Engineering Chemistry S Datta

76 ENGINEERING CHEMISTRY
or,
Thus, for a change from states 1 to 2 we have,
z z z
S2
T2
dT
dS = C
S
v
+ R
1
T1
T
∆S = C v
ln T 2
T1
V2
V1
dV
V
+ R ln V 2
.
V1
Since, for an ideal gas at constant temperature,
V
V
2
1
= P 1
P
2
∴
∆S = C v
ln T 2
+ R ln P 1
.
T P
1
2
At constant temperature i.e., for isothermal expansion,
At constant pressure P,
∆S T
= R ln V 2
V1
= R ln P 1
.
P2
At constant volume V,
∆S p
= C p
ln T 2
T1
∆S v
= C v
ln T 2
.
T1
Entropy change in irreversible process
Entropy is a state function. The change of entropy of a given system from state 1 to state 2
is always the same and is path independent if the path is reversible. So, the change of entropy
(∆S) is given by:
z2
∆S = S 2
– S 1
=
dqr .
1 T
When the flow of heat is irreversible, let a heat reservoir at T 1
(state 1) is brought in
contact with a second reservoir at T 2
(state 2) where T 1
> T 2
, and small quantity of heat q flows
from T 1
to T 2
.
The decrease in entropy of state 1 = q T 1
The decrease in entropy of state 2 = q T 2
q q
∴ Net change of entropy =
q
T
− T1 T2
T
= −
.
2 1 TT 1 2
The difference is a positive quantity, hence, the irreversible heat flow leads to increase
in entropy.
Irreversible isothermal expansion of an ideal gas
Let n moles of an ideal gas is enclosed in a vessel of volume V 1
. The vessel is connected
to a vessel of volume V 2
by a tube containing stop cock, which is evacuated (Fig. 4.4). The
vessels are kept in a thermostat of the temperature T. From the first law of thermodynamics,
we have,
q = ∆E + W