The thermal moments of the grand canonical partition function for solutes are related to the coeffs. of a Taylor-McLaurin expansion because the solutes form a statistical ensemble distributed according to the Boltzmann law. When treating equil. in soln., the solvent is present in large excess and its concn. is in general assumed as const. The mols. of solvent can be considered to form a canonical subsystem. For this subsystem, the change of temp. d ln T produces a change of entropy dS = Cp,w d ln T, where Cp,w is the molar heat capacity of H2O, exactly equiv. to the change of entropy produced by a change of diln. dS = -d ln [W]. The properties of the canonical subsystem combined with those of the grand canonical system explain the variation of the apparent protonation const. of carboxylic acids with the temp. The curve for ethanolic acid plotted as the function of 1/T shows a min. at T = 295.4 K and can be expressed as a polynomial. By changing the ref. temp., θ, a set of values of apparent enthalpy is obtained which plotted against T = θ yields a line of intercept -ΔH0/R and slope nwCp,wθ/R. The no. of H2O mols. involved in the reaction, nw can be calcd. For the protonation of several carboxylic acids that can be represented by a normalized equation, the authors obtain nw = 2.1. By considering the H2O mols. as part of the reaction, the true equil. const. k0 can be calcd. The values of the true enthalpy, -ΔHγ and true std. entropy, -ΔSγ of the protonation-dehydration process come out to be very different from the apparent values, -ΔHappγ and -ΔSappγ, resp. because of enthalpy-entropy compensation concerning the nw H2O mols. involved.
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