Thermodynamics

do the same as the pressure is maintained constant is the specific heat atconstant pressure c p . This is illustrated in Fig. 4–19. The specific heatat constant pressure c p is always greater than c v because at constant pressurethe system is allowed to expand and the energy for this expansion workmust also be supplied to the system.Now we attempt to express the specific heats in terms of other thermodynamicproperties. First, consider a fixed mass in a stationary closed systemundergoing a constant-volume process (and thus no expansion or compressionwork is involved). The conservation of energy principle e in e out e systemfor this process can be expressed in the differential form asThe left-hand side of this equation represents the net amount of energytransferred to the system. From the definition of c v , this energy must beequal to c v dT, where dT is the differential change in temperature. Thus,orde in de out duc v d˛T duat constant volumec v a 0u0T b v(4–19)Similarly, an expression for the specific heat at constant pressure c p can beobtained by considering a constant-pressure expansion or compressionprocess. It yieldsc p a 0h(4–20)0T b pEquations 4–19 and 4–20 are the defining equations for c v and c p , and theirinterpretation is given in Fig. 4–20.Note that c v and c p are expressed in terms of other properties; thus, theymust be properties themselves. Like any other property, the specific heats ofa substance depend on the state that, in general, is specified by two independent,intensive properties. That is, the energy required to raise the temperatureof a substance by one degree is different at different temperatures andpressures (Fig. 4–21). But this difference is usually not very large.A few observations can be made from Eqs. 4–19 and 4–20. First, theseequations are property relations and as such are independent of the type ofprocesses. They are valid for any substance undergoing any process. Theonly relevance c v has to a constant-volume process is that c v happens to bethe energy transferred to a system during a constant-volume process per unitmass per unit degree rise in temperature. This is how the values of c v aredetermined. This is also how the name specific heat at constant volumeoriginated. Likewise, the energy transferred to a system per unit mass perunit temperature rise during a constant-pressure process happens to be equalto c p . This is how the values of c p can be determined and also explains theorigin of the name specific heat at constant pressure.Another observation that can be made from Eqs. 4–19 and 4–20 is that c vis related to the changes in internal energy and c p to the changes inenthalpy. In fact, it would be more proper to define c v as the change in theinternal energy of a substance per unit change in temperature at constantV = constantm = 1 kg∆T = 1°CkJc v = 3.12kg.°C3.12 kJChapter 4 | 179P = constantm = 1 kg∆ T = 1°CkJc p = 5.19kg.°C5.19 kJFIGURE 4–19Constant-volume and constantpressurespecific heats c v and c p(values given are for helium gas).∂u∂T v= the change in internal energywith temperature atconstant volumec v =((∂h∂T p= the change in enthalpy withtemperature at constantpressurec p =((FIGURE 4–20Formal definitions of c v and c p .(2)(1)