Entropy.pdf .

Department of Mechanical & Manufacturing Engineering, MIT, Manipal 1 of 79
THERMODYNAMICS - I
CHAPTER 4
ENTROPY

THERMODYNAMICS - I
Introduction:
The first law of thermodynamics introduces the
concept of the internal energy U, and this term
helps us to understand the nature of energy, as
defined by the first law.
In a similar way the second law introduces the
concept of entropy S, like internal energy it is also a
thermodynamic property and is defined only in
terms of mathematical operations.

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What exactly is entropy?
Entropy is:
• An extensive property.
• A measure of the disorder of a system , the greater
the disorder, the greater the entropy.
• The state of disorder in a thermodynamic system:
the more energy the higher the entropy.
A measure of the dispersal or degradation of energy.

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• A measure of the disorder or randomness in a
closed system.
• Defined as a measure of unusable energy within a
closed or isolated system (the universe for example).
• As usable energy decreases and unusable energy
increases, "entropy" increases.

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• As usable energy decreases and unusable energy
increases, "entropy" increases.
• Entropy is also a gauge of randomness or chaos
within a closed system. As usable energy is
irretrievably lost, disorganization, randomness and
chaos increase.

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Understanding entropy:
Example 1
If the particles represents
gas molecules at normal
temperatures inside a
closed container, which of
the illustration came first ?
One of the ideas involved in
the concept of entropy is that
nature tends from order to
disorder in isolated systems
This tells us that the right
hand box of molecules
happened before the left
Using Newton's laws to
describe the motion of the
molecules would not tell you
which came first.

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Example 2
If you tossed bricks off a truck, which kind of pile of
bricks would you more likely produce ?
Disorder is more probable than order

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Example 3
For a glass of water the
number of molecules is
astronomical
The jumble of ice chips
may look more disordered
in comparison to the glass
of water which looks
uniform and homogeneous
Ice chips place limits on the number of
ways the molecules can be arranged
but not the water
The water molecules in the glass can be arranged in many more
ways They have greater "multiplicity" and therefore greater
entropy

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• Solids have lesser entropy than liquids.
• Statistically, in solids the atoms or molecules are
in their fixed places, whereas in liquids and even
more in gases you never know exactly where to
find them.
• The fixed sequence of atoms in solids reflects a
higher ordered state.

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Prove that two reversible adiabatic paths
cannot intersect each other:
• Let AC and BC be the two
reversible adiabatic
processes, intersect each
other at point C.
• Let AB represents
isothermal process. The
three reversible processes
constitute a thermodynamic
cycle.

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Prove that two reversible adiabatic paths
cannot intersect each other:
• The net area inside the loop is a measure of
work done in the cycle. But such a cycle is
impossible according to Kelvin Planck
Statement of II law.
• It is because, two thermal reservoirs are
necessary for any heat transfer to take place.
• Hence the initial assumption of intersection of
two reversible adiabatic lines is wrong.

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Substitution of a reversible process by the reversible
isothermal and reversible adiabatic processes:

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• Applying the first law for
i-f, i
Qf
= Uf
– Ui
+ i
Wf
• Applying the first law for
i-a-b-f, ( i
Qf
)i-a-b-f
= Uf
– Ui
+ ( i
Wf
)i-a-b-f

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• We know i
Wf
= (i
Wf
)i-a-b-f
(area under the curves are
same).
i
Qf
= (i
Qf
)i-a-b-f
= i
Qa
+ a
Qb
+ b
Qf
Since i
Qa
and b
Qf
are zero
i
Qf
= (i
Qf
)i-a-b-f
= a
Qb
Thus, any reversible path may be substituted by a reversible
adiabatic, a reversible isotherm and a reversible adiabatic
between the same end states.

