Allotropes Of Carbon

To determine the number of six-membered and five-membered rings in Buckminsterfullerene (C₆₀), we need to understand its structure. Buckminsterfullerene is a fullerene molecule with a truncated icosahedron shape, resembling a soccer ball. It consists of 60 carbon atoms arranged in a combination of pentagonal and hexagonal rings.

1. Structure of Buckminsterfullerene:

- Buckminsterfullerene (C₆₀) has a total of 32 faces, with each face being a ring of carbon atoms.

- These faces are composed of pentagons (5-membered rings) and hexagons (6-membered rings).

- Each carbon atom is shared among three rings, and each edge is shared between two rings.

2. Counting the Rings:

- It is a well-established fact in chemistry that C₆₀ has 12 pentagonal rings and 20 hexagonal rings.

- This can be derived using Euler's formula for polyhedra, , where:

- is the number of vertices (60 carbon atoms),
- is the number of edges (90, since each vertex has degree 3, so ),
- is the number of faces (32, including both pentagons and hexagons).

- Let be the number of pentagonal faces and be the number of hexagonal faces. Then:

- (total faces).

- Each pentagon has 5 edges, and each hexagon has 6 edges, but since each edge is shared between two faces, the total number of edges is given by:

.

- Additionally, for a fullerene to be stable, it typically has exactly 12 pentagons (a property of closed carbon cages). So, .

- Substituting into :

.

- Verify with the edge equation:

edges, which is correct.

3. Conclusion:

- Buckminsterfullerene has 12 five-membered rings (pentagons) and 20 six-membered rings (hexagons).

Answer: The correct option is (A) 12, 20.