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A constant-volume gas thermometer is calibrated in dry ice (that is, carbon dioxide in the solid state, which has a temperature of - 80.0°C) and in boiling ethyl alcohol (78.0°C). The two pressures are 0.900 atm and 1.635 atm.
(a) What Celsius value of absolute zero does the calibration yield? What is the pressure at (b) The freezing point of water and
(c) The boiling point of water?
In a constant-volume gas thermometer, the pressure at 20.0°C is 0.980 atm.
(a) What is the pressure at 45.0°C?
(b) What is the temperature if the pressure is 0.500 atm?
Liquid nitrogen has a boiling point of - 195.81°C at atmospheric pressure. Express this temperature
(a) In degrees Fahrenheit and
(b) In kelvins.
Convert the following to equivalent temperatures on the Celsius and Kelvin scales:
(a) The normal human body temperature, 98.6°F;
(b) The air temperature on a cold day, -5.00°F.
The temperature difference between the inside and the outside of an automobile engine is 450°C. Express this temperature difference on
(a) The Fahrenheit scale and
(b) The Kelvin scale.
On a Strange temperature scale, the freezing point of water is -15.0°S and the boiling point is + 60.0°S. Develop a linear conversion equation between this temperature scale and the Celsius scale.
The melting point of gold is 1 064°C, and the boiling point is 2 660°C.
(a) Express these temperatures in kelvins.
(b) Compute the difference between these temperatures in Celsius degrees and kelvins.
The New River Gorge bridge in West Virginia is a steel arch bridge 518 m in length. How much does the total length of the roadway decking change between temperature extremes of - 20.0°C and 35.0°C? The result indicates the size of the expansion joints that must be built into the structure
A copper telephone wire has essentially no sag between poles 35.0 m apart on a winter day when the temperature is - 20.0°C. How much longer is the wire on a summer day when TC # 35.0°C?
The concrete sections of a certain superhighway are designed to have a length of 25.0 m. The sections are poured and cured at 10.0°C. What minimum spacing should the engineer leave between the sections to eliminate buckling if the concrete is to reach a temperature of 50.0°C?
A pair of eyeglass frames is made of epoxy plastic. At room temperature (20.0°C), the frames have circular lens holes 2.20 cm in radius. To what temperature must the frames be heated if lenses 2.21 cm in radius are to be inserted in them? The average coefficient of linear expansion for epoxy is 1.30 X 10-4 (°C)-1.
Each year thousands of children are badly burned by hot tap water. Figure P19.12 shows a cross-sectional view of an anti scalding faucet attachment designed to prevent such accidents. Within the device, a spring made of material with a high coefficient of thermal expansion controls a movable plunger. When the water temperature rises above a preset safe value, the expansion of the spring causes the plunger to shut off the water flow. If the initial length L of the unstressed spring is 2.40 cm and its coefficient of linear expansion is 22.0 X 10-6 (°C)-1, determine the increase in length of the spring when the water temperature rises by 30.0°C. (You will find the increase in length to be small. For this reason actual devices have a more complicated mechanical design, to provide a greater variation in valve opening for the temperature change anticipated.)
The active element of a certain laser is made of a glass rod 30.0 cm long by 1.50 cm in diameter. If the temperature of the rod increases by 65.0°C, what is the increase in
(a) Its length,
(b) Its diameter, and
(c) Its volume? Assume that the average coefficient of linear expansion of the glass is 9.00 X 10-6 (°C)-1.
Inside the wall of a house, an L-shaped section of hot-water pipe consists of a straight horizontal piece 28.0 cm long, an elbow, and a straight vertical piece 134 cm long (Figure P19.14). A stud and a second-story floorboard hold stationary the ends of this section of copper pipe. Find the magnitude and direction of the displacement of the pipe elbow when the water flow is turned on, raising the temperature of the pipe from 18.0°C to 46.5°C.
A brass ring of diameter 10.00 cm at 20.0°C is heated and slipped over an aluminum rod of diameter 10.01 cm at 20.0°C. Assuming the average coefficients of linear expansion are constant,
(a) To what temperature must this combination be cooled to separate them? Is this attainable?
(b) What If? What if the aluminum rod were 10.02 cm in diameter?
