258 ENGINEERING THERMODYNAMICS

dharm

/M-therm/th5-3.pm5

3. It remains unchanged in all adiabatic frictionless processes.


4. It increases if temperature of heat is lowered without work being done as in a throttling
   process.

5.17. Entropy Changes for a Closed System

5.17.1. General case for change of entropy of a gas

Let 1 kg of gas at a pressure p 1 , volume v 1 , absolute temperature T 1 and entropy s 1 , be
heated such that its final pressure, volume, absolute temperature and entropy are p 2 , v 2 , T 2 and s 2
respectively. Then by law of conservation of energy,
dQ = du + dW
where, dQ = Small change of heat,
du = Small internal energy, and
dW = Small change of work done (pdv).
Now dQ = cvdT + pdv
Dividing both sides by T, we get
dQ
T

cdT

T

pdv

T

=+v

But
dQ
T = ds
and as pv = RT

∴

p

T

R

v

=

Hence (^) ds cdT
T
Rdv
v
=+v
Integrating both sides, we get
ds c
dT
T
R dv
s v
s
vT
T
v
v
1
2
1
2
1
2
zzz=+
or (s 2 – s 1 ) = cv loge
T
T
2
1

• 
  

  R loge
  v
  v
  2
  1
  ...(5.28)
  This expression can be reproduced in the following way :
  According to the gas equation, we have
  pv
  T
  pv
  T
  11
  1
  22
  2

  
  

  or T
  T
  p
  p
  v
  v
  2
  1
  2
  1
  2
  1
  =×
  Substituting the value of
  T
  T
  2
  1
  in eqn. (5.28), we get
  s 2 – s 1 = cv loge
  p
  p
  v
  v
  2
  1
  2
  1
  × + R loge v
  v
  2
  1
  = cv loge
  p
  p
  2
  1

  
  


• cv loge
  v
  v
  2
  1


• R loge
  v
  v
  2
  1