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All Textbook Solutions for University Physics Volume 2
Check Your Understanding Does a given reading on a gasoline gauge indicate more gasoline in cold weather or in hot weather, or does the not matter?Check Your Understanding Two objects A and B have the same dimensions identically. A is of a material with a higher thermal expansion coefficient B. If the objects are heated identically, will A feel a greater Stress than B?Check Your Understanding If 25 kJ is necessary to raise the temperature of a rock from 25 to 30 , how much heat is necessary to heat the rock from 45 to 50 ?Check Your Understanding Explain why a cup of water (or soda) with ice cubes stays at 0 , even on a summer day.Check Your Understanding Why does snow Often remain even when daytime temperature are higher than the freezing temperature?Check Your Understanding Name an example from daily life (different from the text) for each mechanism of heat transfer.Check Your Understanding How does the rate of heat transfer by conduction change when all spatial dimensions are doubled?Check Your Understanding Explain why using a fan in the summer feels refreshing.Check Your Understanding How much greater is the rate of heat radiation when body is at the temperature 40 than when it is at the temperature 20 ?What does it mean to say that two systems are in thermal equilibrium?Give an example in which A has some kind of non-thermal equilibrium relationship with B, and B has the same relationship with C, but A does not have that relationship with C.If a thermometer is allowed to come to equilibrium with the air, and a glass of water is not in equilibrium with the air, what will happen to the thermometer reading when it is placed in the water?Give an example of a physical property that varies with temperature and describe how it is used to measure temperature.Pouring cold water into hot glass or ceramic cookware can easily break it. What causes the breaking? Explain Pyrex, a glass with a small coefficient of linear expansion, is less susceptible.One method of getting a tight fit, say of a metal peg in a hole in a metal block, is to manufacture the peg slightly larger than the hole. The peg is then inserted when at a different temperature than the block. Should the block be hotter or colder than the peg during insertion? Explain your answer.Does it really help to run hot water over a tight metal lid on a glass jar before trying to open it? Explain your answer.When a cold alcohol thermometer is placed in a hot liquid, the column of alcohol goes down slightly before going up. Explain why.Calculate the length of a 1-meter rod of a material with thermal expansion coefficient a when the temperature is raised from 300 K to 600 K. Taking your answer as the new initial length, find the length after the rod is cooled back down to 300 K. Is your answer 1 meter? Should it be? How can you account for the result you got?Noting the large stresses that can be caused by thermal expansion, an amateur weapon inventor decides to use it to make a new kind of gun. He plans to jam a bullet against an aluminum rod inside a closed invar tube. When he heats the tube, the rod will expand more than the tube and a very strong force will build up. Then, by a method yet to be determined, he will open the tube in a split second and let the force of the rod launch the bullet at very high speed. What is he overlooking?How is heat transfer related to temperature?Describe a situation in which heat transfer occurs.When heat transfers into a system, is the energy stored as heat? Explain briefly.The brakes in a car increase in temperature by T when bringing the car to rest from a speed v. How much greater would T be if the car initially had twice the speed? You may assume the car stops fast enough that no heat transfers out of the brakes.A pressure cooker contains water and steam in equilibrium at a pressure greater than atmospheric pressure. How does this greater pressure increase cooking speed?As shown below, which is the phase diagram for carbon dioxide, what is the vapor pressure of solid carbon dioxide (dry ice) at -78.5 (Note that the axes in the figure ale nonlinear and the graph is not to scale.)Can carbon dioxide be liquefied at room temperature (20 If so, how? If not, why not? (See the phase diagram in the preceding problem.)What is the distinction between gas and vapor?Heat transfer can cause temperature and phase changes. What else can cause these changes?How does the latent heat of fusion of water help slow the decrease of air temperatures, perhaps preventing temperatures from falling significantly below 0 , in the vicinity of large bodies of water?What is the temperature of ice right after it is formed by freezing water?If you place 0 ice into 0 water in an insulated container, what will the net result be? Will there be less ice and more liquid water, or more ice and less liquid water, or will the amounts stay the same?What effect does condensation on a glass of ice water have on the rate at which the ice melts? Will the condensation speed up the melting process or slow it down?In Miami, Florida, which has a very humid climate and numerous bodies of water nearby, it is unusual for temperatures to rise above about 38 (100 F). In the desert climate of Phoenix, Arizona, however, temperatures rise above that almost every day in July and August. Explain how the evaporation of water helps limit high temperatures in humid climates.In winter, it is often warmer in San Francisco than in Sacramento, 150 km inland. In summer, it is nearly always hotter in Sacramento. Explain how the bodies of water surrounding San Francisco moderate its extreme temperatures.Freeze-dried foods have been dehydrated in a vacuum. During the process, the food freezes and must be heated to facilitate dehydration. Explain both how the vacuum speeds up dehydration and why the food freezes as a result.In a physics classroom demonstration, an instructor inflates a balloon by mouth and then cools it in liquid nitrogen. When cold, the shrunken balloon has a small amount of light blue liquid in it, as well as some snow-like crystals. As it warms up, the liquid boils, and part of the crystals sublime, with some crystals lingering for a while and then producing a liquid. Identify the blue liquid and the two solids in the cold balloon. Justify your identifications using data from Table 1.4.Mechanisms of Heat Transfer What are the main methods of heat transfer from the hot core of Earth to its surface? From Earth's surface to outer space?When our bodies get too warm, they respond by sweating and increasing blood circulation to the surface to transfer thermal energy away from the core. What effect will those processes have on a person in a 40.0 hot tub?Shown below is a cut-away drawing of a thermos bottle (also known as a Dewar flask), which is a device designed specifically to slow down all forms of heat transfer. Explain the functions of the various parts, such as the vacuum, the silvering of the walls, the thin-walled long glass neck, the rubber support, the air layer, and the stopper.Some electric stoves have a flat ceramic surface with heating elements hidden beneath. A pot placed over a heating element will be heated, while the surface only a few centimeters away is safe to touch. Why is ceramic, with a conductivity less than that of a metal but greater than that of a good insulator, an ideal choice for the stove top?Loose-fitting white clothing covering most of the body, shown below, is ideal for desert dwellers, both in the hot Sun and during cold evenings. Explain how such clothing is advantageous during both day and night.One way to make a fireplace more energy-efficient is to have room air circulate around the outside of the file box and back into the room. Detail the methods of heat transfer involved.On cold, clear nights horses will sleep under the cover of large trees. How does this help them keep warm?When watching a circus during the day in a large, dark-colored tent, you sense significant heat transfer from the tent. Explain why this occurs.Satellites designed to observe the radiation from cold (3 K) dark space have sensors that are shaded from the Sun, Earth, and the Moon and are cooled to very low temperatures. Why must the sensors be at low temperature?Why are thermometers that ale used in weather stations shielded from the sunshine? What does a thermometer measure if it is shielded from the sunshine? What does it measure if it is not?Putting a lid on a boiling pot greatly reduces the heat transfer necessary to keep it boiling. Explain why.Your house will be empty for a while in cold weather and want to save energy and money. Should you tum the thermostat down to the lowest level that will protect the house from damage such as freezing pipes, or leave it at the normal temperature? (If you don't like coming back to a cold house, imagine that a timer controls the heating system so the house will be warm when you get back.) Explain your answer.You pour coffee into an unlidded cup, intending to drink it 5 minutes later. You can add cream when you pour the cup or right before you drink it. (The cream is at the same temperature either way. Assume that the cream and coffee come into thermal equilibrium with each other very quickly.) Which way will give you hotter coffee? What feature of this question is different from the previous one?Broiling is a method of cooking by radiation, which produces somewhat different results from cooking by conduction or convection. A gas flame or electric heating element produces a very high temperature close to the food and above it. is radiation the dominant heat-transfer method in this situation?On a cold winter morning, why does the metal of a bike feel colder than the wood of a porch?While traveling outside the United States, you feel sick. A companion gets you a thermometer, which says temperature is 39. What scale is that on? What is your Fahrenheit temperature? Should you seek medical help?What are the following temperatures on the Kelvin scale? (a) 68.0 F, an indoor temperature sometimes recommended for energy conservation in winter (b) 134 F, one of the highest atmospheric temperatures ever recorded on Earth (Death Valley, California, 1913) (c) 9890 F, the temperature of the surface of the Sun(a) Suppose a cold front blows into your locale and drops the temperature by 40.0 Fahrenheit degrees. How many degrees Celsius does the temperature decrease when it decreases by 40.0 F? (b) Show that any change in temperature in Fahrenheit degrees is nine-fifths the change in Celsius degreesAn Associated Press article on climate change said, "Some of the ice shelf 's disappearance was probably during times when the planet was 36 degrees Fahrenheit (2 degrees Celsius) to 37 degrees Fahrenheit (3 degrees Celsius) warmer than it is today." What mistake did the reporter make?