Alternative Approach for Critical-point Calculation

As we have already seen, the critical point can be calculated in a variety of related ways. However the basic expressions that define the critical point involve setting two determinants equal to zero (that is, Eqs. (4.214) and (4.215)). Of the two determinants, one (that is, Eq. (4.215)) requires much effort, especially for multicomponent systems it is necessary to evaluate the derivatives of certain determinants. 

Let us consider the Helmholtz free energy of the closed c-component system at constant T and V sketched in Fig. 4.24. The perturbed two-phase state consists of AT moles of the primed phase and N moles of the double-primed phase. We assume N N. Similar to the U 

If d A — 0, d A should be zero, and the next higher-order even term, that is, d A should be positive, and so on. c A = where 

For d A 0, the matrix A// should be positive definite. The matrix Afj is, positive definite if the principal submatrix is positive definite 

The inequality constraint given by Eq. (4.223) is generally not tested. Heidemann and Khalil (1980) have proposed the use Eqs. (4.225) and (4.226) to calculate the critical point. One distinct feature of the criticality criteria in terms of Eqs, (4.225) and (4.226) in comparison with Eqs. 

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