Engineering Chemistry S Datta

78 ENGINEERING CHEMISTRY
When two gases come in contact, they diffuse and mix-up irreversibly. Such a spontaneous
irreversible process leads to an entropy increase.
Let,
Moles of gas A is n A
Moles of gas B is n B
Temperature is T
Pressure is P
Then, S A
= n A
[C ln T − R ln P + 0
S ]
P
A
S B
= n B [C ln T − R ln P + 0
S ]
After mixing, the total pressure (P)
P = p A
+ p B
.
The entropies of the two gases in the mixture are
P
B
S A
(mix) = n A
[C ln T − R ln + S ]
P p 0
A A
A
S B
(mix) = n B
[C ln T − R ln + S ]
P p 0
B B
So, change of entropy due to mixing,
∆S mix
= (S + S ) – (S A
+ S B
)
Amix
B
Bmix
= SA
+ S
mix B – S
mix A
– S B
A
B
or,
∆S = – n A
R ln p A
P
– n B R ln p B
P
The total moles of mixture = n = n A
+ n B
mol. fraction of A, x A
= nA , x
n B
= n B
n
or,
p A
Now,
P
= x A and p B
P = x B
(As mol. fraction = pressure fraction)
∴ ∆S = – n A
R ln x A
– n B
R ln x B
.
∆S mix
= – n R
L
NM
nA
n
ln x
A
nB
+
n
ln x
= – n R[x A
ln x A
+ x B
ln x B
]
In general, for mixing any number of gases,
B
O
QP
∆S mix
= – n R ∑x i
ln x i
where x i
= mol. fraction of gas i.
Work function (A)
Let us define, A = E – TS
For an isothermal process, ∆A T
= ∆E – T∆S (Q ∆T = 0)
= ∆E – q = – W max
or, – ∆A T
= W max
where W max
is reversible, hence, maximum work is involved. So, work function is such a
thermodynamic property of the system that decreases in work function A i.e., (– ∆A) gives the
maximum amount of work available during isothermal process.