4.3 Solutions 277
ES
ET
=
(∂P/∂V)S
(∂P/∂V)T
=
(∂P/∂V)S
(
∂T
∂V
)
S
(∂P/∂S)T(
∂S
∂V
)
T=
(∂T/∂V)S
(
∂S
∂P
)
T
(∂T/∂P)S(
∂S
∂V
)
T=
(∂P/∂S)V(∂V/∂T)P
(∂V/∂S)P(∂P/∂T)V
from the relations given in Problems 4.21 and 4.22∴ES
ET
=
(∂S/∂T)P
(∂S/∂T)V
=
(∂Q/∂T)P
(∂Q/∂T)V
=
CP
CV
=γ4.37
(∂V/∂T)S
(∂V/∂T)P
=
1
(∂T/∂V)S(∂V/∂T)P
=
1
−
(
∂P
∂S
)
V(
∂V
∂T
)
P
where we have used Eq. (23) of Problem 4.22.
Writing
(
∂P
∂S)
V=
(
∂P
∂T
)
V(
∂T
∂S
)
V=
(∂P/∂T)V
(∂S/∂T)V
(∂V/∂T)S
(∂V/∂T)P
=
(∂S/∂T)V
(∂P/∂T)V(∂V/∂T)P
=
(∂S/∂T)V
−(CP−CV)/T
(by Eq. (4.1) of Problem 4.27T(∂S/∂T)V
−(CP−CV)=
CV
−(CP−CV)
=
1
1 −γ4.38
(∂P/∂T)S
(∂P/∂T)V
=
1
(∂T/∂P)S(∂P/∂T)V
=
1
(
∂V
∂S
)
P(
∂P
∂T
)
V=1
(
∂V
∂T
)
P(
∂T
∂S
)
P(
∂P
∂T
)
V=
(∂S/∂T)P
(
∂V
∂T
)
P(∂P/∂T)V
=
T(∂S/∂T)P
(CP−CV)
=
CP
(CP−CV)
=
γ
γ− 1where we have used Eq. (4.24) of Problem 4.22 and the relationCP=T(
∂S
∂T
)
P