Problem 94 - Chemistry The Molecular Nature of Matter

Question

What is the letter code for a subshell with (a) \(\ell=2\) and (b) \(\ell=4 ?\)

Step-by-Step Solution

Answer

The letter code for a subshell with \(\ell=2\) is d and with \(\ell=4\) is g.

Step 1: Identify Subshell Code

The value of the azimuthal or angular momentum quantum number \(\ell\) determines the subshell letter code. The code is based on a historical naming system: \(\ell=0\) corresponds to s, \(\ell=1\) corresponds to p, \(\ell=2\) corresponds to d, \(\ell=3\) corresponds to f, \(\ell=4\) has historically been assigned g, and so on, continuing alphabetically (omitting j).

Step 2: Find the Letter Code for \(\ell=2\)

Using the naming system, \(\ell=2\) corresponds to the d subshell. Thus, the letter code for a subshell with \(\ell=2\) is d.

Step 3: Find the Letter Code for \(\ell=4\)

Following the same pattern, the naming system gives \(\ell=4\) the letter code g. Therefore, the letter code for a subshell with \(\ell=4\) is g.

Key Concepts

Azimuthal Quantum Number

The azimuthal quantum number, represented by the symbol \(\ell\), plays a crucial role in the quantum mechanical model of an atom. It's essential for determining the angular momentum of an electron within an atom and its subshell level. Each subshell is associated with a different value of \(\ell\), starting from 0 and moving upwards.

For instance, when we talk about an \(\ell=0\), we refer to the 's' subshell, which has a spherical shape around the nucleus. If \(\ell=1\), we have a 'p' subshell that comprises lobed shapes oriented in three-dimensional space. Increasing values of \(\ell\) indicate more complex shapes for the subshell, like the 'd' subshell for \(\ell=2\) and the 'f' subshell for \(\ell=3\). These shapes are crucial as they describe the regions in space where there is a high probability of finding an electron.

Importance of the Azimuthal Quantum Number

The value of \(\ell\) defines not only the shape of the subshell but also the energy level and number of orbitals it contains. The azimuthal quantum number also restricts the number of electrons that a particular subshell can hold, which is given by the formula \(2(2\ell+1)\). This quantum number is intimately linked with the periodic table's structure, providing a way to understand the underlying rules that govern the electron configuration in elements.

Subshell Notation

Subshell notation is a simple way to represent the electron configuration within subshells of an atom. It's part of what we call the electron configuration notation and incorporates the principal quantum number, the azimuthal quantum number, and the number of electrons in the subshell.

The principal quantum number, usually represented by \(n\), indicates the energy level or shell of an electron. The azimuthal quantum number \(\ell\) determines the subshell within that energy level, represented as a letter. For example, \(0 = s, 1 = p, 2 = d, 3 = f\), etc. This is then followed by a superscript number denoting the number of electrons existing within that subshell. For instance, if we have two electrons in the 's' subshell of the first energy level, it would be notated as 1s².

Exercise Application

In the exercise provided, it asks for the letter code of subshells with \(\ell=2\) and \(\ell=4\). According to our quantum number-to-letter correlation, we assign 'd' to \(\ell=2\) and 'g' to \(\ell=4\). Combining this with a principal quantum number, we could then derive notation such as 3d or 5g, telling us the energy level and the type of subshell where the electrons reside.

Electron Subshells

Electron subshells are subdivisions of electron shells, which are energy levels around an atom's nucleus where electrons are likely to be found. The concept of subshells provides a deeper understanding of electron organization within an atom. The distribution of electrons in subshells affects many properties of the atom including its chemical behavior and magnetic properties.

Each electron shell can contain one or more subshells, and these subshells are filled according to energy, from lowest to highest. For instance, the first shell (\(n=1\)) has only one subshell, which is the 1s. The second shell (\(n=2\)) has two subshells, labeled as 2s and 2p. As the principal quantum number \(n\) increases, more subshells become available (e.g., 3s, 3p, and 3d).

Order of Filling Subshells

The order in which subshells are filled is governed by the Aufbau principle, which states that electrons fill orbitals starting with the lowest available energy states before filling higher states. Hence, electrons will fill the 1s subshell before the 2s subshell, and so forth. This order helps determine the electronic configuration of elements and, with it, their placement and behavior in the periodic table.