Conformational analysis

Result

Conformational analysis

T-026Home CU-105Threads organic

Statement

Rotation about a \(\sigma\) (single) bond breaks no bond and creates no new isomer (isomerism), but it does change the molecule's potential energy as a function of the dihedral (torsion) angle, due to torsional strain (repulsion between eclipsing bonding electron pairs) and, where substituents are bulky, additional steric strain between non-bonded groups. Plotting energy against dihedral angle gives a continuous conformational energy landscape: staggered arrangements are energy minima, eclipsed arrangements are energy maxima, and for asymmetrically substituted chains, the staggered minima further split into a lower-energy anti form and higher-energy gauche forms.

Why it matters

isomerism explicitly excluded freely-interconverting rotational shapes from isomer-counting, on the grounds that no bond breaks during rotation; this result explains and quantifies exactly why that exclusion is physically sound, showing that rotational barriers, while real and measurable, are far too low to prevent extremely rapid interconversion at ordinary temperature. It also establishes that a flexible molecule is not equally likely to be found in every possible rotational shape at any instant — a population strongly skewed toward the lowest-energy conformations, governed by the same Boltzmann-distribution logic already used for molecular speeds (maxwell-boltzmann-speeds) earlier in this unit, now applied to a different physical quantity: conformational energy rather than translational kinetic energy.

Hypotheses

At ordinary temperature, the energy barriers separating conformations are low enough that interconversion occurs extremely rapidly (many times per nanosecond) — far too fast for individual conformers to be isolated as separate, stable species.This is precisely what distinguishes a conformer from a true configurational isomer (isomerism): configurational isomers require breaking a covalent bond to interconvert, and are consequently isolable at ordinary temperature and timescales, while conformers interconvert via bond rotation alone, with no such barrier to isolation. Torsional strain (from eclipsing bonding electron pairs) and steric strain (from non-bonded bulky-group repulsion) are treated as approximately independent, additive contributions to a conformation's total energy.This is a simplifying approximation: real potential energy surfaces have some coupling between these effects, and more sophisticated treatments (beyond this introductory level) account for additional contributions such as hyperconjugative electronic effects. The additive picture is nonetheless sufficient to correctly rank the relative energies of the standard named conformations (staggered/eclipsed, anti/gauche) for simple open-chain molecules.

Proof

\text{Newman projection: viewed along the C--C bond axis, front-carbon bonds from the centre, back-carbon bonds from a circle.}

This projection makes the dihedral (torsion) angle between front and back substituents directly visible, and is the standard tool for conformational analysis. A

\text{Staggered (dihedral } 60^\circ,180^\circ,300^\circ\text{ for ethane): front and back bonds maximally separated} \Rightarrow \text{energy minimum.}

Maximal angular separation between the front and back C–H bonding electron pairs minimises their mutual repulsion, the origin of torsional strain. A

\text{Eclipsed (dihedral } 0^\circ,120^\circ,240^\circ\text{): front and back bonds aligned} \Rightarrow \text{energy maximum (torsional strain).}

Direct alignment maximises bonding-electron-pair repulsion; for ethane, the eclipsed-to-staggered energy difference (the full rotational barrier) is experimentally measured at approximately \(12\,\text{kJ/mol}\). A

\text{For a chain with bulky substituents (e.g. butane's C2--C3 bond), staggered conformations split into anti (bulky groups } 180^\circ\text{ apart) and gauche (}60^\circ\text{ apart).}

Anti has no additional steric strain beyond the baseline torsional minimum; gauche introduces extra steric (van der Waals) repulsion between the two nearby bulky groups, raising its energy above anti's even though both remain staggered (torsionally favourable) arrangements. A

E(\theta) \approx E_{\text{torsional}}(\theta) + E_{\text{steric}}(\theta)

The total conformational energy as a function of dihedral angle \(\theta\), per the additive approximation (Hypotheses): a periodic torsional term (minima at staggered angles, maxima at eclipsed angles) plus an additional steric penalty specifically where bulky, non-hydrogen substituents are brought close together. A

Result

\text{ethane: staggered (min)} \leftrightarrow \text{eclipsed (max)},\ \Delta E\approx12\,\text{kJ/mol}

\text{butane: anti (global min)} < \text{gauche (local min, } +3.8\,\text{kJ/mol)} < \text{eclipsed conformations (maxima, up to} \approx19\,\text{kJ/mol)}

Reading. Even the simplest possible rotatable bond (ethane) has a real, measurable, periodic energy landscape; adding bulkier substituents (butane) refines that landscape with an additional steric contribution that further splits the staggered minima into distinct anti and gauche forms.

