How many electrons can occupy a subshell with \(l=\) (a) 0 ; (b) \(1 ;\) (c) \(2 ;\) (d) 3 ?

Quantum numbers play a pivotal role in understanding the arrangement of electrons within an atom, defining the position and energy of electron orbitals. There are four quantum numbers: the principal quantum number (), the azimuthal (angular momentum) quantum number (), the magnetic quantum number (), and the spin quantum number ().

Each electron in an atom is described by this unique set of quantum numbers, following the Pauli exclusion principle, which states that no two electrons can have the same set of all four quantum numbers. The principal quantum number () indicates the energy level and approximate distance of the electron from the nucleus, while the azimuthal quantum number () determines the shape of the atomic orbital.

The azimuthal quantum number, often represented by the symbol (l), plays a crucial part in the electronic structure of an atom. It defines the shape of an atomic orbital and also the energy sublevels within a principal energy level or shell. The value of (l) can range from 0 to (n-1), where (n) is the principal quantum number.

The azimuthal quantum number is integral in determining the electron capacity of a subshell. For instance, when (l) equals 0, 1, 2, or 3, it defines the (s), (p), (d), and (f) subshells respectively, each with its unique shape and varying number of orbitals.

Atomic orbitals are regions around an atom's nucleus where the probability of finding an electron is high. Electrons do not travel in fixed paths, as in the older Bohr model, but in more complex patterns that define an electron cloud. There are different types of orbitals ((s), (p), (d), and (f)), each with a unique shape and orientation in space.

For example, the (s) orbital is spherical, and there is only one (s) orbital in each energy level. The (p) orbitals are dumbbell-shaped and oriented in three-dimensional space along the x, y, and z axes. Higher energy levels contain (d) and (f) orbitals, which have even more complex shapes and are occupied as the atom's energy increases.

Subshell electron configuration describes how electrons are distributed within the subshells of an atom's orbitals. The distribution is governed by the aforementioned quantum numbers and is expressed using notations like 1(s)², 2(p)⁶, and so on, where the numbers indicate the energy level, the letters are the subshell types, and the superscripts represent the number of electrons in those subshells.

The electron capacity of each subshell is directly related to the azimuthal quantum number, (l), using the formula 2(2(l) + 1). This formula yields the maximum number of electrons that a subshell can hold, helping to predict an element's electron arrangement and its chemical properties. Understanding these configurations is essential in fields such as chemistry and materials science, where electron distributions influence how substances interact and behave.