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HW Assignment 36: Chap 10: CQ 10, Problems 30, 40, 50 40 pts Conceptual Question: CQ 10:10 After food is cooked in a pressure cooker, why is it very important to cool the container with cold water before attempting to remove the lid? Answer: At high temperature and pressure, the steam inside exerts large forces on the pot and cover. Strong latches hold them together, but they would explode apart if you tried to open the hot cooker. Problems: Problem 10:30: A tank having a volume of .1 m3 contains helium gas at 150 atm. How many balloons can the tank blow up if each filled balloon is a sphere 0.300 m in diameter at an absolute pressure of 1.2 atm? Solution: 4 3 The volume of helium in each balloon is Vb r. 3 The total volume of the helium at P2 1. at will be 20 m P 150 at V 2 1 V1 P2 m 1. at 20 m 0. m 100 3 12. m 5 3 Thus, the number of balloons that can be filled is V2 12. m 3 5 N 884 baloons l V b 4 3 0. m 150 3 Problem 10:40: The temperature near the top of the atmosphere on Venus is 240 K. a) find the rms speed of hydrogen (H2) at that point in Venus's atmosphere. b) Repeat for carbon dioxide c) It has been found that if the rms speed exceeds 1/6 the planet's escape velocity, the gas eventually leaks out of the atmosphere and into outer space. If the escape velocity on Venus is 10.3 km/s, does hydrogen escape? Does carbon dioxide escape? Solution: 1 3 From KEm olecule m v2 kBT 2 2 3kB T the rms speed of a molecule is vrm s v2 m ar m ol m ass M The mass of the molecule is m NA NA 2. 103 kg m ol 00 a) For hydrogen H 2 , m 3. 1027 kg 32 6. 10 m ol 02 23 e ecul m ol 3 1. 1023 J K 240 K 38 At T 240 K , vrm s H2 3. 10 32 27 kg 1. km s 73 44. 103 kg m ol 0 (b) For carbon dioxide, m 7. 1026 kg , and 31 6. 10 m ol 02 23 e ecul m ol 3 1. 1023 J K 240 K 38 at T 240 K , vrm s CO 2 7. 10 31 26 kg 0. 369 km s 3vescape 10. km s (c) Since, on Venus, 1. km s , we should expect that 71 6 6 hydrogen w iles l cape butcarbon di de w ilnot . Indeed, it is found that carbon dioxide is oxi l the predominant component in the atmosphere of Venus and hydrogen is present only in combination with other elements. Problem 10:50: A vertical cylinder of cross sectional area 0.5 m2 if fitted with a tight fitting frictionless piston of mass 5 kg. If there are 3 mol of an ideal gas in the cylinder at 500 K, determine the height h at which the piston will be in equilibrium under its own weight. Solution: When gas the supports the piston in equilibrium, the gauge pressure of the gas is F m g 5. kg 9. m s 2 0 80 Pgauge 9. 102 Pa , and the absolute pressure is 8 A A 0. m 2 050 P Patm Pgauge 1. 105 9. 102 Pa 013 8 The ideal gas law gives the volume as V nRT P , so the height of the cylindrical space is V nRT 3. m ol 8. J m ol K 500 K 0 31 h 2. m 4 A PA 1. 10 Pa 9. 102 Pa 0. m 2 013 5 8 050 HW Assignment 37: Chap 11: CQ 16 and Prob. 6, 18, 23 40 pts CQ 11:16 Energy is added to ice, raising its temperature from -10 to 5 degrees C. A larger amount of energy is added to the same mass of liquid water, raising its temperature from 15 to 20 degrees C. From these results, we can conclude that a) overcoming the latent heat of fusion of ice requires an input of energy b) the latent heat of fusion of ice delivers some energy to the system, c) the specific heat of ice is less than that of water d) the specific heat of ice is greater than that of water. Answer: c) the specific heat of ice is less than that of water. Problem 11:6 As part of an exercise routine a 50 kg person climbs 10 m up a vertical rope. How many food Calories are expended in a single climb up the rope? (1 food Calorie = 1000 calories). Solution: The internal energy converted to mechanical energy in one ascent of the rope is Q PEg m gh . Since 1 Cal i 1000 cal i 4186 J es , or e or es oul 1 Cal i or e Q 50. kg 9. m s2 10. m 0 80 0 1. Cal i 17 or e 4186 J Problem 11:18 When a driver brakes an automobile the friction between brake drums and the brake shoes converts the car's kinetic energy to thermal energy. If a 1500 kg