The derivs. of the excess grand canonical partition function, ZM, or the abs. grand canonical partition function, ΞM = [M]ZM correspond to means and variances of thermodn. functions. The 1st deriv. with respect to ligand concn., δ ln Zm/δ ln [A], corresponds in thermodn. space to the formation function of Bjerrun, n (mean entropy change), the 2nd deriv. to the buffer capacity, ΔB/pA (entropy variance or dispersion). The latter can be represented in a concn.-dispersion space. In systems where no chem. reaction is taking place, the relations ΔGγ/RT = 0, ZM = 1, and ΞM = [M] hold. Analogs relations hold for each species MAi assocd. 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, ζi whose 1st deriv. with respect to 1/T, δ ln ζi/δ(1/T) is the mean enthalpy -〈ΔHj,i/R〉 of the level i whereas the deriv. δ in ζi-1/δ ln T is the mean entropy 〈ΔSj,i/R〉 of the level. The higher derivs. of ln ζi with respect to 1/T and the higher derivs. of ln ζi-1 with respect to ln T are related to the higher moments of enthalpy and entropy distribution, resp. The 2nd moment (variance) can be exptl. detd. by measurements of the molar isobaric heat capacity, Cp. The diagram Cp = f(ln T) can be considered as thermal-dispersion space. Analogs relations were found between derivs. of ln ΞM or ln ZM with respect to 1/T or ln T and moments of the free energy distribution for grand canonical ensembles. The set of derivs. can be introduced as the coeffs. in a Taylor-McLaurin series reproducing the logarithms of equil. consts. at different temps. up to the limit of stochastic error. The level model with Boltzmann statistical distribution of populations is correct for the description of the properties of systems in equil. in soln. The concn. and thermal dispersion spaces are parallel. Mixed concn.-temp. derivs. can be calcd. for grand canonical ensembles. In particular, new expressions for the apparent isobaric heat capacity, Cp,app can be obtained from mixed concn.-temp. moments.
Probability, thermodynamics and dispersion space for a statistical model of equilibria in solution. 2. Concentration and temperature moments of partition function / A. Braibanti; E. Fisicaro; F. Dallavalle; J.D. Lamb; J. L. Oscarson. - In: THE JOURNAL OF PHYSICAL CHEMISTRY. - ISSN 0022-3654. - 97(1993), pp. 8062-8070. [10.1021/j100132a042]