C
p
−
C
V
=
(
∂
H
∂
T
)
p
−
(
∂
U
∂
T
)
V
=
(
∂
(
U
+
p
V
)
∂
T
)
p
−
(
∂
U
∂
T
)
V
=
(
∂
U
∂
T
)
p
+
p
(
∂
V
∂
T
)
p
−
(
∂
U
∂
T
)
V
{\displaystyle C_{p}-C_{V}=\left({\frac {\partial H}{\partial T}}\right)_{p}-\left({\frac {\partial U}{\partial T}}\right)_{V}=\left({\frac {\partial (U+pV)}{\partial T}}\right)_{p}-\left({\frac {\partial U}{\partial T}}\right)_{V}=\left({\frac {\partial U}{\partial T}}\right)_{p}+p\left({\frac {\partial V}{\partial T}}\right)_{p}-\left({\frac {\partial U}{\partial T}}\right)_{V}}
Entalpie
H
{\displaystyle H}
je definována vztahem
H
=
U
+
p
V
{\displaystyle H=U+pV}
kde
U
{\displaystyle U}
je vnitřní energie soustavy,
p
{\displaystyle p}
je její tlak a
V
{\displaystyle V}
objem.
Vnitřní energie je funkcí teploty a objemu, tudíž
(
∂
U
∂
T
)
p
{\displaystyle \left({\frac {\partial U}{\partial T}}\right)_{p}}
je nutno přepsat jako
(
∂
U
(
T
,
V
(
p
,
T
)
)
∂
T
)
p
{\displaystyle \left({\frac {\partial U(T,V(p,T))}{\partial T}}\right)_{p}}
(
∂
U
(
T
,
V
(
p
,
T
)
)
∂
T
)
p
=
(
∂
U
∂
T
)
V
+
(
∂
U
∂
V
)
T
(
∂
V
∂
T
)
p
{\displaystyle \left({\frac {\partial U(T,V(p,T))}{\partial T}}\right)_{p}=\left({\frac {\partial U}{\partial T}}\right)_{V}+\left({\frac {\partial U}{\partial V}}\right)_{T}\left({\frac {\partial V}{\partial T}}\right)_{p}}
Po dosazení do odvození dostaneme
C
p
−
C
V
=
p
(
∂
V
∂
T
)
p
+
(
∂
V
∂
T
)
p
(
∂
U
∂
V
)
T
=
(
∂
V
∂
T
)
p
[
p
+
(
∂
U
∂
V
)
T
]
{\displaystyle C_{p}-C_{V}=p\left({\frac {\partial V}{\partial T}}\right)_{p}+\left({\frac {\partial V}{\partial T}}\right)_{p}\left({\frac {\partial U}{\partial V}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{p}\left[p+\left({\frac {\partial U}{\partial V}}\right)_{T}\right]}
Z diferenciálu definice vnitřní energie a Maxwellových relací dostaneme
(
∂
U
∂
V
)
T
=
T
(
∂
S
∂
V
)
p
−
p
=
T
(
∂
p
∂
T
)
V
−
p
{\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial S}{\partial V}}\right)_{p}-p=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}
Dalším dosazením do odvození se výraz změní na
C
p
−
C
V
=
T
(
∂
V
∂
T
)
p
(
∂
p
∂
T
)
V
{\displaystyle C_{p}-C_{V}=T\left({\frac {\partial V}{\partial T}}\right)_{p}\left({\frac {\partial p}{\partial T}}\right)_{V}}
Ze vzorce derivace implicitní funkce
(
∂
V
∂
T
)
p
(
∂
T
∂
p
)
V
(
∂
p
∂
V
)
T
=
−
1
{\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{p}\left({\frac {\partial T}{\partial p}}\right)_{V}\left({\frac {\partial p}{\partial V}}\right)_{T}=-1}
vyjádříme
(
∂
V
∂
T
)
p
=
−
(
∂
p
∂
T
)
V
(
∂
p
∂
V
)
T
{\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{p}=-{\frac {\left({\frac {\partial p}{\partial T}}\right)_{V}}{\left({\frac {\partial p}{\partial V}}\right)_{T}}}}
Opět dosadíme
C
p
−
C
V
=
−
T
(
∂
p
∂
T
)
V
2
(
∂
p
∂
V
)
T
{\displaystyle C_{p}-C_{V}=-T{\frac {\left({\frac {\partial p}{\partial T}}\right)_{V}^{2}}{\left({\frac {\partial p}{\partial V}}\right)_{T}}}}
Ze stavové rovnice ideálního plynu
p
V
=
n
R
T
{\displaystyle pV=nRT}
vyjádříme
p
=
n
R
T
V
{\displaystyle p={\frac {nRT}{V}}}
a
(
∂
p
∂
T
)
V
=
n
R
V
;
(
∂
p
∂
V
)
T
=
−
n
R
T
V
2
{\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}={\frac {nR}{V}};\left({\frac {\partial p}{\partial V}}\right)_{T}=-{\frac {nRT}{V^{2}}}}
Znovudosazením do odvození
C
p
−
C
V
=
−
T
(
n
2
R
2
V
2
)
(
−
n
R
T
V
2
)
{\displaystyle C_{p}-C_{V}=-T{\frac {\left({\frac {n^{2}R^{2}}{V^{2}}}\right)}{\left(-{\frac {nRT}{V^{2}}}\right)}}}
dostaneme výsledný Mayerův vztah
C
p
−
C
V
=
n
R
{\displaystyle C_{p}-C_{V}=nR}
c
p
n
−
c
V
n
=
n
R
{\displaystyle c_{p}n-c_{V}n=nR}
c
p
−
c
V
=
R
{\displaystyle c_{p}-c_{V}=R}