Conformations of Ethane and Butane
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You should be comfortable with the tetrahedral geometry of sp3 hybridised carbon, the definition of a dihedral (torsion) angle, and the distinction between constitutional isomers and conformers.
Why this matters
Molecules are not static sculptures; they are dynamic entities that twist and turn, and their shape dictates their reactivity.
Understanding conformational analysis is fundamental to organic chemistry because it explains why certain reactions are faster or slower than others, and why some molecules are more stable than their isomers. While the connectivity of atoms defines the molecule's identity, the spatial arrangement of those atoms in three-dimensional space defines its physical properties and chemical behaviour. In this lecture, we move beyond the simple line-angle structures seen in introductory courses to examine the energy landscapes that govern the rotation of single bonds. We will focus on the simplest hydrocarbons, ethane and butane, to establish the core concepts of torsional strain and steric hindrance, which will serve as the foundation for understanding more complex cyclic systems in subsequent lectures.
The Dynamic Nature of Single Bonds
When we draw a single bond between two carbon atoms, such as in ethane, we often depict it as a rigid line. However, this is a simplification. A single bond, or sigma (\(\sigma\)) bond, is formed by the head-on overlap of two sp3 hybrid orbitals. This overlap creates a cylindrical region of electron density around the internuclear axis. Crucially, this cylindrical symmetry allows the two carbon atoms to rotate freely relative to one another around the axis of the bond. This rotation does not break any bonds; it is a continuous motion that changes the relative positions of the hydrogen atoms attached to each carbon. As one carbon rotates, the hydrogen atoms on that carbon sweep past the hydrogen atoms on the other carbon. This motion is not arbitrary; it is governed by the potential energy of the system. The electrons in the C-H bonds on one carbon repel the electrons in the C-H bonds on the other carbon. When the bonds are aligned directly with each other, this repulsion is maximised. When they are offset, the repulsion is minimised. Therefore, the molecule "prefers" certain orientations over others. These preferred orientations are called conformations. The study of these conformations and the energy differences between them is known as conformational analysis. It is a purely physical phenomenon, distinct from isomerism in the sense that we do not create new compounds; we are simply observing different snapshots of the same molecule as it tumbles through space. The key insight is that while the connectivity remains constant, the energy of the molecule changes continuously as the dihedral angle changes. This creates an energy landscape with valleys (minima) and peaks (maxima). The valleys represent the most stable arrangements, while the peaks represent the least stable, high-energy transition states that the molecule must pass through to rotate from one stable form to another. Understanding this landscape is the first step in predicting the behaviour of larger, more complex organic molecules.
Ethane: The Benchmark of Torsional Strain
Let us begin with ethane, \(\mathrm{C_2H_6}\), the simplest molecule that exhibits conformational isomerism. If we look down the C-C bond axis (a Newman projection), we see two sets of three hydrogen atoms. As the front carbon rotates relative to the back carbon, the dihedral angle between a front C-H bond and a back C-H bond changes from \(0^\circ\) to \(360^\circ\). There are two distinct types of arrangements: staggered and eclipsed. In the staggered conformation, the dihedral angle is \(60^\circ\), \(180^\circ\), or \(300^\circ\). The hydrogen atoms on the front carbon are positioned exactly halfway between the hydrogen atoms on the back carbon. This maximises the distance between the bonding electron pairs, minimising electron-electron repulsion. In the eclipsed conformation, the dihedral angle is \(0^\circ\), \(120^\circ\), or \(240^\circ\). Here, the C-H bonds on the front carbon are directly aligned with the C-H bonds on the back carbon. This alignment brings the electron clouds of the bonds into close proximity, creating a region of high electron density and significant repulsion. This repulsion is known as torsional strain. The energy difference between the staggered minimum and the eclipsed maximum is the barrier to rotation. For ethane, this barrier is approximately \(12\ \mathrm{kJ\,mol^{-1}}\). This value is small enough that at room temperature, the thermal energy (\(kT\)) is sufficient to allow rapid interconversion between these conformations. Therefore, we do not isolate "staggered ethane" and "eclipsed ethane" as separate substances; rather, we observe a rapid equilibrium where the population is heavily weighted towards the lower-energy staggered form. The existence of this barrier confirms that the rotation is not perfectly free; it is hindered by the electronic structure of the bonds themselves.
