Explicit Expression for the Excess Free Energy

The graphical method used in the preceding section for a qualitative discussion can also be used for a quantitative computation of the excess prop ties if the experimental data for pure components are available with a sufficient precision. However for a clear understanding of the excess functions in terms of B, d, p and — cb/j it is useful to derive explicit analytical relations for the excess functions. This is the purpose of the present section. 

The excess free energy (17.4.1) may be expanded in powers of T — Taa and Tbb — Taa- The procedure is exactly the same,as used in Ch. IX for monomer mixtures except that here we expand in T mstead of 1 T. We limit the expansion to terms of the second order in the differences T — Taa or Tbb — Taa- usual the derivatives of X T) are replaced by their macroscopic values (cf. Table 17.5.1). We obtain in this way the excess free energy (at P — 0) 

We may now replace all e and r by their expansion in 6, d and p (cf. 3) but the formula becomes then quite cumbersome. Let us therefore consider only the two most interesting luniting situations. If we take CAjqA — cnlqB, thoi the terms of line 3 in (17.6.1) vanish. Using the expressions of 3 we then obtain the explicit formula for the excess free energy in the absence of structural effects 

We may compare (17.6.2) with the corresponding expression (10.7.4) for monomer mixtures. Apart from the combinatorial entropy and factors qjqA and CA/iA, these formulae are exactly the same, when the mole fractions xa and xb are replaced by Xa and Xb- We must also notice that in our present model the configurational specific heat at constant volume cvA vanishes as a consequence of our assumption that the cell partition functions do not depend on the temperature. Therefore the detailed discussion of the effect of intermolecular forces on excess functions presented in Ch. IX-XI, applies also to polymer mixtures. For example p will again give rise to positive deviations from ideality, positive excess entropy and heat absorption. We shall not go into more detail. 

We shall now discuss the other limiting situation which corresponds to 6 = 5 = /) = 0. Then (17.6.1) reduces to 

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