Problem 46 - Chemistry The Molecular Science

Question

One electron has the set of quantum numbers \(n=3\), \(\ell=1, m_{\ell}=-1,\) and \(m_{s}=+\frac{1}{2} ;\) another electron has the $$ \text { set } n=3, \ell=1, m_{\ell}=1, \text { and } m_{s}=+\frac{1}{2} \text { . } $$ (a) Could the electrons be in the same atom? Explain. (b) Could they be in the same atomic orbital? Explain.

Step-by-Step Solution

Answer

(a) Yes, they can be in the same atom. (b) No, they cannot be in the same orbital.

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the properties of electrons in orbitals. The principal quantum number \( n \) denotes the energy level or shell. The azimuthal quantum number \( \ell \) denotes the subshell or type of orbital (0 for s, 1 for p, etc.). The magnetic quantum number \( m_{\ell} \) specifies the orientation of the orbital, and the spin quantum number \( m_{s} \) describes the electron's spin.

Step 2: Analyzing Quantum Numbers for the First Electron

The first electron has the quantum numbers \( n=3 \), \( \ell=1 \), \( m_{\ell}=-1 \), and \( m_{s}=+\frac{1}{2} \). This means it is in the 3rd energy level, in a p-orbital (since \( \ell=1 \)), in one of the three p-orbital orientations (\( m_{\ell}=-1 \)), and has a spin of +1/2.

Step 3: Analyzing Quantum Numbers for the Second Electron

The second electron has \( n=3 \), \( \ell=1 \), \( m_{\ell}=1 \), and \( m_{s}=+\frac{1}{2} \). Similarly, it is in the 3rd energy level and a p-orbital, albeit in a different orientation (\( m_{\ell}=1 \)), with the same spin direction as the first electron.

Step 4: Determine if They Could Be in the Same Atom

Electrons in the same atom can share the same principal quantum number \( n \) and azimuthal quantum number \( \ell \). The given electrons both have \( n=3 \) and \( \ell=1 \), so they can be in the same atom. There are no restrictions based on their distinct \( m_{\ell} \) values within the same subshell.

Step 5: Determining if They Could Be in the Same Orbital

Two electrons cannot be in the same quantum state within an atom, which is described by the unique set of all four quantum numbers. For them to be in the same atomic orbital, \( m_{\ell} \) would have to be the same, but the spin quantum number \( m_{s} \) would need to differ. Here, they have differing \( m_{\ell} \) values, making them in different orbitals, even though they are in the same subshell.

Key Concepts

Principal Quantum Number

The principal quantum number, represented by the symbol \( n \), is one of the four quantum numbers that describe the unique quantum state of an electron in an atom. It defines the electron's energy level or shell in which the electron resides. As \( n \) increases, the energy and the average distance of the electron from the nucleus increase as well.
In essence, \( n \) can take any positive integer value, starting from 1, 2, 3, and so forth. Each value of \( n \) corresponds to a specific shell. For instance, \( n=1 \) carries the lowest energy and is closest to the nucleus, while higher values indicate higher energy levels that are further away. This principle can be easily linked to the electron's potential energy within the atom's electromagnetic field.

• Energy Level: Indicates the general region through which the electron moves. Lower principal quantum numbers mean lower energy and closer proximity to the nucleus.
• Electron Capacity: The principal quantum number limits the number of electrons a shell can hold, given by the formula \( 2n^2 \).

Understanding the principal quantum number is crucial for visualizing how electrons populate different energy levels and their arrangements in building up the structure of atoms.

Spin Quantum Number

The spin quantum number, denoted as \( m_s \), describes the intrinsic angular momentum, or "spin," of an electron within an atom. Unlike other quantum numbers, the spin quantum number only has two possible values: +1/2 or -1/2. These values are often visualized as "spin up" and "spin down".
This characteristic is essential to the concept of electron pairing within orbitals. According to the Pauli exclusion principle, no two electrons in a single atom can have the same set of four quantum numbers. Therefore, in any given atomic orbital, only two electrons can exist at the most.

• Pairing Electrons: The two possible spin states allow electrons to pair together in an orbital, with one having \( m_s = +1/2 \) and the other \( m_s = -1/2 \).
• Magnetic Property: The spin quantum number contributes to the magnetic moment of an atom, influencing phenomena such as magnetism.

Grasping the concept of the spin quantum number helps explain atomic behavior such as chemical bonding and electron configuration in different orbitals.

Atomic Orbitals

Atomic orbitals represent the regions of space around the nucleus where the probability of finding an electron is highest. Each orbital is defined by a set of three quantum numbers: principal quantum number \( n \), azimuthal quantum number \( \ell \), and magnetic quantum number \( m_\ell \).
Different orbitals have distinct shapes and capacities to hold electrons, determined by these quantum numbers. The azimuthal quantum number \( \ell \), for instance, determines the "shape" of the orbital:

• \( \ell = 0 \): s-orbitals (spherical shape)
• \( \ell = 1 \): p-orbitals (dumbbell-shaped)
• \( \ell = 2 \): d-orbitals (clover-shaped)
• \( \ell = 3 \): f-orbitals (complex shapes)

The magnetic quantum number \( m_\ell \) specifies the orientation of the orbital in space and can take values from \(-\ell\) to \(\ell\). Atomic orbitals are also influenced by electron spin direction, as explained by the spin quantum number \( m_s \).
Taken together, these quantum numbers describe the unique characteristics of orbitals and how electrons fill them, offering a detailed picture of an atom’s electron cloud and chemical properties.