Thermodynamics

Chapter 12 | 667For an ideal gas P RT/v. ThenThus,a 0P0T b Rv v anda 02 P0T 2b c 0 1R>v2d 0v 0T va 0c v0v b 0Twhich states that c v does not change with specific volume. That is, c v is nota function of specific volume either. Therefore we conclude that the internalenergy of an ideal gas is a function of temperature only (Fig. 12–11).(b) For an incompressible substance, v constant and thus dv 0. Alsofrom Eq. 12–49, c p c v c since a b 0 for incompressible substances.Then Eq. 12–29 reduces todu c dTAgain we need to show that the specific heat c depends on temperature onlyand not on pressure or specific volume. This is done with the help ofEq. 12–43:u = u(T)c v = c v (T)c p = c p (T)u = u(T)c = c(T)AIRLAKEa 0c p0P b T a 02 vT 0T b 02Psince v constant. Therefore, we conclude that the internal energy of atruly incompressible substance depends on temperature only.FIGURE 12–11The internal energies and specificheats of ideal gases andincompressible substances depend ontemperature only.EXAMPLE 12–9The Specific Heat Difference of an Ideal GasShow that c p c v R for an ideal gas.Solution It is to be shown that the specific heat difference for an ideal gasis equal to its gas constant.Analysis This relation is easily proved by showing that the right-hand sideof Eq. 12–46 is equivalent to the gas constant R of the ideal gas:Substituting,Therefore,P RTvv RTPT a 0v 20T bS a 0P0v b RTT v P 2 vS a 0v0T b 2Pc p c v T a 0v 20T bP a R P b 2a 0P0v b T a R 2T P b a P v b Rc p c v RPa 0P0v b T