Question 26.7: Identifying Chiral Carbon Atoms in Cycloalkanes. Identify an......

Analyze
A carbon atom is chiral if it is bonded to four different groups. To determine whether a carbon atom in a ring structure is chiral, we must go around the ring in each direction (clockwise and counterclockwise) to see if there is a point of difference. If there is a point of difference, then the carbon atom is chiral.

Solve
When considering carbon atoms in the rings of these molecules, we need only focus on carbon atoms where hydrogen atoms have been replaced by other substituents. If there is no substituent, then the carbon atom has two identical groups (two hydrogen atoms) and thus, cannot be chiral.
Thus, for methylcyclohexane, we focus only on C1, the carbon
bonded to the methyl substituent. The clockwise and counterclockwise paths starting from C1 have identical constitutions (—CH_{2}CH_{2}CH_{2}CH_{2}CH_{2}—) and so, we conclude that C1 in methylcyclohexane is not chiral.

When we compare the clockwise and counterclockwise
paths, starting from C1, in 1,3-dimethylcyclohexane, we
notice that there is a point of difference. The constitution of
the clockwise path (blue) is —CH_{2}CH_{2}CH_{2}CH(CH_{3})CH_{2}— and that of the counterclockwise path (red) is —CH_{2}CH(CH_{3})CH_{2}CH_{2}CH_{2}CH_{2}—.
The two paths are different; thus, C1 is chiral. Following similar reasoning, we find that C3 is also chiral. Thus, in cis-1,3-dimethylcyclohexane, C1 and C3 are chiral.

Assess
In most of the examples we have considered so far, there has been only one chiral atom per molecule. Such molecules are optically active. In this example, we see that 1,3-dimethylcyclohexane has two chiral carbon atoms.
A molecule with two or more chiral carbon atoms may or may not be optically active. Optical activity of molecules containing two or more chiral atoms is discussed in advanced organic chemistry courses.