According to the Pauli exclusion principle, no two electrons within an atom can have the exact same set of quantum numbers, which include the principal quantum number (n), the azimuthal quantum number (ℓ), the magnetic quantum number (mℓ), and the spin quantum number (ms).
For a given atom, the principal quantum number (n) determines the energy level or shell, the azimuthal quantum number (ℓ) determines the subshell or orbital shape, the magnetic quantum number (mℓ) specifies the orientation of the orbital in space, and the spin quantum number (ms) describes the spin of the electron.
Since each set of quantum numbers must be unique, this means that within a particular energy level (n), there can be multiple subshells (ℓ) with different shapes, and each subshell can contain a certain number of orbitals. Each orbital can accommodate a maximum of two electrons, with opposite spins.
The maximum number of electrons that can have the same quantum state, therefore, depends on the values of the quantum numbers. The equation to determine the maximum number of electrons in a given shell is 2n², where n is the principal quantum number.
For example:
- In the first energy level (n = 1), there is only one subshell (s) and a single orbital. Thus, it can accommodate a maximum of 2 electrons.
- In the second energy level (n = 2), there are two subshells (s and p). The s subshell has a single orbital, and the p subshell has three orbitals. Hence, the second energy level can hold a maximum of 2 + 2(3) = 8 electrons.
- In the third energy level (n = 3), there are three subshells (s, p, and d), which can accommodate a total of 2 + 2(3) + 2(5) = 18 electrons.
- And so on.
Therefore, the number of electrons that can have the same quantum state depends on the energy level (n), subshell (ℓ), and the specific orbital within the subshell.