Conformational Analysis of Ethane

Spin the two halves of an ethane molecule against each other. Its energy oscillates six times per revolution. This phenomenon is so small (about 12 kJ/mol at the peak) that nineteenth-century chemists had no way to detect it. Yet it is the seed from which the entire discipline of conformational analysis grows. Ethane looks featureless: two methyl groups joined by a single, freely rotating C–C bond. There are no substituents to clash, no stereocenters, nothing a beginning student would flag as "interesting." That apparent simplicity is exactly why it is the right place to build rigorous intuition. Every complication that shows up later in butane, cyclohexane, and biomolecular torsion angles is already present here in its purest form. This matters because rotation about single bonds is not free in the naive sense implied by "single bonds rotate freely." Ethane's C–C bond does rotate — there is no π-system locking it flat. But the energy is not constant as you turn it. There is a real, physically measurable barrier. Understanding why that barrier exists (not just that it exists) is the entire point of this article. The explanation has evolved over time. Early textbooks blamed simple steric clash between hydrogens. The modern, quantum-mechanically grounded picture instead centers on Torsional Strain and Hyperconjugation. This involves stabilizing donor–acceptor overlap between C–H bonding and antibonding orbitals. That overlap is maximized in the staggered conformation and destroyed in the eclipsed one. The big picture is this: a molecule's total energy as a function of a dihedral (torsion) angle θ defines a conformational energy surface. For ethane this surface is one-dimensional and periodic with period 60°. It has exactly two distinct stationary points per period — a minimum (staggered) and a maximum (eclipsed). Everything about ethane's physical behavior follows from mapping out this curve properly. This includes its NMR-invisible but IR/microwave-detectable internal rotation, its heat capacity contribution, and its role as the textbook reference point for barrier heights. The mapping is done using Newman and Sawhorse Projections as the bookkeeping tool. This article builds the torsional profile from first principles. It shows what staggered and eclipsed actually look like in three dimensions. It explains why the eclipsed form is higher in energy. It gives how much higher, quantitatively, with the real experimental number. It also shows how orbital overlap arguments explain the curve's shape and periodicity. It closes by placing ethane's 12 kJ/mol barrier in context. This context includes butane's much richer landscape. It also includes the general phenomenon of hyperconjugative stabilization that recurs throughout organic chemistry.

Key takeaways

Rotating the C–C bond of ethane by 360° sweeps through six conformations: three staggered (minima) and three eclipsed (maxima), related by the molecule's threefold symmetry.
The staggered conformation is lower in energy by about 12.5 kJ/mol (2.9–3.0 kcal/mol) per eclipsing interaction pair, giving a total barrier to rotation of ~12 kJ/mol.
The modern explanation is hyperconjugation: each staggered C–H bond can donate density from a filled σ(C–H) into an antiperiplanar empty σ*(C–H) across the bond; eclipsing breaks this alignment.
Simple van der Waals steric repulsion between eclipsing hydrogens is real but far too small (H···H distance ~2.55 Å in the eclipsed form) to account for the full barrier alone.
The torsional potential is well modeled by $V(\theta) = \tfrac{V_0}{2}(1-\cos 3\theta)$ , a threefold (period-120°) cosine series consistent with the molecule's $D_{3d}$ / $D_{3h}$ symmetry.
The barrier is directly measured by microwave and far-infrared spectroscopy (torsional splitting) and confirmed by thermodynamic (entropy, heat capacity) data.
Key idea: ethane's rotational barrier is a hyperconjugative electronic effect, not primarily a steric one — staggering is orbital-overlap-optimized, not just space-optimized.

