b ! 0 ¼ O e
½ Š
ð
Þand j T ! 0 ¼ O e
2
 Ã
À
Á
Equation (176) reduces to
dU ¼ N c p À pvb
À
Á
dT þ V pj T À Tb
ð
Þ dV ¼ Nc p dT þ 0 ¼ Nc p dT
ð176BÞ
We have thus the interesting result that the proportionality constant of dU to dT
for ideal gases is c V , while the proportionality constant of dU to dT for “incompressible” substances is c p . The indeterminate term in (175B) is a finite value
accounting for the difference between the value of c V and c p , as shown in the
following.
The following two dS expressions are left as Problem 8.2 of this chapter:
dS ¼
Nc V
T
dT þ
b
j T
dV
dS ¼
Nc p
T
dT À Vbdp
Note again, for incompressible substances, b ! 0 (¼ O[e]) and j T ! 0 (= O[e
2 ]),
the first dS expression is useless involving indeterminate. For such substances, the
second dS yields
dS ¼
Nc p
T
dT;
Correspondingly,
DS ¼ Nc p ln
T 2
T 1
ð71Þ
which was given as Eq. (71) in Sect. 5.3.4 without proof.
We now take the differentiation of the first dS expression with respect to
T holding p constant,
@S
@T
p
¼
Nc V
T
@T
@T
p
þ
b
j T
@V
@T
p
¼
Nc V
T
þ
b
j T
Vb
Note that the LHS term is equal to
Nc p
T . It follows
Nc p ¼ Nc V þ
T Nv
ð Þb
2
j T
9.5 Determination of Thermodynamic Properties …
251