Scope. Numeric values given are approximate, standard literature figures for these specific, frequently cited small-molecule examples; different substituents and chain lengths give quantitatively different (though qualitatively similar) energy landscapes.

Corollaries & converses

At thermal equilibrium, the relative population of two conformations follows the Boltzmann distribution based on their energy difference, exactly the same principle already used for molecular speeds in maxwell-boltzmann-speeds: \(\dfrac{N_2}{N_1}=\dfrac{g_2}{g_1}e^{-\Delta E/RT}\), where \(g_1,g_2\) are the number of equivalent conformations of each type (butane has one anti and two equivalent gauche conformations, so \(g_{\text{anti}}=1\), \(g_{\text{gauche}}=2\)).
Since \(RT\) at room temperature (\(298\,\text{K}\)) is only about \(2.5\,\text{kJ/mol}\) — smaller than even the modest \(3.8\,\text{kJ/mol}\) anti/gauche energy difference, and far smaller than the \(\sim12\)–\(19\,\text{kJ/mol}\) eclipsed-conformation barriers — a real, unequal but non-negligible population of higher-energy conformers persists at any instant, even though the lowest-energy conformation always dominates.
Converse: lowering temperature shifts the equilibrium population further toward the lowest-energy conformation (Problems), since the Boltzmann factor \(e^{-\Delta E/RT}\) shrinks as \(T\) decreases, exactly the same temperature dependence already established for the Maxwell-Boltzmann speed distribution's shape.

Fails without

Treat conformers as isolable, distinct species rather than rapidly interconverting shapes of one molecule (drop Hypotheses): attempting to physically isolate "pure anti-butane" separately from "pure gauche-butane" at ordinary temperature is not experimentally possible, since the barriers separating them are far too low relative to thermal energy to prevent extremely rapid interconversion — directly the same physical reasoning isomerism used, from the opposite direction, to justify excluding conformations from isomer-counting in the first place.
Ignore steric strain entirely, treat all staggered conformations as equally low in energy regardless of substituent bulk: this would incorrectly predict anti and gauche butane are equal in energy, contradicted by the measured \(3.8\,\text{kJ/mol}\) energy difference and by the experimentally observed unequal population distribution between them (Corollaries).

Common errors

Confusing conformers (interconvert by bond rotation alone, not true isomers) with configurational stereoisomers such as cis/trans or enantiomers (require breaking a bond to interconvert, and are true isomers per isomerism) — a frequent terminology confusion, since both involve "different spatial arrangements" in a loose sense.
Assuming every eclipsed conformation of a substituted chain has the same energy; butane's fully eclipsed methyl-over-methyl conformation (\(\theta=0^\circ\)) is measurably higher in energy than its methyl-over-hydrogen eclipsed conformations (\(\theta=120^\circ,240^\circ\)), since the former adds steric strain on top of the shared torsional strain (Problems).
Forgetting to include conformational degeneracy (\(g_1,g_2\)) when computing population ratios via the Boltzmann distribution — butane's two equivalent gauche conformations must be counted together, not as if only one existed.

Discussion

Systematic conformational analysis matured as a distinct sub-field of physical organic chemistry through the mid-20th century, with Derek Barton and Odd Hassel jointly awarded the 1969 Nobel Prize in Chemistry for developing and extending these ideas, notably to cyclic systems (cyclohexane's chair and boat ring conformations, a related but distinct application beyond simple open-chain rotation, not developed further in this introductory treatment).

Conformational preferences have consequences well beyond small molecules like ethane and butane: in much larger and more complex molecules (including proteins, addressed elsewhere in this curriculum), the same underlying physics — rotation about single bonds subject to a torsional-plus-steric energy landscape — is one of the central factors, alongside non-covalent interactions, governing how a long flexible chain folds into a specific, functionally relevant three-dimensional shape.

Common misconception: that a "conformation" is a distinct, separately existing molecule in the same sense an isomer is. A conformation is better understood as one point along a continuous, extremely rapidly traversed energy landscape belonging to a single compound — the population of conformations shifts with temperature and can be characterised statistically (Corollaries), but the underlying molecule, and its molecular formula and connectivity, never change during rotation, directly reinforcing the distinction isomerism already established.

