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, Z(M), represents the probability relative to free M of finding any species MA(i) in a solution containing receptor M and ligand A. On a molecular scale, the partiton 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 dilution or temperature. On a molar scale, the relationship between the partition function, Z(M), and the probability factors for free energy, exp(-DELTAG/RT), enthalpy, exp(-DELTAH/RT), and entropy, exp(DELTAS/R), respectively, can be represented in probability space, which is suited to relate partition function (probability) to the experimental domains of concentration and dilution. The probability space can be transformed into the affinity thermodynamic space suited to the representation of heat exchange (calorimetric domain) and chemical work (cratic domain). This formal analysis is employed to explain why the heat exchanged in a reaction (-DELTAH/RT) in grand canonical ensembles can be measured by means of determinations of concentrations in the cratic domain without any direct calorimetric determination. The heat effect is due to the existence of an intrinsic enthalpy difference in the quantized model of the reaction. Cryscopic (-DELTA(m)H/RT) and ebullioscopic (-DELTA(eb)H/RT) properties are explained by the same principle, in the affinity thermodynamic space. No outstanding enthalpy level is present in canonical ensembles, where no reaction takes place. The analysis shows how the enthalpy and entropy changes upon the temperature are indistinguishable and can be transformed into each other by calculation. Therefore, the isobaric heat capacity C(p) apparently conveys the same thermodynamic information either as C(p) dT = dH or as C(p) d nT = dS, in canonical ensembles. The distinction between grand canonical and canonical ensembles based on the enthalpy difference is a starting point for theoretical studies and for the interpretation of experimental data.