Coexistence slope

An unexpected conclusion from this fonuulation, shown in various degrees of generality in 1970-71, is that for systems that lack tlie synunetry of simple lattice models the slope of the diameter of the coexistence curve... 

Povodyrev et aJ [30] have applied crossover theory to the Flory equation ( section A2.5.4.1) for polymer solutions for various values of N, the number of monomer units in the polymer chain, obtaining the coexistence curve and values of the coefficient p jj-from the slope of that curve. Figure A2.5.27 shows their comparison between classical and crossover values of p j-j for A = 1, which is of course just the simple mixture. As seen in this figure, the crossover to classical behaviour is not complete until far below the critical temperature. 

A one-component system (C = 1) has two independent state variabies (T and p). At the tripie point three phases (soiid, iiquid, vapour) coexist at equiiibrium, so P = 3. From the phase ruie f = 0, so that at the tripie point, T and p are fixed - neither is free but both are uniqueiy determined. If T is free but p depends on T (a sloping line on the phase diagram) then f = 1 and P = 2 that is, two phases, solid and liquid, say, co-exist at equilibrium. If both p and T are free (an area on the phase diagram) F = 2 and P = 1 only one phase exists at equilibrium (see Fig. A1.18). 

Therefore, the calculated coordinates of the triple point for the coexistence of MO, MS and A/SO4 are logPso2 = +2 and logpo = - 12 and the slope of the MO/MSO4 boundary is - y. The straight line from point B having slope — y gives the boundary line (i) between the stability areas of MO and A/SO4. This completes the construction of the phase stability diagram forM-S-O at 1000 K. 

Let us first consider the three-phase equilibrium ( -clathrate-gas, for which the values of P and x = 3/( +3) were determined at 25°C. When the temperature is raised the argon content in the clathrate diminishes according to Eq. 27, while the pressure can be calculated from Eq. 38 by taking yA values following from Eq. 27 and the same force constants as used in the calculation of Table III. It is seen that the experimental results at 60°C and 120°C fall on the line so calculated. At a certain temperature and pressure, solid Qa will also be able to coexist with a solution of argon in liquid hydroquinone at this point (R) the three-phase line -clathrate-gas is intersected by the three-phase line -liquid-gas. At the quadruple point R solid a-hydroquinone (Qa), a hydroquinone-rich liquid (L), the clathrate (C), and a gas phase are in equilibrium the composition of the latter lies outside the part of the F-x projection drawn in Fig. 3. The slope of the three-phase line AR must be very steep, because of the low solubility of argon in liquid hydroquinone. 

Since industrial separation processes operate in the Li L2 region, it is important to determine how the Margules parameters affect the shape of the coexistence curve and the slope of the tie lines. For any liquid-liquid region to exist, at least one of the binary Margules constants must be greater than 2RT(on y positive values are considered here) this is a consequence of the... 

If we now recall the phase rule, it is evident that, at the P-T conditions represented by point D in figure 2.5, slight variations in the P or T values will not induce any change in the structural state of the phase (there are one phase and one component the variance is 2). At point A in the same figure, any change in one of the two intensive variables will induce a phase transition. To maintain the coexistence of kyanite and andalusite, a dP increment consistent with the slope of the univariant curve (there are two phases and one component the variance... 

Figure 5.23 Pressure composition isotherms for critical temperature 7. The construction of the hydrogen absorption in atypical metal (left). The van t Hoff plot is shown on the right. The slope of solid solution (a-phase), the hydride phase the line is equal to the enthalpy of formation (p-phase) and the region ofthe coexistence ofthe divided by the gas constant and the intercept with two phases. The coexistence region is the axis is equal to the entropy of formation...

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).

Figure 7.2 Phase coexistence conditions (circles), showing phase of lowest chemical potential as a function of (a) T, (b) P. The pgsis (heavy dots), p qilid (dashes), and pso id (solid line) curves are plotted with slopes (7.21) [or (7.23)] consistent with the expected order (7.22) [or (7.24)] for the derivative slope (dp/dT)P [or (dp/dP)T].

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). 

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 general requirement for matching chemical potentials thereby determines how the derivatives (slopes) of Giiq(x) and Gsol(x) must be related in order for these phases to coexist, giving rise to a hatched coexistence region. 

Suppose that represents the slope of the coexistence curve in a conventional PT phase diagram, so that, by definition,... 

With the explicit formula (12.75) for the y coefficients, (12.78)-(12.80) become convenient formulas for the slopes of coexistence"6oundaries 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... 

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 ... 

If the solutions contain ion pairs and free ions, the slope should be —0.5 (29). If triple ions coexist with these. ionic species, the slope value shifts toward +0.5. From this, it is suggested that triple ions contribute to the polymerization in these particular solvent systems. 

The solid-liquid boundary, the almost vertical line in the illustration, shows the pressures and temperatures at which solid and liquid coexist in equilibrium. In other words, it shows how the melting point varies with pressure. To show the slopes more clearly, the plots are not to scale, but despite this, the steepness of the lines show that even large changes in pressure result in quite small variations in melting point. 

A remarkable shape is calculated with Eq. (9.35) for (3 > 4. A region is obtained where the 0-versus-P curve has a negative slope (dotted curve in Fig. 9.7). This is physically nonsense The coverage is supposed to decrease with increasing pressure and for one pressure there are three possible values of 6. In reality this is a region of two-phase equilibrium. Single adsorbed molecules and clusters of adsorbed molecules coexist on the surface. The situation is reminiscent of the three-dimensional van der Waals equation of state which can be used to describe condensation. 

In principle, knowledge of a single point on the coexistence curve permits the entire curve to be traced without further calculation of free energies. The key result needed follows simply from Eq. (68), applied to each field and X2 in turn. The slope of the coexistence curve A/ rrra( A, ) = 0 is as follows ... 

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