Conformational Analysis of Butane

The anti-preference established here is the microscopic reason phospholipid and fatty-acid tails pack into extended, parallel zigzag chains in lipid bilayers — every internal C–C bond independently prefers anti, and a chain of such bonds strung together in a row produces the fully extended all-anti zigzag that gives membranes their characteristic thickness and packing density; a single gauche "kink" introduced by unsaturation (a cis double bond) disrupts this packing and is precisely why unsaturated fats have lower melting points than saturated ones. Polyethylene crystallizes with its backbone in the all-anti zigzag for the same reason, and the gauche-content of the amorphous regions (governed by exactly the Boltzmann statistics derived above) controls polymer density and mechanical properties. In drug design, understanding gauche/anti equilibria along flexible linkers is central to bioactive conformation analysis: medicinal chemists routinely lock a flexible chain into its bioactive (often gauche, if that is what the receptor pocket demands) conformation using a ring or a strategically placed substituent, trading the ~3–4 kJ/mol conformational penalty for a large binding-affinity gain by pre-organizing the molecule and avoiding the entropic cost of freezing rotation upon binding. NMR-based conformational analysis of peptides and flexible natural products uses vicinal coupling constants (via the Karplus relationship, itself dependent on dihedral angle) essentially as a direct experimental readout of exactly this anti/gauche population problem, generalized to arbitrary φ. The same logic scales up directly into the analysis of Cyclohexane Chair Conformation, where six such gauche/anti-type interactions around the ring, combined with the geometric constraint of ring closure, single out the chair as uniquely strain-free; and it sets up Steric Strain and Gauche Interactions as the natural next topic, generalizing the methyl-methyl steric penalty derived here to arbitrary substituents of different sizes (A-values), which is the quantitative currency used throughout the rest of conformational analysis.

Ask the tutor: How does this same anti/gauche/eclipsed analysis change when one of the methyl groups in butane is replaced by a much bulkier group like tert-butyl, and what does that predict for the compound's melting and boiling behavior?

Picture butane end-on, sighting down the C2–C3 bond exactly as you would for ethane's Newman projection, but now the front carbon (C2) carries one CH₃ and two H's, and the back carbon (C3) carries one CH₃ and two H's. The dihedral angle $\phi$ is defined as the angle between the front C2–CH₃ bond and the back C3–CH�3 bond, measured clockwise when viewed front-to-back. By convention $\phi = 180°$ is called anti, $\phi = 0°$ is the fully eclipsed (syn) conformer, and $\phi = \pm 60°$ are the two gauche conformers. As $\phi$ sweeps from 0° to 360° in 60° steps, six conformers occur, alternating staggered/eclipsed exactly as in ethane — but because the front and back substituents are no longer all-identical (CH₃, H, H rather than H, H, H), the three staggered forms are not energetically equivalent to one another, and neither are the three eclipsed forms. The staggered forms split into one anti and two (mirror-image, hence degenerate) gauche forms. The eclipsed forms split into one fully syn-eclipsed (Me over Me) form and two degenerate forms in which a methyl eclipses a hydrogen while the other methyl eclipses the opposite hydrogen. Torsional strain — the electronic preference for adjacent C–H (or C–C) σ bonds to be staggered rather than eclipsed, arising from σ(C–H) → σ(C–H) hyperconjugative stabilization being maximized in the staggered geometry — is present in butane exactly as it was in ethane, contributing roughly the same ~12.5 kJ/mol total barrier per eclipsing event pattern. But butane layers a second, purely classical effect on top: <strong>steric strain</strong>, the rise in energy when two bulky, electron-rich groups (here, two methyls, each roughly the size of an ethane CH₃ but now nonbonded to each other) are forced into each other's van der Waals radii. This repulsion is not* a bonding/orbital interaction in the hyperconjugative sense; it is dominated by Pauli exclusion between filled orbitals on the two groups as they approach — the same physics that makes noble gases repel each other at short range.

The key qualitative rule that falls out of this picture: staggered beats eclipsed always (torsional strain), but among staggered forms, anti beats gauche (steric strain), and among eclipsed forms, methyl/H beats methyl/methyl (steric strain again). Both rules are steric in the everyday sense — put the fat groups as far apart as geometry allows — layered on the electronic staggering preference inherited from ethane.