Where this lives today

Conformational analysis built on the ethane torsional model underlies how chemists reason about the shapes of every acyclic and cyclic single-bond framework in organic and biological chemistry. The same threefold hyperconjugative logic explains the preference for the anti conformation in Conformational Analysis of Butane, where the additional methyl–methyl steric term (absent in ethane) reshapes the energy landscape into anti and gauche minima separated by different-height eclipsed and syn-periplanar maxima. In proteins and nucleic acids, backbone dihedral angles (φ, ψ in polypeptides; the sugar-phosphate torsions in DNA/RNA) are governed by conformational energy landscapes that are direct, higher-dimensional generalizations of the single ethane torsional curve, now decorated with additional steric, electrostatic, and hydrogen-bonding terms; molecular mechanics force fields (AMBER, CHARMM, OPLS) used throughout structural biology and drug design explicitly include a $\tfrac{V_n}{2}(1+\cos(n\theta-\gamma))$ torsional term for every rotatable bond, parameterized in large part against exactly the kind of quantum-chemical and spectroscopic data that first characterized ethane. In catalysis and materials chemistry, hyperconjugative σ(C–H)→σ(C–H) or σ(C–H)→σ(C–X) donor–acceptor stabilization — first rigorously dissected using ethane as the minimal test case — is invoked to explain conformational preferences in fluorinated polymers, the anomeric effect in carbohydrate chemistry, and rotational barriers around aryl–aryl single bonds relevant to atropisomeric ligands used in asymmetric catalysis. Even the simplest possible molecule with a rotatable C–C bond, in other words, supplies the conceptual and quantitative template used across the whole of modern conformational chemistry.

Ask the tutor: How does adding a methyl group to each carbon (going from ethane to butane) reshape this torsional curve into distinct anti and gauche minima?

The physical picture — thorough prose

Picture ethane, H₃C–CH₃, viewed down the C–C bond axis using a Newman projection: the front carbon's three C–H bonds project from a central point, the back carbon's three C–H bonds project from the surrounding circle. As the back methyl group rotates relative to the front one, the dihedral angle θ between a chosen front H and a chosen back H sweeps continuously from 0° to 360°. Because each carbon carries three identical hydrogens arranged with local $C_{3v}$ symmetry, the molecule looks physically identical every 120° of rotation, and the front and back sets of hydrogens are staggered relative to each other every 60° in between the eclipsed points. At θ = 60°, 180°, 300° (equivalently, offsets of 60°, 180°, 300° between any front–back H pair) the molecule is in the staggered conformation: each back C–H bond bisects the angle between two front C–H bonds. Viewed down the axis, the six hydrogens are spread as evenly as possible, 60° apart around the circle. This conformation has $D_{3d}$ point-group symmetry (a center of inversion, three $\sigma_d$ planes, an $S_6$ axis along C–C). At θ = 0°, 120°, 240° the molecule is eclipsed: each back C–H bond points directly behind a front C–H bond, so in the Newman projection the six hydrogens appear to overlap in three coincident pairs. This conformation has $D_{3h}$ symmetry (a horizontal mirror plane, no inversion center). Between these two extremes the energy is not constant. Rotating from staggered toward eclipsed costs energy monotonically; rotating from eclipsed back toward the next staggered position releases it. The result is a smooth, perfectly periodic wave with period 120° (six-fold over the full 360°, but each "hump" is identical by symmetry, so the fundamental repeat unit is 120°, corresponding to three humps of true chemical distinctness averaged with the threefold internal symmetry of each methyl rotor — conventionally described with a $\cos 3\theta$ term because there are three equivalent maxima and three equivalent minima per full turn). The energy difference between adjacent staggered and eclipsed forms is the rotational barrier, $V_0 \approx 12.0$ – $12.5$ kJ/mol (2.9 kcal/mol), an extremely well-established experimental number from microwave and far-IR torsional spectroscopy of gaseous ethane. This is a conformational energy difference, not a configurational one: staggered and eclipsed ethane are not isomers you can bottle separately — they interconvert by simple bond rotation at room temperature roughly $10^{10}$ – $10^{11}$ times per second, far faster than any chemical or even most spectroscopic timescales except microwave/IR torsional transitions themselves.

