Page 60 - Entrophy Analysis in Thermal Engineering Systems

52 Entropy Analysis in Thermal Engineering Systems

Fig. 4.4 Mixing equimolar ideal gases in a perfectly insulated chamber.

Consider now isothermal mixing of equimolar ideal gases that initially
occupy an identical volume V i in a perfectly insulated chamber as shown
in Fig. 4.4. From the state equation pV¼nRT, it can be deduced that all
gases have also an identical pressure p i before mixing. The state equation
for the mixture can be expressed as p mix V mix ¼n mix RT, where

k
X
V mix ¼ V j ¼ kV i (4.21)
j¼1
k
X
n mix ¼ n j ¼ kn i (4.22)
j¼1
Substituting Eqs. (4.21) and (4.22) into the state equation we find p mix ¼p i ;
that is, the pressure of the mixture is the same as the pressure of the gases
before mixing. Because the volume of each gas increases from V i to V mix ,
the pressure of the gas decreases from p i to p i /k. The increase in the entropy
is obtained using Eqs. (4.19) and (4.20).
0 1
k k
X X
B p i C
ΔS mix ¼ nR ln A ¼ ð (4.23)
1 nR lnkÞ ¼ n i Rk lnk
@
j
j¼1 p i j¼1
k
j
4.6 Interpretation of entropy
Our investigation of the irreversible phenomena discussed in the pre-
ceding sections reveals that the presence of heat is the sole reason for the
entropy increase in these processes confirming the preposition put forward
in Section 4.1:
Entropy generation may take place only in irreversible processes that include a pas-
sage of heat.
Pressure drop and expansion of ideal gases are common examples of irre-
versible conversion of work into heat. Recall the general entropy balance
equation, Eq. (1.16), which includes heat-to-temperature ratio and entropy