Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 19

Temperature - all with Video Answers

Educators

Chapter Questions

Problem 1

A constant-volume gas thermometer is calibrated in dry ice (that is, evaporating carbon dioxide in the solid state, with a temperature of $-80.0^{\circ} \mathrm{C} )$ and in boiling ethyl alcohol $\left(78.0^{\circ} \mathrm{C}\right) .$ 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?

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

The temperature difference between the inside and the outside of an automobile engine is $450^{\circ} \mathrm{C}$ . Express this temperature difference on (a) the Fahrenheit scale and (b) the Kelvin scale.

Prashant Bana

Numerade Educator

Problem 3

Liquid nitrogen has a boiling point of $-195.81^{\circ} \mathrm{C}$ at atmospheric pressure. Express this temperature (a) in degrees Fahrenheit and (b) in kelvins.

Sean Dougherty

Numerade Educator

Problem 4

The melting point of gold is $1064^{\circ} \mathrm{C},$ and its boiling point is $2660^{\circ} \mathrm{C}$ . (a) Express these temperatures in kelvins. (b) Compute the difference between these temperatures in Celsius degrees and kelvins.

Nathan Prins

Numerade Educator

Problem 5

A copper telephone wire has essentially no sag between poles 35.0 $\mathrm{m}$ apart on a winter day when the temperature is $-20.0^{\circ} \mathrm{C}$ . How much longer is the wire on a summer day when $T_{\mathrm{C}}=35.0^{\circ} \mathrm{C} ?$

Nathan Prins

Numerade Educator

Problem 6

The concrete sections of a certain superhighway are designed to have a length of 25.0 $\mathrm{m}$ . The sections are poured and cured at $10.0^{\circ} \mathrm{C}$ . What minimum spacing should the engineer leave between the sections to eliminate buckling if the concrete is to reach a temperature of
$50.0^{\circ} \mathrm{C}$ ?

Salamat Ali

Numerade Educator

Problem 7

The active element of a certain laser is made of a glass rod 30.0 $\mathrm{cm}$ long and 1.50 $\mathrm{cm}$ in diameter. If the temperature of the rod increases by $65.0^{\circ} \mathrm{C},$ what is the increase in $(\mathrm{a})$ its length, $(\mathrm{b})$ its diameter, and $(\mathrm{c})$ its volume Assume the average coefficient of linear expansion of the glass is $9.00 \times 10^{-6}\left(^{\circ} \mathrm{C}\right)^{-1}$ .

Nathan Prins

Numerade Educator

Problem 8

Review problem. Inside the wall of a house, an L-shaped section of hot water pipe consists of a straight, horizontal piece 28.0 $\mathrm{cm}$ long, an elbow, and a straight vertical piece 134 $\mathrm{cm}$ long (Fig. Pl9.8). A stud and a second-story floor-board 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 $18.0^{\circ} \mathrm{C}$ to $46.5^{\circ} \mathrm{C} .$

Nathan Prins

Numerade Educator

Problem 9

A thin brass ring of inner diameter 10.00 $\mathrm{cm}$ at $20.0^{\circ} \mathrm{C}$ is warmed and slipped over an aluminum rod of diameter 10.01 $\mathrm{cm}$ at $20.0^{\circ} \mathrm{C}$ . Assuming the average coefficients of linear expansion are constant, (a) to what temperature must this combination be cooled to separate the parts? Explain whether this separation is attainable. (b) What If? What if the aluminum rod were 10.02 $\mathrm{cm}$ in diameter?

Nathan Prins

Numerade Educator

Problem 10

At $20.0^{\circ} \mathrm{C},$ an aluminum ring has an inner diameter of 5.0000 $\mathrm{cm}$ and a brass rod has a diameter of $5.0500 \mathrm{cm} .$ (a) If only the ring is warmed, what temperature must it reach so that it will just slip over the rod? (b) What If both the ring and the rod are warmed together, what temperature must they both reach so that the ring barely slips over the rod? Would this latter process work? Explain.

Nathan Prins

Numerade Educator

Problem 11

A volumetric flask made of Pyrex is calibrated at $20.0^{\circ} \mathrm{C} .$ It is filled to the 100 $\mathrm{mL}$ mark with $35.0^{\circ} \mathrm{C}$ acetone. (a) What is the volume of the acetone when it cools to $20.0^{\circ} \mathrm{C}$ ? (b) How significant is the change in volume of the flask?

