Conformations of Ethane: Newman Projections

Conformational analysis built on the Newman projection is not a historical curiosity confined to introductory courses — it is the working vocabulary of modern structural and mechanistic organic chemistry. Every discussion of a reaction's stereochemical outcome that depends on the preferred rotamer of an open-chain intermediate — nucleophilic additions to acyclic ketones analyzed via Felkin–Anh models, radical cyclizations, or SN2 transition-state geometries — is drawn and reasoned about using Newman-style projections down the reacting bond. Pharmaceutical chemists routinely examine Newman projections (or their modern computational analog, a dihedral-angle energy scan from molecular mechanics or DFT) when assessing whether a candidate drug molecule's freely rotatable bonds allow it to adopt the bioactive conformation needed to fit a protein binding pocket; "conformational entropy penalty" upon binding is precisely the cost of freezing a torsional coordinate like the one described here. In structural biology, the same torsional logic scales up: the backbone dihedral angles $\phi$ and $\psi$ of a protein (rotations about the N–Cα and Cα–C(=O) bonds) are graphed on a Ramachandran plot whose allowed and forbidden regions are, at heart, a more elaborate version of ethane's $V(\theta)$ curve, now shaped by steric clash between much bulkier substituents and by competing intramolecular hydrogen bonds. Polymer scientists reasoning about the flexibility and persistence length of a polyethylene chain are, chain segment by chain segment, applying exactly the staggered-versus-eclipsed logic developed here, since each C–C bond along the backbone samples the same three-fold torsional landscape (perturbed by neighboring substituents, as foreshadowed by Conformations of Butane: Gauche and Anti). Even NMR spectroscopy leans on this framework directly: the Karplus equation, relating a vicinal $^3J_{HH}$ coupling constant to the H–C–C–H dihedral angle, is an empirical fit built entirely on the geometric definition of $\theta$ established by the Newman projection, and is used daily to determine unknown conformations and relative stereochemistry of natural products and synthetic intermediates from routine NMR spectra.

Ask the tutor: Why does hyperconjugation into a σ* orbital require the specific anti-periplanar geometry found in the staggered conformer, and how would you show this with an orbital diagram?

Take ethane and orient it so the C–C bond points directly at your eye. The near carbon (call it C1) hides the far carbon (C2) completely — that is the entire point of "looking down the bond." C1's three C–H bonds fan out toward you in a tripod, each separated from its neighbors by $120°$ in the plane perpendicular to your line of sight (this is simply the projection of $sp^3$ tetrahedral geometry onto that plane). Behind C1, invisible in a real ball-and-stick model but drawn explicitly in a Newman projection, C2's three C–H bonds also fan out in their own tripod, also separated by $120°$. The Newman projection converts this line-of-sight view into ink: the front atom becomes a point (or a small "Y" of three lines meeting at a dot) and the back atom becomes a circle, with three lines drawn from the edge of the circle outward (never crossing through the center, since that would look like they belong to the front atom). The dihedral angle between a specified front bond and a specified back bond is simply the angular separation between those two lines as drawn on the page. Because each methyl group is itself $C_{3v}$-symmetric (three equivalent H's, $120°$ apart), the appearance of the Newman projection repeats every $120°$ of rotation. Two special angles matter most:

• Staggered ($\theta = 60°, 180°, 300°$ for the reference pair): back bonds sit exactly between front bonds. Looking down the bond, you see six evenly spaced lines, alternating front/back, each separated by $60°$.
• Eclipsed ($\theta = 0°, 120°, 240°$): each back bond hides directly behind a front bond. Looking down the bond, you see only three lines (though chemists draw the back bond slightly offset, at a small angle, purely so both bonds remain visible on paper).

Note the geometric logic in the drawing: in the staggered projection, the six C–H lines are equally spaced at $60°$ around the full circle — front, back, front, back, alternating — while in the eclipsed drawing the back lines are drawn nearly (not exactly, for clarity) on top of the front lines, at $\theta \approx 0°$. Now consider what physically differs between the two arrangements. In the staggered form, every front C–H bond points between two back C–H bonds — the hydrogens are as far apart as three-fold symmetry allows, minimizing steric crowding of the electron clouds on adjacent atoms. In the eclipsed form, every front C–H bond points directly at a back C–H bond, forcing the C–H bonding electron pairs into closer proximity. There is also a subtler and, it turns out, energetically dominant effect: in the staggered conformation each C–H $\sigma$ bond on the back carbon is anti-periplanar to a C–H $\sigma^*$ antibonding orbital on the front carbon, allowing a small stabilizing donation of electron density (hyperconjugation) that is geometrically impossible in the eclipsed form, where the relevant orbitals are orthogonal rather than aligned. The full quantum mechanical accounting of why the staggered form wins is developed under Torsional Strain and the Origin of the Ethane Barrier; here we take the empirical energy difference as given and build the quantitative torsional potential around it.

