# ps7 - Statistical Physics I(8.044 Spring 2009 Assignment 7...

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Unformatted text preview: Statistical Physics I (8.044) Spring 2009 Assignment 7 April 1, 2009 Due April 8, 2009 Please remember to put your name and section number at the top of what you turn in. Readings The reading for Chapter V of 8.044 is Adkins, Chapters 4, 5.1-5.5 and 7, and Baierlein Chapter 3.1-3.4 and 10. However, you will not be able to follow all of Baierlein’s chapter 10 until we introduce the chemical potential, a few weeks from now. So, I suggest that you read it now, and reread it later. The reading for Chapter VI of 8.044 is the Notes on the Canonical Ensemble, the Notes on Rotational Raman Spectrum of a Diatomic Molecule, which are both available on the 8.044 web page, and Baierlein Chapters 5, 13 and 14. Only the Notes on the Canonical Ensemble and Baierlein Chapter 5 are required for this problem set. Problem Set 7 1. Ideal Refrigerators and Heat Pumps (16 points) (a) A 100 Watt lightbulb is left on inside an ideal Carnot refrigerator. The refriger- ator itself can draw up to 50 Watts of power as it operates its Carnot cycle. (It can run its cycle slower or faster; at its fastest operation, it draws 50 Watts of power. For simplicity we shall approximate the operation of the refrigerator as quasistatic throughout this problem.) The refrigerator dumps heat into a room that is at a temperature T H = 25 ◦ C. i. Can the refrigerator keep its interior at a temperature T C = 5 ◦ C? ii. What is the lowest interior temperature that this Carnot cycle refrigerator could reach? iii. How much power would the refrigerator have to draw if it were required to cool its interior down to 4 K, the temperature at which helium liquefies? [Experimentalists have succeeded in cooling material (many millions of atoms worth) down to temperatures of milli, micro, and even nano-Kelvin. Of course, they do not leave 100 W lightbulbs on inside their refrigerators. However, any real refrigerator allows a certain amount of heat to leak into its interior from outside. The 100 W lightbulb in this problem can be seen as a proxy for any leakage process that lets unwanted heat into the refrigerator. This problem illustrates for you that since the amount of heat leaking into a refrigerator can 1 never be identically zero, even a maximally efficient Carnot refrigerator cannot cool the contents of a refrigerator down to identically zero since to do so would require infinite work.] (b) An ideal Carnot room air conditioner is maintaining a room at temperature T inside = 22 ◦ C (72 ◦ F) by taking heat out of the room and dumping heat into the outside air which has temperature T outside , where T outside > T inside . Heat energy from the outside world leaks back into the room at some rate that is proportional to ( T outside- T inside ). By what factor is the power used by the air conditioner larger on a day when T outside is 32 ◦ C (90 ◦ F) as compared to on a day when T outside is only 27 ◦ C (81 ◦ F)?...
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ps7 - Statistical Physics I(8.044 Spring 2009 Assignment 7...

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