Fullerene (C60)

A molecule shaped exactly like a soccer ball

Pick up a soccer ball and look at the seams. The black patches are pentagons, the white patches are hexagons, and the whole thing is stitched into a sphere. Buckminsterfullerene — C₆₀ — is that pattern rendered in carbon atoms. Put a carbon atom on every vertex where three seams meet, draw a bond along every seam, and you have built a real molecule: sixty carbons, twenty hexagonal faces, twelve pentagonal faces, perfect icosahedral symmetry.

This is not a metaphor or an approximation. The geometric solid is the truncated icosahedron — take a 20-sided icosahedron and slice off each of its twelve corners, and the flat cuts become twelve pentagons while the original triangular faces become twenty hexagons. It is one of the thirteen Archimedean solids, known to geometers for two thousand years before anyone realized carbon would spontaneously assemble into it.

Every carbon in C₆₀ is identical by symmetry — there is exactly one type of atom — yet it sits on the meeting point of two hexagons and one pentagon. That mixed environment is the whole story of fullerene chemistry. A flat graphite sheet is all hexagons; the pentagons are intruders that buckle the sheet, and that buckling is what closes the cage and strains the bonding.

Why exactly twelve pentagons close the cage

You can prove the pentagon count without any chemistry, using Euler's formula for any closed polyhedron:

V − E + F = 2          (Euler's formula)

Let  p = number of pentagons,  h = number of hexagons.

Faces:    F = p + h
Each carbon meets 3 bonds, so:
Vertices: V = (5p + 6h) / 3
Edges:    E = (5p + 6h) / 2     (each edge shared by 2 faces)

Substitute into V − E + F = 2:
  (5p + 6h)/3 − (5p + 6h)/2 + (p + h) = 2
  Multiply through by 6:
  2(5p + 6h) − 3(5p + 6h) + 6(p + h) = 12
  −(5p + 6h) + 6p + 6h = 12
                     p = 12

The hexagon count h drops out entirely — it can be anything (0, 20, 25, hundreds). But the pentagon count is pinned to exactly twelve, every time, for every closed fullerene. Hexagons tile flat (interior angle 120° × 3 = 360°, no curvature). Each pentagon contributes a 60° "angle deficit" that bends the surface; twelve pentagons deliver the total 720° of deficit that any closed surface must carry. C₆₀ is simply the smallest cage that also obeys the isolated-pentagon rule — no two pentagons share an edge, which would pile up unbearable strain.

For C₆₀ the bookkeeping checks out: F = 12 + 20 = 32 faces, V = (5·12 + 6·20)/3 = 180/3 = 60 vertices, E = 180/2 = 90 edges, and 60 − 90 + 32 = 2. Sixty atoms, ninety bonds.

Strained sp² carbon and two kinds of bond

In graphite, each sp²-hybridized carbon is flat: its three σ bonds lie in a plane at 120°, and the leftover p orbital sticks straight up to form the π system. Curving the sheet into a ball forces every carbon to pyramidalize slightly — the three C–C–C bond angles can no longer all be 120° (the two hexagon angles stay near 120° while the pentagon angle drops to about 108°), so the atom puckers out of plane and the p orbital is no longer perpendicular to its three σ bonds. The standard measure of this is the π-orbital axis-vector pyramidalization angle, about 11.6° away from the planar ideal. This is the origin of fullerene's reactivity: the π bonds are pre-bent and strained, like a compressed spring waiting to relieve itself by becoming sp³.

Crucially, the sixty carbons share two distinct bond lengths:

6:6 bond  (fused edge of two hexagons)   ≈ 1.40 Å   "double bond"  (30 of these)
6:5 bond  (edge bordering a pentagon)     ≈ 1.45 Å   "single bond"  (60 of these)

The thirty short 6:6 bonds carry the double-bond character; the sixty longer 6:5 bonds are essentially single bonds. The π electrons deliberately avoid the pentagons — a delocalization pattern that would put double bonds inside a five-membered ring is destabilizing (it looks locally like antiaromatic cyclopentadiene). The result is that C₆₀ is best drawn as a closed-shell polyalkene with thirty localized C=C double bonds, all sitting on hexagon-hexagon junctions, not as an aromatic "superbenzene." That single fact predicts almost all of its chemistry.

