That is,
S T; V
ð
Þ ¼
X n
i¼1
N i s i ¼ N
X n
i¼1
x i s i T; V
ð
Þ
ð78Þ
Despite the similarity in the form of Eq. (78) and the form of Eqs. (76) and (77),
they are quite different because Eqs. (76) and (77) remain the same for either T-V or
T-p as the set of independent variables, while Eq. (78) will assume a different form
if it is expressed in terms of the set of independent variables T-p, as shown below.
5.9.1 Entropy and Specific Gibbs Function of Mixture
in Terms of T-p
According to Dalton’s law, when a gas of specie i alone occupies the volume V at
temperature T, the pressure is equal to the partial pressure p i of the specie in the
mixture. Therefore,
S ¼ N
X n
i¼1
x i s i T; V
ð
Þ¼ N
X n
i¼1
x i s i T; p i
ð
Þ
ð79Þ
Note that this is different from
S ¼ N
X n
i¼1
x i s i T; p
ð
Þ
where p is the pressure of the mixture.
Recall from Eq. (67)
dS ¼ Nc p
dT
T
À NR
dp
p
For ideal gases, therefore,
s T; p
ð
ÞÀs 0 ¼
Z T
T 0
c p T
0
ð Þ
T 0 dT
0
À Rln
p
p 0
The molar entropy of an ideal gas in terms of T-p is
s i T; p i
ð
Þ ¼s i0 þ
Z T
T 0
c pi T
ð Þ
T
dT À Rln
p i
p 0
¼ s i0 þ
Z T
T 0
c pi T
ð Þ
T
dT À Rln
p i
p
p
p 0
¼ s i0 þ
Z T
T 0
c pi T
0
ð Þ
T 0 dT
0
À Rln
p
p 0
À Rln
p i
p
122
5 Entropy and the Entropy Principle