Hydrophobic Hydration Processes. I: Dual-Structure Partition Function for Biphasic Aqueous Systems

[Image: see text] The thermodynamic properties of hydrophobic hydration processes have been analyzed and assessed. The thermodynamic binding functions result to be related to each other by the mathematical relationships of an ergodic algorithmic model (EAM). The active dilution d(A) of species A in solution is expressed as d(A) = 1/(Φ·x(A)) with thermal factor Φ = T(–(C(p,A)/R)) and (1/x(A)) = d(id(A)), where d(id(A)) = ideal dilution. Entropy function is set as S = f(d(id(A)),T). Thermal change of entropy (i.e., entropy intensity change) is represented by the equation (dS)(d) = C(p) dln T. Configuration change of entropy (i.e., entropy density change) is represented by the equation (dS)(T) = (−R dln x(A))(T) = (R dln d(id(A)))(T). Because every logarithmic function in thermodynamic space corresponds to an exponential function in probability space, the sum functions ΔH(dual) = (ΔH(mot) + ΔH(th)) and ΔS(dual) = (ΔS(mot) + ΔS(th)) of the thermodynamic space give birth, in exponential probability space, to a dual-structure partition function {DS-PF}: exp(−ΔG(dual)/RT) = K(dual) = (K(mot)·ζ(th)) = {(exp(−ΔH(mot)/RT))(exp(ΔS(mot)/R))}·{exp(−ΔH(th)/RT) exp(ΔS(th)/R)}. Every hydrophobic hydration process can be represented by {DS-PF} = {M-PF}·{T-PF}, indicating biphasic systems. {M-PF} = f(T,d(id(A))), concerning the solute, is monocentric and produces changes of entropy density, contributing to free energy −ΔG(mot), whereas {T-PF} = g(T), concerning the solvent, produces changes of entropy intensity, not contributing to free energy. Entropy density and entropy intensity are equivalent and summed with each other (i.e., they are ergodic). From the dual-structure partition function {DS-PF}, the ergodic algorithmic model (EAM) can be developed. The model EAM consists of a set of mathematical relationships, generating parabolic convoluted binding functions R ln K(dual) = −ΔG(dual)/T = {f(1/T)*g(T)} and RT ln K(dual) = −ΔG(dual) = {f(T)*g(ln T)}. The first function in each convoluted couple f(1/T) or f(T) is generated by {M-PF}, whereas the second function, g(T) or g(ln T), respectively, is generated by {T-PF}. The mathematical properties of the thermodynamic functions of hydrophobic hydration processes, experimentally determined, correspond to the geometrical properties of parabolas, with constant curvature amplitude C(ampl) = 0.7071/ΔC(p,hydr). The dual structure of the partition function conforms to the biphasic composition of every hydrophobic hydration solution, consisting of a diluted solution, with solvent in excess at constant potential.