Coexistence lines

In general the width of the coexistence line (Ap, Ax, or AM) is proportional to an order parameter s, and its absolute value may be written as... 

Once a point on the coexistence line has been found, one can trace out more of it using the approach of Kofke [177. 178] to numerically integrate die Clapeyron equation... 

Here, A h=hp - is the difference in molar enthalpies of the coexisting phases, and A v is the difference in molar volumes the suffix o indicates that the derivative is to be evaluated along the coexistence line. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Fig. 5. The essential form of the phase diagram for the mesogen GB(3.0, 5.0, 2, 1) the open circles indicate the approximate coexistence lines and the solid circles show the density of the isotropic liquid in equilibrium with the vapour phase...

Equations (1) and (2) hold for all points along the coexistence lines in Figure 6. At the lowest loading, 0.7 molecules/supercage, AE /AEjj = 1.2. Since the two lines are approximately parallel... 

While the main driving force in [43, 44] was to avoid direct particle transfers, Escobedo and de Pablo [38] designed a pseudo-NPT method to avoid direct volume fluctuations which may be inefficient for polymeric systems, especially on lattices. Escobedo [45] extended the concept for bubble-point and dew-point calculations in a pseudo-Gibbs method and proposed extensions of the Gibbs-Duhem integration techniques for tracing coexistence lines in multicomponent systems [46]. 

A second observation relates to calculations near critical points. The coexistence lines in Fig. 10.7 do not extend above a temperature of T = 11.6 because above that temperature significant overlap exists between the liquid and vapor peaks of the histograms. This overlap renders calculations of the liquid and gas densities imprecise. 

Agrawal, R. Mehta, M. Kofke, D. A., Efficient evaluation of three-phase coexistence lines, lnt. J. Thermophys. 1994,15, 1073-1083... 

Kofke, D. A., Semigrand canonical Monte Carlo simulation integration along coexistence lines, Adv. Chem. Phys. 1999,105, 405 142... 

Meijer, E. J. Azhar, F. El, Novel procedure to determine coexistence lines by computer simulation, application to hard-core Yukawa model for charge-stabilized colloids, J. Chem. Phys. 1997,106, 4678-4683... 

Escobedo, F. A., Tracing coexistence lines in multicomponent fluid mixtures by molecular simulation, J. Chem. Phys. 1999,110, 11999-12010... 

Associations within the bulk crystalline phase. The physical property of enantiomeric solids and their mixtures which is cited most often is melting point. Figure 18 gives the melting point versus composition diagram for mixtures of S( + )- and R( — )-SSME. The solid-liquid coexistence line of... 

S - H2O - H+ - e. It is important to note that these fall into two classes. Sulphur will he classified as a type 1 component to signify that we are interested in what compounds it forms. Condensed type 1 compounds, have an activity of 1 if present and > 1 if absent. The activities of the predominant type 1 compounds in solution can he fixed at any desired value. In figure 1 they are set at 10-1. Thus the sulphate ion has an activity of 10 1 in its area and HS- has an activity of 10-1 in its area. On the line between these areas HS and the sulphate ion coexist with equal activities and the position of the coexistence line between these two compounds can be obtained by solution of the equation for the equilibrium ... 

There are 6 sulphur compounds to be considered and therefore 15 possible coexistence lines of which only 9 represent stable equilibria. Moreover, even these 9 lines are valid (i.e. correspond to stable coexistence) along only part of their possible extents. The problem then is to determine the equations for the lines and the range over which they are valid, and to provide a method for plotting them. 

Key features of Pourbaix diagrams are the points of intersection between the coexistence lines. In a simple diagram, three compounds of a dependent component can coexist at these points. Thus, if compounds i, j and k coexist at a point, 3 coexistence lines must radiate from the point, ij, jk and ik. The coordinates of potential triple intersection points can be determined by simultaneous solution of pairs of equations. For example the coordinates of the equilibrium point between sulphur, SOjj- and HS- are determined by solution of the equations... 

The results show the triple intersection is valid because x(1 1) x(1 3) and x(1 5) are higher than the remaining values of x-Thus coexistence lines 1,3, 1,5 and 3,5 do indeed radiate from pH = 7.618, pE = -U.36 5. Valid intersection points are stored together with values of i, j and k. 

For the purpose of computer calculation potential coexistence lines fall into a number of classes. 

The intersection points between coexistence lines and of coexistence lines with the boundaries of the diagram are calculated in the same way as already described for simple diagrams. However, in systems of more than one type 1 component quadrupl intersections can occur at which for example HSOq, SOq2, Cu2 and Cu2S coexist. In such a case four values of x will be found to be equal at the intersection point showing that it is the... 