Nathan Prins

Numerade Educator

Problem 12

On a day that the temperature is $20.0^{\circ} \mathrm{C}$ , a concrete walk is poured in such a way that the ends of the walk are unable to move. (a) What is the stress in the cement on a hot day of $50.0^{\circ} \mathrm{C}$ ? (b) Does the concrete fracture? Take Young's modulus for concrete to be $7.00 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}$ and the compressive strength to be $2.00 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}$ .

Nathan Prins

Numerade Educator

Problem 13

A hollow aluminum cylinder 20.0 $\mathrm{cm}$ deep has an internal capacity of 2.000 $\mathrm{L}$ at $20.0^{\circ} \mathrm{C}$ . It is completely filled with turpentine and then slowly warmed to $80.0^{\circ} \mathrm{C}$ . (a) How much turpentine overflows? (b) If the cylinder is then cooled back to $20.0^{\circ} \mathrm{C},$ how far below the cylinder's rim does the turpentine's surface recede?

Nathan Prins

Numerade Educator

Problem 14

She Golden Gate Bridge in San Francisco has a main span of length $1.28 \mathrm{km},$ one of the longest in the world. Imagine that a taut steel wire with this length and a cross- sectional area of $4.00 \times 10^{-6} \mathrm{m}^{2}$ is laid on the bridge and that on this summer day the temperature of the wire is $35.0^{\circ} \mathrm{C}$ . (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 $-10.0^{\circ} \mathrm{C},$ what is the tension in the wire? Take Young's modulus for steel to be $20.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}$ . (b) Permanent deformation occurs if the stress in the steel exceeds its elastic limit of $3.00 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ . At what temperature would the wire reach its elastic limit? (c) What If? Explain how your answers to parts (a) and (b) would change if the Golden Gate Bridge were twice as long.

Nathan Prins

Numerade Educator

Problem 15

A certain telescope forms an image of part of a cluster of stars on a square silicon charge-coupled detector chip 2.00 $\mathrm{cm}$ on each side. A star field is focused on the chip when it is first turned on, and its temperature is $20.0^{\circ} \mathrm{C}$ . The star field contains 5342 stars scattered uniformly. To make the detector more sensitive, it is cooled to $-100^{\circ} \mathrm{C}$ . How many star images then fit onto the chip? The average coefficient of linear expansion of silicon is $4.68 \times 10^{-6}\left(^{\circ} \mathrm{C}\right)^{-1}$ .

Nathan Prins

Numerade Educator

Problem 16

On your wedding day your lover gives you a gold ring of mass 3.80 g. Fifty years later its mass is 3.35 g. On the average, how many atoms were abraded from the ring during each second of your marriage? The molar mass of gold is $197 \mathrm{g} / \mathrm{mol} .$

Nathan Prins

Numerade Educator

Problem 17

An automobile tire is inflated with air originally at $10.0^{\circ} \mathrm{C}$ and normal atmospheric pressure. During the process, the air is compressed to 28.0$\%$ of its original volume and the temperature is increased to $40.0^{\circ} \mathrm{C} .$ (a) What is the tire pressure? (b) After the car is driven at high speed, the tire's air temperature rises to $85.0^{\circ} \mathrm{C}$ and the tire's interior volume increases by $2.00 \% .$ What is the new tire pressure (absolute) in pascals?

Salamat Ali

Numerade Educator

Problem 18

Gas is contained in an $8.00-\mathrm{L}$ vessel at a temperature of $20.0^{\circ} \mathrm{C}$ and a pressure of $9.00 \mathrm{atm} .$ (a) Determine the number of moles of gas in the vessel. (b) How many molecules are in the vessel?

Sean Dougherty

Numerade Educator

Problem 19

An auditorium has dimensions $10.0 \mathrm{m} \times 20.0 \mathrm{m} \times$ $30.0 \mathrm{m} .$ How many molecules of air fill the auditorium at $20.0^{\circ} \mathrm{C}$ and a pressure of 101 $\mathrm{kPa}$ ?

Nathan Prins

Numerade Educator

Problem 20

A cook puts 9.00 g of water in a $2.00-\mathrm{L}$ pressure cooker and warms it to $500^{\circ} \mathrm{C}$ . What is the pressure inside the container?

