Why are there only two electrons in an s orbital?

An s orbital can only hold two electrons due to the Pauli Exclusion Principle, which states that no two electrons can have the same four quantum numbers.

The Pauli Exclusion Principle is a fundamental principle in quantum mechanics. It was formulated by Austrian physicist Wolfgang Pauli in 1925. According to this principle, no two electrons in an atom can have the same set of quantum numbers. Quantum numbers are a set of four numbers that describe the unique state of an electron in an atom, including its energy level, shape of its orbital, orientation of its orbital, and its spin.

In the context of an s orbital, the first three quantum numbers are the same for both electrons. The principal quantum number (n) defines the energy level and size of the orbital, the azimuthal quantum number (l) defines the shape of the orbital (which is spherical for an s orbital), and the magnetic quantum number (m) defines the orientation of the orbital (which is the same for all s orbitals as they are symmetrical).

However, the fourth quantum number, known as the spin quantum number (s), can have two possible values: +1/2 or -1/2. This means that two electrons in the same s orbital must have opposite spins. Therefore, only two electrons can occupy an s orbital, one with a spin of +1/2 and the other with a spin of -1/2.

This is not only true for s orbitals, but for all types of orbitals. Each orbital in an atom can hold a maximum of two electrons, and these electrons must have opposite spins. This is a fundamental aspect of the electronic structure of atoms and plays a crucial role in determining the chemical behaviour of elements.