The average coefcient of volume expansion for carbon

For each of the following temperatures, nd the equivalent temperature on the indicated scale: (a) 273.15
C on the Fahrenheit scale, (b) 98.6
F on the Celsius scale, and (c) 100 K on the Fahrenheit scale.

The pressure in a constant-volume gas thermometer is 0.700 atm at 100
C and 0.512 atm at 0
C. (a) What is the temperature when the pressure is 0.0400 atm? (b) What is the pressure at 450
C?

Convert the following temperatures to their values on the Fahrenheit and Kelvin scales: (a) the boiling point of liquid hydrogen, 252.87
C; (b) the temperature of a room at 20
C.

Death Valley holds the record for the highest recorded temperature in the United States. On July 10, 1913, at a place called Furnace Creek Ranch, the temperature rose to 134
F. The lowest U.S. temperature ever recorded occurred at Prospect Creek Camp in Alaska on January 23, 1971, when the temperature plummeted to 79.8
F. Convert these temperatures to the Celsius scale.

Show that the temperature 40
is unique in that it has the same numerical value on the Celsius and Fahrenheit scales.

A constant-volume gas thermometer is calibrated in dry ice (80.0
C) and in boiling ethyl alcohol (78.0
C). The respective pressures are 0.900 atm and 1.635 atm. (a) What value of absolute zero does the calibration yield? (b) What pressures would be found at the freezing and boiling points of water? (Note that we have the linear relationship P  A  BT, where A and B are constants.)

Show that if the temperature on the Celsius scale changes by TC, the Fahrenheit temperature changes by TF  (9/5)TC.

The temperature difference between the inside and the outside of an automobile engine is 450
C. Express this difference on (a) the Fahrenheit scale and (b) the Kelvin 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 of the two temperatures in Celsius degrees and in Kelvins.

A cylindrical brass sleeve is to be shrink-tted over a brass shaft whose diameter is 3.212 cm at 0
C. The diameter of the sleeve is 3.196 cm at 0
C. (a) To what temperature must the sleeve be heated before it will slip over the shaft? (b) Alternatively, to what temperature must the shaft be cooled before it will slip into the sleeve?

The New River Gorge bridge in West Virginia is a 518-mlong steel arch. How much will its length change between temperature extremes of 20
C and 35
C?

A grandfather clock is controlled by a swinging brass pendulum that is 1.3 m long at a temperature of 20
C. (a) What is the length of the pendulum rod when the temperature drops to 0.0
C? (b) If a pendulums period is given by , where L is its length, does the change in length of the rod cause the clock to run fast or slow?

A pair of eyeglass frames are made of epoxy plastic (coefcient of linear expansion  1.30  104
C1). 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 into them?

A cube of solid aluminum has a volume of 1.00 m3 at 20
C. What temperature change is required to produce a 100-cm3 increase in the volume of the cube?

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 coefcients of linear expansion are constant, (a) to what temperature must the combination be cooled to separate the two metals? Is that temperature attainable? (b) What if the aluminum rod were 10.02 cm in diameter?

Show that the coefcient of volume expansion, , is re lated to the coefcient of linear expansion,  , through the expression  3  .

A gold ring has an inner diameter of 2.168 cm at a temperature of 15.0
C. Determine its inner diameter at 100
C (

A construction worker uses a steel tape to measure the length of an aluminum support column. If the measured length is 18.700 m when the temperature is 21.2
C, what is the measured length when the temperature rises to 29.4
C? (Note: Dont neglect the expansion of the tape.)

The band in Figure P10.19 is stainless steel (coefcient of linear expansion  17.3  106
C1; Youngs modulus  18  1010 N/m2). It is essentially circular with an initial mean radius of 5.0 mm, a height of 4.0 mm, and a thickness of 0.50 mm. If the band just ts snugly over the tooth when heated to a temperature of 80
C, what is the tension in the band when it cools to a temperature of 37
C?

The Trans-Alaskan pipeline is 1 300 km long, reaching from Prudhoe Bay to the port of Valdez, and is subject to temperatures ranging from 73
C to 35
C. How much does the steel pipeline expand due to the difference in temperature? How can this expansion be compensated for?

