Mayerův vztah

${\displaystyle C_{p}-C_{V}=\left({\frac {\partial H}{\partial T}}\right)_{p}-\left({\frac {\partial U}{\partial T}}\right)_{V}=\left({\frac {\partial (U+pV)}{\partial T}}\right)_{p}-\left({\frac {\partial U}{\partial T}}\right)_{V}=\left({\frac {\partial U}{\partial T}}\right)_{p}+p\left({\frac {\partial V}{\partial T}}\right)_{p}-\left({\frac {\partial U}{\partial T}}\right)_{V}}$

Entalpie ${\displaystyle H}$ je definována vztahem

${\displaystyle H=U+pV}$

kde ${\displaystyle U}$ je vnitřní energie soustavy, ${\displaystyle p}$ je její tlak a ${\displaystyle V}$ objem.

Vnitřní energie je funkcí teploty a objemu, tudíž ${\displaystyle \left({\frac {\partial U}{\partial T}}\right)_{p}}$ je nutno přepsat jako ${\displaystyle \left({\frac {\partial U(T,V(p,T))}{\partial T}}\right)_{p}}$

${\displaystyle \left({\frac {\partial U(T,V(p,T))}{\partial T}}\right)_{p}=\left({\frac {\partial U}{\partial T}}\right)_{V}+\left({\frac {\partial U}{\partial V}}\right)_{T}\left({\frac {\partial V}{\partial T}}\right)_{p}}$

Po dosazení do odvození dostaneme

${\displaystyle C_{p}-C_{V}=p\left({\frac {\partial V}{\partial T}}\right)_{p}+\left({\frac {\partial V}{\partial T}}\right)_{p}\left({\frac {\partial U}{\partial V}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{p}\left[p+\left({\frac {\partial U}{\partial V}}\right)_{T}\right]}$

Z diferenciálu definice vnitřní energie a Maxwellových relací dostaneme

${\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial S}{\partial V}}\right)_{p}-p=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}$

Dalším dosazením do odvození se výraz změní na

${\displaystyle C_{p}-C_{V}=T\left({\frac {\partial V}{\partial T}}\right)_{p}\left({\frac {\partial p}{\partial T}}\right)_{V}}$

Ze vzorce derivace implicitní funkce

${\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{p}\left({\frac {\partial T}{\partial p}}\right)_{V}\left({\frac {\partial p}{\partial V}}\right)_{T}=-1}$

vyjádříme

${\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{p}=-{\frac {\left({\frac {\partial p}{\partial T}}\right)_{V}}{\left({\frac {\partial p}{\partial V}}\right)_{T}}}}$

Opět dosadíme

${\displaystyle C_{p}-C_{V}=-T{\frac {\left({\frac {\partial p}{\partial T}}\right)_{V}^{2}}{\left({\frac {\partial p}{\partial V}}\right)_{T}}}}$

Ze stavové rovnice ideálního plynu

${\displaystyle pV=nRT}$

vyjádříme

${\displaystyle p={\frac {nRT}{V}}}$

a

${\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}={\frac {nR}{V}};\left({\frac {\partial p}{\partial V}}\right)_{T}=-{\frac {nRT}{V^{2}}}}$

Znovudosazením do odvození

${\displaystyle C_{p}-C_{V}=-T{\frac {\left({\frac {n^{2}R^{2}}{V^{2}}}\right)}{\left(-{\frac {nRT}{V^{2}}}\right)}}}$

dostaneme výsledný Mayerův vztah

${\displaystyle C_{p}-C_{V}=nR}$

${\displaystyle c_{p}n-c_{V}n=nR}$

${\displaystyle c_{p}-c_{V}=R}$

• Poissonova konstanta