Problem 19 - Chemistry and Chemical Reactivity

Question

What is the maximum number of electrons that can be identified with each of the following sets of quantum numbers? In one case, the answer is "none." Explain why this is true. (a) \(n=4, \ell=3\) (b) \(n=6, \ell=1, m_{\ell}=-1\) (c) \(n=3, \ell=3, m_{\ell}=-3\)

Step-by-Step Solution

Answer

Sets (a) and (b) allow for 14 and 2 electrons, respectively. Set (c) is invalid.

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the position and energy of an electron in an atom. The principal quantum number \(n\) denotes the energy level, the azimuthal quantum number \(\ell\) describes the subshell, and the magnetic quantum number \(m_\ell\) denotes the orientation of the orbital within the subshell. Lastly, electrons also have a spin quantum number \(m_s\), which can be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Step 2: Analyzing Set (a)

The first set of quantum numbers is \(n=4\), \(\ell=3\). The value of \(\ell=3\) corresponds to the f-subshell, which has magnetic quantum numbers \(m_\ell = -3, -2, -1, 0, 1, 2, 3\). There are 7 possible values for \(m_\ell\), and each orbital can hold 2 electrons (one for each spin direction). Thus, the maximum number of electrons is \(7 \times 2 = 14\).

Step 3: Analyzing Set (b)

The second set of quantum numbers is \(n=6\), \(\ell=1\), \(m_\ell=-1\). The value of \(\ell=1\) corresponds to the p-subshell and \(m_\ell = -1\) signifies one specific orbital in this subshell. An orbital can accommodate 2 electrons (one for each spin direction, \(m_s = +\frac{1}{2}, -\frac{1}{2}\)). Therefore, the maximum number of electrons is 2.

Step 4: Analyzing Set (c)

The third set of quantum numbers is \(n=3\), \(\ell=3\), \(m_\ell=-3\). However, \(\ell\) must be less than \(n\), so the possible values for \(\ell\) with \(n=3\) are 0, 1, and 2. Therefore, \(\ell=3\) is not valid for \(n=3\), and no electrons can have these quantum numbers.

Key Concepts

Electron Configuration

Electron configuration tells us how electrons are arranged in an atom. It's like the blueprint of an atom's structure. By knowing this, we can understand the chemical properties and behavior of an element.

Electrons fill orbitals in a specific order:

• First, they fill lower energy levels before moving to higher ones.
• The sequence of filling is based on the "Aufbau principle," which means "building up."
• Each orbital can hold a maximum of 2 electrons, thanks to the "Pauli exclusion principle."

Understanding electron configuration helps us determine how atoms interact, bond, and form compounds.

Subsidiary Quantum Number

The subsidiary quantum number, also known as the azimuthal or angular momentum quantum number (\(\ell\)), gives us info about the subshell within a principal energy level. It determines the shape of the orbital.

Here's what you need to know:

• The possible values of \(\ell\) range from 0 to \(n-1\), where \(n\) is the principal quantum number.
• The value of \(\ell\) corresponds to different types of orbitals: s (\(\ell=0\)), p (\(\ell=1\)), d (\(\ell=2\)), and f (\(\ell=3\)).
• Each type of orbital has a unique shape, influencing the chemical bonding.

The subsidiary quantum number is key to understanding how electrons are distributed in an atom.

Magnetic Quantum Number

The magnetic quantum number (\(m_\ell\)) tells us about the orientation of an orbital in space. It's essential for knowing which specific orbital an electron occupies within a subshell.

Here’s how it works:

• The values of \(m_\ell\) range from \(-\ell\) to \(+\ell\), creating multiple orientations for each subshell.
• For example, if \(\ell=1\) (a p orbital), then \(m_\ell\) can be \(-1, 0,\) or \(+1\).
• Each orbital defined by a unique \(m_\ell\) can hold 2 electrons with opposite spins.

Knowing the magnetic quantum number helps us identify the exact position of an electron within an atom.

Principal Quantum Number

The principal quantum number (\(n\)) is one of the most vital numbers in quantum mechanics. It's like the address of an electron within an atom, telling us its main energy level.

What you should remember:

• \(n\) is a positive integer \((1, 2, 3,\ldots)\) and indicates the general size and energy of an electron shell.
• Higher \(n\) values mean electrons are further from the nucleus and at a higher energy state.
• The total number of orbitals in an energy level is \(n^2\).

Understanding the principal quantum number is the first step in pinpointing where an electron might be in an atom.