Chapter 8: Thermodynamics
35
From this one can derive Maxwell’s relations:
∂
T
∂
V
S
=

∂
p
∂
S
V
,
∂
T
∂
p
S
=
∂
V
∂
S
p
,
∂
p
∂
T
V
=
∂
S
∂
V
T
,
∂
V
∂
T
p
=

∂
S
∂
p
T
From the total differential and the definitions of
C
V
and
C
p
it can be derived that:
T dS
=
C
V
dT
+
T
∂
p
∂
T
V
dV
and
T dS
=
C
p
dT

T
∂
V
∂
T
p
dp
For an ideal gas also holds:
S
m
=
C
V
ln
T
T
0
+
R
ln
V
V
0
+
S
m
0
and
S
m
=
C
p
ln
T
T
0

R
ln
p
p
0
+
S
m
0
Helmholtz’ equations are:
∂
U
∂
V
T
=
T
∂
p
∂
T
V

p ,
∂
H
∂
p
T
=
V

T
∂
V
∂
T
p
for an enlarged surface holds:
d
W
rev
=

γ
dA
, with
γ
the surface tension. From this follows:
γ
=
∂
U
∂
A
S
=
∂
F
∂
A
T
8.6
Processes
The
efficiency
η
of a process is given by:
η
=
Work done
Heat added
The
Cold factor
ξ
of a cooling down process is given by:
ξ
=
Cold delivered
Work added
Reversible adiabatic processes
For adiabatic processes holds:
W
=
U
1

U
2
. For reversible adiabatic processes holds Poisson’s equation:
with
γ
=
C
p
/C
V
one gets that
pV
γ
=
constant. Also holds:
T V
γ

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 Spring '10
 Ye
 Thermodynamics, Adiabatic process, reversible adiabatic processes, ∂V ∂T

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