21.11:
Entropy Change in Reversible Processes
In the Carnot cycle, the ratio of the heat exchanged during the two reversible and isothermal processes and the temperatures of the isotherms are the same.
Since the other two processes are reversible and adiabatic, there is no heat exchange. Hence, the net entropy change in the cycle, the sum of the entropy change for all the processes, is zero.
Any reversible, cyclic process can be shown to be a sum of many Carnot cycles. Thus, the net entropy change during the process is zero.
Take an arbitrary, closed path for a reversible cycle and break it into two parts, I and II, operating between the points A and B.
Split the cyclic integral of entropy into two parts along I and II. Since the process is reversible, reverse the integral for path II. Thus, it is found that the entropy change between any two points along two independent paths is the same.
Hence, entropy is a state function, similar to internal energy.
21.11:
Entropy Change in Reversible Processes
In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p–V diagram. Then, break the cyclic integral into two parts. By reversing the integral along one path, which is feasible because entropy change is defined for any reversible process, it is shown that the total change in entropy is independent of the path. That is, entropy is a state function.
The statement implies that a unique value of the entropy can describe a system under consideration. Note, however, that the absolute value of entropy is not defined, as the definition gives only the entropy change. Once a reference value of entropy is defined for a particular state of the system, then the absolute value can be obtained for all other states.
Irreversible processes lead to a change in entropy more than reversible processes, thus changing the state of a system differently. Thus, an irreversible cycle leads to a net increase in the entropy of a system and its surroundings. This observation leads to the entropy statement of the second law of thermodynamics.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson; section 20.7; page 672.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 4.6; page 162.