An automobile fuel tank is lled to the brim with 45 L (12 gal) of gasoline at 10
C. Immediately afterward, the vehicle is parked in the sunlight, where the temperature is 35
C. How much gasoline overows from the tank as a result of the expansion? (Neglect the expansion of the tank.)

When the hot water in a certain upstairs bathroom is turned on, a series of 18 ticks is heard as the copper hot-water pipe slowly heats up and increases in length. The pipe runs vertically from the hot-water heater in the basement, through a hole in the floor 5.0 m above the water heater. The ticks are caused by the pipe sticking in the hole in the oor until the tension in the expanding pipe is great enough to unstick the pipe, enabling it to jump a short distance through the hole. If the hotwater temperature is 46
C and room temperature is 20
C, determine (a) the distance the pipe moves with each tick and (b) the force required to unstick the pipe if the cross-sectional area of the copper in the pipe is 3.55  105 m2.

The average coefcient of volume expansion for carbon tetrachloride is 5.81  104 (
C)1. If a 50.0-gal steel container is lled completely with carbon tetrachloride when the temperature is 10.0
C, how much will spill over when the temperature rises to 30.0
C?

On a day when the temperature is 20.0
C, a concrete walk is poured in such a way that its ends are unable to move. (a) What is the stress in the cement when its temperature is 50.0
C on a hot, sunny day? (b) Does the concrete fracture? Take Youngs modulus for concrete to be 7.00  109 N/m2 and the compressive strength to be 2.00  107 N/m2.

Figure P10.25 shows a circular steel casting with a gap. If the casting is heated, (a) does the width of the gap increase or decrease? (b) The gap width is 1.600 cm when the temperature is 30.0
C. Determine the gap width when the temperature is 190
C.

A hollow aluminum cylinder 20.0 cm deep has an internal capacity of 2.000 L at 20.0
C. It is completely lled with turpentine and then warmed to 80.0
C. (a) How much turpentine overows? (b) If it is then cooled back to 20.0
C, how far below the surface of the cylinders rim is the turpentine surface?

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 nal temperature? (b) If the gas is heated so that both the pressure and volume are doubled, what is the nal temperature?

Gas is contained in an 8.0-L vessel at a temperature of 20
C and a pressure of 9.0 atm. (a) Determine the number of moles of gas in the vessel. (b) How many molecules are in the vessel?

(a) An ideal gas occupies a volume of 1.0 cm3 at 20
C and atmospheric pressure. Determine the number of molecules of gas in the container. (b) If the pressure of the 1.0-cm3 volume is reduced to 1.0  1011 Pa (an extremely good vacuum) while the temperature remains constant, how many moles of gas remain in the container?

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 lled balloon is a sphere 0.300 m in diameter at an absolute pressure of 1.20 atm?

A cylinder with a movable piston contains gas at a temperature of 27.0
C, a volume of 1.50 m3, and an absolute pressure of 0.200  105 Pa. What will be its nal temperature if the gas is compressed to 0.700 m3 and the absolute pressure increases to 0.800  105 Pa?

The density of helium gas at T  0
C is  0  0.179 kg/m3. The temperature is then raised to T  100
C, but the pressure is kept constant. Assuming that the helium is an ideal gas, calculate the new density

A weather balloon is designed to expand to a maximum radius of 20 m at its working altitude, where the air pressure is 0.030 atm and the temperature is 200 K. If the balloon is lled at atmospheric pressure and 300 K, what is its radius at liftoff?

A cylindrical diving bell 3.00 m in diameter and 4.00 m tall with an open bottom is submerged to a depth of 220 m in the ocean. The surface temperature is 25.0
C, and the temperature 220 m down is 5.00
C. The density of seawater is 1 025 kg/m3. How high does the seawater rise in the bell when it is submerged?

An air bubble has a volume of 1.50 cm3 when it is released by a submarine 100 m below the surface of a lake. What is the volume of the bubble when it reaches the surface? Assume that the temperature and the number of air molecules in the bubble remain constant during its ascent.

A sealed cubical container 20.0 cm on a side contains three times Avogadros number of molecules at a temperature of 20.0
C. Find the force exerted by the gas on one of the walls of the container.