Nathan Prins

Numerade Educator

Problem 21

A The mass of a hot-air balloon and its cargo (not including the air inside) is 200 $\mathrm{kg}$ . The air outside is at $10.0^{\circ} \mathrm{C}$ and 101 $\mathrm{kPa}$ . The volume of the balloon is $400 \mathrm{m}^{3} .$ To what temperature must the air in the balloon be warmed before the balloon will lift off: (Air density at $10.0^{\circ} \mathrm{C}$ is $1.25 \mathrm{kg} / \mathrm{m}^{3} . )$

Nathan Prins

Numerade Educator

Problem 22

Male-pattern dumbness. Your father and your younger 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 $\mathrm{L}$ (Fig. Pl9.22). Your father puts a negligible amount of concentrated fertilizer into his tank. They both pour in 4.00 $\mathrm{L}$ of water and seal up their tanks, so the tanks 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. To accomplish spraying out all the water, each finds he must pump up the tank three times. Here 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 as the second but result in a disappointingly small amount of water coming out. Account for this phenomenon.

Nathan Prins

Numerade Educator

Problem 23

(a) Find the number of moles in one cubic meter of an ideal gas at $20.0^{\circ} \mathrm{C}$ and atmospheric pressure. (b) For air, Avogadro's number of molecules has mass 28.9 $\mathrm{g}$ . Calculate the mass of one cubic meter of air. State how the result compares with the tabulated density of air.

Nathan Prins

Numerade Educator

Problem 24

At 25.0 m below the surface of the sea (density = $1025 \mathrm{kg} / \mathrm{m}^{3} ),$ where the temperature is $5.00^{\circ} \mathrm{C},$ a diver exhales an air bubble having a volume of $1.00 \mathrm{cm}^{3} .$ If the surface temperature of the sea is $20.0^{\circ} \mathrm{C},$ what is the volume of the bubble just before it breaks the surface?

Nathan Prins

Numerade Educator

Problem 25

A cube 10.0 $\mathrm{cm}$ on each edge contains air (with equivalent molar mass 28.9 $\mathrm{g} / \mathrm{mol} )$ at atmospheric pressure and temperature 300 $\mathrm{K}$ . Find (a) the mass of the gas, (b) the gravitational force exerted on it, and (c) the force it exerts on each face of the cube. (d) Comment on the physical reason such a small sample can exert such a great force.

Nathan Prins

Numerade Educator

Problem 26

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.

Sean Dougherty

Numerade Educator

Problem 27

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 $\left(\mathrm{O}_{2}\right),$ it contains 12.0 $\mathrm{kg}$ 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 the temperature of the tank remains constant.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 28

In state-of-the-art vacuum systems, pressures as low as $10^{-9}$ Pa are being attained. Calculate the number of molecules in a $1.00-\mathrm{m}^{3}$ vessel at this pressure and a temperature of $27.0^{\circ} \mathrm{C}$ .

Nathan Prins

Numerade Educator

Problem 29

How much water will a shearwater shear? To measure how (ar below the ocean's surface a bird dives to catch a fish, Will Mackin used a method originated by Lord Kelvin for soundings by the British Navy. Mackin dusted the interiors of thin plastic tubes with powdered sugar and then sealed one end of each tube. Charging around on a rocky beach at night with a miner's headlamp, he would grab an Audubon's shearwater in its nest and attach a tube to its back. He would then catch the same bird the next night and remove the tube. After hundreds of captures, the birds thoroughly disliked him but were not permanently frightened away from the rookery. Assume in one trial, with a tube 6.50 $\mathrm{cm}$ long, he found that water had entered the tube to wash away the sugar over a distance of 2.70 $\mathrm{cm}$ from the open end. (a) Find the greatest depth to which the shearwater dove, assuming the air in the tube stayed at constant temperature. (b) Must the tube be attached to the bird in any particular orientation for this method to work? (Audubon's shearwater can dive to more than twice the depth you calculate, and larger species can dive nearly ten times deeper.)

Nathan Prins

Numerade Educator

Problem 30

A room of volume $V$ contains air having equivalent molar mass $M$ (in grams per mole). If the temperature of the room is raised from $T_{1}$ to $T_{2},$ what mass of air will leave the room? Assume the air pressure in the room is maintained at $P_{0}$ .

