Of the following sets of four quantum numbers \(\left\\{n, l, m_{l}, m_{s}\right\\}\), identify the ones that are forbidden for an electron in an atom and explain why they are invalid: (a) \(\left\\{4,2,-1,+\frac{1}{2}\right\\}\); (b) \(\left\\{5,0,-1,+\frac{1}{2}\right\\}\); (c) \(\left\\{4,4,-1,+\frac{1}{2}\right\\}\).

The principal quantum number, denoted as n, plays a vital role in the quantum mechanics of atoms. It is a positive integer (1,2,3,...) and is the first of the four quantum numbers that define the state that an electron can be in. Think of it as the main address for electrons in an atom, determining the energy level or shell in which an electron resides. The higher the value of n, the further the electron is from the nucleus, and it also represents higher energy states.

For every principal quantum number, there are n possible values for the azimuthal quantum number, which relates to the shape of the electron's orbit, ranging from 0 to n - 1. This concept is a fundamental one for students to grasp as it lays the groundwork for understanding the more complex quantum numbers and is critical for proper electron configuration.

Closely following the principal quantum number is the azimuthal quantum number, l, which also goes by the name angular momentum quantum number. The possible values of l are integers ranging from 0 up to n - 1, where n is the principal quantum number.

This quantum number gives the shape of the orbital, often associated with subshells designated as s (l=0), p (l=1), d (l=2), and f (l=3). Understanding l is crucial because it helps in pinpointing the exact subshell in which an electron is located within an energy level. Incorrect values of l that do not align with the principal quantum number are not permissible, which explains why certain sets of quantum numbers can be invalid.

Within each subshell denoted by the azimuthal quantum number, there are multiple orientations in space that an orbital can have, and these orientations are specified by the magnetic quantum number, m_l. It is dependent on the value of l and can range from -l to +l, including zero.

This means for a p-orbital where l=1, the m_l can be -1, 0, or +1, corresponding to the three different p-orbitals oriented in space. The wrong assignment of m_l outside this range is a common mistake for students, leading to invalid quantum numbers. For instance, if l=0 for an s-orbital, there can only be a single value for m_l, which is 0.

The final piece of the quantum number puzzle is the spin quantum number, m_s, which represents the intrinsic angular momentum of an electron. Unlike the other quantum numbers, m_s has only two possible values: +1/2 or -1/2. These values signify the two possible spin orientations of the electron: 'spin-up' or 'spin-down'.

An important note for students to remember is that no two electrons in the same atom can have the exact same set of four quantum numbers due to Pauli's Exclusion Principle. Hence, the spin quantum number allows two electrons (with opposite spins) to share the same orbital, as they will have different m_s values.

After understanding the quantum numbers, the next step is to talk about electron configuration. This is the arrangement of electrons in an atom's orbitals, represented by notations that indicate the principal and azimuthal quantum numbers, followed by the number of electrons in those orbitals.

For example, the electron configuration notation '2s²' indicates two electrons in the s subshell of the second energy level. This notation provides a map of how electrons are distributed in an atom and are dictated by the rules established through the quantum numbers. Students must remember that electrons fill the lowest energy orbitals first (Aufbau principle), can only double up in orbitals if they have opposite spins (Pauli), and prefer to fill orbitals singly before pairing up (Hund's rule). The grasp on these concepts ensures a full understanding of how atoms are constructed in the microscopic world.