Thermodynamics

350 | Thermodynamicsis highly irreversible since this generates entropy, and it can cause considerabledamage. A person who gets up in anger is bound to sit down at a loss.Hopefully, someday we will be able to come up with some procedures toquantify entropy generated during nontechnical activities, and maybe evenpinpoint its primary sources and magnitude.INTERACTIVETUTORIALSEE TUTORIAL CH. 7, SEC. 7 ON THE DVD.7–7 ■ THE T ds RELATIONSRecall that the quantity (dQ/T) int rev corresponds to a differential change inthe property entropy. The entropy change for a process, then, can be evaluatedby integrating dQ/T along some imaginary internally reversible pathbetween the actual end states. For isothermal internally reversible processes,this integration is straightforward. But when the temperature varies duringthe process, we have to have a relation between dQ and T to perform thisintegration. Finding such relations is what we intend to do in this section.The differential form of the conservation of energy equation for a closedstationary system (a fixed mass) containing a simple compressible substancecan be expressed for an internally reversible process asdQ int rev dW int rev,out dU(7–21)ButdQ int rev TdSdW int rev,out PdVThus,TdS dU PdV1kJ2(7–22)orTds du Pdv1kJ>kg2(7–23)This equation is known as the first T ds, or Gibbs, equation. Notice that theonly type of work interaction a simple compressible system may involve asit undergoes an internally reversible process is the boundary work.The second T ds equation is obtained by eliminating du from Eq. 7–23 byusing the definition of enthalpy (h u Pv):h u Pv1Eq. 7–232¡¡dh du Pdv vdPf Tds dh vdPTds du Pdv(7–24)ClosedsystemFIGURE 7–28T ds = du + P dvT ds = dh – v dPCVThe T ds relations are valid for bothreversible and irreversible processesand for both closed and open systems.Equations 7–23 and 7–24 are extremely valuable since they relate entropychanges of a system to the changes in other properties. Unlike Eq. 7–4, theyare property relations and therefore are independent of the type of theprocesses.These T ds relations are developed with an internally reversible process inmind since the entropy change between two states must be evaluated alonga reversible path. However, the results obtained are valid for both reversibleand irreversible processes since entropy is a property and the change in aproperty between two states is independent of the type of process the systemundergoes. Equations 7–23 and 7–24 are relations between the propertiesof a unit mass of a simple compressible system as it undergoes a changeof state, and they are applicable whether the change occurs in a closed or anopen system (Fig. 7–28).