What is the maximum number of electrons in an atom that can have the quantum numbers \(n=3\) and \(l=2 ?\) (a) 2 (b) 5 (c) 6 (d) 10

The principal quantum number, denoted as (), is integral to understanding the structure of the atom and electron configurations. It signifies the energy level or shell where an electron resides and greatly influences the electron's energy. With each increase in ()'s value, the electron's energy increases and it occupies a higher energy level, moving further away from the nucleus.

For instance, ()=1 indicates the first and closest energy level to the nucleus, which has the lowest energy. As the () increases, the capacity of the energy levels to hold electrons also increases, following a pattern of 2()^2. So, a principal quantum number of 3 (()=3), as seen in the exercise, implies the third energy level with a higher energy than the first two levels and a capacity to hold more electrons. This concept is foundational for understanding the distribution of electrons within an atom.

The azimuthal quantum number, identified by (l), is also known as the angular momentum quantum number. It ranges from 0 to ()-1 for any given principal quantum number (), defining the shape of the atomic orbitals and splitting the shells into subshells or sublevels. Each of these subshells is labeled with specific letters: s ((l)=0), p ((l)=1), d ((l)=2), and f ((l)=3).

In the context of our problem, (l)=2 corresponds to the d subshell, which has a more complex shape than the s or p subshells. The d subshell plays a crucial role in the electron configuration of transition metals and influences the chemical behavior and bonding of elements that contain d electrons.

Diving deeper into the d subshell electron configuration, it becomes apparent that this subshell can encompass a maximum of 10 electrons. This is due to the existence of five d orbitals, each capable of holding two electrons according to the Pauli exclusion principle. The configuration within the d subshell follows Hund's rule, which states that electrons will occupy empty orbitals singly before pairing up.

This behavior maximizes the total spin and minimizes the repulsion between electrons, keeping them as unpaired as possible. The d subshell is particularly interesting because of its contribution to the unique properties of transition metals, such as variable oxidation states and magnetic behavior, which are influenced by the arrangement of d electrons.

Atomic orbitals form the final piece of our quantum puzzle. These are regions around an atom's nucleus where there is a high probability of finding an electron. They are described by the combination of () and (l) quantum numbers, shaped distinctly for s, p, d, and f subshells.

For (l)=2, as mentioned in the exercise, there are five different d orbitals, often referred to as ddz^2, ddx^2-y^2, ddxy, ddxz, and ddyz, representing the various orientations that these orbitals assume in three-dimensional space. It is fascinating to think about how these seemingly abstract concepts dictate the very real and observable characteristics of atoms, from their structure to how they bond and interact with each other.