Which electron is, on average, farther from the nucleus: an electron in a \(3 p\) orbital or an electron in a \(4 p\) orbital?

The principal quantum number, denoted by the symbol \(n\), plays a crucial role in understanding an electron's behavior within an atom. It defines the energy level or shell in which an electron resides, and it's integral in determining an electron's average distance from the nucleus. An important thing to note is that this number is always a positive integer—1, 2, 3, and so on.

As \(n\) increases, the electron’s energy level and its possible distance from the nucleus increase as well, meaning an electron in a shell with a higher principal quantum number will generally be farther from the nucleus. This explains why in the comparison between a 3p and 4p orbital, the electron in the 4p orbital is farther away—it's residing in an energy level described by a higher principal quantum number (\(n=4\)) compared to the 3p orbital (\(n=3\)).

This also implies that each energy level can hold more electrons than the one before it, related to the formula \(2n^2\), which indicates the maximum number of electrons within a shell.

Closely related to the principal quantum number is the angular momentum quantum number, symbolized by \(l\). This quantum number provides insights into the shape of an atomic orbital and can have any integer value ranging from 0 to \(n-1\). For example, if an electron is in the second shell (\(n=2\)), the possible values of \(l\) can be 0 or 1.

The value of \(l\) is associated with the subshell designation: 0 corresponds to an s orbital, 1 to a p orbital, 2 to a d orbital, and 3 to an f orbital. Thus, a subshell with a higher \(l\) value will generally have a more complex shape, and the electron density will be distributed differently. It's important to remember, however, that \(l\) does not directly indicate the distance from the nucleus; it instead influences the shape of the electron's probability cloud within a given principal energy level.

Atomic orbitals are regions in space where there is a high probability of finding an electron. Each orbital is defined by a unique set of quantum numbers, and these orbitals occupy different shapes such as spherical for s orbitals and dumbbell-shaped for p orbitals.

When visualizing orbitals, keep in mind these are not fixed paths that electrons travel, but rather probability distributions that indicate where an electron is likely to be found. Since each principal energy level can contain one s orbital, three p orbitals, five d orbitals, and seven f orbitals, the complexity and spatial extent of these orbitals increase with higher principal quantum numbers and angular momentum quantum numbers. This complexity contributes to the unique architecture of atoms and how they bond with one another.

Electron configuration is the way in which electrons are distributed among the atomic orbitals of an atom. It's essentially the 'address' of each electron, providing a map of its location in terms of energy levels, sublevels, and specific orbitals.

To understand electron configuration, scientists use the Aufbau principle, the Pauli exclusion principle, and Hund's rule. These principles help determine the electron arrangement that corresponds to the lowest energy state of an atom. Typically, electrons will fill the lowest energy orbitals first (1s before 2s, 2s before 2p, etc.) and each orbital can hold a maximum of two electrons with opposite spins. A fully detailed electron configuration can provide a wealth of information about an atom's reactivity, bonding capabilities, magnetic properties, and more.