The total conformational energy at any dihedral angle is, to a first and very useful approximation, additive over independent through-bond torsional contributions and through-space nonbonded contributions:

where:

• $E_{\text{torsional}}(\phi)$ is the ethane-like periodic term from staggered/eclipsed σ/σ* hyperconjugative differences, roughly a threefold (and smaller sixfold) Fourier series in $\phi$,
• $E_{\text{steric}}(\phi)$ is the extra, non-periodic-in-the-same-way penalty from methyl···methyl or methyl···H van der Waals crowding, largest when the two methyls are closest in space.

A compact and pedagogically standard way to write the torsional part is a truncated Fourier (Pitzer-type) expansion:

where:

• $V_3$ is the threefold barrier parameter (dominant term, same physical origin as ethane's 12.5 kJ/mol threefold barrier),
• $V_1$ is a smaller onefold term needed because the front and back substituents are not all equivalent (CH₃ ≠ H breaks the pure threefold symmetry that ethane has),
• $\phi$ is the C1–C2–C3–C4 dihedral angle.

Reading this equation: the $V_3$ term alone would make all three staggered minima and all three eclipsed maxima equally deep/high, exactly as in ethane. It is the $V_1$ term — present only because butane's front and back "caps" are CH₃ plus two H's rather than three identical H's — that breaks the degeneracy between anti and gauche, and between syn-eclipsed and gauche-eclipsed. Steric strain is what physically justifies including such a term: it is largest at $\phi=0°$ (methyls closest) and essentially zero at $\phi=180°$ (methyls farthest), which is exactly the $\phi$-dependence of $(1-\cos\phi)$. The two central experimental numbers that anchor the whole diagram are the gauche–anti energy difference and the syn-eclipsed barrier height:

where the gauche–anti gap is measured by electron diffraction, Raman/IR band-intensity ratios, and NMR-based population studies, and the syn barrier is obtained from far-IR/Raman torsional spectroscopy and confirmed by high-level ab initio calculation (e.g., CCSD(T)/CBS extrapolations).

Goal. Show, by explicit counting of eclipsing/gauche interactions at each of the six stationary dihedral angles, why the energy ordering must be anti < gauche < gauche-eclipsed < syn-eclipsed, and estimate each gap from additive interaction energies.

Step 1 — catalogue the interaction types. At any staggered or eclipsed conformer of butane, look down the C2–C3 axis. Each of the three front substituents (CH₃, H, H on C2) sits either gauche (60° or 300°, staggered case) or eclipsing (0°, eclipsed case) relative to each of the three back substituents (CH₃, H, H on C3). We need three elementary interaction energies:

• $\varepsilon_{gg}(\text{H,H})$ — gauche H···H, essentially negligible (~0 kJ/mol, hydrogens are small and far apart even gauche),
• $\varepsilon_{gg}(\text{Me,H})$ — gauche Me···H, small but nonzero,
• $\varepsilon_{gg}(\text{Me,Me})$ — gauche Me···Me, the dominant steric term, ≈ +3.8 kJ/mol per such interaction (this is exactly what defines the gauche penalty, since anti has zero such interactions and gauche has exactly one),
• $\varepsilon_{ecl}(\text{H,H})$ — eclipsing H over H, ≈ +4.0 kJ/mol (same torsional unit as in ethane, where three such interactions sum to the ~12.5 kJ/mol barrier, i.e., ≈ 4.2 kJ/mol each),
• $\varepsilon_{ecl}(\text{Me,H})$ — eclipsing Me over H, torsional plus modest steric, ≈ 6 kJ/mol,
• $\varepsilon_{ecl}(\text{Me,Me})$ — eclipsing Me directly over Me, torsional plus severe steric, ≈ 8–11 kJ/mol.

These are empirical increments extracted by fitting the whole set of measured rotational barriers in simple alkanes (this is the same logic used to build force-field "torsional + van der Waals" parameter sets); they are not independently derived from a deeper equation, but once fixed they must reproduce every stationary point self-consistently, which is the check performed below.

