Thermodynamics

Explicit relations for differential changes in entropy are obtained by solvingfor ds in Eqs. 7–23 and 7–24:and(7–25)(7–26)The entropy change during a process can be determined by integratingeither of these equations between the initial and the final states. To performthese integrations, however, we must know the relationship between du ordh and the temperature (such as du c v dT and dh c p dT for ideal gases)as well as the equation of state for the substance (such as the ideal-gasequation of state Pv RT). For substances for which such relations exist,the integration of Eq. 7–25 or 7–26 is straightforward. For other substances,we have to rely on tabulated data.The T ds relations for nonsimple systems, that is, systems that involvemore than one mode of quasi-equilibrium work, can be obtained in a similarmanner by including all the relevant quasi-equilibrium work modes.7–8 ■ ENTROPY CHANGE OF LIQUIDS AND SOLIDSRecall that liquids and solids can be approximated as incompressible substancessince their specific volumes remain nearly constant during a process.Thus, dv 0 for liquids and solids, and Eq. 7–25 for this case reduces to(7–27)since c p c v c and du c dT for incompressible substances. Then theentropy change during a process is determined by integration to be2dTLiquids, solids: s 2 s 1 c 1T2(7–28)T c T 2avg ln 1kJ>kg # K2T 11ds du T PdvTds dh T vdPTds du T cdTTwhere c avg is the average specific heat of the substance over the given temperatureinterval. Note that the entropy change of a truly incompressiblesubstance depends on temperature only and is independent of pressure.Equation 7–28 can be used to determine the entropy changes of solids andliquids with reasonable accuracy. However, for liquids that expand considerablywith temperature, it may be necessary to consider the effects of volumechange in calculations. This is especially the case when the temperaturechange is large.A relation for isentropic processes of liquids and solids is obtained by settingthe entropy change relation above equal to zero. It givesChapter 7 | 351INTERACTIVETUTORIALSEE TUTORIAL CH. 7, SEC. 8 ON THE DVD.T 2Isentropic: s 2 s 1 c avg ln 0S T (7–29)T 2 T 11That is, the temperature of a truly incompressible substance remains constantduring an isentropic process. Therefore, the isentropic process of anincompressible substance is also isothermal. This behavior is closelyapproximated by liquids and solids.