Cyclohexane conformation

Cyclohexane conformations are any of several three-dimensional shapes adopted by cyclohexane. Because many compounds feature structurally similar six-membered rings, the structure and dynamics of cyclohexane are important prototypes of a wide range of compounds.
The internal angles of a regular, flat hexagon are 120°, while the preferred angle between successive bonds in a carbon chain is about 109.5°, the tetrahedral angle. Therefore, the cyclohexane ring tends to assume non-planar conformations, which have all angles closer to 109.5° and therefore a lower strain energy than the flat hexagonal shape.
Consider the carbon atoms numbered from 1 to 6 around the ring. If we hold carbon atoms 1, 2, and 3 stationary, with the correct bond lengths and the tetrahedral angle between the two bonds, and then continue by adding carbon atoms 4, 5, and 6 with the correct bond length and the tetrahedral angle, we can vary the three dihedral angles for the sequences,, and. The next bond, from atom 6, is also oriented by a dihedral angle, so we have four degrees of freedom. But that last bond has to end at the position of atom 1, which imposes three conditions in three-dimensional space. If the bond angle in the chain should also be the tetrahedral angle then we have four conditions. Normally this would mean that there are no degrees of freedom of conformation, giving a finite number of solutions. With atoms 1, 2, and 3 fixed, there are two solutions, called chair, but it turns out that there is also a continuum of solutions, a topological circle where angle strain is zero, including the twist boat and the boat conformations. All the conformations on this continuum have a twofold axis of symmetry running through the ring, whereas the chair conformations do not. It is because of the symmetry of the conformations on this continuum that it is possible to satisfy all four constraints with a range of dihedral angles at. On this continuum the energy varies because of Pitzer strain related to the dihedral angles. The twist-boat has a lower energy than the boat. In order to go from the chair conformation to a twist-boat conformation or the other chair conformation, bond angles have to be changed, leading to a high-energy half-chair conformation. So the relative energies are: with chair being the most stable and half-chair the least. All relative conformational energies are shown below. At room temperature the molecule can easily move among these conformations, but only chair and twist-boat can be isolated in pure form, because the others are not at local energy minima.
The boat and twist-boat conformations, as said, lie along a continuum of zero angle strain. If there are substituents that allow the different carbon atoms to be distinguished, then this continuum is like a circle with six boat conformations and six twist-boat conformations between them, three "right-handed" and three "left-handed". But if the carbon atoms are indistinguishable, as in cyclohexane itself, then moving along the continuum takes the molecule from the boat form to a "right-handed" twist-boat, and then back to the same boat form, then to a "left-handed" twist-boat, and then back again to the achiral boat. The passage from boat → right-twist-boat → boat → left-twist-boat → boat constitutes a full pseudorotation.

Coplanar carbons

Another way to compare the stability within two molecules of cyclohexane in the same conformation is to evaluate the number of coplanar carbons in each molecule. Coplanar carbons are carbons that are all on the same plane. Increasing the number of coplanar carbons increases the number of eclipsing substituents. This increases the overall torsional strain and decreases the stability of the conformation. Cyclohexane diminishes the torsional strain from eclipsing substituents through adopting a conformation with fewer coplanar carbons. For example, if a half-chair conformation contains four coplanar carbons and another half-chair conformation contains five coplanar carbons, the conformation with four coplanar carbons will be more stable.

Principal conformers

The different conformations are called "conformers", a blend of the words "conformation" and "isomer".