Figure

Newman projections of staggered and eclipsed ethane, with the dihedral angle θ marked

Figure

The torsional energy profile of ethane: V(θ) = (V₀/2)(1 − cos 3θ)

The core ideas — the governing relationship

The torsional potential of ethane, to lowest order consistent with its symmetry, is written as a truncated Fourier (cosine) series in the dihedral angle θ:

V(\theta) = \frac{V_0}{2}\left(1 - \cos 3\theta\right)

where:

$\theta$ is the H–C–C–H dihedral angle (degrees or radians) between a reference front hydrogen and a reference back hydrogen,
$V_0$ is the barrier height, the energy difference between the eclipsed maximum and staggered minimum (experimentally $V_0 \approx 12.5$ kJ/mol $= 2.9$ kcal/mol),
the factor 3 in $\cos 3\theta$ encodes the threefold periodicity forced by the $C_{3v}$ local symmetry of each methyl rotor — the potential must return to itself every 120°, and $\cos 3\theta$ is the lowest-order term satisfying that.

Reading the formula: at $\theta = 0°$ (eclipsed), $\cos 3\theta = \cos 0° = 1$ , so $V = 0$ — wait, check the sign convention carefully. We want the eclipsed conformation to be the maximum. Define θ so that θ = 0 is eclipsed; then we need $V(0)=V_0$ and $V(60°)=0$ . Since $\cos(3\times 60°) = \cos 180° = -1$ , we get $V(60°) = \tfrac{V_0}{2}(1-(-1)) = V_0$ — that puts the maximum at 60°, the opposite of what we want. The correct assignment (and the one universally used) is:

V(\theta) = \frac{V_0}{2}\left(1 + \cos 3\theta\right), \qquad \theta = 0° \text{ defined as eclipsed}

Check: at $\theta=0°$ , $\cos 0° = 1$ , so $V(0°) = V_0$ (eclipsed = maximum, correct). At $\theta = 60°$ , $\cos 180° = -1$ , so $V(60°) = 0$ (staggered = minimum, correct). At $\theta = 120°$ , $\cos 360° = 1$ , so $V(120°) = V_0$ (next eclipsed maximum, correct — matches the 120° periodicity of the methyl rotor). The gradient (torque) driving rotation is

-\frac{dV}{d\theta} = \frac{3V_0}{2}\sin 3\theta

where the torque vanishes exactly at every staggered and eclipsed geometry (θ a multiple of 60°), confirming these are the only stationary points of the one-term model, with staggered as minima ( $d^2V/d\theta^2 > 0$ ) and eclipsed as maxima ( $d^2V/d\theta^2 < 0$ ).

Derivation AMechanism 1 — why eclipsing costs energy: the hyperconjugation argument

Goal. Show, using orbital overlap and curved-arrow-style donor–acceptor reasoning, why the staggered conformation is stabilized relative to eclipsed, and why the effect is periodic with the observed magnitude scale.

Step 1 — identify the interacting orbitals. Each C–H bond has both a filled bonding orbital, $\sigma_{CH}$ , and an empty antibonding orbital, $\sigma^*_{CH}$ , of much higher energy. Across the central C–C bond, a $\sigma_{CH}$ bond on the front carbon and a $\sigma^*_{CH}$ orbital on the back carbon (or vice versa) can overlap through space and through the intervening C–C σ-framework, provided their local axes are close to anti-periplanar (dihedral near 180°) — more precisely, for adjacent-carbon C–H bonds the useful alignment is when the front C–H bond is anti to a specific back C–H bond, which for a freely rotating methyl-methyl system reduces to: overlap is maximized when a front-carbon C–H bond and a back-carbon C–H bond are staggered (offset by 60°), because this is the geometry in which the $\sigma_{CH}$ lobe on the front carbon points most directly at the back-lobe of a $\sigma^*_{CH}$ on the rear carbon along the C–C axis.

Step 2 — the donor-acceptor interaction (curved arrow). In the staggered conformation, draw a curved arrow from the filled $\sigma_{C-H}$ bonding orbital (front carbon) into the empty $\sigma^*_{C-H}$ antibonding orbital (back carbon), representing a two-electron stabilizing interaction: $\sigma_{CH} \rightarrow \sigma^*_{CH}$ . This is a second-order perturbation (hyperconjugative donation), lowering the total electronic energy by an amount proportional to

\Delta E \propto \frac{|\langle \sigma_{CH} | \hat{H} | \sigma^*_{CH}\rangle|^2}{\varepsilon_{\sigma^*} - \varepsilon_{\sigma}}

where the numerator (overlap-dependent interaction matrix element) is maximized precisely when the two orbitals are aligned antiperiplanar-like along the C–C axis — which in ethane's local geometry is realized in the staggered form. There are six such donor–acceptor pairs around the ring in the staggered conformation (each of the three back C–H σ bonds donates into an antiperiplanar-ish front σ, and vice versa, though the precise pairwise counting gives three dominant σ→σ stabilizations at any instant, restated below).

