for

3.  The Entropy is a thermodynamic property of a
working substance and serves as a valuable tool in
the second law analysis of engineering devices.
 Entropy is a function of a quantity of heat which
shows the possibility of conversion of that into
work.
 Entropy is a thermodynamic property; it can be
viewed as a measure of disorder i.e. More
disorganized a system the higher its entropy.

4.  The Clausius Theorem is a mathematical
explanation of the Second Law of
Thermodynamics.
 Also referred to as the “Inequality of Clausius”,
the theorem was developed by Rudolf Clausius
who intended to explain the relationship
between the heat flow in a system and the
entropy of the system and its surroundings.
 The Clausius Theorem was first published in
1862in Clausius’ sixth memoir, “On the
Application of the Theorem of the Equivalence
of Transformations to Interior Work”.

6.  “ When a system undergoes a complete cyclic
process, the integral of around the cycle is less
than zero.”
 Mathematically : ( ) ≤ 0
 δQ is energy flow into the system due to heating
and T being absolute temperature of the body
when that energy is absorbed.
 The following equation must be found true for
any cyclical process that is possible, reversible or
not.

7.  Consider a reversible engine R and irreversible engine I
working between two thermal reservoirs at temperatures TH
and TL.
 Efficiency of reversible engine is :
where δQH = heat added, δQL = heat rejected.
 Efficiency of irreversible engine is :
 We know that efficiency of reversible engine is more than
that of irreversible engine under same temperature limit.
∴ ηR > ηI ∴ ( )R > ( )I
∴ ( )R > ( )I (∵ for reversible engines )

8.  ∴ ( ) < ( )I
 ∴ ( )I < ( )I
 ∴ ( )I - ( )I < 0
 We know that, heat added ( δQH ) should be positive
and heat rejected( δQL ) should be negative.
 ∴ ( )I - (- )I < 0

9.  ∴ ( )I + ( )I < 0
 Considering complete original irreversible cycle :
 ∴
 ∴ ∮ ( ) < 0 for an irreversible cycle.
 According to Clausius Theorem ∮ = 0 for reversible
cycle.
 Combining results for reversible and irreversible cycle,
we get :
This expression is known as
Clausius Inequality.

11. On a T-S diagram, the area under the
process curve represents the heat transfer
for internally reversible processes.

12. δQint, rev = T dS
Qint, rev =
Qint, rev = T0 ΔS
δqint, rev = T ds
qint, rev =
qint, rev = T0 Δs

13. Mollier diagram: The h-s diagram
For adiabatic steady-flow devices, the vertical
distance Δh on an h-s diagram is a measure of
work, and the horizontal distance Δs is a
measure of irreversibility's.