A square hole 8.00 cm along each side is cut in a sheet of copper.
(a) Calculate the change in the area of this hole if the temperature of the sheet is increased by 50.0 K.
(b) Does this change represent an increase or a decrease in the area enclosed by the hole?
The average coefficient of volume expansion for carbon tetrachloride is 5.81 X 10-4 (°C)-1. If a 50.0-gal steel container is filled completely with carbon tetrachloride when the temperature is 10.0°C, how much will spill over when the temperature rises to 30.0°C?
At 20.0°C, an aluminum ring has an inner diameter of 5.000 0 cm and a brass rod has a diameter of 5.050 0 cm.
(a) If only the ring is heated, what temperature must it reach so that it will just slip over the rod?
(b) What If? If both are heated together, what temperature must they both reach so that the ring just slips over the rod? Would this latter process work?
A volumetric flask made of Pyrex is calibrated at 20.0°C. It is filled to the 100-mL mark with 35.0°C acetone.
(a) What is the volume of the acetone when it cools to 20.0°C?
(b) How significant is the change in volume of the flask?
A concrete walk is poured on a day when the temperature is 20.0°C in such a way that the ends are unable to move.
(a) What is the stress in the cement on a hot day of 50.0°C?
(b) Does the concrete fracture? Take Young’s modulus for concrete to be 7.00 X 109 N/m2 and the compressive strength to be 2.00 X 109 N/m2.
A hollow aluminum cylinder 20.0 cm deep has an internal capacity of 2.000 L at 20.0°C. It is completely filled with turpentine and then slowly warmed to 80.0°C. (a) How much turpentine overflows? (b) If the cylinder is then cooled back to 20.0°C, how far below the cylinder’s rim does the turpentine’s surface recede?
A beaker made of ordinary glass contains a lead sphere of diameter 4.00 cm firmly attached to its bottom. At a uniform temperature of - 10.0°C, the beaker is filled to the brim with 118 cm3 of mercury, which completely covers the sphere. How much mercury overflows from the beaker if the temperature is raised to 30.0°C?
A steel rod undergoes a stretching force of 500 N. Its cross-sectional area is 2.00 cm2. Find the change in temperature that would elongate the rod by the same amount as the 500-N force does. Tables 12.1 and 19.1 are available to you.
The Golden Gate Bridge in San Francisco has a main span of length 1.28 km—one of the longest in the world. Imagine that a taut steel wire with this length and a cross-sectional area of 4.00 X 10-6 m2 is laid on the bridge deck with its ends attached to the towers of the bridge, on a summer day when the temperature of the wire is 35.0°C.
(a) When winter arrives, the towers stay the same distance apart and the bridge deck keeps the same shape as its expansion joints open. When the temperature drops to -10.0°C, what is the tension in the wire? Take Young’s modulus for steel to be 20.0 x 1010 N/m2.
(b) Permanent deformation occurs if the stress in the steel exceeds its elastic limit of 3.00 X 108 N/m2. At what temperature would this happen?
(c) What If? How would your answers to (a) and (b) change if the Golden Gate Bridge were twice as long?
Gas is contained in an 8.00-L vessel at a temperature of 20.0°C and a pressure of 9.00 atm.
(a) Determine the number of moles of gas in the vessel.
(b) How many molecules are there in the vessel?
A certain telescope forms an image of part of a cluster of stars on a square silicon charge-coupled detector (CCD) chip 2.00 cm on each side. A star field is focused on the CCD chip when it is first turned on and its temperature is 20.0°C. The star field contains 5 342 stars scattered uniformly. To make the detector more sensitive, it is cooled to -100°C. How many star images then fit onto the chip? The average coefficient of linear expansion of silicon is 4.68 X 10-6 (°C)-1.
An automobile tire is inflated with air originally at 10.0°C and normal atmospheric pressure. During the process, the air is compressed to 28.0% of its original volume and the temperature is increased to 40.0°C.
(a) What is the tire pressure?
(b) After the car is driven at high speed, the tire air temperature rises to 85.0°C and the interior volume of the tire increases by 2.00%. What is the new tire pressure (absolute) in pascals?
A tank having a volume of 0.100 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 1.20 atm?
An auditorium has dimensions 10.0 m X 20.0 m X 30.0 m. How many molecules of air fill the auditorium at 20.0°C and a pressure of 101 kPa?
