Phase boundary, slope

The phase diagram for water is unusual. The solid/liquid phase boundary slopes to the left with increasing pressure because the melting point of water decreases with increasing pressure. Note that the normal melting point of water is lower than its triple point. The diagram is not drawn to a uniform scale. 

We shall see in the following section that for water, the solid-liquid phase boundary slopes in the opposite direction, and then only the second of these... 

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 ...

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.)... 

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,... 

From the slopes of the phase boundaries, one can judge [using the Clapeyron mnemonic (7.32)] that pSoiid > Pi, Pii (i.e., a high-pressure ice cube of frozen helium will sink in either He-I or He-II) and that pn> Pi (i.e., the low-T He-II superfluid floats on the high-T He-I normal fluid). One can also judge from its placement at lower T that He-II is more highly ordered than He-I (5n < Si), despite its superfluid proclivities. 

Despite the complexities introduced by metastable solid and liquid phases, topological features of the phase diagram can be thermodynamically interpreted in a standard manner from Clapeyron-type equations. Thus, from the forward slopes of a-fi, /3-liquid (stable), and a-liquid (metastable) phase boundaries, we can infer from (7.32) that... 

Similar calculations were also performed in the strong segregation limit. In this case, the two-phase and disordered homogeneous regions were found to be smaller, and the phase boundaries were more vertical (see Fig. 6.49) (Shi and Noolandi 1995). This phase diagram was interpreted on the basis of interfacial curvature. If the diblocks are completely segregated, the phase boundaries are determined by the total composition / (=

phase boundaries are parallel to this line (dashed line sloping to the left in Fig. 6.49) ( one-component approximation ). This explains the approximately parallel... 

Figures 7.11a,b are arbitrary examples of the depths of hydrate phase stability in permafrost and in oceans, respectively. In each figure the dashed lines represent the geothermal gradients as a function of depth. The slopes of the dashed lines are discontinuous both at the base of the permafrost and the water-sediment interface, where changes in thermal conductivity cause new thermal gradients. The solid lines were drawn from the methane hydrate P-T phase equilibrium data, with the pressure converted to depth assuming hydrostatic conditions in both the water and sediment. In each diagram the intersections of the solid (phase boundary) and dashed (geothermal gradient) lines provide the lower depth boundary of the hydrate stability fields.

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. 

Clearly, the phase boundaries lie on top of each other. A closer look reveals a slight decrease in the slope r0 of Eq. (26) according to Ca2+>Sr2+>-Ba2+, which indicates that the larger the bivalent earth alkaline cation is, the smaller is the stoichiometric amount of M2+ necessary to precipitate NaPA. Although based on a different method, the present results suggest a comparison with data from Pochard et al. [73]. In doing so, we have to keep in mind that our own r0 values are of fair accuracy at best because they were evaluated from slopes based on a few data points only. Still, this trend in rQ is the opposite to the observation of Pochard et al. [73]. They found a decrease of the amount of M2+ per COO function at the precipitation threshold, if Ca2+... 

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