(a) At what temperature do the Fahrenheit and Celsius scales have the same numerical value? (b) At what temperature do the Fahrenheit and Kelvin scales have same numerical value?A person taking a reading of the temperature in a freezer in Celsius makes two mistakes: first omitting the negative sign and then thinking the temperature is Fahrenheit. That is, the person reads xC as xF . Oddly enough, the result is the correct Fahrenheit temperature. What is the original Celsius reading? Round your answer to three significant figures.The height of the Washington Monument is measured to be 170.00 m on a day when the temperature is 35.0 . What will its height be on a day when the temperature falls to -10.0 ? Although the monument is made of limestone, assume that its coefficient of thermal expansion is the same as that of marble. Give your answer to five significant figures.How much taller does the Eiffel Tower become at the end of a day when the temperature has increased by 15 ? Its original height is 321 m and you can assume it is made of steel.What is the change in length of a 3.00-cm-long column of mercury if its temperature changes from 37.0 to 40.0 , assuming the mercury is constrained to a cylinder but unconstrained in length? Your answer will show why thermometers contain bulbs at the bottom instead of simple columns of liquid.How large an expansion gap should be left between steel railroad rails if they may reach a maximum temperature 35.0 greater than when they were laid? Their original length is 10.0 m.You are looking to buy a small piece of land in Hong Kong. The price is "only"' $60,000 per square meter. The land title says the dimensions ale 20 m 30 m. By how much would the total price change if you measured the parcel with a steel tape measure on a day when the temperature was 20 above the temperature that the tape measure was designed for? The dimensions of the land do not change.Global warming will produce rising sea levels partly due to melting ice caps and partly due to the expansion of water as average ocean temperatures rise. To get some idea of the size of this effect, calculate the change in length of a column of water 1.00 km high for a temperature increase of 1.00 C. Assume the column is not free to expand sideways. As a model of the ocean, that is a reasonable approximation, as only parts of the ocean very close to the surface can expand sideways onto land, and only to a limited degree. As another approximation, neglect the fact that ocean wan-ling is not uniform with depth.(a) Suppose a meter stick made of steel and one made of aluminum ale the same length at 0 C. What is their difference in length at 22.0 C? (b) Repeat the calculation for two 30.0-m-long surveyor's tapes.(a) If a 500-mL glass beaker is filled to the brim with ethyl alcohol at a temperature of 5.00C, how much will overflow when the alcohol's temperature reaches the room temperature of 22.0 C? (b) How much less water would overflow under the same conditions?Most cars have a coolant reservoir to catch radiator fluid that may overflow when the engine is hot. A radiator is made of copper and is filled to its 16.0-L capacity when at 10.0 . What volume of radiator fluid will overflow when the radiator and fluid reach a temperature of 95.0 , given that the fluid's volume coefficient of expansion is =400106/C? (Your answer will be a conservative estimate, as most car radiators have operating temperatures greater than 95.0 ).A physicist makes a cup of instant coffee and notices that, as the coffee cools, its level drops 3.00 mm in the glass cup. Show that this decrease cannot be due to thermal contraction by calculating the decrease in level if the 350 cm3 of coffee is in a 7.00-cm-diameter cup and decreases in temperature from 95.0 to 45.0 . (Most of the drop in level is actually due to escaping bubbles of air.)The density of water at 0 C is very nearly 1000 kg/m3 (it is actually 999.84 kg/m3), whereas the density of ice at 0 C is 917 kg/m3. Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.)Show that =3a , by calculating the infinitesimal change in volume dT of a cube with sides of length L when the temperature changes by dT .Calorimetry On a hot day, the temperature of an 80,000-L swimming pool increases by 1.50 . What is the net heat transfer during this heating? Ignore any complications, such as loss of water by evaporation.To sterilize a 50.0-g glass baby bottle, we must raise its temperature from 22.0 to 95.0 . How much heat transfer is required?The same heat transfer into identical masses of different substances produces different temperature changes. Calculate the final temperature when 1.00 kcal of heat transfers into 1.00 kg of the following, originally at 20.0 : (a) water; (b) concrete; (c) steel; and (d) mercury.Rubbing your hands together warms them by converting work into thermal energy. If a woman rubs her hands back and forth for a total of 20 rubs, at a distance of 7.50 cm per nub, and with an average frictional force of 40.0 N, what is the temperature increase? The mass of tissues warned is only 0.100 kg, mostly in the palms and fingers.A 0.250-kg block of a pule material is heated from 20.0 to 65.0 by the addition of 4.35 kJ of energy. Calculate its specific heat and identify the substance of which it is most likely composed.Suppose identical amounts of heat transfer into different masses of copper and water, causing identical changes in temperature. What is the ratio of the mass of copper to water?(a) The number of kilocalories in food is determined by calorimetry techniques in which the food is burned and the amount of heat transfer is measured. How many kilocalories per gram ale there in a 5.00-g peanut if the energy from burning it is transferred to 0. 500 kg of water held in a 0.100-kg aluminum cup, causing a 54.9- temperature increase? Assume the process takes place in an ideal calorimeter, in other words a perfectly insulated container. (b) Compare your answer to the following labeling information found on a package of dry roasted peanuts: a sewing of 33 g contains 200 calories. Comment on whether the values are consistent.Following vigorous exercise, the body temperatureof an 80.0 kg person is 40.0 C. At what rate in wattsmust the person transfer thermal energy to reduce the bodytemperature to 37.0 C in 30.0 min, assuming the bodycontinues to produce energy at the rate of 150 W? (1 watt =1 joule/secondor 1 W = J/s)In a study of healthy young men[1], doing 20 push-ups in 1 minute burned an amount of energy per kg that for a 70.0-kg man corresponds to 8.06 calories (kcal). How much would a 70.0-kg man's temperature rise if he did not lose any heat during that time?A 1.28-kg sample of water at 10.0 is in a calorimeter. You drop a piece of steel with a mass of 0.385 kg at 215 into it. After the sizzling subsides, what is the final equilibrium temperature? (Make the reasonable assumptions that any steam produced condenses into liquid water during the process of equilibration and that the evaporation and condensation don't affect the outcome, as we'll see in the next section.)Repeat the preceding problem, assuming the water is in a glass beaker with a mass of 0.200 kg, which in tum is in a calorimeter. The beaker is initially at the same temperature as the water. Before doing the problem, should the answer be higher or lower than the preceding answer? Comparing the mass and specific heat of the beaker to those of the water, do you think the beaker will make much difference?How much heat transfer (in kilocalories) is required to thaw a 0.450-kg package of frozen vegetables originally at 0 if their heat of fusion is the same as that of water?A bag containing 0 ice is much more effective in absorbing energy than one containing the same amount of 0 water. (a) How much heat transfer is necessary to raise the temperature of 0.800 kg of water from 0 to 30.0 ? (b) How much heat transfer is required to first melt 0.800 kg of 0 ice and then raise its temperature? (c) Explain how your answer supports the contention that the ice is more effective.(a) How much heat transfer is required to raise the temperature of a 0.750-kg aluminum pot containing 2.50 kg of water from 30.0 to the boiling point and then boil away 0.750 kg of water? (b) How long does this take if the rate of heat transfer is 500 W?Condensation on a glass of ice water causes the ice to melt faster than it would otherwise. If 8.00 g of vapor condense on a glass containing both water and 200 g of ice, how many grams of the ice will melt as a result? Assume no other heat transfer occurs. Use Lvfor water at 37 as a better approximation than Lvfor water at 100 .)On a trip, you notice that a 3.50-kg bag of ice lasts an average of one day in your cooler. What is the average power in watts entering the ice if it staffs at 0 and completely melts to 0 water in exactly one day?On a certain dry sunny day, a swimming pool 's temperature would rise by 1.50 if not for evaporation. What fraction of the water must evaporate to carry away precisely enough energy to keep the temperature constant?