Imagine a baby alien playing with a spherical balloon the size of the Earth in the outer solar system. Helium gas inside the balloon has a uniform temperature of 50.0 K due to radiation from the Sun. The uniform pressure of the helium is equal to normal atmospheric pressure on Earth.
(a) Find the mass of the gas in the balloon.
(b) The baby blows an additional mass of 8.00 X 1020 kg of helium into the balloon. At the same time, she wanders closer to the Sun and the pressure in the balloon doubles. Find the new temperature inside the balloon, whose volume remains constant.
Just 9.00 g of water is placed in a 2.00-L pressure cooker and heated to 500°C. What is the pressure inside the container?
One mole of oxygen gas is at a pressure of 6.00 atm and a temperature of 27.0°C.
(a) If the gas is heated at constant volume until the pressure triples, what is the final temperature?
(b) If the gas is heated until both the pressure and volume are doubled, what is the final temperature?
The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kg. The air outside is at
10.0°C and 101 kPa. The volume of the balloon is 400 m3. To what temperature must the air in the balloon be heated before the balloon will lift off? (Air density at 10.0°C is 1.25 kg/m3.)
Your father and your little brother are confronted with the same puzzle. Your father’s garden sprayer and your brother’s water cannon both have tanks with a capacity of 5.00 L (Figure P19.34). Your father inserts a negligible amount of concentrated insecticide into his tank. They both pour in 4.00 L of water and seal up their tanks, so that they also contain air at atmospheric pressure. Next, each uses a hand-operated piston pump to inject more air, until the absolute pressure in the tank reaches 2.40 atm and it becomes too difficult to move the pump handle. Now each uses his device to spray out water—not air— until the stream becomes feeble, as it does when the pressure in the tank reaches 1.20 atm. Then he must pump it up again, spray again, and so on. In order to spray out all the water, each finds that he must pump up the tank three times. This is the puzzle: most of the water sprays out as a result of the second pumping. The first and the third pumping-up processes seem just as difficult, but result in a disappointingly small amount of water coming out. Account for this phenomenon.
(a) Find the number of moles in one cubic meter of an ideal gas at 20.0°C and atmospheric pressure.
(b) For air, Avogadro’s number of molecules has mass 28.9 g. Calculate the mass of one cubic meter of air. Compare the result with the tabulated density of air
The void fraction of a porous medium is the ratio of the void volume to the total volume of the material. The void is the hollow space within the material; it may be filled with a fluid. A cylindrical canister of diameter 2.54 cm and height 20.0 cm is filled with activated carbon having a void fraction of 0.765. Then it is flushed with an ideal gas at 25.0°C and pressure 12.5 atm. How many moles of gas are contained in the cylinder at the end of this process?
A cube 10.0 cm on each edge contains air (with equivalent molar mass 28.9 g/mol) at atmospheric pressure and temperature 300 K. Find
(a) The mass of the gas, (b) its weight, and
(c) The force it exerts on each face of the cube.
(d) Comment on the physical reason why such a small sample can exert such a great force.
At 25.0 m below the surface of the sea (p =1 025 kg/m3), where the temperature is 5.00°C, a diver exhales an air bubble having a volume of 1.00 cm3. If the surface temperature of the sea is 20.0°C, what is the volume of the bubble just before it breaks the surface?
The pressure gauge on a tank registers the gauge pressure, which is the difference between the interior and exterior pressure. When the tank is full of oxygen (O2), it contains 12.0 kg of the gas at a gauge pressure of 40.0 atm. Determine the mass of oxygen that has been withdrawn from the tank when the pressure reading is 25.0 atm. Assume that the temperature of the tank remains constant.
Estimate the mass of the air in your bedroom. State the quantities you take as data and the value you measure or estimate for each.
On his honeymoon James Joule traveled from England to Switzerland. He attempted to verify his idea of the interconvertibility of mechanical energy and internal energy by measuring the increase in temperature of water that fell in a waterfall. If water at the top of an alpine waterfall has a temperature of 10.0°C and then falls 50.0 m (as at Niagara Falls), what maximum temperature at the bottom of the falls could Joule expect? He did not succeed in measuring the temperature change, partly because evaporation cooled the falling water, and also because his thermometer was not sufficiently sensitive.
