The Clausius Inequality
• Expressions of inequality/equality relating to heat flow at
a fixed temperature.
• The expression is required for the derivation of an
equation for entropy – which is our next main topic.
• Derived from a “thought experiment” using Carnot
engines acting in a series.
Carnot Cycle
Pressure
a
•
Q1
P =
nRT 1
V
b
•
Q=0
P =
nRT
V
Q=0
•
2
T1
P =
const .
V
d
Q2
•c T2
Volume

Heat Flows in a Carnot Cycle
Hot Reservoir, T1
T1
Q1
T2
=
Q1
Q2
C
W
Q2
Cold Reservoir, T2
Q 1= Q2
T1
T2
For a Carnot cycle, some of the heat into the cycle is converted
to work so that Q1 > Q2. We also know that
Q
Q
1
T1
Such that
Q1
T1

Q2

2
T2
0
T2
One could also consider the small amount of reversible heat flow
dQrev that flows at a temperature T at each point in the cycle.
The net heat flow is equal to the sum of the differential flows:

cycle
Q
T
rev

Q1
T1

 Q2
0
T2
From the definition of an integral, we find for the entire cycle
rev
that
Q
 T 0
But is this generally true for any cycle?
~
T~
Principal reservoir
~
Q
~
Q
dW1
d Q1
~
T
d Q1
T
T1
~
T
T1
C1
dQ1
T1
• dQ1
1 2 •
Working
substance in
a cycle
Derivation of the Clausius Inequality
Auxiliary
reservoirs
Principal reservoir
d Q1
~
T
dW1
d Q1
~
T
T1
~
T
dQ 2
~
T
~
T
dQ 2
T1
C1
T2
~
T
T2
C2
dQ1
dQ2
T1
T2
dQ1
•
dQ2
2 •
Working 3 • Add N Carnot
engines in total
substance in
a cycle
1
Composite
System
~
T
Derivation of the Clausius Inequality
dW2
Analysis of Composite Device
The total heat supplied by the reservoir to the
composite device in a cycle with N engines is:
N
Q =
∑
~
T
dQi T
i
i =1
The total work done by the composite device in a
cycle is
N
W =
∑d W
i
i =1
In an entire cycle, DU for the working substance and
the composite device is 0. Then the First Law says:
DU = 0 = W + Q
An important conclusion is that Q = -W.
Principal reservoir
Q = -W > 0
~
T
N T~
Q =
i
T1
dQ i
Violates Kelvin
statement of the
2nd Law!
N
W =  dW i
i
What can be
allowed?
Q= - W
Composite
system
Derivation of the Clausius Inequality
~
T
Principal reservoir
Allowed:
Q
Q= - W < 0
W
Composite
system
Derivation of the Clausius Inequality
~
T
Principal reservoir
Allowed:
Q
Q=-W =0
(Sum of +ve and
–ve dQ and dW)
W
Composite
system
Derivation of the Clausius Inequality
Results Allowed by Second Law
N
~
Q =T
Q < 0 or Q = 0, thus
∑T
i =1
~
But T ≠0, so we can divide by
N
dQi
∑
i 1
dQi
i
≤0
~
T
≤0
Ti
In the limit of small dQi, we can integrate over the entire
cycle:
dQ
∫T
≤0
Clausius Inequality
o
where To is the temp. of the reservoir (external heat source) and
the circle represents integration over the entire cycle.
Reversible Cycles
If the cycle at the centre of the composite device is
reversible, then it can be run in reverse.
The opposite result is then obtained. In the composite device,
work is done ON the system, and heat is given OFF to the
reservoir.
~
T
Q
Allowed:
0Q=-W
W
Reversible Cycles
•Thus, there are restrictions on Q when the cycle is operated in
reverse.
• But the cycle can still also be operated in the forward
N
direction such that:
dQi
~
W =Q =T
∑T
i =1
≥0
i
• Only one solution satisfies both requirements for
N
forward and reverse cycles:
dQi
~
Q =T
• Integrating:
dQ R
∫T
∑T
i =1
= 0
=0
i
Clausius Equality
In a reversible process, the temperature of the system and
reservoir are equal: T.