Page 156 - Advanced Thermodynamics for Engineers, Second Edition

7.1 THE MAXWELL RELATIONSHIPS 143

These expressions will now be used to consider relationships derived previously. The following
functional relationships have already been obtained.

du ¼ Tds pdv (7.10)
dh ¼ Tds þ vdp (7.11)
df ¼ pdv sdT (7.12)
dg ¼ vdp sdT (7.13)
Consider the expression for du, given in Eqn (7.10) then, by analogy with Eqn (7.3),

vu vu vT vp
T ¼ ; p ¼ and ¼ (7.14)
vs vv vv vs
v s s v
In a similar manner the following relationships can be obtained, for constant composition or single
component systems:

vh vh vT vv
T ¼ ; v ¼ ; ¼ (7.15)
vs vp vp vs
p s s p

vf vf vp vs
p ¼ ; s ¼ ; ¼ (7.16)
vv vT vT vv
T v v T

vg vg vv vs
v ¼ ; s ¼ ; ¼ : (7.17)
vp vT vT vp
T p p T
If the pairs of relationships for T are equated then,
vu vh

¼ (7.18a)
vs vs
v p
and, similarly
vg vh

¼ (7.18b)
vp vp
T s

vu vf
¼ (7.18c)
vv vv
s T

vg vf
¼ (7.18d)
vT vT
p v
In addition to these equivalences, Eqns (7.14)–(7.17) also show that

vT vp
¼ (7.19a)
vv vs
s v

vT vv
¼ (7.19b)
vp vs
s p