A popular brand of cola contains 6.50 g of carbon dioxide dissolved in 1.00 L of soft drink. If the evaporating carbon dioxide is trapped in a cylinder at 1.00 atm and 20.0°C, what volume does the gas occupy?
In state-of-the-art vacuum systems, pressures as low as 10-9 Pa are being attained. Calculate the number of molecules in a 1.00-m3 vessel at this pressure if the temperature is 27.0°C.
A room of volume V contains air having equivalent molar mass M (in g/mol). If the temperature of the room is raised from T1 to T2, what mass of air will leave the room? Assume that the air pressure in the room is maintained at P0.
A diving bell in the shape of a cylinder with a height of 2.50 m is closed at the upper end and opens at the lower end. The bell is lowered from air into sea water (p =| 1.025 g/cm3). The air in the bell is initially at 20.0°C. The bell is lowered to a depth (measured to the bottom of the bell) of 45.0 fathoms or 82.3 m. At this depth the water temperature is 4.0°C, and the bell is in thermal equilibrium with the water.
(a) How high does sea water rise in the bell?
(b) To what minimum pressure must the air in the bell be raised to expel the water that entered?
A student measures the length of a brass rod with a steel tape at 20.0°C. The reading is 95.00 cm. What will the tape indicate for the length of the rod when the rod and the tape are at (a) - 15.0°C and (b) 55.0°C?
Consider Joule’s apparatus described in Figure 20.1. The mass of each of the two blocks is 1.50 kg, and the insulated tank is filled with 200 g of water. What is the increase in the temperature of the water after the blocks fall through a distance of 3.00 m?
The temperature of a silver bar rises by 10.0°C when it absorbs 1.23 kJ of energy by heat. The mass of the bar is 525 g. Determine the specific heat of silver.
The density of gasoline is 730 kg/m3 at 0°C. Its average coefficient of volume expansion is 9.60 X 10-4 /°C. If 1.00 gal of gasoline occupies 0.003 80 m3, how many extra kilograms of gasoline would you get if you bought 10.0 gal of gasoline at 0°C rather than at 20.0°C from a pump that is not temperature compensated?
A 50.0-g sample of copper is at 25.0°C. If 1 200 J of energy is added to it by heat, what is the final temperature of the copper?
Systematic use of solar energy can yield a large saving in the cost of winter space heating for a typical house in the north central United States. If the house has good insulation, you may model it as losing energy by heat steadily at the rate 6 000 W on a day in April when the average exterior temperature is 4°C, and when the conventional heating system is not used at all. The passive solar energy collector can consist simply of very large windows in a room facing south. Sunlight shining in during the daytime is absorbed by the floor, interior walls, and objects in the room, raising their temperature to 38°C. As the sun goes down, insulating draperies or shutters are closed over the windows. During the period between 5:00 P.M. and 7:00 A.M. the temperature of the house will drop, and a sufficiently large “thermal mass” is required to keep it from dropping too far. The thermal mass can be a large quantity of stone (with specific heat 850 J/kg . °C) in the floor and the interior walls exposed to sunlight. What mass of stone is required if the temperature is not to drop below 18°C overnight?
A mercury thermometer is constructed as shown in Figure P19.47. The capillary tube has a diameter of 0.004 00 cm, and the bulb has a diameter of 0.250 cm.
A liquid with a coefficient of volume expansion ' just fills a spherical shell of volume Vi at a temperature of Ti (see Fig. P19.47). The shell is made of a material that has an average coefficient of linear expansion &. The liquid is free to expand into an open capillary of area A projecting from the top of the sphere.
(a) If the temperature increases by ΔT, show that the liquid rises in the capillary by the amount Δh given by Δh = (Vi/A)(B – 3a) ΔT. (b) For a typical system, such as a mercury thermometer, why is it a good approximation to neglect the expansion of the shell?
A cylinder is closed by a piston connected to a spring of constant 2.00 X 103 N/m (see Fig. P19.50). with the spring relaxed, the cylinder is filled with 5.00 L of gas at a pressure of 1.00 atm and a temperature of 20.0°C. (a) If the piston has a cross-sectional area of 0.010 0 m2 and negligible mass, how high will it rise when the temperature is raised to 250°C? (b) What is the pressure of the gas at 250°C?
