entropyh, thermo notes,thermodynamics note chapter 6

1. [R Gnyawali / P Timilsina] Page 1
Chapter-6: Entropy
When the first law of thermodynamics was stated, the existence of property, the internal energy,
was found. Similarly, Second law also leads to definition of another property, known as entropy. If first
law is said to be law of internal energy, then second law may be stated to be the law of entropy.
1 Clausius Inequality
Whenever a system undergoes a cycle, ∫ )(
T
Qδ
is zero if the cycle is reversible and negative if the cycle is
irreversible, i.e. in general, 0)( ≤∫ T
Qδ
Proof: Consider a reversible heat engine cycle operating between reservoirs at temperatures TH and TL.
For this cycle, the cyclic integral of the heat transfer, ∫ Qδ , is greater than 0.
0>−=∫ LH QQQδ
Since, TH and TL are constant, from the definition of absolute temperature scale, and for a reversible cycle,
∫ =−= 0
L
L
H
H
T
Q
T
Q
T
Qδ
0=∴ ∫rev
T
Qδ
Now, let’s consider an irreversible cyclic heat engine operating between the same TH and TL and receiving
the same quantity of heat QH. Comparing the irreversible cycle with the reversible one,
revirr WW <
Since QH – QL = W; for both the reversible and irreversible cycles,
revLHirrLH QQQQ ,, −<−
Consequently,
0
0
0
,
,
<∴
<−=
>−=
∫
∫
∫
irr
L
irrL
H
H
irrLH
T
Q
T
Q
T
Q
T
Q
QQQ
δ
δ
δ
Thus, we conclude that 0)( ≤∫ T
Qδ
for all heat engine cycle and similarly it can be proved that 0)( ≤∫ T
Qδ
for refrigerator cycles.
2 Entropy - A Property
There exists a property of a closed system such that a change in its value is equal to ∫
2
1
T
Qδ
for any
reversible process undergone by the system between state 1 and state 2.
revLirrL QQ ,, >∴

2. [R Gnyawali / P Timilsina] Page 2
Let a system undergo a reversible process from state 1 to state 2 along a path A, and let the cycle be
completed along path B, which is also reversible. For reversible cycle, we can write,
1
2
A
B
C
)1(.........0)()(
0
1
2
2
1
eqn
T
Q
T
Q
T
Q
BA =+
=
∫∫
∫
δδ
δ
Now consider another reversible cycle, which proceeds first along path C and is then completed along path
B. For this cycle we can write,
)2(.......0)()(
1
2
2
1
eqn
T
Q
T
Q
T
Q
BC =+= ∫∫ ∫
δδδ
From these equations (1) and (2), we can write,
∫∫ =
2
1
2
1
)()( CA
T
Q
T
Q δδ
Since the quantity
T
Qδ
is same for all reversible paths between states 1 and 2, we conclude that this
quantity is independent of path and it is a function of the end states only. Therefore, it is a property. This
property is called entropy and is denoted by S.
So, dS
T
Q
rev =)(
δ
And for irreversible process,
Let the cycle is made of irreversible process C and reversible process B. Thus this is an irreversible cycle.
So,
)3(......0)()(
1
2
2
1
eqn
T
Q
T
Q
T
Q
BC <+= ∫∫ ∫
δδδ
From above equations (1) and (3)
∫∫ >
2
1
2
1
)()( CA
T
Q
T
Q δδ
Since path A is reversible, and it is a property ∫∫∫ ==
2
1
2
1
2
1
)( CAA dSdS
T
Qδ

3. [R Gnyawali / P Timilsina] Page 3
Thus,
C
C
T
Q
dS ∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
>
2
1
2
1
δ
So Entropy change in an irreversible process,
dS
T
Q
irr <)(
δ
Thus in general, Entropy change
)(
T
Q
dS
δ
≥
Entropy and Second Law of Thermodynamics for an Isolated System
The microscopic disorder of a system is prescribed by a system property is called entropy.
“The entropy S, an extensive equilibrium property, must always increase or remain constant for an
isolated system.”
This is expressed mathematically as,
0≥isolateddS ……………………………..eq(1)
Or,
0)( ≥− IsolatedInitialFinal SS …………………………………eq(2)
Entropy, like our other thermodynamic properties, is defined only at equilibrium states or for quasi-
equilibrium processes. Equation (2) shows that the entropy of the final state is never less than that of the
initial state for any process which an isolated system undergoes.
Entropy is a measure of the molecular disorder of the substance. Larger values of entropy imply larger
disorder or uncertainty and lower values imply more microscopically organized states.
The term entropy production or entropy generation Sgen is considered in eq (1) to eliminate the inequality
sign.
0)( =− isolatedgenSdS δ
Or
0)( =−− IsolatedInitialgenFinal SSS
Here, genSδ is the entropy generated during a change in system state and is always positive or zero.
Principal of increase of entropy (Entropy generation)
The entropy of an isolated system increases in all real processes and is conserved in ideal processes. As a
result of natural processes the entropy of the universe steadily increases.
0=genSδ for reversible process.
0>genSδ for irreversible process. Entropy gets increasing in irreversible process.

