PROBABILITY, THERMODYNAMICS, AND DISPERSION SPACE FOR A STATISTICAL-MODEL OF EQUILIBRIA IN SOLUTION .3. AQUEOUS-SOLUTIONS OF MONOCARBOXYLICACIDS AT DIFFERENT TEMPERATURES

The thermal moments of the grand canonical partition function for solu tes are related to the coefficients of a Taylor-MacLaurin expansion be cause the solutes form a statistical ensemble distributed according to the Boltzmann law. When treating equilibria in solution, the solvent is present in large excess and its concentration is in general assumed as constant. The molecules of solvent can be considered to form a can onical subsystem. For this subsystem, the change of temperature d In T produces a change of entropy dS = C(p,w) d ln T, where C(p,w) is the molar heat capacity of water, exactly equivalent to the change of entr opy produced by a change of dilution dS = -d ln [W]. The properties of the canonical subsystem combined with those of the grand canonical sy stem explain the variation of the apparent protonation constant of car boxylic acids with the temperature. The curve for ethanoic acid plotte d as the function of 1/T shows a minimum at T = 295.4 K and can be exp ressed as a polynomial: ln k(app) = ln k(theta) + (-DELTAH(app)/R)thet a(1/T = 1/theta) + 1/2(DELTACp,app/R)theta(1/T - 1/theta)2 + 1/6{part ial derivative(DELTACp,app/R)/partial derivative(1/T}theta(1/T - 1/th eta)3 + 1/24{theta2(DELTACp,app/R)/partial derivative(1/T2}theta(1/T - 1/theta)4 + ... with (-DELTAH(app)THETA/R)theta = -DELTAH-degrees/R - n(w)C(p,w) theta/R. By changing the reference temperature, 0, a set of values of apparent enthalpy is obtained which plotted against T = t heta yields a line of intercept DELTAH-degrees/R and slope n(w)C(p,w)t heta/R. The number of water molecules involved in the reaction, n(w) c an be calculated. For the protonation of several carboxylic acids that can be represented by a normalized equation, we obtain n(w) = 2.1. By considering the water molecules as part of the reaction, the true equ ilibrium constant k-degrees can be calculated. The values of the true enthalpy, DELTAH(THETA) and true standard entropy, DELTAS(THETA) of th e protonation-hydration process come out to be very different from the apparent values, DELTAH(app)THETA and DELTAS(app)THETA, respectively because of enthalpy-entropy compensation concerning the n(w) water mol ecules involved.