An aluminum pipe, 0.655 m long at 20.0°C and open at both ends, is used as a flute. The
pipe is cooled to a low temperature but then is filled with air at 20.0°C as soon as you start to play it. After that, by how much does its fundamental frequency change as the metal rises in temperature from 5.00°C to 20.0°C?
A liquid has a density p.
(a) Show that the fractional change in density for a change in temperature
ΔT is Δp/ = -BΔT. What does the negative sign signify?
(b) Fresh water has a maximum density of 1.000 0 g/cm3 at 4.0°C. At 10.0°C, its density is 0.999 7 g/cm3. What is ' for water over this temperature interval?
Long-term space missions require reclamation of the oxygen in the carbon dioxide exhaled by the crew. In one method of reclamation, 1.00 mol of carbon dioxide produces
1.00 mol of oxygen and 1.00 mol of methane as a byproduct. The methane is stored in a tank under pressure and is available to control the attitude of the spacecraft by controlled venting. A single astronaut exhales 1.09 kg of carbon dioxide each day. If the methane generated in the respiration recycling of three astronauts during one week of flight is stored in an originally empty 150-L tank at -45.0°C, what is the final pressure in the tank?
A vertical cylinder of cross-sectional area A is fitted with a tight-fitting, frictionless piston of mass m (Fig. P19.53)
(a) If n moles of an ideal gas are in the cylinder at a temperature of T, what is the height h at which the piston is in equilibrium under its own weight?
(b) What is the value for h if n = 0.200 mol, T = 400 K, A = # 0.008 00 m2, and m # 20.0 kg?
A bimetallic strip is made of two ribbons of dissimilar metals bonded together.
(a) First assume the strip is originally straight. As they are heated, the metal with the greater average coefficient of expansion expands more than the other, forcing the strip into an arc, with the outer radius having a greater circumference (Fig. P19.54a). Derive an expression for the angle of bending - as a function of the initial length of the strips, their average coefficients of linear expansion, the change in temperature, and the separation of the centers of the strips (Δr = r2 - r1).
(b) Show that the angle of bending decreases to zero when ΔT decreases to zero and also when the two average coefficients of expansion become equal.
(c) What If? What happens if the strip is cooled?
(d) Figure P19.54b shows a compact spiral bimetallic strip in a home thermostat. The equation from part
(a) Applies to it as well, if Ө is interpreted as the angle of additional bending caused by a change in temperature. The inner end of the spiral strip is fixed, and the outer end is free to move. Assume the metals are bronze and Invar, the thickness of the strip is 2Δr = 0.500 mm, and the overall length of the spiral strip is 20.0 cm. Find the angle through which the free end of the strip turns when the temperature changes by one Celsius degree. The free end of the strip supports a capsule partly filled with mercury, visible above the strip in Figure P19.54b. When the capsule tilts, the mercury shifts from one end to the other, to make or break an electrical contact switching the furnace on or off.
The Nova laser at Lawrence Livermore National Laboratory in California is used in studies of initiating controlled nuclear fusion (Section 45.4). It can deliver a power of 1.60 X 1013 W over a time interval of 2.50 ns. Compare its energy output in one such time interval to the energy required to make a pot of tea by warming 0.800 kg of water from 20.0°C to 100°C.
The rectangular plate shown in Figure P19.55 has an area Ai equal to If the temperature increases by ΔT, each dimension increases according to the equation ΔL = aLi ΔT, where & is the average coefficient of linear expansion. Show that the increase in area is ΔA = 2aAi ΔT. What approximation does this expression assume?
A clock with a brass pendulum has a period of 1.000 s at 20.0°C. If the temperature increases to 30.0°C,
(a) By how much does the period change, and
(b) How much time does the clock gain or lose in one week?
(a) Derive an expression for the buoyant force on a spherical balloon, submerged in water, as a function of the depth below the surface, the volume of the balloon at the surface, the pressure at the surface, and the density of the water. (Assume water temperature does not change with depth)
(b) Does the buoyant force increase or decrease as the balloon is submerged?
(c) At what depth is the buoyant force half the surface value?
