The Carnot cycle
qr
T
T2
T1
2
a
b
d
c
qr
1
s
Entropy changes in reversible processes
đqr  du  Pdv
đqr du P
ds 

 dv
T
T T
Various cases:
•
Adiabatic process: đqr = 0, ds = 0, s = constant. A reversible
adiabatic process is isentropic. THIS IS NOT TRUE
FOR AN IRREVERSIBLE PROCESS!
•
Isothermal process:
s2  s1   ds  
2
1
đqr qr

T
T
Entropy changes in reversible processes
đqr  du  Pdv
đqr du P
ds 

 dv
T
T T
Various cases:
•
Isothermal: ideal gas case (du = 0; đq = -đw; Pv = RT )
2
s2  s1   ds  
1
2
1
2 R
 v2 
P
dv   dv  R ln  
1 v
T
 v1 
Entropy changes in reversible processes
đqr  du  Pdv
đqr du P
ds 

 dv
T
T T
Various cases:
•
Isochoric process: We assume u = u(v,T) in general, so
that u = u(T) in an isochoric process. Therefore, as in
the case for an ideal gas, du = cvdT. Thus,
s2  s1  
2
1
 T2 
dT
cv
 cv ln   ,
T
 T1 
provided cv is independent of T over the integration
(really only true for ideal gas, but often good approx.).
What about reversible paths?
P
DS > 0
(1)
1. Isobaric (P = const)
(2)
2. Isothermal (PV = const)
3. Adiabatic (PVg = const)
(3)
(4)
DS < 0
4. Isochoric (V = const)
DS = 0
V
•For a given reversible path, there is some associated physics.
The combined 1st and 2nd Laws
The 2nd law need not be restricted to reversible processes:
dU  đQ  đW
  đQr      đWr   
 TdS  PdV
• đQ is identifiable with TdS, as is đW with PdV, but only
for reversible processes.
• However, the last equation is valid quite generally, even
for irreversible processes, albeit that the correspondence
between đQ & TdS, and đW & PdV, is lost in this case.
Entropy associated with irreversible processes
Universe at
temperature T2
T1
Q
T2 > T1
Entropy associated with irreversible processes
Isobaric
Cp1, T1i
dQ
Cp2, T2i
Entropy associated with irreversible processes
Isobaric
Cp1, T1i
Cp2, T2i
dQ2
dQ1
Tf
Infinite heat reservoir
Entropy associated with irreversible processes
Isobaric
Cp1, T1i
Cp2, T2i
dQ2
dQ1
Tf
Finite heat reservoir
Qin = Qout, otherwise Tf would not remain constant
Entropy of mixing
final state f
initial state i
n1
T, P
n2
T, P
n = n1 + n2
T, P = P1 + P2
Definition of partial pressure Pj of a constituent in a mixture:
Pj  x j P.
Here, P is the pressure of the mixture, and xj is the kilomole fraction
of the jth constituent gas:
n
n
xj 
j
n
i
i

j
n
Entropy of mixing
final state f
initial state i
n1
T, P
n2
T, P
n = n1 + n2
T, P = P1 + P2
P is the pressure of the mixture; Pj is the partial pressure and xj the
kilomole fraction of the jth constituent gas, where
Pj  x j P
Then,
and
xj 
nj
n
i
i
DS  nR  x1 ln x1  x2 ln x2 

nj
n
Entropy of mixing
0.8
xlnx-[(1-x)ln(1-x)]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.3
x
0.6
0.9
Maxwell’s demon