Thermodynamics

2s 2 s 1 c v 1T2 dT T R v 2ln v 1(7–31)Chapter 7 | 355The entropy change for a process is obtained by integrating this relationbetween the end states:1A second relation for the entropy change of an ideal gas is obtained in asimilar manner by substituting dh c p dT and v RT/P into Eq. 7–26 andintegrating. The result iss 2 s 1 21dTc p 1T2T R P 2ln P 1(7–32)The specific heats of ideal gases, with the exception of monatomic gases,depend on temperature, and the integrals in Eqs. 7–31 and 7–32 cannot beperformed unless the dependence of c v and c p on temperature is known.Even when the c v (T) and c p (T) functions are available, performing longintegrations every time entropy change is calculated is not practical. Thentwo reasonable choices are left: either perform these integrations by simplyassuming constant specific heats or evaluate those integrals once and tabulatethe results. Both approaches are presented next.Pv = RTdu = C v dTdh = C p dTFIGURE 7–31A broadcast from channel IG.© Vol. 1/PhotoDiscConstant Specific Heats (Approximate Analysis)Assuming constant specific heats for ideal gases is a common approximation,and we used this assumption before on several occasions. It usuallysimplifies the analysis greatly, and the price we pay for this convenience issome loss in accuracy. The magnitude of the error introduced by thisassumption depends on the situation at hand. For example, for monatomicideal gases such as helium, the specific heats are independent of temperature,and therefore the constant-specific-heat assumption introduces noerror. For ideal gases whose specific heats vary almost linearly in the temperaturerange of interest, the possible error is minimized by using specificheat values evaluated at the average temperature (Fig. 7–32). The resultsobtained in this way usually are sufficiently accurate if the temperaturerange is not greater than a few hundred degrees.The entropy-change relations for ideal gases under the constant-specificheatassumption are easily obtained by replacing c v (T) and c p (T) in Eqs.7–31 and 7–32 by c v,avg and c p,avg , respectively, and performing the integrations.We obtainandT 2 v 2s 2 s 1 c v,avg ln R ln 1kJ>kg # K2T 1 v 1T 2 P 2s 2 s 1 c p,avg ln R ln 1kJ>kg # K2T 1 P 1(7–33)(7–34)Entropy changes can also be expressed on a unit-mole basis by multiplyingthese relations by molar mass:c pc p,avgAverage c pT 1Actual c pT avgT 2FIGURE 7–32Under the constant-specific-heatassumption, the specific heat isassumed to be constant at someaverage value.TT 2 v 2s 2 s 1 c v,avg ln RT u ln 1kJ>kmol # K21 v 1(7–35)