Consider an object with any one of the shapes displayed in Table 10.2. What is the percentage increase in the moment of inertia of the object when it is heated from 0°C to 100°C if it is composed of (a) copper or (b) aluminum? Assume that the average linear expansion coefficients shown in Table 19.1 do not vary between 0°C and 100°C.
A copper wire and a lead wire are joined together, end to end. The compound wire has an effective coefficient of linear expansion of 20.0 X 10-6 (°C)-1. What fraction of the length of the compound wire is copper?
Following a collision in outer space, a copper disk at 850°C is rotating about its axis with an angular speed of 25.0 rad/s. As the disk radiates infrared light, its temperature falls to 20.0°C. No external torque acts on the disk.
(a) Does the angular speed change as the disk cools off? Explain why.
(b) What is its angular speed at the lower temperature?
Two concrete spans of a 250-m-long bridge are placed end to end so that no room is allowed for expansion (Fig. P19.61a). If a temperature increase of 20.0°C occurs, what is the height y to which the spans rise when they buckle (Fig P19.61b)?
Two concrete spans of a bridge of length L are placed end to end so that no room is allowed for expansion (Fig. P19.61a). If a temperature increase of ΔT occurs, what is the height y to which the spans rise when they buckle (Fig P19.61b)
A 1.50-kg iron horseshoe initially at 600°C is dropped into a bucket containing 20.0 kg of water at 25.0°C. What is the final temperature? (Ignore the heat capacity of the container, and assume that a negligible amount of water boils away.)
(a) Show that the density of an ideal gas occupying a volume V is given by p = PM/RT, where M is the molar mass.
(b) Determine the density of oxygen gas at atmospheric pressure and 20.0°C
(a) Use the equation of state for an ideal gas and the definition of the coefficient of volume expansion, in the form β = (1/V) dV/dT, to show that the coefficient of volume expansion for an ideal gas at constant pressure is given by β = 1/T, where T is the absolute temperature.
(b) What value does this expression predict for β at 0°C? Compare this result with the experimental values for helium and air in Table 19.1. Note that these are much larger than the coefficients of volume expansion for most liquids and solids
Starting with Equation 19.10, show that the total pressure P in a container filled with a mixture of several ideal gases is P = P1 + P2 + P3 + ., where P1, P2, . . . , are the pressures that each gas would exert if it alone filled the container (these individual pressures are called the partial pressures of the respective gases). This result is known as Dalton’s law of partial pressures
A sample of dry air that has a mass of 100.00 g, collected at sea level, is analyzed and found to consist of the following gases:
Nitrogen (N2) = 75.52 g
Oxygen (O2) = 23.15 g
Argon (Ar) = 1.28 g
Carbon dioxide (CO2) # 0.05 g
Plus trace amounts of neon, helium, methane, and other gases.
(a) Calculate the partial pressure of each gas when the pressure is 1.013 X 105 Pa.
(b) Determine the volume occupied by the 100-g sample at a temperature of 15.00°C and a pressure of 1.00 atm. What is the density of the air for these conditions?
(c) What is the effective molar mass of the air sample?
A 1.50-kg iron horseshoe initially at 600°C is dropped into a bucket containing 20.0 kg of water at 25.0°C. What is the final temperature? (Ignore the heat capacity of the container, and assume that a negligible amount of water boils away.
Helium gas is sold in steel tanks. If the helium is used to inflate a balloon, could the balloon lift the spherical tank the helium came in? Justify your answer. Steel will rupture if subjected to tensile stress greater than its yield strength of 5 X 108 N/m2. Suggestion: You may consider a steel shell of radius r and thickness t containing helium at high pressure and on the verge of breaking apart into two hemispheres.
A cylinder that has a 40.0-cm radius and is 50.0 cm deep is filled with air at 20.0°C and 1.00 atm (Fig. P19.68a). A 20.0-kg piston is now lowered into the cylinder, compressing the air trapped inside (Fig. P19.68b). finally a 75.0-kg man stands on the piston, further compressing the air, which remains at 20°C (Fig. P19.68c)
(a) How far down ("h) does the piston move when the man steps onto it?
(b) To what temperature should the gas be heated to raise the piston and man back to hi?