Butane: Introducing Steric Strain
When we move to butane, \(\mathrm{C_4H_{10}}\), the situation becomes more complex because the substituents on the central C-C bond are no longer just hydrogens; they are methyl groups (\(\mathrm{-CH_3}\)). Looking down the C2-C3 bond, we have two methyl groups and two hydrogen atoms on each carbon. The staggered conformations are no longer all equivalent. There are two distinct staggered forms: the anti conformation and the gauche conformation. In the anti conformation, the two bulky methyl groups are \(180^\circ\) apart. This maximises the distance between the largest groups, minimising steric repulsion. This is the global energy minimum. In the gauche conformation, the two methyl groups are \(60^\circ\) apart. While the C-H bonds are still staggered relative to each other, the methyl groups are now in close proximity. The electron clouds of the C-H bonds within the methyl groups repel each other more strongly than in the anti form. This additional repulsion is called steric strain. The energy of the gauche conformation is higher than that of the anti conformation by approximately \(3.8\ \mathrm{kJ\,mol^{-1}}\). The eclipsed conformations also split into two types. The fully eclipsed conformation, where the two methyl groups are \(0^\circ\) apart, is the highest energy state due to severe steric and torsional strain. The other eclipsed conformations, where a methyl group eclipses a hydrogen, are lower in energy than the fully eclipsed form but still higher than the staggered forms. Thus, butane has a more complex energy landscape with two distinct minima (anti and gauche) and two distinct maxima (fully eclipsed and partially eclipsed). The population of butane at equilibrium will consist predominantly of the anti form, with a smaller fraction of gauche forms, and a negligible fraction of eclipsed forms.
Energy Landscapes and Population
The relationship between energy and population is governed by the Boltzmann distribution. The ratio of molecules in a higher energy state (\(E_2\)) to those in a lower energy state (\(E_1\)) is given by \(\frac{N_2}{N_1} = e^{-\Delta E / RT}\), where \(\Delta E = E_2 - E_1\), \(R\) is the gas constant, and \(T\) is the temperature. This equation tells us that even if a conformation is higher in energy, it will still be present to some extent, provided the energy difference is not too large. For butane, the energy difference between anti and gauche is \(3.8\ \mathrm{kJ\,mol^{-1}}\). At \(298\ \mathrm{K}\), \(RT \approx 2.48\ \mathrm{kJ\,mol^{-1}}\). Therefore, each gauche form is present at \(e^{-3.8 / 2.48} \approx e^{-1.53} \approx 0.22\) of the anti population. There are two gauche conformations (dihedral angles of \(+60^\circ\) and \(-60^\circ\)), so the gauche total is about 0.43 of the anti population: roughly 70% anti and 30% gauche. The anti form is the major contributor to the equilibrium mixture, but the gauche form is not negligible. In contrast, the energy difference between the anti form and the fully eclipsed form is much larger, approximately \(19\ \mathrm{kJ\,mol^{-1}}\). The ratio of fully eclipsed to anti is \(e^{-19 / 2.48} \approx e^{-7.66} \approx 0.00047\). This is a very small number, indicating that the fully eclipsed conformation is rarely observed. This quantitative analysis allows us to predict the average properties of a sample of butane, such as its dipole moment or its heat of combustion, by weighting the properties of each conformation by its population. It also explains why reactions that require a specific conformation, such as the anti-periplanar arrangement for elimination reactions, are favoured when that conformation is the most stable or most accessible.
The result
Conformational analysis
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.
A common misconception is that the eclipsed conformations are conformers that could be isolated. They are maxima on the energy profile, the transition states for rotation, and molecules pass through them in about a picosecond. Only the minima (anti and the two gauche forms of butane) are conformers with measurable populations. A second subtlety: a staggered conformer is not automatically the most stable, since gauche butane is staggered yet 3.8 kJ mol-1 above anti because of steric strain between the methyl groups. C
Worked example
Problem. The gauche conformation of butane lies 3.8 kJ mol-1 above the anti conformation. Remembering that there are two gauche conformations, find the percentage of butane molecules that are gauche at 298 K.
- Thermal energy: \(RT = 8.314 \times 298 = 2477.6\ \mathrm{J\,mol^{-1}} = 2.478\ \mathrm{kJ\,mol^{-1}}\).
- Boltzmann factor for one gauche conformer relative to anti: \(e^{-3.8/2.478} = e^{-1.534} = 0.2157\).
- There are two equivalent gauche conformers (\(+60^\circ\) and \(-60^\circ\)), so \(N_{\text{gauche}}/N_{\text{anti}} = 2 \times 0.2157 = 0.4314\).
- Fraction gauche: \(\dfrac{0.4314}{1 + 0.4314} = 0.301\), that is 30.1% gauche and 69.9% anti.
- Forgetting the factor of 2 would give 17.7%, a common error: degeneracy matters as much as energy.
Units check. \(\Delta E\) and \(RT\) are both in kJ mol-1, so the exponent is dimensionless and the result is a pure fraction.
Three quick tests
- Why is the staggered conformation of ethane more stable than the eclipsed conformation?
Answer
Staggered minimises torsional strain by maximising the distance between bonding electron pairs, reducing repulsion. - In butane, which staggered conformation is more stable, anti or gauche?
Answer
The anti conformation is more stable because the bulky methyl groups are \(180^\circ\) apart, minimising steric strain. - If the energy barrier to rotation in ethane is \(12\ \mathrm{kJ\,mol^{-1}}\), is this barrier high or low compared to typical bond dissociation energies?
Answer
It is very low; bond dissociation energies are typically hundreds of \(\mathrm{kJ\,mol^{-1}}\), so rotation is rapid at room temperature.
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Lecture notes checked line by line by the editor; every number in the worked example was recomputed.