d subshell: \(2(2(2)+1)=2(5)=10\) Ultimately, the 5d subshell can hold a maximum of 10 electrons. #c) 2s Subshell Maximum Electrons#

Step 1: Identify the quantum numbers

For a 2s subshell, the principle quantum number (n) is 2 and the angular momentum quantum number (l) is 0 (since s corresponds to l=0).

Step 2: Calculate the maximum number of electrons

Using the formula 2(2l+1), we can calculate the number of electrons in the 2s subshell: \(2(2(0)+1)=2(1)=2\) Ultimately, the 2s subshell can hold a maximum of 2 electrons. #d) 4f Subshell Maximum Electrons#

Step 1: Identify the quantum numbers

For a 4f subshell, the principle quantum number (n) is 4 and the angular momentum quantum number (l) is 3 (since f corresponds to l=3).

Step 2: Calculate the maximum number of electrons

Using the formula 2(2l+1), we can calculate the number of electrons in the 4f subshell: \(2(2(3)+1)=2(7)=14\) Ultimately, the 4f subshell can hold a maximum of 14 electrons.

Key Concepts

Understanding Quantum Numbers

Quantum numbers are essential to the quantum mechanical model of atoms, helping us understand the arrangement of electrons within an atom. Each electron in an atom is described by a set of four quantum numbers.

• **Principal Quantum Number (n):** This number denotes the electron shell or energy level of an electron within an atom. The larger the value of \( n \), the higher the energy level, and consequently, the farther the electron is from the nucleus.
• **Angular Momentum Quantum Number (l):** This number, also known as the azimuthal quantum number, dictates the shape of the orbital. It can take values from 0 to \( n-1 \), with each value corresponding to a specific type of subshell: \( l=0 \) (s), \( l=1 \) (p), \( l=2 \) (d), and \( l=3 \) (f).
• **Magnetic Quantum Number (m_l):** This number describes the orientation of the orbital in space. It ranges from \(-l\) to \(+l\).
• **Spin Quantum Number (m_s):** Describes the spin of the electron, which can either be \(+1/2\) or \(-1/2\).

Knowing these numbers allows chemists and physicists to predict electron configurations and understand chemical properties.

Determining Maximum Electrons

The maximum number of electrons that can occupy a specific subshell is determined by the angular momentum quantum number \( l \). The relationship is given by the formula \( 2(2l + 1) \), which calculates the number of possible electron states within a subshell.

• For \( l=0 \) (s subshell), this results in a maximum of 2 electrons.
• For \( l=1 \) (p subshell), the maximum is 6 electrons.
• For \( l=2 \) (d subshell), the maximum is 10 electrons.
• For \( l=3 \) (f subshell), the maximum is 14 electrons.

These values derive from the number of orbitals per subshell and how each orbital can hold two electrons with opposite spins. This systematic approach helps us predict electron capacity in various subshells, such as those mentioned in the exercise: 3p with 6 electrons, 5d with 10 electrons, 2s with 2 electrons, and 4f with 14 electrons.

Understanding Subshells

Subshells are divisions of electron shells, with each subshell corresponding to different energy levels and numbers of orbitals based on their quantum numbers. When analyzing subshells, it's important to comprehend how **electrons are distributed** across them.

• **s Subshells**: Consist of only one orbital and can hold up to two electrons.
• **p Subshells**: Comprising three orbitals, they can accommodate up to six electrons.
• **d Subshells**: Featuring five orbitals, these subshells can hold a maximum of ten electrons.
• **f Subshells**: With seven orbitals, they reach a capacity of fourteen electrons.

Subshells provide a more detailed view beyond the simple electron shell model, allowing us to understand electron movement and chemical bonding better. Understanding these divisions is key to predicting and explaining the chemical behaviors of different elements.