Question

Each of the following four sets contains two electrons, each of which is defined by four quantum numbers (n, l, ml, ms). For each set, indicate whether the quantum numbers for the two electrons are valid for the two electrons in the highest energy subshell of a neutral titanium atom. If they are not valid, list the principle that the quantum numbers violate (Pauli Exclusion, Aufbau, or Hund’s Rule) and explain below.

(the numbers are shown below in the order stated below and need to be proven valid or invalid based on the rules above)

Set #: Electron 1: Electron 2: Valid or invalid?: Violation of what rule?:

1 (3, 2, +2, +½)] (4, 1, +1, +½)

2 (3, 2, +1, +½) (3, 2, +1, +½)

3 (3, 2, +2, +½) (3, 2, +1, +½)

4 (3, 2, +1, +½) (3, 2, +1, –½)

Answer 1

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Set #1: (3, 2, +2, +½)] (4, 1, +1, +½)

First electron [(3, 2, +2, +½)] corresponds to an electron in the level 3, sub-level d (l=2). ml=+2 indicates the spatial orientation of the d orbital and ms=+½ indicates the spin of the electron. The second electron [(4, 1, +1, +½)] corresponds to an electron in the level 4, sublevel p (l=1), with a spatial orientation of ml=+1 and spin ms=+½. This distribution violates the Aufbau principle. The Aufbau principle tells us that to build the electronic configuration of a given atom we have to place the electrons into orbitals in order of increasing energy. In this problem we are analyzing a neutral titanium atom and the last two highest energy subshells would be 4s and 3d. 4s should be filled before electrons enter in the 4p orbitals because level 4s is lower in energy.

Set #2: (3, 2, +1, +½) (3, 2, +1, +½)

Both electrons have the same four quantum numbers so this distribution violates the Pauli Exclusion principle which states that two electrons cannot share the same four quantum numbers.

Set #3: (3, 2, +2, +½) (3, 2, +1, +½)

Electron one is a 3d electron placed in one of the five d orbitals with orientation ml=+2 and spin ms=+½. Electron number two is also a 3d electron placed in a second d orbitals with orientation ml=+1 and spin ms=+½. This distribution is possible (valid).

Set #4: (3, 2, +1, +½) (3, 2, +1, –½)

Electron one is a 3d electron placed in one of the five d orbitals with orientation ml=+1 and spin ms=+½. Electron number two is also a 3d electron placed in the same d orbitals with orientation ml=+1 and spin ms=-½. This distribution is in violation of Hund’s Rule. Hund's Rule tells us how to fill a set of degenerate orbitals. It states that the ground state should have the maximum multiplicity, which means not to pair electrons until all of the degenerate set is filled with one electron. Set #4 pairs two electrons in the same d orbital.

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