Cyclohexane's ring puckers into a chair to avoid the eclipsing (torsional) strain of a flat hexagon. Each carbon carries one axial bond (perpendicular to the ring's mean plane) and one equatorial bond (pointing outward, near the plane). Ring atom height is modelled with a two-mode pucker:
y_i(t) = A·cos(πi + πt) + B·sin(πt)·cos(2θ_i)
θ_i = i·60°, i = 0..5, t ∈ [0,1] = flip progress
t=0 and t=1 → both chairs (energy minima)
t=0.5 → boat, flagpole atoms at i=0,3
A full ring flip (chair → half-chair → twist-boat/boat → half-chair → chair) swings every substituent through ~90°, interconverting axial ⇄ equatorial. The barrier comes from eclipsing strain in the half-chair transition states (~10-11 kcal/mol above the chair), modelled here as
E_ring(t) = E_barrier · sin²(πt) [~10.5 kcal/mol peak]
E_subst(t) = A_value · cos²(θ(t)), θ(t) = (π/2)·t
E_total(t) = E_ring(t) + E_subst(t)
The A-value is the real experimental ΔG° (kcal/mol) a substituent pays for sitting axial instead of equatorial — 1,3-diaxial steric clash with the other axial hydrogens. Bigger groups (tert-butyl, A ≈ 4.9) push the equilibrium almost entirely toward the chair that puts them equatorial. The equilibrium axial/equatorial split away from any flip animation obeys a simple two-state Boltzmann law:
K = [eq]/[ax] = exp(A / R·T), R = 1.987×10⁻³ kcal/(mol·K)
%eq = 100·K/(1+K), %ax = 100/(1+K)
- Substituent selector — swaps the group on C1 and its A-value.
- Flip progress slider / Auto-flip — scrubs or animates the chair↔chair pathway; watch the colored substituent swing from axial to equatorial.
- Temperature slider — recomputes the Boltzmann axial/equatorial population at that A-value.
Real-world relevance: this is the same axial/equatorial preference that fixes the shape of glucose, steroids and countless drug scaffolds — A-values predict which face of a cyclohexane ring reacts first.