Animated Solution for Chemistry - s and p-Block Elements: The number of pentagons in $C_{60}$ and trigons (triangles) in white phosphorus, respectively, are

The beauty of chemistry often lies in the profound connection between the microscopic geometry of molecules and their macroscopic physical properties. In this problem, we are asked to recall and analyze the structural features of two fascinating allotropes: Buckminsterfullerene ($C_{60}$) and White Phosphorus ($P_4$).

While this might seem like a simple memory-based question, the reasons behind these structures are deeply rooted in chemical bonding, atomic size, and even pure mathematics! Let's embark on a journey to understand why these molecules look the way they do.

Buckminsterfullerene ($C_{60}$)

Imagine a hollow sphere made entirely of carbon atoms. For decades, scientists believed carbon only existed in two major allotropic forms: the hard, tetrahedral network of diamond, and the soft, layered sheets of graphite. But in 1985, a new form of carbon was discovered, consisting of 60 carbon atoms arranged in a perfect sphere. It was named Buckminsterfullerene, after the architect Richard Buckminster Fuller, who designed geodesic domes that looked strikingly similar to this molecule.

But how exactly are these 60 atoms arranged? If you've ever played with a standard soccer ball, you already know the answer! The structure is a truncated icosahedron.

To create a closed spherical shell using carbon atoms (which prefer to form three bonds in an $s{p}^2$-like hybridized state), you cannot use only hexagons. A sheet of pure hexagons is perfectly flat—that's graphite! To make the sheet curl and eventually close into a sphere, you must introduce pentagons.

This is where a beautiful piece of mathematics comes in, known as Euler's Polyhedron Formula: Where $V$ is the number of vertices (atoms), $E$ is the number of edges (bonds), and $F$ is the number of faces (rings).

Through a rigorous mathematical proof using Euler's formula, it can be shown that any closed fullerene structure made entirely of hexagons and pentagons must contain exactly 12 pentagons, regardless of how large the molecule is!

For $C_{60}$, the structure consists of: 32 Total Faces 20 Hexagons (6-membered rings) 12 Pentagons* (5-membered rings)

Therefore, the number of pentagons in $C_{60}$ is exactly 12.

Now, let's shift our focus from the elegant sphere of carbon to the highly reactive, glowing-in-the-dark allotrope of phosphorus: White Phosphorus.

Phosphorus sits right below nitrogen in Group 15 of the periodic table. Nitrogen gas exists as a highly stable diatomic molecule ($N_2$), held together by a robust triple bond ($N\equiv N$). You might wonder, why doesn't phosphorus do the same and form $P_2$?

The answer lies in atomic size. Phosphorus atoms are significantly larger than nitrogen atoms. Because of this larger size, their $3p$ orbitals are more diffuse and cannot overlap effectively sideways to form strong $p\pi -p\pi$ multiple bonds.

Since phosphorus cannot form stable double or triple bonds with itself, it must satisfy its valency (needing 3 more electrons to complete its octet) by forming three single bonds.

To achieve this, four phosphorus atoms come together to form a discrete molecule, $P_4$. Geometrically, the most symmetric way for four atoms to bond to each other is by placing them at the corners of a regular tetrahedron.

In this tetrahedral $P_4$ molecule: Every phosphorus atom is bonded to the other three. Every phosphorus atom has one lone pair of electrons. * The bond angles are exactly $60^{\circ }$.

The question asks for the number of trigons in white phosphorus. "Trigon" is simply a geometric term for a triangle.

If we look at the tetrahedral structure of $P_4$, we need to count the number of triangular faces. A tetrahedron is a pyramid with a triangular base. It consists of: 1. One triangular base. 2. Three triangular sides meeting at the top apex.

Adding these up, a tetrahedron has exactly 4 triangular faces. Therefore, the number of trigons in white phosphorus is 4.

A Note on Reactivity: It is worth mentioning that the $60^{\circ }$ bond angle in the $P_4$ tetrahedron is much smaller than the ideal $p$-orbital angle of $90^{\circ }$ (or the $s{p}^3$ angle of $109.5^{\circ }$). This creates massive angular strain within the molecule, making white phosphorus incredibly unstable and highly reactive—so much so that it spontaneously catches fire in the air!

Bringing our two analyses together: The number of pentagons in $C_{60}$ is 12. The number of trigons in $P_4$ is 4.

Looking at our options, this perfectly matches Option (b): 12 and 4.

By understanding the geometric constraints and bonding preferences of these elements, what seems like a rote-memorization fact transforms into a logical and beautiful consequence of nature's rules.