Quantum numbers are used to determine the number of electrons in an atom indirectly by specifying the allowed energy levels and orbital shapes within an atom. Each electron in an atom is described by a unique set of quantum numbers, which provide information about its energy, angular momentum, orientation, and spin.
The principal quantum number (n) indicates the energy level or shell in which an electron resides. It defines the size of the orbital and the average distance of the electron from the nucleus. The values of n are positive integers starting from 1 (for the innermost shell) and increase as you move farther from the nucleus.
The azimuthal quantum number (l) determines the shape of the electron's orbital. It specifies the subshell or sublevel within a given energy level and provides information about the orbital angular momentum of the electron. The values of l depend on the principal quantum number (n) and can range from 0 to (n-1). The corresponding subshells are denoted by the letters s, p, d, f, and so on, representing different orbital shapes.
The magnetic quantum number (m_l) defines the specific orientation or spatial distribution of the orbital within a subshell. It indicates the number of allowed orbitals within a given subshell and can take on integer values ranging from -l to +l, including zero.
Lastly, the spin quantum number (m_s) describes the spin of the electron. It can have two possible values: +1/2 or -1/2, representing the two opposite spin directions.
The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers. Therefore, each combination of the quantum numbers (n, l, m_l, m_s) can only be occupied by two electrons, with opposite spins.
To determine the total number of electrons in an atom, one needs to consider the filling order of the energy levels and subshells according to the aufbau principle, Hund's rule, and the Pauli exclusion principle. These rules guide the arrangement of electrons in the orbitals, starting from the lowest energy level and filling them in a specific order until all electrons are accounted for.
By understanding the principles governing the quantum numbers and their relationship to the electronic structure of an atom, scientists can determine the number of electrons in each shell, subshell, and orbital of an atom.