Step 2 — anti (φ = 180°). Looking down C2→C3: front CH₃ is directly opposite (anti to) back CH�3; front H's are each gauche to one back H and gauche to the back CH₃... actually in the pure staggered anti geometry, each front substituent is gauche to two of the three back substituents (since staggered means 60° offsets throughout, with the one substituent directly opposite, i.e. 180°, being the anti pair). Concretely: CH₃(front)···CH₃(back) are anti (180°, not gauche — no penalty), CH₃(front) is gauche to each of the two back H's (negligible), and each front H is gauche to the back CH₃ and to one back H. Tallying only the interactions that matter energetically (Me···Me gauche pairs = 0 here, since the methyls are anti to each other, not gauche):

Step 3 — gauche (φ = 60° or 300°). Now the front CH₃ and back CH₃ are gauche to each other (60° apart) instead of anti. This introduces exactly one Me···Me gauche interaction that anti did not have:

This reproduces the experimental gauche–anti gap by construction — but the value 3.8 kJ/mol is not merely fitted air; it is corroborated independently by the observed 1,3-diaxial-type destabilization seen throughout cyclohexane and open-chain conformational analysis, where one gauche Me···Me interaction consistently costs 3.4–4.0 kJ/mol across many different molecules, strong evidence that this is a transferable, physically meaningful increment rather than an artifact of one measurement.

Step 4 — syn-eclipsed (φ = 0°). Here the front CH₃ eclipses the back CH₃ directly (the worst possible interaction, $\varepsilon_{ecl}(\text{Me,Me})$), and each front H eclipses a back H ($\varepsilon_{ecl}(\text{H,H})$, twice):

Substituting representative increments $\varepsilon_{ecl}(\text{Me,Me}) \approx 11$ kJ/mol and $\varepsilon_{ecl}(\text{H,H}) \approx 4.2$ kJ/mol (the same value that, summed three times, gives ethane's 12.5–12.6 kJ/mol barrier):

This lands squarely inside the experimentally bracketed range of 19–25 kJ/mol, with the spread in literature values reflecting different spectroscopic/computational methods and the fact that the true potential is not perfectly additive (there is also a smaller "cross term" from the changed C–C–C bond angle as the methyls repel each other, which softens the geometry and slightly lowers the barrier from a naively pairwise estimate).

Step 5 — gauche-eclipsed (φ = 120° or 240°). Here one front substituent eclipses the back CH₃ and vice versa in a "crossed" arrangement: specifically, front CH₃ eclipses a back H, and back CH₃ eclipses a front H, while the remaining front H eclipses the remaining back H:

This is comfortably between the anti/gauche minima and the syn-eclipsed maximum, consistent with spectroscopic estimates placing this barrier near 14–16 kJ/mol — confirming the ordering

Step 6 — sanity check by symmetry. The profile must be periodic with period 360° and must satisfy $E(\phi) = E(-\phi)$ because gauche⁺ (φ=+60°) and gauche⁻ (φ=−60°=300°) are related by the mirror symmetry of the molecule (reflecting swaps which face of C3 you view from, turning one gauche rotamer into its mirror image, which for the achiral butane skeleton is degenerate in energy). The additive model above indeed gives identical $E_{\text{gauche}}$ at φ = 60° and φ = 300°, and identical $E_{\text{gauche-ecl}}$ at φ = 120° and φ = 240°, exactly as required — a necessary (though not sufficient) consistency check on the whole construction.

Goal. Convert the anti/gauche energy gap into an actual room-temperature population ratio, and show explicitly how the degeneracy of the two gauche wells enters the statistics — a step students very commonly get wrong by forgetting the factor of two.

Step 1 — write the Boltzmann ratio for a single gauche well versus anti. For two non-degenerate states differing in Gibbs energy by $\Delta G^\circ$, the equilibrium population ratio is

where $\Delta G^\circ = G_{\text{gauche}} - G_{\text{anti}} = +3.8$ kJ/mol, $R = 8.314$ J mol⁻¹K⁻¹, and $T = 298$ K.

Step 2 — evaluate the exponent.

