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QUESTION

# State the Pauli exclusion principle in your own words. Explain why the Pauli exclusion principle is important to the understanding of the periodic table?

The Pauli Exclusion Principle is in fact the main reason why we have the idea of an "" and why some transition metals like Chromium have access to \mathbf(12) electrons, and sometimes up to even \mathbf(18).

The Pauli Exclusion Principle essentially states:

No two electrons may have entirely identical quantum states; at least one quantum number must be different.

I've given a formal explanation of the octet rule . Please read that before proceeding, as I will be furthering that discussion.

Following that, we then realize that the octet rule is centered around the Pauli Exclusion Principle.

EXCEPTIONS TO THE OCTET RULE

We can then determine how Chromium, for example, can use 12 , sometimes, for the same reason, and not just 8. Here's what I mean.

Chromium's is:

(the original diagram is different, but it was wrong, because it had electrons in the 4p, which would mean 30 electrons, not 24. It also has the 3d higher in energy than the 4s, but it's actually not, for Chromium, according to Eric Scerri, replying to David Talaga.)

1s^2 2s^2 2p^6 3s^2 3p^6 color(blue)(3d^5 4s^1)

where the blue atomic orbitals are the valence orbitals.

QUINTUPLE BONDS?!

The energy levels are so close together, however, that Chromium actually sometimes has access to 12 , rather than just 6. That's why Chromium can sometimes make SIX bonds. Just take a look at this!

One single bond and one quintuple bond, and one interaction (dashed bond)! Okay, so how in the world?!

QUANTUM NUMBER CONSIDERATIONS

We can realize that Chromium sometimes has access to its 3p orbitals as well, as that would give it 12 electrons, and the 3p is closest in energy to the 3d orbital (when moving downwards in energy).

So, we can consider the following :

n = 3:

l = 1, 2 m_l = -2, -1, 0, +1, +2 m_s = pm"1/2"

(covering the 3p and 3d orbitals)

n = 4:

l = 0 m_l = 0 m_s = pm"1/2"

(covering the 4s orbital)

In the same type of atomic orbital (examining only the 3d, for instance, or examining only the 4s, etc), all quantum numbers are the same, except for m_l and m_s, which CAN be different.

UNIQUE QUANTUM STATES

As a result, for the 3d orbital, we have five unique quantum states for the five available spin-up electrons (m_l = -2, -1, 0, +1, +2 with m_s = +"1/2"). m_l just ultimately tells us that there are five different 3d orbitals (3d_(z^2), 3d_(x^2 - y^2), 3d_(xy), 3d_(xz), and 3d_(yz)).

This, however, doesn't include the spin-down electrons due to Hund's rule of favoring the maximum spin state, which, for Chromium's 3d orbitals, is +"5/2", and due to how there are exactly 5 electrons here.

Next, for the 3p orbitals, we have three pairs of electrons. We have m_l = -1, 0, +1 with m_s = -"1/2", as well as m_l = -1, 0, +1 with m_s = +"1/2". That makes for a total of six unique quantum states, and thus six possible electrons that can exist in the 3p orbitals.

Finally, the 4s orbital has only one valence electron, which obviously can only exist in one possible way at a time. The quantum numbers corresponding to it are l = 0, m_l = 0 with m_s = +"1/2". Whether you believe that a single electron can flip its spin or not, either way, that means one unique quantum state.

TAKE-HOME MESSAGE

Hence, Chromium could sometimes have 5 + 6 + 1 = \mathbf(12) unique quantum states for each of the 12 electrons available for bonding, following the Pauli Exclusion Principle.

Each electron can only occupy one state at a time (like how one twin can only be that twin for all time), so with 12 electrons, 12 states are implicitly possible. That's how I would rationalize why in the world Chromium can make SIX bonds sometimes. :)