Entropy metastable

Many substances exist in two or more solid allotropic fomis. At 0 K, the themiodynamically stable fomi is of course the one of lowest energy, but in many cases it is possible to make themiodynamic measurements on another (metastable) fomi down to very low temperatures. Using the measured entropy of transition at equilibrium, the measured heat capacities of both fomis and equation (A2.1.73) to extrapolate to 0 K, one can obtain the entropy of transition at 0 K. Within experimental... 

Figure 4.7 Heat capacity and phase transitions in phosphine. The same entropy at point (a) is obtained by going the stable (lower) route or by going the metastable (upper) route. Point (b) is the normal boiling temperature.

Cp is the specific heat at constant pressure, k is the compressibility at constant temperature. The conversion process of a second-order phase transition can extend over a certain temperature range. If it is linked with a change of the structure (which usually is the case), this is a continuous structural change. There is no hysteresis and no metastable phases occur. A transformation that almost proceeds in a second-order manner (very small discontinuity of volume or entropy) is sometimes called weakly first order . 

Figure 5.3 Entropy of liquid and crystalline aluminium in stable, metastable and unstable temperature regions [12]. The temperatures where the entropy of liquid and crystalline aluminium are equal are denoted Tf and 7 jm crySt, respectively.

This law is particularly useful as it allows the total entropy of a substance to be obtained if sufficiently low-temperature enthalpy or heat-capacity measurements are available. However, questions remain as to its validity when considering metastable crystalline forms and the law, as stated, would not apply to defect-stabilised structures and amorphous phases. 

In addition to the melting point of the P phase and the a/P allotropic transfonnation temperature in Fig. 6.1(b), there is a fluther intersection between the Gibbs energy of a and liquid phases. This corresponds to the metastable melting point of the a phase. A linear model will then dictate that the entropy of melting for a is defined by the entropies of melting and transformation at the two other critical points (Ardell 1963),... 

Figure 6.4. (a) Variation of the entropy of fusion with melting point for different crystal structures (from Saunders el at. 1988) and (b) schematic illustration of the possibility of a change in value and sign for the entropy of transformation if the metastable structure has a low melting point (from Miodownik 1992). 

Table 6.5. Maximum values of the magnetic enthalpy and entropy for various allotropes of Fe, Co and Ni based on data and methodology drawn from Mio wnik (197 and additional data from de Fontaine el al. (1995). Structures in brackets correspond to metastable forms that have not been observed in the TP diagram of the element...

Depending on the data available, Eqs (6.17)-(6.23) reproduce experimental pressure effects with considerable accuracy in many cases. In particular, Eq. (6.18) can be used to confirm entropy data derived using more conventional techniques and can also provide data for metastable allotropes. Ti again provides a leading example, as pressure experiments revealed that the u -phase, previously only detected as a metastable product on quenching certain Ti alloys, could be stabilised under pressure (Fig. 6.14). Extrapolation of the P/w transus line yields the metastable allotropic transformation temperature at which the / -phase would transform to w in the absence of the a-phase, while die slope of the transus lines can be used to extract a value for the relevant entropy via Eq. (6.18). 

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