Step 3 — why eclipsing destroys the overlap. Rotating by 60° to the eclipsed geometry moves every C–H bond directly in front of another C–H bond (dihedral 0°) rather than gauche-offset (dihedral 60°). At dihedral 0°, the $\sigma_{CH}$ and $\sigma^*_{CH}$ orbitals on opposite carbons are no longer oriented for effective through-space/through-bond donation — the interaction matrix element drops sharply (it depends on the angular overlap integral, which falls off away from the optimal antiperiplanar-like alignment characteristic of the staggered geometry). The stabilizing donor–acceptor interactions present in the staggered form are largely switched off, so the eclipsed conformation sits higher in energy — not because of new repulsion appearing, but because a stabilization present elsewhere is lost.

Step 4 — quantify the periodicity. Because each methyl group has three equivalent C–H bonds spaced 120° apart, the hyperconjugative stabilization pattern must repeat every 120° (returning to an equivalent set of six σ→σ alignments), and by the up–down symmetry of maxima and minima within that repeat unit, the dominant* Fourier component of $V(\theta)$ is the $n=3$ term, $\cos 3\theta$ , exactly as required by symmetry in the equation above. Higher harmonics ( $\cos 6\theta$ , etc.) exist but are much smaller corrections.

Step 5 — connect to the number. High-level ab initio and localized-orbital (NBO) energy decomposition analyses assign roughly 10–12 kJ/mol of the ~12.5 kJ/mol barrier to the loss of hyperconjugative stabilization on eclipsing, with steric (exchange-repulsion) and electrostatic contributions being comparatively minor and partially cancelling. The hyperconjugative picture, not classical steric clash, is therefore the load-bearing explanation.

Go deeper · Derivation B · 2 minMechanism 2 — the steric (Pauli repulsion) argument and why it is insufficient alone

Derivation BMechanism 2 — the steric (Pauli repulsion) argument and why it is insufficient alone

Goal. Evaluate the historically older explanation — that eclipsed hydrogens repel each other sterically — and show quantitatively why it cannot be the dominant term.

Step 1 — geometry of the clash. In eclipsed ethane, each pair of eclipsing H atoms sits at an H···H distance of about 2.29–2.55 Å depending on the level of theory/geometry used (compare: the sum of van der Waals radii for two hydrogens is roughly 2.4 Å, using $r_{vdW}(H) \approx 1.2$ Å). This distance is only modestly shorter than in the staggered form, where the analogous (nearest) H···H distances are larger, around 2.55–3.1 Å depending on which pair you track (vicinal gauche-like H···H distances in the staggered rotamer are longer because the substituents are spread 60° apart rather than eclipsing at 0°).

Step 2 — classical steric (exchange-repulsion) energy estimate. Modeling the H···H interaction with a simple Lennard-Jones-type repulsive term,

E_{rep}(r) \propto \left(\frac{r_0}{r}\right)^{12}

where $r_0$ is a reference van der Waals contact distance, a decrease in $r$ from ~2.9 Å (staggered-like separation) to ~2.3–2.5 Å (eclipsed) produces a real but modest increase in repulsive energy — careful force-field and ab initio partitioning analyses (Pophristic–Goodman, Bickelhaupt–Baerends, and related studies) generally find that steric (Pauli/exchange) repulsion contributes only a few kJ/mol to the total barrier, and in some partitioning schemes is largely compensated by other terms.

