process, Cy does represent the energy increase per unit of temperature(per unit of mass)and consequently has been called the"specific heat at constant volume". However, it is more useful to think of Cy in terms of its definition as a certain partial derivative, which is a thermodynamic property, rather than a quantity related to energy transfer as heat in the special constant volume ocess the enthalpy as a function of T and P, that is view the imple compressible substance we can regard that define the state. Thus h=h(, P) Taking the differential dh d7 dp The derivative(ah/aT)p is called the specific heat at constant pressure, denoted by Cp The derivatives cy and cp constitute two of the most important thermodynamic derivative functions. Values of these properties have been experimentally determined as a function of the thermodynamic state for an enormous number of simple compressible substances 14)Ideal gases The equation of state for an ideal gas is PV= NRT where n is the number of moles of gas in the volume V. Ideal gas behavior furnishes an extremely good approximation to the behavior of real gases for a wide variety of aerospace applications. It should be remembered, however, that describing a substance as an ideal gas constitutes a model of the actual physical situation, and the limits of model validity must always be kept in mind One of the other important features of an ideal gas is that its internal energy depends only upon its temperature.( For now, this can be regarded as another aspect of the model of actual systems that the perfect gas represents, but it can be shown that this is a consequence of the form of the equation of state Since u depends only on T du= Cy(r)dT In the above equation we have indicated that Cy can depend on t. Like the internal energy, the enthalpy is also only dependent on T for an ideal gas. (If u is a function of T, then, using the perfect gas equation of state, u Pv is also Therefore dh= cp(T)dT. Further, dh= du d(Pv)=CvdT+r dt. Hence, for an ideal gas, Cu=C In general, for other substances, u and h depend on pressure as well as on temperature. In this respect, the ideal gas is a very special model 0-70-7 process, cv does represent the energy increase per unit of temperature (per unit of mass) and consequently has been called the "specific heat at constant volume". However, it is more useful to think of cv in terms of its definition as a certain partial derivative, which is a thermodynamic property, rather than a quantity related to energy transfer as heat in the special constant volume process. The enthalpy is also a function of state. For a simple compressible substance we can regard the enthalpy as a function of T and P, that is view the temperature and pressure as the two variables that define the state. Thus, h = h(T,P). Taking the differential, dh = ∂h ∂T P dT + ∂h ∂P T dP The derivative ∂h ∂T P is called the specific heat at constant pressure, denoted by cp. The derivatives cv and cp constitute two of the most important thermodynamic derivative functions. Values of these properties have been experimentally determined as a function of the thermodynamic state for an enormous number of simple compressible substances. 14) Ideal Gases The equation of state for an ideal gas is PV = NRT, where N is the number of moles of gas in the volume V. Ideal gas behavior furnishes an extremely good approximation to the behavior of real gases for a wide variety of aerospace applications. It should be remembered, however, that describing a substance as an ideal gas constitutes a model of the actual physical situation , and the limits of model validity must always be kept in mind. One of the other important features of an ideal gas is that its internal energy depends only upon its temperature. (For now, this can be regarded as another aspect of the model of actual systems that the perfect gas represents, but it can be shown that this is a consequence of the form of the equation of state.) Since u depends only on T, du = cv (T)dT In the above equation we have indicated that cv can depend on T. Like the internal energy, the enthalpy is also only dependent on T for an ideal gas. (If u is a function of T, then, using the perfect gas equation of state, u + Pv is also.) Therefore, dh = cP(T)dT. Further, dh = du + d(Pv) = cv dT + R dT. Hence, for an ideal gas, cv = cP - R. In general, for other substances, u and h depend on pressure as well as on temperature. In this respect, the ideal gas is a very special model