Thermodynamic models for fluid phase equilibria calculations, such as equations of state and activity coefficients, are being challenged by the need to describe complex and/or strongly oxygenated molecules. In this context, previous papers proposed ‘discrete modeling’ as a novel approach to incorporate a more detailed molecular picture into thermodynamics from scratch. The approach is characterized by the rigorous use of Shannon information as thermodynamic entropy. As a proof of concept, the thermal and caloric equations of state, heat capacity and Maxwell-Boltzmann distribution for ideal gas were derived on the basis of discrete states of individual molecules [1-2]. To further extend this approach to strongly interacting condensed-phase systems [3], in this paper a previous application of discrete Markov-chains to thermodynamic modeling [4] is modified and extended from a flat lattice towards a three-dimensional, Ising-type lattice model to be compared to the well-known quasi-chemical (pair) approximation by Guggenheim.
From the whole lattice system, a small, three-dimensional subgroup resp. cluster of sites is picked out as a representative part of the system and the basis for thermodynamic modeling. Its stepwise formation is described by starting from one lattice site and successively adding further nearest-neighbor sites using conditional probabilities in terms of discrete Markov-chains. Such clusters can be treated as statistically independent subsystems, yet account sufficiently for cooperative effects due to molecular interactions inside the cluster. The according probability of occurrence of clusters can more vividly be rewritten in terms of probabilities of pairwise interactions which are also used by the quasi-chemical approximation by Guggenheim. Next, the internal energy and the Shannon entropy of the system are formulated on the basis of these pairwise probabilities. Using Shannon entropy equivalently to thermodynamic entropy, constrained maximization of entropy then yields the equilibrium distribution for the probabilities of pairwise interactions.
For given system parameters like composition and interchange energy, the resulting equilibrium properties of the system, i.e. internal energy and entropy, are compared to those derived from Monte-Carlo simulations and to the Guggenheim model. It is shown that the new approach considerably improves representation of Monte-Carlo data, particularly at strong molecular interactions, and can therefore be considered as a promising basis for future gE-model development.
References:
[1] Pfleger M., Wallek T., Pfennig A. ‘Constraints of Compound Systems: Prerequisites for Thermodynamic Modeling Based on Shannon Entropy’. Entropy 2014, 16, 2990-3008
[2] Pfleger M., Wallek T., Pfennig A. ‘Discrete Modeling: Thermodynamics Based on Shannon Entropy and Discrete States of Molecules’. Ind. Eng. Chem. Res. 2015, 54, 4643-4654
[3] Wallek, T., Pfleger, M., Pfennig, A. ‘Discrete Modeling of Lattice Systems: The Concept of Shannon Entropy Applied to Strongly Interacting Systems’. Ind. Eng. Chem. Res. 2016, 55, 2483–2492
[4] Vinograd V.L., Perchuk L.L. ‘Informational Models for the Configurational Entropy of Regular Solid Solutions: Flat Lattices’. J. Phys. Chem. 1996, 100, 15972-15985
