Thermodynamics

Two of the Gibbs relations were derived in Chap. 7 and expressed asdu T ds P dv(12–10)dh T ds v dP(12–11)The other two Gibbs relations are based on two new combination properties—theHelmholtz function a and the Gibbs function g, defined asa u Ts(12–12)g h Ts(12–13)Differentiating, we getda du T ds s dTdg dh T ds s dTSimplifying the above relations by using Eqs. 12–10 and 12–11, we obtainthe other two Gibbs relations for simple compressible systems:da s dT P dv(12–14)dg s dT v dP(12–15)A careful examination of the four Gibbs relations reveals that they are of theformwithdz M dx N dya 0M0y b a 0Nx 0x b y(12–4)(12–5)since u, h, a, and g are properties and thus have exact differentials. ApplyingEq. 12–5 to each of them, we obtaina 0T0v b a 0Ps 0s b va 0T0P b a 0vs 0s b Pa 0s0v b a 0PT 0T b va 0s0P b a 0vT 0T b P(12–16)(12–17)(12–18)(12–19)These are called the Maxwell relations (Fig. 12–8). They are extremelyvaluable in thermodynamics because they provide a means of determiningthe change in entropy, which cannot be measured directly, by simply measuringthe changes in properties P, v, and T. Note that the Maxwell relationsgiven above are limited to simple compressible systems. However, othersimilar relations can be written just as easily for nonsimple systems such asthose involving electrical, magnetic, and other effects.Chapter 12 | 657∂T( = – ( ∂P–– ––∂v ) ∂s )s v∂T( –– ) = ––∂v∂P ( )s ∂s P( )T( )∂P––∂s= ––∂v ∂T v––∂s ∂v( ∂P) = – (––)T ∂T PFIGURE 12–8Maxwell relations are extremelyvaluable in thermodynamic analysis.