In the second law analysis, it is useful to plot the process on diagrams for which has one coordinate is entropy. The two diagrams commonly used in second law analysis are temperature-entropy (T-s) and enthalpy-entropy (h-s) diagrams. For some pure substance, like water, the entropy is tabulated with other properties.

The Total Heat Transfer Equals the Total Area under the Process Curve on the T-s Diagram

On a P-v diagram, the area under the process curve is equal, in magnitude, to the work done during a quasi-equilibrium expansion or compression process of a closed system. On a T-s diagram, the area under an internally reversible process curve is equal, in magnitude, to the heat transferred between the system and its surroundings. That is,

Note the area has no meaning for irreversible processes.

The T-s diagram of a Carnot cycle is shown on the left. The area under process curve 1-2 (area 1-2-B-A-1) equals the heat input from a source (Q_H). The area under process curve 3-4 (area 4-3-B-A-4) equals the heat rejected to a sink (Q_L). The area enclosed by the 4 processes (area 1-2-3-4-1) equals the net heat gained during the cycle, which is also the net work output.

How to Use the Mollier Diagram

The enthalpy-entropy (h-s) diagram is valuable in the analysis of steady-flow devices such as nozzle, turbine, and compressor.

The h-s diagram is also called Mollier diagram after the German scientist R. Mollier. The Mollier diagram of water is shown on the left. The Mollier diagram for water contains constant-quality lines, constant-pressure lines, and constant-temperature lines. The temperature lines in the mixture region are straight.

The Tds Relations for Closed System

In the previous section, 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.