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Unformatted text preview: Heat Homework Problems 1. How much higher is the Eiffel tower on the hottest day in the year compared wit h the coldest night? (You can, of course, find all the info you need using Google.) 2. An old fashio ned mercury in glass fever thermometer has a bulb at the end containing say 0.25 ml o f mercury. Assume the mercury goes 3 cm further along the tube for a temperature of 104°F compared with 98.6°F, what is the radius of the capillary tube (the thin ho llow in the middle of the glass that the mercury goes up? (You need to find the coefficient of expansion o f mercury, of course.)
3. A lead bullet is fired into a block of wood at 300 m sec 1 . Assume the frict ional heat ing as the bullet is stopped in the wood all goes into heating the bullet (the wood is a fairly good insulator). Does the bullet melt ? You’ll need to find its latent heat of melt ing as well as its specific heat and melt ing temperature. 4. When Joule was on his honeymoon in Switzerland, he wielded a large thermo meter with which to measure the temperature difference between the water at the top and that at the bottom of a waterfall near Chamo nix. Let’s assume it was: “The Arpenaz Waterfall in Sallanches, … leaping fro m a height of 270 m, … , is very spectacular” (thank you, Google). I take it the “leap” is outwards and downwards. What temperature difference did Joule expect to find? 5. A carbon dioxide fire extinguisher ho lds three liters of compressed carbon dio xide, and weighs 3 kg more when full than when empt y. At room temperature (20°C) what is the pressure inside the ext inguisher? (Assume the gas obeys the ideal gas law in this range of pressures.) 6. Cass Sackett in our Department can cool a gas of Rubidium atoms into a BoseEinstein
condensate. He tells me this weird pheno menon takes place as the gas cools below 10 7 K. What is the average (rms) speed of the atoms at that temperature? 7. In the derivation o f the pressure an ideal gas exerts on the sides of a cubical box, gravit y was neglected. Now include it: compare the pressure on the top of the box wit h that on the bottom, thinking in terms of the mo lecules bouncing back and forth between the top and bottom, but being accelerated by gravit y on the way down, and prove that the pressure difference is the same as a simple hydrostatic picture would give. (Take the box to be small, so that the fract ional velocit y change is small.) 8. If the Calories in a Big Mac could be used wit h 100% efficiency to raise it vert ically, how high would it go? (We write Calorie meaning the energy to raise 1 kg of water by 1°C.)
9. If a cyclist maintains a speed of 10 m sec 1 on level ground, and her muscles can turn chemical energy (Calories) into work with 40% efficiency, how many Calories an hour does she burn? Assume C = 0.35 in the inert ial drag formula, but you have to estimate her cross sectional area. 10. In the reactor of a nuclear power plant, water under pressure is heated to 330°C. The plant 9
discharges waste heat into river water at 15°C. The electrical power produced is 1 gigawatt (10 watts). If the plant were an ideal Carnot heat engine, what would be the rate of production of waste heat? Actually, such plants only operate at about 60% of Carnot efficiency—so what is the real waste heat production? Assume now the cooling water from the river runs through at a rate of 40,000 kg/sec. (40 tons of water per second! This is a typical figure.) What is the temperature rise o f the cooling water? (These figures are fro m: http://dspace.mit.edu/bitstream/1721.1/35222/1/MITEL7904009531570.pdf) 11. (a) Give rough estimates of the fraction o f oxygen mo lecules in the upper atmosphere that are mo ving at escape velocit y or above. (The temperature can reach 1000K.) (Hint: the only important factor is e
written e -3 vescap e / 2 v 2 2 2 - mvesca pe / 2 kT and in fact since 1 mv 2 = 3 kT this factor can be 2
where v is the root mean square velocit y at the gas temperature.) (b) Do the same for any helium atoms in the upper atmosphere. Note that any fract ion that contains a significant number of atoms will be big enough for all to escape eventually. (c) Do the same for hydrogen mo lecules. (d) Mars is at about 0°C. Would you expect it to have any at mosphere? What of the commo n gases would you expect to find, if any? Give reasons. (e) We know the Moon has no atmosphere—is that consistent with what we know of its gravit y and temperature? 12. Joule was the first person to suggest that meteors shone bright ly because as they entered the atmosphere, frict ion converted their kinet ic energy to heat. Assuming a meteor is an iron ball, and it enters the earth’s at mosphere at a speed similar to that of the earth’s orbital speed around the sun. Estimate what temperature it might reach. Would it melt ? 13. A Brownian motion problem: recall the Brownian motion arises from a tiny but visible object, like a grain of po llen, being buffeted about by rando m hits fro m mo lecules on all sides. Let’s make a simple mathematical model: represent its path by a sequence of vectors, all of unit length, but each new step is in a completely random direct ion. Let’s call these unitlength rr
vectors a1 , a2 , K a . We’ll restrict the motion to two dimensio ns for simplicit y. Draw a few ,N
sample paths. Using the fact that the direct ion of each vector is completely random relat ive to any other vector, prove that the average distance traveled after N steps is proportional to N . Hint: consider the square of the distance from the beginning to the end of the N step path, and see what happens to it when you average over all possible step direct ions for the sequence of steps making up the path. 14. A standard gasoline engine is quite well approximated by the Otto cycle, which has two fixed vo lume sides, and adiabats top and bottom. Heat is absorbed alo ng the 4 to 1 leg, heat is dumped alo ng the 2 to 3 leg. There is of course no heat exchange alo ng the adiabat ic legs 1 to 2 and 3 to 4. P
T1 (a) Find the heat that goes in and the heat dumped and figure out the effic iency o f the engine just fro m those figures. Your answer should be a funct ion of the four temperatures only. T2 T4 T3 V1 V2 V (b) Find the work the engine does from 1 to 2 in terms o f the two temperatures and CV. (c) Using the equat ion for an adiabat, find the work done from 1 to 2 in terms o f T1, V1, V2. Next, do the same for the work done on the engine fro m 3 to 4, in terms of T4, V1, V2. (d) Relate the net work done by the engine in one cycle to the heat input along the 4 to 1 leg to find the efficiency in terms o f the two volumes V1, V2 only. The ratio of thesetwo volumes is the compression ratio. ...
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