Problem 9 - College Physics

Question

Make a list of the four quantum numbers \(n , l , m _ { l } ,\) and \(s\) for each of the 12 electrons in the ground state of the magnesium atom.

Step-by-Step Solution

Answer

Identify the electron configurations and quantum numbers for each electron.

Step 1: Understand the Problem

Magnesium (Mg) has an atomic number of 12, which means it has 12 electrons in its ground state. Our task is to find the four quantum numbers \(n, l, m_l, \) and \(s\) for each of these electrons.

Step 2: Identify the Electron Configuration

The electron configuration for magnesium is \([\text{Ne}] 3s^2\). This can be expanded to \(1s^2 2s^2 2p^6 3s^2\). From this, we can see that the first 2 electrons fill the 1s subshell, the next 2 fill the 2s subshell, the next 6 fill the 2p subshell, and the final 2 fill the 3s subshell.

Step 3: Determine Quantum Numbers for 1s Electrons

For the 1s subshell (2 electrons):- Principal quantum number \(n = 1\).- Azimuthal quantum number \(l = 0\) (s subshell).- Magnetic quantum number \(m_l = 0\) (only one orientation for \(l = 0\)).- Spin quantum numbers \(s = +\frac{1}{2}\) and \(-\frac{1}{2}\).

Step 4: Determine Quantum Numbers for 2s Electrons

For the 2s subshell (2 electrons):- Principal quantum number \(n = 2\).- Azimuthal quantum number \(l = 0\) (s subshell).- Magnetic quantum number \(m_l = 0\).- Spin quantum numbers \(s = +\frac{1}{2}\) and \(-\frac{1}{2}\).

Step 5: Determine Quantum Numbers for 2p Electrons

For the 2p subshell (6 electrons):- Principal quantum number \(n = 2\).- Azimuthal quantum number \(l = 1\) (p subshell).- Magnetic quantum numbers \(m_l = -1, 0, +1\). Each \(m_l\) value can have two electrons: - For \(m_l = -1\): \(s = +\frac{1}{2}\) and \(-\frac{1}{2}\). - For \(m_l = 0\): \(s = +\frac{1}{2}\) and \(-\frac{1}{2}\). - For \(m_l = +1\): \(s = +\frac{1}{2}\) and \(-\frac{1}{2}\).

Step 6: Determine Quantum Numbers for 3s Electrons

For the 3s subshell (2 electrons):- Principal quantum number \(n = 3\).- Azimuthal quantum number \(l = 0\) (s subshell).- Magnetic quantum number \(m_l = 0\).- Spin quantum numbers \(s = +\frac{1}{2}\) and \(-\frac{1}{2}\).

Key Concepts

Magnesium Electron Configuration

To understand the quantum numbers associated with magnesium, we first need to look at its electron configuration. Magnesium, which has the atomic number 12, houses 12 electrons. The electron configuration describes how these electrons are organized within the atom's shells and subshells. For magnesium, this arrangement can be denoted as \( [\text{Ne}] \, 3s^2 \). To express it in a more detailed manner, we consider it as \( 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \).
This configuration shows us that:
\(1s\) subshell holds 2 electrons, \(2s\) subshell contains another 2 electrons, \(2p\) subshell has 6 electrons, and the \(3s\) subshell carries the last 2 electrons.
This arrangement relates directly to the effective distribution of quantum numbers, which are characteristics that describe each electron's position and behavior.

Principal Quantum Number

The principal quantum number, expressed as \( n \), plays a pivotal role in atomic structure. It indicates the electron shell level or the energy level in which electrons reside. Increasing \( n \) means electrons are situated further from the atomic nucleus.
For magnesium:

• The electrons in the \(1s\) orbital have a principal quantum number \(n = 1\).
• Those in the \(2s\) and \(2p\) orbitals have \(n = 2\).
• The electrons in the \(3s\) orbital have \(n = 3\).

As \(n\) increases, the energy level ascends, meaning electrons in higher levels require more energy to be removed from the atom.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized by \( l \), defines the subshell or the shape of the orbital in which the electron is located. Its values range from \(0\) to \(n-1\).

• For \(s\) orbitals, like \(1s\), \(2s\) and \(3s\), the azimuthal quantum number \(l\) is \(0\). These are spherical.
• For \(2p\) orbitals, the quantum number \(l\) is \(1\). These have a dumbbell shape.

Each different value of \( l \) pertains to a specific subshell (\(s, p, d, f\)), delineating the shape and energy level differences within a principal level.

Spin Quantum Number

The spin quantum number, denoted as \( s \), represents the intrinsic angular momentum of the electrons, commonly known as 'spin' and can either be \( +\frac{1}{2} \) or \( -\frac{1}{2} \). This characteristic describes the orientation of an electron's spin in its magnetic field.
Considering magnesium's electron configuration:

• Each orbital can hold 2 electrons, with opposing spins, making use of \(s = +\frac{1}{2}\) and \(s = -\frac{1}{2}\).
• This is true for all orbitals: \(1s, 2s, 2p,\) and \(3s\).

The pairing of opposite spins in an orbital reduces repulsion between electrons due to their negative charges, thus stabilizing the atom's structure.