Problem 83 - Chemistry: An Atoms-Focused Approach

Question

How many electrons could occupy orbitals with the following quantum numbers? a. \(n=2, \ell=0\) b. \(n=3, \ell=1, m_{\ell}=0\) c. \(n=4, \ell=2\) d. \(n=1, \ell=0, m_{\ell}=0\)

Step-by-Step Solution

Answer

Based on the given quantum numbers: a. For n=2 and ℓ=0, there can be 2 electrons. b. For n=3, ℓ=1, and mℓ=0, there can be 2 electrons. c. For n=4 and ℓ=2, there can be 10 electrons. d. For n=1, ℓ=0, and mℓ=0, there can be 2 electrons.

Step 1: Identify Quantum Numbers

For this exercise, we have n=2 and ℓ=0. Since the exercise doesn't give us an exact value for mℓ, we need to find all possible values.

Step 2: Find mℓ values

The magnetic quantum number, mℓ, can have integer values between -ℓ and +ℓ, inclusive. Here, we have ℓ=0, which means mℓ can only be 0.

Step 3: Calculate Number of Electrons

For each value of mℓ, there can be two possible values of electron spin quantum number, mₛ, which can be either +½ or -½. So in this case, for mℓ = 0, there will be 2 electrons possible. Answer (a): 2 electrons #b. n=3, ℓ=1, mℓ=0#

Step 1: Identify Quantum Numbers

For this part, we have n=3, ℓ=1, and mℓ=0.

Step 2: Calculate Number of Electrons

As mℓ is already given and since we have a single value, we can go directly to calculate the number of electrons. As mentioned above, for each mℓ value, there can be two possible values of mₛ. So, here, there will be 2 electrons possible. Answer (b): 2 electrons #c. n=4, ℓ=2#

Step 1: Identify Quantum Numbers

For this part, we have n=4 and ℓ=2. We have to find all possible mℓ values.

Step 2: Find mℓ values

Since we have ℓ=2, the possible values of mℓ are -2, -1, 0, +1, and +2.

Step 3: Calculate Number of Electrons

For each mℓ value, there are two possible mₛ values. So there will be 2 electrons for each mℓ value. As there are 5 possible mℓ values, there will be 2 × 5 = 10 electrons in total. Answer (c): 10 electrons #d. n=1, ℓ=0, mℓ=0#

Step 1: Identify Quantum Numbers

For this part, we have n=1, ℓ=0, and mℓ=0.

Step 2: Calculate Number of Electrons

As mℓ is already given and since we have a single value, we can go directly to calculate the number of electrons. As mentioned above, for each mℓ value, there can be two possible values of mₛ. Here, there will be 2 electrons possible. Answer (d): 2 electrons

Key Concepts

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's orbitals. Electrons are distributed among different energy levels, or shells, which are labeled with the principal quantum number, \( n \). Within each level, there are sub-levels or orbitals, defined by the azimuthal quantum number, \( \ell \). These sub-levels correspond to shapes like s (\( \ell = 0 \)), p (\( \ell = 1 \)), d (\( \ell = 2 \)), and f (\( \ell = 3 \)).
Understanding electron configurations helps predict chemical properties and behavior:

• The number of electrons that can occupy an orbital is determined by the magnetic quantum number, \( m_\ell \), and the spin quantum number, \( m_s \).
• Each orbital can hold a maximum of two electrons with opposite spins (\( +\frac{1}{2} \) and \( -\frac{1}{2} \)).

Recognizing these patterns is key to understanding how atoms bond and interact.

Magnetic Quantum Number

The magnetic quantum number, \( m_\ell \), indicates the orientation of an orbital in space. For a given value of \( \ell \), \( m_\ell \) can range from \( -\ell \) to \( +\ell \), yielding \( 2\ell + 1 \) possible values.
This quantum number determines how many orbitals are available within a sub-shell:

• \( \ell = 0 \) (s orbitals) leads to \( m_\ell = 0 \)
• \( \ell = 1 \) (p orbitals) allows \( m_\ell = -1, 0, +1 \)
• \( \ell = 2 \) (d orbitals) includes \( m_\ell = -2, -1, 0, +1, +2 \)

The magnetic quantum number plays a crucial role in defining the specific shape and path an electron will likely follow within its sub-level. This is essential for understanding the electron's impact on the atom's magnetic properties.

Spin Quantum Number

The spin quantum number, \( m_s \), describes the intrinsic spin of an electron within an orbital. It can have one of two possible values: \( +\frac{1}{2} \) or \( -\frac{1}{2} \). This does not mean the electron is literally spinning like a top, but this property accounts for certain magnetic behaviors of electrons.
Here's why \( m_s \) is essential:

• It ensures that each orbital can hold up to two electrons.
• The electrons must have opposite spins, leading to a pair with \( m_s = +\frac{1}{2} \) and \( m_s = -\frac{1}{2} \).

This concept of spin is fundamental to the Pauli exclusion principle, stating no two electrons in an atom can have the same set of quantum numbers. Understanding \( m_s \) helps in explaining how atoms fill their orbitals and how they interact with magnetic fields.