Thermodynamics

170 | ThermodynamicsEXAMPLE 4–3Isothermal Compression of an Ideal GasA piston–cylinder device initially contains 0.4 m 3 of air at 100 kPa and 80°C.The air is now compressed to 0.1 m 3 in such a way that the temperatureinside the cylinder remains constant. Determine the work done during thisprocess.Solution Air in a piston–cylinder device is compressed isothermally. Theboundary work done is to be determined.Analysis A sketch of the system and the P-V diagram of the process areshown in Fig. 4–8.Assumptions 1 The compression process is quasi-equilibrium. 2 At specifiedconditions, air can be considered to be an ideal gas since it is at a high temperatureand low pressure relative to its critical-point values.Analysis For an ideal gas at constant temperature T 0 ,where C is a constant. Substituting this into Eq. 4–2, we have22W b P dV 11PV mRT 0 CorP C VCV dV C 21dVV C ln V 2(4–7)In Eq. 4–7, P 1 V 1 can be replaced by P 2 V 2 or mRT 0 . Also, V 2 /V 1 can bereplaced by P 1 /P 2 for this case since P 1 V 1 P 2 V 2 .Substituting the numerical values into Eq. 4–7 yieldsW b 1100 kPa2 10.4 m 3 2aln 0.10.4 ba 1 kJ1 kPa # m3 b 55.5 kJV 2 PV 1 V 1 ln1Discussion The negative sign indicates that this work is done on the system(a work input), which is always the case for compression processes.V 1P2T 0 = 80°C = const.AIRV 1 = 0.4 m 3P 1 = 100 kPaT 0 = 80°C = const.10.1FIGURE 4–8Schematic and P-V diagram for Example 4–3.0.4V, m 3