Thermodynamics

356 | ThermodynamicsandT 2 P 2s 2 s 1 c p,avg ln R u ln 1kJ>kmol # K2T 1 P 1(7–36)Variable Specific Heats (Exact Analysis)When the temperature change during a process is large and the specificheats of the ideal gas vary nonlinearly within the temperature range, theassumption of constant specific heats may lead to considerable errors inentropy-change calculations. For those cases, the variation of specific heatswith temperature should be properly accounted for by utilizing accuraterelations for the specific heats as a function of temperature. The entropychange during a process is then determined by substituting these c v (T) orc p (T) relations into Eq. 7–31 or 7–32 and performing the integrations.Instead of performing these laborious integrals each time we have a newprocess, it is convenient to perform these integrals once and tabulate theresults. For this purpose, we choose absolute zero as the reference temperatureand define a function s° asT, K.300310320 . .(Table A-17)s°, kJ/kg • K.1.702031.734981.76690.FIGURE 7–33The entropy of an ideal gas dependson both T and P. The function srepresents only the temperaturedependentpart of entropy.(7–37)Obviously, s° is a function of temperature alone, and its value is zero atabsolute zero temperature. The values of s° are calculated at various temperatures,and the results are tabulated in the appendix as a function of temperaturefor air. Given this definition, the integral in Eq. 7–32 becomeswhere s° 2 is the value of s° at T 2 and s° 1 is the value at T 1 . Thus,P 2s 2 s 1 s° 2 s° 1 R ln 1kJ>kg # K2P 1It can also be expressed on a unit-mole basis as 21s° T0dTc p 1T2TdTc p 1T2T s° 2 s° 1P 2s 2 s 1 s° 2 s° 1 R u ln 1kJ>kmol # K2P 1(7–38)(7–39)(7–40)Note that unlike internal energy and enthalpy, the entropy of an ideal gasvaries with specific volume or pressure as well as the temperature. Therefore,entropy cannot be tabulated as a function of temperature alone. The s°values in the tables account for the temperature dependence of entropy (Fig.7–33). The variation of entropy with pressure is accounted for by the lastterm in Eq. 7–39. Another relation for entropy change can be developedbased on Eq. 7–31, but this would require the definition of another functionand tabulation of its values, which is not practical.