Temperature Entropy Diagram

Entropy change of a system is given by . During the reversible process, the energy transfer as heat to the system from the surroundings is given by

(24.1)

Figure 24.1

Refer to figure 24.1. Here T and S are chosen as independent variables. The is the area under the curve. The first law of thermodynamics gives . Also for a reversible process, we can write,

and

(24.2)

Therefore,

(24.3)

For a cyclic process, the above equation reduces to

(24.4)

For a cyclic process, the above equation reduces to

For a cyclic process represents the net heat interaction which is equal to the net work done by the system. Hence the area enclosed by a cycle on a T − S diagram represents the net work done by a system. For a reversible adiabatic process, we know that

(24.5)

or,

(24.6)

or,

(24.7)

Hence a reversible adiabatic process is also called an isentropic process. On a T − S diagram, the Carnot cycle can be represented as shown in Fig 24.1. The area under the curve 1-2 represents the energy absorbed as heat by the system during the isothermal process. The area under the curve 3-4 is the energy rejected as heat by the system. The shaded area represents the net work done by the system.

We have already seen that the efficiency of a Carnot cycle operating between two thermal reservoirs at temperatures T₁and T₂ is given by

(24.8)

This was derived assuming the working fluid to be an ideal gas. The advantage of T − S diagram can be realized by a presentation of the Carnot cycle on the T − S diagram. Let the system change its entropy from to during the isothermal expansion process 1-2. Then,

(24.9)

and,

(24.10)

and,

or,

(24.11)

This demonstrates the utility of T − S diagram.