Isobaric phase coexistence

Area within the coexistence curve of an isobaric phase diagram (temperature vs. composition) or an isothermal phase diagram (pressure vs. composition). 

A straightforward, but tedious, route to obtain information of vapor-liquid and liquid-liquid coexistence lines for polymeric fluids is to perform multiple simulations in either the canonical or the isobaric-isothermal ensemble and to measure the chemical potential of all species. The simulation volumes or external pressures (and for multicomponent systems also the compositions) are then systematically changed to find the conditions that satisfy Gibbs phase coexistence rule. Since calculations of the chemical potentials are required, these techniques are often referred to as NVT- or NPT- methods. For the special case of polymeric fluids, these methods can be used very advantageously in combination with the incremental potential algorithm. Thus, phase equilibria can be obtained under conditions and for chain lengths where chemical potentials cannot be reliably obtained with unbiased or biased insertion methods, but can still be estimated using the incremental chemical potential ansatz [47-50]. 

The melting lines, sublimation curves, and branches of the vapor-pressure curve all terminate at the horizontal broken line in Figure 8.10. That line, which is both an isobar and an isotherm, contains the triple point an equilibrium situation in which three phases coexist simultaneously. A triple point occurs at one pressure and one temperature, but at three different molar volumes—one for each phase hence, the triple point is marked by three filled circles on Figure 8.10. For liquid water in contact with water vapor and the normal phase of ice, the triple point occurs at 0.01 °C and 0.0061 bar. 

B. Vekhter, R.S. Berry, Phase coexistence in clusters an experimental isobar and an elementary model. J. Chem. Phys. 106(15), 6456-6459 (1997)... 

At a given temperature and pressure eqs. (4.7) and (4.8) must be solved simultaneously to determine the compositions of the two phases a and P that correspond to coexistence. At isobaric conditions, a plot of the composition of the two phases in equilibrium versus temperature yields a part of the equilibrium T, x-phase diagram. 

Point in the isobaric temperature-composition plane for a binary mixture where the compositions of all coexisting phases become identical. 

Equation (11.163) shows how the isochoric heat capacity of a heterogeneous two-phase system can be evaluated from known isobaric properties (CP, aP) of the individual phases and the direction y(T of the coexistence coordinate cr. 

Cooling the system is continued until the temperature of Point 2, where the hydrate phase (vertical area that begins at Point 7) forms from the vapor (Point 8) and liquid (Point 6). At Point 2 three phases (Lw-H-V) coexist for two components, so Gibbs Phase Rule (F = 2 — 3+2) indicates that only the isobaric pressure of the entire diagram is necessary to obtain the temperature and the concentrations of the three phases (Fw, H, and V) in equilibrium. 

Here, x and x" are the mole fractions in the coexisting phases Tc. pc and Xc are the temperature, the pressure and the mole fraction at the critical point PP and Pt are the critical indices Bp and Bt are the amplitudes of the equilibrium curves. For the T,x binodal at atmospheric pressure the exponent pp has a universal value close to 1/3. An analysis made for several isobars has shown that within the experimental error the quantity Pp retains its value along the line of critical points including the double critical point. The critical exponent pT for isotherms does not considerably change its value at pressures up to 70-80 MPa, with px=pp. In the vicinity of the DCP, one can observe an anomalous increase of pT. The behaviour of the exponents Pp and pT along the line of critical points at pressures from atmospheric to 200 MPa is shown in figure 3. 

An important consideration in the existence of a spinodal is the prescribed experimental conditions. In a monodisperse melt, liquid liquid coexistence can only occur along a line in the pressure-temperature/>—T plane. Hence, liquid liquid phase separation under isobaric conditions can only be transient, before the entire phase reverts to the dense liquid. On the other hand, an isochoric quench would be expected to yield true spinodal-like behaviour. The true system is probably something between the two extremes, with volume leaving the system on some timescale. Based on estimates of thermal diffusivity in melts, the time to shrink is of order 10 s (based on a 1 m sample thickness). If... 

These two theorems are general and include as particular cases the theorems established in chap. XVIII, 6 and in chap. XXIII. They do not however apply to mono variant or invariant systems. Thus the eutectic point, which is certainly an indifferent point, does not represent, mathematically, an extreme value of or p for it is the point of intersection of two curves each of which refers to a two-phase system e.g. solution + ice or solution-f salt) under constant pressure. Only at the eutectic do three phases (solution + salt + ice) coexist. A mono variant three-phase system does not have an isobaric curve. 

