Question

An electron is known to be in the \(n=4\) shell and the \(l=2\) subshell. How many possible combinations of quantum numbers could this electron have? a. 1 b. 2 c. 5 d. 10

Short answer

The electron can have 10 possible combinations of quantum numbers.

Step-by-step solution

- Understand the Quantum Numbers

Quantum numbers describe the properties of electrons in atoms. The principal quantum number () represents the shell, and the angular momentum quantum number () represents the subshell. For = 4 and = 2, this is a d-subshell in the fourth shell.

- Determine the Magnetic Quantum Number ()

The magnetic quantum number () can take on integer values ranging from to . So for = 2, the possible values of are -2, -1, 0, 1, and 2. This means there are 5 possible values.

- Determine the Spin Quantum Number ()

The spin quantum number () can be either +1/2 or -1/2, indicating the two possible spin states of the electron.

- Calculate Total Combinations

Each of the 5 possible values can each pair with either of the 2 possible values. Thus, there are combinations, so we multiply 5 and 2 to get: 5 x 2 = 10.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by the letter \( n \), defines the main energy level or shell in which the electron resides. It is a key quantum number that determines the size and energy of an orbital. Higher values of \( n \) indicate orbitals that are further from the nucleus. For instance:

• \( n = 1 \) denotes the first shell
• \( n = 2 \) denotes the second shell
• And so on...

This quantum number can take on any positive integer value starting from 1. In the given problem, the electron is known to be in the \( n = 4 \) shell, meaning it is in the fourth energy level. This provides an idea of where the electron is located in relation to the nucleus and the energy associated with it.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted by \( l \), defines the subshell or orbital shape where the electron is found. This quantum number can take any integer value from 0 to \( n-1 \). Each value of \( l \) corresponds to a specific type of subshell:

• \( l = 0 \) corresponds to an s subshell
• \( l = 1 \) corresponds to a p subshell
• \( l = 2 \) corresponds to a d subshell
• \( l = 3 \) corresponds to an f subshell

In the exercise, the electron is indicated to be in the \( l = 2 \) subshell, which is a d subshell. Therefore, within the fourth shell (\( n = 4 \)), it is the fourth energy level's d subshell. This tells us about the type of orbital that the electron occupies.

Magnetic Quantum Number

The magnetic quantum number, signified as \( m_l \), describes the orientation of the orbital in which the electron is present. This quantum number can hold integer values ranging from \( -l \) to \( l \). For instance, if \( l = 2 \):

• Possible \( m_l \) values are: -2, -1, 0, 1, and 2

There are five possible orientations an electron can adopt in a d subshell (as \( l = 2 \)). Each possible value of \( m_l \) represents a different spatial orientation of the d orbital.
In simpler terms, \( m_l \) helps to elucidate where on the three-dimensional axis the electron's orbital lies. In our problem, since \( l = 2 \), \( m_l \) can take one of five values, indicating there are five possible orientations for the electron's orbital.

Spin Quantum Number

The spin quantum number, denoted by \( m_s \), accounts for the electron’s spin, an intrinsic form of angular momentum. Unlike the previous quantum numbers, \( m_s \) can only take on two possible values:

• +1/2 (spin-up)
• -1/2 (spin-down)

This implies that for each possible combination of the principal, angular momentum, and magnetic quantum numbers, there are two possible orientations of the electron's spin.
In the context of our problem, for each of the five possible \( m_l \) values, the spin quantum number \( m_s \) can be either +1/2 or -1/2. This results in a total of 5 possibilities (from \( m_l \)) multiplied by 2 (from \( m_s \)), yielding 10 possible combinations of quantum numbers. Hence, the correct answer for the exercise is option d: 10 possible combinations.