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Claussius Theorem:
The cyclic integral of for a reversible cycle is
zero. In symbols

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is known as ENTROPY
Definition:
Entropy, S is a property of system such that its
increase S2
- S1
as the system changes from state 1
to state 2 is given by,
In differential form equation (1) can be written as

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Total entropy is denoted by capital letter S and lower
case ‘s’ represents the specific entropy, i.e. entropy /
unit mass.
The Clausius Inequality:
When any system undergoes a cyclic process, the
integral around the cycle of is less than or
equal to zero.
In symbols,

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Proof:
For any reversible cycle from Clausius theorem, we
have
From the Carnot’s theorem we know that
Hence,
For a reversible engine

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Using sign conventions of +ve for absorption of heat
and –ve for the rejection of heat , we get,

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From this we see that the algebraic sum of the ratios of
the amounts of heat transferred to the absolute
temperature for a cyclic irreversible process is always
less than zero,
Combining equations (2) and (7), we get
This is known as Clausius
Inequality

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Clausius Inequality provides the criterion of the
reversibility of a cycle

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Illustration of Clausius Inequality:
Example 1: Consider the flow of heat from the reservoir at
temp T1
to that at T2
across the conductor as shown.
Conductor is the system. In the steady state there is no
change in the state of the system
Since Q1
= 1000 kJ ,δQ2
= -1000 kJ

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Example 2. E is the system which executes a
cyclic process.
However, if E were a reversible
engine, then work δW would
have been,

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In symbols, for arbitrary paths
A and B,
Consider a system which
executes a reversible cyclic
process 1A2C1, we have
Entropy is a property:

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Similarly, for the reversible cyclic process 1B2C1, we
can write,
Equations (1) – (2) gives

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Path A and path B are arbitrary and has the
same value for any reversible path between (1) and (2)
, hence from the definition of entropy we may write (
S2
- S1
) has the same value for any reversible path
between 1 and 2 .
Therefore ENTROPY is a property.

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Causes of Entropy Change:
1. Heat Transfer to or from a system
• Heat transfer to a system increases the entropy of
that system.
• Heat transfer from a system decreases the
entropy of that system.
2. Mass flow into or out of a control volume.
3. Irreversibilities such as friction, heat transfer
through a finite temperature difference, fast
expansion or compression always cause the
entropy to increase

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Calculation of entropy change for different
process:
Entropy change in IRREVERSIBLE process:
For a process that occurs
irreversibly, the change in
entropy is greater than the heat
change divided by the absolute
temperature.
In symbols,

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Consider an arbitrary irreversible cycle 1-A-2-B-1,
From the Clausius Inequality
Since the entropy is a thermodynamic property,
we can write

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For a reversible process
Using equation (1), for an irreversible cycle,

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For small changes in states, we write
The subscript I
represents the
irreversible process
The equation (7) states that in an irreversible process
the change in entropy is greater than

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Where equality sign is for reversible process and
inequality sign is for irreversible process.

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IMPORTANT:
The effect of irreversibility is always to increase the
entropy of the system. If an isolated system is
considered, from the first law of thermodynamics the
internal energy of the system will remain constant.
From the above expression
The entropy of an isolated system either
increases or remains constant. This is a corollary
of the second law of thermodynamics and this
explains the principle of increase in entropy.

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Mathematical Expression of the Second Law:
for reversible processes
for irreversible processes
The above equations may be regarded as the
analytical expressions of the second law of
thermodynamics.

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Important Relations for Entropy
The first law for a closed system is given by,
If the kinetic and potential energies are very small, the
equation (1) can be written as,

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For a reversible process,
Therefore equation (2) becomes
The work done at the boundary of a system during a
reversible process is,

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Substituting this in equation (3), we get
Equation (5) involves only changes in properties and
involves no path functions.
This equation is valid for all processes, both reversible
and irreversible and that it applies to the substance
undergoing a change of state as the result of flow
across the boundary of the open and closed systems.

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Since H = U + PV and
dH = dU + d(PV)
We get, dU = dH – d(PV)
On substitution this in equation (5), we obtain

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First and Second Law Combined Relations:
Holds good for any process, reversible or
irreversible and for any system
Holds good for any process undergone by
a closed stationary system
Holds good for a reversible process
undergone by a closed system when pdV
work is present
True only for a reversible process

THERMODYNAMICS - I
This equation holds good for any process reversible or
irreversible undergone by a closed system, since it is a
relation among properties which are independent of
the path.
This equation also relates only properties of the closed
system. There is no path function term in the equation.
Hence the equation holds good for any process.