What is the average kinetic energy of a molecule of oxygen at a temperature of 300 K?

(a) What is the total random kinetic energy of all the molecules in 1 mole of hydrogen at a temperature of 300 K? (b) With what speed would a mole of hydrogen have to move so that the kinetic energy of the mass as a whole would be equal to the total random kinetic energy of its molecules?

Use Avogadros number to nd the mass of a helium atom.

The temperature near the top of the atmosphere on Venus is 240 K. (a) Find the rms speed of hydrogen (H2) at that point in Venuss atmosphere. (b) Repeat for carbon dioxide (CO2). (c) It has been found that if the rms speed exceeds one-sixth of the planets escape velocity, the gas eventually leaks out of the atmosphere and into outer space. If the escape velocity on Venus is 10.3 km/s, does hydrogen escape? Does carbon dioxide?

A cylinder contains a mixture of helium and argon gas in equilibrium at a temperature of 150
C. (a) What is the average kinetic energy of each type of molecule? (b) What is the rms speed of each type of molecule?

Three moles of nitrogen gas, N2, at 27.0
C are contained in a 22.4-L cylinder. Find the pressure the gas exerts on the cylinder walls.

Superman leaps in front of Lois Lane to save her from a volley of bullets. In a 1-minute interval, an automatic weapon res 150 bullets, each of mass 8.0 g, at 400 m/s. The bullets strike his mighty chest, which has an area of 0.75 m2. Find the average force exerted on Supermans chest if the bullets bounce back after an elastic, head-on collision.

In a period of 1.0 s, 5.0  1023 nitrogen molecules strike a wall of area 8.0 cm2. If the molecules move at 300 m/s and strike the wall head on in a perfectly elastic collision, nd the pressure exerted on the wall. (The mass of one N2 molecule is 4.68  1026 kg.)

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 (Fig. P10.45). A stud and a second-story oorboard hold the ends of this section of copper pipe stationary. Find the magnitude and direction of the displacement of the pipe elbow when the water ow is turned on, raising the temperature of the pipe from 18.0
C to 46.5
C.

The active element of a certain laser is an ordinary glass rod 20 cm long and 1.0 cm in diameter. If the temperature of the rod increases by 75
C, nd its increases in (a) length, (b) diameter, and (c) volume.

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?

A 1.5-m-long glass tube that is closed at one end is weighted and lowered to the bottom of a freshwater lake. When the tube is recovered, an indicator mark shows that water rose to within 0.40 m of the closed end. Determine the depth of the lake. Assume constant temperature.

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, with 1.00 mol of methane as a by-product. 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 recycling of three astronauts respiration during one week of flight is stored in an originally empty 150-L tank at 45.0
C, what is the nal pressure in the tank?

A vertical cylinder of cross-sectional area 0.050 m2 is tted with a tight-tting, frictionless piston of mass 5.0 kg (Fig. P10.50). If there are 3.0 mol of an ideal gas in the cylinder at 500 K, determine the height h at which the piston will be in equilibrium under its own weight.

A liquid with a coefcient of volume expansion of just lls a spherical ask of volume V0 at temperature T (Fig. P10.51). The ask is made of a material that has a coefcient of linear expansion of  . The liquid is free to expand into a capillary of cross-sectional area A at the top. (a) Show that if the temperature increases by T, the liquid rises in the capillary by the amount h  (V0/A)(  3  )T. (b) For a typical system, such as a mercury thermometer, why is it a good approximation to neglect the expansion of the ask?

A hollow aluminum cylinder is to be tted over a steel piston. At 20
C, the inside diameter of the cylinder is 99% of the outside diameter of the piston. To what common temperature should the two pieces be heated in order that the cylinder just t over the piston?

A steel measuring tape was designed to read correctly at 20
C. A parent uses the tape to measure the height of a 1.1-m-tall child. If the measurement is made on a day when the temperature is 25
C, is the tape reading longer or shorter than the actual height, and by how much?

Before beginning a long trip on a hot day, a driver inates an automobile tire to a gauge pressure of 1.80atm at 300K. At the end of the trip, the gauge pressure has increased to 2.20 atm. (a) Assuming that the volume has remained constant, what is the temperature of the air inside the tire? (b) What percentage of the original mass of air in the tire should be released so the pressure returns to its original value? Assume that the temperature remains at the value found in (a) and the volume of the tire remains constant as air is released.

