Foundations — σ vs π bonds — overlap, strength

Before you can enjoy the parent note σ vs π bonds, you must own every word it uses. This page builds each one from nothing, in an order where each idea leans on the one before it. Never trust a symbol you have not seen defined here first.

1. The nucleus, the electron, and "sharing"

The picture: think of the nucleus as a small hard bead with a "+" on it, and electrons as a fuzzy negative mist hovering around it.

Why the topic needs it: a covalent bond happens when two nuclei end up glued together. Two "+" beads would normally push each other apart (they repel!). The only thing that can hold them is negative electron mist sitting between them, pulling both inward. So the whole topic is really the question: where does the shared electron mist sit, and how much of it is between the nuclei?

2. The orbital — the shape of the electron mist

We only need two shapes for this topic:

• $s$ orbital — a sphere centred on the nucleus. Same in every direction (no "pointing").
• $p$ orbital — a dumbbell: two lobes on opposite sides of the nucleus, pointing along one axis. There are three of them, pointing along three perpendicular directions, named $p_x$, $p_y$, $p_z$.

The picture: an $s$ orbital is a fuzzy ball. A $p$ orbital is two fuzzy balloons tied together nose-to-nose with the knot at the nucleus.

Why the topic needs it: a sphere has no direction, but a dumbbell points. Because $p$ orbitals point, they can meet a neighbour either tip-first (head-on) or side-by-side. That single freedom is what creates two kinds of bond.

3. The bond axis — the reference line

The picture: two beads with a ruler laid between their centres. The ruler is the axis.

Why the topic needs it: "head-on" means the orbitals meet on this line; "sideways" means they meet off this line. Without agreeing on the reference line, the words head-on and sideways have no meaning. By convention we usually call this line the $z$-axis, which is why the $p_z$ orbital is the one that overlaps head-on.

4. Overlap — orbitals sharing the same space

The picture: two soap bubbles pushed into each other. The lens-shaped region where both bubbles exist is the overlap.

Why the topic needs it: overlap is the bond. More overlap between the nuclei → more shared mist in the binding zone → stronger bond. This is the master rule of the whole topic:

$\text{Bond strength}\propto \text{amount of overlap}$

Read $\propto$ as "grows together with"/"is proportional to": if the right side gets bigger, so does the left.

5. Head-on vs sideways — the two overlap geometries

Now that "orbital", "axis" and "overlap" exist, the two bond types are just two ways to overlap:

Why weaker for π: re-read §1 — the π mist is off the axis, so it pulls the nuclei sideways instead of together. Same distance, less useful glue.

6. The nodal plane — the empty sheet inside a π bond

The picture: the two π lobes are like bread slices above and below; the nodal plane is the empty gap between them where there is no filling.

Why the topic needs it: the nodal plane is the fingerprint of a π bond. A σ bond has no nodal plane through the axis (its mist is a solid cylinder around the line); a π bond always has one. If someone asks "is this σ or π?", checking for a nodal plane containing the axis answers it instantly.

7. Cylindrical symmetry — why σ can spin and π cannot

The picture: a σ bond's mist is a rolling pin around the axis — twist the top atom and nothing changes. A π bond's mist has a definite up/down (the two lobes), so twisting the top atom by 90° swings its dumbbell out of alignment and the sideways overlap breaks.

Why the topic needs it: this single property explains the whole "free rotation" row of the parent's table — σ rotates freely, π does not, which is the reason double bonds are locked (see cis-trans isomerism).

8. The integral sign $\int$ — turning "amount of overlap" into a number

The parent note writes the overlap integral $S=\int {\psi }_A{\psi }_{B}d\tau$. Let us earn every symbol.

The picture: lay a fine grid over the whole molecule. At each cell, multiply cloud-A-strength by cloud-B-strength. The integral sweeps the grid and sums every product. Head-on ($p_z$–$p_z$): big products right between the nuclei → big $S$. Sideways ($p_x$–$p_x$): products are smaller and half-cancel across the nodal plane → small $S$.

Why we use an integral and not just "look": "amount of overlap" is a fuzzy word, but the bond needs a number so we can say $S_{\sigma }>S_{\pi }$ and mean it precisely. The integral is the only honest way to total a quantity that is spread continuously over space. It is the same tool used everywhere in Molecular Orbital Theory.

9. Bond order and bond energy — the scoreboard

Why the topic needs it: the parent proves "π < σ" using numbers: a C=C double bond (615 kJ/mol) is less than two C–C singles (2 × 348 = 696), so the added π contributes only 267 kJ/mol < 348. You cannot follow that argument without knowing what these numbers mean. Full detail lives in Bond order, length and energy.

The prerequisite map

This feeds forward into Hybridisation (sp, sp2, sp3) (directional hybrids always make σ), Resonance and conjugation (many π clouds sharing), and VSEPR and molecular geometry (σ framework sets the shape).

Equipment checklist

Test yourself — cover the right side and answer before revealing.