Imagine an alternate universe where all of the quantum number rules were identical to ours except m_{s} had three allowed values (rather than two as it does in our universe). If this were the case, and the Pauli Exclusion Principle still applies, how many electrons would be allowed in each orbital

so in a given orbital there can be 3 electrons.

Explanation:

The Pauli exclusion principle states that all the quantum numbers of an electron cannot be equal, if the spatial part of the wave function is the same, the spin part of the wave function determines how many electrons fit in each orbital.

In the case of having two values, two electrons change. In the case of three allowed values, one electron fits for each value, so in a given orbital there can be 3 electrons.

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