4. [R Gnyawali / P Timilsina] Page 4
It is thus proved that the entropy of an isolated system can never decrease. It always increases and remains
constant only when the process is reversible. This is known as the principle of increase of entropy (entropy
generation) or simply entropy principle.
3 Entropy
The entropy of a system is a thermodynamic property which is a measure of the degree of molecular
disorder existing in the system. It describes the randomness or uncertainty of the system. It is a function of
a quantity of heat which shows the possibility of conversion of heat into work. Thus, for maximum
entropy, there is minimum availability for conversion into work and for minimum entropy there is a
maximum availability for conversion into work.
Characteristics:
1. It increases when heat is supplied irrespective of the fact whether temperature changes or not.
2. It decreases when heat is removed whether the temperature changes or not.
3. It remains unchanged in all adiabatic reversible processes.
4. The increase in entropy is small when heat is added at a high temperature and is greater when heat
addition is made at a lower temperature.
4 Lost Work
For an infinitesimal reversible process by a closed system,
)1(............................... eqPdVdUdQ RR +=
In reversible process only,
PdVdW = , But in irreversible process PdVdW ≠
If the process is irreversible,
)2(............................... eqdWdUdQ II +=
Since U is property,
IR dUdU =
From equation (1) and (2),
dWPdVdQdQ
dWdQPdVdQ
IR
IR
−+=
−=−
Dividing on both sides by T
T
dWPdV
T
dQ
T
dQ
IR
−
+⎟
⎠
⎞
⎜
⎝
⎛
=⎟
⎠
⎞
⎜
⎝
⎛
The difference ( )dWPdV − indicates the work that is lost due to irreversibility, and is called the lost work.
The lost work approach zero as the process approaches reversibility as a limit.
5 Entropy-Property and Relation for an ideal gas and Incompressible substances
The Gibbs equation, an important relation in thermodynamics, is given by:
PdVTdSdU −= ……………………eq(1)
This relation relates the equilibrium thermodynamics properties.
PVUH +=
VdPPdVdUdH ++= ………………….eq(2)
From eq(1) and eq(2) yields,
VdPTdSdH += ……………………..eq(3)

5. [R Gnyawali / P Timilsina] Page 5
These two equations are also represented on an intensive basis as
PdvTdsdu −=
vdPTdsdh += …………………eq(4)
The changes in entropy are obtained directly from these equations as
dP
T
V
T
dH
dS
dV
T
P
T
dU
dS
−=
+=
……………………………..eq(5)
Or
dP
T
v
T
dh
ds
dv
T
P
T
du
ds
−=
+=
…………………………………..eq(6)
Ideal Gas Relations:
The internal energy and enthalpy can be expressed as
dTmCdH p= And dTmCdU V=
mRTPV =
From above, eq(4) reduces to
P
dP
mR
T
dTmC
dS
V
dV
mR
T
dTmC
dS
p
V
−=
+=
…………………………eq(7)
For Common Process:
1. Isochoric Process
)ln(
1
2
2
1
12
T
T
mC
T
dT
mCSS
T
dT
mCdS
dTmCQ
VV
V
V
==−
=
=
∫
δ
2. Isobaric Process
)ln(
1
2
2
1
12
T
T
mC
T
dT
mCSS
T
dT
mCdS
dTmCQ
pp
p
P
==−
=
=
∫
δ
3. Isothermal Process 4. Adiabatic Process
0
0
==
=
T
Q
dS
Q
δ
δ
5. Polytropic Process