(a) How much heat transfer is necessary to raise the temperature of a 0.200-kg piece of ice from -20.0 to 130.0 , including the energy needed for phase changes? (b) How much time is required for each stage, assuming a constant 20.0 kJ/s rate of heat transfer? (c) Make a graph of temperature versus time for this process.In 1986, an enormous iceberg broke away from the Ross Ice Shelf in Antarctica. It was an approximately rectangular prism 160 km long, 40.0 kin wide, and 250 m thick. (a) What is the mass of this iceberg, given that the density of ice is 917 kg/m3? (b) How much heat transfer (in joules) is needed to melt it? (c) How many years would it take sunlight alone to melt ice this thick, if the ice absorbs an average of 100 W/m2, 12.00 h per day?How many grams of coffee must evaporate from 350 g of coffee in a 100-g glass cup to cool the coffee and the cup from 95.0 to 45.0 ? Assume the coffee has the same thermal properties as water and that the average heat of vaporization is 2340 kJ/kg (560 kcal/g). Neglect heat losses through processes other than evaporation, as well as the change in mass of the coffee as it cools. Do the latter two assumptions cause your answer to be higher or lower than the true answer?(a) It is difficult to extinguish a fire on a crude oil tanker, because each liter of crude oil releases 2.80107 J of energy when burned. To illustrate this difficulty, calculate the number of liters of water that must be expended to absorb the energy released by burning 1.00 L of crude oil, if the water's temperature rises from 20.0 C to 100 C, it boils, and the resulting steam's temperature rises to 300 C at constant pressure. (b) Discuss additional complications caused by the fact that crude oil is less dense than water.The energy released from condensation in thunderstorms can be very large. Calculate the energy released into the atmosphere for a small storm of radius 1 km, assuming that 1.0 cm of rain is precipitated uniformly over this area.To help prevent frost damage, 4.00 kg of water at 0 is sprayed onto a fruit tree. (a) How much heat transfer occurs as the water freezes? (b) How much would the temperature of the 200-kg tree decrease if this amount of heat transferred from the tree? Take the specific heat to be 3.35k J/kg. , and assume that no phase change occurs in the tree.A 0.250-kg aluminum bowl holding 0.800 kg of soup at 25.0 is placed in a freezer. What is the final temperature if 388 kJ of energy is transferred from the bowl and soup, assuming the soup's thermal properties are the same as that of water?A 0.0500-kg ice cube at 30.0 is placed in 0.400 kg of 35.0- water in a very well-insulated container. What is the final temperature?If you pour 0.0100 kg of 20.0 water onto a 1.20-kg block of ice (which is initially at 15.0 what is the final temperature? You may assume that the water cools so rapidly that effects of the surroundings are negligible.Indigenous people sometimes cook in watertight baskets by placing hot locks into water to bring it to a boil. What mass of 500- granite must be placed in 4.00 kg of 15.0- water to bring its temperature to 100 , if 0.0250 kg of water escapes as vapor from the initial sizzle? You may neglect the effects of the surroundings.What would the final temperature of the pan and water be in Example 1.7 if 0.260 kg of water were placed in the pan and 0.0100 kg of the water evaporated immediately, leaving the remainder to come to a common temperature with the pan?(a) Calculate the rate of heat conduction through house walls that are 13.0 cm thick and have an average thermal conductivity twice that of glass wool. Assume there are no windows or doors. The walls' surface area is 120 m2 and their inside surface is at 18.0 , while their outside surface is at 5.00 . (b) How many 1-kW room heaters would be needed to balance the heat transfer due to conduction?The rate of heat conduction out of a window on a winter day is rapid enough to chill the air next to it To see just how rapidly the windows transfer heat by conduction, calculate the rate of conduction in watts through a 3.00-m2 window that is 0.634 cm thick (1/4 in.) if the temperatures of the inner and outer surfaces are 5.00 and 10.0 , respectively. (This rapid rate will not be maintained—the inner surface will cool, even to the point of frost formation.)Calculate the rate of heat conduction out of the human body, assuming that the core internal temperature is 37.0 , the skin temperature is 34.0 , the thickness of the fatty tissues between the core and the skin averages 1.00 cm, and the surface area is 1.40 m2.Suppose you stand with one foot on ceramic flooring and one foot on a wool carpet, making contact over an area of 80.0 cm2 with each foot. Both the ceramic and the carpet are 2.00 cm thick and ale 10.0 on their bottom sides. At what rate must heat transfer occur from each foot to keep the top of the ceramic and carpet at 33.0 ?A man consumes 3000 kcal of food in one day, converting most of it to thermal energy to maintain body temperature. If he loses half this energy by evaporating water (through breathing and sweating), how many kilograms of water evaporate?A firewalker runs across a bed of hot coals without sustaining burns. Calculate the heat transferred by conduction into the sole of one foot of a firewalker given that the bottom of the foot is a 3.00-mm-thick callus with a conductivity at the low end of the range for wood and its density is 300 kg/m3. The area of contact is 25.0 cm2 the temperature of the coals is 700 , and the time in contact is 1.00 s. Ignore the evaporative cooling of sweat.(a) What is the rate of heat conduction through the 3.00-cm-thick fur of a large animal having a I .40-m surface area? Assume that the animal's skin temperature is 32.0 , that the air temperature is 5.00 , and that has the same thermal conductivity as air. (b) What food intake will the animal need in one day to replace this heat transfer?A warms transfers energy by conduction through its blubber at the rate of 150 W when immersed in 1.00 water. The walrus's internal core temperature is 37.0 and it has a surface area of 2.00 m2. What is the average thickness of its blubber, which has the conductivity of fatty tissues without blood?Compare the rate of heat conduction through a 13.0-cm-thick wall that has an area of 10.0 m2 and a thermal conductivity that of glass wool with the rate of heat conduction through a 0.750-cm-thick window that has an area of 2.00 m2, assuming the same temperature difference across each.Suppose a person is covered head to foot by wool clothing with average thickness of 2.00 cm and is transferring energy by conduction through the clothing at the rate of 50.0 W. What is the temperature difference across the clothing, given the surface area is 1.40 m2?Some stove tops are smooth ceramic for easy cleaning. If the ceramic is 0.600 cm thick and heat conduction occurs through the same area and at the same rate as computed in Example 1.11, what is the temperature difference across it? Ceramic has the same thermal conductivity as glass and brick.One easy way to reduce heating (and cooling) costs is to add extra insulation in the attic of a house. Suppose a single-story cubical house already had 15 cm of fiberglass insulation in the attic and in all the exterior surfaces. If you added an extra 8.0 cm of fiberglass to the attic, by what percentage would the heating cost of the house drop? Take the house to have dimensions 10 m by 15 m by 3.0 m. Ignore air infiltration and heat loss through windows and doors, and assume that the interior is uniformly at one temperature and the exterior is uniformly at another.Many decisions are made on the basis of the payback period: the time it will take through savings to equal the capital cost of an investment. Acceptable payback times depend upon the business or philosophy one has. For some industries, a payback period is as small as 2 yeas) Suppose you wish to install the extra insulation in the preceding problem. If energy cost $1.00 per million joules and the insulation was $4.00 per square meter, then calculate the simple payback time. Take the average T for the 120-day heating season to be 15.0 .In 1701, the Danish astronomer Ole Rømer proposed a temperature scale with two fixed points, freezing water at 7.5 degrees, and boiling water at 60.0 degrees. What is the boiling point of oxygen, 90.2 K, on the Rømer scale?What is the percent error of thinking the melting point of tungsten is 3695 instead of the correct value of 3695 K?An engineer wants to design a structure in which the difference in length between a steel beam and an aluminum beam remains at 0.500 m regardless of temperature, for ordinary temperatures. What must the lengths of the beams be?How much stress is cleated in a steel beam if its temperature changes from 15 to 40 but it cannot expand? For steel, the Young's modulus Y=210109N/m2 from Stress, Strain, and Elastic Modulus (http://cnx.org/content/m58342/latest/#fs-id1163713086230). (Ignore the change in area resulting from the expansion.)A brass rod (Y=90109N/m2), with a diameter of 0.800 cm and a length of 1.20 m when the temperature is 25 , is fixed at both ends. At what temperature is the force in it at 36,000 N?A mercury thermometer still in use for meteorology has a bulb with a volume of 0.780 cm3 and a tube for the mercury to expand into of inside diameter 0.130 mm. (a) Neglecting the thermal expansion of the glass, what is the spacing between marks 1 apart? (b) If the thermometer is made of ordinary glass (not a good idea), what is the spacing?Even when shut down after a period of normal use, a large commercial nuclear reactor transfers thermal energy at the rate of 150 MW by the radioactive decay of fission products. This heat transfer causes a rapid increase in temperature if the cooling system fails (1 watt = 1 joule/second or 1 W = 1 J/s and 1 MW = 1 megawatt). (a) Calculate the rate of temperature increase in degrees Celsius per second (C/s) if the mass of the is 1.60105 kg and it has an average specific heat of 0.3349 kJ/kg . C. (b) How long would it take to obtain a temperature increase of 2000 C, which could cause some metals holding the radioactive materials to melt? (The initial rate of temperature increase would be greater than that calculated here because the heat transfer is concentrated in a smaller mass. Later, however, the temperature