The curve has period $120°$, maxima of $12.0\ \text{kJ/mol}$ at the eclipsed angles $0°, 120°, 240°, 360°$, and minima of $0$ at the staggered angles $60°, 180°, 300°$.

The torsional energy of ethane, as a function of the dihedral angle $\theta$ between a reference front C–H bond and a reference back C–H bond, is well described by a single cosine term:

where:

• $V(\theta)$ is the torsional potential energy at dihedral angle $\theta$,
• $V_0$ is the full barrier height (the energy difference between eclipsed and staggered conformers),
• the factor $3$ inside the cosine encodes the three-fold periodicity forced by the $C_{3v}$ symmetry of each methyl rotor,
• $\theta = 0°$ is chosen, by convention, to correspond to an eclipsed conformation, so that $\cos(0) = 1$ gives $V(0) = V_0$, the maximum.

Reading this equation: at $\theta = 0°, 120°, 240°$ the cosine equals $+1$ and $V = V_0$ (eclipsed maxima); at $\theta = 60°,180°,300°$ the cosine equals $-1$ and $V=0$ (staggered minima). The period of $\cos(3\theta)$ is $120°$, matching exactly the fact that rotating one methyl group by $120°$ relative to the other simply relabels which hydrogen is which — the molecule (and hence its energy) is unchanged. A second essential relationship converts an energy difference between conformers into a population ratio at thermal equilibrium, via the Boltzmann distribution:

where:

• $N_{\text{eclipsed}}/N_{\text{staggered}}$ is the ratio of molecules instantaneously found near the eclipsed versus staggered geometry,
• $\Delta V = V_0$ is the barrier height (energy of eclipsed minus energy of staggered),
• $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$ is the gas constant,
• $T$ is the absolute temperature in kelvin.

This equation makes explicit something conceptually crucial: even though the eclipsed conformation is a strained, high-energy geometry, it is not "forbidden" — it is simply thermally disfavored, and it is transiently populated as every ethane molecule rotates through it on the way from one staggered well to the next.

Example 1 (foundational) — Reading and constructing a Newman projection. Draw the Newman projection of ethane looking from C1 toward C2, in the staggered conformation, and label the dihedral angle between a specific front hydrogen $H_{1a}$ and its nearest back hydrogen $H_{2a}$. Reasoning. Place the front carbon's three C–H bonds, by convention, pointing straight up and to the two lower corners: $\phi_1(H_{1a}) = 90°$, $\phi_1(H_{1b}) = 210°$, $\phi_1(H_{1c}) = 330°$ (measuring counterclockwise from the positive $x$-axis, a common drawing convention). For the staggered conformer, each back C–H bond must bisect a pair of front bonds. The gaps between front bonds are centered at $90°+60°=150°$, $210°+60°=270°$, and $330°+60°=30°$ (mod $360°$). So the back bonds sit at $\phi_2 = 30°, 150°, 270°$. The nearest back hydrogen to $H_{1a}$ (at $90°$) is the one at $150°$ (a separation of $60°$) rather than the one at $30°$ (also $60°$ away, by symmetry — both are equally "nearest," which is the hallmark of a perfectly staggered arrangement). Either choice gives:

This matches Step 4 of the second derivation exactly, confirming the drawing is self-consistent. Limiting case check. If instead we had drawn the eclipsed conformer, the back bonds would sit at $\phi_2 = 90°, 210°, 330°$ (directly behind the front ones), giving $\theta = 0°$ for the aligned pair — recovering the eclipsed maximum of $V(\theta)$, exactly as expected, with no arbitrary choice needed since the eclipsing bond is unambiguous.