Fullerene vs the other carbon allotropes

	Diamond	Graphite	Fullerene (C₆₀)
Hybridization	sp³	sp² (flat)	sp² (curved/pyramidalized)
Bonds per C	4	3	3
Structure	Infinite 3D network	Infinite stacked sheets	Discrete 60-atom molecule
Holds together by	Covalent (all directions)	Covalent in-plane, van der Waals between sheets	Covalent in cage, van der Waals between cages
Electrical behavior	Insulator (5.5 eV gap)	Conductor in-plane	Semiconductor (~1.6–1.9 eV gap); K₃C₆₀ superconducts at 19 K
Solubility	Insoluble	Insoluble	Soluble — ~2.9 mg/mL in toluene; deep magenta
C–C bond length	1.54 Å (one type)	1.42 Å (one type)	1.40 & 1.45 Å (two types)
Density	3.51 g/cm³	2.27 g/cm³	1.65 g/cm³ (fcc crystal)

The standout entry is solubility. Diamond and graphite are giant covalent solids — there is no "molecule" to dissolve. C₆₀ is a genuine molecule held to its neighbors only by van der Waals forces, so it dissolves in nonpolar solvents to give a striking magenta solution. That single property is what made fullerene chemistry possible: you can do solution reactions on it the way you would on any other organic compound.

The numbers that pin it down

• Diameter: the carbon nuclei sit on a sphere ~7.1 Å across; including the π electron cloud the van der Waals diameter is ~10 Å (1 nm). The hollow interior is ~4 Å — wide enough to swallow a metal atom.
• Molar mass: 60 × 12.011 = 720.66 g/mol. The mass-spectrum peak at 720 is what tipped off the discoverers.
• Heat of formation: ΔH_f ≈ +2280 kJ/mol (about +38 kJ per mole of carbon above graphite). C₆₀ is strained and metastable — graphite is the thermodynamic ground state — but the kinetic barrier to ripping the cage open is enormous, so it is perfectly stable on the shelf.
• Sublimation: it sublimes around 400 °C without melting at atmospheric pressure, and that volatility is exploited to purify it.
• Electron affinity: the triply-degenerate LUMO is low-lying, so C₆₀ is a good electron acceptor. Cyclic voltammetry (in cold MeCN/toluene) resolves up to six reversible one-electron reductions, generating C₆₀⁻ all the way to C₆₀⁶⁻. The six waves step down roughly evenly from about −1.0 V to −3.3 V vs Fc/Fc⁺.
• Symmetry: point group I_h, the highest of any molecule — 120 symmetry operations. Its ¹³C NMR shows a single line at 143 ppm because all sixty carbons are equivalent, a beautiful confirmation of the cage structure.

How C60 actually reacts

Because the strained 6:6 double bonds are electron-poor and ready to relieve their pyramidalization strain, fullerene behaves like an exceptionally reactive alkene. Its signature reactions add a group across a 6:6 bond, turning two cage carbons from sp² to sp³:

• Bingel reaction (cyclopropanation). A halomalonate carbanion — generated from a malonate diester with a base such as DBU, the α-halogen often installed in situ with CBr₄ — attacks a 6:6 bond and closes a three-membered ring across it. This is the workhorse for making soluble, cyclopropane-bridged methanofullerene diester derivatives, the same C–C(R)(R)–C bridge motif found in the PCBM electron-acceptor used in organic solar cells (PCBM itself is made by a related diazoalkane addition rather than the malonate route).
• Prato reaction (1,3-dipolar cycloaddition). An azomethine ylide, generated in situ from an amino acid (e.g. sarcosine) and an aldehyde, adds across a 6:6 bond to give a fulleropyrrolidine. It is the standard route to water-soluble and bioactive fullerenes.
• Diels–Alder cycloaddition. A 6:6 bond acts as the dienophile (it's electron-poor, exactly what a normal-demand Diels–Alder wants), adding a diene across it.
• Reduction to fullerides. Reacting C₆₀ with three equivalents of potassium gives K₃C₆₀, a salt of C₆₀³⁻. Below 19 K it becomes a superconductor — electrons flow with zero resistance through a lattice of charged buckyballs.
• Hydrogenation. Forcing conditions add up to 36 or even 60 hydrogens (C₆₀H₃₆, C₆₀H₆₀), progressively converting the strained sp² cage toward a saturated sp³ ball.

What C₆₀ resists is electrophilic aromatic substitution — the reaction that defines benzene. There is no ring current to attack, and the molecule has no hydrogens to substitute in the first place. Every transformation instead works by addition, which is the chemical fingerprint of a strained polyalkene rather than an aromatic.

Where buckyballs show up

• Organic photovoltaics. PCBM — a soluble cyclopropane-bridged methanofullerene of C₆₀ — is the canonical electron acceptor in bulk-heterojunction solar cells, paired with a polymer donor. The buckyball's low-lying, triply-degenerate LUMO and its near-spherical shape make it an ideal three-dimensional electron sink.
• The interstellar medium. C₆₀ and C₇₀ were detected in space by the Spitzer telescope in 2010, and ionized C₆₀⁺ was confirmed in 2015 as the carrier of two long-mysterious diffuse interstellar absorption bands. Fullerene is one of the largest molecules ever firmly identified in space — exactly the soot-like carbon chemistry Kroto was originally trying to model.
• Soot and flames. Trace C₆₀ forms naturally in sooty flames and even in candle smoke; it occurs geologically in shungite and in the mineral fulgurite where lightning has struck carbon-rich rock.
• Quantum information. N@C₆₀ — a single nitrogen atom rattling inside the cage — has a remarkably long electron-spin coherence time because the cage shields the spin from its environment, making it a candidate molecular qubit.
• Lubricants and tribology. Because solid C₆₀ is a soft molecular crystal of near-spherical units, the buckyballs can act like molecular ball bearings, a property explored for low-friction coatings.

Common misconceptions and pitfalls

• "C₆₀ is aromatic like a 3D benzene." No. The double bonds localize on the hexagon-hexagon bonds and avoid the pentagons; there is no benzene-like ring current. C₆₀ reacts by addition, not substitution, which is the defining behavior of a strained alkene, not an arene.
• "All sixty C–C bonds are equal." All sixty atoms are equivalent by symmetry, but the bonds come in two flavors — thirty short 6:6 bonds (~1.40 Å) and sixty longer 6:5 bonds (~1.45 Å). Equivalent atoms do not imply equivalent bonds.
• "You could just keep adding hexagons to make any size ball." The hexagon count is free, but the pentagon count is fixed at twelve by Euler's formula for every closed fullerene, from C₂₀ up to giant cages. There is no such thing as an eleven- or thirteen-pentagon closed fullerene.
• "Fullerene is the most stable form of carbon." The opposite — it is metastable, about 38 kJ/mol per carbon higher in energy than graphite. It survives because the activation barrier to unzip the cage is huge, not because it is the thermodynamic minimum.
• "Carbon nanotubes and graphene are fullerenes." Loosely they are all sp²-carbon "fullerene-family" materials, but a strict fullerene is a closed cage. A nanotube is an open cylinder (often capped by fullerene hemispheres) and graphene is a single flat sheet with zero pentagons. Reserve "fullerene" for the closed cages.
• "It's named after a chemist." It's named after the architect-engineer Buckminster Fuller, whose geodesic domes share the pentagon-and-hexagon geometry — hence "buckminsterfullerene," casually "buckyball."