Worked examples

\text{Ethane: eclipsed} - \text{staggered} \approx 12\,\text{kJ/mol}, \text{ arising entirely from torsional (bond-bond) repulsion.}

Since every substituent on ethane is a hydrogen (no bulky groups), no additional steric contribution is present; the entire rotational barrier is torsional strain, the simplest possible case of the general energy landscape (Step 5). A

\text{Butane at 298 K, } \Delta E(\text{gauche}-\text{anti})=3.8\,\text{kJ/mol},\ RT\approx2.478\,\text{kJ/mol}: \quad \frac{N_{\text{anti}}}{N_{\text{total}}}\approx69.9\%, \quad \frac{N_{\text{gauche,combined}}}{N_{\text{total}}}\approx30.1\%

Applying the Boltzmann population formula (Corollaries) with degeneracies \(g_{\text{anti}}=1\), \(g_{\text{gauche}}=2\): despite gauche's real, measurable energy penalty, nearly a third of butane molecules are found in a gauche conformation at any instant at room temperature, since the \(3.8\,\text{kJ/mol}\) barrier is comparable to (not enormously larger than) \(RT\). A

\text{Ethane barrier}\approx12\,\text{kJ/mol (torsional only)}; \qquad \text{Butane at 298 K: }\approx70\%\text{ anti, }\approx30\%\text{ gauche}

Reading. The same energy-landscape framework both explains the simplest possible rotational barrier (ethane) and, once combined with Boltzmann statistics, predicts a quantitatively verifiable population split for a more complex, sterically differentiated case (butane).

Scope. The Boltzmann population calculation applies to any pair of conformations with a known energy difference and degeneracy, at any specified temperature.

Problems

Repeat the butane anti/gauche population calculation from Worked example 2, but at \(T=200\,\text{K}\) instead of \(298\,\text{K}\). Compute the new anti and gauche population fractions and comment on the direction of the shift.
Solution
At \(200\,\text{K}\), \(RT=\dfrac{(8.314)(200)}{1000}\approx1.663\,\text{kJ/mol}\). Boltzmann factor: \(e^{-3.8/1.663}\approx0.100\); combined gauche weight \(=2\times0.100=0.200\); total weight \(=1+0.200=1.200\). Anti fraction: \(1/1.200\approx83.1\%\); gauche fraction: \(0.200/1.200\approx16.9\%\). The population shifts further toward anti at lower temperature (from \(\approx70\%\) at \(298\,\text{K}\) to \(\approx83\%\) at \(200\,\text{K}\)), exactly the Corollaries' predicted direction: cooling suppresses access to the higher-energy conformation.
Explain why butane's fully eclipsed conformation with the two methyl groups directly overlapping (\(\theta=0^\circ\)) is higher in energy than its other two eclipsed conformations, where a methyl group eclipses a hydrogen instead (\(\theta=120^\circ,240^\circ\)), even though all three are torsionally eclipsed.
Solution
By the additive energy model (Step 5), every eclipsed conformation shares the same baseline torsional strain contribution (bond-bond electron repulsion, present at any eclipsed angle regardless of which groups are involved). The \(\theta=0^\circ\) conformation additionally places the two bulky methyl groups directly over one another, adding a substantial steric-strain contribution on top of the shared torsional strain; the other two eclipsed conformations pair a methyl with a much smaller hydrogen, adding comparatively little extra steric strain. The methyl-over-methyl eclipsed conformation is therefore the highest-energy point on the entire rotational energy landscape.
A student is shown two molecular structures and told they represent "different spatial arrangements" of the same formula. One pair requires physically breaking a bond to interconvert; the other interconverts freely by bond rotation alone. Classify each pair using the terminology established in isomerism and this result.
Solution
The pair requiring bond-breaking to interconvert is a pair of true configurational isomers (stereoisomers, per isomerism) — potentially enantiomers, or geometric cis/trans isomers, depending on the specific structural feature involved. The pair interconverting freely by bond rotation alone is a pair of conformers (this result) — not isomers at all, but different transient shapes of the identical single compound.
Using Worked example 2's result (\(\approx70\%\) anti at \(298\,\text{K}\)), predict qualitatively (without recalculating) whether raising the temperature well above \(298\,\text{K}\) would increase or decrease the gauche population fraction, and explain why using the Boltzmann factor's temperature dependence.
Solution
Raising temperature increases the gauche population fraction. The Boltzmann factor \(e^{-\Delta E/RT}\) increases toward \(1\) as \(T\) increases (since \(\Delta E/RT\) shrinks), meaning the energy penalty for occupying the higher-energy gauche conformation becomes relatively less significant compared to the larger available thermal energy — the opposite direction of the shift found in Problem 1, where lowering \(T\) suppressed the gauche population instead.