increase would slow down because the 500,000-kg steel containment vessel would also begin to heat up.)You leave a pastry in the refrigerator on a plate and ask your roommate to take it out before you get home so you can eat it at room temperature, the way you like it Instead, your roommate plays video games for hours. When you return, you notice that the pastry is still cold, but the game console has become hot. Annoyed, and knowing that the pastry w-ill not be good if it is microwaved, you warm up the pastry by unplugging the console and putting it in a clean trash bag (which acts as a perfect calorimeter) with the pastry on the plate. After a while, you find that the equilibrium temperature is a nice, warm 38.3 . You know that the game console has a mass of 2.1 kg. Approximate it as having a uniform initial temperature of 45 . The pastry has amass of 0.16 kg and a specific heat of 3.0 k J/(kg . ), and is at a uniform initial temperature of 4.0 . The plate is at the same temperature and has a mass of 0.24 kg and a specific heat of 0.90 J/(kg . ). What is the specific heat of the console?Two solid spheres, A and B, made of the same material, are at temperatures of , and 100 , respectively. The spheres are placed in thermal contact in an ideal calorimeter, and they reach an equilibrium temperature of 20 . Which is the bigger sphere? What is the ratio of their diameters?In some countries, liquid nitrogen is used on daily trucks instead of mechanical refrigerators. A 3.00-hour delivery trip requires 200 L of liquid nitrogen, which has a density of 808 kg/m3. (a) Calculate the heat transfer necessary to evaporate this amount of liquid nitrogen and raise its temperature to 3.00 . (Use cp and assume it is constant over the temperature range.) This value is the amount of cooling the liquid nitrogen supplies. (b) What is this heat transfer rate in kilowatt-hours? (c) Compare the amount of cooling obtained from melting an identical mass of 0- ice with that from evaporating the liquid nitrogen.Some gun fanciers make their own bullets, which involves melting lead and casting it into lead slugs. How much heat transfer is needed to raise the temperature and melt 0.500 kg of lead, stating from 25.0 ?A 0.800-kg iron cylinder at a temperature of 1.00103 is dropped into an insulated chest of 1.00 kg of ice at its melting point. What is the final temperature, and how much ice has melted?Repeat the preceding problem with 2.00 kg of ice instead of 1.00 kg.Repeat the preceding problem with 0.500 kg of ice, assuming that the ice is initially in a copper container of mass 1.50 kg in equilibrium with the ice.A 30.0-g ice cube at its melting point is dropped into an aluminum calorimeter of mass 100.0 g in equilibrium at 24.0 with 300.0 g of an unknown liquid. The final temperature is 4.0 . What is the heat capacity of the liquid?(a) Calculate the rate of heat conduction through a double-paned window that has a 150-m2 area and is made of two panes of 0.800 cm-thick glass separated by a 1.00 cm air gap. The inside surface temperature is 15.0 C, while that on the outside is 10.0 OC. (Hint: There are identical temperature drops across the two glass panes. First find these and then the temperature drop across the air gap. This problem ignores the increased heat transfer in the air gap due to convection.) (b) Calculate the rate of heat conduction through a 1.60-cm-thick window of the same area and with the same temperatures. Compare your answer with that for part (a).(a) An exterior wall of a house is 3 m tall and 10 m wide. It consists of a layer of drywall with an R factor of 0.56, a layer 3.5 inches thick filled with fiberglass batts, and a layer of insulated siding with an R factor of 2.6. The wall is built so well that there are no leaks of air through it. When the inside of the wall is at 22 and the outside is at 2 , what is the rate of heat flow through the wall? (b) More realistically, the 3.5-inch space also contains 2-by-4 studs—wooden boards 1.5 inches by 3.5 inches oriented so that 3.5-inch dimension extends from the drywall to the siding. They are "on 16-inch centers," that is, the centers of the studs ale 16 inches apart. What is the heat current in this situation? Don't worry about one stud mole or less.For the human body, what is the rate of heat transfer by conduction through the body's tissue with the following conditions: the tissue thickness is 3.00 cm, the difference in temperature is 2.00 , and the skin area is 1.50 m2. How does this compare with the average heat transfer rate to the body resulting from an energy intake of about 2400 kcal per day? (No exercise is included.)You have a Dewar flask (a laboratory vacuum flask) that has an open top and straight sides, as shown below. You fill it with water and put it into the freezer. It is effectively a perfect insulator, blocking all heat transfer, except on the top. After a time, ice forms on the surface of the water. The liquid water and the bottom surface of the ice, in contact with the liquid water, are at 0 . The top surface of the ice is at the same temperature as the air in the freezer, 18 . Set the rate of heat flow through the ice equal to the rate of loss of heat of fusion as the water freezes. When the ice layer is 0.700 cm thick, find the rate in m/s at which the ice is thickening.An infrared heater for a sauna has a surface area of 0.050 m2 and an emissivity of 0.84. What temperature must it run at if the required power is 360 W? Neglect the temperature of the environment.(a) Deter-nine the power of radiation from the Sun by noting that the intensity of the radiation at the distance of Earth is 1370 W/m2. Hint: That intensity will be found everywhere on a spherical surface with radius equal to that of Earth's orbit. (b) Assuring that the Sun's temperature is 5780 K and that its emissivity is 1, find its radius.A pendulum is made of a rod of length L and negligible mass, but capable of thermal expansion, and a weight of negligible size. (a) Show that when the temperature increases by dT, the period of the pendulum increases by a fraction aLdT/2 . (b) A clock controlled by a brass pendulum keeps time correctly at 10 C. If the room temperature is 30 C, does the clock run faster or slower? What is its error in seconds per day?At temperatures of a few hundred kelvins the specific heat capacity of copper approximately follows the empirical formula c=a+T+T2, where a=349J/kgK, =0.107J/kgK2, and =4.58105JkgK. How much heat is needed to raise the temperature of a 2.00-kg piece of copper from 20 to 250 ?In a calorimeter of negligible heat capacity, 200 g of steam at 150 and 100 g of ice at 40 are mixed. The pressure is maintained at 1 atm. What is the final temperature, and how much steam, ice, and water are present?An astronaut performing an extra-vehicular activity (space walk) shaded from the Sun is wearing a spacesuit that can be approximated as perfectly white ( e=0 ) except for a 5 cm × 8 cm patch in the form of the astronaut's national flag. The patch has emissivity 0.300. The spacesuit under the patch is 0.500 cm thick, with a thermal conductivity k 0.0600 W/m , and its inner surface is at a temperature of 20.0 . What is the temperature of the patch, and what is the rate of heat loss through it? Assume the patch is so thin that its outer surface is at the same temperature as the outer surface of the spacesuit under it. Also assume the temperature of outer space is 0 K. You will get an equation that is very hand to solve in closed form, so can solve it numerically with a graphing calculator, with software, or even by trial and error with a calculator.The goal in this problem is to find the growth of an ice layer as a function of time. Call the thickness of the ice layer L. (a) Derive an equation for dL/dt in terms of L , the temperature T above the ice, and the properties of ice (which can leave in symbolic form instead of substituting the numbers). (b) Solve this differential equation assuming that at t=0 , you have L=0 . If you have studied differential equations, you will know a technique for solving equations of this type: manipulate the equation to get dL/dt multiplied by a (very simple) function of L on one side, and integrate both sides with respect to time. Alternatively, you may be able to use your knowledge of the derivatives of various functions to guess the solution, which has a simple dependence on t. (c) Will the water eventually freeze to the bottom of the flask?As the very first rudiment of climatology, estimate the temperature of Earth. Assume it is a perfect sphere and its temperature is uniform. Ignore the greenhouse effect. Thermal radiation from the Sun has an intensity (the "solar constant" S) of about 1370 W/m2 at the radius of Earth's orbit. (a) Assuming the Sun's rays are parallel, what area must S be multiplied by to get the total radiation intercepted by Earth? It will be easiest to answer in tens of Earth's radius, R. (b) Assume that Earth reflects about 30% of the solar energy it intercepts. In other words, Earth has an albedo with a value of A=0.3 . In terms of S, and R, what is the rate at which Earth absorbs energy from the Sun? (c) Find the temperature at which Earth radiates energy at the same rate. Assume that at the infrared wavelengths where it radiates, the emissivity e is 1. Does your result show that the greenhouse effect is important? (d) How does your answer depend on the area of Earth?Let's stop ignoring the greenhouse effect and incorporate it into the previous problem in a very rough way. Assume the atmosphere is a single layer, a spherical shell around Earth, with an emissivity e 0.77 (chosen simply to give the light answer) at infrared wavelengths emitted by Earth and by the atmosphere. However, the atmosphere is transparent to the Sun's radiation (that is, assume the radiation is at visible wavelengths with no infrared), so the Sun's radiation leaches the surface. The greenhouse effect comes from the difference between the atmosphere transmission of visible light and its rather strong absorption of infrared. Note that the atmosphere's radius is not significantly different