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1. Lecture 3: The Second Law of Thermodynamics

   Schroeder Ch. 3.1, 3.2, 3.4 Gould and Tobochnik Ch. 2.12 – 2.20
2. Outline The Kelvin and Clausius Statements Macroscopic View of Entropy Entropy and Temperature Entropy and Pressure Entropy and the Ideal Gas Integrability Conditions
3. Introduction The only restriction that the first law of thermodynamics places on a process is that energy should be conserved. However, there are many processes that do not spontaneously occur in nature. Perpetual motion machines A lake spontaneously freezing This demonstrates that the first law is only a necessary condition for the occurrence of a thermodynamic process, not a sufficient one. The impossibility of these processes are described in the second law of thermodynamics
4. Kelvin and Clausius Statements There are two original statements of the second law of thermodynamics by Lord Kelvin and Rudolf Clausius The Kelvin Statement:A transformation whose only effect is to transform energy extracted from a body in the form of heat into useful work is impossible. The Clausius Statement:A transformation whose only effect is to transfer energy in the form of heat from a body at a given temperature to a body at a higher temperature is impossible.
5. Equivalence of Kelvin and Clausius Statements Violation of Kelvin statement leads to violation of Clausius statement
6. Equivalence of Kelvin and Clausius Statements Violation of Clausius statement leads to violation of Kelvin statement
7. Carnot’s Theorem An important corollary of the second law is Carnot’s theorem. All heat engines between two heat reservoirs are less efficient than a Carnot engine operating between the same reservoirs. Every Carnot engine between a pair of heat reservoirs is equally efficient.
8. Proof of Carnot’s Theorem Let C be a Carnot engine operating between the temperatures Th and Tl and let M represent any other heat engine operating between the same two temperatures. By the first law, we have that Consider a process consisting of n’ cycles of M and n cycles of C in reverse.
9. Proof of Carnot’s Theorem The net external work done by our system is then The net heat removed from is The net heat added to By the first law of thermodynamics, we have
10. Proof of Carnot’s Theorem To a good approximation, we can write This implies that Kelvin’s statement implies that , which produces Therefore, we have
11. Reversible Processes If M acts as a refrigerator and C as a heat engine, then it can also be proved that Therefore, the efficiencies of the two engines are identical and thus, the cycle M is called a reversible cycle. A reversible process is a process in which both the system and the environment can be returned to the original state by means of quasi-static processes. Carnot’s theorem shows that the Carnot engine is reversible and thus, all engines will have an efficiency less than the efficiency of the Carnot engine.
12. Reversible Engines Let’s consider an arbitrary reversible engine and divide the cycle into a large number of Carnot engines. For each Carnot cycles, heat is absorbed and rejected by the cycle during isothermal processes and thus we can write For the set of cycles shown, we can say that
13. Introduction to Entropy Because for a reversible cycle, then there exists a function of state S, called entropy,such that Similar to potential energy in classical mechanics, only changes in entropy are relevant
14. Irreversible Engines Let’s consider an arbitrary irreversible engine. This implies that there are one of more portions of the engines that cannot be accurately described by Carnot cycles. To reproduce the cycle, we have Since the second sum is less than or equal to zero then
15. Entropy and the Second Law Suppose that the cycle S was irreversible. Let’s choose two points on the cycle so that path I from ito f is not quasi-static and path II from f to iis quasi-static. Then Since path II is quasi-static, then The entropy of a thermally isolated system undergoing a thermodynamic process may never decrease. It follows that thermally isolated systems achieve equilibrium at the maximum of the entropy
16. Entropy and Heat Engines For an arbitrary heat engine, we have that is minimum when the cycle is reversible and . For these conditions, we find that the maximum thermal efficiency is Any efficiency greater than this would lead to a decrease in entropy, which is a restatement of the Kelvin statement
17. Entropy and Temperature Consider an isolated composite system that is partitioned into two subsystems by a fixed, impermeable, insulating wall. We can write the total entropy as For thermal equilibrium, we replace the insulating wall with a conducting wall. According to the 2nd law of thermodynamics, the internal energy in A and B will be such that the entropy is maximized. Thus
18. Entropy and Temperature Since the total internal energy of the system is conserved, we have Because the temperatures of the two systems are equal in thermal equilibrium, we conclude that must be associated with temperature. We define the thermodynamic temperature as Temperature can be interpreted as the response of the entropy to a change in the internal energy of the system.
19. Entropy and Temperature As the system approaches thermal equilibrium, the change of entropy of the composite system will increase If , then , which implies that energy is transferred from a system with a higher value of T to a system with a lower value of T. Thus, no process exists in which a cold body becomes cooler while a hotter body becomes still hotter while the constraints on the bodies and the state of its environments remain unchanged. This is a restatement of the Clausius statement
20. Entropy and Pressure Consider an isolated system that is partitioned into two subsystems Assume that the system is in thermal equilibrium. Therefore, the total entropy of the system is maximized and we have
21. Entropy and Pressure Since the total volume of the system is fixed of the system is conserved, we have Because the pressures of the two subsystems are equal in mechanical equilibrium, we conclude that must be associated with pressure. We define the thermodynamic pressure as Pressure can be interpreted as the response of the entropy to a change in the volume of the system.
22. Entropy Change in a Free Expansion Consider an ideal gas in a closed, insulated container divided into two chambers by an impermeable partition. Since the temperature of the gas does not change, then we have In this process, the entropy of the universe increased even though there was no change in internal energy of the system, consistent with the second law of thermodynamics. To restore the gas to its original state requires work
23. Entropy Change Upon Heating and Cooling A solid with heat capacity at temperature is placed in contact with another solid with heat capacity at a lower temperature . The total change in the entropy of the system is given by It can be shown that the equilibrium temperature is given by CA CB TA TB
24. Entropy Change Upon Heating and Cooling Suppose that subsystem B becomes a heat reservoir. Since a heat bath doesn’t change its temperature when energy is added/subtracted from it, then This implies that CA CB TA TB
25. Entropy Change Upon Heating and Cooling The change in entropy becomes The first term gives The second term gives Therefore, the change in entropy becomes
26. Entropy and the Second Law The general expression for entropy can be written as The first law of thermodynamics in terms of entropy is given as This statement and the statement that for an isolated system, together summarize the content of the first two laws of thermodynamics.