Bond rotation (ethane, butane)
The rotating C–C bond is held at 1.53 Å with tetrahedral groups on each carbon, and only the dihedral angle changes. The strain at 0°, 60°, 120° and 180° is the sum of the interaction costs in the table. The energy curve is a three-term cosine series, E = c + a cos φ + b cos 2φ + d cos 3φ, fit exactly through those four points. For butane, both gauche forms (g⁺ and g⁻) are counted, so K = [gauche]/[anti] = 2e−ΔG°/RT. Rates use the Eyring equation, k = (kBT/h)e−ΔG‡/RT, treating the barrier from the curve as ΔG‡ with a transmission coefficient of 1, so they are order-of-magnitude estimates.
Ring geometry
Ring shapes come from Cremer–Pople puckering coordinates (amplitude Q, angles θ and φ). θ = 0° is the chair and θ = 180° the flipped chair. At θ = 90° the boats (φ = 0°, 60°, …) and twist-boats (φ = 30°, 90°, …) sit on a pseudorotation circle. The path runs chair → half-chair (θ ≈ 51°) → twist-boat → boat → twist-boat → half-chair → chair′, with every C–C bond scaled to an average of 1.53 Å and C–H at 1.09 Å. Substituent directions are tetrahedral at each ring carbon. A bond is drawn axial when it lies within about 35° of the ring normal and equatorial when it lies more than 60° from it.
Energy profile
The curve is a smooth schematic through the stationary points below. Its maxima and minima sit at the named conformations, but its shape between them is not a computed potential energy surface. For substituted rings, the difference in chair strain is blended in across the path.
| Bond-rotation interaction | Strain used |
|---|
| H/H eclipsed | 1.0 kcal/mol (4.0 kJ/mol) |
| CH₃/H eclipsed | 1.4 kcal/mol (6.0 kJ/mol) |
| CH₃/CH₃ eclipsed | 2.6 kcal/mol (11 kJ/mol) |
| Ethane barrier (3 × H/H) | 3.0 kcal/mol (12 kJ/mol) |
| Butane 120° eclipsed / 0° fully eclipsed | 3.8 / 4.6 kcal/mol (16 / 19 kJ/mol) |
| Cyclohexane conformation | Energy above chair |
|---|
| Half-chair (transition state) | 10.8 kcal/mol (45 kJ/mol) |
| Boat | 6.9 kcal/mol (29 kJ/mol) |
| Twist-boat (intermediate) | 5.5 kcal/mol (23 kJ/mol) |
| Interaction | Strain used |
|---|
| CH₃/H 1,3-diaxial (each) | 0.87 kcal/mol (3.6 kJ/mol) |
| Axial CH₃ (A-value, 2 × above) | 1.74 kcal/mol (7.3 kJ/mol) |
| CH₃/CH₃ gauche | 0.9 kcal/mol (3.8 kJ/mol) |
These are standard teaching values. Measured barriers and A-values vary slightly with method and solvent. Gas-phase measurements put the butane anti–gauche difference somewhat lower, near 0.6–0.7 kcal/mol. Sources: J. D. Kemp & K. S. Pitzer,
J. Chem. Phys. 1936, 4, 749 (ethane barrier); W. A. Herrebout et al.,
J. Phys. Chem. 1995, 99, 578 (butane); J. McMurry,
Organic Chemistry (Cengage), interaction increments; E. L. Eliel & S. H. Wilen,
Stereochemistry of Organic Compounds (Wiley, 1994); F. A. L. Anet & A. J. R. Bourn,
J. Am. Chem. Soc. 1967, 89, 760; M. Squillacote et al.,
J. Am. Chem. Soc. 1975, 97, 3244.