Thermodynamics

Since u and h depend only on temperature for an ideal gas, the specificheats c v and c p also depend, at most, on temperature only. Therefore, at agiven temperature, u, h, c v , and c p of an ideal gas have fixed values regardlessof the specific volume or pressure (Fig. 4–23). Thus, for ideal gases,the partial derivatives in Eqs. 4–19 and 4–20 can be replaced by ordinaryderivatives. Then the differential changes in the internal energy and enthalpyof an ideal gas can be expressed asand(4–23)(4–24)The change in internal energy or enthalpy for an ideal gas during a processfrom state 1 to state 2 is determined by integrating these equations:anddu c v 1T2 dTdh c p 1T2 dT2¢u u 2 u 1 c v 1T2 dT1kJ>kg212¢h h 2 h 1 c p 1T2 dT1kJ>kg21(4–25)(4–26)To carry out these integrations, we need to have relations for c v and c p asfunctions of temperature.At low pressures, all real gases approach ideal-gas behavior, and thereforetheir specific heats depend on temperature only. The specific heats of realgases at low pressures are called ideal-gas specific heats, or zero-pressurespecific heats, and are often denoted c p0 and c v 0 . Accurate analytical expressionsfor ideal-gas specific heats, based on direct measurements or calculationsfrom statistical behavior of molecules, are available and are given asthird-degree polynomials in the appendix (Table A–2c) for several gases. Aplot of c – p0(T) data for some common gases is given in Fig. 4–24.The use of ideal-gas specific heat data is limited to low pressures, but thesedata can also be used at moderately high pressures with reasonable accuracyas long as the gas does not deviate from ideal-gas behavior significantly.The integrations in Eqs. 4–25 and 4–26 are straightforward but rathertime-consuming and thus impractical. To avoid these laborious calculations,u and h data for a number of gases have been tabulated over small temperatureintervals. These tables are obtained by choosing an arbitrary referencepoint and performing the integrations in Eqs. 4–25 and 4–26 by treatingstate 1 as the reference state. In the ideal-gas tables given in the appendix,zero kelvin is chosen as the reference state, and both the enthalpy and theinternal energy are assigned zero values at that state (Fig. 4–25). The choiceof the reference state has no effect on u or h calculations. The u and hdata are given in kJ/kg for air (Table A–17) and usually in kJ/kmol for othergases. The unit kJ/kmol is very convenient in the thermodynamic analysis ofchemical reactions.Some observations can be made from Fig. 4–24. First, the specific heatsof gases with complex molecules (molecules with two or more atoms) arehigher and increase with temperature. Also, the variation of specific heatsChapter 4 | 181u = u(T )h = h(T )c v = c v (T )c p = c p (T )FIGURE 4–23For ideal gases, u, h, c v , and c p varywith temperature only.C p0kJ/kmol · K60 CO 2H 2 O50403020AirO 2H 2Ar, He, Ne, Kr, Xe, Rn1000 2000 3000Temperature, KFIGURE 4–24Ideal-gas constant-pressure specificheats for some gases (see Table A–2cfor c p equations).