Question

What is the maximum number of electrons that can be identified with each of the following sets of quantum numbers? In some cases, the answer may be "none." In such cases, explain why "none" is the correct answer. (a) \(n=3\) (b) \(n=3\) and \(\ell=2\) (c) \(n=4, \ell=1, m_{\ell}=-1,\) and \(m_{\mathrm{s}}=+1 / 2\) (d) \(n=5, \ell=0, m_{\ell}=-1, m_{\mathrm{s}}=+1 / 2\)

Short answer

(a) 18, (b) 10, (c) 1, (d) none.

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers help portray the position and energy of an electron within an atom. The principle quantum number, \(n\), defines the electron's energy level. The azimuthal quantum number, \(\ell\), determines the orbital's shape. The magnetic quantum number, \(m_\ell\), gives the spatial orientation of the orbital. Finally, the spin quantum number, \(m_s\), indicates the electron's spin orientation.

Analyze n=3

The principal quantum number \(n=3\) can support the following \(\ell\) values: 0 (s), 1 (p), and 2 (d). Each \(\ell\) value plays host to various orbitals defined by \(m_\ell\). The possible orbitals for all of these are 1 for \(\ell=0\), 3 for \(\ell=1\), and 5 for \(\ell=2\). Each orbital can hold 2 electrons. Thus, for \(n=3\), the total maximum electrons = \(2(1 + 3 + 5) = 18\).

Checking n=3 and ℓ=2

With \(n=3\) and \(\ell=2\), we're referring specifically to the 3d subshell. The possible \(m_\ell\) values are -2, -1, 0, 1, and 2, representing five orbitals. Each orbital can hold 2 electrons, allowing a total of \(5 \times 2 = 10\) electrons in the 3d subshell.

Examine n=4, ℓ=1, m_ℓ=-1, m_s=+1/2

The given values \(n=4\), \(\ell=1\), \(m_\ell=-1\), and \(m_s=+1/2\) refer to one electron in a specific orbital (4p subshell, p orbital). Since the spin direction \(m_s\) indicates only one electron, the maximum number is 1.

Evaluate n=5, ℓ=0, m_ℓ=-1, m_s=+1/2

The condition \(n=5\), \(\ell=0\), \(m_\ell=-1\), and \(m_s=+1/2\) is problematic. In the case of \(\ell=0\), \(m_\ell\) should be 0. Here, \(m_\ell=-1\) is not valid for \(\ell=0\). Therefore, no electrons can have such quantum numbers.

Key concepts

Principle Quantum Number

The principle quantum number, represented as \(n\), plays a crucial role in defining an electron’s energy level in an atom. It is the very first number in a set of four that helps describe the unique state of an electron.
Understanding the principle quantum number is like understanding the basics of a ladder.

• Each successive value of \( n \) signifies an additional energy level further from the nucleus.
• For instance, \( n=1 \) indicates the closest energy level to the nucleus, while \( n=3 \) would indicate a level further away.

The value of \( n \) determines the maximum number of electrons within a particular energy level. If we take \( n=3 \) as an example, it can host a total of 18 electrons. This calculation comes from the sum of electrons that each azimuthal quantum number can contribute to within the \( n=3 \) level.

In summary, \( n \) is a fundamental number highlighting where an electron is likely to be found and how much energy it possesses within an atom.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( \ell \), is a key player in determining the shape of the electron's orbital. While the principle quantum number sets the stage, \( \ell \) adds the specifics on orbital types.
Here’s how the azimuthal quantum number works:

• The value of \( \ell \) ranges from 0 to \( n-1 \).
• Each \( \ell \) corresponds to a specific type of orbital shape: spherical (\( \ell=0 \)), polar (\( \ell=1 \)), or cloverleaf (\( \ell=2 \)), among others.

For example, if \( n=3 \) and \( \ell=2 \), it refers to the d subshell, which has a more complex shape than a p or s orbital. Each of these has designated subshapes and orientations, defining the geometry of where an electron can be found inside the atom.

Understanding \( \ell \) is crucial as it guides scientists in predicting how electrons move within atoms, hinting at the overall structure and reactivity of an element.

Magnetic Quantum Number

The magnetic quantum number, symbolized as \( m_\ell \), further refines our understanding of an electron's orbital by specifying its orientation in space. This quantum number depends on \( \ell \), adding another layer of detail to orbital information.
To grasp the function of \( m_\ell \), consider these points:

• For each value of \( \ell \), the magnetic quantum number \( m_\ell \) can take on values ranging from \( -\ell \) to \( +\ell \).
• This means if \( \ell=2 \), \( m_\ell \) can be any of the five values: -2, -1, 0, +1, or +2.
• Each value of \( m_\ell \) is associated with a unique orientation of the orbital in 3D space, allowing for multiple electrons within different orientations of the same subshell.

The presence of \( m_\ell \) confirms that not only can orbitals vary in general shape, but also in specific orientations, thus influencing elements’ chemical properties and spatial configurations.

Understanding \( m_\ell \) is indispensable for accurately painting the picture of electron arrangements within atoms.

Spin Quantum Number

The spin quantum number, denoted as \( m_s \), is essential in describing an electron's intrinsic spin—essentially the direction it rotates on its own axis. It’s the final piece in the puzzle of quantum numbers that define an electron’s state.
Let’s break down why \( m_s \) is fundamental:

• \( m_s \) can have only two possible values: +1/2 or -1/2.
• These values refer to an electron's spin direction, making it either "spin up" or "spin down".

This simple distinction significantly affects the electron's interaction with other electrons and magnetic fields, adhering to the Pauli exclusion principle: no two electrons in the same atom can have identical sets of quantum numbers.
Finally, the spin quantum number ensures that even within the same orbital, electrons can coexist with opposite spins, thus allowing atoms to maintain stability and complex structures.