chapter21 entropy and the second law of thermodynamics
TRANSCRIPT
Chapter21Chapter21
Entropy and the Second Entropy and the Second Law of ThermodynamicsLaw of Thermodynamics
21-1 Some One – Way Processes21-1 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 Entropy21-2 Change in Entropy
The change in entropy of a system is
defined)entropy in (change f
iif T
dQSSS
f
iif dQT
SSS1
To apply Eq.21-1 to the isothermal expansion
process) isothermalentropy,in (change T
QSSS if
Q is the total energy transferred as heat during the process
avg
if T
QSSS
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.
Sample Problem 21-1Sample Problem 21-1
i
f
V
VnRTQ ln
i
fifrev V
VnR
T
VVnRT
T
QS ln
)/ln(
Substituting n=1.00 mol and Vf/Vi=2
KJ
KmolJmolV
VnRS
i
frev
/76.5
)2)(ln/31.8)(00.1(ln
KJSS revirrev /76.5
Sample Problem 21-2Sample Problem 21-2Step 1.
Step 2.
iL
f
T
T
T
T
f
iL
T
Tmc
T
dTmc
T
mcdT
T
dQS
f
iL
f
iL
ln
KJK
KKkgJkgSL
/86.35333
313ln)/386)(5.1(
KJK
KKkgJkgSR
/23.38293
313ln)/386)(5.1(
KJKJKJ
SSS RLrev
/4.2/23.38/86.35
KJSS irrevrev /4.2
Entropy as a State FunctionThere are related by the first law of thermodynamics in differential form(Eq.19-27)
dWdQdE int
Solving for dQ then leads to
dTnCpdVdQ V
T
dTnC
V
dVnR
T
dQV
f
i V
f
i
f
i T
dTnC
V
dVnR
T
dQ
The entropy change is
i
fV
i
fif T
TnC
V
VnRSSS lnln
21-3 The Second Law of Thermodynamics21-3 The Second Law of ThermodynamicsWe can calculate separately the entropy changes for the gas and the reservoir:
T
QSgas
||
T
QSres
||
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
amics) thermodynof law (second 0S
21-4 Entropy in the Real World: Engines21-4 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.
|||| LH QQW
L
L
H
HLH T
Q
T
QSSS
||||
L
L
H
H
T
Q
T
Q ||||
We must have for a complete cycle
Efficiency of a Carnot Engine
engine)any y,(efficienc ||
||
forpay energy we
get energy we
HQ
W
||
||1
||
||||
H
L
H
LHC Q
Q
Q
engine)carnot y,(efficienc 1H
LC T
T
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.
Led to the following alternative version of the second law of thermodynamics:
Stirling Engine
Sample Problem 21-3
(a)
KJK
J
T
QS
L
LL /18.2
300
655
%65647.0850
30011
K
K
T
T
H
L
(b) kWWs
J
t
WP 8.44800
25.0
1200
(c) JJW
QH 1855647.0
1200||
(d) JJJWQQ HL 65512001855||||
(e) KJK
J
T
QS
H
HH /18.2
850
1855
Sample Problem 21-4Sample Problem 21-4
%27268.0)273100(
)2730(11
K
K
T
T
H
L
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: 21-5 Entropy in the Real World: RefrigeratorsRefrigerators
In an ideal refrigerator,all processes are reversible and no wasteful energy transfers occur due to,say, friction and turbulence.
An ideal refrigerator:
||
||
forpay what we
wantwhat we
W
QK L
||||
||
LH
LC QQ
QK
(coefficient of performance,any refrigerator)
The net entropy change for the entire system is
HL T
Q
T
QS
||||
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.
Another formulation of the second law of thermodynamics:
LH
LC TT
TK
(coefficient of performance,Carnot refrigerator.)
21-6 The Efficiencies of Real Engines21-6 The Efficiencies of Real Engines
claim) (a CX
||
||
|'|
||
HH Q
W
Q
W
|'||| HH QQ
|'||'||||| LHLH QQQQ
QQQQQ LLHH |'||||'|||
An efficiency is greater than :
If Eq.21-15 is true,from the definition of efficiency
From the first law of thermodynamics:
21-7 A Statistical View of Entropy21-7 A Statistical View of EntropyExtrapolating from six molecules to the general case of N molecules
ion)configurat ofity (multiplic !!
!
21 nn
NW
The basic assumption of statistical mechanics is:
All microstates are equally probable
Sample Problem 21-5Sample Problem 21-5
29
6464
157
21
1001.1
)1004.3)(1004.3(
1033.9
!50!50
!100
!!
!
nn
NW
1!0!100
!100
!!
!
21
nn
NW
NNNN )(ln!ln
equation)entropy s'(Boltzmann lnWkS
Probability and EntropyA relationship between the entropy S of a configuration of a gas and the multiplicity W of that configuration.
The Stirling’s approximation is :
Sample Problem 21-6Sample Problem 21-6
1!0!
!
N
NWi
)!2/()!2/(
!
NN
NW f
01lnln kWkS ii
2lnnRS f
2ln02ln nRnRSS if
2ln)]2ln(ln)(ln[
])2/ln()(ln[
)]2/()2/ln()2/[(2])(ln[
])!2/ln[(2)!ln(
NkNNNNk
NNNNNNk
NNNkNNNk
NkNkS f
])!2/ln[(2)!ln(ln NkNkWkS ff
bab
aln2lnln
2
REVIEW & SUMMARYREVIEW & SUMMARYCalculating Entropy ChangeThe change in entropy of a system is
defined)entropy in (change f
iif T
dQSSS
process) isothermalentropy,in (change T
QSSS if
Q is the total energy transferred as heat during the process
avg
if T
QSSS
The entropy change is
i
fV
i
fif T
TnC
V
VnRSSS lnln
The Second Law of ThermodynamicsThe second law of thermodynamics and can be written as
amics) thermodynof law (second 0S
Engines
engine)any y,(efficienc ||
||
forpay energy we
get energy we
HQ
W
H
L
H
LC T
T
Q
Q 1
||
||1
Refrigerators
||
||
forpay what we
wantwhat we
W
QK L
LH
L
LH
LC TT
T
QK
||||
||
Entropy from a Statistical ViewExtrapolating from six molecules to the general case of N molecules
ion)configurat ofity (multiplic !!
!
21 nn
NW
equation)entropy s'(Boltzmann lnWkS
A relationship between the entropy S of a configuration of a gas and the multiplicity W of that configuration.
NNNN )(ln!ln
The Stirling’s approximation is :