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 first 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 second derivative to the buffer capacity, DELTAB/pA (entropy variance or dispersion). The latter can be represented in a concentration-dispersion space. In systems where no chemical reaction is taking place, the relations DELTAG(THETA)/RT = 0, Z(M) = 1, and XI(M) = [M] hold. Analogous relations hold for each species MA(i) associated to energy level i. The distribution of the population among the sublevels j of each level i can be represented by intralevel canonical partition function, zeta(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 derivative 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 derivatives 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. The second moment (variance) can be experimentally determined by measurements 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 respect to 1/T or ln T and moments of the free energy distribution for grand 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 distribution of populations is correct for the description of the properties of systems in equilibrium in solution. The concentration and thermal dispersion spaces are parallel. Mixed concentration-temperature derivatives can be calculated for grand canonical ensembles. In particular, new expressions for the apparent isobaric heat capacity, C(p,app) can be obtained from mixed concentration-temperature moments.