It is true that reversible processes and purely mechanical processes are idealized pro- cesses that cannot occur in practice, but a spontaneous process can bepracticallyreversible if carried out sufficiently slowly, orpracticallypurely mechanical if friction and temperature gradients are negligible. In that sense, they are not impossible processes. This book will reserve the term “impossible” for a process that cannot be approached by any spontaneous process, no matter how slowly or how carefully it is carried out.
4.2 STATEMENTS OF THE SECOND LAW
A description of themathematical statement of the second lawis given in the box below.
dS Dảq=Tbfor a reversible change of a closed system;
dS > ảq=Tbfor an irreversible change of a closed system;
whereS is an extensive state function, the entropy, and
ảqis an infinitesimal quantity of energy transferred by heat at a portion of the boundary where the thermodynamic temperature isTb.
The box includes three distinct parts. First, there is the assertion that a property called entropy, S, is an extensive state function. Second, there is an equation for calculating the entropy change of a closed system during a reversible change of state: dS is equal to
ảq=Tb.1 Third, there is a criterion for spontaneity: dS is greater thanảq=Tb during an irreversible change of state. The temperature Tb is a thermodynamic temperature, which will be defined in Sec. 4.3.4.
Each of the three parts is an essential component of the second law, but is somewhat abstract. What fundamental principle, based on experimental observation, may we take as the starting point to obtain them? Two principles are available, one associated with Clausius and the other with Kelvin and Planck. Both principles are equivalent statements of the second law. Each asserts that a certain kind of process is impossible, in agreement with common experience.
Consider the process depicted in Fig. 4.1(a) on the next page. The system is isolated, and consists of a cool body in thermal contact with a warm body. During the process, energy is transferred by means of heat from the cool to the warm body, causing the temperature of the cool body to decrease and that of the warm body to increase. Of course, this process is impossible; we never observe heat flow from a cooler to a warmer body. (In contrast, the reverse process, heat transfer from the warmer to the cooler body, is spontaneous and irreversible.) Note that this impossible process does not violate the first law, because energy is conserved.
Suppose we attempt to bring about the same changes in the two bodies by interposing a device of some sort between them, as depicted in Fig. 4.1(b). Here is how we would like the device to operate in the isolated system: Heat should flow from the cool body to the device,
1During a reversible process, the temperature usually has the same valueTthroughout the system, in which case we can simply write dSDảq=T. The equation dSDảq=Tballows for the possibility that in an equilibrium state the system has phases of different temperatures separated by internal adiabatic partitions.
Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version:www.chem.umd.edu/thermobook
CHAPTER 4 THE SECOND LAW
4.2 STATEMENTS OF THESECONDLAW 103
(a) cool
q
warm
(b) cool
q
device
q
warm
Figure 4.1 Two impossible processes in isolated systems.
(a) Heat transfer from a cool to a warm body.
(b) The same, with a device that operates in a cycle.
an equal quantity of heat should flow from the device to the warm body, and the final state of the device should be the same as its initial state. In other words, we want the device to transfer energy quantitatively by means of heat from the cool body to the warm body while operating in acycle. If the device could do this, there would be no limit to the quantity of energy that could be transferred by heat, because after each cycle the device would be ready to repeat the process. But experience shows thatit is impossible to build such a device! The proposed process of Fig. 4.1(b) is impossible even in the limit of infinite slowness.
The general principle was expressed by Rudolph Clausius2 in the words: “Heat can never pass from a colder to a warmer body without some other change, connected there- with, occurring at the same time.” For use in the derivation to follow, the statement can be reworded as follows.
The Clausius statement of the second law: It is impossible to construct a device whose only effect, when it operates in a cycle, is heat transfer from a body to the device and the transfer by heat of an equal quantity of energy from the device to a warmer body.
Next consider the impossible process shown in Fig. 4.2(a) on the next page. A Joule paddle wheel rotates in a container of water as a weight rises. As the weight gains potential energy, the water loses thermal energy and its temperature decreases. Energy is conserved, so there is no violation of the first law. This process is just the reverse of the Joule paddle- wheel experiment (Sec. 3.7.2) and its impossibility was discussed on page 66.
We might again attempt to use some sort of device operating in a cycle to accomplish the same overall process, as in Fig. 4.2(b). A closed system that operates in a cycle and does net work on the surroundings is called aheat engine. The heat engine shown in Fig.
4.2(b) is a special one. During one cycle, a quantity of energy is transferred by heat from a heat reservoir to the engine, and the engine performs an equal quantity of work on a weight, causing it to rise. At the end of the cycle, the engine has returned to its initial state. This would be a very desirable engine, because it could convert thermal energy into an equal quantity of useful mechanical work with no other effect on the surroundings.3 The
2Ref. [32], page 117.
3This hypothetical process is called “perpetual motion of the second kind.”
CHAPTER 4 THE SECOND LAW
4.2 STATEMENTS OF THESECONDLAW 104
b
(a)
heat engine q
b
(b) Figure 4.2 Two more impossible processes.
(a) A weight rises as a liquid becomes cooler.
(b) The same, with a heat engine.
engine could power a ship; it would use the ocean as a heat reservoir and require no fuel.
Unfortunately,it is impossible to construct such a heat engine!
The principle was expressed by William Thomson (Lord Kelvin) in 1852 as follows:
“It is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.” Max Planck4gave this statement: “It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.” For the purposes of this chapter, the principle can be reworded as follows.
The Kelvin–Planck statement of the second law: It is impossible to construct a heat en- gine whose only effect, when it operates in a cycle, is heat transfer from a heat reservoir to the engine and the performance of an equal quantity of work on the surroundings.
Both the Clausius statement and the Kelvin–Planck statement assert that certain pro- cesses, although they do not violate the first law, are neverthelessimpossible.
These processes would not be impossible if we could control the trajectories of large numbers of individual particles. Newton’s laws of motion are invariant to time re- versal. Suppose we could measure the position and velocity of each molecule of a macroscopic system in the final state of an irreversible process. Then, if we could somehow arrange at one instant to place each molecule in the same position with its velocity reversed, and if the molecules behaved classically, they would retrace their trajectories in reverse and we would observe the reverse “impossible” process.
The plan of the remaining sections of this chapter is as follows. In Sec. 4.3, a hypo- thetical device called a Carnot engine is introduced and used to prove that the two physical statements of the second law (the Clausius statement and the Kelvin–Planck statement) are equivalent, in the sense that if one is true, so is the other. An expression is also derived for the efficiency of a Carnot engine for the purpose of defining thermodynamic temperature.
Section 4.4 combines Carnot cycles and the Kelvin–Planck statement to derive the existence
4Ref. [133], p. 89.
Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version:www.chem.umd.edu/thermobook
CHAPTER 4 THE SECOND LAW