4.3 Solutions 273
Use (2) and (3) in (4)CP−CV=−T
(
∂P
∂V
)
T(
∂V
∂T
) 2
P(5)
CP−CV=−T
(
∂V
∂P
)
T(
∂P
∂T
) 2
V(6)
Equation (5) can be written in terms of the bulk modulusEat constant tem-
perature and the coefficient of volume expansion∝.E=−(
∂P
∂V/V
)
; α=1
V
(
∂V
∂T
)
(7)
Cp−Cν=TEα^2 V (8)4.30 TakingTandVas independent variables
S=f(T,V)dS=(
∂S
∂T
)
VdT+T(
∂S
∂V
)
TdVMultiplying byT,TdS=T(
∂S
∂T
)
VdT+T(
∂S
∂V
)
TdV=CVdT+T(
∂S
∂V
)
TdVBut(
∂S
∂V
)
T=
(
∂P
∂T
)
ν∴TdS=CVdT+T(
∂P
∂T
)
VdVAlso,
(
∂P
∂T)
V=−
(
∂P
∂V
)(
∂V
∂T
)
P∴TdS=CVdT−T(
∂P
∂V
)(
∂V
∂T
)
PdVIntroducing relationsα=V^1 (∂V/∂T)PandET=−V(∂P/∂V)Tfor volume
coefficient of expansion and isothermal elasticity
TdS=CVdT+TαETdV