Step 3 — the historical over-attribution. Early 20th-century treatments (pre-modern quantum chemistry) attributed the entire ~12 kJ/mol barrier to H···H steric clash, reasoning that eclipsed hydrogens are "closer together" and must repel more strongly. This picture correctly predicts the sign of the effect (eclipsed higher than staggered) but badly overestimates how much of the magnitude comes from sterics alone, and it cannot naturally explain related phenomena such as the sizeable barriers found in molecules where the "eclipsing" substituents are lone pairs or small groups with negligible steric bulk (e.g., barriers in fluoromethane–type and heteroatom-substituted rotors), which hyperconjugation explains via σ→σ or lone-pair→σ donation instead.

Step 4 — reconciling the two pictures. Modern energy-decomposition analyses (using methods such as NBO deletion, Block-Localized Wavefunction, or Kohn–Sham energy decomposition) consistently find that removing hyperconjugative σ→σ* interactions computationally (by deleting them from the wavefunction) removes most of the calculated barrier, while removing only the steric/exchange term removes comparatively little — evidence that hyperconjugative donor-acceptor stabilization, not steric repulsion, is the dominant physical origin of the ethane rotational barrier, even though a nonzero, non-negligible steric destabilization coexists with it.

Step 5 — the synthesis used in this article. The total barrier is best understood as

V_0 \approx \Delta E_{hyperconj} + \Delta E_{steric} + \Delta E_{electrostatic}

where the hyperconjugative term is the largest single contributor (roughly 70–90% depending on method), the steric term is small and destabilizing toward eclipsed, and electrostatic contributions are smaller still and can go either way depending on the specific analysis — but all three sum to reproduce the experimental $V_0 \approx 12.5$ kJ/mol.

Worked examples

Example 1 (foundational) — locating stationary points and classifying them. Given $V(\theta) = \tfrac{V_0}{2}(1+\cos 3\theta)$ with $V_0 = 12.5$ kJ/mol, find all stationary points in $0° \le \theta < 360°$ and classify each. Set the derivative to zero:

\frac{dV}{d\theta} = -\frac{3V_0}{2}\sin 3\theta = 0

This requires $\sin 3\theta = 0$ , i.e. $3\theta = 0°, 180°, 360°, 540°, 720°, 900°, \dots$ , giving

\theta = 0°, 60°, 120°, 180°, 240°, 300°

Second derivative:

\frac{d^2V}{d\theta^2} = -\frac{9V_0}{2}\cos 3\theta

At $\theta = 0°$ : $\cos 0° = 1$ , so $d^2V/d\theta^2 = -\tfrac{9V_0}{2} < 0$ — a maximum (eclipsed), value $V(0°) = V_0 = 12.5$ kJ/mol. At $\theta = 60°$ : $\cos 180° = -1$ , so $d^2V/d\theta^2 = +\tfrac{9V_0}{2} > 0$ — a minimum (staggered), value $V(60°) = 0$ . Repeating, $\theta = 120°, 240°$ are maxima (eclipsed, each $V=12.5$ kJ/mol) and $\theta = 180°, 300°$ are minima (staggered, each $V=0$ ), confirming three equivalent maxima and three equivalent minima per full turn, exactly matching the physical picture.

Example 2 (applied) — estimating the population ratio of staggered vs. eclipsed geometries at room temperature. Although "eclipsed ethane" is not a stable species you can isolate, it is meaningful to ask what fraction of a Boltzmann ensemble of molecules sits instantaneously near the top of the barrier versus near the bottom, using the Boltzmann factor at $T = 298$ K. Using $V_0 = 12.5$ kJ/mol and $R = 8.314$ J/(mol·K):

\frac{P(\text{eclipsed})}{P(\text{staggered})} \approx e^{-V_0/RT}

Compute the exponent:

\frac{V_0}{RT} = \frac{12{,}500 \text{ J/mol}}{(8.314 \text{ J mol}^{-1}\text{K}^{-1})(298 \text{ K})} = \frac{12{,}500}{2477.6} = 5.045

So:

\frac{P(\text{eclipsed})}{P(\text{staggered})} \approx e^{-5.045} \approx 6.4\times10^{-3}

