# Thermodynamic identities

1a) Write the thermodynamic identity for dU, including the chemical potential term.

b) From this identity, find an expression for the pressure in terms of partial derivative of U.

c) Find the entropy anduse this to write the identities from 1b)

$dU=TdS-PdV+μdN$     or $TdS=dU+PdV-μdN$

$1/T=(∂S/∂U)_(V,N)$
$p/T=(∂S/∂V) |_(U,N)$

$μ/T=-(∂S/∂N)_(U,V)$

$S=Nk*ln⁡{(VU/N^2 )+1}$

$1/T=(∂S/∂U)_(V,N)=Nk*(V/N^2 )/((VU/N^2 )+1)=$

$=Nk 1/(U+N^2/V)=Nk V/(UV+N^2 )$

$p/T=(∂S/∂V) |_(U,N)=Nk*(U/N^2 )/((VU/N^2 )+1)=Nk U/(UV+N^2 )$

$μ/T=-(∂S/∂N)_(U,V)=-Nk (-2VU/N^3 )/((VU/N^2 )+1)=Nk 2/(N+N^3/UV)$