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 sec-
ond relation is called the cyclic relation,and it is frequently used in ther-
modynamics (Fig. 12–7).656 | Thermodynamics
Function: z + 2xy – 3y^2 z = 01) z ===Thus, =––––––^1—––––^2 xy
3 y^2 – 1—––––^2 y
3 y^2 – 1––z( )x (^) y
2) x ==^3 y =
(^2) z – z
—––––
2 y
—––––^3 y^2 – 1
2 y
––x
( ) zy
––x
( ) zy
––z
( )x (^) y
FIGURE 12–6
Demonstration of the reciprocity
relation for the function
z 2 xy 3 y^2 z0.
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.
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.
Solution The cyclic and reciprocity relations are to be verified for an ideal gas.
Analysis The ideal-gas equation of state PvRTinvolves the three vari-
ables P, v, and T. Any two of these can be taken as the independent vari-
ables, with the remaining one being the dependent variable.
(a) Replacing x, y, and zin Eq. 12–9 by P, v, and T, respectively, we can
express the cyclic relation for an ideal gas as
where
Substituting yields
which is the desired result.
(b) The reciprocity rule for an ideal gas at Pconstant can be expressed as
Performing the differentiations and substituting, we have
Thus the proof is complete.
R
P
1
P>R
S
R
P
R
P
a
0 v
0 T
b
P
1
10 T> 0 v (^2) P
a
RT
v^2
ba
R
P
ba
v
R
b
RT
Pv
1
TT 1 P, v 2
Pv
R
Sa
0 T
0 P
b
v
v
R
vv 1 P, T 2
RT
P
Sa
0 v
0 T
b
P
R
P
PP 1 v, T 2
RT
v
Sa
0 P
0 v
b
T
RT
v^2
a
0 P
0 v
b
T
a
0 v
0 T
b
P
a
0 T
0 P
b
v
1
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 rela-
tions. They are obtained from the four Gibbs equations by exploiting the
exactness of the differentials of thermodynamic properties.