Chemistry Electron configurations

Question

Construct a possible explanation for why true configurations would be a lower potential energy state for the elements in Model 4. HINT: What types of atomic interactions cause a lower potential energy or higher potential energy? How far apart in energy are electrons in the higher energy levels?

MODEL 4 - Predicted versus True Configuration

Predicted Configuration

Cr 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^4

Cu 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^9

Ce 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^10 4p^6 5s^2 4d^10 5p^6 6s^2 4f^2

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True Configuration

Cr 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^5

Cu 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^10

Ce 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^10 4p^6 5s^2 4d^10 5p^6 6s^2 4f^1 5d^1

Answer 1

Short answers first. The true configurations sit lower in potential energy because the levels being shuffled - 4s against 3d for Cr and Cu, 4f against 5d for Ce - are almost the same energy, so promoting an electron between them costs nearly nothing. What the move buys is a much larger reduction in electron-electron repulsion. The Aufbau prediction only optimizes one of the two energy terms in an atom; the real ground state optimizes their sum.

1. The two interactions your hint is pointing at. Potential energy in an atom comes from exactly two sources, and they pull in opposite directions. Electron-nucleus attraction lowers it. Electron-electron repulsion raises it. Filling orbitals from the bottom up only minimizes the first term, which is why it sometimes gets the answer wrong.

Repulsion has two pieces worth naming separately, because they do different work in these three cases:

(a) Pairing. Two electrons in the same orbital occupy the same region of space, so they repel harder than any other pair in the atom. Every doubly occupied orbital carries that penalty.

(b) Exchange. Two electrons with parallel spins in different orbitals of the same subshell are forbidden by the Pauli principle from ever being at the same point. Each one effectively carries a small exclusion zone around it, so parallel electrons stay farther apart on average and repel less. This is a real energy saving, and it grows fast: N parallel electrons give N(N-1)/2 parallel pairs.

2. Chromium. Both effects push the same way here.

Cr predicted:  [Ar] 4s^2 3d^4
   parallel d electrons: 4  ->  4*3/2  =  6 exchange pairs
   doubly occupied orbitals: 1  (the 4s)

Cr true:       [Ar] 4s^1 3d^5
   parallel d electrons: 5  ->  5*4/2  = 10 exchange pairs
   doubly occupied orbitals: 0

   net: +4 exchange pairs AND one pairing penalty removed

In the true configuration every valence electron in Cr is unpaired - six orbitals, six electrons, all spins parallel. That is the maximum repulsion saving available, and it is bought by moving one electron up a step that is barely a step at all.

3. Copper, where the popular explanation quietly fails. Count the same two quantities and something interesting happens.

Cu predicted:  [Ar] 4s^2 3d^9
   d exchange:  5 up, 4 down  ->  10 + 6  = 16 pairs
   doubly occupied orbitals: 1 (4s) + 4 (3d) = 5

Cu true:       [Ar] 4s^1 3d^10
   d exchange:  5 up, 5 down  ->  10 + 10 = 20 pairs
   doubly occupied orbitals: 5 (3d)

   net: +4 exchange pairs, pairing count UNCHANGED

Copper gains the same four exchange pairs, but it does not avoid a single pairing penalty - five doubly occupied orbitals before, five after. So for Cu the saving is exchange, plus the fact that by Z = 29 the 3d orbitals have contracted and dropped below 4s in energy, making 3d the cheaper place to put the electron in the first place. If you had explained Cr with pairing alone, Cu would have caught you out.

4. Cerium, which works for a completely different reason. This is the most useful one in Model 4, because neither slogan applies to it.

Ce predicted:  [Xe] 6s^2 4f^2
   both f electrons crowded into the tiny, compact 4f orbitals

Ce true:       [Xe] 6s^2 4f^1 5d^1
   one electron moved out into the much larger, more diffuse 5d

Notice that f^2 and f^1 d^1 both have two unpaired electrons, so nothing here is about half-filling or about avoiding a pair. The 4f orbitals are physically tiny - they are buried inside the xenon core, closer to the nucleus than the 5s and 5p electrons that surround them. Two electrons confined in that small a volume repel each other severely. The 5d orbital is far more spread out, so moving one electron there sharply reduces the repulsion between them, and because 4f and 5d are nearly degenerate at the start of the lanthanides the move is essentially free. Same principle as Cr and Cu - trade a tiny orbital-energy cost for a large repulsion saving - but a different repulsion term.

5. Now the second half of your hint: how far apart are these levels? This is the part that makes the whole explanation work.

gap from 1s to 2s in Cr:            of order 10^2 to 10^3 eV
gap from 4s to 3d in Cr:            of order 1 eV (roughly 100 kJ/mol)
typical pairing / exchange energy:  also of order 1 eV

The outer levels are crowded into a window about the same size as the repulsion energies you are trading against - which is exactly why repulsion can outvote the orbital ordering there. Deep in the core the gaps are hundreds of times larger than any repulsion term, so nothing can overturn them. That is the reason you never see an anomalous 1s or 2p configuration, and why every exception in the periodic table lives in the d and f blocks where two levels have drifted close together.

One honest caveat, worth writing into your answer. Half-filled and filled subshells are a rationalization of a near-degeneracy, not a law. Molybdenum copies chromium (5s^1 4d^5) but tungsten does not (6s^2 5d^4). Palladium goes further than copper, all the way to 4d^10 5s^0. If a filled or half-filled subshell carried a fixed bonus, none of those would happen. The statement that always holds is the general one: the ground state is whichever whole configuration minimizes the total energy, and when two levels fall within about an eV of each other, electron-electron repulsion is what decides.

The habit to take away. When you are asked why a true configuration beats the predicted one, do not stop at the phrase about half-filled shells. Say which repulsion term went down, and say why the level spacing was small enough to let it matter. That answer works for Cr, for Cu and for Ce - the slogan only works for two of the three.