Thermodynamics

An alternative form of this relation is obtained by using the cyclic relation:Substituting the result into Eq. 12–45 gives(12–46)This relation can be expressed in terms of two other thermodynamic propertiescalled the volume expansivity b and the isothermal compressibility a,which are defined as (Fig. 12–10)anda 0P0T b a 0Tv 0v b a 0vP 0P b 1 S a 0PT0T b a 0vv 0T b a 0PP 0v b Tc p c v T a 0v0T b 2b 1 v a 0v0T b P(12–47)a 1 (12–48)v a 0v0P b TSubstituting these two relations into Eq. 12–46, we obtain a third generalrelation for c p c v :c p c v vTb2a(12–49)It is called the Mayer relation in honor of the German physician and physicistJ. R. Mayer (1814–1878). We can draw several conclusions from this equation:1. The isothermal compressibility a is a positive quantity for all substancesin all phases. The volume expansivity could be negative for somesubstances (such as liquid water below 4°C), but its square is always positiveor zero. The temperature T in this relation is thermodynamic temperature,which is also positive. Therefore we conclude that the constant-pressure specificheat is always greater than or equal to the constant-volume specific heat:c p c v(12–50)2. The difference between c p and c v approaches zero as the absolutetemperature approaches zero.3. The two specific heats are identical for truly incompressible substancessince v constant. The difference between the two specific heats isvery small and is usually disregarded for substances that are nearly incompressible,such as liquids and solids.Pa 0P0v b T20°C100 kPa1 kg20°C100 kPa1 kgChapter 12 | 665––∂v( ∂T )P21°C100 kPa1 kg(a) A substance with a large β∂(––v∂T )P21°C100 kPa1 kg(b) A substance with a small βFIGURE 12–10The volume expansivity (also calledthe coefficient of volumetricexpansion) is a measure of the changein volume with temperature atconstant pressure.EXAMPLE 12–7Internal Energy Change of a van der Waals GasDerive a relation for the internal energy change as a gas that obeys the vander Waals equation of state. Assume that in the range of interest c v variesaccording to the relation c v c 1 c 2 T, where c 1 and c 2 are constants.Solution A relation is to be obtained for the internal energy change of avan der Waals gas.