Maria Diss
Bryan Smith
The
Remarkable Properties of C60: in Chemistry, Symmetry and Graph
Theory
INTRODUCTION
Among all the elements, Carbon is the backbone to life. An entire branch of chemistry, Organic Chemistry, is devoted to this element and its bonding natures with other elements. Since Carbon has only four valent electrons it is able to produce long homoatomic stable chains, molecular chains containing one element or atom. Up until 1985 diamond and graphite were the only known forms of molecules entire composed of Carbon; however, a new form of an entirely Carbon composed molecule exists. Its shape is a spherical closed pentagonal/hexagonal homoatomic shell, which is based from geodesic domes that are based on hexagons and pentagons. Hence, it is no surprise that these homoatomic shells are called Fullerenes, after the American architect Richard Buckminster Fuller, who was renowned for his geodesic domes (Figure 1).
Figure 1. Richard Buckminster Fuller[1]
Although there are other fullerenes, the primary focus of this paper will be of the fullerene C60 (Figure 2). C60, given the nickname Bucky Ball, has remarkable properties in chemistry, rotational symmetry, and graph theory.
Figure 2. The Fullerene C60: a Bucky ball[2]
HISTORY
In 1985, Harold Kroto of the University of Sussex and Richard Smalley of Rice University reported the discovery of stable structures consisting only of carbon atoms. The most abundant and stable of these new structures was the C60 molecule. For their discovery of the Buckyball, Kroto, Smalley, and their colleague, Robert Curl, won the 1996 Nobel Prize in Chemistry. Initially, C60 could only be produced in tiny amounts, but soon things began to change dramatically. In 1990, Wolfgang Kratschmer, Lowell Lamb, Konstantinos Fostiropoulos, and Donald Huffman discovered how to produce pure C60 in much larger quantities. With the discovery of C60, new branches of Fullerene-Chemistry have opened, instantly capturing the imagination of chemists and have since been the subject of intensive research. By 1997, 9,000 Fullerene compounds were known.
CHEMISTRY
The C60 species is a new form of carbon, adding to the two other forms known, graphite and diamond (Figure 3). The new form of carbon, C60, is different from graphite and diamond in that it consists of discrete molecules rather than networks of carbon atoms. In graphite, the carbon atom layers are held in place with van der Waals interactions, which causes them to slip past each other under pressure. Diamond, on the other hand, consists of carbon atoms that are covalently bonded to each other by s (single) bonds in a three-dimensional network, which gives extraordinary rigidity and hardness to the structure. C60, which has a molecular weight of 720 amu (atomic mass units), is a mustard-colored solid that is soluble in common organic solvents, especially solvents containing benzene or toluene. Although C60, is soluble in organic solvents, it dissolves relatively slowly, indicating that the molecules pack together very well in their crystals, and has a very high melting point.
Figure 3. The three-dimensional structures of graphite (left) and diamond (right).[3]
STRUCTURE AND SYMMETRY
The 13C nuclear magnetic resonance spectrum of C60 shows a single peak at 142.68 ppm in the aromatic region of the spectrum. Since only one peak appeared in the nuclear magnetic resonance spectrum, all the carbons in the atoms in C60 are equivalent to each other. This is possible only with a highly symmetrical structure, also indicated by the high melting point and close packing of the crystal structure. The structure proposed for C60 (Figure 2) is that of a truncated icosahedron, or more familiarly that of a soccer ball (Figure 4). The truncated icosahedron is believed to have been conceived by Archimedes as one of his 13 solids known as the “Archimedean solids.” The molecule itself consists of 60 carbons atoms, arranged as 12 pentagons and 20 hexagons, which were formed by cutting off or truncating the 12 vertices of the icosahedron and thus replacing them with a pentagonal face. Thus in this structure, every carbon atom is at the juncture of two six-membered rings, which contain three double bonds, and one five-membered ring. The most striking property of its structure is its high symmetry.
Figure 4. Truncating an icosahedron.[4]
For the C60 molecule there are three rotation axes (Figure 5). Perhaps, the most intriguing of these is the 5-fold axis, which is through the centers of two facing pentagons. The term 5-fold refers to the number of times that the molecule can be rotated around its axis, which in this case is 5, and look the same as it did previous to rotation. By looking down on one of the pentagons, one can see that the molecule is symmetric under rotations of 360°/5 = 72 degrees. Next, the rotation axis for the 3-fold model is through the center of two facing hexagons. Unlike the 5-fold axis, the 3-fold axis takes a rotation of 120 degrees to map the molecule onto itself. Lastly, there is a 2-fold axis, which is through the centers of the edges between two hexagons. This axis takes a rotation of 180 degrees to map the molecule onto itself.
