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Unformatted text preview: S a 0x b 0z y 1 1 0 z> 0 x 2 y 1
(12–8) 0z 0x ba b 0x y 0y z 0y 0x 0x 0z b Sa ba b a b 0y x 0y z 0z x 0x y (12–9) cen84959_ch12.qxd 4/5/05 3:58 PM Page 656 656 | Thermodynamics
The first relation is called the reciprocity relation, and it shows that the inverse of a partial derivative is equal to its reciprocal (Fig. 12–6). The second relation is called the cyclic relation, and it is frequently used in thermodynamics (Fig. 12–7). Function: z + 2xy – 3y2z = 0 2xy 1) z = —–––– 3y2 – 1 3y2z – z 2) x = —–––– 2y Thus, () 2y z –– = —–––– xy 3y2 – 1
2 EXAMPLE 12–3 Verification of Cyclic and Reciprocity Relations Using the ideal-gas equation of state, verify (a) the cyclic relation and (b) the reciprocity relation at constant P. 3y – 1 ( ––xz )y = —–––– 2y 1 –––––– x –– zy 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 variables P, 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 can express the cyclic relation for an ideal gas as ( ––xz )y = () a 0P 0v 0T ba ba b 0v T 0T P 0P v RT 0P Sa b v 0v T RT 0v Sa b P 0T P 0T Pv Sa b R 0P v RT Pv 1 FIGURE 12–6 Demonstration of the reciprocity relation for the function z 2xy 3y2z 0. where P v T
Substituting yields P 1 v, T 2 v 1 P, T 2 T 1 P, v 2 RT v2 R P v R a RT R v ba ba b P R v2 1 which is the desired result. (b) The reciprocity rule for an ideal gas at P constant can be expressed as a 0v b 0T P 1 1 0 T> 0 v 2 P R P Performing the differentiations and substituting, we have R P FIGURE 12–7 Partial differentials are powerful tools that are supposed to make life easier, not harder.
© Reprinted with special permission of King Features Syndicate. 1 R S P> R P Thus the proof is complete. 12–2 ■ THE MAXWELL RELATIONS The equations that relate the partial derivatives of properties P, v, T, and s of a simple compressible system to each other are called the Maxwell relations. They are obtained from the four Gibbs equations by exploiting the exactness of the differentials of thermodynamic properties. cen84959_ch12.qxd 4/5/05 3:58 PM Page 657 Chapter 12
Two of the Gibbs relations were derived in Chap. 7 and expressed as
du dh T ds T ds P dv v dP
(12–10) (12–11) | 657 The other two Gibbs relations are based on two new combination properties—the Helmholtz function a and the Gibbs function g, defined as
a g u h Ts Ts
(12–12) (12–13) Differentiating, we get
da dg du dh T ds T ds s dT s dT Simplifying the above relations by using Eqs. 12–10 and 12–11, we obtain the other two Gibbs relations for simple compressible systems:
da dg s dT s dT P dv v dP
(12–14) (12–15) A careful examination of the four Gibbs relations reveals that they are of the form
dz M dx N dy
a 0M b 0y x a 0N b 0x y
(12–5) since u, h, a, and g are properties and thus have exact differentials. Applying Eq. 12–5 to each of them,...
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