Page 213 - Advanced Thermodynamics for Engineers, Second Edition

200 CHAPTER 9 THERMODYNAMIC PROPERTIES OF IDEAL GASES

The total amount of substance in the gases is
n T ¼ n a þ n b : (9.72)
The entropies of the gases before mixing are

h i
; (9.73)
S m a ¼ n a s m a ¼ n a s m a ðTÞ < ln p r a þ s 0;m a
and
h i
; (9.74)
S m b ¼ n b s m b ¼ n b s m b ðTÞ < ln p r b þ s 0;m b
giving the sum of the entropies before mixing (i.e. separated) as
h i

Þ
Þ
ðS m T sep ¼ðn T s m T sep ¼ n T x a s m a ðTÞþ x b s m b ðTÞþ x a s 0;m a þ x b s 0;m b < x a ln p r a þ x b ln p r b
(9.75)
Since the gases are separated and both at the pressure, p r , then

Þ
ðS m T sep ¼ n T x a s m a ðTÞþ x b s m b ðTÞþ x a s 0;m a þ x b s 0;m b < ln p (9.76)
Hence, the molar entropy, when the gases are separate, is

Þ
ðs m T sep ¼ x a s m a ðTÞþ x b s m b ðTÞþ x a s 0; m a þ x b s 0; m b < ln p
(9.77)
X X
¼ x i s m i þ x i s 0; m i < ln p
The entropy of the gases after mixing is
X
Þ
ðS m T mix ¼ n T s m T ¼ ns m ¼ n a s m a þ n b s m b (9.78)
are the molar entropies if each component occupied the whole volume.
where S m a and S m b
Thus, substituting for s m from Eqn (9.20a) gives
Þ
ðn T s m T mix ¼ n a s m a ðTÞþ n b s m b ðTÞ <ðn a ln p r a þ n b ln p r b Þþ n a s 0;m a þ n b s 0;m b (9.79)
Þ (9.80)
ðs m T mix ¼ x a s m a ðTÞþ x b s m b ðTÞ <ðx a ln p r a þ x b ln p r b Þþ x a s 0;m a þ x b s 0;m b
¼ p i =p 0 ; and p i ¼ partial pressure of constituent i.
where p r i
Equation (9.80) may be written
X X X
Þ
ðs m T mix ¼ x i s m i ðTÞþ x i s 0;m i < x i ln p r i (9.81)
Now, consider the final term in Eqn (9.81), based on two constituents to simplify the mathematics,

x a ln pr a þ x b ln pr b ¼ x a ln x a p r þ x b ln x b p r
¼ x a ln x a þ x a ln p r þ x b ln x b þ x b ln p r
X X
(9.82)
¼ x i ln x i þ ln p r x i
X X 1
¼ x i ln x i þ ln p r ¼ ln p r x i ln
x i