Thermodynamics - Solutions

In an effort to keep up with everyone in their affluent neighborhood, your friends have decided they must have a new luxury build-in refrigerator. The fact that they never eat at home is simply not the point. But despite their need to be trendy, they remain committed to energy conservation and ask your advice about how to make their new toy as energy efficient as possible.

1. If they locate the new refrigerator in the central column of their enormous kitchen, the refrigerator will be completely surrounded by the kitchen and its air. What effects, if any, will the operation of this refrigerator have on the room air's thermal energy and temperature? Explain.

Answer: The room air's thermal energy and temperature will both increase.        

Why: The refrigerator will consumer ordered energy as it pumps heat from the colder food compartment to its condenser coils. That ordered energy will become thermal energy during the pumping process and will be released from the condenser coils along with heat transferred from the food compartment. Since both the food compartment and the condenser coils are located in the room air, the refrigerator will have the useless effect of moving heat from one part of the room air to another. However, it will also add extra thermal energy to the room air--the thermal energy produced from ordered energy during the pumping process. As a result, the total thermal energy in the room air will increase and the room air will become hotter.

2. You suggest installing the new refrigerator in an outer wall of the house, so the refrigerator's doors open into the kitchen but its back has access to the outdoor air. By sliding a panel to one side or the other, your friends can choose whether to open the refrigerator's compressor and condenser to outdoor air or to indoor air. During which season of the year would having those components open to the outdoor air make the refrigerator most energy efficient? Explain.

Answer: The refrigerator would be most energy efficient during the winter, because then the temperature difference between the cold food compartment and the "hot" outdoor air would be as small as possible and relatively little ordered energy would be needed to pump heat from the food compartment to the outdoor air.

Why: Removing heat from a cold object and adding that heat to a hot object reduces the total entropy of the two objects. That's because a joule of heat is more disordering to a cold object than to a hot object. The bigger the temperature difference between those two objects, the larger the entropy decrease. To avoid violating the second law of thermodynamics, the refrigerator pumping the heat must convert enough ordered energy into thermal energy to ensure that the total entropy of the world doesn't decrease. The bigger the temperature difference the heat goes through during the pumping, the more ordered energy the refrigerator needs to convert into thermal energy to satisfy the second law. But when the temperature difference between the food compartment and the outdoor air is small, then the refrigerator doesn't need to consume much ordered energy. That's why the refrigerator would be most energy efficient during the winter.

3. Insulating the refrigerator's doors is always a good idea because it keeps the food in the refrigerator from exchanging heat with the room air. But insulating the refrigerator's compressor and condenser is always a bad idea. Why?

Answer: Heat extracted from the food is released to the room or outdoor air by the compressor and condenser. If they can't release that heat, the food can't be cooled.

Why: The refrigerator can't destroy the food's thermal energy, it must transfer that thermal energy elsewhere. In fact, it converts ordered energy into extra thermal energy as it operates and this thermal energy must also be transferred elsewhere. If you prevent the compressor and condenser from transferring this thermal energy to the room or outdoor air, they will become hotter and hotter until the refrigerator stops working altogether.

4. The refrigerator has been installed as you suggested but is presently operating so that its compressor and condenser are exposed to room air from the kitchen. A six-pack of root beer has been sitting for hours on the counter next to the refrigerator. Your friend opens the refrigerator door, puts the six-pack on the ice-cold shelf, and closes the door. The refrigerator hums quietly as it begins to do its job of cooling the root beer. As the refrigerator cools the root beer, how is entropy changing, if at all, in (a) the root beer, (b) the refrigerator's food compartment (excluding the root beer itself), (c) the room air, and (d) in total for the entire house? (Note: assume that the food compartment's temperature remains constant while the root beer is being chilled. You don’t have to explain your answers to these 4 questions.)

Answer: Entropy is (a) decreasing in the root beer, (b) not changing in the food compartment, (c) increasing in the room air, and (d) increasing in the entire house.

Why: Since only the thermal energy contents of these items are changing (they aren't breaking or changing size), entropy is directly related to temperature. As the root beer becomes colder, its thermal energy content and entropy both decrease. Since the food compartment isn't changing temperature, its thermal energy content and entropy are remaining constant. The room air is becoming hotter, so its thermal energy content and entropy are increasing. And we know that the heat transfer process requires that ordered energy become thermal energy, so the total thermal energy content of the entire house is increasing and with it, the entropy of the entire house.

The only thing perpetual about perpetual motion machines is people's interest in trying to invent them despite any history of success. Because of your strong background in physics and thermodynamics, you decide to sell short when the new high-tech startup company "perpetualmotion.com" conducts its initial public stock offering. Naturally, you make a fortune and wind up in perpetual motionlessness on a beach somewhere in the Caribbean.

