Slope of phase boundaries

There are some limitations to the use of DSC in phase studies in addition of its inability to identify phases [5], They are that (1) there are difficulties in locating very steep phase boundaries in heterogeneous systems from DSC data alone because heat capacity depends strongly on the slope of phase boundaries and so the heat capacity jumps are small [10] and (2) the determination of phase boundaries in systems with slow nucleation rate, interfacial transport problems, or inherently slow phase changes may not be possible [11], This is why it is hard to discover a liquid miscibility gap by DSC measurements. 

FIGURE 8-7 The phase diagram for carbon dioxide (not to scale). The liquid can exist only at pressures above 5.1 atm. Note the slope of the boundary between the solid and liquid phases it shows that the freezing point rises as pressure is applied. 

Now, a question arises, Is there a way to quantitatively describe the phase boundaries in terms of P and T The phase rule predicts the existence of the phase boundaries, but does not give any clue on the shape (slope) of the boundaries. To answer the above question, we make use of the fact that at equilibrium the chemical potential of a substance is the same in all phases present. 

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... 

Figure 1. Schematic representation of isobaric and isothermal measnre-ments of a phase transition with a negative slope, dTjdP, for the trans-ition boundary. The same equations apply to transitions with positive slopes of the boundary.

Figure 1 shows the stability boundaries of s-H hydrates helped by CH4. The solid circle, open triangle, open circle, open diamond, open reverse-triangle, and solid reverse-triangle stand for the phase equilibria for the CH4+1,1-DMCH , CH4+MCH , CH4+c/ -1,2-DMCH CH4+MCP, CH4+c-Octane, and CH4+CW-1,4-DMCH s-H hydrate systems, respectively. The three-phase equilibrium data of pure CH4 hydrate are also plotted with solid square . The equilibrium pressures of s-H hydrates decrease from that of pure CH4 hydrate. The slope of stability boundary for each s-H hydrate system (the plot of In p vs. T is almost linear) is somewhat steeper than that of pure CH4 hydrate. As reported in the literatures , these equilibrium curves would cross at high temperature where the s-H hydrate is dissociated and the pure CH4 s-I hydrate is reconstructed. 

Based on their hydrothermal experiments, Aja et al. (1991a, 1991b) and Aja and Rosenberg (1992) have concluded that the I/S clays are not two-phase mixtures, but have several defined compositions. They have inferred these compositions from the solution chemistry and the slopes of reaction boundaries. For example, their data in Fig. 9.12, suggest the most stable I/S solids are Ko.48 io( H)2 Ko.69 U o(OF1)2 at 25°C, and Ko.3i/Oio(UH)2 and Ko,g5/0 o(OH)2 at 125°C. The... 

A change in slope of a cooling curve is an indication that the system is passing across a phase boundary, irrespective of the complexity of the system. Cooling curves are therefore usefiil in mapping out the presence of phase boundaries and in the construction of phase diagrams. 

We now discuss the behavior of phase boundaries. When t critical line w = LOc(u,t) which is the phase boundary between the DS and AS phases, is almost linear with positive slope. Beyond the multicritical point, slope of the line separating the AS and the DC phases rises rather sharply. In a region specified hy Uc < u < Uci (the value of Mq depends on t) we have the coexistence between the adsorbed SAW and the collapsed globule phase. This region is shown in Fig 18 by a dotted line. For u > Uq (t) the line w = w u, t) becomes almost flat. The value of Uq (t) decreases as t is increased and becomes equal to that of Uc at t = t = 0.34115. Att = t the multicritical point becomes a symmetric desorbed and collapsed pentacritical point having four eigenvalues greater than one. 

Fig. 4.1. Pseudo-equilibrium pressure-temperature phase diagram of cerium. The phase boundaries involving 5-Ce and liquid are true equilibrium boundaries. The letters C.P. mean critical point. The question mark for the ala phase boundary indicates that there is considerable doubt about the slope of this boundary, see text for further discussion.

The results of an investigation of the electric-field-induced transition between these two phases is shown in Fig. 9 [56]. At any given temperature the short pitch helix of TGBA is unwound by the field and either a uniform smectic C structure (I) or a modulated one in the form of stripes (II) or parquet (III) appears. The slope of the boundary between the TGBA and SmC phases can be explained by the Clausius-Clapey-ron equation (Eq. (9)) where, in this case, AP=Es(C ) as Ps(TGBA)=0. Note that the transition temperature from TGBA to the isotropic phase is, in fact, field independent due to the much higher enthalpy of that transition. 

Fig. 2. PT diagram for a pure substance that expands on melting (not to scale). For a substance that contracts on melting, eg, water, the fusion curve. A, has a negative slope point / is a triple state point c is the gas—Hquid critical state (—) are phase boundaries representing states of two-phase equiUbrium ...

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. 

The general theoretical treatment of ion-selective membranes assumes a homogeneous membrane phase and thermodynamic equilibrium at the phase boundaries. Obvious deviations from a Nemstian behavior are explained by an additional diffusion potential inside the membrane. However, allowing stationary state conditions in which the thermodynamic equilibrium is not established some hitherto difficult to explain facts (e.g., super-Nemstian slope, dependence of the selectivity of ion-transport upon the availability of co-ions, etc.) can be understood more easily. 

The normal melting, boiling, and triple points give three points on the phase boundary curves. To construct the curves from knowledge of these three points, use the common features of phase diagrams the vapor-liquid and vapor-solid boundaries of phase diagrams slope upward, the liquid-solid line is nearly vertical, and the vapor-solid line begins at P = 0 and P = 0 atm. 

Here Raoultian standard states are used for both the pure metal and the impurity. The slope dxB/dr of the phase boundaries can now be derived by differentiation with respect to temperature. Let/(xB) denote the left-hand side of eq. (4.35) or (4.36) then (see Lupis, Further reading)... 

In this case the equations are greatly simplified and the ratio of the slopes of the two phase boundaries at xA =1 is given by the activity coefficients of B at infinite dilution in the liquid and solid phases [11] ... 

We now look at the phase diagram for water in Figure 5.10. Ice melts at 0 °C if the pressure is p° (as represented by T and Pi respectively on the figure). If the pressure exerted on the ice increases to P2, then the freezing temperature decreases to 7). (The freezing temperature decreases in response to the negative slope of the liquid-solid phase boundary (see the inset to Figure 5.10), which is most unusual virtually all other substances show a positive slope of (lp/dT.)... 

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