Torsional Strain and the Origin of the Ethane Barrier

The ethane barrier is the atomic-scale seed of essentially all conformational chemistry. In drug design, computing accurate torsional energy profiles for rotatable bonds is essential for predicting bioactive conformations and building reliable molecular-mechanics force fields (AMBER, CHARMM, OPLS); these force fields explicitly parameterize torsional terms as Fourier series of exactly the $\frac{V_3}{2}(1-\cos n\varphi)$ form derived here, with coefficients fit (in modern force-field development) to reproduce quantum-mechanically computed hyperconjugative barriers rather than purely steric estimates. In materials science, the same physics governs rotational barriers about C–C and C–Si bonds in polymer backbones, influencing chain flexibility, glass transition temperatures, and mechanical properties of plastics and elastomers. In structural biology, hyperconjugative preferences for anti-periplanar alignment underlie the preferred backbone and side-chain torsion angles (χ angles) that stabilize specific protein conformations and are tabulated in Ramachandran-adjacent rotamer libraries used in crystallographic refinement and homology modeling. The anomeric effect in carbohydrate chemistry — the preference for an electronegative substituent at the anomeric carbon of a pyranose ring to sit axial rather than equatorial, contrary to naive steric expectation — is mechanistically the same phenomenon as the ethane barrier, generalized: an oxygen lone pair donating into an anti-periplanar σ\(C–X) orbital, exactly analogous to the σ(C–H)→σ\(C–H) donation described here but with a lone pair as donor. This connects directly forward to how substituted alkane chains prefer Conformations of Butane: Gauche and Anti arrangements, and to why cyclohexane rings and other cyclic systems distort away from idealized polygon geometries in Ring Strain in Cycloalkanes: Baeyer's Model and Reality, both of which layer additional steric (1,3-diaxial, gauche-butane) strain on top of this same underlying torsional/hyperconjugative foundation. Modern computational instrumentation — high-level ab initio methods (CCSD(T), MP2) combined with NBO deletion experiments — routinely dissects torsional barriers in exactly the manner outlined in Mechanism 1 for far more complex molecules, from pharmaceutical leads to polymer monomers, making the ethane barrier not a historical curiosity but the calibration standard against which every more complicated torsional analysis is still checked.

Ask the tutor: How does adding a methyl group (as in butane) combine this torsional/hyperconjugative barrier with genuine steric gauche strain to produce the anti/gauche energy difference?

Picture ethane down the C–C axis, exactly as you would draw a Newman projection. In the staggered conformation, each front C–H bond bisects the angle between two back C–H bonds; the H–C–C–H dihedral angles are 60°, 180°, and 300° (≡ −60°) for the three pairs. In the eclipsed conformation, front and back C–H bonds line up directly, with dihedral angles of 0°, 120°, and 240°. The classical (steric) account says: in the eclipsed form, each pair of eclipsing hydrogens sits only about 2.29 Å apart (closer than the sum of their van der Waals radii, ~2.4 Å), producing exchange-repulsion between the electron clouds. Rotating to staggered increases the H···H distance to about 2.55 Å (since staggered H's are gauche-like, not directly aligned), relieving this repulsion. This model correctly predicts that staggered is lower in energy and that the potential has three-fold symmetry (three equivalent staggered minima, three equivalent eclipsed maxima per 360°). But when you compute the actual van der Waals repulsion energy between two H atoms at 2.29 Å using standard nonbonded potentials, you get only a fraction of a kJ/mol per pair — nowhere near enough to build a 12 kJ/mol barrier from three such pairs. The orbital picture explains the missing energy. Each C–H bond has a bonding combination (σ, filled, two electrons, lower energy) and an antibonding combination (σ\, empty, higher energy) built from the carbon sp³ hybrid and the hydrogen 1s orbital. A filled σ(C–H) orbital on one carbon and an empty σ\(C–H) orbital on the neighboring carbon can interact — donating electron density from the filled orbital into the empty one — exactly as a lone pair donates into an antibonding orbital in classic hyperconjugation or anomeric effects. This donation is a second-order perturbative stabilization, and its magnitude depends critically on the dihedral angle between the donor σ bond and the acceptor σ\* bond, because orbital overlap (and hence donation) is maximal when the two are collinear (anti-periplanar, 180°) and minimal when they are orthogonal (90°) or eclipsed (0°, where overlap is small but nonzero due to geometry).