Step 3 — exponentiate.

So each individual gauche well is populated at only 21.6% the level of the anti well — but there are two gauche wells (gauche⁺ and gauche⁻), degenerate by the mirror symmetry established in Step 6 above, and only one anti well.

Step 4 — include the degeneracy explicitly. The total gauche population (summed over both wells) relative to anti is

Step 5 — convert to mole fractions. Since anti + total gauche = 1 (ignoring the negligibly populated eclipsed maxima, which are transition-state-like saddle regions, not stable wells, and contribute vanishingly to equilibrium population at 298 K):

So the population is ≈ 70% anti : 30% gauche (roughly 15% in each individual gauche well). Some textbooks quote figures closer to 64:36 or 68:32 depending on whether $\Delta G^\circ$ is taken as 2.7, 3.3, or 3.8 kJ/mol (different spectroscopic methods and different temperatures give a genuine spread); the calculation method above is the part that must not be forgotten, and the degeneracy factor of 2 is the single most common source of a wrong final ratio in this exact problem.

Step 6 — limiting-case check. As $T \to \infty$, $\exp(-\Delta G^\circ/RT) \to 1$ for any finite $\Delta G^\circ$, so $n_{\text{gauche,total}}/n_{\text{anti}} \to 2$, i.e. $x_{\text{anti}} \to 1/3$ and $x_{\text{gauche}} \to 2/3$ — simply the statistical-weight ratio (one anti well versus two gauche wells) once thermal energy swamps the 3.8 kJ/mol energetic preference. This is exactly the correct high-temperature limit: with no energetic bias at all, population should track degeneracy alone, 1:2, which is what the formula gives. Conversely as $T \to 0$, the ratio correctly collapses to $x_{\text{anti}} \to 1$: only the single global minimum survives.

Example 1 (foundational) — locating and naming the stationary points from a given structure. Draw the Newman projection of butane looking from C2 to C3 with the front C2–CH₃ bond pointing straight up (φ reference = 0° defined at the back CH₃ also pointing straight up, i.e., eclipsing). Rotate the back carbon 150° clockwise. What is the dihedral angle, is the conformer staggered or eclipsed, and which named conformer is it closest to? Reasoning. Starting from φ = 0° (syn-eclipsed, back CH₃ eclipsing front CH₃) and rotating the back group 150° clockwise gives φ = 150°. Since 150° is not a multiple of 60°, this is not a stationary point at all — it is an intermediate geometry between gauche-eclipsed (120°) and anti (180°), specifically 30° short of full anti. It will relax (if the rotation is not artificially frozen) toward the nearest minimum, φ = 180° (anti), because the torque from both torsional strain (favoring staggered) and steric strain (favoring methyls apart) points that way for any φ between 120° and 240°. This example emphasizes that the six named conformers are the stationary points of a continuous $E(\phi)$ curve, not the only allowed geometries — real molecules sample the whole curve, they simply spend most of their time near the minima.

Example 2 (applied) — estimating a mixture's average dipole-moment-sensitive property. Suppose a spectroscopic technique reports that a sample of 1,2-dichloroethane-like reasoning is being tested on butane's gauche/anti equilibrium using a property $P$ that takes value $P_{\text{anti}} = 0$ (by symmetry, anti butane's skeletal torsional environment is centrosymmetric-like in the relevant local sense) and $P_{\text{gauche}} = 1.00$ (arbitrary units) in each gauche well. Using the 298 K mole fractions from the Boltzmann derivation above, predict the population-weighted observable $\langle P \rangle$. Reasoning. This is a direct application of $\langle P \rangle = \sum_i x_i P_i$:

So a bulk measurement would report an averaged signal corresponding to roughly 30% "gauche character," even though no individual molecule is ever partway between anti and gauche at equilibrium (interconversion between wells is fast on all but the shortest spectroscopic timescales, of order picoseconds, since the barrier of ~15–16 kJ/mol at 298 K, $RT \approx 2.5$ kJ/mol, is easily surmounted many times per nanosecond). This is exactly the reasoning used experimentally to extract $\Delta G^\circ_{\text{gauche-anti}}$ in the first place — from Raman intensity ratios or vicinal NMR coupling constants, which are population-weighted averages, run in reverse through the Boltzmann relation to recover the energy gap.

