I. Combinatorial Theory of Nonlinear Graphs, Applied to the Virial Equation of State. II. Chemical Thermodynamics of Open Systems

Citation

Larsen, Alvin Henry (1969) I. Combinatorial Theory of Nonlinear Graphs, Applied to the Virial Equation of State. II. Chemical Thermodynamics of Open Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/SCDN-4B19. https://resolver.caltech.edu/CaltechETD:etd-08312006-091634

Abstract

PART I:

By extension of the concept of a linear graph, as a topological configuration of vertexes with lines (2-bonds) connecting certain pairs of them, a nonlinear graph is defined to include also associations of certain triplets of the vertexes, or 3-bonds, representable by topological "areas" of triangles; quadruplets, or 4-bonds, by "volumes" of tetrahedra; etc. The configurational part of the partition function, including nonadditivity effects or three-body and higher interactions in the potential energy of a configuration of molecules, may be expanded as a sum of integrals over products of cluster functions, corresponding to a sum of nonlinear graphs. Certain special types of graphs, called trees and stars, figure prominently in the combinatorial analysis. The nth virial coefficient corresponds to the sum of all stars on n vertexes. Since topologically equivalent graphs correspond to integrals which yield the same result upon integration, the nth virial coefficient corresponds also to the sum of all topologically distinct stars on n vertexes, each multiplied by the appropriate combinatorial coefficient.

Development of the combinatorial theory for nonlinear graphs, trees, and stars proceeds similarly to that forlinear graphs. An explicit formula for counting nonlinear graphs on distinguishable vertexes is obtained, and generating functions relate the numbers of graphs to the numbers of corresponding nonlinear trees and stars. A novel term, the cycle function, is defined; the triplet cycle function, for 3-bonds, is derived; and cycle functions are used in generalizing Polya's theorem to apply to nonlinear graphs with more than one type of bond present. A theorem is thus obtained which solves the problem of counting nonlinear graphs on indistinguishable vertexes, and a relationship between generating functions permits the number of corresponding trees to be calculated. All of these techniques are extended to apply to multicomponent systems and to rooted graphs and trees. Then the problem of counting stars on indistinguishable or multicomponent vertexes is solved by a systematic procedure. The numbers of such stars are also closely approximated by a simple formula. Including 2- and 3-bonds only, the number of distinct stars on n = 3 indistinguishable vertexes is 5; for n = 4, there are 72 topologically distinct stars; and for n = 5, 10,346.

Because of the rapidly increasing number and complexity of the calculations, a practical limit for actual calculations involving any nonadditivity effects for a pure substance is the fourth virial coefficient. For multicomponent systems, the numbers of topologically distinct stars are always greater than for systems of a single component; hence even more compelling reasons would then be necessary to justify calculation of the fourth virial coefficient including three-body nonadditivity effects.

PART II:

General thermodynamic expressions for partial derivatives of extent of reaction with respect to external composition perturbations by a single species, for multicomponent, multiple-reaction systems constrained to paths of chemical equilibrium under various conditions, are obtained as the solution of a set of simultaneous, linear algebraic equations. The corresponding heat and temperature effects follow immediately. Derivatives of the extent of reaction are evaluated for ideal solutions. For multicomponent composition perturbations, the derivatives result from a linear combination of those for perturbations by a single species, each weighted by the net mole fraction of the given species in the streams crossing the boundary. Possible applications of the thermodynamic expressions include behavior of open systems under externally introduced composition perturbations, error analysis, and optimization of yield.

Thesis Files