Page 199 - Advanced thermodynamics for engineers

9.3 TABLES OF u(T) AND h(T) AGAINST T 185

Van’t Hoff equation, which will be derived in section (12.8) where dissociation is introduced. This
states that

h m ¼ T 2 d m 0 m (9.33)
T
dT
where, for T 0 ¼ 0K,
0
m ¼ chemical potential term ¼ g m ðTÞþ g 0;m ¼ g m ðTÞþ h 0;m (9.34)
m
and
h m ¼ h m ðTÞþ h 0;m : (9.13)
0
0
(Note: the term m ; is similar to g ; which was introduced in Eqn (9.28). At this stage it is
m m
sufficient to note that chemical potential has the same numerical value as the specific Gibbs energy.
The chemical potential will be defined in the Section 12.2.)
By definition, in Eqn (9.31),
h m ðTÞ h m h 0;m 2 3 4
¼ ¼ a 1 þ a 2 T þ a 3 T þ a 4 T þ a 5 T ; (9.35)
<T <T
giving
h m ðTÞ a 1 2 3
¼ þ a 2 þ a 3 T þ a 4 T þ a 5 T : (9.36)
<T 2 T
Substituting Eqns (9.34), (9.13) and (9.36) into Eqn (9.33) and rearranging gives

a 1 2 3 h 0;m 1 g m ðTÞþ h 0;m
þ a 2 þ a 3 T þ a 4 T þ a 5 T þ 2 dT ¼ d ; (9.37)
T <T < T
which can be integrated to

2 3 4
a 3 T a 4 T a 5 T h 0;m g m ðTÞþ h 0;m
a 1 ln T þ a 2 T þ þ þ þ A ¼ : (9.38)
2 3 4 <T <T
It is conventional to let A ¼ a 1 a 6 (9.39)
and then

2 3 4
g m ðTÞ a 3 T a 4 T a 5 T
¼ a 1 ð1 ln TÞ a 2 T þ þ þ a 6 : (9.40)
<T 2 3 4
The value of entropy can then be obtained from

h m ðTÞ g m ðTÞ
s m ðTÞ¼ ; (9.41)
T
giving

3 2 4 3 5 4
s m ðTÞ¼< a 1 lnT þ 2a 2 T þ a 3 T þ a 4 T þ a 5 T þ a 6 : (9.42)
2 3 4