PROBABILITY, THERMODYNAMICS, AND DISPERSION SPACE FOR A STATISTICAL-MODEL OF EQUILIBRIA IN SOLUTION .2. CONCENTRATION AND TEMPERATURE MOMENTS OF PARTITION-FUNCTION

The derivatives of the excess grand canonical partition function, Z(M) , or the absolute grand canonical partition function, XI(M) = [M]Z(M) correspond to means and variances of thermodynamic functions. The firs t derivative with respect to ligand concentration, partial derivative ln Z(M)/partial derivative In [A], corresponds in thermodynamic space to the formation function of Bjerrum, h (mean entropy change), the sec ond derivative to the buffer capacity, DELTAB/pA (entropy variance or dispersion). The latter can be represented in a concentration-dispersi on space. In systems where no chemical reaction is taking place, the r elations DELTAG(THETA)/RT = 0, Z(M) = 1, and XI(M) = [M] hold. Analogo us relations hold for each species MA(i) associated to energy level i. The distribution of the population among the sublevels j of each leve l i can be represented by intralevel canonical partition function, zet a(i) whose first derivative with respect to 1/T, partial derivative ln zeta(i)/partial derivative(1/T) is the mean enthalpy - (DELTAH(j,i)/R ) of the level i whereas the derivative a In zeta(i)-1/partial derivat ive ln T is the mean entropy (DELTAS(j,i)/R) of the level. The higher derivatives of ln zeta(i) with respect to 1/T and the higher derivativ es of In zeta(i)-1 with respect to In T are shown to be related to the higher moments of enthalpy and entropy distribution, respectively. Th e second moment (variance) can be experimentally determined by measure ments of the molar isobaric heat capacity, C(p). The diagram Cp = f(ln 7) can be considered as thermal-dispersion space. Analogous relations have been found between derivatives of In XI(M) or ln Z(M) with respe ct to 1/T or ln T and moments of the free energy distribution for gran d canonical ensembles. The set of derivatives can be introduced as the coefficients in a Taylor-MacLaurin series reproducing the logarithms of equilibrium constants at different temperatures up to the limit of stochastic error. The level model with Boltzman statistical distributi on of populations is correct for the description of the properties of systems in equilibrium in solution. The concentration and thermal disp ersion spaces are parallel. Mixed concentration-temperature derivative s can be calculated for grand canonical ensembles. In particular, new expressions for the apparent isobaric heat capacity, C(p,app) can be o btained from mixed concentration-temperature moments.