The key geometric fact, made precise below, is that hyperconjugative stabilization from a single donor–acceptor σ/σ\ pair scales with the square of the overlap, which itself scales as $\cos^2\theta$ where $\theta$ is the dihedral angle between the donor and acceptor bond axes (measuring 180° as perfectly anti-periplanar). Summing over all nine such vicinal σ(C–H)→σ\(C–H) interactions around the C–C bond (three on each side interacting with three on the other, though only the near-anti and near-eclipsed ones matter most) reproduces both the three-fold periodicity and, semi-quantitatively, the magnitude of the real barrier.

The rotational potential energy of ethane as a function of the C–C dihedral angle φ is, to leading order, a pure cosine with three-fold periodicity:

where:

• $E(\varphi)$ is the torsional energy at dihedral angle $\varphi$ (defining $\varphi = 60°$ as staggered, $\varphi = 0°$ as eclipsed),
• $V_3$ is the barrier height, the full energy difference between eclipsed maximum and staggered minimum,
• the factor $\tfrac12(1-\cos3\varphi)$ oscillates between 0 (staggered, $\varphi = 60°,180°,300°$) and 1 (eclipsed, $\varphi = 0°,120°,240°$).

Reading this equation: at $\varphi = 0°$ (fully eclipsed), $\cos(0) = 1$, so $E = 0$ shifted appropriately — by convention we set the eclipsed conformer as the energy zero and staggered as $-V_3$, or vice versa; either way the difference between maxima and minima is exactly $V_3$. Experimentally (from microwave and far-infrared torsional spectroscopy of ethane),

The hyperconjugative contribution to this barrier can be estimated from second-order perturbation theory applied to the donor σ(C–H) and acceptor σ\*(C–H) natural bond orbitals. The stabilization energy from one donor–acceptor pair is:

where:

• $n_\sigma$ is the occupation number of the donor σ orbital (2, since it's a filled bond),
• $\langle \sigma|\hat F|\sigma^*\rangle$ is the off-diagonal Fock (or Kohn–Sham) matrix element coupling donor and acceptor — proportional to their spatial overlap,
• $\varepsilon_{\sigma^*} - \varepsilon_{\sigma}$ is the energy gap between the antibonding and bonding orbital,
• the minus sign shows this term always lowers energy (donation is always stabilizing).

Because the coupling matrix element is proportional to orbital overlap, and overlap between two σ-type bond orbitals separated by dihedral angle $\theta$ falls off approximately as $\cos\theta$ (with $\theta$ measured from perfect anti-alignment), the stabilization energy — being proportional to the square of the coupling — scales as:

reading: donation is maximal (most negative, most stabilizing) when $\theta = 0°$ (perfectly anti-periplanar donor/acceptor pair) and vanishes at $\theta = 90°$ (orthogonal, zero overlap by symmetry).

Example 1 (foundational): Converting the barrier between units and finding the barrier per H···H eclipsing interaction. The experimental barrier is $V_3 = 12.0$ kJ/mol. Convert to kcal/mol and to a per-eclipsing-pair quantity, given that there are three equivalent H···H eclipsing interactions in the fully eclipsed conformer.

If (in the crude, purely additive steric picture) the entire barrier were divided evenly among the three equivalent eclipsing H···H interactions,

This is the number often quoted in introductory courses as "the cost of one eclipsing H–H interaction," roughly 4.0 kJ/mol (about 1.0 kcal/mol) per pair. It is worth being explicit that this is an operational bookkeeping number — useful for predicting barriers in other eclipsed systems by counting interactions — not a direct measurement of van der Waals repulsion at 2.29 Å, which independent nonbonded-potential estimates put far lower (a few tenths of a kJ/mol). The discrepancy between 4.0 kJ/mol "as if steric" and the much smaller true van der Waals contribution is exactly the gap that hyperconjugation fills, as shown in Mechanism 1.

Limiting-case check: as the dihedral angle sweeps continuously from $\varphi=0°$ to $360°$, $E(\varphi)$ from the formula in the core-ideas section must return exactly to its starting value and pass through exactly three maxima and three minima — consistent with the three-fold symmetry of a staggered/eclipsed/staggered/eclipsed... sequence every 60°, which is what the cosine form guarantees by construction.