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Principle of the Increase of Entropy:
Entropy Change for the System + Surroundings
Using the principle of increase
in entropy

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The total change of entropy for the combined system

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The change in the entropy of the surroundings
would be,
The same conclusion can be had for an open
system, because the change in the entropy of the
system would be

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This means that processes involving an interaction of
a system and its surroundings will take place only if
the net entropy change is greater than zero or in the
limit remains constant. The entropy attains its
maximum value when the system reaches a stable
equilibrium state from a non equilibrium state.

THERMODYNAMICS - I
Entropy for an Ideal Gas :
Let the ideal gas undergoes a change of state from
state 1 to state 2. Let T1
, V1
and T2
, V2
be the
temperatures and volumes at state 1 and 2
respectively.

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Integrating,

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For general case ( process ), change of entropy is
given by,
or in specific values,

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Special Cases:
• Constant temperature process ( Isothermal process),
T = Constant
• Constant volume process , V = Constant

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• Constant pressure process , P = Constant
• Reversible adiabatic process or isentropic process,
S = Constant

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• Polytropic process
Carnot cycle on a T-s diagram:
Processes:
Process a-b: Isentropic Process
Process b-c: Isothermal Process
Process c-d: Isentropic Process
Process d-a: Isothermal Process

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Efficiency of the Carnot engine is given by,

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Application of Entropy
Mixing of two fluids
System 1 is consisting of a
fluid of mass m1
, specific
heat c1
and temperature t1
.
System 2 is comprising of
fluid of mass m2
, specific
heat c2
and temperature t2
.

THERMODYNAMICS - I
Application of Entropy
Mixing of two fluids
These two systems
together constitute a
composite system in an
adiabatic enclosure.
When the partition is
removed, two fluids mix
together and at equilibrium,
let tf
be the final
temperature such that t1
>
tf
> t2
.

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Since energy interaction is at confined space,
system is isolated. Heat lost by system 1 will be
equal to heat gained by system 2.
Entropy change for the system 1 is
(Negative)

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Entropy change for the system 2 is
(Positive)
The mixing process is irreversible. To determine
net entropy change, the irreversible paths are
replaced by reversible paths. If

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Maximum work obtainable from two finite bodies
at temperature T1
and T2
Consider two finite bodies 1 and 2
at T1
and T2
respectively.
The heat engine produces work.
The temperature of body 1 reduces
whereas that of body 2 increases.
When both the bodies attain the
final temperature Tf
, the heat
engine stops operating.

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Let the bodies remain at constant temperature and
undergo no change of phase.
Total heat drawn from body 1 is
Q1
= mCp
(T1
– Tf
)
where Cp
= specific heat for two bodies at constant
temperature.
Heat rejected to body 2,
Q2
= mCp
(Tf
– T2
)

THERMODYNAMICS - I
Hence, total work done by the heat engine,
W = Heat added – Heat rejected
W = mCp
(T1
– Tf
) - mCp
(Tf
– T2
)
W = mCp
[ T1
+ T2
– 2Tf
]
For given values of Cp
, constant values of T1
and T2
,
the work obtained will be maximum when Tf
is
minimum.

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Entropy change for the body 1 is
(Negative)
Entropy change for the body 2 is
(Positive)

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By the principle of increase of entropy,
For minimum value of Tf
Hence, maximum work output is given by

THERMODYNAMICS - I
Maximum work obtainable from a finite body
and TER
The finite body is at
temperature T and TER is at T0
.
The finite body has a thermal
capacity C such that T > T0
.
When the heat Q is taken by
HE, temperature of body
decreases, but that of TER will
remain unchanged at T0
.

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Maximum work obtainable from a finite body
and TER
Heat engine stops working
when T approaches T0
.
The amount of work delivered is
W and the heat rejected to TER
is Q – W.

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For the body,
For TER,
By the principle of increase of entropy,

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For reversible process,
Therefore,
But

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