Mathematics and Chemistry: Partners in Understanding Our WorldIn a series of columns I will give a sketch of some of the issues arising from using mathematics in fields outside of mathematics, but for this column I will use examples from the area of chemistry...
Which came first, the chicken or the egg? Trying to answer this question is a bit like trying to ask for mathematics, which came first, theoretical mathematics or applied mathematics?
When people think about theoretical mathematics they often think of the Pythagoreans and Euclid's Elements. For example, the Pythagoreans are reputed to have been the earliest to think about the issue of irrational numbers. A number which is the ratio of two integers a and b, where b is not zero, is called rational. For measuring lengths, areas, and volumes in practical terms the way a chemist might need to, one can get away with only rational numbers. Despite the fact that one can conceptualize about numbers, the real numbers, which are not all rational, these numbers and lengths can be approximated "arbitrarily" close using only rational quantities. Famous examples of irrational numbers are √ 2 , π, and e (the base of the natural logarithm system -although base 2 and 10 are used more often). A chemist might be interested in knowing the volume of the region in space bounded by a Euclidean sphere whose exact value is (4/3)πr3, but for any particular radius it is good enough to work with rational numbers. Chemists also don't particularly care if one represents numbers in base 10 or with some other system. The units for measurements that are chosen vary from the metrical system to the one still used, with many disadvantages, in the United States.
The world of chemistry involves the creation of molecules from the atoms that occur naturally in the world. There are 92 such elements and they have complex properties - some are gases, some are liquid, and some are solid at "room" temperature (although there is some debate about the number of naturally occurring elements, 92 is the number most often cited). From the very beginnings of chemistry, mathematics was used to create quantitative and qualitative models for helping comprehend the world of chemistry by understanding the elements that make up molecules. An atom is made up of particles which are known as protons, neutrons, and electrons. Measurement issues concerning these particles are a big part of what chemistry is about. Protons, neutrons, and electrons have mass and they have electrical charge (or the lack thereof) and mass and charge can be measured. Patterns in the mass and charge of atomic particles helped chemists get insight into the nature of atoms and the molecules these atoms can form.
Herbert Aaron Hauptman (1917-2011) (Courtesy of Wikipedia)
One interaction of mathematics and chemistry that is so familiar now that it is taken for granted, is the way certain quantities are balanced or preserved when a chemical reaction occurs. In a chemical reaction the masses of the inputs to the reaction must be the same as the masses of the products produced by the reaction.
Figure 1 (Balancing equations of molecular reactions) (Courtesy of Wikipedia)
It turns out that one can use ideas from the algebra learned in high school (or in a linear algebra class) to use systems of linear equations to "balance" chemical reactions in this spirit. The idea is to introduce letters for the molecules involved and then to use chemical principles for producing these equations. For example, in the equation above we have four molecules CH4, O2, CO2, and H2O. Suppose we use the letters x, y, z, w for the numbers of these molecules, respectively. Can we deduce relationships between the values of x, y, z and w? Since C appears on the left and right of the "equation" (bottom of Figure 1) we can say that x = z. What about oxygen? We have 2y = 2z + w because there are two oxygen atoms on the left, and two atoms in the carbon dioxide and one in the water. Finally for H, we have 4x = 2w. Notice here we have more "unknowns" than equations and if there is a way to balance the this reaction we will need to use nonnegative integers as the values for x, y, and z. Why not try a solution where we find the unknowns, y, z, and w in terms of x? Using some simple algebra we see that:
From early on chemists have used diagrams to help them think through issues involving molecules. For example, here is the way a methane molecule might be drawn.
Figure 2 (Diagram of methane molecule. Courtesy of Wikipedia)
Figure 3 (Diagram which tries to show 3-dimensional information about the methane molecule. Courtesy of Wikipedia)
Figure 4 (Representation of the methane molecule. Courtesy of Wikipedia)
While the material I am about to discuss is probably less important in the grand scheme of things to chemistry than some other ways mathematics is helping the subject, it has the advantage of being accessible with less mathematical background. One of the things mathematics typically does is formalize "ad hoc" tools and use the general theory that evolves from the mathematical approach to advantage. Diagrams such as the one chemists might draw can be considered in the framework of geometrical diagrams called graphs. Graph theory involves studying the properties of diagrams that make use of dots and line segments. The dots of a graph are called vertices and the line segments are called edges. When chemists use these diagrams they are usually called molecular graphs or structural formulas. A structural formula for methane is shown in Figure 5. The dots represent atoms, labeled to show what kind of atom is involved, and the edges represent bonds formed between the atoms. Different authors use the same or similar terms in different ways. Here the diagram does not have metric (distance) information, and the angles between bonds and the lengths of bonds is not part of the information that the diagram is conveying. So graph theory is a kind of geometry but geometry that does not use "metrical" information directly.
