Towards a comprehensive non-ergodic treatment of H-bonds and hydrophobicity in real solutions: The mobile order and disorder theory

Abstract

The theory of mobile order and disorder (MOD) in H-bonded liquids is a completely new approach of the thermodynamics of H-bonds in liquids, and starts from the realistic analysis of the essential features which differentiate both the solid and liquid states. Unlike crystals, the ‘structure’ of liquids is always changing and the intermolecular contacts are never fixed: in liquids, all molecular contacts including H-bonds are transient. The continuous changes of the environment of a given molecule constitute an essential entropic characteristic of liquids which precludes, from a thermodynamic point of view, to treat them as deformed lattices. Molecular associations in crystals also differ from those in the liquid state by their stoichiometry. In the solid state, the stoichiometry of the associates is fixed and ruled by the crystal packing requirements, whereas the association state in liquids is perpetually changing. In the course of time, an amphiphilic molecule will be found in all possible states, such as monomolecule, head or tail of an open chain association, or bonded at both ends to neighbors. As a result, the thermodynamic entities to be considered at equilibrium are not the spectroscopic ones as assumed by the classical multicomponent theories. Finally, the presence of ephemeral open chain associations confers to the overall liquid system a ‘non-ergodic’ behavior: the probability in time for a molecule to escape from H-bonding is much larger than the proportion of molecules which do not form H-bonds at a given time to one or more neighbors (the time average and the ensemble average are not the same). The very anti-ergodic nature of open H-bond chains has dramatic consequences for the thermodynamics. It renders questionable almost all thermodynamic treatments on H-bonding in liquids based on the usual Boltzmann expression (which equates the thermodynamic probability of a system with the static probability of distribution of the various species at a given time). As suggested by Einstein, the classical Boltzmann equation for non-ergodic systems has to be replaced by another Boltzmann relation relating the thermodynamic probability of a state to the fraction of time during which the system is found in that state. The thermodynamics of MOD theory accordingly expresses the equilibrium condition in terms of time fractions for the time schedule that a given molecule is ‘free’ or ‘bonded’, and not in terms of the concentrations of the various associated species presumed to be present in the ensemble. These principles yield fundamental new equations for the entropy of mixing and for the effect of H-bonds on the chemical potential of dissolved substances. These equations constitute the basis of different MOD-theory-based solubility or partition models. Derived on a strictly thermodynamic basis and unspoiled by any adjustable parameter, these models enable correct predictions of solubilities in H-bonded solvents and organic solvent/water partition coefficients while providing a completely new quantitative treatment of the hydrophobic effect.