1)
d
T
=
(
∂
T
∂
P
)
S
,
N
,
.
.
.
d
P
{\displaystyle dT=\left({\frac {\partial T}{\partial P}}\right)_{S,N,...}dP}
comme
(
∂
T
∂
P
)
S
.
(
∂
P
∂
S
)
T
.
(
∂
S
∂
T
)
P
=
−
1
{\displaystyle \left({\frac {\partial T}{\partial P}}\right)_{S}.\left({\frac {\partial P}{\partial S}}\right)_{T}.\left({\frac {\partial S}{\partial T}}\right)_{P}=-1}
donc
(
∂
T
∂
P
)
S
=
−
(
∂
S
∂
P
)
T
(
∂
S
∂
T
)
P
=
+
(
∂
V
∂
T
)
P
(
∂
S
∂
T
)
P
=
α
.
V
(
C
p
T
)
{\displaystyle \left({\frac {\partial T}{\partial P}}\right)_{S}=-{\frac {\left({\frac {\partial S}{\partial P}}\right)_{T}}{\left({\frac {\partial S}{\partial T}}\right)_{P}}}=+{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}}{\left({\frac {\partial S}{\partial T}}\right)_{P}}}={\frac {\alpha .V}{\left({\frac {{\mathcal {C}}_{p}}{T}}\right)}}}
comme on a pour N moles :
C
p
=
N
.
c
p
e
t
V
=
N
.
v
{\displaystyle {\mathcal {C}}_{p}=N.c_{p}\ \ \ \ et\ \ \ \ V=N.v}
alors
d
T
=
(
∂
T
∂
P
)
S
d
P
=
T
.
α
.
v
c
p
.
d
P
{\displaystyle dT=\left({\frac {\partial T}{\partial P}}\right)_{S}dP={\frac {T.\alpha .v}{c_{p}}}\ .\ dP}
Pour une petite variation δP, on aura une variation δT telle que:
δ
T
≈
T
.
α
.
v
c
p
.
δ
P
{\displaystyle \delta T\ \ \approx \ \ {\frac {T.\alpha .v}{c_{p}}}\ .\ \delta P}
de même, on aura:
d
μ
=
(
∂
μ
∂
P
)
S
d
P
{\displaystyle d\mu =\left({\frac {\partial \mu }{\partial P}}\right)_{S}dP}
et
d
μ
=
v
.
d
P
−
s
.
d
T
{\displaystyle d\mu =v.dP-s.dT}
on en déduit:
(
∂
μ
∂
P
)
S
=
v
−
s
.
(
∂
T
∂
P
)
S
=
v
−
s
.
T
.
α
.
v
c
p
{\displaystyle \left({\frac {\partial \mu }{\partial P}}\right)_{S}=v-s.\left({\frac {\partial T}{\partial P}}\right)_{S}=v-s.{\frac {T.\alpha .v}{c_{p}}}}
donc
δ
μ
≈
{
v
−
s
.
T
.
α
.
v
c
p
}
.
δ
P
{\displaystyle \delta \mu \ \ \approx \ \ \ \left\{v-s.{\frac {T.\alpha .v}{c_{p}}}\right\}\ .\ \delta P}
2)
χ
S
=
−
1
V
.
(
∂
V
∂
p
)
S
{\displaystyle \chi _{S}=-{\frac {1}{V}}.{\Bigl (}{\frac {\partial V}{\partial p}}{\Bigr )}_{S}}
(
∂
V
∂
P
)
S
=
−
(
∂
S
∂
P
)
V
(
∂
S
∂
V
)
P
=
−
(
∂
S
∂
T
)
V
.
(
∂
T
∂
P
)
V
(
∂
S
∂
T
)
P
.
(
∂
T
∂
V
)
P
=
−
c
v
c
p
.
(
∂
T
∂
P
)
V
.
1
α
.
V
{\displaystyle \left({\frac {\partial V}{\partial P}}\right)_{S}=-{\frac {\left({\frac {\partial S}{\partial P}}\right)_{V}}{\left({\frac {\partial S}{\partial V}}\right)_{P}}}=-{\frac {\left({\frac {\partial S}{\partial T}}\right)_{V}.\left({\frac {\partial T}{\partial P}}\right)_{V}}{\left({\frac {\partial S}{\partial T}}\right)_{P}.\left({\frac {\partial T}{\partial V}}\right)_{P}}}=-{\frac {c_{v}}{c_{p}}}.\left({\frac {\partial T}{\partial P}}\right)_{V}.{\frac {1}{\alpha .V}}}
(
∂
T
∂
P
)
V
=
−
(
∂
V
∂
P
)
T
(
∂
V
∂
T
)
P
=
−
χ
T
.
V
α
.
V
=
−
χ
T
α
{\displaystyle \left({\frac {\partial T}{\partial P}}\right)_{V}=-{\frac {\left({\frac {\partial V}{\partial P}}\right)_{T}}{\left({\frac {\partial V}{\partial T}}\right)_{P}}}=-{\frac {\chi _{T}.V}{\alpha .V}}=-{\frac {\chi _{T}}{\alpha }}}
donc
:
χ
S
=
+
1
V
.
c
v
c
p
.
χ
T
α
.
(
α
.
V
)
=
c
v
c
p
.
χ
T
{\displaystyle :\ \ \ \ \chi _{S}=+{\frac {1}{V}}.{\frac {c_{v}}{c_{p}}}.{\frac {\chi _{T}}{\alpha }}.(\alpha .V)={\frac {c_{v}}{c_{p}}}.\chi _{T}}
et
c
P
c
v
=
χ
T
χ
S
{\displaystyle {\frac {c_{P}}{c_{v}}}={\frac {\chi _{T}}{\chi _{S}}}}
comme
:
c
p
−
c
v
=
T
.
v
.
α
2
/
χ
T
{\displaystyle :\ \ \ \ \ \ c_{p}-c_{v}=T.v.\alpha ^{2}/\chi _{T}}
On en déduit que:
χ
T
−
χ
S
=
T
.
v
.
α
2
c
P
{\displaystyle \chi _{T}-\chi _{S}={\frac {T.v.\alpha ^{2}}{c_{P}}}}