from Earth's, but since the atmosphere is a layer above Earth, it emits radiation both upward and downward, so it has twice Earth's area. There are three radiative energy transfers in this problem: solar radiation absorbed by Earth's surface; infrared radiation from the surface, which is absorbed by the atmosphere according to its emissivity; and infrared radiation from the atmosphere, half of which is absorbed by Earth and half of which goes out into space. Apply the method of the previous problem to get an equation for Earth 's surface and one for the atmosphere, and solve them for the two unknown temperatures, surface and atmosphere. a. In terms of Earth's radius, the constant , and the unknown temperature Ts of the surface, what is the power of the infrared radiation from the surface? b. What is the power of Earth 's radiation absorbed by the atmosphere? c. In terms of the unknown temperature Te of the atmosphere, what is the power radiated from the atmosphere? d. Write an equation that says the power of the radiation the atmosphere absorbs from Earth equals the of the radiation it emits. e. Half of the power radiated by the atmosphere hits Earth. Write an equation that says that the power Earth absorbs from the atmosphere and the Sun equals the power that it emits. f. Solve your two equations for the unknown temperature of Earth. For steps that make this model less crude, see for example (https://openstaxcollege.org/l/21paulgormlec) by Paul O'Gorrnan.Check Your Understanding The recommended daily amount of vitamin B3 or niacin, C6NH5 O2, for women who are not pregnant or nursing, is 14 mg. Find the number of molecule of niacin in that amount.Check Understanding The density of in a Classroom ( p=1.00 and T=20C ) is 1.28 kg/m3. At what pressure is the density 0.600 kg/m3 if the temperature is kept constant?Check Your Understanding Liquids and solids have densities on the order of 1000 greater than gases. Explain how this implies that the distances between molecules in gases are on the order of 10 times greater than the size of their molecules.Check Your Understanding If you consider a very small object, such as a grain of pollen, in a gas, then the number of molecules striking its surface would also be relatively small. Would you expect the grain of pollen to experience any fluctuations in pressure due to statistical fluctuations in the number of gas molecules striking it in a given amount of time?Check Your Understanding Which has a longer mean free path, liquid water or water vapor in the air?Check Your Understanding Suppose 2 moles of helium gas at 200 K are mixed with 2 moles gas at 400 K in a calorimeter. What is the final temperature?Two H2 molecules can react with one O2 molecule to produce two H2O molecules. How many moles of hydrogen molecules ale needed to react with one mole of oxygen molecules?Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why?In the last chapter, free convection was explained as the result of buoyant forces on hot fluids. Explain the upward motion of air in flames based on the ideal gas law.How is momentum related to the pressure exerted by a gas? Explain on the molecular level, considering the behavior of molecules.If one kind of molecule has double the radius of another and eight times the mass, how do their mean free paths under the same conditions compare? How do their mean free times compare?What is the average velocity of the air molecules in the room where you are right now?Why do the atmospheres of Jupiter, Saturn, Uranus, and Neptune, which are much more massive and farther from the Sun than Earth is, contain large amounts of hydrogen and helium?Statistical mechanics says that in a gas maintained at a constant temperature through thermal contact with a bigger system (a "reservoir") at that temperature, the fluctuations in internal energy are typically a fraction 1/N of the internal energy. As a fraction of the total internal energy of a mole of gas, how big are the fluctuations in the internal energy? Are we justified in ignoring them?Which is more dangerous, a closet where tanks of nitrogen are stored, or one where tanks of carbon dioxide are stored?Experimentally it appears that many polyatomic molecules' vibrational degrees of freedom can contribute to some extent to their energy at room temperature. Would you expect that fact to increase or decrease their heat capacity from the value R? Explain.One might think that the internal energy of diatomic gases is given by Eint=5RT/2 . Do diatomic gases near room temperature have more or less internal energy than that? Hint: Their internal energy includes the total energy added in raising the temperature from the boiling point (very low) to room temperature.You mix 5 moles of H2 at 300 K with 5 moles of He at 360 K in a perfectly insulated calorimeter. Is the final temperature higher or lower than 330 K?One cylinder contains helium gas and another contains krypton gas at the same temperature. Mark each of these statements true, false, or impossible to determine from the given information. (a) The rms speeds of atoms in the two gases are the same. (b) The average kinetic energies of atoms in the two gases are the same. (c) The internal energies of 1 mole of gas in each cylinder are the same. (d) The pressures in the two cylinders ale the same.Repeat the previous question if one gas is still helium but the other is changed to fluorine, F2.An ideal gas is at a temperature of 300 K. To double the average speed of its molecules, what does the temperature need to be changed to?The gauge pressure in your car tires is 2.50105N/m2 at a temperature of 35.0 when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to 40.0 ? Assume the tires have not gained or lost any air.Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0 . (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is 60.0 (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. Is this effect significant?People buying food in sealed bags at high elevations often notice that the bags are puffed up because the air inside has expanded. A bag of pretzels was packed at a pressure of 1.00 atm and a temperature of 22.0 . When opened at a summer picnic in Santa Fe, New Mexico, at a temperature of 32.0 , the volume of the air in the bag is 1.38 times its original volume. What is the pressure of the air?How many moles are there in (a) 0.0500 g of N2 gas (M = 28.0 g/mol)? (b) 10.0 g of CO2 gas (M = 44.0 g/mol)? (c) How many molecules are present in each case?A cubic container of volume 2.00 L holds 0.500 mol of nitrogen gas at a temperature of 25.0 . What is the net force due to the nitrogen on one wall of the container? Compare that force to the sample's weight.Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at 37.0 (body temperature) and that the total volume in the lungs is several times the amount inhaled in a typical breath as given in Example 2.2.An airplane passenger has 100 cm3 of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to 7.50104N/m2 ?A company advertises that it delivers helium at a gauge pressure of 1.72107 Pa in a cylinder of volume 43.8 L. How many balloons can be inflated to a volume of 4.00 L with that amount of helium? Assume the pressure inside the balloons is 1.72105 Pa and the temperature in the cylinder and the balloons is 25.0 .According to http:hyperphysics.phy-astr.gsu.edu/hbase/solar/venusenv.html, the atmosphere of Venus is approximately 96.5% CO2 and 3.5% N2 by volume. On the surface, where the temperature is about 750 K and the pressure is about 90 atm, what is the density of the atmosphere?An expensive vacuum system can achieve a pressure as low as 1.00107 N/m2 at 20.0 . How many molecules ale there in a cubic centimeter at this pressure and temperature?The number density N/V of gas molecules at a certain location in the space above our planet is about 1.001011 m3 , and the pressure is 2.751010 N/m2 in this space. What is the temperature there?A bicycle tire contains 2.00 L of gas at an absolute pressure of 7.00105 N/m2 and a temperature of 18.0 . What will its pressure be if you let out an amount of air that has a volume of 100cm3 at atmospheric pressure? Assume tire temperature and volume remain constant.In a common demonstration, a bottle is heated and stoppered with a hard-boiled egg that's a little bigger than the bottle's neck. When the bottle is cooled, the pressure difference between inside and outside forces the egg into the bottle. Suppose the bottle has a volume of 0.500 L and the temperature inside it is raised to 80.0 while the pressure remains constant at 1.00 atm because the bottle is open. (a) How many moles of air are inside? (b) Now the egg is put in place, sealing the bottle. What is the gauge pressure inside after the air cools back to the ambient temperature of 25 but before the egg is forced into the bottle ?A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of 14107 N/m2 and a temperature of 25.0 . The cylinder is cooled to dry ice temperature ( 78.5 ) to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank as in part (c) appear to be a practical solution?Find the number of moles in 2.00 L of gas at 35.0 and under 7.41107 N/m2 of pressure.Calculate the depth to which Avogadro's number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.