Coexistence boundary

The determination of the character and location of phase transitions has been an active area of research from the early days of computer simulation, all the way back to the 1953 Metropolis et al. [59] MC paper. Within a two-phase coexistence region, small systems simulated under periodic boundary conditions show regions of apparent thermodynamic instability [60] simulations in the presence of an explicit interface eliminate this at some cost in system size and equilibration time. The determination of precise coexistence boundaries was usually done indirectly, through the... 

In contrast, a heterogeneous solution of noncritical composition (e.g., v < xc, as shown by the arrow and dashed line in Fig. 7.11) shows a qualitatively different behavior as it is rises through the coexistence boundary and into the homogeneous region near and above Tc. For each increase in temperature along the dashed line in Fig. 7.11, a horizontal tie-line yields both the compositions of the A-rich and B-rich liquids (from the two ends of the tie-line), as well as the relative amounts of each phase (from the lever rule). Clearly, the critical composition xc remains near the middle of the tie-line as T increases toward Tc, whereas a noncritical composition x xc moves toward one or other terminus of the tie-line as the temperature is raised. 

It only remains to follow the heterogeneous phase-coexistence boundary to its critical limit, where the two phases become identical In this limit, (11.163) becomes... 

With the explicit formula (12.75) for the y coefficients, (12.78)-(12.80) become convenient formulas for the slopes of coexistence"Boundaries in various phase-diagram representations (including those with an extensive axis). Notice in particular that the derivatives (12.78) involving only intensive variables (as plotted in conventional phase diagrams) can be evaluated solely in terms of the 7 coefficients (i.e., in terms of extensive... 

So far we have simply noted the existence of curves on a typical P,T diagram as signaling the loci of conditions under which two phases coexist. No fundamental analytical relations characterizing these coexistence boundaries have yet been developed. We now attend to this matter. 

The simple Quadratic Representation 4 of f(p T) is definitely applicable in a large neighborhood of the coexistence boundary, and evolves to a form equivalent to a virial expansion with second and third virial coefficients at supercritical temperatures. 

An isochoric equation has been developed for computing thermodynamic functions of pure fluids. It has its origin on a given liquid-vapor coexistence boundary, and it is structured to be consistent with the known behavior of specific heats, especially about the critical point. The number of adjustable, least-squares coefficients has been minimized to avoid irregularities in the calculated P(p,T) surface by using selected, temperature-dependent functions which are qualitatively consistent with isochores and specific heats over the entire surface. Several nonlinear parameters appear in these functions. Approximately fourteen additional constants appear in auxiliary equations, namely the vapor-pressure and orthobaric-densities equations, which provide the boundary for the P(p,T) equation-of-state surface. 

With the above objectives in mind, we constrain the equation to the liquid-vapor coexistence boundary. For any density, the coexistence temperature, Ta(p), is obtained by iteration from equations for the ortho-baric densities. Thus the vapor pressure, P[Pg.351]

According to some researchers, the two states are in fact two distinct phases, with real coexistence boundaries. If this claim can ever be verified, then it follows that there should exist a second critical point for water, akin to the critical point at 374°C, where the liquid and gas phases coalesce. It may, however, be impossible to confirm the existence of such a point by experiment, because it would lie well below Thom where freezing cannot be avoided. 

Figure 14. Proposed low-coverage melting phase diagram of N2 on graphite according to an improved incipient triple-point model [261] (solid lines) and the density functional treatment [305] (dashed lines). Dots indicate the location of the coexistence boundary from heat capacity experiments. (Adapted from Fig. 3 of Ref. 261.)...

Figure 9.1 Phase diagram for a typical pure substance showing the hquid-gas, solid-hquid, and solid-gas phase coexistence boundaries. Point A where these three coexistence lines meet is the triple point. Point B at which the hquid-gas coexistence line terminates is the critical point.

Far away from the critical temperature, the mean of the vapour and liquid densities is represented in first approximation by a rectilinear diameter. However, close to the eritical point, the critical fluctuations modify not only the shape of the coexistence boundary, which becomes... 

With the pre-selected polymer concentrations, isobaric critical lines can be constructed from the isopleths in Figure 1. The thus obtained critical lines from 100 bar to 800 bar, respectively, show in Figure 2 on a T- plane. In Figure 2 the coexistence boundaries are described on a T- plane at the indicated pressures (bar)(the boundary under 1 bar was extrapolated from Figure 1). The shape of the coexistence curve depends only slightly on pressure, which can be recognized by comparing the curve at lOObar with the curve at 800 bar, and this system shows that it is an upper critical solution temperature (UCST) behavior system... 

---