Chapter21
Entropy and the Second
Law of Thermodynamics
21-1 Some One – Way Processes
If an irreversible process occurs in a closed system,
the entropy S of the system always increase;it
never decreases.
There are two equivalent ways to define the change
in entropy of a system:
(1) In terms of the system’s temperature and the
energy it gains or loses as heat
(2) By counting the ways in which the atoms or
molecules that make up the system can be arranged.
21-2 Change in Entropy
The change in entropy of a system is
S  S f  Si  
f
i
dQ
(change in entropy defined)
T
To apply Eq.21-1 to the isothermal expansion
1
S  S f  Si 
T

i
f
dQ
Q is the total energy transferred as heat during
the process
S  S f  Si 
Q
(change in entropy, isothermal process)
T
To find the entropy change for an
irreversible process occurring in a
closed system,replace that process
with any reversible process that
connects the same initial and final
states.Calculate the entropy change
for this reversible process with
Eq.21-1.
Q
S  S f  Si 
Tavg
Sample Problem 21-1
Q  nRT ln
S rev
Vf
Vi
Vf
Q nRT ln( V f / Vi )
 
 nR ln
T
T
Vi
Substituting n=1.00 mol and Vf/Vi=2
S rev  nR ln
Vf
Vi
 (1.00mol )(8.31J / mol  K )(ln 2)
 5.76 J / K
Sirrev  Srev  5.76 J / K
Sample Problem 21-2
Step 1.
S L  
f
i
 mc ln
T f mcdT
T f dT
dQ

 mc
T
TiL T
iL
T
T
Tf
TiL
S L  (1.5kg)(386 J / kg  K ) ln
313K
333K
 35.86 J / K
Step 2.
313K
S R  (1.5kg)(386 J / kg  K ) ln
293K
 38.23 J / K
Srev  S L  S R
 35.86 J / K  38.23J / K  2.4 J / K
Srev  Sirrev  2.4 J / K
Entropy as a State Function
There are related by the first law of thermodynamics in
differential form(Eq.19-27)
dEint  dQ  dW
Solving for dQ then leads to
dQ  pdV  nCV dT
dQ
dV
dT
 nR
 nCV
T
V
T

i
f
f
f
dQ
dV
dT
  nR
  nCV
i
i
T
V
T
The entropy change is
S  S f  Si  nR ln
Vf
Vi
 nCV ln
Tf
Ti
21-3 The Second Law of Thermodynamics
We can calculate separately the entropy changes for the gas
and the reservoir:
S gas  
S res
|Q|
T
|Q|

T
We can modify the entropy postulate of Section 21-1 to
include both reversible and irreversible processes:
If a process occurs in a closed system,the entropy
of the system increases for irreversible processes
and remains constant for reversible processes.It
never decreases.
The second law of thermodynamics and can be written as
S  0 (second law of thermodyn amics)
21-4 Entropy in the Real World: Engines
A Carnot Engine
In an ideal engine,all processes are reversible and
no wasteful energy transfers occur due to,say,
friction and turbulence.
W | QH |  | QL |
| QH | | QL |
S  S H  S L 

TH
TL
We must have
| QH | | QL |

TH
TL
for a complete cycle
Efficiency of a Carnot Engine
energy we get
|W |


(efficienc y, any engine)
energy we pay for | QH |
C 
| QH |  | QL |
|Q |
 1 L
| QH |
| QH |
C  1
TL
(efficienc y, carnot engine)
TH
Led to the following alternative version of the second law of
thermodynamics:
No series of processes is possible whose sole result
is the transfer of energy as heat from a thermal
reservoir and the complete conversion of this energy
to work.
Stirling Engine
Sample Problem 21-3
TL
300 K
 1
 0.647  65%
TH
850 K
(a)
  1
(b)
W 1200 J
P

