Critical point inflection properties

The horizontal segments of the isotherms in the two-phase region become progressively shorter at higher temperatures, being ultimately reduced to a point at C. Thus, the critical isotherm, labeled T exhibits a horizontal inflection at the critical point C at the top of the dome. Here the liquid and vapor phases cannot be distinguished from one another, because their properties are the same. 

Recall that the principle of corresponding state scales property data to that at the critical point. We can relate the van der Waals parameters to the temperature and pressure at the critical point by noting that there is an inflection point on the critical isotherm, as shown in Figure 4.12. Mathematically, we can say ... 

We can then use the property of the critical point in this point value (dpIdV) is zero since its tangent inherits a horizontal lines sv, and (fiFpIdV ) is zero because it is a point of inflection. For the critical point it is fair to say that... 

We can use thermodynamics to predict the arithmetic sign of the excess molar properties above above the temperature (UCST) of the upper critical end point (UCEP), and below the temperature (LCST) of a lower critical end point (LCEP). At an (UCEP), the chemical potential goes through a point of inflection. The result is that... 

We note from Figures 10.3-2 and 10.3-3 that the shapes of the critical loci of mixtures are complicated and that, in general, the critical temperature and/or pressure of a binary mixture is not intermediate to those properties of the pure fluids. It is of interest to note that, in analogy with the properties of a pure fluid, a pseudocritical point of a mixture of a fixed composition is defined by the mechanical stability inflection point,... 

In Fig. 11.5.C-2 the locus of the partial pressure and temperature in the maximum of the temperature profile and the locus of the inflection points before the hot spot are shown as p and (pj), respectively. Two criteria were derived from this. The first criterion is based on the observation that extreme sensitivity is found for trajectories—the p-T relations in the reactor—intersecting the maxima curve p beyond its maximum. Therefore, the trajectory going through the maximum of the p -curve is considered as critical. This is a criterion for runaway based on an intrinsic property of the system, not on an arbitrarily limited temperature increase. The second criterion states that runaway will occur when a trajectory intersects (Pi)i, which is the locus of inflection points arising before the maximum. Therefore, the critical trajectory is tangent to the (pi)i-curve. A more convenient version of this criterion is based on an approximation for this locus represented by p in... 

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