The spin quantum number is a crucial concept in quantum mechanics that describes the intrinsic angular momentum of electrons within atomic orbitals. According to the Pauli exclusion principle, each orbital can accommodate a maximum of two electrons, provided they possess opposite spins. This principle asserts that no two electrons in the same orbital can share the same set of four quantum numbers, which include the principal quantum number, angular momentum quantum number, magnetic quantum number, and spin quantum number.

The spin quantum number, denoted as \( m_s \), specifically addresses the orientation of an electron's spin. Electrons can have a spin that is either "up" or "down." When an electron is described as having an "up" spin, it is assigned a spin quantum number of \( m_s = +\frac{1}{2} \). Conversely, an electron with a "down" spin has a spin quantum number of \( m_s = -\frac{1}{2} \). This arrangement ensures that the two electrons in an orbital have opposite spins, which is essential for maintaining the stability of the atom.

In summary, when filling an orbital, the first electron is placed with an "up" spin, followed by a second electron with a "down" spin. This systematic approach to electron configuration highlights the importance of the spin quantum number in understanding the behavior of electrons in atomic structures.