Question

\(P_{1}, P_{2}\)-dimethyl cyclohexene contains exactly three optical isomers. Then \(\left(P_{1}+P_{2}\right)\) is, where \(P_{1}\) and \(P_{2}\) represent position of methyl according to IUPAC (A) Even (B) 3 (C) Variable (D) 9

Short answer

The correct answer is (C) Variable, as the sum of \(P_{1}\) and \(P_{2}\) values for the three optical isomers of \(P_{1}, P_{2}\)-dimethyl cyclohexene are different, resulting in the calculated values 3, 4, and 5.

Step-by-step solution

Draw the structure of \(P_{1}, P_{2}\)-dimethyl cyclohexene

Draw a cyclohexene ring with hydrogen atoms (not necessary to draw all to avoid clutter) and two methyl groups attached to the numbered carbon atoms as \(P_{1}\) and \(P_{2}\).

Identify the three optical isomers

Find the combinations of \(P_{1}\) and \(P_{2}\) positions that result in three different optical isomers. Remember that optical isomers are non-superimposable mirror images of each other. The three possible optical isomers are: 1. \(cis-P_{1}, cis-P_{2}\) configuration: both methyl groups are on the same side of the ring 2. \(trans-P_{1}, trans-P_{2}\) configuration: both methyl groups are on opposite sides of the ring 3. \(cis-P_{1}, trans-P_{2}\) configuration: one methyl group is on one side of the ring and the other is on the opposite side

Determine \(P_{1}\) and \(P_{2}\) values for each isomer

Assign specific \(P_{1}\) and \(P_{2}\) values to each of the three optical isomers: 1. \(cis-P_{1}, cis-P_{2}\) configuration: \(P_{1} = 1\), \(P_{2} = 2\) 2. \(trans-P_{1}, trans-P_{2}\) configuration: \(P_{1} = 1\), \(P_{2} = 4\) 3. \(cis-P_{1}, trans-P_{2}\) configuration: \(P_{1} = 1\), \(P_{2} = 3\) Notice that the group placements are important as cyclohexene is not symmetrical.

Calculate \(\left(P_{1}+P_{2}\right)\) for each isomer

Sum the \(P_{1}\) and \(P_{2}\) values for each of the three optical isomers: 1. \(cis-P_{1}, cis-P_{2}\) configuration: \(\left(P_{1}+P_{2}\right) = 1+2 = 3\) 2. \(trans-P_{1}, trans-P_{2}\) configuration: \(\left(P_{1}+P_{2}\right) = 1+4 = 5\) 3. \(cis-P_{1}, trans-P_{2}\) configuration: \(\left(P_{1}+P_{2}\right) = 1+3 = 4\)

Check the given answer options

Examine the given answer choices and match them with the calculated \(\left(P_{1}+P_{2}\right)\) values: (A) Even - False, as there is only one even value, 4. (B) 3 - False, as only one of the isomers has this value. (C) Variable - True, as the sum differs for each isomer. (D) 9 - False, as none of the isomers have this value. Therefore, the correct answer is (C) Variable.

Key concepts

Stereochemistry and Optical Isomers

Stereochemistry refers to the study of the spatial arrangements of atoms in molecules and how this arrangement affects the properties and reactions of those molecules. A crucial concept in stereochemistry is optical isomerism, which arises when molecules have the same molecular formula but differ in the way the atoms are arranged in space. Optical isomers, also known as enantiomers, are non-superimposable mirror images of each other, much like one's left and right hands.

For instance, considering the exercise about dimethyl cyclohexene, the position of the methyl groups on the cyclohexene ring can create different stereoisomers or optical isomers. These isomers have the same chemical formula but different physical properties, such as the direction they rotate plane-polarized light, which is a defining characteristic of optical isomers. When we have multiple substituents on a cyclic compound, orientations such as 'cis' (same side) and 'trans' (opposite sides) become significant in creating different isomers.

Importance of Non-superimposable Structures

Understanding non-superimposable structures is fundamental to recognizing how these molecules can be optical isomers. Even if two molecules look similar or have similar compositions, the spatial arrangement can change their interactions with biological systems, making this concept particularly important in the field of pharmaceuticals, where the different isomers of a drug can have drastically different effects.

IUPAC Nomenclature for Organic Compounds

The International Union of Pure and Applied Chemistry (IUPAC) nomenclature is a systematic method of naming organic chemical compounds as recommended by the IUPAC. It is designed to give an unambiguous name to a compound, based on a set of agreed-upon rules, which allows chemists to avoid confusion and ensure clear communication.

In the case of cycloalkenes like the one in our exercise, the IUPAC nomenclature becomes instrumental for accurately describing the location of the substituents. The ring structure is numbered in such a way that the double bond gets the lowest possible numbers, and substituents like methyl groups are named with prefixes like 'cis' and 'trans' to indicate their relative positions. Each distinct arrangement represents a different compound and thus has to have a unique name that reflects its structure.

Steps in Naming Cycloalkenes

To apply IUPAC rules, first identify the main functional groups, like the alkene part in cyclohexene. Then number the longest continuous chain containing the highest priority functional group. In a ring, you start numbering close to the functional group or substituent so that these get the lowest possible numbers. Finally, you use 'cis' and 'trans' descriptors if necessary, to show the spatial configuration of the substituents. Proper nomenclature is key to avoiding mistakes and making sure that everyone is speaking the same scientific language.

Cycloalkenes and Isomerism

Cycloalkenes are a type of cyclic hydrocarbons with one or more double bonds within the ring structure. These compounds are interesting because the double bond introduces rigidity, which restricts rotation and leads to the possibility of cis-trans isomerism. This is when two even-carbon-numbered cycloalkenes can exist in two different forms based on the relative positioning of the substituents around the double bond.

Drawing from our textbook exercise, a molecule such as dimethyl cyclohexene can exhibit different isomeric forms. These isomers can have substantially different chemical and physical properties. The simplest cycloalkene is cyclobutene, where the ring strain and bond angles significantly impact the compound's reactivity.

Ring Strain and Reactivity

In cycloalkenes, the smaller the ring, the greater the ring strain due to bond angle deviations from the ideal sp2 hybridized 120-degree bond angles, thus affecting their stability and reactivity. This means that smaller cycloalkenes are much more reactive and may undergo reactions that open up the ring structure to relieve strain. As the ring gets larger, the strain decreases, and the properties become more like those of their acyclic counterparts. This has major implications in synthesis and the potential applications of cycloalkenes in various chemical industries.