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

The thermal moments of the grand canonical partition function for solutes are related to the coefficients of a Taylor-MacLaurin expansion because 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 canonical 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 entropy produced by a change of dilution 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 constant of carboxylic acids with the temperature. The curve for ethanoic acid plotted as the function of 1/T shows a minimum at T = 295.4 K and can be expressed as a polynomial: ln k(app) = ln k(theta) + (-DELTAH(app)/R)theta(1/T = 1/theta) + 1/2(DELTAC*p,app/R)theta(1/T - 1/theta)2 + 1/6{partial derivative(DELTAC*p,app/R)/partial derivative(1/T}theta(1/T - 1/theta)3 + 1/24{theta2(DELTAC*p,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 = theta yields a line of intercept DELTAH-degrees/R and slope n(w)C(p,w)theta/R. The number of water molecules involved in the reaction, n(w) can 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 equilibrium constant k-degrees can be calculated. The values of the true enthalpy, DELTAH(THETA) and true standard entropy, DELTAS(THETA) of the 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 molecules involved.