Functions, excess

Suppose for example we have determined yg as a function of composition, then (1.6.16) permits us to calculate 72- 

In this book, we shall always use the activity coefficients y as defined by (1.6.1) (except in Ch. XV, 4). 

The difierence between the thermodynamic function of mixing (denoted by superscript ) studied in 4-6 for an actual system, and the value corresponding to a perfect solution at the same T, p and composition, will be called the thermodynaipc exocoa (denoted by super- 

The excess functions permit the direct representation of the deviation from the laws of perfect solutions and will be used systematically in this book. They have been introduced by Scatchard [1931]. 

The thermodynamic excess functions differ from the thermodynamic functions of mixing only for quantities which involve the entropy. For example, the excess enthalpy A is identical with the enthalpy of mixing given by (1.6.6). Furthermore the excess volume v is identical with the volume of mixing given by (1.6.7). The excess entropy (in terms of activity coefficients) is given by (cf. 1.6.5) 

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Similarly, molar excess functions have been determined for various thiazole-solvent binary mixtures (Table 1-46) (307-310). 

From this equation, the temperature dependence of is known, and vice versa (21). The ideal-gas state at a pressure of 101.3 kPa (1 atm) is often regarded as a standard state, for which the heat capacities are denoted by CP and Real gases rarely depart significantly from ideaHty at near-ambient pressures (3) therefore, and usually represent good estimates of the heat capacities of real gases at low to moderate, eg, up to several hundred kPa, pressures. Otherwise thermodynamic excess functions are used to correct for deviations from ideal behavior when such situations occur (3). 

The extent of deviation from ideal solution behavior and hence, the magnitude and arithmetic sign of the excess function, depend upon the nature of the interactions in the mixture. We will now give some representative examples. 

The other two approaches divide the excess functional into a hard-core and an attractive part with different approximations for the two. Rosinberg and coworkers [126-129] have derived a functional from Wertheim s first-order perturbation theory of polymerization [130] in the limit of complete association. Woodward, Yethiraj, and coworkers [39,131-137] have used the weighted density approximation for the hard-core contribution to the excess free energy functional, that is,... 

Most real solutions cannot be described in the ideal solution approximation and it is convenient to describe the behaviour of real systems in terms of deviations from the ideal behaviour. Molar excess functions are defined as... 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

A great many of the difficulties (and sometimes the misunderstandings) arise from point (c). It is however important to notice that the APM describes the properties of solutions as finite differences between suitable composition-dependent averages and the properties of the pure components. Series expansions in powers of 6, p, 6, and a were introduced afterwards for the purpose of qualitative discussion and comparison with other treatments, e.g., the theory of conformal solutions.34>85>36 They introduce artificial difficulties due to their slow convergencef which have nothing to do with the physical ideas of the APM. Therefore expansions of this type should be proscribed for all quantitative applications one should instead use the compact expressions of the excess functions. 

The thermodynamic excess functions of these four versions of the APM have been investigated by Prigogine and his co-workers10-11 and it turns out that nearly equivalent results are respectively found for (a) the crude version and the refined version I (which both assume an equal sharing of V between the molecules),... 

It has been stressed in Section I that it is essentially these four parameters d, p, 0, and a which determine the values of the excess properties of the mixture A + B should all these parameters be equal to zero then all excess functions vanish. 

V. DISCUSSION OF THE QUALITATIVE PREDICTIONS OF THE APM CONCERNING THE MAIN EXCESS FUNCTIONS... 

Our aim in this section is to investigate the qualitative behavior of the excess functions of the APM and to compare it with the available experimental data. We accept the combination rules (63) as first approximations so that the excess functions are essentially related to the two parameters 8 and p. 

These conclusions are valid for both the crude and the refined versions the only difference is that the negative domain of the excess functions is somewhat smaller for the crude version, which probably overemphasizes the p-effect (size difference of A and B components). 

We report in Table VII the signs of the excess functions reported in the literature for eight binary liquid mixtures of simple molecules the corresponding values of 8 and p for each mixture are given (first component = reference component A) as well as the temperature and TAA. These values of 8 andp have been deduced from Tables V and VI, and the reference component has been chosen in such a way that all the <5 s are positive. 

TABLE VII. Experimental Signs of the Excess Functions of Several Mixtures of Simple Molecules in Relation to the d and p Values... 

We notice first that gE and hE are positive for all mixtures in agreement with the predictions of the APM. To analyze the behavior of the other excess functions we proceed in the following... 

When comparing the predictions of the APM with the experimental data one should always keep in mind that d and p are subject to errors amounting to 0.02 and 0.01 respectively this fact may introduce quite large uncertainties in the calculated excess functions. For example, assuming the validity of the combinations rules, one has the approximate form for g 32... 

This rather slight modification in 8 and a brings the calculated excess functions into almost perfect agreement with the experimental data. 

Fig. 10. Experimental and theoretical excess functions gEjkT, hE/kT, and vE of the system N2-02 at 77°K.

All the theoretical excess functions are practically equal to zero, in marked disagreement with the experimental data. 

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