Nathan Prins

Numerade Educator

Problem 31

A student measures the length of a brass rod with a steel tape at $20.0^{\circ} \mathrm{C}$ . The reading is $95.00 \mathrm{cm} .$ What will the tape indicate for the length of the rod when the rod and the tape are at (a) $-15.0^{\circ} \mathrm{C}$ and $(\mathrm{b}) 55.0^{\circ} \mathrm{C} ?$

Sean Dougherty

Numerade Educator

Problem 32

The density of gasoline is 730 $\mathrm{kg} / \mathrm{m}^{3}$ at $0^{\circ} \mathrm{C}$ . Its average coefficient of volume expansion is $9.60 \times 10^{-4}\left(^{\circ} \mathrm{C}\right)^{-1}$ . Assume 1.00 gal of gasoline occupies $0.00380 \mathrm{m}^{3} .$ How many extra kilograms of gasoline would you get if you bought 10.0 gal of gasoline at $0^{\circ} \mathrm{C}$ rather than at $20.0^{\circ} \mathrm{C}$ from a pump that is not temperature compensated?

Nathan Prins

Numerade Educator

Problem 33

A mercury thermometer is constructed as shown in Figure Pl9.33. The capillary tube has a diameter of 0.00400 $\mathrm{cm}$ , and the bulb has a diameter of $0.250 \mathrm{cm} .$ Ignoring the expansion of the glass, find the change in height of the mercury column that occurs with a temperature change of $30.0^{\circ} \mathrm{C}$ .

Vipender Yadav

Numerade Educator

Problem 34

A liquid with a coefficient of volume expansion $\beta$ just fills a spherical shell of volume $V_{i}$ at a temperature of $T_{i}$ (Fig. $\mathrm{P} 19.33 ) .$ The shell is made of a material with an average coefficient of linear expansion $\alpha .$ The liquid is free to expand into an open capillary of area $A$ projecting from the top of the sphere. (a) Assuming the temperature increases by $\Delta T$ , show that the liquid rises in the capillary by the amount $\Delta h$ given by the equation $\Delta h=$ $\left(V_{i} / A\right)(\beta-3 \alpha) \Delta T$ (b) For a typical system such as a mercury thermometer, why is it a good approximation to ignore the expansion of the shell?

Nathan Prins

Numerade Educator

Problem 35

Review problem. An aluminum pipe, 0.655 $\mathrm{m}$ long at $20.0^{\circ} \mathrm{C}$ and open at both ends, is used as a flute. The pipe is cooled to a low temperature, but then filled with air at $20.0^{\circ} \mathrm{C}$ 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 $5.00^{\circ} \mathrm{C}$ to $20.0^{\circ} \mathrm{C}$ ?

Nathan Prins

Numerade Educator

Problem 36

Two metal bars are made of invar and one is made of aluminum. At $0^{\circ} \mathrm{C}$ , each of the three bars is drilled with two holes 40.0 $\mathrm{cm}$ apart. Pins are put through the holes to assemble the bars into an equilateral triangle. (a) First ignore the expansion of the invar. Find the angle between the invar bars as a function of Celsius temperature. (b) Is your answer accurate for negative as well as positive temperatures? Is it accurate for $0^{\circ} \mathrm{C}$ ? (c) Solve the problem again, including the expansion of the invar. (d) Aluminum melts at $660^{\circ} \mathrm{C}$ and invar at $1427^{\circ} \mathrm{C}$ . Assume the tabulated expansion coefficients are constant. What are the greatest and smallest attainable angles between the invar bars?

Dominador Tan

Numerade Educator

Problem 37

A liquid has a density $\rho .$ (a) Show that the fractional change in density for a change in temperature $\Delta T$ is $\Delta \rho / \rho=-\beta \Delta T$ . What does the negative sign signify? (b) Fresh water has a maximum density of 1.0000 $\mathrm{g} / \mathrm{cm}^{3}$ at $4.0^{\circ} \mathrm{C}$ . At $10.0^{\circ} \mathrm{C},$ its density is $0.9997 \mathrm{g} / \mathrm{cm}^{3} .$ What is $\beta$
for water over this temperature interval?