Interpretation: at room temperature the population sitting at the exact eclipsed geometry (per unit dihedral angle, in a crude two-state approximation) is less than 1% of the population sitting at the staggered geometry — the molecule spends the overwhelming majority of its time near the staggered minima and only fleetingly passes through the eclipsed maxima during rotation. (This two-state Boltzmann ratio is a simplification — the real system is a continuous torsional oscillator/rotor, not a two-level system — but it correctly conveys the strong preference for staggered geometries and is the standard back-of-envelope estimate taught alongside the barrier value.) As a limiting-case check: if $V_0 \to 0$ (imagine a hypothetical rotor with no torsional preference at all), the ratio $e^{-V_0/RT} \to e^0 = 1$ , correctly predicting equal population of all rotational angles — i.e., genuinely free rotation, consistent with the definition of $V_0$ as the entire origin of the preference.

Example 3 (exam/competition-level) — from spectroscopic torsional splitting to the barrier height, and comparing two isotopologues. The internal rotation of ethane is quantized: instead of a continuum, torsional motion in the shallow-barrier limit is described by a hindered-rotor Schrödinger equation,

-\frac{\hbar^2}{2I_r}\frac{d^2\psi}{d\theta^2} + \frac{V_0}{2}(1+\cos 3\theta)\,\psi = E\,\psi

where $I_r$ is the reduced moment of inertia for internal rotation of one methyl group against the other. Far-infrared and Raman spectroscopy resolve torsional energy level splittings that, combined with a known (computed) reduced moment of inertia, allow $V_0$ to be extracted by fitting observed transition frequencies to the eigenvalues of this equation (a Mathieu-type equation once you substitute $x = 3\theta/2$ ). Historically, this is exactly how the barrier height of 1030–1040 cm⁻¹ (≈ 12.3–12.4 kJ/mol, in close agreement with the thermodynamically derived value) was pinned down for normal ethane (C₂H₆), and a smaller, mass-dependent shift is observed for the fully deuterated isotopologue C₂D₆. Task: given that the reduced moment of inertia for internal rotation scales with the mass of the rotating groups, and that deuterium has roughly twice the mass of hydrogen, predict qualitatively whether the torsional energy level spacings (not the barrier height itself, which is an electronic/potential-energy quantity and is isotope-independent to a very good approximation within the Born–Oppenheimer approximation) should be larger or smaller for C₂D₆ compared to C₂H₆. Reasoning: the torsional level spacings for a hindered rotor scale roughly as

\Delta E \propto \sqrt{\frac{V_0}{I_r}}

(by analogy with a harmonic oscillator built from the quadratic expansion of $V(\theta)$ near its minimum, where the "force constant" is set by $V_0$ and the "mass" by $I_r$ ). Since $I_r$ for C₂D₆ is larger than for C₂H₆ (heavier rotating atoms, same geometry, same $V_0$ because the barrier is an electronic-structure property nearly independent of nuclear mass under the Born–Oppenheimer approximation), the torsional level spacings for C₂D₆ must be smaller than for C₂H₆ — deuteration compresses the torsional spectrum without changing the barrier height itself. This is exactly the qualitative behavior observed experimentally, and it is a clean, checkable limiting-case-style prediction: as isotopic mass increases with the potential held fixed, quantized level spacings shrink, consistent with the general particle-in-a-well scaling relation.

Limiting cases, scope & common traps

Case / Question	Correct statement
Is eclipsed ethane a real, isolable molecule?	No — it is a transition-state-like maximum on the torsional potential, not a local minimum; it cannot be isolated, only inferred spectroscopically and computationally.
Does the C–C bond in ethane "not rotate" because of the barrier?	No — 12.5 kJ/mol is easily overcome by thermal energy at room temperature (RT ≈ 2.5 kJ/mol, but rotation is a large-amplitude motion sampling the whole potential, and rotational interconversion rates are ~10¹⁰–10¹¹ s⁻¹); rotation is rapid, just not perfectly "free" (uniform-energy) rotation.
Is the barrier mainly steric (H···H clash)?	No — modern energy decomposition shows hyperconjugative σ(C–H)→σ*(C–H) stabilization loss dominates; steric repulsion is a real but minor contributor.
Does $V(\theta)=\tfrac{V_0}{2}(1+\cos3\theta)$ have only one Fourier term in reality?	No — it is the leading term; real potentials include small $\cos 6\theta$ , $\cos 9\theta$ corrections, but the threefold term dominates by far.
Is the barrier the same for all ethane-like rotors (e.g., CH₃–CF₃, CH�3–CCl₃)?	No — replacing H by more electronegative or bulkier groups changes both hyperconjugative donor/acceptor energies and steric terms substantially; barriers can be several times larger or occasionally smaller.
Does the barrier height depend on isotopic substitution (H → D)?	Essentially no, to a very good Born–Oppenheimer approximation — the electronic potential energy surface is unchanged; only the quantized torsional levels (which depend on reduced mass) shift.

