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Critical inflection temperature

Figure 8.17 Vapor fugacity for component 2 in a liquid mixture. At temperature T, large positive deviations from Raoult s law occur. At a lower temperature, the vapor fugacity curve goes through a point of inflection (point c), which becomes a critical point known as the upper critical end point (UCEP). The temperature Tc at which this happens is known as the upper critical solution temperature (UCST). At temperatures less than Tc, the mixture separates into two phases with compositions given by points a and b. Component 1 would show similar behavior, with a point of inflection in the f against X2 curve at Tc, and a discontinuity at 7V... Figure 8.17 Vapor fugacity for component 2 in a liquid mixture. At temperature T, large positive deviations from Raoult s law occur. At a lower temperature, the vapor fugacity curve goes through a point of inflection (point c), which becomes a critical point known as the upper critical end point (UCEP). The temperature Tc at which this happens is known as the upper critical solution temperature (UCST). At temperatures less than Tc, the mixture separates into two phases with compositions given by points a and b. Component 1 would show similar behavior, with a point of inflection in the f against X2 curve at Tc, and a discontinuity at 7V...
In Figure 2F-1 the composition where d2( G)/d 22 s equal to zero, or at the inflection point on the Gibbs energy surface, is defined as the spinodal composition. This corresponds to the boundary between an unstable solution and a metastable solution. If the necessary amount of free energy is supplied to the metastable system, the solution will phase separate into two phases with binodal compositions unstable system will always phase separate into the two phases. The temperature where the two points of inflection on the energy surface merge into a single point is defined as the critical solution temperature. [Pg.20]

Figure 2.16 Analysis of phase behaviour of a binary blend of polymer and solvent or two polymers exhibiting an upper critical solution temperature, Top variation of Gibbs free energy with composition, 0(0 = 0i or 02) at four temperatures. The tie line CC defines the compositions on the binodal curve. The locus of points defined by the points of inflection (9 G/90 )t,p = 0 define the spinodal curve. At point A (inside the spinodal curve), the mixture will spontaneously phase separate (into domains with compositions 0 and 0") via spinodal decomposition. However, at point B (outside the spinodal curve) there is an energy barrier to phase separation, which then occurs by nucleation and growth... Figure 2.16 Analysis of phase behaviour of a binary blend of polymer and solvent or two polymers exhibiting an upper critical solution temperature, Top variation of Gibbs free energy with composition, 0(0 = 0i or 02) at four temperatures. The tie line CC defines the compositions on the binodal curve. The locus of points defined by the points of inflection (9 G/90 )t,p = 0 define the spinodal curve. At point A (inside the spinodal curve), the mixture will spontaneously phase separate (into domains with compositions 0 and 0") via spinodal decomposition. However, at point B (outside the spinodal curve) there is an energy barrier to phase separation, which then occurs by nucleation and growth...
Another term, the critical solution temperature (CST), was introduced to designate the temperature beyond which the solubility of nonionic surfactants in organic solvents increases markedly, as marked by an inflection in the solubility curve. Mazer and Benedek used the critical micellar temperature (CMT) to refer to the phase boundary between a hydrated solid phase and a micellar phase." The CMT value was taken as the midpoint of the temperature range over which the hydrated solid phase clarified on slow warming with vigorous shaking. [Pg.115]

Above a certain temperature, the upper critical solution temperature (UCST), the single-phase polymer solution is stable at all polymer concentrations. The critical point is reached when the optima of the chanical potential versus Vj curve merges with the inflection point (represented by the curve in Figure 3.3). Mathanatically, the critical conditions are given by... [Pg.57]

For the phase separation problem, the maximum and minima in Fig. 8.2b and the inflection points between them must also merge into a common point at the critical temperature for the two-phase region. This is the mathematical criterion for the smoothing out of wiggles, as the critical point was described above. [Pg.531]

At the temperature of the critical isotherm (71 = 304.19 K for C02), the coexistence region has collapsed to a single point and represents a point of inflection in the isotherm. From calculus we know that at an inflection point, the first and second derivatives are equal to zero so that... [Pg.397]

Some isotherms corresponding to the Van der Waals equation are shown in figure 5. At a certain critical temperature Tc the isotherm has an intermediate form that goes through an inflection point at critical pressure and volume of Pc and Vc. To satisfy this requirement the first and second derivatives must vanish,... [Pg.507]

There are several possibilities for the determination of the critical micellar concentration. If the micelles are formed from charged surfactants, a plot of the electrophoretic current at constant high voltage against the surfactant concentration shows an inflection point at the ccmc. It should be noted that the critical micellar concentration changes with temperature, the kind and concentration of counterions, and other buffer ingredients. [Pg.54]

Isothermals for temperatures above the critical temperatures (qv) possess neither inflection points nor extrema, but for T [Pg.270]

For temperatures below the vapor—liquid critical temperature, T, isotherms to the left of the liquid saturation curve (see Fig. 3) represent states of subcooled liquid isotherms to the right of the vapor saturation curve are for superheated vapor. For sufficiently large molar volumes, V, all isotherms are approximated by the ideal gas equation, P = RTjV. Isotherms in the two-phase liquid—vapor region are horizontal. The critical isotherm at temperature T exhibits a horizontal inflection at the critical state, for which... [Pg.484]

In Figure 2.2-7a the bubble-point curve shows a horizontal point of inflection at the critical point l2=h and in Figure 2.2-7d the binodal shows a horizontal point of inflection at the critical point lj-g. At temperatures lower than TLcep and temperatures higher than Tucep the P c-sections are the same as for type I systems. [Pg.30]

Figure 2-10 shows a more nearly complete pressure-volume diagram.2 The dashed line shows the locus of all bubble points and dew points. The area within the dashed line indicates conditions for which liquid and gas coexist. Often this area is called the saturation envelope. The bubble-point line and dew-point line coincide at the critical point. Notice that the isotherm at the critical temperature shows a point of horizontal inflection as it passes through the critical pressure. [Pg.59]

Already we have seen that the critical temperature isotherm on a pressure-volume diagram for a pure substance has a horizontal point of inflection as it passes through the critical pressure. The data of Figure 2-10 clearly show this. Thus, for a pure substance at the critical point... [Pg.131]

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... [Pg.292]

Thomas and Bowes [11] observed that, under runaway conditions, two inflection points exist before the maximum in the temperature-time plane, while they are missing in slow reaction conditions. Critical conditions are, then, defined as those where the inflection points first appear before the temperature maximum, i.e.,... [Pg.79]

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. [Pg.36]

FIG. 7.7 Typical curves of free enthalpy G vs. volume fraction ip as a function of temperature, from above the critical temperature Tc to temperatures below Tc. The diagram below is obtained with the aid of the double tangent curves and the inflection points to construct the binodal curves (phase diagram) and spinodal curves (stability diagram), respectively. It has to be emphasised that this is the result for low molecular weight liquids, for polymer solutions the critical point would be very close to ip = 0. [Pg.212]


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See also in sourсe #XX -- [ Pg.325 , Pg.326 , Pg.327 ]




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