On the Geometrical Thermodynamics of Chemical Reactions

Abstract. The formal structure of geometrical thermodynamics is reviewed with particular emphasis on the geometry of equilibria submanifolds. On these submanifolds thermodynamic metrics are defined as the Hessian of thermodynamic potentials. Links between geometry and thermodynamics are explored for single and multiple component, closed and open systems. For single component, closed thermodynamic systems a detailed exposition is given which establishes a clear connection between the degeneracy and the scalar curvature of the Weinhold metric (Journal of Chemical Physcis, I-V, vol 63) and physical properties such as phase transition and non-ideal inter-particle interactions. Compelling evidence for the relationship is provided by several specific applications. These include the Ideal gas, the van der Waals and the Berthelot gases. For these cases, the degeneracy and the scalar curvature of the Weinhold metric are entirely consistent with the actual physical situation. That is, for an Ideal gas no phase transition occurs and there are no interparticle interactions meanwhile the Weinhold metric is never degenerate and has zero scalar curvature. For the van der Waals and Berthelot gases, both display a phase transition and experience interparticle interactions, the Weinhold metric is generally degenerate along a sub-manifold of co-dimension one and has non-zero scalar curvature. For multi-component closed and open systems the Gibbs free energy is employed as the thermodynamic potential to investigate the connection between geometry and thermodynamics. The Gibbs free energy is chosen for the analysis of multicomponent systems and, in particular, chemical reactions. Major emphasis is focused on a detailed examination of single chemical reactions in a multi-component closed system. Then the approach is extended to consideration of l independent chemical reactions. For single chemical reactions, the general Gibbs metric for the Ideal gas mixture is provided. Specific applications include the isothermal and isothermal-isobaric cases for a simple synthesis and single displacement chemical reactions. For these simple systems, results suggest an intriguing relationship between non-ideal interparticle interactions and phase transitions. Finally, the Gibbs metric is provided for multi-component open systems, including both ideal and non-ideal solutions.