Entropy deficiency theory

Information theory shows that if the assumption of complete energy randomization breaks down, then Equation [1] remains valid provided it is interpreted as a ratio between an effective number of states divided by an effective density of states. Both the numerator and the denominator are then reduced (but not necessarily to the same extent) by a so-called entropy deficiency factor. A mechanism of cancellation of errors arises, which accounts for the success of the simple theory. 

The maximum entropy method and the associated surprisal theory are an outgrowth of information theory. They involve a comparison between the actual shape of the KERD and the hypothetical, most statistical, so-called prior distribution . Two precious pieces of information can be derived from this comparison (i) an identification of the constraint that operates on the dynamics and prevents it from being statistical and (ii) the magnitude of the entropy deficiency which can be related to the fraction of phase space effectively sampled by the transition state. Values of 75-80% have been obtained in the case of the halogenobenzene ions. 

Information theory has been applied to the F + HD reaction. An impressive feat was the prediction of a large difference in the rotational entropy deficiency for the HF and DF channels, based upon the known ratio of = (1.45) and the difference in vibrational entropy deficiencies,... 

It will be observed even for the limited data in Table 3.1 that entropies of dilution (as indicated by ip) are highly variable from one polymer-solvent system to another and from one solvent to another for the same polymer depending on the geometrical character of the solvent. This is contrary to the theory developed from consideration of lattice arrangements according to which Ip should be approximately and nearly independent of the system. It may be noted that theories of polymer solutions fail to take into account the specific geometrical character of the solvent in relation to the polymer segment. This is a serious deficiency which must be borne in mind in applications of these theories. 

A serious deficiency in the classical theory, as exemplified by equation (4.5), is the assumption of incompressibility. This deficiency can easily be remedied by the addition of free volume in the form of holes to the system. These holes will be about the size of a mer and occupy one lattice site. In materials science and engineering, holes are frequently called vacancies. Imagine that a multicomponent mixture is mixed with No holes of volume fraction vo. Then the entropy of mixing is... 

The assumptions that the volume of the solution remains constant upon mixing (Aymix = 0) and thus it is incompressible are the reasons generally put forward to explain the deficiencies of this theory. These assumptions indeed imply no variation of a variable such as the pressure and more generally no variation of the equation of state. For the calculation of the enthalpy and the entropy of mixing, a theory that would include relations between the temperature, the pressure, and the volume of the system considered is thus a necessity. 

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