Critical endpoints

J. Putz, R. Holyst, M. Schick. Nonmonotonic behavior of a contact angle on approching critical endpoints. Phys Rev A 46 3369-3312, 1992. Phys Rev E 46 3035, 1993. 

NOAEL In this example, 18r NOAEL is the critical endpoint for which an intermediate inhalation exposure MRL is based. As you can see from the LSE figure key, the open-circle symbol indicates to a NOAEL for the test species-rat. The key number 18 corresponds to the entry in the LSE table. The dashed descending arrow indicates the extrapolation from the exposure level of 3 ppm (see entry 18 in the Table) to the MRL of 0.005 ppm (see footnote "b" in the LSE table). 

NOAEL In this example, 18r NOAEL is the critical endpoint for which an intermediate inhalation exposure MRL is based. As you can see from the LSE figure key, the open-circle symbol indicates to a... 

T Tanaka. Collapse of gels and the critical endpoint. Phys Rev Lett 40 820-823, 1978. 

Clinical trials generate vast quantities of data, most of which are processed by the sponsor. Assessments should be kept to the minimum that is compatible with the safety and comfort of the subject. Highest priority needs to be given to assessment and recording of primary endpoints, as these will determine the main outcome of the study. The power calculation for sample size should be based on the primary critical endpoint. Quite frequently, trials have two or more evaluable endpoints. It must be stated clearly in the protocol whether the secondary endpoints are to be statistically evaluated, in which case power statements will need to be given, or are simply... 

Often the essentials of phase diagrams in P,7,x-space are represented in a P,7-projection. In this type of diagrams only non-variant (F=0) and monovariant (F=l) equilibria can be represented. Since pressure and temperature of phases in equilibrium are equal, a four-phase equilibrium is now represented by one point and a three-phase equilibrium with one curve. Also the critical curve and the azeotropic curve are projected as a curve on the P, 7-plane. A four-phase point is the point of intersection of four three-phase curves. The point of intersection of a three-phase curve and a critical curve is a so-called critical endpoint. In this intersection point both the three-phase curve and the critical curve terminate. 

In Figure 2.2-3 the curves Ig are the vapour pressure curves of the pure components which end in a critical point l=g. The curves l=g, h=g and h=g are vapour-liquid critical curves and the curves h=h are curves on which two liquid phases become critical. The points of intersection of a critical curve with a three-phase curve hhg is a critical endpoint. Distinction can be made between upper critical endpoints (UCEP) and lower critical endpoints (LCEP). The UCEP is highest temperature of a three-phase curve, the LCEP is the lowest temperature of a three-phase curve. The point of intersection of the hhg curve with a l/=g curve is a critical endpoint in which the li liquid phase and the vapour phase are critical in the presence of a non-critical l2 phase (h+(h=g)) and the point of intersection of the hhg curve with a h=h curve is a critical endpoint in which the two liquid phases h and // are critical in the presence of a non-critical vapour phase (h=h)+g)-... 

Type IV fluid phase behaviour is a combination of type II and type V behaviour. These systems show two branches of the l2lig curve, three branches of the critical curve and three critical endpoints. At low temperature the P c-sections for this type of systems are similar to Figure 2.2-6, at temperatures close to the critical temperature of the pure component A PResections similar to Figure 2.2-7 are found. 

In type HI phase behaviour the two branches of the l2lig equilibrium are combined and also the two branches of the l2=li critical curve are united. Only one critical endpoint is left. The h=h/h=g branch of the critical curve can have the shape as is shown in Figure 2.2-3, but it is also possible that this curve goes from the critical point of component B to high pressure via a temperature minimum or that dP/dT is always positive [10]. 

In type VI phase behaviour a three-phase curve l2hg is found with an LCEP and an UCEP. Both critical endpoints are of the type (l2=li)+g and are connected by a l2=h critical curve which shows a pressure maximum. For this type of phase behaviour at constant pressure closed loop isobaric regions of l2+li equilibria are found with a lower critical solution temperature and an upper critical solution temperature. 

In Figure 2.2-8 the critical endpoint temperatures for the family 0f CO2 + n-alkanes systems are plotted as a function of the carbon number n. If in a particular binary system the three-phase curve hhg is followed to low temperature then at a certain temperature a solid phase is formed (solid n-alkane or solid C02 at low carbon numbers). This occurs at one unique temperature because we now have four phases in equilibrium in a binary system, so according to the phase rule F= 0. Below this so-called quadruple point temperature the hhg curve is metastable. 

In Figure 2.2-8 also the quadruple point temperatures of the different systems is plotted and the curve through this points intersects the curve through the critical endpoints at a carbon number 23< <24. This means that for carbon numbers larger than 23 no stable hhg and h+h... 

Figure 2.2-8. The family of carbon dioxide with n-alkanes. Critical endpoint temperatures and quadruple point temperatures as a function of carbon number n. quadruple point hhgscoi, quadruple point 5n-aikaneW/g A UCEP l2=lig LCEP /2=//g A UCEP hh=g-...

As discussed for CO2 + n-alkane systems at carbon numbers n<24 the three-phase curve hhg ends a low temperature in a quadruple point s2l2lig. This is shown schematically in Figure 2.2-9a and b. In the quadrupel point three other three-phase curves terminate. The s2hh curve runs steeply to high pressure and ends in a critical endpoint where this curve intersects the critical curve. The s2l2g curve runs to the triple point of pure component B and the s/l/g curve runs to lower temperature and ends at low temperature in a second quadruple point s2silig (not shown). 

With increasing carbon number of the n-alkane the melting curve of the n-alkane shifts to higher temperature. As a result also the quadruple point s2lil2g shifts to higher temperature (Figure 2.2-9b) and eventually coincides with the critical endpoint l2+(li=g) of the l2l/g curve... 

The phase behaviour of many polymer-solvent systems is similar to type IV and type HI phase behaviour in the classification of van Konynenburg and Scott [5]. In the first case, the most important feature is the presence of an Upper Critical Solution Temperature (UCST) and a Lower Critical Solution Temperature (LCST). The UCST is the temperature at which two liquid phases become identical (critical) if the temperature is isobarically increased. The LCST is the temperature at which two liquid phases critically merge if the system temperature is isobarically reduced. At temperatures between the UCST and the LCST a single-phase region is found, while at temperatures lower than the UCST and higher than the LCST a liquid-liquid equilibrium occurs. Both the UCST and the LCST loci end in a critical endpoint, the point of intersection of the critical curve and the liquid liquid vapour (hhg) equilibrium line. In the two intersection points the two liquid phases become critical in the presence of a... 

Three-phase, SLG equilibrium temperatures and pressures for binary mixtures of pentane and toluene with TPP are given in Tables II and III, respectively. A lower critical endpoint (LCEP) was observed for pentane-TPP mixtures, and is also denoted in Table II. 

The four-phase line shows an upper critical endpoint (UCEP) at 313.10 K and 8.232 MPa. At higher temperatures an interesting phenomenum can be oberseved. With increasing temperature the two critical lines (L2=L3)V and L2(L3=V), which bound the three-phase region L2L3V, appproach and finally meet at a common endpoint, a tricritical point where phases L2, L3, and V become critical simultaneously. The procedure to determine the tricritical point has been described previously [4], The tricritical point TCP was determined to 320.75 K and 9.26 MPa. 

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