Question

The total possible stereoisomer for \(1,2,3,4,5,6\)-hexachloro cyclohexane and \(1,2,3,4,5,6,7\)-heptachloro cycloheptane is \(x\) and \(y\) respectively, then correct relation is (A) \(x>y\) (B) \(x=y+1\) (C) \(y=x+1\) (D) \(x=y\)

Short answer

The total possible stereoisomers for 1,2,3,4,5,6-hexachloro cyclohexane (x) is 63, and for 1,2,3,4,5,6,7-heptachloro cycloheptane (y) is 128. However, none of the given options (A, B, C, and D) show the correct relation between x and y, which is x = y - 65.

Step-by-step solution

1: Determine the number of chiral centers for hexachloro cyclohexane

: Since all the carbons in cyclohexane have two hydrogen atoms and one carbon atom attached, substituting one hydrogen with a chlorine atom introduces chirality. Therefore, all 6 carbon atoms in 1,2,3,4,5,6-hexachloro cyclohexane are chiral centers.

2: Calculate the number of stereoisomers for hexachloro cyclohexane

: Using the formula 2^n, where n = 6 (number of chiral centers), we can calculate the number of stereoisomers. 2^6 = 64 However, we need to subtract the meso compounds. There is one meso compound in this case, so we have: 64 - 1 = 63. Therefore, x = 63.

3: Determine the number of chiral centers for heptachloro cycloheptane

: Similar to hexachloro cyclohexane, substituting one hydrogen with a chlorine atom introduces chirality in cycloheptane. Therefore, all 7 carbon atoms in 1,2,3,4,5,6,7-heptachloro cycloheptane are chiral centers.

4: Calculate the number of stereoisomers for heptachloro cycloheptane

: Using the formula 2^n, where n = 7 (number of chiral centers), we can calculate the number of stereoisomers. 2^7 = 128 The meso compounds for heptachloro cycloheptane are the same as hexachloro cyclohexane. Therefore, we do not subtract any meso compounds in this case. So, y = 128.

5: Compare the values of x and y, and find the correct relation among the given options

: We have x = 63 and y = 128. Comparing these values, we see that y > x. However, none of the given options in the exercise has this relation. So, we need to check for the difference between x and y. y - x = 128 - 63 = 65 Comparing the options with this difference, we find the correct relation as follows: x = y - 65 However, none of the given options matches this relation. Thus, there is no correct relation among the given options (A, B, C, and D).

Key concepts

Chiral Centers

Chiral centers are atoms within a molecule that have four different substituents, leading to molecules that are non-superimposable on their mirror images. These molecules are known as 'chiral molecules'.

In the context of the given exercise, chiral centers can be identified when a carbon atom in a cyclohexane or cycloheptane ring is bonded to four different groups – in this case, due to the substitution of hydrogen atoms with chlorine atoms. Each chiral center can give rise to two different configurations (R or S), which contributes to the molecule's overall stereochemistry.

With each chiral center added, the total number of possible stereoisomers (distinct spatial arrangements of atoms) theoretically doubles according to the formula 2^n, where 'n' is the number of chiral centers. However, the presence of symmetry in the molecule can reduce this number due to the formation of meso compounds.

Understanding the concept of chiral centers is critical, as it plays a fundamental role in determining the number of stereoisomers a molecule can have, which is essential in fields such as pharmaceuticals where different stereoisomers can have different biological activities.

Meso Compounds

Meso compounds are a special type of stereoisomer that contain multiple chiral centers but are achiral themselves due to an internal plane of symmetry. This symmetry results in parts of the molecule being mirror images of each other, leading to the cancellation of optical activity.

Even though meso compounds have chiral centers, they do not contribute to the count of stereoisomers in the traditional sense because they do not exist as pairs of enantiomers (non-superimposable mirror images). In our exercise, identifying meso compounds is crucial in accurately calculating the total number of stereoisomers for a molecule such as hexachloro cyclohexane.

Meso compounds serve as an excellent demonstration of why the simple calculation based on the formula 2^n may not always be correct. They remind us that the real molecular structure and symmetry must be considered to get the true count of distinct stereoisomers.

Stereoisomerism in Organic Chemistry

Stereoisomerism encompasses the different spatial arrangements of atoms in molecules that have the same molecular formula. In organic chemistry, this concept is vital as it explains the existence of molecules that exhibit varied chemical and physical properties despite having identical connectivity.

There are two primary types of stereoisomerism - enantiomerism and diastereomerism. Enantiomers are stereoisomers that are non-superimposable mirror images of each other, while diastereomers are not mirror images but still have different spatial arrangements of atoms.

In the example of hexachloro cyclohexane and heptachloro cycloheptane, we deal with these complexities. The process requires accounting not only for the number of chiral centers but also factors like meso compounds and potential symmetry that affects the total number of distinguishable stereoisomers.

The ability to predict and understand stereoisomerism is fundamental for the synthesis of compounds with desired properties, especially in the pharmaceutical industry, where different stereoisomers can lead to drugs with different efficacy and safety profiles.