Question

Cp - Cv = R how to solve it ?

Answer 1

PV=nRTPV=nRT

CP−CV=T(∂P∂T)V,n(∂V∂T)P,nCP−CV=T(∂P∂T)V,n(∂V∂T)P,n

P=nRTV⇒(∂P∂T)V,n=nRVP=nRTV⇒(∂P∂T)V,n=nRV

V=nRTP⇒(∂V∂T)P,n=nRPV=nRTP⇒(∂V∂T)P,n=nRP

substituting

T(∂P∂T)V,n(∂V∂T)P,n=T(nRV)(nRP)=(nRTV)(nRP)=P(nRP)=nRT(∂P∂T)V,n(∂V∂T)P,n=T(nRV)(nRP)=(nRTV)(nRP)=P(nRP)=nR

this equation reduces simply to Mayer's relation,

CP,m−CV,m=R