Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

Two of the Gibbs relations were derived in Chap. 7 and expressed as

(12–10)

(12–11)

The other two Gibbs relations are based on two new combination proper-
ties—the Helmholtz functionaand the Gibbs functiong, defined as

(12–12)

(12–13)

Differentiating, we get

Simplifying the above relations by using Eqs. 12–10 and 12–11, we obtain
the other two Gibbs relations for simple compressible systems:

(12–14)

(12–15)

A careful examination of the four Gibbs relations reveals that they are of the
form

(12–4)

with

(12–5)

since u,h,a, and gare properties and thus have exact differentials. Apply-
ing Eq. 12–5 to each of them, we obtain

(12–16)

(12–17)

(12–18)

(12–19)

These are called the Maxwell relations(Fig. 12–8). They are extremely
valuable in thermodynamics because they provide a means of determining
the change in entropy, which cannot be measured directly, by simply mea-
suring the changes in properties P,v, and T. Note that the Maxwell relations
given above are limited to simple compressible systems. However, other
similar relations can be written just as easily for nonsimple systems such as
those involving electrical, magnetic, and other effects.

a

0 s

0 P

b

T

a

0 v

0 T

b

P

a

0 s

0 v

b

T

a

0 P

0 T

b

v

a

0 T

0 P

b

s

a

0 v

0 s

b

P

a

0 T

0 v

b

s

a

0 P

0 s

b

v

a

0 M

0 y

b

x

a

0 N

0 x

b

y

dzM dxN dy

dgs dTv dP

das dTP dv

dgdhT dss dT

daduT dss dT

ghTs

auTs

dhT dsv dP

duT dsP dv

Chapter 12 | 657

( ) ––∂T = –( ) ∂––∂sP

∂ vs

( ) ∂––T =

∂ Ps

∂––s

( ) ∂ PT

( ) ––∂s = ( ) ∂∂––TP

∂ vT

= –

∂––v

( ) ∂s (^) P
∂––v
( ) ∂ TP
v
v
FIGURE 12–8
Maxwell relations are extremely
valuable in thermodynamic analysis.