the definition of entropy is given by

      

Rearranging the above equation gives

                      (1)

The entropy change during an internally reversible process (1-2) is

      

Only when the relation between δQ and T is known, the entropy change can be determined. The relations between δQ and T can be found by considering the energy balance of a closed system.

The differential form of the energy balance for a closed system, which contains a simple substance and undergoes an internally reversible process, is given by

      dU = δQ_rev - δW_rev             (2)

The boundary work of a closed system is

      δW_rev = PdV                     (3)

Substituting equations (1) and (3) into equation (2) gives

      dU = TdS- PdV
      TdS = dU + PdV

or

      Tds = du +Pdv                   (4)

where
      s = entropy per unit mass

Equation (4) is known as the first relation of Tds, or Gibbs equation.

The definition of enthalpy gives       h = u + Pv differential the above equation yields       dh = du +Pdv + vdP Replacing du + Pdv with Tds yields       dh = Tds + vdP       Tds = dh -vdP                   (5) Equation (5) is known as the second relation of Tds. Although the Tds equations are obtained through an internally reversible process, the results can be used for both reversible or irreversible processes since entropy is a property. Rewriting equations (4) and (5) in the following form       ds = du/T + Pdv/T       ds = dh/T + vdP/T The entropy change during a process can be determined by integrating the above equations between the initial and the final states.