Probability, thermodynamics, and dispersion space for a statistical model of equilibria in solution. 1. Quantum levels and thermodynamic functions in grand canonical and canonical ensembles

The relative or excess grand canonical partition function, ZM, represents the probability relative to free M of finding any species MAi in a soln. contg. receptor M and ligand A. On a mol. scale, the partition function can be seen as the distribution of population among levels i of a quantized model. The properties of the model are here defined. The distribution of species can be modulated from outside either by changing diln. or temp. On a molar scale, the relation between the partition function, ZM, and the probability factors for free energy, exp(-ΔG/RT), enthalpy, exp(-Δ4H/RT), and entropy, exp(ΔS/R), resp., can be represented in probability space, which is suited to relate partition function (probability) to the exptl. domains of concn. and diln. The probability space can be transformed into the affinity thermodn. space suited to the representation of heat exchange (calorimetric domain) and chem. work (cratic domain). This formal anal. is employed to explain why the heat exchanged in a reaction (-ΔH/RT) in grand canonical ensembles can be measured by detns. of concns. in the cratic domain without any direct calorimetric detn. The heat effect is due to the existence of an intrinsic enthalpy difference in the quantized model of the reaction. Cryscopic (-ΔmH/RT) and ebullioscopic (-ΔebH/RT) properties are explained by the same principle, in the affinity thermodn. space. No outstanding enthalpy level is present in canonical ensembles, where no reaction takes place. The anal. shows how the enthalpy and entropy changes upon the temp. are indistinguishable and can be transformed into each other by calcn. Therefore, the isobaric heat capacity Cp apparently conveys the same thermodn. information either as Cp dT = dH or as Cp d nT = dS, in canonical ensembles. The distinction between grand canonical and canonical ensembles based on the enthalpy difference is a starting point for their studies and for the interpretation of exptl. data.