Treat butane as having one anti conformer (energy 0) and two equivalent gauche conformers (each 3.8 kJ/mol above anti), ignoring all other conformers. Use $R = 8.314 J K^{- 1} mo l^{- 1}$ and neglect the temperature dependence of the energies. At what temperature will the total gauche population equal the anti population?

Answer: C

Equal populations require

2 e^{- Δ E / RT} = 1

, so

Δ E / RT = ln 2

and

T = 3800/ (8.314 \times 0.6931) = 659

K. Ignoring the two-fold degeneracy would require

Δ E / RT = 0

(infinite T) or give 457 K =

Δ E / R

unrelated to the condition.