For two spin-paired electrons that are in the same orbital, describe how the values for the first three quantum numbers (n, L, and mL) are related for the two electrons.

Is this because no other electron has the same quantum number?

Yes. n, l, and mL are the same BECAUSE one of the electrons is ms = +1/2 and the other one is ms = -1/2.

The values for the first three quantum numbers (n, L, and mL) describe the energy level, orbital shape, and orientation of an electron in an atom, respectively.

For two spin-paired electrons that are in the same orbital, their values for the first three quantum numbers are related as follows:

1. Principal Quantum Number (n): This describes the energy level or shell of an electron. Only electrons in the same energy level have the same principal quantum number. In the case of two spin-paired electrons, both electrons will have the same value for n since they are in the same orbital.

2. Azimuthal Quantum Number (L): This describes the orbital shape or subshell of an electron. For a given value of n, the possible values for L range from 0 to (n-1). However, for two spin-paired electrons in the same orbital, their azimuthal quantum numbers must have different values to ensure opposite spins. This means that one of the electrons will have mL = +L and the other will have mL = -L.

3. Magnetic Quantum Number (mL): This describes the orientation or magnetic orientation of an electron. The magnetic quantum number can take any value between -L and +L, including zero. For two spin-paired electrons in the same orbital, one electron will have a positive value of mL and the other will have a negative value of mL. This ensures that their spins are opposite to each other.

So, to summarize, for two spin-paired electrons in the same orbital:
- They will have the same principal quantum number (n) since they are in the same energy level.
- They will have different azimuthal quantum numbers (L) to ensure opposite spins.
- They will have opposite magnetic quantum numbers (mL) to ensure opposite spins.