Question

What are the possible values for \(\mathbf{m}_{\ell}\) for (a) the d sublevel? (b) the s sublevel? (c) all sublevels where \(\mathbf{n}=5\) ?

Short answer

Answer: (a) For the d sublevel, the possible values for \(m_\ell\) are -2, -1, 0, 1, 2. (b) For the s sublevel, the only possible value for \(m_\ell\) is 0. (c) For all sublevels where \(n=5\), the possible values of \(m_\ell\) are: s sublevel: 0; p sublevel: -1, 0, 1; d sublevel: -2, -1, 0, 1, 2; f sublevel: -3, -2, -1, 0, 1, 2, 3; g sublevel: -4, -3, -2, -1, 0, 1, 2, 3, 4.

Step-by-step solution

(a) d sublevel

For the d sublevel, the azimuthal quantum number \(\mathbf{l}=2\). Therefore, the possible values for \(\mathbf{m}_{\ell}\) are: \(-\mathbf{l}, -\mathbf{l}+1, \dots, \mathbf{l}-1, \mathbf{l}\) Substituting \(\mathbf{l}=2\), we get: \(-2, -1, 0, 1, 2\) So, the possible values for \(\mathbf{m}_{\ell}\) for the d sublevel are \(-2, -1, 0, 1, 2\).

(b) s sublevel

For the s sublevel, the azimuthal quantum number \(\mathbf{l}=0\). Therefore, the possible values for \(\mathbf{m}_{\ell}\) are: \(-\mathbf{l}, -\mathbf{l}+1, \dots, \mathbf{l}-1, \mathbf{l}\) Substituting \(\mathbf{l}=0\), we get: \(0\) So, the only possible value for \(\mathbf{m}_{\ell}\) for the s sublevel is \(0\).

(c) all sublevels where \(\mathbf{n}=5\)

For an energy level \(\mathbf{n}=5\), we must first identify the possible values for the azimuthal quantum number \(\mathbf{l}\), which range from \(0\) to \(\mathbf{n}-1\). In this case, the possible values for \(\mathbf{l}\) are \(0, 1, 2, 3, 4\). Now, let's find the possible values for \(\mathbf{m}_{\ell}\) for each \(\mathbf{l}\): 1. For \(\mathbf{l}=0\) (s sublevel), \(\mathbf{m}_{\ell}=0\). 2. For \(\mathbf{l}=1\) (p sublevel), \(\mathbf{m}_{\ell}=-1, 0, 1\). 3. For \(\mathbf{l}=2\) (d sublevel), \(\mathbf{m}_{\ell}=-2, -1, 0, 1, 2\). 4. For \(\mathbf{l}=3\) (f sublevel), \(\mathbf{m}_{\ell}=-3, -2, -1, 0, 1, 2, 3\). 5. For \(\mathbf{l}=4\) (g sublevel), \(\mathbf{m}_{\ell}=-4, -3, -2, -1, 0, 1, 2, 3, 4\). So, for all sublevels where \(\mathbf{n}=5\), the possible values of \(\mathbf{m}_{\ell}\) are: s sublevel: \(\mathbf{m}_{\ell}=0\). p sublevel: \(\mathbf{m}_{\ell}=-1, 0, 1\). d sublevel: \(\mathbf{m}_{\ell}=-2, -1, 0, 1, 2\). f sublevel: \(\mathbf{m}_{\ell}=-3, -2, -1, 0, 1, 2, 3\). g sublevel: \(\mathbf{m}_{\ell}=-4, -3, -2, -1, 0, 1, 2, 3, 4\).

Key concepts

Azimuthal Quantum Number

Understanding the azimuthal quantum number, also known as the angular momentum quantum number, is crucial to comprehending the layout of electrons within an atom. This quantum number, represented by the symbol , takes on integer values ranging from 0 up to -1, where is the principal quantum number associated with the energy level of the electron.

Determines the shape of the electron cloud for an electron within a particular orbital. For instance, when is 0, the electron is in an s orbital, which is spherical. As the value of increases, the complexity of the orbital shapes also increases; equals 1 for p orbitals, 2 for d orbitals, and so on. These shapes are significant as they dictate the electron cloud's spatial distribution and how electrons will interact with each other as well as with external fields and particles.

The exercise specifically addressed the d sublevel, which corresponds to equal to 2. Each increment in brings about a new sublevel with its unique set of orbitals, leading to a more complex electron arrangement and behavior within atoms.

Magnetic Quantum Number

The magnetic quantum number, denoted as , arises from the quantization of an electron's angular momentum component in the direction of an external magnetic field. This number can have integer values ranging from - up to +, including zero. So, for each value of the azimuthal quantum number, , there are 2 + 1 possible values for , representing different orientations of an orbital in space.

For example, when considering a d sublevel (where equals 2), the possible values which determine the orientation of the orbital, are -2, -1, 0, 1, and 2. This results in five distinct orientations for d orbitals, such as dxz, dyz, and so on. The exercise demonstrates how the number of possible magnetic quantum numbers increases with the value of , which is directly tied to the variety of orbital orientations available within a sublevel.

This variability in the magnetic quantum number is critical for predicting the electron configuration, especially when identifying the atomic and magnetic properties of an element, as it impacts how atoms will align themselves in magnetic fields and how they will bond with other atoms.

Electron Sublevels

Electron sublevels are divisions within an energy level of an atom and are defined by the azimuthal quantum number, . These sublevels help classify orbitals into different shapes, which plays a substantial role in the chemical bonding and properties of elements. The sublevels are labeled as s, p, d, f, and so on, depending on the value of , starting from 0 for s, 1 for p, 2 for d, and increasing with the energy level of the electrons.

The exercise touches on the understanding that each principal energy level (designated by the principal quantum number, ) incorporates various sublevels. For example, for equal to 5, there are five sublevels ranging from s to g. Every sublevel consists of orbitals where electrons are likely to be found, and the number of these orbitals increases with higher values. Thus, the complexity and possibilities for electron configuration grow with higher energy levels.

These sublevels are also tied to the periodic table's structure, aiding in the prediction of an element's behavior during chemical reactions. Knowing the distribution of electrons across various sublevels is fundamental to the study of atomic and molecular properties.