Chair conformation

The chair conformation is the most stable conformer. At, 99.99% of all molecules in a cyclohexane solution adopt this conformation.
File:Chair comformation of methylcyclohexane.png|thumb|350x350px|Chair comformation of methylcyclohexane with ring flip
The C–C ring of the chair conformation has the same shape as the 6-membered rings in the diamond cubic lattice. This can be modeled as follows. Consider a carbon atom to be a point with four half-bonds sticking out towards the vertices of a tetrahedron. Place it on a flat surface with one half-bond pointing straight up. Looking from directly above, the other three half-bonds will appear to point outwards towards the vertices of an equilateral triangle, so the bonds will appear to have an angle of 120° between them. Arrange six such atoms above the surface so that these 120° angles form a regular hexagon. Reflecting three of the atoms to be below the surface yields the desired geometry.
All carbon centers are equivalent. They alternate between two parallel planes, one containing C1, C3 and C5, and the other containing C2, C4, and C6. The chair conformation is left unchanged after a rotation of 120° about the symmetry axis perpendicular to these planes, as well as after a rotation of 60° followed by a reflection in the midpoint plane, resulting in a symmetry group of D_3d. While all C–C bonds are tilted relative to the plane, diametrically opposite bonds are parallel to each other.
Six of the twelve C–H bonds are axial, pointing upwards or downwards almost parallel to the symmetry axis. The other six C–H bonds are equatorial, oriented radially outwards with an upwards or downwards tilt. Each carbon center has one axial C–H bond and one equatorial C–H bond, enabling each X–C–C–Y unit to adopt a staggered conformation with minimal torsional strain. In this model, the dihedral angles for series of four carbon atoms going around the ring alternate between exactly +60° and −60°.
The chair conformation cannot be deformed without changing bond angles or lengths. It can be represented as two linked chains, C1–C2–C3–C4 and C1–C6–C5–C4, each mirroring the other, with opposite dihedral angles. The C1–C4 distance depends on the absolute value of this dihedral angle, so in a rigid model, changing one angle requires changing the other angle. If both dihedral angles change while remaining opposites of each other, it is not possible to maintain the correct C–C–C bond angles at C1 and C4.
The chair geometry is often preserved when the hydrogen atoms are replaced by halogens or other simple groups. However, when these hydrogens are substituted for a larger group, additional strain may occur due to diaxial interactions between pairs of substituents occupying the same-orientation axial position, which are typically repulsive due to steric crowding.

Boat and twist-boat conformations

The boat conformations have higher energy than the chair conformations. The interaction between the two flagpole hydrogens, in particular, generates steric strain. Torsional strain also exists between the C2–C3 and C5–C6 bonds, which are eclipsed — that is, these two bonds are parallel one to the other across a mirror plane. Because of this strain, the boat configuration is unstable.
The molecular symmetry is C_2v.
The boat conformations spontaneously distorts to twist-boat conformations. Here the symmetry is D₂, a purely rotational point group with three twofold axes. This conformation can be derived from the boat conformation by applying a slight twist to the molecule so as to remove eclipsing of two pairs of methylene groups. The twist-boat conformation is chiral, existing in right-handed and left-handed versions.
The xyz positions of the carbon atoms for one of the two enantiomers of the twist-boat giving bond lengths of 1 and tetrahedral bond angles are:
In this model, the dihedral angles for the chains 1-2-3-4 and 4-5-6-1 are approximately +70.64°, and for the other four −33.16°, and are the opposites for the other enantiomer. This is a slightly twisted version of a boat conformation:
It is also a twisted version of the boat obtained by negating the y and z coordinates of the above.
The concentration of the twist-boat conformation at room temperature is less than 0.1%, but at it can reach 30%. Rapid cooling of a sample of cyclohexane from to will freeze in a large concentration of twist-boat conformation, which will then slowly convert to the chair conformation upon heating.

Half-chair conformation

The half-chair conformation of cyclohexane is an intermediate stage in the "ring flip" between two chair conformations. It's a high-energy state due to torsional and angle strain, making it less stable than both the chair and boat conformations.

Dynamics

Chair to chair

The interconversion of chair conformers is called ring flipping or chair-flipping. Carbon–hydrogen bonds that are axial in one configuration become equatorial in the other, and vice versa. At room temperature the two chair conformations rapidly equilibrate. The proton NMR spectrum of cyclohexane is a singlet at room temperature, with no separation into separate signals for axial and equatorial hydrogens.
In one chair form, the dihedral angle of the chain of carbon atoms is positive whereas that of the chain is negative, but in the other chair form, the situation is the opposite. So both these chains have to undergo a reversal of dihedral angle. When one of these two four-atom chains flattens to a dihedral angle of zero, we have the half-chair conformation, at a maximum energy along the conversion path. When the dihedral angle of this chain then becomes equal to that of the other four-atom chain, the molecule has reached the continuum of conformations, including the twist boat and the boat, where the bond angles and lengths can all be at their normal values and the energy is therefore relatively low. After that, the other four-carbon chain has to switch the sign of its dihedral angle in order to attain the target chair form, so again the molecule has to pass through the half-chair as the dihedral angle of this chain goes through zero. Switching the signs of the two chains sequentially in this way minimizes the maximum energy state along the way — having the dihedral angles of both four-atom chains switch sign simultaneously would mean going through a conformation of even higher energy due to angle strain at carbons 1 and 4.
The detailed mechanism of the chair-to-chair interconversion has been the subject of much study and debate. The half-chair state is the key transition state in the interconversion between the chair and twist-boat conformations. The half-chair has C₂ symmetry. The interconversion between the two chair conformations involves the following sequence: chair → half-chair → twist-boat → half-chair′ → chair′.