Thermodynamic property fractions, same components

TABLE 14.1. Thermodynamic Properties of Two Ideal Solutions (A and A ) of Different Mole Fractions, A,- and A/, Prepared from the Same Components... 

This is the same as (5.2) for the case where f = 0. For a closed ensemble at equilibrium, C + 2 variables (T, P, and C) determine the state of the entire thermodynamic system, including its size. Because of the intensive properties are independent of the size, they are specified by C + 1 independent intensive variables T, P, and C - 1 mole fractions of components. This is easily understood by the Gibbs-Duhem equation from (5.2 ) any intensive property (T, P, ju,) is a function of C + 1 intensive variables. The number of degrees of freedom of the system is therefore identical to that of the ususal macroscopic system, that is, C + 1 for C components in one phase. 

Changing the distance between the critical points requires a new variable (in addition to the three independent fractional concentrations of the four-component system). As illustrated by Figure 5, the addition of a fourth thermodynamic dimension makes it possible for the two critical end points to approach each other, until they occur at the same point. As the distance between the critical end points decreases and the height of the stack of tietriangles becomes smaller and smaller, the tietriangles also shrink. The distance between the critical end points (see Fig. 5) and the size of the tietriangles depend on the distance from the tricritical point. These dependencies also are described scaling theory equations, as are physical properties such as iuterfacial... 

A naive expectation is that each thermodynamic mixture property is a sum of the analogous property for the pure components at the same temperature and pressure weighted with their fractional compositions. That is, if U- is the internal energy per mole of pure species i at temperature T and pressure P, then it would be convenient if, for a C-component mixture,... 

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