 4800W  4.8kW
t
0.25s
W
(c)
1200 J
| QH |

 1855 J

0.647
(d)
| QL || QH | W  1855J 1200J  655J
(e)
QH 1855 J
S H 

 2.18 J / K
TH
850 K
S L 
QL  655 J

 2.18 J / K
TL
300 K
Sample Problem 21-4
  1
TL
(0  273) K
 1
 0.268  27%
TH
(100  273) K
PROBLEM - SOLVING TACTICS
Heat is energy that is transferred from one body
to another body owing to a difference in the
temperatures of the bodies.
Work is energy that is transferred from one body
to another body owing to a force that acts between
them.
21-5 Entropy in the Real World:
Refrigerators
An ideal refrigerator:
In an ideal refrigerator,all processes
are reversible and no wasteful
energy transfers occur due to,say,
friction and turbulence.
what we want
| QL |
K

what we pay for | W |
(coefficient of performance,any refrigerator)
| QL |
KC 
| QH |  | QL |
TL
KC 
TH  TL
(coefficient of performance,Carnot refrigerator.)
The net entropy change for the entire system is
|Q| |Q|
S  

T L TH
Another formulation of the second law of
thermodynamics:
No series of processes is possible
whose sole result is the transfer of
energy as heat from a reservoir at a
given temperature to a reservoir at a
higher temperature.
21-6 The Efficiencies of Real Engines
An efficiency
is greater than
:
 X   C (a claim)
If Eq.21-15 is true,from the definition of efficiency
|W |
|W |

| Q' H | | QH |
| QH || Q'H |
From the first law of thermodynamics:
| QH |  | QL || Q'H |  | Q'L |
| QH |  | Q'H || QL |  | Q'L | Q
21-7 A Statistical View of Entropy
Extrapolating from six molecules to the
general case of N molecules
W
N!
(multiplic ity of configurat ion)
n1!n2 !
The basic assumption of statistical
mechanics is:
All microstates are equally probable
Sample Problem 21-5
N!
100!
W

n1!n2 ! 50!50!

9.33 10
(3.04 1064 )(3.04 1064 )
 1.0110 29
157
W
N!
100!

1
n1!n2 ! 100!0!
Probability and Entropy
A relationship between the entropy S of a configuration of
a gas and the multiplicity W of that configuration.
S  k ln W (Boltzmann ' s entropy equation)
The Stirling’s approximation is :
ln N! N (ln N )  N
Sample Problem 21-6
N!
Wi 
1
N !0!
N!
Wf 
( N / 2)!( N / 2)!
Si  k ln Wi  k ln 1  0
S f  k ln W f  k ln( N!)  2k ln[( N / 2)!]
a
ln 2  ln a  2 ln b
b
S f  k ln( N !)  2k ln[( N / 2)!]
 k[ N (ln N )  N ]  2k[( N / 2) ln( N / 2)  ( N / 2)]
 k[ N (ln N )  N  N ln( N / 2)  N ]
 k[ N (ln N )  N (ln N  ln 2)]  Nk ln 2
S f  nR ln 2
S f  Si  nR ln 2  0  nR ln 2
REVIEW & SUMMARY
Calculating Entropy Change
The change in entropy of a system is
S  S f  Si  
f
i
dQ
(change in entropy defined)
T
Q is the total energy transferred as heat during the process
S  S f  Si 
Q
(change in entropy, isothermal process)
T
Q
S  S f  Si 
Tavg
The entropy change is
S  S f  Si  nR ln
Vf
Vi
 nCV ln
Tf
Ti
The Second Law of Thermodynamics
The second law of thermodynamics and can be written as
S  0 (second law of thermodyn amics)
Engines

energy we get
|W |

(efficienc y, any engine)
energy we pay for | QH |
C  1
| QL |
T
 1 L
| QH |
TH
Refrigerators
K
what we want
|Q |
 L
what we pay for | W |
KC 
| QL |
TL

| QH |  | QL | TH  TL
Entropy from a Statistical View
Extrapolating from six molecules to the general case of N
molecules
W
N!
(multiplic ity of configurat ion)
n1!n2 !
A relationship between the entropy S of a configuration of
a gas and the multiplicity W of that configuration.
S  k ln W (Boltzmann ' s entropy equation)
The Stirling’s approximation is :
ln N! N (ln N )  N