Thermodynamic Models in Negative Space

In the preceding discussions, we have presented the idea of viewing the entire composition space, both positive and negative. However, it is reasonable to question whether these negative compositions are thermodynamically consistent and if 

All phase equilibrium thermodynamic models have to obey, and are constructed according to, the Gibbs Duhem relationship shown in Equation 2.25 [8]. 

Intuitively, all vapor liquid equilibrium models obey this relationship in positive composition space, but in order to be sure that it is useful, and valid, to analyze negative compositions, the Gibbs Duhem relationship should be maintained for negative composition too. Since 7/= I for the ideal and constant relative volatility cases, this condition is always valid for the entire composition spectrum, since the derivative of a constant is zero. Fortunately, it can be shown that all other models where y, is a function of composition (NRTL, Wilson, and so on) obey this rule, except in areas of discontinuity. Therefore, it is safe and thermodynamically sound, to analyze and interpret these negative compositions as they still obey fundamental thermodynamic rules. 

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