Figure 5. The rotational axes of the C60 molecule.[5]
Since each axis passes through two pentagons, and since there are 12 pentagons in the molecule, there are only 6 different 5-fold axes. Similarly, since there are 20 hexagons, there are only 10 different 3-fold axes. The number of differing 2-fold axes is 15, since each hexagon is neighbored by three other hexagons, so there are 30 edges between two hexagons, i.e. 15 different 2-fold axes. Thus, by combining all those transformations, 120 different symmetry operations can be found. This makes C60 the molecule with the largest known number of symmetry operations, the most symmetric of molecules.
GRAPH THEORY
In order to use graph theory to represent a molecule, it is necessary to define a representation for points and lines. The point of a graph represents each atom of a chemical structure and lines represent the atomic bonds as shown in Figure 5. The two graphs in this figure have the same chemical formula of C3H7OH, yet they are different molecules that exhibit different physical properties. These chemical compounds are isomers. In 1857, the mathematician A. Cayley observed that many of these graphs have theoretic representations, which have the property that there is a unique path of lines between each pair of points. A graph with this property is called a tree. Cayley attempted to use this work to enumerate the possibilities for certain isomers, which are trees.[6] Furthermore, the number of edges from a given point, in this case a given atom, is dependent on the number of valance electrons the atom has. For example in figure 6, Carbon has four vertices since it has four valence electrons. Hydrogen, on the other hand, has only one vertex, which corresponds to its one valence electron.
Figure 6. Chemical Isomers of C3H7OH. [7]
For more geometrically complicated molecules, a theorem of the mathematician Leonhard Euler can be used to show that the spherical surface of a molecule that is entirely built from pentagons and hexagons must have exactly 12 pentagons. Since the Euler characteristic is a number, C, which characterizes the various classes of geometric figures, it can be calculated from the number of vertices V, edges E, and faces F.
C = V – E + F (1)
For simple polyhedra, Euler showed that the Euler characteristic is C = 2. This is easily seen to be true for a cube. In a cube V = 8, E = 12 and F = 6, hence C = 8 – 12 + 6 = 2. Showing C is an invariant for all simple polyhedra is much harder.[8]
For the case of the geometry of C60, the Euler formula (1) can be greatly simplified. Since the structure is made only from pentagons and hexagons, let P be the number of pentagons, and H be the number of hexagons. Thus the number of faces is
F = P + H.
Since each edge is shared by two polygons the number of edges is
E= (5P + 6H)/2.
For the number of vertices, we know that at least three polygons share a vertex and that for a convex polyhedron the sum of the internal angles of the polygons must not exceed 360 degrees. Since the internal angle for a pentagon is 108 degrees and for a hexagon is 120 degrees, there must not be more than three polygons sharing a vertex. Thus, there are exactly three polygons meeting at each vertex, so
V = (5P + 6H)/3.
Inserting all this into (1) and assuming C = 2, we end up with
P = 12
REFERENCES
1. Douglas B. West, Introduction to Graph Theory, Prentice-Hall Inc., Upper Saddle River, NJ,
1996, 255.
2. http://www.bfi.org/bucky_balls.htm
3. http://cst-www.nrl.navy.mil/lattice/struk/carbon.html
4. http://insite.com.br/rodrigo/bucky/buckminster_fuller.html
5. http://linc.cs.vt.edu/schools/bhs/projects/John%20Gabrysch/conclusion.htm
6. http://www.mpi-stuttgart.mpg.de/andersen/fullerene.html
7. http://www.psyclops.com/bucky.shtml
8. http://www.spacedaily.com/news/carbon-01e.html
9. L.R. Foulds, Graph Theory Applications, Springer-Verlag, New York, NY, 1992.
10. Nenad Trinajstic, Chemical Graph Theory, Vol. 1, CRC Press Inc., Boca Raton, Florida, 1983.
11. Seyhan Ege, Organic Chemistry, Houghton Mifflin Company, Boston, MA, 1999, 766.