Hydrophobic Hydration Processes: Intensity Entropy and Null Thermal Free Energy and Density Entropy and Motive Free Energy.

The processes at the molecule level, which are the source of the ergodic properties of thermodynamic systems, are analyzed with special reference to entropy. The entropy change produced by increasing the temperature depends on the increase of velocity of the particles with a decrease of the squared mean sojourn time (τ ) and gradual loss of instant energy intensity. The diminution, which is due to dilution, of the number of terms in the summation of cumulative sojourn time (τ ) produces loss of energy density, thus generating a gradual increase of density entropy, d . The ergodic property of thermodynamic systems consists of the equivalence of density entropy (dependent on dilution) with intensity entropy (dependent on temperature). This equivalence has been experimentally verified in every hydrophobic hydration process as thermal equivalent dilution. An ergodic dual-structure partition function {DS-PF} represents the state probability of every hydrophobic hydration process, corresponding to the biphasic composition of these systems. The dual-structure partition function {DS-PF} ( ·ζ) is the product of a motive partition function {M-PF} ( ) multiplied by a thermal partition function {T-PF} (ζ = 1). {M-PF} gives rise to changes of density entropy, whereas {T-PF} gives rise to changes of intensity entropy. {M-PF} is referred to a reacting mole ensemble (reacting solute) composed of few elements (moles), ruled by binomial distribution, whereas {T-PF} is referred to a nonreacting molecule ensemble (NoremE) (nonreacting solvent), which is composed of a very large population of elements (molecules), ruled by Boltzmann statistics. Statistical thermodynamic methods cannot be applied to {M-PF} that can be calculated by numerical methods from the experimental titration data. By development of the dual-structure partition function {DS-PF}, parabolic convoluted binding functions are obtained. The tangents to the binding functions represent the dual enthalpy, -Δ = (-Δ - Δ ), and the dual entropy, Δ = (Δ + Δ ). The connections between canonical and grand-canonical partition functions of statistical thermodynamics with thermal and motive partition functions of chemical thermodynamics, respectively, are discussed. Special attention has been devoted to the equality Δ / + Δ = 0, typical of NoremEs, as an entropy-enthalpy compensation with Δ / = 0. The thermodynamic potential change Δμ, as proposed by potential distribution theorem (PDT) for iceberg formation from {T-PF} of the solvent, is nonexistent because the excess solvent is at a constant potential (Δμ = 0). The information level offered by the ergodic algorithmic model (EAM) is more complete and correct than that offered by the potential distribution theorem (PDT): the stoichiometry of the water reaction in hydrophobic hydration processes is determined by the EAM as the function of the number ±ξ. Quasi-chemical approximation, renamed the chemical molecule/mole scaling function (Che. m/M. sF), is a fundamental breakthrough in the application of statistical thermodynamics to chemical reactions. Boltzmann statistical molecule distribution of the thermal partition function {T-PF} is scaled with binomial mole distribution of the motive partition function {M-PF}. For computer-assisted drug design, the alternative calculation procedure of Talhout, based on the previous experimental determination of binding functions, is recommended. The ergodic algorithmic model (EAM), applied to the experimental convoluted binding functions, can recover the distinct terms of intensity entropy (Δ /) and density entropy (Δ ), together with other essential information elements, lost by computer simulations.