Example 2 (applied): Estimating the rotational barrier of 1,1,2,2-tetrafluoroethane and explaining the deviation from simple ethane using both models. 1,1,2,2-Tetrafluoroethane (F₂HC–CHF₂) has a measured internal rotation barrier of about 18–20 kJ/mol between its staggered conformers — noticeably higher than plain ethane's 12.0 kJ/mol, even though replacing H with the larger F atom might naively be expected to increase steric repulsion in the eclipsed form dramatically (van der Waals radius of F, ~1.47 Å, is substantially larger than H's ~1.20 Å). If sterics were the whole story, an F···F eclipsing interaction should be dramatically more repulsive than an H···H one, correctly predicting a higher barrier — so far consistent. But the size and pattern of barriers across the halogenated ethanes (F, Cl, Br substitution patterns) do not scale simply with atomic radius; they correlate much better with the number and energy of available σ\(C–F) or σ\(C–H) acceptor orbitals. Fluorine's high electronegativity lowers the energy of the σ\(C–F) orbital (making it a better acceptor, since $\varepsilon_{\sigma^*}$ is smaller in the denominator of the hyperconjugation formula), and simultaneously the C–H bonds are somewhat better donors when attached to a carbon bearing electronegative F substituents (rehybridization increases s-character in the C–H bonds, per Bent's rule, strengthening σ(C–H) as a donor). The result is that σ(C–H)→σ\(C–F) anti-periplanar donation across the C–C bond is stronger than plain σ(C–H)→σ\*(C–H) donation in ethane, mechanistically raising the barrier well beyond what a naive "just bigger atoms, more steric clash" argument would predict on its own. Quantitatively:

This roughly 7 kJ/mol enhancement is understood as arising predominantly from the improved σ(C–H)→σ\(C–F) hyperconjugative pathway (lower-energy σ\ acceptor orbital), not from a doubling of steric bulk — a clean, checkable illustration of why the orbital picture, not the ball-bumping picture, is needed to rationalize real substituent trends.

Example 3 (competition-level): Estimating the population ratio of staggered to eclipsed-like conformations and the implied rotational barrier crossing rate at 298 K, then checking against the observed IR torsional frequency. Given $V_3 = 12.0$ kJ/mol, estimate (a) the equilibrium population ratio of molecules sitting exactly at a staggered minimum versus exactly at an eclipsed maximum using a naive Boltzmann factor (recognizing this is a crude two-state cartoon of a continuous rotor, useful for building intuition about barrier-crossing statistics), and (b) compare the thermal energy $RT$ at 298 K to $V_3$ to judge whether rotation is "fast" on the NMR/vibrational timescale.

(a) The Boltzmann ratio of populations at the potential maximum versus minimum is

At $T = 298$ K, $R = 8.314$ J/(mol·K), so $RT = 8.314 \times 298 = 2478$ J/mol $= 2.478$ kJ/mol. Then:

So even at the exact top of the barrier, the population is suppressed by roughly a factor of 126 relative to the bottom of the well — but because $V_3$ is comparable to $RT$ (not many multiples larger), thermal energy is easily enough to populate the barrier top transiently and the molecule crosses back and forth extremely rapidly; this is not a case (like biphenyl atropisomers with barriers of 100+ kJ/mol) where distinct rotamers could ever be isolated at room temperature.

(b) Comparing directly: $RT$ (2.48 kJ/mol at 298 K) is about one-fifth of $V_3$ (12.0 kJ/mol) — small enough that thermal population of near-barrier-top geometries is easy and rotation is correspondingly fast, but large enough that the potential well is not "washed out": ethane spends the overwhelming majority of its time near the three staggered minima, consistent with a well-defined equilibrium geometry and with observed torsional vibrational fine structure (the C–C torsional mode is observed near 289 cm⁻¹ in the Raman/IR spectrum of ethane, and standard treatment of this mode as a hindered rotor using the $V_3$ value above reproduces the observed torsional level spacings to good approximation).

Limiting case: as $V_3 \to 0$ (imagine artificially switching off hyperconjugation and sterics), the exponential ratio $\to 1$, meaning every dihedral angle becomes equally populated — free internal rotation, exactly the qualitative behavior expected if there really were no preference between staggered and eclipsed, confirming the formula's sanity at the trivial limit.

The rotational barrier of ethane was first inferred not from a single "discoverer" in the Newtonian sense but through a convergence of thermodynamic and spectroscopic evidence in the late 1930s. Its interpretation was substantially reshaped decades later by the development of natural bond orbital theory.