(a) What is the gauge pressure in a 25.0 cc car tire containing 3.60 mol of gas in a 30.0-L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and 25.0 ? Assume the temperature remains at 25.0 cc and the volume remains constant.A person hits a tennis ball with a mass of 0.058 kg against a wall. The average component of the ball's velocity perpendicular to the wall is 11 m/s, and the ball hits the wall every 2.1 s on average, rebounding with the opposite perpendicular velocity component. (a) What is the average force exerted on the wall? (b) If the part of the wall the person hits has an area of 3.0 m2, what is the average pressure on that area?A person is in a closed room (a racquetball court) with v=453 m3 hitting a ball (m 42.0 g) around at random without any pauses. The average kinetic energy of the ball is 2.30 J. (a) What is the average value of vx2 ? Does it matter which direction you take to be x ? (b) Applying the methods of this chapter, find the average pressure on the walls? (c) Aside from the presence of only one "molecule" in this problem, what is the main assumption in Pressure, Temperature, and RMS Speed that does not apply here?Five bicyclists are riding at the following speeds: 5.4 m/s, 5.7 m/s, 5.8 m/s, 6.0 m/s, and 6.5 m/s. (a) What is their average speed? (b) What is their rms speed?Some incandescent light bulbs are filled with argon gas. What is vrms for argon atoms near the filament, assuming their temperature is 2500 K?Typical molecular speeds (vrms) are large, even at low temperatures. What is vrms for helium atoms at 5.00 K, less than one degree above helium's liquefaction temperature ?What is the average kinetic energy in joules of hydrogen atoms on the 5500 surface of the Sun? (b) What is the average kinetic energy of helium atoms in a legion of the solar corona where the temperature is 6.00105 K ?What is the ratio of the average translational kinetic energy of a nitrogen molecule at a temperature of 300 K to the gravitational potential energy of a nitrogen-molecule-Earth system at the ceiling of a 3-m-tall room with respect to the same system with the molecule at the floor?What is the total translational kinetic energy of the air molecules in a room of volume 23 m3 if the pressure is 9.5104 Pa (the room is at fairly high elevation) and die temperature is 21 ? Is any item of data unnecessary for the solution?The product of the pressure and volume of a sample of hydrogen gas at 0.00 is 80.0 J. (a) How many moles of hydrogen are present? (b) What is the average translational kinetic energy of the hydrogen molecules? (c) What is the value of the product of pressure and volume at 200 ?What is the gauge pressure inside a tank of 4.86104 mol of compressed nitrogen with a volume of 6.56 m3 if the rms speed is 514 m/s?If the rms speed of oxygen molecules inside a refrigerator of volume 22.0 ft.3 is 465 m/s, what is the partial pressure of the oxygen? There are 5.71 moles of oxygen in the refrigerator, and the molar mass of oxygen is 32.0 g/mol.The escape velocity of any object from Earth is 11.1 km/s. At what temperature would oxygen molecules (molar mass is equal to 32.0 g/mol) have root-mean-square velocity vrms equal to Earth's escape velocity of 11.1 km/s.The escape velocity from the Moon is much smaller than that from the Earth, only 2.38 km/s. At what temperature would hydrogen molecules (molar mass is equal to 2.016 g/mol) have a root-mean-square velocity vrms equal to the Moon's escape velocity?Nuclear fusion, the energy solute of Sun, hydrogen bombs, and fusion reactors, occurs much more readily when the average kinetic energy of the atoms is high—that is, at high temperatures. Suppose you want the atoms in your fusion experiment to have average kinetic energies of 6.41014 J . What temperature is needed?Suppose that the typical speed (vrms) of carbon dioxide molecules (molar mass is 44.0 g/mol) in a flame is found to be 1350 m/s. What temperature does this indicate?(a) Hydrogen molecules (molar mass is equal to 2.016 g/mol) have vrms equal to 193 m/s. What is the temperature? (b) Much of the gas near the Sun is atomic hydrogen (H rather than H2). Its temperature would have to be 1.5107 K for the speed vrms to equal the escape velocity from the Sun. What is that velocity?There are two important isotopes of uranium, U235 and U238 ; these isotopes are nearly identical chemically but have different atomic masses. Only U235 is very useful in nuclear reactors. Separating the isotopes is called uranium enrichment (and is often in the news as of this writing, because of concerns that some countries are enriching uranium with the goal of making nuclear weapons.) One of the techniques for enrichment, gas diffusion, is based on the different molecular speeds of uranium hexafluoride gas, UF6 . (a) The molar masses of U235 and U238F6 are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their typical speeds vrms ? (b) At what temperature would their typical speeds differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?The partial pressure of carbon dioxide in the lungs is about 470 Pa when the total pressure in the lungs is 1.0 atm. What percentage of the air molecules in the lungs is carbon dioxide? Compare your result to the percentage of carbon dioxide in the atmosphere, but 0.033%.Dry air consists of approximately 78% nitrogen, 21% oxygen, and 1% argon by mole, with trace amounts of other gases. A tank of compressed dry air has a volume of 1.76 cubic feet at a gauge pressure of 2200 pounds per square inch and a temperature of 293 K. How much oxygen does it contain in moles?(a) Using data from the previous problem, find the mass of nitrogen, oxygen, and argon in 1 mol of dry air. The molar mass of N2 is 28.0 g/mol, that of O2 is 32.0 g/mol, and that of argon is 39.9 g/mol. (b) Dry air is mixed with pentane 5 H12, molar mass 72.2 g/mol), an important constituent of gasoline, in an air-fuel ratio of 15:1 by mass (roughly typical for car engines). Find the partial pressure of pentane in this mixture at an overall pressure of 1.00 atm.(a) Given that air is 21% oxygen, find the minimum atmospheric pressure that gives a relatively safe partial pressure of oxygen of 0.16 atm. (b) What is the minimum pressure that gives a partial pressure of oxygen above the quickly fatal level of 0.06 atm? (c) The air pressure at the summit of Mount Everest (8848 m) is 0.334 atm. Why have a few people climbed it without oxygen, while some who have tried, even though they had trained at high elevation, had to tum back?(a) If the partial pressure of water vapor is 8.05 torr, what is the dew point? (760 torr = I atm 101, 325 Pa) (b) On a warn day when the air temperature is 35 and the dew point is 25 , what are the partial of the water in the air and the relative humidity?To give a helium atom nonzero angular momentum requires about 21.2 eV of energy (that is, 21.2 eV is the difference between the energies of the lowest-energy or ground state and the lowest-energy state with angular momentum). The electron-volt or eV is defined as 1.601019 J. Find the temperature T where this amount of energy equals kTB/2 . Does this explain why we can ignore the rotational energy of helium for most purposes? (The results for other monatomic gases, and for diatomic gases rotating around the axis connecting the two atoms, have comparable orders of magnitude.)(a) How much heat must be added to raise the temperature of 1.5 mol of air 25.0 to 33.0 at constant volume? Assume air is completely diatomic. (b) Repeat the problem for the same number of moles of xenon, Xe.A sealed, rigid container of 0.560 mol of an unknown ideal gas at a temperature of 30.0 is cooled to 40.0 . In the process, 980 J of heat are removed from the gas. Is the gas monatomic, diatomic, or polyatomic?A sample of neon gas (Ne, molar mass M=20.2 g/mol) at a temperature of 13.0 is put into a steel container of mass 47.2 g that's at a temperature of 40.0 . The final temperature is 28.0 . (No heat is exchanged with the surroundings, and you can neglect any change in the volume of the container.) What is the mass of the sample of neon?A steel container of mass 135 g contains 24.0 g of ammonia, NH3, which has a molar mass of 17.0 g/mol. The container and gas are in equilibrium at 12.0 . How much heat has to be removed to reach a temperature of 20.0 ? Ignore the change in volume of the steel.A sealed room has a volume of 24 m3. It's filled with air, which may be assumed to be diatomic, at a temperature of 24 and a pressure of 9.83104 Pa. A 1.00-kg block of ice at its melting point is placed in the room. Assume the walls of the room transfer no heat. What is the equilibrium temperature?Heliox, a mixture of helium and oxygen, is sometimes given to hospital patients who have trouble breathing, because the low mass of helium makes it easier to breathe than air. Suppose helium at 25 is mixed with oxygen at 35 to make a mixture that is 70% helium by mole. What is the final temperature? Ignore any heat flow to or from the surroundings, and assume the final volume is the sum of the initial volumes.Professional divas sometimes use heliox, consisting of 79% helium and 21% oxygen by mole. Suppose a perfectly rigid scuba tank with a volume of 11 L contains heliox at an absolute pressure of 2.1107 Pa at a temperature of 31 . (a) How many moles of helium and how many moles of oxygen are in the tank? (b) The diver goes down to a point where the sea temperature is 27 while using a negligible amount of the mixture. As the gas in the tank reaches this new temperature, how much heat is removed from it?In car racing, one advantage of mixing liquid nitrous oxide (N2O) with air is that the boiling of the "nitrous" absorbs latent heat of vaporization and thus cools the air and ultimately the fuel-air mixture, allowing more fuel-air mixture to go into each cylinder. As a very rough look at this process, suppose 1.0 mol of nitrous oxide gas at its boiling point, 88 , is mixed with 4.0 mol of air (assumed diatomic) at 30 . What is the final temperature of the mixture? Use the measured heat capacity of N2O at 25 , which is 30.4J/mol . (The primary advantage of nitrous oxide is that it consists of 1/3 oxygen, which is more than air contains, so it supplies more oxygen >to bum the fuel. Another advantage is that its decomposition into nitrogen and oxygen releases energy in the cylinder.)In a sample of hydrogen sulfide ( M=34.1 g/mol) at a temperature of 3.00102 K, estimate the ratio of the number of molecules that have speeds very close to vrms to the number that have speeds very close to 2vrms .Using the approximation v1v1+v f(v)dvf(v1)v for small v , estimate the fraction of nitrogen molecules at a temperature of 3.00102 K that have speeds between 290 m/s and 291 m/s.Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between 3.451021 J and 3.501021 J. `By counting squares in the following figure, estimate the fraction of argon atoms at T = 300 K that have speeds between 600 m/s and 800 m/s. The curve is correctly normalized. The value of a square is its length as measured on the x-axis times its height as measured on the y-axis, with the units given on those axes. `Using a numerical integration method such as Simpson's rule, find the fraction of molecules in a sample of oxygen gas at a temperature of 250 K that have speeds between 100 m/s and 150 m/s. The molar mass of oxygen (O2) is 32.0 g/mol. A precision to two significant digits is enough.Find (a) the most probable speed, (b) the average speed, and (c) the rms speed for nitrogen molecules at 295 K.Repeat the preceding problem for nitrogen molecules at 295 K.At what temperature is the average speed of carbon dioxide molecules ( M=44.0 g/mol) 510 m/s?The most probable speed for molecules of a gas at 296 K is 263 m/s. What is the molar mass of the gas? (You might like to figure out what the gas is likely to be.)a) At what temperature do oxygen molecules have the same average speed as helium atoms ( M=4.00 g/mol) have at 300 K? b) What is answer to the same question about most probable speeds? c) What is the answer to the same question about rms speeds?In the deep space between galaxies, the density of molecules (which are mostly single atoms) can be as low as 106 atoms/m3, and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in m3) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?(a) Find the density in SI units of air at a pressure of 1.00 atm and a temperature of 20 , assuming that air is 78% N2, 21% O2, and 1% Ar, (b) Find the density of the atmosphere on Venus, assuming that it's 96% CO2 and 4% N2, with a temperature of 737 K and a pressure of 92.0 atm.The air inside a hot-air balloon has a temperature of 370 K and a pressure of 101.3 kPa, the same as that of the air outside. Using the composition of air as 78% N2, 21% O2, and 1% Ar, find the density of the air inside the balloon.When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 80%. If the temperatures at the bottom and the top of the lake are 4.0 and 10 , respectively, how deep is the lake? `(a) Use the ideal gas equation to estimate the temperature at which 1.00 kg of steam (molar mass M=18.0 g/mol) at a pressure of 1.50106 Pa occupies a volume of 0.220 m3. (b) The van der Waals constants for water are a=0.5537 Pa m6/mol2 and b=3.049105 m3/mol. Use the Van der Waals equation of state to estimate the temperature under the same conditions. (c) The actual temperature is 779 K. Which estimate is better? `One process for decaffeinating coffee uses carbon dioxide ( M=44.0 g/mol) at a molar density of about 14,0 mol/m3 and a temperature of about 60 . (a) Is CO2 a solid, liquid, gas, or supercritical fluid under those conditions? (b) The van der Waals constants for carbon dioxide are a=0.3658 Pa m6/mol2 and b=4.286105 m3/mol. Using the van der Waals equation, estimate pressure of CO2 at that temperature and density. `On a winter day when the air temperature is 0 , the relative humidity is 50%. Outside air comes inside and is heated to a room temperature of 20 . What is the relative humidity of the air inside the room. (Does this problem show why inside air is so dry in winter?)On a warm day when the air temperature is 30 , a metal can is slowly cooled by adding bits of ice to liquid water in it. Condensation first appears when the can reaches 15 . What is the relative humidity of the air? `(a) People often think of humid air as "heavy." Compare the densities of air with 0% relative humidity and 100% relative humidity when both are at 1 atm and 30 . Assume that the dry air is an ideal gas composed of molecules with a molar mass of 29.0 g/mol and the moist air is the same gas mixed with water vapor. (b) As discussed in the chapter on the applications of Newton's laws, the air resistance felt by projectiles such as baseballs and golf balls is approximately FD=CpAv2/2 , where p is the mass density of the air, A is the cross-sectional area of the projectile, and C is the projectile's drag coefficient. For a fixed air pressure, describe qualitatively how the range of a projectile changes with the relative humidity. (c) When a thunderstorm is coming, usually the humidity is high and the air pressure is low. Do those conditions give an advantage or disadvantage to home-run hitters?The mean flee path for helium at a certain temperature and pressure is 2.10107 m. The radius of a helium atom can be taken as 1.101011 m. What is the measure of the density of helium under those conditions (a) in molecules per cubic meter and (b) in moles per cubic meter? `The mean free path for methane at a temperature of 269 K and a pressure of 1.11105 Pa is 4.81108 m. Find the effective radius r of the methane molecule.In the chapter on fluid mechanics, Bernoulli's equation for the flow of incompressible fluids was explained in terms of changes affecting a small volume dV of fluid. Such volumes are a fundamental idea in the study of the flow of compressible fluids such as gases as well. For the equations of hydrodynamics to apply, the mean free path must be much less than the linear size of such a volume, adV1/3 . For air in the stratosphere at a temperature of 220 K and a pressure of 5.8 kPa, how big should a be for it to be 100 times the mean free path? Take the effective radius of air molecules to be 1.881011 m, which is roughly correct for N2.Find the total number of collisions between molecules in 1.00 s in 1.00 L of nitrogen gas at standard temperature and (0 , 1.00 atm). Use 1.881010 m as the effective radius of a nitrogen molecule. (The number of collisions per second is the reciprocal of the collision time.) Keep in mind that each collision involves two molecules, so if one molecule collides once in a certain period of time, the collision of the molecule it hit cannot be counted. `(a) Estimate the specific heat capacity of sodium from the Law of Dulong and Petit. The molar mass of sodium is 23.0 g/mol. (b) What is the percent error of your estimate from the known value, 1230 J/kg ? `A sealed, perfectly insulated container contains 0.630 mol of air at 20.0 and an iron stirring bar of mass 40.0 g. ne stilling bar is magnetically driven to a kinetic energy of 50.0 J and allowed to slow down by air resistance. What is the equilibrium temperature? `Find the ratio f(vp)/f(vrms) for hydrogen gas ( M=2.02 g/mol) at a temperature of 77.0 K.Unreasonable results. (a) Find the temperature of 0.360 kg of water, modeled as an ideal gas, at a pressure of 1.01105 Pa if it has a volume of 0.615 m3. (b) What is unreasonable about this answer? How could you get a better answer?Unreasonable results. (a) Find the sped of hydrogen sulfide, H2S, molecules at a temperature of 250 K. Its molar mass is 31.4 g/mol (b) The result isn't very unreasonable, but why is it less reliable than those for, say, neon or nitrogen?An airtight dispenser for drinking water is 25 cm × 10 cm in horizontal dimensions and 20 cm tall. It has a tap of negligible volume that opens at the level of the bottom of the dispenser. Initially, it contains Water to a level 3.0 cm from the top and air at the ambient pressure, 1.00 atm, from there to the top. When the tap is opened, water will flow out until the gauge pressure at the bottom of dispenser, and thus at the opening of the tap, is 0. What volume of water flows out? Assume the temperature is constant, the dispenser is perfectly rigid, and the water has a constant density of 1000 kg/m3.Eight bumper cars, each with a mass of 322 kg. are running in a room 21.0 m long and 130 m wide. They have no driver, so they just bounce around on their own. The rms speed of the cars is 2.50 m/s. Repeating the arguments of Pressure, Temperature, and RMS Speed, find the average force per unit length (analogous to Pressure) that the cars exert the walls.Verify that vp=2kBTm.`Verify the normalization equation 0f(v)dv=1 In doing the integral, first make substitution u=m2kBTv=vvp. This "scaling" transformation gives you all features of the answer except for the integral, which is a dimensionless numerical factor. You'll need the formula 0x2e x 2dx= 4 to find the numerical factor and verify the normalization.`Verify that v=8kBTm. Make the same scaling transformation as in the preceding problem.`Verify that vrms=v2=3kBTm.`The paths ABC, AC, and ADC represent three different quasi-static transitions between the equilibrium states A and C . Check Your Understanding How much work is done by gas, as given in Figure 3.6, when it expands quasi-statically along the path ADC?Check Your Understanding The quantities below represent four different transitions between same initial and final state. Fill in the blanks.Check Your Understanding Why was it necessary to state that the process of Example 3.5 is quasi-static.Check Your Understanding When 1.00 g of ammonia boils at atmospheric pressure and 330 . its volume changes from 1.47 to 1130 cm3. Its heat of vaporization at this pressure is 1.37106 J/kg. What is the change in the internal energy of the ammonia when it vaporizes?Consider these scenarios and state whether work is done by the system on the environment (SE) or by the environment on the system (ES): (a) opening a carbonated beverage; (b) filling a flat tire; (c) a sealed empty gas can expands on a hot day, bowing out the walls.Is it possible to determine whether a change in internal energy is caused by heat transferred, by work performed, or by a combination of the two? `When a liquid is vaporized, its change in internal energy is not equal to the heat added. Why?Why does a bicycle pump feel warm as you inflate your tire?Is it possible for the temperature of a system to remain constant when heat flows into or out of it? If so, give examples.What does the first law of thermodynamics tell us about the energy of the universe? `Does adding heat to a system always increase its internal energy?A great deal of effort, time, and money has been spent in the quest for a so-called perpetual-motion machine, which is defined as a hypothetical machine that operates or produces useful work indefinitely and/or a hypothetical machine that produces mole work or energy than it consumes. Explain, in