The chemist behind it

The concept of restricted internal rotation about single bonds was not the work of a single individual but crystallized over several decades. Early recognition that C–C single bonds might not permit fully free rotation came from thermodynamic anomalies: measured heat capacities and third-law entropies of simple hydrocarbons like ethane did not match values calculated assuming completely unrestricted internal rotation, a discrepancy noted and analyzed carefully in the 1930s. This forced physical chemists to introduce a hindering potential into the statistical-mechanical treatment of ethane's thermodynamics, effectively "discovering" the rotational barrier from bulk thermodynamic measurements before any spectroscopic method could resolve it directly. The theoretical apparatus — treating internal rotation as a hindered rotor with a Fourier-series potential, and extracting barrier heights by fitting partition functions to measured entropies and heat capacities — is closely associated with the broader development of statistical thermodynamics applied to molecular structure in the early-to-mid twentieth century, a period in which chemists such as Kenneth Pitzer made foundational contributions to treating hindered internal rotation quantitatively and systematically across many small hydrocarbons, putting the ethane barrier and its analogues in longer-chain alkanes on firm quantitative footing.

Origins · 1 minKenneth S. Pitzer and how this was discovered

Kenneth S. Pitzer

1914–1997 · American physical chemist and pioneer of statistical thermodynamics of hindered internal rotation

Pitzer's work in the late 1930s and beyond established the standard framework for treating internal rotation about single bonds as a hindered rotor problem, reconciling calculated and measured thermodynamic properties (entropy, heat capacity) of ethane and its homologues by introducing a threefold cosine barrier into the partition function — precisely the $V(\theta)$ used throughout this article. His methodology became the standard tool by which rotational barriers throughout organic chemistry are extracted from bulk thermodynamic data, well before high-resolution microwave and far-infrared spectroscopy could measure such barriers with comparable or greater precision directly from torsional transition frequencies.

History

1936

— Thermodynamic anomalies in ethane's measured entropy/heat capacity indicate hindered, not free, internal rotation.

1937–1940

s — Pitzer and coworkers develop the hindered-rotor statistical mechanical framework, fitting a threefold cosine barrier (~12 kJ/mol) to thermodynamic data for ethane.

1950

s–1960s — Microwave and far-infrared spectroscopy directly resolve torsional energy level splittings, confirming and refining the barrier height independently of thermodynamic fitting.

1970

s–1990s — Ab initio quantum chemistry calculations reproduce the barrier and begin decomposing its physical origin into steric, electrostatic, and hyperconjugative contributions.

2000

s–present — Modern energy decomposition analyses (NBO deletion, Bickelhaupt–Baerends EDA, Pophristic–Goodman studies) firmly establish hyperconjugative σ→σ* stabilization loss as the dominant contributor to the ethane rotational barrier.

Why is ethane, with no substituents, considered "interesting" for conformational analysis?

Ethane is the simplest possible molecule with a freely rotatable single bond flanked by more than one substituent on each carbon, making it the cleanest system in which to isolate the intrinsic electronic origin of rotational barriers, uncomplicated by steric bulk from larger groups. Because both carbons carry only hydrogens, any energy difference between conformations must arise from H···H interactions and orbital effects alone — there is no methyl-methyl steric clash to muddy the picture, as there will be starting with butane. This makes ethane the ideal minimal test case for comparing steric and hyperconjugative explanations quantitatively, and its ~12.5 kJ/mol barrier is the reference number against which every more complex rotor's barrier is compared.

