Thermodynamics

Chapter 4 | 167This integral can be evaluated only if we know the functional relationship Pbetween P and V during the process. That is, P f (V) should be 1available. Note that P f (V) is simply the equation of the process path onProcess patha P-V diagram.The quasi-equilibrium expansion process described is shown on a P-Vdiagram in Fig. 4–3. On this diagram, the differential area dA is equal to2PdV, which is the differential work. The total area A under the processdA = P dVcurve 1–2 is obtained by adding these differential areas:V 12Area A dA (4–3) 2dV V 2 VP dV1A comparison of this equation with Eq. 4–2 reveals that the area underthe process curve on a P-V diagram is equal, in magnitude, to the workdone during a quasi-equilibrium expansion or compression process of aclosed system. (On the P-v diagram, it represents the boundary work doneper unit mass.)A gas can follow several different paths as it expands from state 1 to state2. In general, each path will have a different area underneath it, and sincethis area represents the magnitude of the work, the work done will be differentfor each process (Fig. 4–4). This is expected, since work is a path function(i.e., it depends on the path followed as well as the end states). If workwere not a path function, no cyclic devices (car engines, power plants)could operate as work-producing devices. The work produced by thesedevices during one part of the cycle would have to be consumed duringanother part, and there would be no net work output. The cycle shown inFig. 4–5 produces a net work output because the work done by the systemduring the expansion process (area under path A) is greater than the workdone on the system during the compression part of the cycle (area underpath B), and the difference between these two is the net work done duringthe cycle (the colored area).If the relationship between P and V during an expansion or a compressionprocess is given in terms of experimental data instead of in a functionalform, obviously we cannot perform the integration analytically. But we canalways plot the P-V diagram of the process, using these data points, and calculatethe area underneath graphically to determine the work done.Strictly speaking, the pressure P in Eq. 4–2 is the pressure at the innersurface of the piston. It becomes equal to the pressure of the gas in thecylinder only if the process is quasi-equilibrium and thus the entire gas inthe cylinder is at the same pressure at any given time. Equation 4–2 canalso be used for nonquasi-equilibrium processes provided that the pressureat the inner face of the piston is used for P. (Besides, we cannot speak ofthe pressure of a system during a nonquasi-equilibrium process since propertiesare defined for equilibrium states only.) Therefore, we can generalizethe boundary work relation by expressing it as2W b P i dV1(4–4)where P i is the pressure at the inner face of the piston.Note that work is a mechanism for energy interaction between a systemand its surroundings, and W b represents the amount of energy transferredfrom the system during an expansion process (or to the system during a1PFIGURE 4–3The area under the process curve on aP-V diagram represents the boundarywork.P1V 1CW A = 10 kJW B = 8 kJW C = 5 kJBA2V 2FIGURE 4–4The boundary work done during aprocess depends on the path followedas well as the end states.P2BW netV 2 V 1FIGURE 4–5The net work done during a cycle isthe difference between the work doneby the system and the work done onthe system.A1VV