Allowed values for the quantum numbers of electrons are as follows: \\[ \begin{aligned} n &=1,2,3, \ldots \\ l &=0,1,2,3, \ldots, n-1 \\ m_{l} &=0,\pm 1,\pm 2,\pm 3, \ldots, \pm l \\ m_{s} &=\pm \frac{1}{2} \end{aligned} \\] The relationships between \(n\) and the shell designations are noted in Table \(2.1 .\) Relative to the subshells, \(l=0\) corresponds to an \(s\) subshell \(l=1\) corresponds to a \(p\) subshell \(l=2\) corresponds to a \(d\) subshell \(l=3\) corresponds to an \(f\) subshell For the \(K\) shell, the four quantum numbers for each of the two electrons in the 1 s state in the order of \(n l m_{l} m_{s},\) are \(100\left(\frac{1}{2}\right)\) and \(100\left(-\frac{1}{2}\right) .\) Write the four quantum numbers for all of the electrons in the \(L\) and \(M\) shells, and note which correspond to the \(s, p,\) and \(d\) subshells.

Understanding electronic configuration is fundamental in chemistry, as it describes how electrons are distributed in the orbitals of an atom. The distribution follows a set of principles – the Aufbau principle, Pauli exclusion principle, and Hund's rule. The Aufbau principle states that electrons fill orbitals starting with the lowest energy levels first. The Pauli exclusion principle asserts that no two electrons can have the same set of four quantum numbers, meaning each orbital can hold a maximum of two electrons with opposite spins. Hund's rule affirms that electrons will fill an equal energy orbital (degenerate) singly before filling them in pairs.

The electronic configuration can be written using the nomenclature 'ns', 'np', 'nd', and 'nf', where 'n' is the principal quantum number indicating the energy level, and 's', 'p', 'd', and 'f' are the subshell labels derived from the angular momentum quantum number 'l'. An example is carbon, with an electronic configuration of 1s² 2s² 2p², meaning it has 2 electrons in both its 1s and 2s subshells, and 2 electrons in its 2p subshell.

Atomic orbitals represent regions in space where electrons are likely to be found. Each orbital is defined by three quantum numbers: the principal quantum number 'n', the angular momentum quantum number 'l', and the magnetic quantum number 'm_l'. The value of 'n' determines the size and energy level of the orbital, while 'l' determines its shape. There are different shapes for orbitals: 's' is spherical, 'p' is dumbbell-shaped, 'd' is cloverleaf-shaped, and 'f' orbitals have more complex shapes.

The magnetic quantum number 'm_l' gives information about the orientation of the orbital in space. For example, p-orbitals have three possible orientations, corresponding to 'm_l' values -1, 0, and 1. These properties affect the chemical bonding and properties of an atom. When assigning electrons to orbitals, it's important to know that electrons will fill the lowest energy orbitals first, a practice that can predict the reactive nature and bonding capabilities of an element.

Electron shells are areas that surround the nucleus of an atom where electrons move about. These shells are labeled with integer numbers starting from 1, which correlate to the principal quantum number 'n'. The value of 'n' not only represents the shell (like K, L, M) but also denotes the energy level of an electron – the larger the number, the higher the energy and the farther from the nucleus the shell is.

Each shell can hold a specific number of electrons: the first shell (n=1) can hold up to 2 electrons, the second shell (n=2) can hold up to 8, and so on, following the pattern 2n². These shells are divided into subshells (s, p, d, f) based on the angular momentum quantum number 'l'. As the principal quantum number increases, the number of subshells also increases, accommodating more electrons. The filling of these shells and subshells follows a set order dictated by the rules mentioned in the electronic configuration. It's important to note how the chemical properties and periodicity of elements are deeply connected to the arrangement of their electron shells.