TdS Equations
For a closed system containing a pure compressible substance undergoing a reversible process
dU = ?Qrev - ?Wrev = TdS - PdV
TdS = dU + PdV, or Tds = du + pdv ( per unit mass)
This is the famous Gibbsian equation
Eliminate du by using the definition of enthalpy h=u+pv
dh = du + pdv + vdp, thus du + pdv = dh - vdp
Tds = du + pdv, also Tds = dh - vdp
Important: these equations relate the entropy change of a system to the changes in other properties: dh, du, dp, dv. Therefore, they are independent of the processes. These relations can be used for reversible as well as irreversible processes. ( Even their derivation is based on a reversible process.)