Thermodynamics

672 | Thermodynamicsfinally along the T 2 constant line to P 2 , as we did for the enthalpy. Thisapproach is not suitable for entropy-change calculations, however, since itinvolves the value of entropy at zero pressure, which is infinity. We canavoid this difficulty by choosing a different (but more complex) pathbetween the two states, as shown in Fig. 12–17. Then the entropy changecan be expressed asTT 2T 1Actualprocess path22*P 2P 1Alternativeprocess path11*P 0a*b*FIGURE 12–17An alternative process path to evaluatethe entropy changes of real gasesduring process 1-2.ss 2 s 1 1s 2 s b*2 1s b* s 2*2 1s 2* s 1*2 1s 1* s a*2 1s a* s 1 2 (12–61)States 1 and 1* are identical (T 1 T 1* and P 1 P 1*) and so are states 2and 2*. The gas is assumed to behave as an ideal gas at the imaginary states1* and 2* as well as at the states between the two. Therefore, the entropychange during process 1*-2* can be determined from the entropy-changerelations for ideal gases. The calculation of entropy change between anactual state and the corresponding imaginary ideal-gas state is moreinvolved, however, and requires the use of generalized entropy departurecharts, as explained below.Consider a gas at a pressure P and temperature T. To determine how muchdifferent the entropy of this gas would be if it were an ideal gas at the sametemperature and pressure, we consider an isothermal process from the actualstate P, T to zero (or close to zero) pressure and back to the imaginary idealgasstate P*, T* (denoted by superscript *), as shown in Fig. 12–17. Theentropy change during this isothermal process can be expressed as1s P s P*2 T 1s P s 0*2 T 1s 0* s P*2 Twhere v ZRT/P and v* v ideal RT/P. Performing the differentiationsand rearranging, we obtainP11 Z2R1s P s P*2 T cPBy substituting T T cr T R and P P cr P R and rearranging, the entropydeparture can be expressed in a nondimensionalized form asZ s 1 s * s 2 T,PR u P00 P R0a 0v00T b dP P a 0v*0T b dPPP RTP a 0Zr0T b d dPPc Z 1 T R a 0Z0T RbP Rd d 1ln P R 2(12–62)The difference (s – * s – ) T,P is called the entropy departure and Z s is calledthe entropy departure factor. The integral in the above equation can beperformed by using data from the compressibility charts. The values of Z sare presented in graphical form as a function of P R and T R in Fig. A–30.This graph is called the generalized entropy departure chart, and it isused to determine the deviation of the entropy of a gas at a given P and Tfrom the entropy of an ideal gas at the same P and T. Replacing s* by s idealfor clarity, we can rewrite Eq. 12–61 for the entropy change of a gas duringa process 1-2 ass2 s 1 1 s 2 s 12 ideal R u 1Z s2 Z s12(12–63)