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The Pauli Exclusion Principle states that, in an atom or molecule, no two electrons can have the same four electronic quantum numbers. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. This means if one is assigned an up-spin ( +1/2), the other must be down-spin (-1/2).
Electrons in the same orbital have the same first three quantum numbers, e.g., \(n=1\), \(l=0\), \(m_l=0\) for the 1s subshell. Only two electrons can have these numbers, so that their spin moments must be either \(m_s = -1/2\) or \(m_s = +1/2\). If the 1s orbital contains only one electron, we have one \(m_s\) value and the electron configuration is written as 1s1 (corresponding to hydrogen). If it is fully occupied, we have two \(m_s\) values, and the electron configuration is 1s2 (corresponding to helium). Visually these two cases can be represented as
As you can see, the 1s subshell can hold only two electrons and when filled the electrons have opposite spins.
Dr. Craig Fisher (Japan Fine Ceramics Center)