Allotropes of Carbon (Diamond, Graphite, Fullerenes)

What allotropy means for carbon

Allotropes are different structural forms of the same element existing in the same physical state. Carbon provides one of the best examples of allotropy because two factors converge in it: its strong tendency towards catenation (the linking of carbon atoms into chains and rings through robust $\ce{C-C}$ bonds) and its ability to form $p\pi\text{-}p\pi$ multiple bonds with itself. Together these allow carbon to adopt several distinct lattices, both crystalline and amorphous.

Diamond and graphite are the two long-known crystalline forms. In 1985 a third form, the fullerenes, was discovered by H. W. Kroto, R. E. Smalley and R. F. Curl — work for which they received the Nobel Prize in 1996. The decisive variable across all three is the hybridisation of the carbon atom: $sp^3$ in diamond, $sp^2$ in graphite and in fullerenes. Once you fix the hybridisation, the geometry, and hence the macroscopic properties, follow inevitably.

Diamond — the sp³ network

In diamond every carbon atom undergoes $sp^3$ hybridisation and is linked to four other carbon atoms through these hybrid orbitals in a tetrahedral arrangement. The $\ce{C-C}$ bond length is 154 pm. Because each atom is bonded tetrahedrally to four neighbours, and each of those to four more, the structure extends through space as a single rigid three-dimensional network of carbon atoms — effectively one giant molecule.

Directional covalent bonds run throughout the entire lattice. To deform or cleave diamond you must break extremely strong $\ce{C-C}$ bonds in every direction at once, and there is no plane of weakness to exploit. This is precisely why diamond is the hardest substance on earth, and why — despite being a purely covalent solid rather than an ionic or metallic one — it has a very high melting point. All four valence electrons of every carbon are committed to localised sigma bonds, so none are free to move; diamond is therefore a non-conductor (electrical insulator).

The hardness of diamond is exploited directly: it is used as an abrasive for sharpening hard tools, in making dies, and in the manufacture of tungsten filaments for electric bulbs. As a precious stone it is used in jewellery, where it is measured in carats (1 carat = 200 mg).

Graphite — the sp² sheets

Graphite has a layered structure. Each layer is composed of planar hexagonal rings of carbon atoms, and within a layer the $\ce{C-C}$ bond length is 141.5 pm — shorter than in diamond, reflecting partial double-bond character. Successive layers are held to one another only by weak van der Waals forces, and the distance between two adjacent layers is 340 pm, far larger than any bonding distance.

Within each sheet, every carbon atom is $sp^2$ hybridised and makes three sigma bonds with three neighbouring carbon atoms. The fourth valence electron of each carbon enters a $\pi$ bond, and these $\pi$ electrons are delocalised over the whole sheet. Being mobile, they allow graphite to conduct electricity along the plane of the layers. This single fact — mobile delocalised electrons — is what separates graphite from diamond electrically.

The mechanical behaviour follows from the layering. Because adjacent sheets are bound only by weak van der Waals forces, the layers slide over one another readily; graphite therefore cleaves easily between the layers and is soft and slippery. For this reason it serves as a dry lubricant in machines running at high temperature, where ordinary oil cannot be used.

Graphite has further uses that flow from its conductivity and inertness. Graphite fibres embedded in plastic form high-strength, lightweight composites used in tennis rackets, fishing rods, aircraft and canoes. Being a good conductor, graphite is used for electrodes in batteries and in industrial electrolysis, and crucibles made of graphite are inert to dilute acids and alkalies.

Fullerenes and the C₆₀ cage

Fullerenes are made by heating graphite in an electric arc in the presence of inert gases such as helium or argon. The sooty material formed by condensation of vapourised $\ce{C}_n$ small molecules consists mainly of $\ce{C60}$, with a smaller quantity of $\ce{C70}$ and traces of fullerenes containing even numbers of carbon atoms up to 350 or above. Unusually, fullerenes are the only pure form of carbon, because their smooth closed structure has no "dangling" bonds at the edges — unlike a graphite sheet, which terminates somewhere.

The $\ce{C60}$ molecule is cage-like and shaped like a soccer ball; it is named Buckminsterfullerene. It contains twenty six-membered rings and twelve five-membered rings. The fusion rule is specific and examinable: a six-membered ring is fused with six- or five-membered rings, but a five-membered ring can only fuse with six-membered rings. All sixty carbon atoms are equivalent and all undergo $sp^2$ hybridisation.

Each carbon forms three sigma bonds with three other carbons, and the remaining electron at each carbon is delocalised in molecular orbitals, which imparts aromatic character to the molecule. The ball has 60 vertices, each occupied by one carbon atom, and it contains both single and double bonds with $\ce{C-C}$ distances of 143.5 pm (single) and 138.3 pm (double). Spherical fullerenes are also called bucky balls.

Master comparison table

The single table below collapses the entire subtopic into a structure-to-property map. Reading it left to right reproduces nearly every objective question asked on carbon allotropes: hybridisation, geometry, the consequence for conductivity and hardness, and the use that follows.

Stability and impure forms

Although diamond looks the most "permanent" of the three, it is graphite that is thermodynamically the most stable allotrope of carbon. By convention the standard enthalpy of formation $\Delta_f H^{\circ}$ of graphite is taken as zero. Relative to it, diamond and fullerene $\ce{C60}$ lie higher in energy, with $\Delta_f H^{\circ}$ values of 1.90 and 38.1 kJ mol⁻¹ respectively. Diamond persists in everyday conditions only because its conversion to graphite is kinetically extremely slow, not because it is the lower-energy form.

Several familiar "forms" of carbon are not separate allotropes at all but impure variants of graphite or fullerenes. Carbon black is obtained by burning hydrocarbons in a limited supply of air; charcoal and coke are obtained by heating wood or coal respectively at high temperature in the absence of air. Each finds use according to its texture: activated charcoal, being highly porous, adsorbs poisonous gases and is used in water filters and air-conditioning systems to control odour; carbon black is a black pigment in inks and a filler in tyres; and coke is used as a fuel and largely as a reducing agent in metallurgy.