[pnicogen1]

What is the difference between Coloumbs energy vs Exchange energy in the context of the filling order of orbitals. My notes say that Coloumbs energy is the energy required to put 2e- in the same orbital. The exchange energy has something to do with the spin of the electrons. I'm just not sure how to distinguish the two when deciding whether these energies are favorable between two filling models. The problems give two filling models and asks whether the promotion, coloumb, and exchange energies are favorable from one model to the next.

For the promotion energy, I just simply look for an electron and where it is promoted. This is usually unfavorable (e.g. there is an expenditure of energy when promoting and electron from s to p)

Then I'm not sure how to decide favorability with the coloumbs and exchange energies....

[Corribus]

I'm not sure really what the question is.  Coulomb integral describes electrostatic interactions; the exchange integral doesn't really have a classical analogue and comes about because the wavefunction has to be antisymmetric with respect to electron exchange.  Bonding orbitals are primarily stable because of the exchange integral.

[pnicogen1]

I guess the specific question would help...

So the electron config for Mo is 5s14d5 instead of 5s24d4. Indicate whether the following factors are unfavorable, favorable, approx. zero, or not applicable in favoring the observed ground config over the alternative ground configuration given.

The answers were
Coloumb energy: favorable
Exchange energy: favorable
Promotion energy: not favorable.

I'm wasn't really sure how to tell whether the couloumb energy and exchange energies were favorable for the configurations....

[blaisem]

First observe the s and d orbitals (might help to draw this on a piece of paper).  The s orbital has one sub-orbital; the d orbital has five.

5s²4d⁴ has the s orbital filled (of course, with the electrons having antiparallel spin).  The d-orbital has 4 half filled sub-orbitals (of course, with all electrons having parallel spin).

Now fill in the orbitals for the case of 5s¹4d⁵

Coulomb Energy: Coulomb energy is based on electrostatic interactions; namely, like charges repel (ie. positive potential energy) and unlike charges attract (ie. negative potential energy).  So, it makes sense that two electrons, which have like charge, have a positive (greater) energy when they are near each other.

Exchange Energy: The question is probably referring to Hund's first rule, which you can read about here (and it is probably mentioned in your book, as well as many other areas)

http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/hund.html#c2
http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/atstruct.html#c2

Promotion Energy:  A bound electron by definition has a negative potential energy.  The more tightly bound the electron, the lower the potential energy.  In order to promote an electron, you have to raise its energy to a higher level.  If you have, hypothetically, an electron bound at -12 eV, and an electron more tightly bound at a lower energy of -13 eV, promotion to a certain energy higher than either of them will require more energy for the more tightly bound electron.

Compare the definitions to the two different diagrams you drew for the configurations.  Hopefully, this helps you make sense of it!