The key to understanding trends in ring strain is that the atoms in a ring do not necessarily lie flat in a plane. We begin by studying the most stable conformation of cyclohexane, which has completely staggered dihedral angles at each of the six C-C bonds. This conformation is not flat but is folded into the shape of a lawn chair, so it is called the chair conformation.
While the chair is typically drawn from such a perspective view, keep in mind that the chair actually has three-fold rotational symmetry. That is, it can be rotated by 120 or 240 degrees and look identical. The best way to visualize the chair conformation is to build one for yourself using a molecular model kit. You should verify that the chair has no eclipsing interactions. This can also be seen in a Newman projection down a set of two C-C bonds. Notice how this resembles two Newman projections of ethane joined together by - CH 2 - groups:
There are two distinctive types of C-H bonds on the chair. One set is comprised of C-H bonds that extend vertically up and down and are called axial bonds. The other set consists of C-H bonds that extend out to the periphery of the ring and are called equatorial bonds. Each carbon has one axial bond and one equatorial bond.
Recall from our discussion of cis-trans isomerism that each carbon has a bond that points up to the top face of the ring and a bond that points down to the bottom face. Don't get these confused with the axial/equatorial classification. One equatorial bond isn't necessarily cis to another equatorial bond, and the same applies for axial bonds. The up/down orientation of an axial position changes from one carbon to its neighbors. If axial is up on one carbon, axial will be down on its two neighbors. The same is true for equatorial positions.
The chair conformation is used so frequently that you should become comfortable with drawing them. The most important feature of chair conformations is that they consist of three sets of parallel C-C bonds. Begin by drawing the six C-C bonds that comprise the skeleton of the chair:
Like other conformations we have studied, chair conformations are in a state of constant flux. Because all the C-C bonds are interconnected, they cannot rotate independently but have to move together. For example, one end of the chair could "flip up" to put the cyclohexane ring in a boat conformation.
The boat conformation is less stable than the chair conformation because it experiences a number of eclipsing interactions. Whereas the chair conformation resembles two staggered ethanes, the boat conformation resembles two eclipsed ethanes. In addition, there is considerable repulsion between hydrogens on the two "tips" of the boat. These hydrogens are called flagpole hydrogens. The combined effects of torsional strain and steric hindrance between flagpole hydrogens makes the boat conformation less stable than the chair by 6.9 kcal/mol.
We could flip either end of the boat down to regain a chair conformation. The two possible chair conformations that can be obtained are distinct; all of the axial bonds in one chair become equatorial in the other and vice versa. These two chair conformations can be interconverted by going through the boat intermediate. Such a chair-chair interconversion is sometimes called a chair flip. Build a model of cyclohexane with distinct colors for the axial and equatorial hydrogens. Try the chair flip yourself to verify that the colors really do change positions. The effect is really quite startling the first time you see it!
When substituents are placed on the cyclohexane ring, they prefer to take equatorial positions over axial positions. This positional preference is shown for methylcyclohexane. When the methyl group occupies the axial position, there is steric hindrance between it and the axial hydrogens three carbons away. These repulsive effects are called 1,3-diaxial interactions. 1,3-diaxial interactions can also be understood in terms of gauche butane. The highlighted bonds indicate the butane-like structures in the axial conformation of methylcyclohexane. It turns out that the axial methyl conformation is less stable by 1.8 kcal/mol, precisely the cost of two gauche butane interactions.
The amount of energy it "costs" to move a substituent group into the axial position is sometimes referred to as the A-value of that substituent group. For instance, the A-value of a methyl group is 1.8 kcal/mol. The A-values of several substituent groups are listed below. A-values can be useful for estimating the energy difference between the two conformations of a substituted cyclohexane. However, a simple summation of A-values does not always give the right answer, as Problem 9 will show.
It is useful to know the energy difference between the two chair conformations because it enables you to calculate the relative abundance of each conformation. For instance, the 1.8 kcal/mol A-value of methyl allows us to predict that less than 1 in 20 molecules of methylcyclohexane will occupy the axial position at room temperature. The A-value of the tert-butyl group is so large that any molecule with a tert-butyl substituent is "locked" into the conformation that places the tert-butyl group in the equatorial position.