Mayukh Banik

Numerade Educator

Problem 38

A cylinder is closed by a piston connected to a spring of constant $2.00 \times 10^{3} \mathrm{N} / \mathrm{m}$ (Fig. Pl9.38). With the spring relaxed, the cylinder is filled with 5.00 $\mathrm{L}$ of gas at a pressure of 1.00 atm and a temperature of $20.0^{\circ} \mathrm{C} .$ (a) If the piston has a cross-sectional area of 0.0100 $\mathrm{m}^{2}$ and negligible mass, how high will it rise when the temperature is raised to $250^{\circ} \mathrm{C}^{2}$ (b) What is the pressure of the gas at $250^{\circ} \mathrm{C} ?$

Nathan Prins

Numerade Educator

Problem 39

A vertical cylinder of cross-sectional area $A$ is fitted with a tight-fitting, frictionless piston of mass $m$ (Fig. Pig. 39 ) (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 $n=0.200 \mathrm{mol}, T=400 \mathrm{K}, A=0.00800 \mathrm{m}^{2}$ , and $m=20.0 \mathrm{kg}$ ?

Nathan Prins

Numerade Educator

Problem 40

A bimetallic strip is made of two ribbons of different metals bonded together. (a) First assume the strip is originally straight. As the strip is warmed, 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. Pl9.40a, page 550 ). Derive an expression for the angle of bending $\theta$ 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 $\left(\Delta r=r_{2}-r_{1}\right)$ (b) Show that the angle of bending decreases to zero when $\Delta T$ 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 $\mathrm{P} 19.40 \mathrm{b}$ shows a compact spiral bimetallic strip in a home thermostat. If $\theta$ is interpreted as the angle of additional bending caused by a change in temperature, the equation from part (a) applies to it as well. 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 $2 \Delta r=0.500 \mathrm{mm},$ and the overall length of the spiral strip is $20.0 \mathrm{cm} .$ Find the angle through which the free end of the strip turns when the temperature changes by 1'C. The free end of the strip supports a capsule partly filled with mercury, visible above the strip in Figure Pl9.40b. 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.

Nathan Prins

Numerade Educator

Problem 41

She rectangular plate shown in Figure P19.41 has an area $A_{i}$ equal to $\ell w$ . If the temperature increases by $\Delta T$ , each dimension increases according to the equation $\Delta L=$ $\alpha L_{i} \Delta T,$ where $\alpha$ is the average coefficient of linear expansion. Show that the increase in area is $\Delta A=2 \alpha A_{i} \Delta T$ . What approximation does this expression assume?

Nathan Prins

Numerade Educator

Problem 42

The measurement of the average coefficient of volume expansion for a liquid is complicated because the container also changes size with temperature. Figure P19.42 shows a simple means for overcoming this difficulty. With this apparatus, one arm of a U-tube is maintained at $0^{\circ} \mathrm{C}$ in an ice-water bath, and the other arm is maintained at a different temperature $T_{\mathrm{C}}$ in a constant-temperature bath. The connecting tube is horizontal. (a) Explain how use of this equipment permits determination of $\beta$ for the liquid from measurements of the column heights $h_{0}$ and $h_{t}$ of
the liquid columns in the U-tube, without having to correct for expansion of the apparatus. (b) Derive the expression for $\beta$ in terms of $h_{0}, h_{t},$ and $T_{\mathrm{C}}$

Nathan Prins

Numerade Educator

Problem 43

A copper rod and a steel rod are different in length by 5.00 $\mathrm{cm}$ at $0^{\circ} \mathrm{C}$ . The rods are warmed and cooled together. Is it possible that the length difference remains constant at all temperatures? Explain. Describe the lengths at $0^{\circ} \mathrm{C}$ as precisely as you can. Can you tell which rod is longer? Can you tell the lengths of the rods?

Nathan Prins

Numerade Educator

Problem 44

Review problem. A clock with a brass pendulum has a period of 1.000 s at $20.0^{\circ} \mathrm{C}$ . If the temperature increases to $30.0^{\circ} \mathrm{C},$ (a) by how much does the period change and (b) how much time does the clock gain or lose in one week?

Nathan Prins

Numerade Educator

Problem 45

Review problem. 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 warmed from $0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ if it is composed of (a) copper or (b) aluminum? Assume the average linear expansion coefficients shown in Table 19.1 do not vary between $0^{\circ} \mathrm{C}$ and $100^{\circ} \mathrm{C}$ .