automobile traveling at 30 m/s comes to a halt, how much does the temperature rise in each of the four 8 kg iron brake drums? (specific heat of iron is 448 J/kg-C) Solution: The kinetic energy given up by the car is absorbed as internal energy by the four brake drums (a total mass of 32 kg of iron). Thus, KE Q m drum scFe T or 2 1500 kg 30 m s 47C 2 2 1 m carvi 1 T 2 m drum scFe 32 kg 448 J kg °C Problem 11:23 What mass of steam that is initially at 120 degrees C is needed to warm 350 g of water and its 300 g aluminum container from 20 to 50 degrees C? Solution: In order to come to equilibrium at 50°C, the steam must: cool to 100°C, condense, and then cool (as condensed water) to 50°C. Thus, the conservation of energy equation is m steam cteam 120C 100C Lv c 100C 50C s w m w c m cupc l 50C- C w A 20 or m steam m c m cupcA l 30C w w . csteam 20C Lv cw 50C This gives 0. kg 4186 J kg C 0. kg 900 J kg C 30C 350 300 m steam , 2010 J kg C 20C 2. 10 J kg 4186 J kg C 50C 26 6 and m steam 2. 102 kg 21 g 1 HW Assignment 38: Chap 11: CQ 12 and Prob. 36, 42, 66 40 pts CQ 11:12 You need to pick up a very hot cooling pot in your kitchen. You have a pair of hot pads. Should you soak them in cold water or keep them dry in order to pick up the pot most comfortably? Answer: Keep them dry. The air pockets in the pad conduct energy slowly. Wet pads absorb some energy in warming up themselves, but the pot would still be hot and the water would quickly conduct a lot of energy to your hand. Problem 11:36 A box with a total surface area of 1.2 sq. m and a wall thickness of 4 cm is made of insulating material. A 10 W electric heater inside the box maintains the inside temperature at 15 degrees C above the outside temperature. Find the thermal conductivity k of the insulating material. Solution: Since the air temperature inside the box remains constant, the power input from the heater T must equal the energy transfer to the exterior. Thus, Ã kA , giving L Ã L 10. W 4. 102 m 0 00 2 k 2. 10 W m C 22 20 2 A T 1. m 15. C 0 Problem 11:42 The surface temperature of the Sun is about 5800 K. Taking the radius of the Sun to be 6.96 x 108 m, calculate the total energy radiated by the Sun each second. Assume e = 0.965. Solution: With an emissivity of e 0. , temperature of T 5800 K , and radius of r 6. 108 m , the 965 96 total power radiated by the spherical Sun is W 4 6.96 10 m 0.965 5 800 K 2 P AeT 4 5.67 108 2 4 8 4 m K or P 3.77 1026 W Problem 11:66 A wood stove is used to heat a single room. The stove is cylindrical in shape with a diameter of 40 cm and a length of 50 cm and operates at a temperature of 400 F. a) If the temperature of the room is 70 F determine the amount of radiant energy delivered to the room by the stove each second if the emissivity is 0.920. b) If the room is square with walls that are 8 ft high and 25 feet long, determine the R value needed in the walls and ceiling to maintain the inside temperature at 70 F if the outside temperature is 32 F. Solution: (a) The surface area of the stove is A stove A ends A cylindrical 2 r2 2 r , or h i s de A stove 2 0. m 2 0. m 0. m 0. m 2 2 200 200 500 880 5 The temperature of the stove is Ts 400°F 32. F 204C 477 K while that 0 9 5 of the air in the room is Tr 70. F 32. F 21. C 294 K . If the emissivity of 0 0 1 9 the stove is e 0. , the net power radiated to the room is 920 Ã A stovee Ts4 Tr4 5. 108 W m 2 K 4 0. m 67 880 2 0. 477 K 920 4 294 K 4 or Ã 2. 103 W 03 (b) The total surface area of the walls and ceiling of the room is A 4A wal A ceii 48. f 25. f 25. f 1. 103 f 2 2 l lng 00 t 0 t 0 t 43 t If the temperature of the room is constant, the power lost by conduction through the walls and ceiling must equal the power radiated by the stove. Thus, from thermal conduction equation, Ã A Th Tc R i , the net R value needed in the walls and ceiling is Ri A Th Tc 1. 10 43 3 f 2 70. F 32. F 1054 J 1 h t 0° 0° Ã 2. 10 J s 03 3 1 Bt 3600 s u or Ri 7. f 2 F h Bt 78 t u in US units 2 ft 2 F h 1m 1o C 3600s 1Btu m2 C s or Ri 7.78 o 1.37 in SI units Btu 3.281 ft 1.8 F 1h 1054 J J