What is the value of \(\ell\) for (a) a \(p\) orbital and (b) a \(g\) orbital?

Orbital notation is a way to represent the electron configuration of atoms. It articulates not only the energy levels and types of orbitals that electrons occupy but also their spin states. Each orbital is denoted by a combination of letters and numbers that correspond to its energy level, type, and number of electrons. For instance, the notation 2p\(^4\) indicates there are four electrons in a p-type orbital in the second energy level.

The angular momentum quantum number, represented by \( \ell \), is integral to understanding orbital notation. This number determines the shape of the orbital and ranges from 0 to \(n-1\), where \(n\) is the principal quantum number or the energy level. The value of \( \ell \) for different types of orbitals follows an order, with 0 for s orbital, 1 for p, 2 for d, and so on. The specific value of angular momentum quantum number allows us to determine the type of orbital and vice versa. Thus, orbital notation serves as a visual guide to the arrangement and types of orbitals that electrons inhabit in an atom.

A p orbital, corresponding to the angular momentum quantum number \( \ell = 1 \) is distinctive for its dumbbell-like shape. There are three p orbitals at each energy level, starting from the second energy level upwards, and they are oriented perpendicularly to each other along the x, y, and z axes, named px, py, and pz, respectively.

The p orbitals are significant in explaining the geometry and bonding of molecules. For example, the formation of a water molecule involves the overlap of oxygen's p orbitals with hydrogen's s orbitals, leading to the bent structure of H\(_2\)O. A full p orbital can hold up to six electrons, following the Pauli exclusion principle and Hund's rule for electron distribution. Understanding the p orbital's relationship to angular momentum is crucial for predicting molecular structures and the behavior of atoms during chemical reactions.

Though often not discussed in introductory chemistry due to its lack of occupancy in ground-state atoms of elements currently known, the g orbital is a hypothetical region of space that an electron could occupy. It is represented by the angular momentum quantum number \( \ell = 4 \).

G orbitals come into play when discussing very heavy elements, or in certain high-energy states where electrons may be excited enough to occupy these orbitals. There would be five different types of g orbitals, each with varying complex shapes beyond the commonly recognized s, p, and d orbital shapes. Each g orbital can theoretically hold a maximum of two electrons with opposite spins, similar to other orbitals following the Pauli exclusion principle. Discussions about g orbitals are typically more theoretical but contribute to the expanded understanding of quantum chemistry and the potential chemical behavior of super-heavy elements.