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), repre sents 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 h ere defined. The distribution of species can be modulated from outside either by changing dilution or temperature. On a molar scale, the rel ationship between the partition function, Z(M), and the probability fa ctors for free energy, exp(-DELTAG/RT), enthalpy, exp(-DELTAH/RT), and entropy, exp(DELTAS/R), respectively, can be represented in probabili ty space, which is suited to relate partition function (probability) t o the experimental domains of concentration and dilution. The probabil ity space can be transformed into the affinity thermodynamic space sui ted to the representation of heat exchange (calorimetric domain) and c hemical work (cratic domain). This formal analysis is employed to expl ain why the heat exchanged in a reaction (-DELTAH/RT) in grand canonic al ensembles can be measured by means of determinations of concentrati ons in the cratic domain without any direct calorimetric determination . The heat effect is due to the existence of an intrinsic enthalpy dif ference 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 e nthalpy level is present in canonical ensembles, where no reaction tak es place. The analysis shows how the enthalpy and entropy changes upon the temperature are indistinguishable and can be transformed into eac h other by calculation. Therefore, the isobaric heat capacity C(p) app arently conveys the same thermodynamic information either as C(p) dT = dH or as C(p) d nT = dS, in canonical ensembles. The distinction betw een grand canonical and canonical ensembles based on the enthalpy diff erence is a starting point for theoretical studies and for the interpr etation of experimental data.