Nathan Prins

Numerade Educator

Problem 46

Review problem. (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 the 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 one-half the surface value?

Nathan Prins

Numerade Educator

Problem 47

Two concrete spans of a 250 -m-long bridge are placed end to end so that no room is allowed for expansion (Fig. P19.47a). If a temperature increase of $20.0^{\circ} \mathrm{C}$ occurs, what is the height $y$ to which the spans rise when they buckle (Fig. Pl9.47b)?

Sean Dougherty

Numerade Educator

Problem 48

Two concrete spans that form a bridge of length $L$ are placed end to end so that no room is allowed for expansion (Fig. Pl9.47a). If a temperature increase of $\Delta T$ occurs, what is the height $y$ to which the spans rise when they buckle (Fig. Pl9.47b)?

Nathan Prins

Numerade Educator

Problem 49

(a) Show that the density of an ideal gas occupying a volume $V$ is given by $\rho=P M / R T,$ where $M$ is the mola mass. (b) Determine the density of oxygen gas at atmspheric pressure and $20.0^{\circ} \mathrm{C}$ .

Sean Dougherty

Numerade Educator

Problem 50

(a) Take the definition of the coefficient of volume expansion to be
$$\beta=\left.\frac{1}{V} \frac{d V}{d T}\right|_{P=\text { constant }}=\frac{1}{V} \frac{\partial V}{\partial T}$$
Use the equation of state for an ideal gas to show that the coefficient of volume expansion for an ideal gas at constant pressure is given by $\beta=1 / T,$ where $T$ is the absolute temperature. (b) What value does this expression predict for $\beta$ at $0^{\circ} \mathrm{C}$ ? State how this result compares
with the experimental values for helium and air in Table 19.1. Notice that these values are much larger than the coefficients of volume expansion for most liquids and solids.

Nathan Prins

Numerade Educator

Problem 51

Starting with Equation 19.10 , show that the total pressure $P$ in a container filled with a mixture of several ideal gases is $P=P_{1}+P_{2}+P_{3}+\ldots,$ where $P_{1}, P_{2}, \ldots,$ 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.

Sean Dougherty

Numerade Educator

Problem 52

Review problem. Following a collision in outer space, a copper disk at $850^{\circ} \mathrm{C}$ is rotating about its axis with an angular speed of 25.0 $\mathrm{rad} / \mathrm{s}$ . As the disk radiates infrared light, its temperature falls to $20.0^{\circ} \mathrm{C} .$ No external torque acts on the disk. (a) Does the angular speed change as the disk cools? Explain how it changes or why it does not. (b) What is its angular speed at the lower temperature?

Nathan Prins

Numerade Educator

Problem 53

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 $5 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ . Suggestion: You may consider a steel shell of radius $r$ and thickness $t$ having the density of iron and containing helium at high pressure and on the verge of breaking apart into two hemispheres.

Rashmi Sinha

Numerade Educator

Problem 54

A cylinder that has a 40.0 -cm radius and is 50.0 $\mathrm{cm}$ deep is filled with air at $20.0^{\circ} \mathrm{C}$ and 1.00 $\mathrm{atm}$ (Fig. Pl9.54a). A 20.0 -kg piston is now lowered into the cylinder, compressing the air trapped inside as it takes equilibrium height $h_{i}$ (Fig. Pl9.54b). Finally, a $75.0-\mathrm{kg}$ dog stands on the piston, further compressing the air, which remains at $20^{\circ} \mathrm{C}$ (Fig. $\mathrm{P} 19.54 \mathrm{c} ) .$ (a) How far down $(\Delta h)$ does the piston move when the dog steps onto it? (b) To what temperature should the gas be warmed to raise the piston and dog back to $h_{i} ?$

Nathan Prins

Numerade Educator

Problem 55

The relationship $L_{f}=L_{i}(1+\alpha \Delta T)$ is a valid approxima-tion when the average coefficient of expansion is small. If $\alpha$ is large, one must integrate the relationship $d L / d T=$ $\alpha L$ 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 $\mathrm{m}$ and a temperature change of $100.0^{\circ} \mathrm{C}$ , determine the error caused by the approximation when $\alpha=2.00 \times 10^{-5} \mathrm{C}^{\circ} \mathrm{C} )^{-1}$ (a typical value for a metal) and when $\alpha=0.0200\left(^{\circ} \mathrm{C}\right)^{-1}$ (an unrealistically large value for comparison).

