This is a useful document

1. Heat reservoir at
temperature T2 > T1
Cold reservoir at
temperature T1 < T2
Heat
Engine
Q2
Q1
Q  heat
W  work
both in Joules
Conversion of Heat to Work (a heat engine)
W

2. Environment at
temperature Th > Tl
Refrigerator, inside
temperature Tl < Th
Refrig-
erator
Qh
Ql
W
Cooling via Work (Carnot Refrigerator)
/ l
R l
h l
l
h l
Q
Q W
Q Q
T
T T
  




3. Environment
(Home) Th > Tl
Resevoir (ground)
Tl < Th
Heat
Pump
Qh
Ql
W
Carnot Heat Pump
/
1 1
h
HP h
h l
h l
h l h l
Q
Q W
Q Q
T T
T T T T
  

   
 
Always > 100%

4. The Clausius Inequality and the 2nd Law
2
đQ
1
đQ
Divide any reversible cycle into a
series of thin Carnot cycles, where
the isotherms are infinitesimally
short:
P
v
• We have proven that the quantity dS = dQr/T is a state
variable, since its integral around a closed loop is equal to
zero, i.e. the integration of differential entropy, dS, is
path independent!


5. The Clausius Inequality and the 2nd Law
2
đQ
1
đQ
Divide any reversible cycle into a
series of thin Carnot cycles, where
the isotherms are infinitesimally
short:
P
v
For a reversible process!
Leads to the definition of
entropy for a reversible
process:
r
đQ
dS =
T


6. The Clausius Inequality and the 2nd Law
2
đQ
1
đQ
Divide any reversible cycle into a
series of thin Carnot cycles, where
the isotherms are infinitesimally
short:
P
v
• There is one major caveat: the cycle must be reversible.
In other words, the above assumes only configuration
work (PdV) is performed.
• If the cycle additionally includes dissipative work, it is
not clear how to include this in the above diagram.


7. 0
1 1 1
1
0
1 1
1
0 1
1 1
0 1 0
0
1
1
1
/ 1
1
/
T
Q W Q
T
T
Q W
T
T T
W T
T T T
T
Q
T
  
 



 
 
 
 
 

   
 
 
 
0
st nd
nd
0
0
and W
1 law implies W 0 [violates 2 law]
Run backwards and set W 0 [satisfies 2 law]
Therefore W 0
and in the limit of infinetesimal chan
i i
i i
i
i
i i
i i i
i i
T
Q Q W
T
Q
Q
T Q
W Q Q T
T T
 

 
 
 
 
    
 
  
ges 0
dQ
T


The Clausius Inequality