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.