Example 3 (exam/competition level) — asymmetric substituted analogue and its full $E(\phi)$ ordering. Consider 2-methylpentane's C3–C4 bond, i.e., a Newman projection where the front carbon (C3) bears one isopropyl-fragment-defining CH(CH₃)₂ branch as its largest substituent, one CH₃(from C2 wait — construct concretely: 2,2-dimethylbutane rotated about C3–C4) — to keep this tractable and rigorous, use instead the C2–C3 bond of 2-methylbutane (isopentane), CH₃–CH(CH₃)–CH₂–CH₃, where front carbon C2 bears substituents {CH₃, CH₃, H} (two methyls, since C2 is the branch point) and back carbon C3 bears {CH₃(C4), H, H}. Rank the six stationary conformers by energy and identify the global minimum, justifying each ranking with an explicit interaction count as in Derivation 1. Reasoning. Front carbon C2 now carries two methyls and one H (call them Mₐ, M_b, H_f); back carbon C3 carries one methyl and two H's (M_back, H₁, H₂), exactly as in plain butane's back carbon. Because the front carbon has two big groups instead of one, there is no longer a conformer with zero Me···Me gauche interactions available at every staggered position — we must check each of the three staggered rotamers individually. Define φ as the angle between Mₐ (front) and M_back (back). At φ = 180°, Mₐ is anti to M_back (good, no gauche Me···Me from this pair), but now M_b (front, at φ_offset = −60° from Mₐ, i.e. at φ = 120°) sits gauche to M_back, generating exactly one Me···Me gauche interaction — the same count that plain gauche-butane had. At φ = 60° (Mₐ gauche to M_back on one side) M_b sits at φ = 300°, anti to nothing useful — check directly: with Mₐ at 60° from M_back, M_b (120° further around the front carbon) sits at 60+120=180° from M_back, i.e. anti; so this rotamer has exactly one Me···Me gauche interaction (Mₐ···M_back) and zero from M_b. At φ = 300° (Mₐ at −60°/300° from M_back), by the mirror argument M_b again picks up one gauche Me···Me interaction with M_back. Tallying:

Remarkably, in this branched case all three staggered rotamers carry exactly one Me···Me gauche interaction — there is no interaction-free staggered conformer, because the front carbon's two methyls can never simultaneously both avoid being gauche to the single back methyl (with three positions 120° apart and only one "anti" slot available to one substituent at a time). The true energetic ordering among the three then comes from second-order effects (Me···Me distance differs slightly conformer to conformer due to the extra methyl crowding its neighbors, and there is additional Me(front)···Me(front) 1,2-steric strain that is φ-independent and cancels in the ranking). This worked example is a genuine exam-style trap: it shows that "anti is always the global minimum" is a butane-specific result of having only one bulky group per carbon, not a universal law — the general principle is "minimize the number and severity of gauche and eclipsing interactions between the bulkiest groups," which for plain butane happens to single out φ = 180° uniquely, but for more heavily branched neighbors requires the explicit interaction count shown here.

The modern quantitative account of butane's rotational profile is the direct experimental and theoretical extension of the ethane torsional-barrier story, and the person most associated with turning "staggered is favored" into a rigorous, spectroscopically measured energy landscape for butane specifically is the physical chemist Kenneth S. Pitzer, whose statistical-thermodynamic treatment of internal rotation in hydrocarbons (from the 1930s–1940s) provided the theoretical machinery — partition functions over hindered rotors — that later spectroscopists used to extract the actual gauche–anti and eclipsed-barrier numbers for butane from far-infrared and Raman torsional spectra. Pitzer's original insight, developed with collaborators including J. E. Kilpatrick and W. D. Gwinn, was that treating internal rotation as a classical free rotor badly overestimated molar entropies and heat capacities of hydrocarbons; only by introducing a hindering potential — precisely the $V_3, V_1$ Fourier picture used above — could statistical thermodynamics reproduce measured thermodynamic data for ethane, propane, and butane simultaneously.