Figure 5 (Graph representing methane)
While it may seem like a small step to use Figure 5 instead of Figure 2, it is larger than appears at first glance. Using the mathematical results and ideas from the mathematics of "graph theory" opens many new perspectives for how to think about the chemistry of molecules which might not have occurred otherwise. It is natural to object that diagram in Figure 2 has been drawn in the plane, surely the methane molecule exists in three dimensional space, that it might be helpful to use a more "accurate" diagram. However, what mathematical modeling does is to make simplifying assumptions and in this case it is very helpful to work with a plane diagram even though it loses information.
Figure 6 (Diagram showing double bond. Courtesy of Wikipedia)
Figure 7 (Hydrocarbon with a tree structure. Courtesy of Wikipedia)
While they differ in the size of the print and the lengths of the line segments shown, they share the feature that only carbon and hydrogen atoms are present, and, thus, such molecules are known as hydrocarbons. Also notice that one of the diagrams indicates that two carbon atoms are joined with two bonds though the two molecules have the same number of carbon atoms, they differ in the number of hydrogen atoms. Thus, you will not be surprised if these molecules had different chemical and physical properties. What aspects of such diagrams make them "legal" from the point of view of a chemist? What about the diagram:
Figure 8 (Graph which can't represent a hydrocarbon)
Chemists know that the diagram in Figure 8 cannot be the structural diagram for a hydrocarbon. On the other hand, the diagram below (Figure 9):
Figure 9 (Graph used to represent a hydrocarbon with double bond)
can be labeled with C's and H's so that the result is a "legal" hydrocarbon - one with two carbon atoms and four hydrogen atoms. Thus, this is a graph theory version of the diagram in Figure 6. Notice that if we count the number of edges at each vertex there are two vertices which have four edges at them and 4 vertices that have one edge at them. If we add up these numbers we get 4 + 4 + 1 + 1 + 1 + 1 = 12, which is twice the number of edges. This is a general fact, that since each edge has exactly two end points, adding up the number of edges at a vertex will always yield twice the number of edges.
What governs such diagrams from a chemist's point of view is that different kinds of atoms can form only specific allowable numbers of "bonds" with other atoms. Use of the word valence in graph theory and the theory of polytopes is "borrowed" from the chemical notation of valence. Graph theorists capture the idea of "bonding" between atoms with the concept of degree or valence of a vertex of a graph. Consider the graph below (Figure 10). It has 6 vertices and 18 edges. However, some of these edges join a vertex to itself, called a loop or self-loop, and for some vertices there is more than one pair of edges that connect the same pair of vertices. When the graph has such vertices, joined by several edges, the edges involved are called multiple edges. Some definitions of graphs don't allow for loops and multiple edges. (Some authors call graphs where loops as well as multiple edges are allowed pseudographs, while the term multigraph is reserved for the case where there are no loops but multiple edges are allowed.) But for the kind of models we want to use here, chemistry applications, we definitely want to allow multiple edges because structural formulas with "double bonds" are not uncommon. While some authors restrict the word graph to mean dot-line diagrams without loops or multiples edges, here we will allow both.
Figure 10 (Diagram of a graph with loops and multiple edges. Courtesy of Wikipedia)
In mathematics one has the "freedom" to define terms so that they capture important applications ideas (like valence in chemistry) or because they have mathematical appeal. It would be tempting to define the valence or degree of the vertices in the graph in Figure 10 as the number of edges at a vertex. However, when one does this, it is not true that the sum of the valences of the vertices of the graph adds to twice the number of edges. The "culprits" are the loops, which contribute not one edge at a vertex but two "ends" at a vertex. With this in mind we will use the following definition of valence or degree of a vertex.