For the hydrocarbon--CO2 systems studied here, at pressures above the critical pressure (7.383 MPa) and above the critical temperature (304.21 K) of C02 the isobaric x,T coexistence plots of liquid and vapor phases form simple closed loops. The minimum occurs at the lower consolute point or the Lower Critical Solution Temperature (LCST). Since pressure is usually uniform in the vicinity of a heat transfer surface, such diagrams serve to display the equilibrium states possible in a heat transfer experiment. 

A phase diagram of a one-component system can be plotted readily in two dimensions, with the pressure and the temperature as coordinates. If we connect the outer lines in Fig. 7.1 mutually, then we could identify a pulped tetrahedron in the figure. However, this model is not sufficient even in a binary system, as besides temperature and pressure the composition emerges as an additional variable. Mostly, common binary phase diagrams are an isobaric section of a general three-dimensional phase diagram with the mole fraction as third variable. However, the mole fractions may be different in the various coexistent phases. In the sense of intensive variables, instead of the mole fraction, the chemical potential ii is the logical pendant to temperature T and pressure p. 

Eutectic A system consisting of two or more solid phases and a liquid that coexist at an invariant point (constant temperature in an isobaric system). This temperature is the minimum melting temperature for the assemblage of solids. Any flux of heat into or out of the system will not affect the temperature until one of the phases is exhausted. 

In a few cases a different behavior is observed. In particular, this can happen if the system, for example, shows negative deviation from Raoult s law or a pressure maximum azeotrope. For the isobaric data of the system ethane-heptane and the isothermal data of the system COz-ethane this is shown in Figure 5.10. As can be seen for the system ethane-heptane, closed curves like islands appear at pressures of 68.9 and 86.2 bar. The reason is that at these pressures both components are supercritical (ethane Pc = 48.8 bar, Tc = 305.4 K heptane Pc = 27.3 bar, Tc = 540.3 K) but the mixture is subcritical, which means coexisting liquid and vapor phase. For the system CO2-ethane, the isotherms at 293 K and 298 K show... 

In P-r space, we see only two remarkable features the vapor pressure curve, indicating the conditions under which the vapor and liquid coexist, and the critical point, at which the distinction between vapor and liquid disappears. We indicate in this figure the critical isotherm 7 = Tc and the critical isobar P = Pc. If the liquid is heated at a constant pressure exceeding the critical pressure, it expands and reaches a vapor-like state without undergoing a phase transition. Andrews and Van der Waals called this phenomenon the continuity of states. 

Fig. 4 shows schematic pTx diagrams (upper row) and the corresponding p T) projections (lower row) for a binary system here p = total pressure, T = temperature, x = mole fraction. Most important for the classification of fluid phase equilibria is the shape of the different critical curves. A critical curve connects all binary critical points in the pTx space here the two coexisting fluid phases become identical, namely at the extreme values of the isothermal p x) or isobaric T x) sections. Details can be found in the earlier publications [2-6,10,12]. 

It has been found that in the region of concentrations where two hydrogen-palladium alloys coexist (the perpendicular sections of the curves of Fig. 40) it is not possible to trace out the same isobaric curve on sorption and desorption. There is a hysteresis effect illustrated by Fig. 40. It is interesting to find in the study by Lombard, Eichner and Albert (96) that the permeability-temperature curve follows in an inverse manner the absorption isobar, there being a great increase in the permeability in the region 180-200° C. It may be that this rapid alteration in permeability marks the change from )ff-phase to a-phase alloy. 

At T < Tc, LDL and HDL can be interconverted by, for example, isothermal compresslon/decompression, isobaric heating/cooling, or by following any thermodynamic path in the P-T plane that crosses the LLPT line (Fig. 1). As the LLPT line is crossed, coexistence of LDL and HDL occurs. From the thermodynamic point of view, the LLPT is analogous to the well-known LGPT the liquid and gas phases in the LGPT correspond respectively, to LDL and HDL in the LLPT. The order parameter in both phase transitions is the density difference between the two phases involved. At the LG first-order transition, liquid and gas domains coexist. Similarly, at the LLPT, LDL and HDL domains coexist. 

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