6. [R Gnyawali / P Timilsina] Page 6
)ln()ln(
)ln(
)ln()ln(
2
1
1
2
12
1
2
2
1
1
2
P
P
mR
V
V
mRSS
V
V
mR
T
Q
dS
P
P
mRT
V
V
mRTQ
==−
==
==
δ
δ )ln()
1
(
1
2
T
T
n
n
mCdS V
−
−
=
γ
Entropy Change for Incompressible Fluid or Solid Substances
Since volume is constant, change in dV is zero.
mCdTdU =
The eq(6) now becomes as
T
mCdT
dS =
)ln(
1
2
2
1
12
T
T
mC
T
dT
mCSS ==− ∫
Isentropic Process for an Ideal Gas and an Incompressible Substances
An isentropic process is a constant-entropy process. If a control mass undergoes a process which is both
reversible and adiabatic, then the second law specifies the entropy change to be zero. Although the
isentropic process might be an idealization of an actual process, this process serves as a limiting process,
for particular applications.
Isentropic Process for an Incompressible Fluid or Solid
The entropy change for incompressible Fluid or Solid is given as:
T
mCdT
dS = ……………..eq(1)
For isentropic process dS = 0, so dT = 0. Thus, an isentropic process is an isothermal process for an
incompressible fluid or solid.
Also, the internal energy is given as;
mCdTdU =
So dU = 0 for an isentropic process. The change in enthalpy is
PVUH +=
VdPPdVdUdH ++= ……………….eq(2)
But dU = 0 for this process and dV = 0 for incompressible fluids or solids.
So, eq(2) becomes as
VdPdH =
In intensive basis, vdPdh = ………………….eq(3)
This last expression is integrated to yield
)( 1212 PPvhh −=− ……………..eq(4)
Since v = constant.
This last expression is particularly useful in adiabatic work considerations of liquid pumps.

7. [R
C
R
is
A
us
en
un
re
A
W
sy
co
m
If
eq
eq
W
is
Th
w
Th
th
Ir
Th
be
Th
Fo
If
R Gnyawali
Concept o
Reversible w
called rever
Available En
seful work b
Let us
ngine is calle
navailable en
Thus,
Wmax =
For gi
For a
ejection is th
Availability:
When a syste
ystem will n
ompletely re
maximum rev
f a system i
quilibrium w
quilibrium w
With a given
in equilibriu
he theoretic
when operatin
hus, the ava
he surroundin
rreversibilit
he actual wo
etween the tw
hus,
or any proce
f real process
/ P Timilsin
of reversi
work: The m
rsible work.
nergy: Avai
by ideal proc
s take an ex
ed available
nergy.
QH = A.E. +
= A.E. = QH
ven tempera
given TH,
he temperatur
m is in equi
not be capa
eversible pro
versible work
is in equilib
with the surr
with the surro
state for any
um with the
al maximum
ng with a res
ailability for
ngs.
ty
ork which a
wo is called
ess, I ≥ 0
ses could tak
na]
ibility, irre
aximum amo
ilable energy
cesses which
ample of a c
energy (AE
+ U.E.
H - U.E.
ature, TH & T
will in
re of the sur
ilibrium with
able of doin
ocess until it
k will have b
brium with
oundings. T
oundings.
y system, the
surrounding
m amount of
servoir at the
a system is
system doe
the process.
ke place in a
reversibili
ount of work
y is the max
h reduces the
cycle heat e
E) and the m
TL
ncrease with
rroundings. T
h the environ
ng any work
reaches a st
been done by
the surroun
That is at pre
e reversible
gs.
f work whic
e constant pr
equal to thi
s is always
.
I =
a completely
ity and A
k that can be
ximum porti
e system to a
engine. The
minimum ene
h decrease of
TO
nment, no sp
k. Therefore
tate in which
y the system
ndings, it mu
essure PO an
work will b
ch can be ob
ressure and t
is maximum
less than the
= Wrev – W
reversible m
vailability
e developed
ion of energ
a dead state.
maximum a
ergy that has
f TL, The lo
pontaneous c
e, if a syste
h it is in equ
m.
ust certainly
nd temperatu
e maximum
btained from
temperature
m reversible w
e idealized r
manner, the r
Pa
ty
by a system
gy which co
amount of w
to be reject
west practic
change of st
em in a giv
uilibrium wit
y be in pres
ure TO. It mu
m when the fi
m a system
p0 and T0 is
work minus
reversible wo
reversibility
age 7
undergoing
ould be conv
work obtaina
ted to the sin
cal temperatu
tate will occ
ven state un
th the enviro
ssure and te
ust also be in
inal state of
at any state
called avail
the work do
ork, and the
would be ze
a problem
verted into
able by this
nk is called
ure of heat
ur, and the
ndergoes a
onment, the
emperature
n chemical
the system
p1 and T1
ability.
one against
difference
ero.