Graph theory in chemistry
The value of graph theory to chemistry started to become apparent in the 19th century. Work by two British mathematicians, Arthur Cayley (1821-1895) and James Joseph Sylvester (1814 -1897), laid the ground for a long tradition of successful use of graph-theoretical ideas in chemistry.
James Joseph Sylvester
These two individuals, very different in their backgrounds and temperaments, were also remarkable for having been good friends. Sylvester never married but Cayley together with his wife came to America to visit Sylvester during his stay at Johns Hopkins University in Baltimore. For better or worse, when the term graph is used in mathematics it can mean "graph" of a function like y = x2 or a dots and lines diagram such as that in Figure 10. It was Sylvester that we have to thank for introducing this second sense of the word "graph." Perhaps surprisingly, the first use of the word "graph" for a curve is relatively recent, too.
Figure 11 (A connected graph used to model a hydrocarbon)
Graph theorists have developed terms for how to move around in graphs to get from one vertex to another by moving along edges. For example there are several paths from vertex e to vertex i in graph H in Figure 11. Ecbfglhi, ecflghi, and ecbglhi are all examples of different paths. These paths have lengths 7, 6, and 6 respectively because they use this number of edges in getting from e to i. However, there are "shorter" paths from e to i in the sense that they use fewer edges. Both ecbghi and ecflhi have length 5. We can define the distance between two vertices in a connected graph, (a graph having one piece) as the length of the shortest path between them. The distance between vertices e and k is 4. There is a unique path which has this distance, while the distance between e and g is 3 but there are two that give this shortest distance, ecfg and ecbg. Note that we can interpret this diagram as that of a hydrocarbon, since all its vertices are either 4-valent or 1-valent.
One approach to explaining in the "center" of a graph is to use these definitions.
Remember that the distance between two vertices is the length of a shortest path between them. In Figure 11, for example, vertices d, e, i and j have eccentricity 5, vertices a, c, h and k have eccentricity 4, and vertices b, f, g and l have eccentricity 3. The central vertices in Figure 11 are b, f, g, and l.
Figure 12 (Two different representations of a 3-dimensional cube)
Figure 13 (Tree with a center consisting of an edge)
Figure 14 (Pruned hydrocarbon tree in red; center a single vertex)
When Cayley first began doing his work on isomers of hydrocarbons which were trees, he had to contend with making a decision about what graph theory model to use for a molecule. Tree molecule hydrocarbons have 4-valent and 1-valent vertices and we have just seen that the center of such a tree molecule is the same for the graph where the 1-valent vertices are deleted. Cayley looked at two different approaches. In one he drew the diagram he called a plerogram which includes the 1-valent and 4-valent vertices, and in the other, he drew the graph which only consisted of the carbon atoms - the structure he called a kenogram. Butane C4H10 has two isomers and the kenographs for these are n-butane (4-vertex path) and isobutane. These two "kerograms" are shown in Figure 15.
Figure 15 (Two isomers of butane, carbon atoms only)
Can you draw the plerograms for these two isomers?
Harry Wiener was a more recent pioneer in trying to develop an "index" for the graphs of chemical molecules which would help one grasp the physical properties of a molecule by computing information from the graph of the molecule. Wiener's idea was to look at the distances between the vertices of the graph that represented the molecule. If the molecule is a tree, what graph would best capture the properties of the molecule? One could use the graph of the connections between the carbon molecules only (kenogram), or one could use the full graph with carbons and hydrogen atoms (plerogram).
Other mathematics used in chemistry
Sometimes issues of "mathematical appeal" govern attempts to discover molecules with what might have valuable properties or whose synthesis will develop new chemical "assembly" methods. For example three of the platonic solids, the tetrahedron, the cube, and the dodecahedron have vertices which are 3-valent. Can one construct hydrocarbons which have their carbon atoms at the vertices of these solids, and additional hydrogen atoms to achieve 4-valence at all of the vertices of the original solid? In particular, Figure 12 shows a 3-cube, but the right angles at the vertices made chemists wonder if such angles could be achieved for the carbon atoms of an actual molecule. Responding to this challenge, Philip Eaton and Thomas Cole constructed cubane and related compounds. Similarly, dodecahedrane has also been synthethesized but "tetrahedrane" has still not been made!
Cayley, A., On the mathematical theory of isomers, Phil. Mag., 67 (1874), 444–446.
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