Figure 2.9 Phase diagram for C02, showing solid-gas (S + G, sublimation ), solid-liquid (S + L, fusion ), and liquid-gas (L + G, vaporization ) coexistence lines as PT boundaries of stable solid, liquid, or gaseous phases. The triple point (triangle), critical point (x), and selected 280K isotherm of Fig. 2.8 (circle) are marked for identification. Note that the fusion curve tilts slightly forward (with slope 75 atm K-1) and that the sublimation and vaporization curves meet with slightly discontinuous slopes (angle < 180°) at the triple point. The dotted and dashed half-circle shows two possible paths between a liquid (cross-hair square) and a gas (cross-hair circle) state, one discontinuous (dashed) crossing the coexistence line, the other continuous (dotted) encircling the critical point (see text).

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

What does it mean that (25°C, 23.8 Torr) is a point on the liquid-vapor coexistence line Consider a beaker of liquid water at 25°C, covered with a lid and allowed to come into equilibrium with its own vapor ... 

Figure 7.1 also shows the critical point (circle-x, dashed lines) of water, the terminus of the liquid-vapor coexistence line. Beyond this point (which occurs at Tc = 374°C, Pc = 217.7 atm), there is no longer a sensible distinction between liquid and vapor, so one should only speak of a supercritical fluid (or simply fluid ) beyond the dashed lines. A sample of water above Pc can never exhibit a boiling point, no matter how far the temperature is increased, nor can a sample above Tc exhibit condensation, no matter how far the pressure is increased. 

The critical state is evidently an invariant point (terminus of a line) in this case, because it lies at a dimensional boundary between states of / =2 (p = 1) and /= 1 (p = 2). The critical point is therefore a uniquely specified state for a pure substance, and it plays an important role (Section 2.5) as a type of origin or reference state for description of all thermodynamic properties. Note that a limiting critical terminus appears to be a universal feature of liquid-vapor coexistence lines, whereas (as shown in Fig. 7.1) solid-liquid and solid-vapor lines extend indefinitely or form closed networks with other coexistence lines. 

As will be illustrated in Section 7.2.3, this provides a powerful mnemonic for judging the relative densities of adjacent phases from the qualitative features (slopes) of coexistence lines in the phase diagram (or vice versa). 

Vaporization Transition Clausius-Clapeyron Equation For the liquid-vapor coexistence line ( vapor-pressure curve ), the Clapeyron equation (7.29) becomes... 

In accordance with the Clapeyron equation and Le Chatelier s principle, the more highly ordered (low-entropy) phases tend to lie further to the left (at lower 7), whereas the higher-density phases tend to lie further upward (at higher 7). The mnemonic (7.32) allows us to anticipate the relative densities of adjacent phases. From the slope, for example, of the ice II-ice III coexistence line (which tilts forward to cover ice III), we can expect that ice II is denser than ice III (pn > pm). Similarly, from the forward slopes of the liquid coexistence lines with the high-pressure ices II, V, and VI, we can expect that cubes of ice II, ice V, and ice VI would all sink in a glass of water, whereas ice I floats (in accord with the backward tilt of its phase boundary). Many such inferences can be drawn from the slopes of the various phase boundaries in Fig. 7.3, all consistent with the measured phase densities Pphase (in gL 1), namely,... 

The principal features of elemental sulfur in the displayed T, P range are the usual liquid and vapor phases and two solid forms, a-sulfur ( red sulfur, of orthorhombic crystalline form) and /3-sulfur ( yellow sulfur, monoclinic needle-like crystals), both of which are available as common stockroom species. The stable phase ranges for each elemental form are shown by the solid lines in Fig. 7.5. The liquid-vapor coexistence line terminates in a critical point at 1041°C, and will not be discussed further. 

The first of these can be recognized as the ordinary Clapeyron equation for a pure two-phase system (usually written for equimolar phases Af(1) = Af(2) = 1 cf. Sections 7.2.2 and 11.11), and the second is an analogous equation determining the slope of the coexistence curve in the pi-T plane. These equations in turn determine the slope of the coexistence line in the pi-P plane ... 

For example for a two-phase (p = 2) water-ice (c = 1) system we have only one degree of freedom (/ = 1). Thus, we can change the temperature of water and still have coexisting ice, but only at one given pressure. This is the melting point (temperature) A (Fig. A.2) that lies on the coexistence line. If we move to point A we are in the water phase (p = 1) and according to (A.35) we now have two degrees of freedom (/ = 2), temperature and pressure. This triple point is where all three phases coexist (p = 3) it is uniquely defined (/ = 0). This temperature is the official zero of the Celsius scale. 

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