Intuition The one core ideaA single C–C bond is a swivel joint : the two carbons can spin relative to each other, and each twist-angle is a different snapshot of the same molecule. Everything in this topic is just measuring how crowded each snapshot is and how much time the molecule spends there.
Before we can talk about "eclipsed", "gauche", or that scary E ( ϕ ) = 2 E barrier ( 1 + cos 3 ϕ ) nothing and builds each tool from a picture.
Ethane is two carbons stuck together, each carbon also holding some hydrogens. To understand why carbons can spin at all , we need to know the shape around each carbon .
Definition sp³ carbon — a little tripod-plus-oneA carbon in an alkane spreads its four bonds as far apart as possible in 3D, pointing to the corners of a tetrahedron (a 4-cornered pyramid). The angle between any two bonds is about 109.5°
Picture: stand at the centre of the carbon; the four bonds shoot out like the legs of a camera tripod plus one leg straight up.
Why we need it: this fixed tetrahedral spread is why the three back-bonds can either hide behind or peek between the three front-bonds as we rotate. See Alkanes — structure and sp3 hybridisation .
Definition σ (sigma) bond — the spinnable bondA σ bond is a bond where the shared electrons sit directly along the line joining the two atoms (a symmetric sausage around the axis).
Picture: a cylinder of electron-glue wrapped around the C–C line.
Why we need it: because the electron cloud is symmetric around the axis, you can twist one end without tearing the bond — that is exactly what makes rotation possible. (A double bond would have a second, sideways cloud that locks rotation.)
Everything hinges on one number : how far we have rotated the back carbon relative to the front one.
Definition Dihedral (torsion) angleϕ down the C–C bond so the two carbons overlap. Pick one bond on the front carbon and the nearest bond on the back carbon. The angle you sweep from one to the other is ==ϕ
ϕ = 0° same way (they overlap on paper).ϕ = 60° gap between two front bonds.
angle and not a distance?
Rotation is circular — after a full spin you return to the start. Angles are the natural language of "going round in a circle", and they let one number capture the whole motion.
Two special names for two special angles come straight from this picture:
Definition Eclipsed vs Staggered
Eclipsed (ϕ = 0° , 120° , 240° hide behind front bonds. Crowded.Staggered (ϕ = 60° , 180° , 300° gaps . Roomy.Picture: eclipsed = spokes lined up (a 3-pointed star); staggered = spokes offset (a clean 6-pointed star).
We cannot see ϕ
"Energy goes up" is vague. It splits into two named discomforts , each with its own picture.
Definition Torsional strainExtra energy when bonds (electron pairs) are forced to overlap in the eclipsed position — like poles of magnets pushed together. Present even for tiny H atoms.
Picture: two spokes stacked → their electron clouds shove apart.
Deep-dived in Hyperconjugation (the stabilising overlap that is best when staggered).
Extra energy when bulky groups (like CH 3 van der Waals repulsion .
Picture: two tennis balls glued to spokes, bumping.
Only matters when groups are fat — that is why ethane (only H's) barely feels it but butane (two methyls) does. See Steric strain and van der Waals repulsion .
Common mistake Don't confuse the two strainsTorsional strain is about bonds lining up (any size). Steric strain is about bulky groups colliding . Staggered kills torsional strain — it does not kill steric strain if fat groups are still close (that is butane's whole story).
E relative to the comfiest snapshot, which we set to 0
Unit: kJ/mol = kilojoules per mole = energy per big batch (6 × 1 0 23
Intuition Why the cosine, of all functions?As we spin, energy must repeat and must be smooth (no sudden jumps). The cosine is the simplest wave that is + 1 − 1
Picture: a smooth hill-valley-hill curve.
cos θ
cos 0° = + 1 cos 180° = − 1
The parent uses cos 3 ϕ 3 three full hill-valley cycles into one 360° 120°
The last formula, N 2 / N 1 = e − Δ E / R T
Definition The symbols in the Boltzmann ratio
Δ E difference in energy between two conformers (higher minus lower).R = 8.314 J mol − 1 K − 1 T kelvin (room temp ≈ 298 K e − x
e − Δ E / R T up an energy hill Δ E exponentially as the hill grows or the temperature drops. Small hill or hot day → ratio near 1 0 Boltzmann distribution .
Worked example Sanity-check the exponential's behaviour
If Δ E = 0 e 0 = 1
If Δ E e − huge → 0 Why this step: testing the extremes proves the formula behaves sensibly before we trust its middle values.
delta E between conformers
Boltzmann exp gives populations
ethane and butane conformers
Test yourself — you are ready for the main note only if you can answer each without peeking.
What shape do the four bonds around an sp³ carbon make, and what angle separates them? A tetrahedron; about 109.5° between bonds.
Why can a σ (single) bond rotate but a double bond cannot? A σ bond's electron cloud is symmetric around the axis, so twisting doesn't tear it; a double bond's second (sideways) cloud locks rotation.
What does the dihedral angle ϕ The angle between a front-carbon bond and the nearest back-carbon bond, viewed straight down the C–C axis.
How are the front and back carbons drawn in a Newman projection? Front = a dot with three lines to the centre; back = a circle with three lines from its rim.
At what ϕ Eclipsed at 0°, 120°, 240°; staggered at 60°, 180°, 300°.
What is the difference between torsional and steric strain? Torsional = bonds/electron pairs lined up (any size); steric = bulky groups' clouds colliding (van der Waals).
Why does a cosine appear in the ethane energy formula, and why cos 3 ϕ Energy repeats smoothly as we spin (cosine wave); the 3 gives three cycles per turn, matching three identical H's.
What is 1 mole and why measure energy in kJ/mol? A mole is 6 × 1 0 23
In e − Δ E / R T Δ E R T Δ E R T
What happens to e − Δ E / R T Δ E = 0 Δ E Equals 1 (states equally populated) when 0; approaches 0 (high state empty) when huge.
Parent topic — the main note these foundations feed.Alkanes — structure and sp3 hybridisation — where the tetrahedral carbon comes from.Hyperconjugation — the electronic reason staggered wins.Steric strain and van der Waals repulsion — the "bulky groups bump" strain.Boltzmann distribution — the origin of the population formula.Stereochemistry — isomerism overview — how conformers relate to other isomers.Cyclohexane conformations chair and boat — the same ideas applied to rings. 
