Animated Solution for Chemistry - s and p-Block Elements: The incorrect statement regarding the structure of ${\text{C}}_{60}$ is

Welcome to the fascinating world of fullerenes! When we talk about carbon allotropes, diamond and graphite usually steal the spotlight. However, in 1985, scientists discovered a completely new, spherical form of carbon: Buckminsterfullerene, or ${\text{C}}_{60}$.

Imagine a molecular soccer ball made entirely of carbon atoms. This highly symmetrical molecule is a truncated icosahedron, a shape that has captivated mathematicians and chemists alike. To solve problems related to ${\text{C}}_{60}$, we must deeply understand its geometric rules and bonding nature.

Let's break down the structural rules of this molecular sphere. ${\text{C}}_{60}$ is constructed from a specific arrangement of $5$-membered rings (pentagons) and $6$-membered rings (hexagons).

If you look closely at the surface of a standard soccer ball, you will notice a strict pattern: a pentagon is never adjacent to another pentagon. Every single $5$-membered ring is completely surrounded by $6$-membered rings. This is known in chemistry as the Isolated Pentagon Rule (IPR). Fusing two pentagons together would create too much local curvature and angle strain, destabilizing the molecule.

Because pentagons are only surrounded by hexagons, it logically follows that the hexagons must act as the bridge. Therefore, a $6$-membered ring is fused to both $5$-membered rings and other $6$-membered rings. This confirms that statements (a) and (c) in our problem are perfectly correct.

The $s{p}^2$ Network

Now, let's zoom in on a single carbon atom within this spherical cage. Unlike diamond, where carbon is $s{p}^3$ hybridized and bonded to four others, every carbon atom in ${\text{C}}_{60}$ is connected to exactly three neighboring carbon atoms.

This means each carbon forms three $\sigma$ bonds. What happens to the fourth valence electron? It participates in a delocalized $\pi$ bond network that spans the entire surface of the sphere. Because it forms three $\sigma$ bonds and one $\pi$ bond, the hybridization of every carbon atom is $s{p}^2$. This makes statement (b) absolutely true.

Finally, we arrive at the numbers. How many rings are there? According to Euler's polyhedron formula ($V-E+F=2$), any closed spherical structure made exclusively of pentagons and hexagons must contain exactly $12$ pentagons to close the sphere, regardless of how many hexagons it has.

For ${\text{C}}_{60}$, the magic numbers are exactly $12$ five-membered rings and $20$ six-membered rings.

When we look at statement (d), it claims that the molecule contains $12$ six-membered rings and $24$ five-membered rings. This is completely backwards and mathematically impossible for a stable fullerene.

By systematically verifying the geometry, bonding, and mathematical constraints of the fullerene cage, we can confidently conclude that statement (d) provides the wrong count of rings, making it the incorrect statement we were looking for.