The Polytropic Process – Ideal Gas
Definition of a polytropic process:
p2  V1 
 
p1  V2 
n
If the fluid is an ideal gas,
n
p2  V1   mRT1 / p1 
  

p1  V2   mRT2 / p2 
This leads to two additional relationships for ideal gases,
T2  p2 
 
T1  p1 
1
 n 1 / n
1 n
and
T2  V2 
 
T1  V1 
The Polytropic Process – Ideal Gas
The work done during a polytropic process is,
W12 

V2
V1
p1V1 n
p2 V2  p1V1
dV 
n
V
1 n
for n  1
If the fluid is an ideal gas,
W12 
mR T2  T1 
1 n
For the case where n = 1,
W12 
2

V2
V1
for n  1
p1V1
V2
dV  p1V1 ln
V
V1
Ideal Gas Entropy  f(T,V)
For the ideal gas, recall that
Pv  RT , du  cv dT , and dh  c p dT
Then from the first Gibbs equation,
s2  s1 

u2
u1
du

T

v2
v1
P
dv 
T

T2
T1
dT
cv

T

v2
v1
RT dv

v T

T2
T1
v2
dT
cv
R ln
T
v1
Furthermore, if the heat capacity can be assumed constant,
s2  s1  cv ln
T2
v
 R ln 2
T1
v1
Notice that the entropy of an ideal gas is a function of both
temperature and specific volume (or pressure).
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Ideal Gas Entropy f(T,P)
For the ideal gas, recall that
Pv  RT , du  cv dT , and dh  c p dT
Then from the second Gibbs equation,
s2  s1 

h2
h1
dh

T

P2
P1
v
dP 
T

T2
T1
dT
cp

T

P2
P1
RT dP

P T

T2
T1
P2
dT
cp
 R ln
T
P1
Furthermore, if the heat capacity can be assumed constant,
s2  s1  c p ln
T2
P
 R ln 2
T1
P1
Notice that the entropy of an ideal gas is a function of both
temperature and pressure.
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