terms of the first law of thermodynamics, why or why not such a machine is likely to be constructed.When a gas expands isothermally, it does work. What is the source of energy needed to do this work?If the pressure and volume of a system are given, is the temperature always uniquely determined?It is unlikely that a process can be isothermal unless it is a very slow process. Explain why. Is the same true for isobaric and isochoric processes? Explain your answer.How can an object transfer heat if the object does not possess a discrete quantity of heat?Most materials expand when heated. One notable exception is water between 0 and 4 , which actually decreases in volume with the increase in temperature. Which is greater for water in this temperature region, Cp or Cv ?Why are there two specific heats for gases Cp and Cv , yet only one given for solid?Is it possible for to be smaller than unity? `Would you expect to be larger for a gas or a solid?There is no change in the internal of an ideal gas undergoing an isothermal process since the internal energy depends only on the temperature. Is it therefore correct to say that an isothermal process is the same as an adiabatic process for an ideal gas? Explain your answer. `Does a gas do any work when it expands adiabatically? If so, what is the source of the energy needed to do this work?A gas follows on an isothermal curve, where p is the pressure, V is the volume, b is a constant, and c is a function of temperature. Show that a temperature scale under an isochoric process can be established with this gas and is identical to that of an ideal gas.A mole of gas has isobaric expansion coefficient dV/dT=R/p and isochoric pressure-temperature coefficient dp/dT=p/T . Find the equation of state of the gas.Find the equation of state of a solid that has an isobaric expansion coefficient dv/dT=2cTbp and an isothermal pressure-volume coefficient dv/dp=bT.A gas at a pressure of 2.00 atm undergoes a quasi-static isobaric expansion from 3.00 to 5.00 L. How much work is done by the gas? `It takes 500 J of work to compress quasi-statically 0.50 mol of an ideal gas to one-fifth its original volume. Calculate the temperature of the gas, assuming it remains constant during the compression.It is found that, when a dilute gas expands quasi-statically from 0.50 to 4.0 L, it does 250 J of work. Assuming that the gas temperature remains constant at 300 K, how many moles of gas are present?In a quasi-static isobaric expansion. 500 J of work are done by the gas. If the gas pressure is 0.80 atm, what is the fractional increase in the volume of the gas, assuming it was originally at 20.0 L?When a gas undergoes a quasi-static isobaric change in volume from 10.0 to 2.0 L, 15 J of work from an external source are required. What is the pressure of the gas? `An ideal gas expands quasi-statically and isothermally from a state with pressure p and volume V to a state with volume 4V. Show that the work done by the gas in the expansion is pV(ln 4).As shown below, calculate the work done by the gas in the quasi-static processes represented by the paths (a) AB; (b) ADB; (c) ACB; and (d) ADCB. `(a) Calculate the work done by the gas along the closed path shown below. The curved section between R and S is semicircular. (b) If the process is carried out in the opposite direction, what is the work done by the gas?An ideal gas expands quasi-statically to three times its original volume. Which process requires more work from the gas, an isothermal process or an isobaric one? Determine the ratio of the work done in processes.A dilute gas at a pressure of 2.0 atm and a volume of 4.0 L is taken through the following quasi-static steps: (a) an isobaric expansion to a volume of 10.0 L, (b) an isochoric change to a pressure of 0.50 atm, (c) an isobaric compression to a volume of 4.0 L, and (d) an isochoric change to a pressure of 2.0 atm. Show these steps on a PV diagram and determine from your graph the net work done by the gas.What is the average mechanical energy of the atoms of an ideal monatomic gas at 300 K?What is the internal energy of 6.00 mol of an ideal monatomic gas at 200 ?Calculate the internal energy of 15 mg of helium at a temperature of 0 . `Two monatomic ideal gases A and B are at the same temperature. If 1.0 g of gas A has the same internal energy as 0.10 g of gas B, what are (a) the ratio of the number of moles of each gas and (b) the ration of the atomic masses of the two gases?The van der Waals coefficients for oxygen are a=0.138Jm3/mol2 and b=3.18105m3/mol. Use these values to draw a van der Waals isotherm of oxygen at 100 K. On the same graph, draw isotherms of one mole of an ideal gas.Find the work done in the quasi-static processes shown below. The states are given as (p, V) values for the points in the PV plane: 1 (3 atm, 4 L), 2 (3 atm, 6 L), 3 (5 atm, 4 L), 4 (2 atm, 6 L), 5 (4 atm, 2 L), 6 (5 atm, 5 L) and 7 (2 atm, 5 L).When a dilute gas expands quasi-statically from 0.50 to 4.0 L, it does 250 J of work. Assuming that the gas temperature remains constant at 300 K, (a) what is the change in the internal energy of the gas? (b) How much heat is absorbed by the gas in this process?In a quasi-static isobaric expansion, 500 J of work are done by the gas. The gas pressure is 0.80 atm and it was originally at 20.0 L. If the internal energy of the gas increased by 80 J in the expansion, how much heat does the gas absorb?An ideal gas quasi-statically and isothermally from a state with pressure p and volume V to a state with volume 4V. How much heat is added to the expanding gas?As shown below, if the heat absorbed by the gas along AB is 400 J, determine the quantities of heat absorbed along (a) ADB; (b) ACB; and (c) ADCB.During the isobaric expansion from A to B represented below, 130 J of heat are removed from the gas. What is the change in its internal energy?(a) What is the change in internal energy for the process represented by the closed path shown below? (b) How much heat is exchanged? (c) If the path is traversed in the opposite direction, how much heat is exchanged?When a gas expands along path AC shown below, it does 400 J of work and absorbs either 200 or 400 J of heat. (a) Suppose you are told that along path ABC, the gas absorbs either 200 or 400 J of heat. Which of these values is correct? (b) Give the correct answer from part (a), how much work is done by the gas along ABC? (c) Along CD, the internal energy of the gas decreases by 50 J. How much heat is exchanged by the gas along this path?When a gas expands along AB (see below), it does 500 J of work and absorbs 250 J of heat. When the gas expands along AC, it does 700 J of work and absorbs 300 J of heat. (a) How much heat does the gas exchange along BC? (b) When the gas makes the transmission from C to A along CDA, 800 J of work are done on it from C to D. How much heat does it exchange along CDA?A dilute gas is stored in the left chamber of a container whose walls are perfectly insulating (see below), and the right chamber is evacuated. Wien the partition is removed, the gas expands and fills the entire container. Calculate the done by the gas. Does the internal energy of the gas change in this process?Ideal gases A and B are stored in the left and right chambers of an insulated container, as shown below. The partition is removed and the gases mix. Is any done in this process? If the temperatures of A and B are initially equal, what happens to their common temperature after they are mixed?An ideal monatomic gas at a pressure of 2.0105N/m2 and a temperature of 300 K undergoes a quasi-static isobaric expansion from 2.0103 to 4.0103 cm3. (a) What is the work done by the gas? (b) What is the temperature of the gas after the expansion? (c) How many moles of gas are there? (d) What is the change in internal energy of the gas? (e) How much heat is added to the gas?Consider the process for steam in a cylinder shown below. Suppose the change in the internal energy in this process is 30 kJ. Find the heat entering the system.The state of 30 moles of steam in a cylinder is changed in a cyclic manner from a-b-c-a, where the pressure and volume of the states are: a (30 atm, 20 L), b (50 atm, 20 L), and c (50 atm, 45 L). Assume each change takes place along the line connecting the initial and final states in the pV plane. (a) Display the cycle in the pV plane. (b) Find the net work done by the steam in one cycle. (c) Find the net amount of heat flow in the steam over the course of one cycle.A monatomic ideal gas undergoes a quasi-static process that is described by the function pV=p1+3(vv1) , where the stating state is (p1,v1) and the final state (p2,v2) . Assume the system consists of n moles of the gas in a container that can exchange heat with the environment and whose volume can change freely. (a) Evaluate the work done by the gas during the change in the state. (b) Find the change in internal energy of the gas. (c) Find the heat input to the gas during the change. (d) What ale initial and final temperatures?A metallic container of fixed volume of 2.5103 m3 immersed in a large tank of temperature 27 contains two compartments separated by a freely movable wall. Initially, the wall is kept in place by a stopper so that there are 0.02 mol of the nitrogen gas on one side and 0.03 mol of the oxygen gas on the other side, each occupying half the volume. When the stopper is removed, the wall moves and comes to a final position. The movement of the wall is controlled so that the wall moves in infinitesimal quasi-static steps. (a) Find the final volumes of the two sides assuming the ideal gas behavior for the two gases. (b) How much work does each gas do on the other? (c) What is the change in the internal energy of each gas? (d) Find the amount of heat that enters or leaves each gas.A gas in a cylindrical closed container is adiabatically and quasi-statically expanded from a state A (3 MPa, 2 L) to a state B with volume of 6 L along the path 1.8pV= constant. (a) Plot the path in the pV plane. (b) Find the amount of work done by the gas and the change in the internal energy of the gas during the process.