Foundations — Why O₂ is paramagnetic (MOT prediction)

This is the prerequisites page for the parent topic. The parent throws around symbols like ${\sigma }_{2p_z}$, ${\pi }^{\ast }$, $N_b$, $\mu$, "degenerate", "Hund". If any of those look like alien runes, start here. We define each one in plain words, tie it to a picture, and say why the topic cannot proceed without it.

0. What is an electron, and why does it act like a magnet?

Before any bonding, we need the raw ingredient.

The picture: a spinning charged ball is a tiny loop of moving charge — and moving charge makes a magnetic field. So one electron = one microscopic bar magnet.

Why the topic needs this: the entire claim "O₂ is paramagnetic" reduces to "O₂ contains electrons whose tiny magnets do NOT cancel." If you don't picture an electron as a magnet, the word "paramagnetic" is meaningless.

1. Atomic orbital — the "parking space" for one or two electrons

The picture: an $s$ orbital is a round ball; a $p$ orbital is a dumbbell (two lobes) pointing along one axis. There are three $p$ orbitals — one along each axis — called $p_x$, $p_y$, $p_z$.

Why the topic needs this: oxygen's electrons start in these atomic parking spaces. When two oxygen atoms meet, these AOs combine into new molecular parking spaces. No AOs → no MOs.

2. Molecular orbital — parking spaces that belong to the whole molecule

When two atoms bond, their AOs merge. Two AOs going in always produce two MOs coming out.

The picture: think of two water ripples meeting. Crest-meets-crest reinforces (bonding, cloud builds up in the middle). Crest-meets-trough cancels (antibonding, a bald spot appears in the middle).

Why the topic needs this: the two lonely electrons of O₂ live in an antibonding orbital, the ${\pi }^{\ast }$. If you don't know what the star means, you can't locate them. See Molecular Orbital Theory for the full framework.

3. σ vs π — two SHAPES of overlap

The picture: $p_z$ orbitals (pointing along the O–O axis) meet nose-to-nose → σ overlap → ${\sigma }_{2p_z}$. The $p_x$ and $p_y$ orbitals (pointing sideways) meet flank-to-flank → π overlap → ${\pi }_{2p_x}$ and ${\pi }_{2p_y}$.

Why the topic needs this: there are two sideways directions ($x$ and $y$), so there are two π orbitals of identical energy. That "two identical" fact is the seed of the whole paramagnetism story (next section).

4. Degenerate — "same energy," the magic word

The picture: two parking spots at the same floor of a car park. Neither is "better."

Why the topic needs this: ${\pi }_{2p_x}^{\ast }$ and ${\pi }_{2p_y}^{\ast }$ are degenerate. When 2 electrons must fill two equal-energy spots, a special rule decides how — and that rule (next) is what forces them apart. Without degeneracy, they'd just pile into the lower one and pair up (diamagnetic). Degeneracy is why O₂ is special.

5. Aufbau + Hund — the two filling rules

See Aufbau Principle.

The picture: passengers on a bus prefer an empty double-seat to squeezing next to a stranger. Only once every double-seat has one person do they start doubling up.

Why the topic needs this: O₂'s last 2 electrons face the two degenerate ${\pi }^{\ast }$ spots. Hund says: one each, same spin → two lonely ↑ ↑ electrons → paramagnetic. This single rule is the answer. See Hund's Rule.

6. The symbols on the energy ladder

Now that every idea is built, here is what each parent-note symbol means:

The subscript = which AOs made it. A star = antibonding. σ or π = head-on or side-on shape. That's the entire alphabet.

7. Symbols in the formulas

See Bond Order.

See Paramagnetism and Diamagnetism.

Worked sanity-check (this is the parent's headline number): O₂ has $n=2$, so $\mu =\sqrt{2(2+2)}=\sqrt{8}\approx 2.83\text{BM}.$

Prerequisite map

Everything on the left feeds the single conclusion on the right.

Equipment checklist

Test yourself — reveal only after answering aloud:

Connections

• Parent topic — the payoff
• Molecular Orbital Theory — where these parking-space ideas come from
• Aufbau Principle — fill lowest first
• Hund's Rule — the rule that creates the two loners
• Bond Order — uses $N_b$ and $N_a$ defined here
• Paramagnetism and Diamagnetism — the magnetic classification
• Valence Bond Theory limitations — why the dot-picture misses the loners
• N2 vs O2 MO diagram — where the ladder order shifts
• Superoxide and Peroxide ions — same framework, different electron counts