Nathan Prins

Numerade Educator

Problem 56

A steel wire and a copper wire, each of diameter $2.000 \mathrm{mm},$ are joined end to end. At $40.0^{\circ} \mathrm{C}$ , each has an unstretched length of $2.000 \mathrm{m} .$ The wires are connected between two fixed supports 4.000 $\mathrm{m}$ apart on a tabletop. The steel wire extends from $x=-2.000 \mathrm{m}$ to $x=0$ , the copper wire extends from $x=0$ to $x=2.000 \mathrm{m},$ and the tension is negligible. The temperature is then lowered to $20.0^{\circ} \mathrm{C}$ . At this lower temperature, find the tension in the wire and the $x$ coordinate of the junction between the wires. (Refer to Tables 12.1 and $19.1 . )$

Mayukh Banik

Numerade Educator

Problem 57

Review problem. A guitar string made of steel with a diameter of 1.00 $\mathrm{mm}$ is stretched between supports 80.0 $\mathrm{cm}$ apart. The temperature is $0.0^{\circ} \mathrm{C}$ . (a) Find the mass per unit length of this string. (Use the value $7.86 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ for the density.) (b) The fundamental frequency of transverse oscillations of the string is 200 $\mathrm{Hz}$ . What is the tension in the string? (c) The temperature is raised to $30.0^{\circ} \mathrm{C}$ . Find the resulting values of the tension and the fundamental frequency. Assume both the Young's modulus (Table 12.1$)$ and the average coefficient of expansion (Table 19.1$)$ have constant values between $0.0^{\circ} \mathrm{C}$ and $30.0^{\circ} \mathrm{C} .$

Nathan Prins

Numerade Educator

Problem 58

In a chemical processing plant, a reaction chamber of fixed volume $V_{0}$ is connected to a reservoir chamber of fixed volume 4$V_{0}$ by a passage containing a thermally insulating porous plug. The plug permits the chambers to be at different temperatures. It 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 $27.0^{\circ} \mathrm{C}$ . Intake and exhaust valves to the pair of chambers are closed. The reservoir is maintained at $27.0^{\circ} \mathrm{C}$ while the reaction chamber is warmed to $400^{\circ} \mathrm{C}$ . What is the pressure in both chambers after these temperatures are achieved?

Nathan Prins

Numerade Educator

Problem 59

A $1.00-\mathrm{km}$ steel railroad rail is fastened securely at both ends when the temperature is $20.0^{\circ} \mathrm{C}$ . As the temperature increases, the rail buckles, taking the shape of an arc of a vertical circle. Find the height $h$ of the center of the rail when the temperature is $25.0^{\circ} \mathrm{C}$ . You will need to solve a transcendental equation.

Sean Dougherty

Numerade Educator

Problem 60

Review problem. A perfectly plane house roof makes an angle $\theta$ with the horizontal. When its temperature changes, between $T_{c}$ before dawn each day and $T_{h}$ in the middle of each afternoon, the roof expands and contracts uniformly with a coefficient of thermal expansion $\alpha_{1}$ . Resting on the roof is a flat, rectangular metal plate with expansion coefficient $\alpha_{2},$ greater than $\alpha_{1}$ . The length of the plate is $L,$ measured along 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 $\mu_{k}$ . 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 called 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 frictional 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
$$\frac{L}{2}\left(1-\frac{\tan \theta}{\mu_{k}}\right)$$
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
$$\frac{L}{\mu_{k}}\left(\alpha_{2}-\alpha_{1}\right)\left(T_{h}-T_{c}\right) \tan \theta$$
(d) Evaluate the distance an aluminum plate moves each day if its length is 1.20 $\mathrm{m}$ , the temperature cycles between $4.00^{\circ} \mathrm{C}$ and $36.0^{\circ} \mathrm{C},$ and the roof has slope $18.5^{\circ},$ coeffcient of linear expansion $1.50 \times 10^{-5}\left(^{\circ} \mathrm{C}\right)^{-1},$ and coeffcient 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?

Dominador Tan

Numerade Educator