Thermodynamics

656 | ThermodynamicsFunction: z + 2xy – 3y 2 z = 0( )y2xy1) z = —–––– ––z=3y 2 – 1 x( )y2) x =3y 2 z – z x—–––– ––2y z=Thus,z 1(––= ––––––x) y x–– z( )yFIGURE 12–6Demonstration of the reciprocityrelation for the functionz 2xy 3y 2 z 0.2y—––––3y 2 – 13y 2 – 1—––––2yFIGURE 12–7Partial differentials are powerful toolsthat are supposed to make life easier,not harder.© Reprinted with special permission of KingFeatures Syndicate.The first relation is called the reciprocity relation, and it shows that theinverse of a partial derivative is equal to its reciprocal (Fig. 12–6). The secondrelation is called the cyclic relation, and it is frequently used in thermodynamics(Fig. 12–7).EXAMPLE 12–3Verification of Cyclic and Reciprocity RelationsUsing the ideal-gas equation of state, verify (a) the cyclic relation and (b)the reciprocity relation at constant P.Solution The cyclic and reciprocity relations are to be verified for an ideal gas.Analysis The ideal-gas equation of state Pv RT involves the three variablesP, v, and T. Any two of these can be taken as the independent variables,with the remaining one being the dependent variable.(a) Replacing x, y, and z in Eq. 12–9 by P, v, and T, respectively, we canexpress the cyclic relation for an ideal gas aswhereSubstituting yieldswhich is the desired result.(b) The reciprocity rule for an ideal gas at P constant can be expressed asPerforming the differentiations and substituting, we haveThus the proof is complete.a 0P0v b a 0vT 0T b a 0TP 0P b 1vP P 1v, T2 RTvv v 1P, T2 RTPT T 1P, v2 PvRa RTv 2 baR P bav R b RT Pv 1a 0v0T b PS a 0P0v b RTT v 2S a 0v0T b P R PS a 0T0P b v v R110T>0v2 PRP 1P>R S R P R P12–2 ■ THE MAXWELL RELATIONSThe equations that relate the partial derivatives of properties P, v, T, and sof a simple compressible system to each other are called the Maxwell relations.They are obtained from the four Gibbs equations by exploiting theexactness of the differentials of thermodynamic properties.