An aluminum cup of mass 200 g contains 800 g of water in thermal equilibrium at 80.0°C. The combination of cup and water is cooled uniformly so that the temperature decreases by 1.50°C per minute. At what rate is energy being removed by heat? Express your answer in watts
The relationship Lf = Li (1 + αΔT) is an approximation that works when the average coefficient of expansion is small. If α is large, one must integrate the relationship dL/dT =αL to determine the final length.
(a) Assuming that the coefficient of linear expansion is constant as L varies, determine a general expression for the final length.
(b) Given a rod of length 1.00 m and a temperature change of 100.0°C, determine the error caused by the approximation when α = 2.00 X 10-5 (°C)-1 (a typical value for a metal) and when & = 0.020 0 (°C)-1 (an unrealistically large value for comparison).
A steel wire and a copper wire, each of diameter 2.000 mm, are joined end to end. At 40.0°C, each has an unscratched length of 2.000 m; they are connected between two fixed supports 4.000 m apart on a tabletop, so that the steel wire extends from x = -2.000 m to x = 0, the copper wire extends from x = 0 to x = 2.000 m, and the tension is negligible. The temperature is then lowered to 20.0°C. At this lower temperature, find the tension in the wire and the x coordinate of the junction between the wires.
An aluminum calorimeter with a mass of 100 g contains 250 g of water. The calorimeter and water are in thermal equilibrium at 10.0°C. Two metallic blocks are placed into the water. One is a 50.0-g piece of copper at 80.0°C. The other block has a mass of 70.0 g and is originally at a temperature of 100°C. The entire system stabilizes at a final temperature of 20.0°C.
(a) Determine the specific heat of the unknown sample.
(b) Guess the material of the unknown, using the data in Table 20.1.
A combination of 0.250 kg of water at 20.0°C, 0.400 kg of aluminum at 26.0°C, and 0.100 kg of copper at 100°C is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final temperature of the mixture.
A steel guitar string with a diameter of 1.00 mm is stretched between supports 80.0 cm apart. The temperature is 0.0°C.
(a) Find the mass per unit length of this string. (Use the value 7.86) 103 kg/m3 for the density.)
(b) The fundamental frequency of transverse oscillations of the string is 200 Hz. What is the tension in the string?
(c) If the temperature is raised to 30.0°C, find the resulting values of the tension and the fundamental frequency. Assume that both the Young’s modulus (Table 12.1) and the average coefficient of expansion (Table 19.1) have constant values between 0.0°C and 30.0°C.
In a chemical processing plant, a reaction chamber of fixed volume V0 is connected to a reservoir chamber of fixed volume 4V0 by a passage containing a thermally insulating porous plug. The plug permits the chambers to be at different temperatures. The plug allows gas to pass from either chamber to the other, ensuring that the pressure is the same in both. At one point in the processing, both chambers contain gas at a pressure of 1.00 atm and a temperature of 27.0°C. Intake and exhaust valves to the pair of chambers are closed. The reservoir is maintained at 27.0°C while the reaction chamber is heated to 400°C. What is the pressure in both chambers after this is done?
A 1.00-km steel railroad rail is fastened securely at both ends when the temperature is 20.0°C. As the temperature increases, the rail begins to buckle. If its shape is an arc of a vertical circle, find the height h of the center of the rail when the temperature is 25.0°C. You will need to solve a transcendental equation.
A perfectly plane house roof makes an angle - with the horizontal. When its temperature changes, between Tc before dawn each day to Th in the middle of each afternoon, the roof expands and contracts uniformly with a coefficient of thermal expansion α1. Resting on the roof is a flat rectangular metal plate with expansion coefficient α2, greater than α1. The length of the plate is L, measured up the slope of the roof. The component of the plate’s weight perpendicular to the roof is supported by a normal force uniformly distributed over the area of the plate. The coefficient of kinetic friction between the plate and the roof is μk. The plate is always at the same temperature as the roof, so we assume its temperature is continuously changing. Because of the difference in expansion coefficients, each bit of the plate is moving relative to the roof below it, except for points along a certain horizontal line running across the plate. We call this the stationary line. If the temperature is rising, parts of the plate below the stationary line are moving down relative to the roof and feel a force of kinetic friction acting up the roof. Elements of area above the stationary line are sliding up the roof and on them kinetic friction acts downward parallel to the roof. The stationary line occupies no area, so we assume no force of static friction acts on the plate while the temperature is changing. The plate as a whole is very nearly in equilibrium, so the net friction force on it must be equal to the component of its weight acting down the incline.
