Structural, geometrical, and optical isomers can be drawn having the formula \(\mathrm{C}_{6} \mathrm{H}_{12}\) . Give examples to illustrate these types of isomerism for \(\mathrm{C}_{6} \mathrm{H}_{12}\)

Structural isomers have the same molecular formula but different arrangements of atoms in the structure. In order to find a structural isomer for the formula C6H12, we can try different ways of connecting the six carbon atoms. One possible structural isomer of C6H12 is hex-1-ene: \(CH_{2}=CHCH_{2}CH_{2}CH_{2}CH_{3}\) Another structural isomer could be 2-methylpent-1-ene: \(CH_{2}=C(CH_{3})CH_{2}CH_{2}CH_{3}\) These two molecules have the same molecular formula, but they have different connectivity of carbon atoms which makes them structural isomers.

Geometrical isomers occur in compounds with restricted rotation around a bond, usually a double bond. They are different in the arrangement of substituent groups around the double bond. To illustrate geometrical isomerism in C6H12, let's focus on hex-2-ene: \(CH_{3}CH_{2}CH=CHCH_{2}CH_{3}\) Here, the double bond between the 2nd and 3rd carbon atoms leads to the possibility of two different geometrical isomers: 1. (Z)-hex-2-ene (cis isomer): \(CH_{3}CH_{2}CH=CHCH_{2}CH_{3}\), in which the two methyl groups are on the same side. 2. (E)-hex-2-ene (trans isomer): \(CH_{3}CH_{2}CH=CHCH_{2}CH_{3}\), in which the two methyl groups are on opposite sides. These two molecules are geometrical isomers because they have different spatial configurations of the substituent groups around the double bond.

Optical isomers are stereo-isomers that differ in the spatial arrangement of atoms, specifically they have chiral centers in their structures. Optical isomers are non-superimposable mirror images of each other. To illustrate optical isomerism in C6H12, let's focus on 2-methylpent-2-ene as an example: \(CH_{3}C(CH_{3})=CHCH_{2}CH_{3}\) Here, the second carbon atom has four different groups attached to it, making it a chiral center: 1. Methyl group 2. Hydrogen 3. Ethyl group 4. \(\mathrm{C}_{2}\mathrm{H}_{3}\) group We can thus have two possible optical isomers (enantiomers): 1. (R)-2-methylpent-2-ene: \(CH_{3}C(^{*}\mathrm{H})CH_{3}=CHCH_{2}CH_{3}\) 2. (S)-2-methylpent-2-ene: \(CH_{3}C(^{*}\mathrm{H})CH_{3}=CHCH_{2}CH_{3}\) The (*) symbol is used to indicate that the carbon atom is chiral. The (R) or (S) notation is used to indicate the specific configuration around the chiral center. These two molecules are optical isomers (enantiomers) because they have different spatial configurations around the chiral center.