Chemical combinatorics. Part I. Chemical kinetics, graph theory, and combinatorial entropy

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Abstract
Chemical species may be considered in their ‘real’ state in three-dimensional space, or formally in a (highly flexible)‘graph-like’ state. On this basis, a clearer separation than hitherto can be achieved between the contributions of metrical geometry and of topology to molecular properties. The problem of this separation is basically one of combinatorial theory, and progress is here made by the use of elementary properties of permutation groups in the framework of combinatorial graph theory. Three theorems are derived which are potentially of direct interest to graph-theoreticians. Their principal appeal, however, lies in unifying various ad hoc combinatorial procedures of several domains of chemical physics. Chemical kinetics provides the natural framework for expounding this combinatorial theory to chemists. A phenomenological rate constant may be factorised into the fundamental rate constant, which depends on the driving force urging the reactants through a transition state to form products, and the statistical factor, arising from the number of sites available within a given reactant for entering into the transition state. The statistical factors are simply related to the symmetry numbers of the rotational partition functions of the molecules involved. Methods for calculating molecular weight distributions of polymers, involving enumerations of ordered rooted trees, fit into a general formalism for combinatorial entropies of molecules. The combinatorial entropy effect which favours straight chain over branched structures in high temperature equilibria can be more naturally explained in graph-theoretical terms than on the basis of rotational symmetry of ‘real-state’ molecules.

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