5. One of the products that perpetualmotion.com announced at its press conference, held just before it started selling shares, was a device that the company claimed could produce electric power endlessly without having to come in contact with any external objects or materials. According to the press release, all you had to do was connect wires to its two terminals and it would send electric power out through those wires forever. At the conference, they used the device to light a light bulb for an hour or two. This device looked exactly like a suitcase full of D batteries. In fact, when the Justice Department eventually raided the company's headquarters, is was discovered that the device actually was a suitcase full of D batteries. However, you knew immediately that the device was fraudulent because it violated one of the laws of thermodynamics. Which law did it violate and why?

Answer: It violates the first law of thermodynamics—it is creating energy from nothing and that simply isn't possible.

Why: For a device to produce electric power endlessly, without any external input at all would violate conservation of energy. The form of energy conservation that is most appropriate here is the first law of thermodynamics, since it explicitly includes both heat and work as mechanisms for transferring energy.

6. A second perpetualmotion.com device was said to be able to produce electricity endlessly so long as you immersed it completely in a bath of hot water at a uniform temperature. The device supposedly used heat from the hot water to generate electricity. Oddly enough, this device looked exactly like a waterproof suitcase containing... you guessed it: D batteries. To make a long story short, you knew it was a fraud because it violated another law of thermodynamics. Which law was the problem this time and why?

Answer: It violates the second law of thermodynamics—it is converting thermal energy (from the hot water) directly into ordered energy and therefore decreasing the total entropy of the world.

Why: The energy this device is taking in is thermal energy from the water and the energy it is putting out is electricity. That process is converting disordered thermal energy into ordered electrical energy, resulting in a decreasing in disorder and entropy. The second law of thermodynamics forbids such a result on statistical grounds. It is fantastically unlikely.

7. The company's most sophisticated unit for generating electricity consisted of a heat pump and a heat engine. An electrically operated air conditioner pumped heat from one container to another, creating a hot container and a cold container. A heat engine then used the hot and cold containers to generate electricity. While we haven't yet studied heat engines in class, you are aware that heat will flow naturally from the hot container to the cold container and should not be surprised to learn that some of that heat can be diverted before it reaches the cold object and converted into useful work instead. In the present case, the work takes the form of electric power that is then split in half. Half of this power is used to operate the air conditioner mentioned earlier and the other half is available as the output of this wonderful power source. As usual, the device looked like luggage, but this time it was a garment bag with a matching carry on. Despite the complexity of this machine, you were sure it was fraudulent, too. Why couldn’t the output of the heat engine be enough to power the heat pump and still have enough power to do other things as well? (Assume that the device operated in an environment at a single uniform temperature. I'm not looking for a detailed analysis of the parts here, just a statement about its overall behavior and the impossibility that the machine can do what it claims to do.)

Answer: Two possible answers: (1) If this device is thermally isolated from its environment, then it would be violating the first law of thermodynamics (conservation of energy) because it would have the overall effect of producing energy without consuming anything or obtaining energy from anywhere else, and (2) If this device is able to exchange heat with its environment, then it would be violating the second law of thermodynamics (entropy never decreases) because it would have the overall effect of converting disordered (thermal energy) continuously into ordered energy.

Why: Without an external source of ordered energy or an external source of heat plus an external dumping ground for excess entropy (that's what a heat engine has), no device can release ordered energy indefinitely. The complicated arrangement here doesn't fulfill those requirements. While it is possible for a heat pump to create a hot and cold pair of objects, and for a heat engine to use that pair to generate power, the power will be at most just enough to make the heat pump operate. In any real system, the two machines will gradually lose order and come to a stop. There is no way that that you'll be able to extract ordered energy from the two machines without speeding that stopping process.

8. In a last ditch effort to keep the company afloat after the feds began to go after it, the company started producing power the old fashion way: by burning fuel. They used the only fuel they could find: tens and twenties. The crisp green bills burned nicely but didn't provide enough power to keep the company out of bankruptcy. In an effort to recover their investments, the creditors carefully collected all the burned bills, the smoke that was produced, and the energy that was generated. But try as they might, they could not get this collection of stuff to turn back into unburned bills and fresh air. The laws of motion didn’t stop this unburning process, so what is? Why don't the bills unburn?

Answer: The unburning is incredibly unlikely (equivalently, it would result in a decrease in overall entropy).         

Why: The laws of motion don't forbid objects from unburning. As long as you keep everything released by the burning process, the laws of motion can't stop the burning process from going backward and unburning the bills. But that unburning would convert disorder into order and decrease entropy overall. Such a decrease in disorder and entropy is incredibly unlikely.