## T K

Fig. 3.6 Molar entropies of selected elements at 1 atm pressure.

### 3.7.4 Free Energy of phase transitions

The temperature effect on the Gibbs free energy must be different for liquids and solids in order that there be a unique melting temperature at which both phases coexist at equilibrium. The source of this behavior lies in the entropies of the two states, as can be demonstrated by the fundamental differential dg = -sdT + vdp (Eq (1.18a)). At constant pressure, the second term vanishes, and dividing what remains by dT gives:

This equation (in integrated form) is shown schematically in the left-hand graph of Fig 3.7 for a substance with no solid-solid phase transitions. Because entropy is a measure of order (or disorder) of a system, the entropy of the liquid is always greater than that of the solid at the same temperature. Since the entropy is always positive, Eq (3.26) requires that the slope of the plot of gL Vs T be steeper than the corresponding plot for the solid. At the melting point the equilibrium criterion of minimum free energy (Eq (1.20a)) is satisfied and the two phases coexist. For T < TM, the free energy of the solid is lower than that of the liquid, so minimization of the system's free energy causes the liquid to disappear. Conversely, for T > TM, the solid phase is unstable with respect to the liquid. The analysis of binary (two-component) phase diagrams discussed in Chap. 8 requires as input the free energy difference AgM = gL - gs for each species at temperatures other than their melting points.

Combining the enthalpy and entropy of melting given by Eqs (3.21) and (3.24), with AsM(TM) in Eq (3.24) replaced by AhM(TM)/TM, the free energies of melting is:

This equation assumes a temperature-independent heat-capacity difference between the two phases. The first term on the right hand side of Eq (3.27a) is the lowest-order approximation: 