Problem 2 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\(\mathrm{C}_{60}\), an allotrope of carbon cantains: (a) 12 hexagons and 20 pentagons. (b) 18 hexagons and 14 pentagons. (c) 16 hexagons and 16 pentagons. (d) 20 hexagons and 12 pentagons.

Step-by-Step Solution

Answer

The correct answer is (a) 12 hexagons and 20 pentagons.

Step 1: Understanding the Structure of C60

The molecule \( \mathrm{C}_{60} \) is known as a buckyball or Buckminsterfullerene. This molecule has a spherical shape, similar to a soccer ball, and is composed of 60 carbon atoms arranged into a combination of hexagons and pentagons. Specifically, \( \mathrm{C}_{60} \) is known to have exactly 12 pentagonal faces since Euler's polyhedron formula only allows exactly 12 pentagons in a closed structure.

Step 2: Applying Euler's Polyhedron Formula

Euler's formula for polyhedra, \( V - E + F = 2 \), where \( V \) is the number of vertices, \( E \) is the number of edges, and \( F \) is the number of faces. Since each carbon atom forms 3 bonds, \( V = 60 \). Using the fact that each edge is shared by two faces, you can calculate \( E \). However, the key point is that any fullerene structure must have exactly 12 pentagons to form a closed 3D shape.

Step 3: Verifying the Number of Hexagons for C60

Given the requirement of 12 pentagons, we determine the number of hexagons as follows. The total number of faces \( F = H + P \), where \( H \) is hexagons and \( P \) is pentagons. \(P = 12\), so the leftover faces are hexagons. To fill the 60 carbon framework in a symmetric structure, it is known that \( H = 20 \) and \( P = 12 \).

Step 4: Conclusion

Since \( \mathrm{C}_{60} \) needs 12 pentagons and Euler's formula requires a certain number of hexagons to complete the structure, the correct composition is having 20 hexagons. Therefore, the only choice that satisfies both conditions is option (a), which states 12 hexagons and 20 pentagons.

Key Concepts

Euler's polyhedron formula

Euler's polyhedron formula is a fascinating concept in the world of geometry, especially when applied to three-dimensional shapes like polyhedra. The formula is given by \( V - E + F = 2 \), where \( V \) stands for the number of vertices, \( E \) represents the edges, and \( F \) denotes the faces of the polyhedron. This formula is particularly useful in solving problems related to polyhedral structures, such as fullerenes.

In the context of buckminsterfullerene, or \( \mathrm{C}_{60} \), Euler's polyhedron formula helps establish the necessary conditions for forming a closed three-dimensional shape. For example, in any fullerene, it's crucial to maintain exactly 12 pentagons to form a stable spherical structure. This constraint arises because once we plug in the known values for vertices \( V \) and pentagonal faces (a type of face), the arrangement of hexagons naturally follows to satisfy the equation.

Thus, Euler's formula is not just an abstract numerical expression, it serves as a guiding principle in discovering the intricate details of molecular geometry present in fullerenes.

carbon allotropes

Carbon allotropes are various structural forms of carbon atoms bonded in different ways. The versatility of carbon comes from its ability to form four strong covalent bonds, leading to a wide range of structures with distinct properties.

Common carbon allotropes include:

• Graphite: Comprised of layers of hexagonally arranged carbon atoms, allowing it to slide easily past one another, making it an excellent lubricant.
• Diamond: Each carbon atom is tethered to four others in a tetrahedral lattice, giving diamond its renowned hardness and optical clarity.
• Fullerenes: Structures like \( \mathrm{C}_{60} \), featuring atoms arranged in mixed hexagons and pentagons, often forming closed hollow shapes.

The distinctive arrangement of carbon in fullerenes like buckminsterfullerene (\( \mathrm{C}_{60} \)) also gives rise to unique chemical and physical properties. These structures provide interesting opportunities in materials science, especially for applications in nanotechnology and electronics.

fullerene structure

The fullerene structure is a captivating molecular geometry that combines the attributes of geometric elegance and chemical intrigue. A prominent example of a fullerene is Buckminsterfullerene, or \( \mathrm{C}_{60} \), which is a symmetrical arrangement of 60 carbon atoms.

The structure of \( \mathrm{C}_{60} \) resembles a truncated icosahedron—a shape similar to a common soccer ball. It features 12 pentagons and 20 hexagons, conforming to the geometrical constraints dictated by Euler's polyhedron formula. The arrangement of these hexagons and pentagons ensures a stable, closed configuration, essential for such structures.

This unique molecular architecture allows \( \mathrm{C}_{60} \) to exhibit properties that are beneficial for various scientific and technological fields:

• Exceptional strength and resilience.
• Conduction electricity and serving as super-capacitors.
• Potential use in drug delivery systems due to their ability to enclose other molecules.

Understanding the fullerene structure not only helps in appreciating this fascinating allotrope of carbon but also in expanding its utility in innovative applications.