(a) Prove that the stationary line is at a distance of below the top edge of the plate
(b) Analyze the forces that act on the plate when the temperature is falling, and prove that the stationary line is at that same distance above the bottom edge of the plate.
(c) Show that the plate steps down the roof like an inchworm, moving each day by the distance
(d) Evaluate the distance an aluminum plate moves each day if its length is 1.20 m, if the temperature cycles between 4.00°C and 36.0°C, and if the roof has slope 18.5°, coefficient of linear expansion 1.50 X 10-5 (°C)-1, and coefficient of friction 0.420 with the plate.
(e) What If? What if the expansion coefficient of the plate is less than that of the roof? Will the plate creep up the roof?
Is it possible for two objects to be in thermal equilibrium if they are not in contact with each other? Explain
If water with a mass mh at temperature Th is poured into an aluminum cup of mass mAl containing mass mc of water at Tc, where Th > Tc , what is the equilibrium temperature of the system?
A water heater is operated by solar power. If the solar collector has an area of 6.00 m2 and the intensity delivered by sunlight is 550 W/m2, how long does it take to increase the temperature of 1.00 m3 of water from 20.0°C to 60.0°C?
Two thermally insulated vessels are connected by a narrow tube fitted with a valve that is initially closed. One vessel, of volume 16.8 L, contains oxygen at a temperature of 300 K and a pressure of 1.75 atm. The other vessel, of volume 22.4 L, contains oxygen at a temperature of 450 K and a pressure of 2.25 atm. When the valve is opened, the gases in the two vessels mix, and the temperature and pressure become uniform throughout.
(a) What is the final temperature?
(b) What is the final pressure?
How much energy is required to change a 40.0-g ice cube from ice at -10.0°C to steam at 110°C?
A 50.0-g copper calorimeter contains 250 g of water at 20.0°C. How much steam must be condensed into the water if the final temperature of the system is to reach 50.0°C?
A 3.00-g lead bullet at 30.0°C is fired at a speed of 240 m/s into a large block of ice at 0°C, in which it becomes embedded. What quantity of ice melts?
Steam at 100°C is added to ice at 0°C.
(a) Find the amount of ice melted and the final temperature when the mass of steam is 1.0 g and the mass of ice is 50.0 g.
(b) What If? Repeat when the mass of steam is 1.00 g and the mass of ice is 50.0 g.
A 1.00-kg block of copper at 20.0°C is dropped into a large vessel of liquid nitrogen at 77.3 K. How many kilograms of nitrogen boil away by the time the copper reaches 77.3 K? (The specific heat of copper is 0.092 0 cal/g . °C. The latent heat of vaporization of nitrogen is 48.0 cal/g).
Assume that a hailstone at 0°C falls through air at a uniform temperature of 0°C and lands on a sidewalk also at this temperature. From what initial height must the hailstone fall in order to entirely melt on impact?
In an insulated vessel, 250 g of ice at 0°C is added to 600 g of water at 18.0°C.
(a) What is the final temperature of the system?
(b) How much ice remains when the system reaches equilibrium?
Two speeding lead bullets, each of mass 5.00 g, and at temperature 20.0°C, collide head-on at speeds of 500 m/s each. Assuming a perfectly inelastic collision and no loss of energy by heat to the atmosphere, describe the final state of the two-bullet system.
A sample of ideal gas is expanded to twice its original volume of 1.00 m3 in a quasi-static process for which P =αV 2, with α = 5.00 atm/m6, as shown in Figure P20.23. How much work is done on the expanding gas?
(a) Determine the work done on a fluid that expands from i to f as indicated in Figure P20.24.
(b) What If? How much work is performed on the fluid if it is compressed from f to i along the same path?
An ideal gas is enclosed in a cylinder with a movable piston on top of it. The piston has a mass of 8 000 g and an area of 5.00 cm2 and is free to slide up and down, keeping the pressure of the gas constant. How much work is done on the gas as the temperature of 0.200 mol of the gas is raised from 20.0°C to 300°C?