12.3: Energy-Conversion Cycle
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This system can act as a heat engine if the interaction of the system with its environments, and the externally applied magnetic field, are both controlled appropriately. The idea is to make the system change in a way to be described, so that is goes through a succession of states and returns to the starting state. This represents one cycle, which can then be repeated many times. During one cycle heat is exchanged with the two environments, and work is exchanged between the system and the agent controlling the magnetic field. If the system, over a single cycle, gets more energy in the form of heat from the environments than it gives back to them, then energy must have been delivered to the agent controlling the magnetic field in the form of work.
The cycle of the heat engine is shown below in Figure 12.2. Without loss of generality we can treat the case where \(H\) is positive. Assume that the left environment has a temperature \(T_1\) which is positive but less (i.e., a higher value of \(\beta\)) than the temperature \(T_2\) for the right environment (the two temperatures must be different for the device to work). This cycle is shown on the plane formed by axes corresponding to \(S\) and \(T\) of the system, and forms a rectangle, with corners marked \(a, b, c\), and \(d\), and sides corresponding to the values \(S_1, S_2, T_1\), and \(T_2\).
Since the temperatures are assumed to be positive, the lower energy levels have a higher probability of being occupied. Therefore, the way we have defined the energies here, the energy \(E\) is negative. Thus as the field gets stronger, the energy gets more negative, which means that energy actually gets delivered from the system to the magnetic apparatus. Think of the magnetic field as increasing because a large permanent magnet is physically moved toward the system. The magnetic dipoles in the system exert a force of attraction on that magnet so as to draw it toward the system, and this force on the magnet as it is moved could be
Figure 12.2: Temperature Cycle
used to stretch a spring or raise a weight against gravity, thereby storing this energy. Energy that moves into the system (or out of the system) of a form like this, that can come from (or be added to) an external source of energy is work (or negative work).
First consider the bottom leg of this cycle, during which the temperature of the system is increased from \(T_1\) to \(T_2\) without change in entropy. An operation without change in entropy is called adiabatic. By Equation 12.15 above, increasing \(T\) is accomplished by increasing \(H\), while not permitting the system to interact with either of its two environments. (In other words, the barriers preventing the dipoles in the system from interacting with those in either of the two environments are in place.) The energy of the system goes down (to a more negative value) during this leg, so energy is being given to the external apparatus that produces the magnetic field, and the work done on the system is negative.
Next, consider the right-hand leg of this cycle, during which the entropy is increased from \(S_1\) to \(S_2\) at constant temperature \(T_2\). This step, at constant temperature, is called isothermal. According to Equation 12.15, this is accomplished by decreasing \(H\), while the system is in contact with the right environment, which is assumed to be at temperature \(T_2\). (In other words, the barrier on the left in Figure 12.1 is left in place but that on the right is withdrawn.) During this leg the change in energy \(E\) arises from heat, flowing in from the high-temperature environment, and work from the external magnetic apparatus. The heat is \(T_2(S_2 − S_1)\) and the work is positive since the decreasing \(H\) during this leg drives the energy toward 0.
The next two legs are similar to the first two except the work and heat are opposite in direction, i.e., the heat is negative because energy flows from the system to the low-temperature environment. During the top leg the system is isolated from both environments, so the action is adiabatic. During the left-hand isothermal leg the system interacts with the low-temperature environment.
After going around this cycle, the system is back where it started in terms of its energy, magnetic field, and entropy. The two environments are slightly changed but we assume that they are each so much larger than the system in terms of the number of dipoles present that they have not changed much. The net change is a slight loss of entropy for the high-temperature environment and a gain of an equal amount of entropy for the low-temperature environment. Because these are at different temperatures, the energy that is transferred when the heat flow happens is different—it is proportional to the temperature and therefore more energy leaves the high-temperature environment than goes into the low-temperature environment. The difference is a net negative work which shows up as energy at the magnetic apparatus. Thus heat from two environments is converted to work. The amount converted is nonzero only if the two environments are at different temperatures.
Table 12.2 summarizes the heat engine cycle.
| Leg | Start | End | Type | dS | dT | H | E | Heat in | Work in |
|---|---|---|---|---|---|---|---|---|---|
| bottom | a | b | adiabatic | 0 | positive | increases | decreases | 0 | negative |
| right | b | c | isothermal | positive | 0 | decreases | increases | positive | positive |
| top | c | d | adiabatic | 0 | negative | decreases | increases | 0 | positive |
| left | d | a | isothermal | negative | 0 | increases | decreases | negative | negative |
| Total | a | a | complete cycle | 0 | 0 | no change | no change | positive | negative |
For each cycle the energy lost by the high-temperature environment is \(T_2(S_2 − S_1)\) and the energy gained by the low-temperature environment is \(T_1(S_2 − S_1)\) and so the net energy converted is the difference \((T_2 − T_1)(S_2 − S_1)\). It would be desirable for a heat engine to convert as much of the heat lost by the high-temperature environment as possible to work. The machine here has efficiency
This ratio is known as the Carnot efficiency, named after the French physicist Sadi Nicolas Léonard Carnot (1796 - 1832).\(^3\) He was the first to recognize that heat engines could not have perfect efficiency, and that the efficiency limit (which was subsequently named after him) applies to all types of reversible heat engines.
The operations described above are reversible, i.e., the entire cycle can be run backwards, with the result that heat is pumped from the low-temperature environment to the one at high temperature. This action does not occur naturally, and indeed a similar analysis shows that work must be delivered by the magnetic apparatus to the magnetic dipoles for this to happen, so that more heat gets put into the high-temperature environment than is lost by the low-temperature environment. Heat engines run in this reverse fashion act as refrigerators or heat pumps.
\(^3\)For a biography check out http://www-groups.dcs.st-andrews.ac.uk/∼history/Mathematicians/Carnot Sadi.html