Example 2 (applied) — Population ratio of eclipsed versus staggered ethane at room temperature. Using $V_0 = 12.0\ \text{kJ/mol}$ for the ethane rotational barrier, estimate the fraction of molecules found instantaneously in the eclipsed geometry relative to the staggered geometry at $T = 298\ \text{K}$. Reasoning. Apply the Boltzmann ratio derived above:

Compute the exponent first:

The denominator is:

So:

Then:

Interpretation: for roughly every 126 molecules found at (or extremely near) a staggered geometry at any instant, only about 1 sits at the exact eclipsed maximum — a strong but not absolute preference. This is why ethane's rotation is described as "essentially free": with a barrier this low, thermal energy at room temperature ($RT \approx 2.5\ \text{kJ/mol}$) is enough for the molecule to cross the barrier roughly $10^{10}$–$10^{11}$ times per second (an estimate obtainable from transition-state theory, using $V_0/RT\approx4.8$ in an Arrhenius-type prefactor of $\sim 10^{13}\,\text{s}^{-1}$), so essentially no population is ever trapped in a single well. Limiting-case check. As $T \to \infty$, $V_0/RT \to 0$ and the ratio $\to \exp(0) = 1$: at very high temperature the populations of eclipsed and staggered geometries become equal, exactly as expected since thermal energy overwhelms the modest torsional preference and the molecule samples all $\theta$ nearly uniformly. As $T\to 0$, the ratio $\to 0$: the molecule freezes into the staggered well, consistent with quantum-mechanical localization at absolute zero.

Example 3 (exam / competition level) — Distinguishing ethane's barrier from a substituted analog and predicting a Newman diagram for a general dihedral. 1,2-Difluoroethane, $\text{FCH}_2\text{–CH}_2\text{F}$, is not ethane, but consider a hypothetical thought experiment: suppose a symmetric, ethane-like rotor had a torsional potential with two significant Fourier terms,

with $V_0 = 12.0\ \text{kJ/mol}$ and $V_2 = 1.50\ \text{kJ/mol}$ (a small second-harmonic correction). Find the exact angle nearest $\theta=60°$ at which the true minimum occurs, to first order. Reasoning. Differentiate and set the derivative to zero:

Near $\theta = 60°$, write $\theta = 60° + \delta$ with small $\delta$. Then $3\theta = 180° + 3\delta$, so $\sin(3\theta) = \sin(180°+3\delta) = -\sin(3\delta) \approx -3\delta$ (in radians, for small $\delta$). Also $6\theta = 360°+6\delta$, so $\sin(6\theta) = \sin(6\delta) \approx 6\delta$. Substituting:

Since $\frac{9(12.0)}{2} - 18(1.50) = 54.0 - 27.0 = 27.0 \neq 0$, the only solution is $\delta = 0$: to first order in the small-second-harmonic limit, the minimum stays exactly at $\theta = 60°$, undisplaced. This makes physical sense — a pure $\cos(6\theta)$ correction shares the same symmetry axis as $\cos(3\theta)$ at $\theta=60°$ (both have zero slope there by symmetry, since $60°$ is a symmetry point for both harmonics separately), so a small admixture cannot shift the extremum's location, only its curvature (and hence the vibrational frequency of the torsional oscillation about that minimum). This is exactly the kind of reasoning used, in more advanced treatments, to explain why substituted ethanes (e.g., 1,2-dihaloethanes, where gauche/anti populations shift measurably) require genuinely new physics — electrostatics, dipole–dipole terms, larger steric bulk — layered on top of the basic three-fold pattern rather than a simple change in the coefficient of the same harmonic; that richer story is the subject of Conformations of Butane: Gauche and Anti. Limiting case check. Setting $V_2 = 0$ recovers pure ethane: $\delta = 0$ trivially, minimum exactly at $\theta=60°$, matching the geometric derivation above term-for-term.

The Newman projection is named for Melvin Spencer Newman (1908–1993), an American organic chemist at Ohio State University who introduced the "front atom/back atom" projection formalism in a short 1955 paper aimed squarely at making conformational analysis teachable. Newman's broader career centered on physical organic chemistry and synthetic methodology — steric effects in aromatic systems, the synthesis of polycyclic aromatic hydrocarbons, and reactive intermediates — but it is this single pedagogical device, born from the practical need to represent three-dimensional torsional geometry unambiguously on a flat page, for which his name is now attached to nearly every organic chemistry textbook diagram of a rotating C–C bond. The conceptual foundation Newman's projection formalized — that rotational conformers of ethane have measurably different energies — rests on much earlier physical chemistry. Rotational barriers in ethane-like molecules were inferred indirectly from heat capacity and spectroscopic anomalies starting in the 1930s, well before quantum chemical calculations could assign a clean orbital-level cause. The size of the ethane barrier itself, close to $12\ \text{kJ/mol}$, was pinned down by microwave and infrared spectroscopy in the mid-20th century, and its microscopic origin — steric repulsion versus hyperconjugative stabilization — remained (and to a real extent still remains) an active topic of computational and conceptual debate, addressed in depth under Torsional Strain and the Origin of the Ethane Barrier.