The slope of phase boundaries

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence 

The variation of the phase transition temperature with pressure can be calculated from the knowledge of the volume and enthalpy change of the transition. Most often both the entropy and volume changes are positive and the transition temperature increases with pressure. In other cases, notably melting of ice, the density of the liquid phase is larger than of the solid, and the transition temperature decreases 

Thus the pT slope is for a second-order transition is given as 

A phase diagram displays the regions of the potential space where the various phases of the system are stable. The potential space is given by the variables of the 

For a single-component systemp and T can be varied independently when only one phase is present. When two phases are present in equilibrium, pressure and temperature are not independent variables. At a certain pressure there is only one temperature at which the two phases coexist, e.g. the standard melting temperature of water. Hence at a chosen pressure, the temperature is given implicitly. A point 

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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. 

An equation similar to the Clausius-Clapeyron equation can be obtained for the slope of the phase boundary... 

The slope of the phase boundary curve dpidt can be estimated from the Ttr - Pit plot obtained by using a PVT or high-pressure differential thermal analysis (DTA) method [118,119]. The volume-dependent entropy (correction for the volume change) ASy, the transition entropy (AStr)p under ordinary pressure, and the constant-volume entropy (AStr)v obtained therefrom are arranged in this order in Tables 2,3, and 4. 

The relation between the slope of the phase boundary curve and properties of the substance is discussed in Section 9.2... 

The second confirmation comes from the results of direct conversion of calcite to aragonite. As summarized by Crawford et al. (1972) the 2°C intercept of the phase boundary established in this way is 3110 atmospheres (corresponding to a AG of 210 cal and a solubility ratio of l.i -7). As direct conversion has been accomplished at temperatures as low as 60 C and as the slope of the phase boundary is well established, the extrapolation to 2 C is firmly based. 

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)... 

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. 

Thus the effect of applying pressure is expected to be two-fold. First, the transition temperature is changed hyIXPlaK. Therefore the slope of the phase transition boundary 5T,/8P is 2 XjaK, and Equation (2) can be re-written ... 

This is the Clausius-Clapeyron equation. From it, the slope (dpjdT) of the phase boundary and the observed volume difference between the two phases, the entropy of the transition and hence its latent heat can be found. These quantities, evaluated at the various triple points, are shown for ordinary water in table 3.1. 

The precipitation boundary may or may not coincide with the solubility curve since its position in the diagram depends on the time and method of detection of the onset of precipitation. In effect, it establishes the metastable zone for the given system. If a stable precipitate is formed at low levels of supersaturation the precipitation boundary and solubility curve may be assumed to be virtually coincident. In cases where they do not coincide, the composition of the critical nuclei may be determined from the slopes of the precipitation boundary while the compositions of the corresponding bulk equilibrium solid phases may be obtained from the solubility curve. Hence, comparison of the two curves can yield information as to whether the composition of both nuclei and precipitate is the same or if the bulk solid phase is formed by a solid-state transformation from a metastable precursor. A useful account of the significance of the zones on a precipitation diagram has also been given by Nielsen (1979). 

Here s a general derivation of the slope of a phase boundary, illustrated for the equilibrium between gas (G) and liquid (L) (see Figure 14.7). We want... 

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