T

3.7.3 Entropy

In Sect. 2.5.2, it was shown that except at high temperatures, Cp and Cv are approximately equal for most solids and liquids. The chief reason for this is the small specific volumes of condensed phases. If the difference between the two heat capacities is ignored, du and dh in Eqs (3.7a) and (3.8a) are about equal as well. In addition, the second terms on the right hand sides of these equations can be neglected because v is small. Hence, both of these equations are effectively only the single equation ds = dh/T = CpdT/T. Assuming temperature-independent heat capacities and integrating between a reference temperature Tref and temperature T the analogs of Eqs (3.19) - (3.21) for the entropies of solids and liquids are:

According to the 3rd law of thermodynamics, if Tref is chosen to be zero Kelvin, then the reference entropy is zero. However, this assignment of the reference state is not necessary in order to use Eq (3.19) in thermodynamic calculations. For entropy calculations to be consistent with the enthalpy analysis, the same reference temperature must be chosen for each.

By analogy to Eq (3.18), the entropy of fusion at the normal melting point is equal to AhM(TM)/TM. Figure 3.6 shows the entropies of a number of common elements.

The entropy of the diamond form of carbon shown in Fig. 3.6 is lower than that of graphite. This difference favors graphite as the stable form of carbon, but the relative stability of the two forms are reversed at ultrahigh pressures. The graphite-diamond conversion is analyzed in Sect. 5.8.

Discontinuities in the curves for Au, Al, and Cu represent the entropies of fusion of these elements. The curve for iron contains four jumps, the last of which represents melting. The other three at lower temperatures reflect the thermal effects of crystal structure transitions. Each is accompanied by an enthalpy change and an entropy change. These solid-solid transition effects are included in Eqs (3.20) - (3.25) in the same way as the solid-liquid phase change is treated.

The curve for gaseous O2 is also shown in Fig. 3.6. The entropy of the gas is several times larger than those of the condensed phases, reflecting the more highly ordered states of the latter. Changing pressure from the 1-atm value to which Fig. 3.6 applies would have little effect on the entropy curves for the condensed phase. However, such a sL(T) = sref,s + CPs ln(TM / Tref ) + AsM(TM) + CPL ln(T/ TM )

sL(T) = sref,s + CPs ln(TM / Tref ) + AsM(TM) + CPL ln(T/ TM )

change would significantly affect the entropy of O2. Equation (3.10) shows that increasing the O2 pressure from 1 atm to, say, 10 atm would lower the entropy by Rln10 ~ 20 J/mole-K. 