Critical region coexistence curve

Below T, liquid and vapour coexist and their densities approach each other along the coexistence curve in the T-Vplane until they coincide at the critical temperature T. The coexisting densities in the critical region are related to T-T by the power law... 

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

Finally, we consider the isothennal compressibility = hi V/dp)y = d hi p/5p) j, along tlie coexistence curve. A consideration of Figure A2.5.6 shows that the compressibility is finite and positive at every point in the one-phase region except at tlie critical point. Differentiation of equation (A2.5.2) yields the compressibility along the critical isochore ... 

At the critical pohit (and anywhere in the two-phase region because of the horizontal tie-line) the compressibility is infinite. However the compressibility of each conjugate phase can be obtained as a series expansion by evaluating the derivative (as a fiuictioii of p. ) for a particular value of T, and then substituting the values of p. for the ends of the coexistence curve. The final result is... 

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

Integral equation approaches with improved self-consistency were reviewed recently by Caccamo [55]. Unfortunately, in the case of almost all approaches, their accuracy begins to decrease as one leaves the liquid state region located shghtly above the triple point in temperature and follows the liquid-gas coexistence curve in the density-temperature plane up to the critical region. In particular, the shape of the coexistence curve and location... 

Case III. As the pressure increases still further, the solubility curve intersects larger liquid-liquid regions until the critical solution pressure of the system has been reached. Above this critical pressure, no vapor phase exists, and the phase diagram consists of only the coexistence curve, as shown in Fig. 28c. In Fig. 28, L, and L2 stand for the two liquid phases and F stands for a fluid phase. 

Studies of liquid-vapor coexistence are, generally, best addressed in the framework of an open ensemble thus the state variables here comprise both the particle coordinates r and the particle number N. A path with the appropriate credentials can be constructed by identifying pairs of values of the chemical potential p and the temperature T which trace out some rough approximation to the coexistence curve in the p—7 plane, but extend into the one-phase region beyond the critical point. Once again there is some circularity here to which we shall return. Making the relevant variables explicit, the sampling distribution [Eq. (26)] takes the form... 

Consider a sudden temperature jump that brings a homogeneous mixture at the critical composition 0c into the two-phase region. The system will spontaneously phase separate into two phases with compositions given by the values on the coexistence curve at that new temperature. This spontaneous phase separation, called spinodal decomposition, occurs because the mixture is locally unstable. Any small composition fluctuation is sufficient to initiate the phase separation process. At any point inside the-... 

For pure fluids, the critical region is complicated by behavior that cannot be easily correlated. The most obvious instance is the curvature of the temperature-density coexistence curve. Classical equations tend to overestimate the critical point when fit to data outside the critical region. In other words, the true curvature is flatter than the curvature of a classical equation. Mathematically, (p — p ) Tc - instead of the... 

The model coexistence curves for our choice of s — 3.0 are shown in Fig. 4.18. The shaded areas indicate regions of phase coexistence of the confined system. The remarkable ehange of the phase diagram relative to that of the bulk system is caused by the strong confinement together with the strong selectivity of the pore for water. As expected, the critical temperature of the pore fluid is shifted downward. The critical composition has moved toward the water-rich side because of the selective character of the substrates. 

Figure 5. Evolution of isotherms in the P - p phase diagram for the core softened potential with third critical point in metastable region. Cl - gas + hquid, C2 - LDL + HDL, and C3 - HDL + VHDL critical points. Red lines (online) are coexistence curves green lines (online) are spinodals. Critical point location rcci = 0.0064, xa = 0.1189, yci =0.0998 7tc2 = 0.1423, Xc2 = 0.3856, yc2 = 0.33 ttcs = 0.07487, xcs = 0.2398, yes =0.6856. Model parameter set a = 6.962, bh =2.094, WUa=3, b,=7.0686.

Fig. 40. Schematic description of unstable thermodynamic fluctuations in the two-phase regime of a binary mixture AB at a concentration cb (a) in the unstable regime inside the two branches tp of the spinodal curve and (b) in the metastable regime between the spinodal curve tp and the coexistence curve The local concentration c(r) at a point r = (x. y, z.) in space is schematically plotted against the spatial coordinate x at some time after the quench. In case (a), the concentration variation at three distinct times t, ti, u is indicated. In case (b) a critical droplet is indicated, of diameter 2R , the width of the interfacial regions being the correlation length Note that the concentration profile of the droplet reaches the other branch ini, of the coexistence curve in the droplet center only for weak supersaturations of the mixture, where cb -

The fluorescence and isotherm measurements are consistent with the generalized phase diagram for fatty acids shown schematically in Fig. 12. Presumably the G-LE coexistence region ends in a critical point but none of the measurements that have been carried out can fix and define the asymptotic shape of the coexistence curve with any precision. The measurements show a clear narrowing of the range of LE-LC coexistence, which suggests that this may end at some sort of critical point as well, but here again experiments do not yet provide much direct information. 

It seems likely that the G-LE region ends in a critical point, but this has not yet been demonstrated by direct experiment. The detailed shape of the G-LE coexistence curve is still unknown. Although the LE-LC coexistence region in PDA has been reasonably well mapped, there have not yet been... 

Fig. 3. Equation of state obtained from the energy equation. The curves are isotherms and are labeled with appropriate t. The broken portion of the isotherms occur in the two-phase region which bounded by the liquid vapor coexistence curve. The solid dot is the critical point. The quantity Po =...

Obtain this envelopte for the van der Waals fluid. This.is the spinodai curve. The region between the coexistence curve and the curve just obtained is the metastable region of the fluid. Notice also that the critical point of the fluid is metastable. 

Some binary systems show a minimum at a lower critical-solution temperature a few systems show closed-loop two-phase regions with a maximum and a minimum.) As the temperature is increased at any composition other than the critical composition x = x, the compositions of the two coexisting phases adjust themselves to keep the total mole fraction unchanged until the coexistence curve is reached, above which only one phase... 

---