Two concrete spans of a 250-m-long bridge are placed end to end so that no room is allowed for expansion (Fig. P10.55a). If the temperature increases by 20.0
C, what is the height y to which the spans rise when they buckle (Fig. P10.55b)?

A copper rod and a steel rod are heated. At 0
C, the copper rod has a length LC and the steel one has a length LS. When the rods are being heated or cooled, a difference of 5.00 cm is maintained between their lengths. Determine the values of LC and LS.

If 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?

An expandable cylinder has its top connected to a spring with force constant 2.00  103 N/m. (See Fig. P10.58.) The cylinder is lled with 5.00 L of gas with the spring relaxed at a pressure of 1.00 atm and a temperature of 20.0
C. (a) If the lid has a cross-sectional area of 0.010 0 m2 and negligible mass, how high will the lid rise when the temperature is raised to 250
C? (b) What is the pressure of the gas at 250
C?

A swimmer has 0.820 L of dry air in his lungs when he dives into a lake. Assuming the pressure of the dry air is 95% of the external pressure at all times, what is the volume of the dry air at a depth of 10.0 m? Assume that atmospheric pressure at the surface is 1.013  105 Pa.

Two small containers, each with a volume of 100 cm3, contain helium gas at 0
C and 1.00 atm pressure. The two containers are joined by a small open tube of negligible volume, allowing gas to ow from one container to the other. What common pressure will exist in the two containers if the temperature of one container is raised to 100
C while the other container is kept at 0
C?

A bimetallic bar is made of two thin strips of dissimilar metals bonded together. As they are heated, the one with the larger average coefcient of expansion expands more than the other, forcing the bar into an arc, with the outer strip having both a larger radius and a larger circumference. (See Fig. P10.61.) (a) Derive an expression for the angle of bending,  , as a function of the initial length of the strips, their average coefcients of linear expansion, the change in temperature, and the separation of the centers of the strips (r  r2  r1). (b) Show that the angle of bending goes to zero when T goes to zero or when the two coefcients of expansion become equal. (c) What happens if the bar is cooled?

A 250-m-long bridge is improperly designed so that it cannot expand with temperature. It is made of concrete with   12  106
C1. (a) Assuming that the maximum change in temperature at the site is expected to be 20
C, nd the change in length the span would undergo if it were free to expand. (b) Show that the stress on an object with Youngs modulus Y when raised by T with its ends rmly xed is given by  YT. (c) If the maximum stress the bridge can withstand without crumbling is 2.0  107 Pa, will it crumble because of this temperature increase? Youngs modulus for concrete is about 2.0  1010 Pa.

The density of gasoline is 730 kg/m3 at 0
C. Its volume expansion coefcient is 9.6  104
C1. If 1.00 gal of gasoline occupies 0.003 8 m3, how many extra kilograms of gasoline are obtained when 10 gallons of gasoline are bought at 0
C rather than at 20
C?

Fill one basin with hot tap water (not to exceed about 100
F). Fill another basin with cold tap water, and add ice until about one-third of the mixture is ice. Fill the third basin with an equal mixture of hot and cold tap water. Place your left hand in the hot water and your right hand in the cold water for about 15 s. Then place both hands in the basin of lukewarm water for 15 s. Describe whether the water feels hot or cold to either of your hands and why this effect occurs.

Tape two plastic straws tightly together along their entire length, but with a 2-cm offset. Hold them in a stream of very hot water from a faucet so that the water pours through one straw but not the other. Quickly hold the straws up and sight along their length. You should be able to see a very slight curvature in the tape. The effect is small, so look closely. Running cold water through the same straw and again sighting along the length will help you see the small change in shape more clearly. Explain these observations.

You can study the thermal expansion of air with a Florence ask and a balloon. To do so, boil a small amount of water in the ask. Then quickly remove the ask from the heat source and place the mouth of the balloon over the mouth of the ask. Observe what happens to the balloon when you place the ask in a basin of cold water. Explain your observation. Reheat the water in the ask with the balloon still in place, and explain your observations.