Question

The total number of cyclic isomer of molecular formula \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{Cl}_{4}\) having centre of symmetry is (A) 3 (B) 5 (C) 6 (D) 8

Short answer

The total number of cyclic isomers of molecular formula \(\text{C}_{6}\text{H}_{8}\mathrm{Cl}_{4}\) having center of symmetry is 6, as calculated through the analysis of molecular structure, degrees of unsaturation, and symmetry considerations. The correct answer is (C) 6.

Step-by-step solution

Interpret the molecular formula

Given the molecular formula \(\text{C}_{6}\text{H}_{8}\mathrm{Cl}_{4}\), it has 6 carbon atoms, 8 hydrogen atoms, and 4 chlorine atoms. We also know that it forms a cyclic compound made of carbon atoms.

Calculate degrees of unsaturation

Degrees of unsaturation (DU) can be calculated using the formula: \[DU = \frac{(2 × C + 2) - (H + X - N)}{2}\] Where, C = number of carbon atoms (6 in our case), H = number of hydrogen atoms (8), X = number of halogen atoms (Cl = 4), and N = number of nitrogen atoms (0). Substituting the values, we get: \[DU = \frac{(2 × 6 + 2) - (8 + 4 - 0)}{2}\] \[DU = \frac{(12 + 2) - (12)}{2}\] \[DU = \frac{14 - 12}{2}\] \[DU = 1\] Degrees of unsaturation represent the number of pi bonds or rings present in the compound. In our case, the DU is 1, which indicates that the molecule is a cyclic hydrocarbon (one ring).

Structure of the cyclic hydrocarbon

The cyclic compound with 6 carbon atoms is a hexagon with alternating single and double bonds between carbon atoms. The number of possible positions for placing the four chlorine atoms will be 6.

Determine the isomers with a center of symmetry

A molecule with a center of symmetry can be divided into two equal parts, which are identical to each other. If we place a chlorine atom at one position, we must place another chlorine atom at exactly the opposite position to maintain symmetry. There are 3 different ways to do this: - Position the chlorine atoms at the 1 and 4 positions on the hexagon. - Position the chlorine atoms at the 2 and 5 positions on the hexagon. - Position the chlorine atoms at the 3 and 6 positions on the hexagon. For each arrangement mentioned above, you will find two additional positions for the other chlorine atoms that will maintain the center of symmetry. So, for each arrangement, there are 2 isomers with a center of symmetry.

Total number of isomers with a center of symmetry

For each of the three ways to position chlorine atoms, there are 2 possible isomers with the center of symmetry, resulting in a total of 6 isomers. Therefore, the correct answer is (C) 6.

Key concepts

Degrees of Unsaturation

Understanding the 'degrees of unsaturation' is essential in predicting and interpreting the structure of organic compounds. In organic chemistry, a degree of unsaturation represents the total number of rings and pi bonds within a molecule. Each degree of unsaturation can be either a ring or a pi bond (found in double or triple bonds).

The formula for calculating degrees of unsaturation is given by:
\[\begin{equation}DU = \frac{{(2 \times C + 2) - (H + X - N)}}{2}\end{equation}\]where C stands for the number of carbon atoms, H for hydrogen, X for halogen atoms (such as Chlorine, Bromine), and N for nitrogen atoms. For a compound like \( \mathrm{C}_{6}\mathrm{H}_{8}\mathrm{Cl}_{4} \), where no nitrogen is present, the DU comes out to be 1.

This tells us that the compound can either form one ring or contain one pi bond. Since we're considering cyclic isomers with a center of symmetry in our problem, this degree of unsaturation indicates the presence of one cyclic ring. Understanding and calculating degrees of unsaturation are a key skill in problems like those found in Organic Chemistry for JEE and other competitive exams.

Molecular Formula Interpretation

In organic chemistry, correctly interpreting the molecular formula is the first critical step towards understanding a molecule's possible structures and characteristics. The molecular formula provides crucial information on the types and numbers of atoms present in a molecule. For instance, the formula \( \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{Cl}_{4} \) reveals that the compound includes 6 carbon atoms, 8 hydrogen atoms, and 4 chlorine atoms.

When approaching a molecular formula, it's essential to consider how the atoms might connect, given the valency rules and possible structures that emerge from the degrees of unsaturation. From the given molecular formula, we can discern that since carbon typically forms four bonds and hydrogen forms one, there must be a cyclic structure or pi bonds to satisfy the valencies. For cyclic structures, the placement of substituents (in this case, chlorine atoms) may impact the molecule's symmetry, leading to different isomers with unique physical and chemical properties, which is a point of focus for preparatory exams like JEE in India.

Organic Chemistry JEE

Organic Chemistry is a fundamental cornerstone for students preparing for the Joint Entrance Examination (JEE), which is an engineering entrance examination in India. The subject challenges students to utilize various principles, including the degrees of unsaturation and molecular formula interpretation, to deduce molecular structures and their properties. These concepts are not just academic; they have practical implications in the synthesis and identification of organic compounds.

For example, understanding the concept of a 'center of symmetry' in isomers is crucial. A center of symmetry in a molecule implies it can be divided into two equal and congruent halves. In the exam context, questions often present molecular formulas and require the student to calculate possible isomers, focusing on symmetry and unsaturation degrees, just like in the given problem.Strategies for solving such problems include determining the degrees of unsaturation to predict the backbone of the molecule (rings and double bonds), then considering the spatial arrangement of substituents, like in the case of our chlorinated cyclic compound. It's this type of problem-solving that is not only tested in JEE but is also foundational for real-world organic chemistry challenges.