Question

For the shell with \(n=6\), what are the possible values of \(\ell ?\)

Short answer

The possible values of \(\ell\) for \(n=6\) are 0, 1, 2, 3, 4, and 5.

Step-by-step solution

Recall the quantum number rules

Quantum numbers are sets of numerical values that give information about the properties of electrons in atoms. For any principal quantum number, n, the angular momentum quantum number, \(\ell\), can take on any integer value from 0 to \(n - 1\).

Apply the rule to the given principal quantum number

Since the given shell has a principal quantum number (n) of 6, we apply the rule to find the possible values of \(\ell\). \(\ell\) can be any integer from 0 to \(n - 1\), which in this case is 0 to 5.

List all possible values of \(\ell\)

Based on the rule, the possible values of \(\ell\) for \(n=6\) are 0, 1, 2, 3, 4, and 5.

Key concepts

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as \(\ell\), is integral to understanding the shape of electron orbitals within atoms. It tells us about the shape of the electron cloud associated with an atomic orbital. For instance, \(\ell = 0\) corresponds to s-orbitals, which have a spherical shape, while \(\ell = 1\) corresponds to p-orbitals, which are dumbbell-shaped.

These orbitals are not just abstract concepts; they dictate how electrons behave and interact with other particles, as their shape defines the volume in space where there's a high probability of finding an electron. As seen in the exercise solution, for a given principal quantum number \(n\), the values of \(\ell\) range from 0 up to \(n - 1\). Thus, for \(n = 6\), \(\ell\) can take on the values 0, 1, 2, 3, 4, or 5, representing various types of orbitals (s, p, d, f, and so on) available to the electrons in that shell.

Quantum Number Rules

Quantum number rules are a set of principles that govern the allowed values for quantum numbers, which ultimately describe the arrangement of electrons in atoms. Besides the angular momentum quantum number, there are other quantum numbers such as the principal quantum number (\(n\)), magnetic quantum number (\(m_\ell\)), and spin quantum number (\(m_s\)).

Let's break down these rules:

• The principal quantum number \(n\) can be any positive integer, signifying the shell or energy level.
• The angular momentum quantum number \(\ell\) follows the condition \(0 \leq \ell < n\), dictating the shape of the orbital.
• The magnetic quantum number \(m_\ell\) takes values from \(\ell\) to \(\ell\) (including negatives), indicating the orientation of the orbital in space.
• The spin quantum number \(m_s\) can be either +1/2 or -1/2, which shows the two possible spin states of an electron.

These rules are not merely mathematical constructs but are deeply connected to the physical properties of atoms and reflect the restrictions imposed by the laws of quantum mechanics.

Electron Properties in Atoms

The properties of electrons in atoms are intrinsic to the behavior of atoms and dictate how they connect to form molecules. Quantum numbers provide a detailed description of an electron's characteristics.

An electron's property includes its energy level, spatial distribution, magnetic field interaction, and intrinsic angular momentum (spin). Electrons occupy discrete energy levels or shells, which can be thought of as distinct distances from the nucleus where electrons are likely to be found. These levels are quantified by the principal quantum number (\(n\)).

The angular momentum quantum number (\(\ell\)) affects the electron's angular momentum and also contributes to the shape of the region in space where finding an electron is most probable. Meanwhile, the magnetic quantum number (\(m_\ell\)) determines the orientation of the electron's orbital in a magnetic field. Lastly, the spin quantum number (\(m_s\)) represents the electron's spin, which contributes to the magnetic properties of the atom.

Understanding these properties aids in explaining chemical bonding, the periodic table arrangement, and the magnetic properties of materials, illustrating that these quantum numbers are not only crucial for theoretical chemistry but also for practical applications in material science and technology.