How is the ethane rotational barrier actually measured experimentally?

Two independent routes converge on the same number. First, thermodynamic measurements (heat capacity, third-law entropy) of gaseous ethane deviate from values predicted assuming completely free internal rotation; fitting a hindered-rotor partition function to these deviations yields a barrier around 12 kJ/mol. Second, and more directly, far-infrared and Raman spectroscopy resolve torsional energy level splittings — the quantized vibrational-like levels of the hindered internal rotor — and fitting these transition frequencies to the eigenvalues of the corresponding Schrödinger equation yields a barrier of about 1035 cm⁻¹ (≈12.3–12.4 kJ/mol), in excellent agreement with the thermodynamic estimate.

Is the staggered conformation of ethane chiral or does it have any stereocenters?

No. Staggered ethane has $D_{3d}$ symmetry, which includes an inversion center and multiple mirror planes, making it achiral; neither carbon is a stereocenter because each bears three identical hydrogens plus the other carbon. Conformational analysis of ethane is entirely about energy as a function of a single rotatable dihedral angle, not about configuration or chirality — those concepts become relevant once you move to substituted, stereogenic systems, but ethane itself is the conformational (not configurational) starting point for the whole subject.

Why does the torsional potential use cos 3θ rather than cos θ or cos 2θ?

The exponent in the cosine term must match the rotational symmetry of the two halves of the molecule being rotated against each other. Each methyl group in ethane has three equivalent hydrogens arranged with $C_3$ local symmetry, so the potential energy must return to an identical value every 120° of rotation — a full period of $2\pi/3$ radians. A cosine term $\cos(n\theta)$ has period $2\pi/n$ , so $n=3$ is the correct (lowest-order, dominant) choice; $\cos\theta$ or $\cos 2\theta$ would incorrectly predict a period of 360° or 180°, inconsistent with the molecule's actual threefold symmetry and with the observed three staggered/three eclipsed conformations per full turn.

How does the ethane barrier compare to the barriers seen in butane conformational analysis?

Ethane's single ~12.5 kJ/mol barrier, arising purely from H···H hyperconjugative and steric effects, is the simplest case. In butane, replacing one hydrogen on each carbon with a methyl group breaks the full threefold symmetry: rotation now passes through a lower-energy anti minimum, two equivalent (higher-energy) gauche minima, and eclipsed maxima of different heights depending on whether methyl groups eclipse each other (highest, syn-periplanar) or eclipse hydrogens (lower). The ethane torsional curve is essentially the "background" hyperconjugative wave onto which butane adds an extra, substituent-dependent steric term — the natural next step covered under Conformational Analysis of Butane.

Does the rotational barrier mean ethane molecules are "locked" in one shape at room temperature?

No — quite the opposite conclusion is correct. Because $V_0 \approx 12.5$ kJ/mol is comparable to, though larger than, the thermal energy scale $RT \approx 2.5$ kJ/mol at 298 K, ethane molecules rotate rapidly and continuously through all dihedral angles, spending most of their time near the staggered minima but readily passing over the eclipsed maxima roughly $10^{10}$ – $10^{11}$ times per second. This is fast enough that on the timescale of ordinary chemistry (and even most spectroscopy other than dedicated torsional/microwave spectroscopy), ethane behaves as if it were a single, rapidly averaged species rather than a mixture of distinguishable rotamers.

What role does the reduced moment of inertia play in the quantum treatment of ethane's internal rotation?

When internal rotation is treated quantum mechanically as a hindered rotor rather than classically, the relevant "mass" term in the kinetic energy operator is the reduced moment of inertia $I_r$ for relative rotation of one methyl group against the other, analogous to reduced mass in a two-body vibration problem. This quantity, combined with the barrier height $V_0$ , sets the spacing of quantized torsional energy levels, with spacings scaling roughly as $\sqrt{V_0/I_r}$ near the bottom of each well. Heavier isotopologues (like C₂D₆) have larger $I_r$ and hence more closely spaced torsional levels, even though $V_0$ itself is essentially isotope-independent.