Thermodynamics

v ZRT/P and simplifying Eq. 12–56, we can write the enthalpy departureat any temperature T and pressure P as1h* h2 T RT 2 PThe above equation can be generalized by expressing it in terms of the reducedcoordinates, using T T cr T R and P P cr P R . After some manipulations, theenthalpy departure can be expressed in a nondimensionalized form as0a 0Z0T b dPP PChapter 12 | 671Z h 1h* h2 TR u T cr T 2 R P R(12–57)where Z h is called the enthalpy departure factor. The integral in the aboveequation can be performed graphically or numerically by employing data fromthe compressibility charts for various values of P R and T R . The values of Z h arepresented in graphical form as a function of P R and T R in Fig. A–29. Thisgraph is called the generalized enthalpy departure chart, and it is used todetermine the deviation of the enthalpy of a gas at a given P and T from theenthalpy of an ideal gas at the same T. By replacing h* by h ideal for clarity, Eq.12–53 for the enthalpy change of a gas during a process 1-2 can be rewritten as0a 0Z0T RbP Rd 1ln P R 2h 2 h 1 1h 2 h 1 2 ideal R u T cr 1Z h2 Z h12(12–58)orh 2 h 1 1h 2 h 1 2 ideal RT cr 1Z h2 Z h12(12–59)where the values of Z h are determined from the generalized enthalpy departurechart and (h – 2 h– 1 ) ideal is determined from the ideal-gas tables. Noticethat the last terms on the right-hand side are zero for an ideal gas.Internal Energy Changes of Real GasesThe internal energy change of a real gas is determined by relating it to theenthalpy change through the definition h – u – Pv – u – ZR u T:u2 u 1 1h 2 h 1 2 R u 1Z 2 T 2 Z 1 T 1 2(12–60)Entropy Changes of Real GasesThe entropy change of a real gas is determined by following an approachsimilar to that used above for the enthalpy change. There is some differencein derivation, however, owing to the dependence of the ideal-gas entropy onpressure as well as the temperature.The general relation for ds was expressed as (Eq. 12–41)s 2 s 1 T 2T 1c pT dT P 2where P 1 , T 1 and P 2 , T 2 are the pressures and temperatures of the gas at theinitial and the final states, respectively. The thought that comes to mind atthis point is to perform the integrations in the previous equation first along aT 1 constant line to zero pressure, then along the P 0 line to T 2 , andP 1a 0v0T b dPP