Solved: Two small containers, each with a volume of 100

On a very cold day in upstate New York, the temperature is 225C, which is equivalent to what temperature in Fahrenheit? (a) 246F (b) 277F (c) 18F (d) 298 K (e) 213F

A hole is drilled in a metal plate. When the metal is heated, what happens to the diameter of the hole? (a) It decreases. (b) It increases. (c) It remains the same. (d) The answer depends on the initial temperature of the metal. (e) None of these

The average coefficient of linear expansion of copper is 17 3 1026 (C)21. The Statue of Liberty is 93 m tall on a summer morning when the temperature is 25C. Assume the copper plates covering the statue are mounted edge to edge without expansion joints and do not buckle or bind on the framework supporting them as the day grows hot. What is the order of magnitude of the statues increase in height if the temperature increases by 5C? (a) 0.1 mm (b) 1 mm (c) 1 cm (d) 10 cm (e) 1 m

Convert 162F to the equivalent temperature in Kelvin. (a) 373 K (b) 288 K (c) 345 K (d) 201 K (e) 308 K

One way to cool a gas is to let it expand. When a certain gas under a pressure of 5.00 3 106 Pa at 25.0C is allowed to expand to 3.00 times its original volume, its final pressure is 1.07 3 106 Pa. What is its final temperature? (a) 177C (b) 233 K (c) 212 K (d) 191 K (e) 115 K

If the volume of an ideal gas is doubled while its temperature is quadrupled, does the pressure (a) remain the same, (b) decrease by a factor of 2, (c) decrease by a factor of 4, (d) increase by a factor of 2, or (e) increase by a factor of 4?

A container holds 0.50 m3 of oxygen at an absolute pressure of 4.0 atm. A valve is opened, allowing the gas to drive a piston, increasing the volume of the gas until the pressure drops to 1.0 atm. If the temperature remains constant, what new volume does the gas occupy? (a) 1.0 m3 (b) 1.5 m3 (c) 2.0 m3 (d) 0.12 m3 (e) 2.5 m3

What is the internal energy of 26.0 g of neon gas at a temperature of 152C? (a) 2 440 J (b) 6 830 J (c) 3 140 J (d) 5 870 J (e) 5 020 J

Suppose you empty a tray of ice cubes into a bowl partly full of water and cover the bowl. After one-half hour, the contents of the bowl come to thermal equi- MULTIPLE-CHOICE QUESTIONS The multiple-choice questions in this chapter may be assigned online in Enhanced WebAssign. librium, with more liquid water and less ice than you started with. Which of the following is true? (a) The temperature of the liquid water is higher than the temperature of the remaining ice. (b) The temperature of the liquid water is the same as that of the ice. (c) The temperature of the liquid water is less than that of the ice. (d) The comparative temperatures of the liquid water and ice depend on the amounts present.

Which of the assumptions below is not made in the kinetic theory of gases? (a) The number of molecules is very small. (b) The molecules obey Newtons laws of motion. (c) The collisions between molecules are elastic. (d) The gas is a pure substance. (e) The average separation between molecules is large compared with their dimensions.

Suppose for a brief moment the gas molecules hitting a wall stuck to the wall instead of bouncing off the wall. How would the pressure on the wall be affected during that brief time? (a) The pressure would be zero. (b) The pressure would be halved. (c) The pressure would remain unchanged. (d) The pressure would double. (e) The answer would depend on the area of the wall.

If the temperature of an ideal gas is increased from 200 K to 600 K, what happens to the rms speed of the molecules? (a) It increases by a factor of 3. (b) It remains the same. (c) It is one-third of the original speed. (d) It is !3 times the original speed. (e) It increases by a factor of 6.

A rubber balloon is filled with 1 L of air at 1 atm and 300 K and is then cooled to 100 K. The rubber remains flexible as it cools. (i) What happens to the volume of the balloon? (a) It decreases to 16 L. (b) It decreases to 13 L. (c) It decreases to L/!3. (d) It is constant. (e) It increases. (ii) What happens to the pressure of the air in the balloon? (a) It decreases to 16 atm. (b) It decreases to 13 atm. (c) It decreases to 1/!3 atm. (d) It is constant. (e) It increases.

Two cylinders A and B at the same temperature contain the same quantity of the same kind of gas. Cylinder A has three times the volume of cylinder B. What can you conclude about the pressures the gases exert? (a) We can conclude nothing about the pressures. (b) The pressure in A is three times the pressure in B. (c) The pressures must be equal. (d) The pressure in A must be one-third the pressure in B.

A brass ring of diameter 10.00 cm at 20.0C is heated and slipped over an aluminum rod of diameter 10.01 cm at 20.0C. Assuming the average coefficients 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?

A solid substance has a density r0 at a temperature T0. If its temperature is increased by an amount DT, show that its density at the higher temperature is given by r 5 r0 1 1 bDT

Lead has a density of 11.3 3 103 kg/m3 at 0C. (a) What is the density of lead at 90C? (b) Based on your answer to part (a), now consider a situation in which you plan to invest in a gold bar. Would you be better off buying it on a warm day? Explain.

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 steel wire with this length and a cross-sectional area of 4.00 3 1026 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 35.0C. (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 210.0C, what is the tension in the wire? Take Youngs modulus for steel to be 20.0 3 1010 N/m2. (b) Permanent deformation occurs if the stress in the steel exceeds its elastic limit of 3.00 3 108 N/m2. At what temperature would the wire reach its elastic limit? (c) Explain how your answers to (a) and (b) would change if the Golden Gate Bridge were twice as long.

An underground gasoline tank can hold 1.00 3 103 gallons of gasoline at 52.0F. If the tank is being filled on a day when the outdoor temperature (and the gasoline in a tanker truck) is 95.0F, how many gallons from the truck can be poured into the tank? Assume the temperature of the gasoline quickly cools from 95.0F to 52.0F upon entering the tank.

Show that the coefficient of volume expansion, b, is related to the coefficient of linear expansion, a, through the expression b 5 3a.

A hollow aluminum cylinder 20.0 cm deep has an internal capacity of 2.000 L at 20.0C. It is completely filled with turpentine at 20.0C. The turpentine and the aluminum cylinder are then slowly warmed together to 80.0C. (a) How much turpentine overflows? (b) What is the volume of the turpentine remaining in the cylinder at 80.0C? (c) If the combination with this amount of turpentine is then cooled back to 20.0C, how far below the cylinders rim does the turpentines surface recede?

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 21.2C, what is the measured length when the temperature rises to 29.4C? Note: Dont neglect the expansion of the tape.

The band in Figure P10.23 is stainless steel (coefficient of linear expansion 5 17.3 3 1026 C21; Youngs modulus 5 18 3 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 fits snugly over the tooth when heated to a temperature of 80C, what is the tension in the band when it cools to a temperature of 37C?

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 273C to 135C. (a) How much does the steel pipeline expand due to the difference in temperature? (b) How can one compensate for this expansion?

The average coefficient of volume expansion for carbon tetrachloride is 5.81 3 1024 (C)21. If a 50.0-gal steel container is filled completely with carbon tetrachloride when the temperature is 10.0C, how much will spill over when the temperature rises to 30.0C?

The density of gasoline is 7.30 3 102 kg/m3 at 0C. Its average coefficient of volume expansion is 9.60 3 1024(C)21, and note that 1.00 gal 5 0.003 80 m3. (a) Calculate the mass of 10.0 gal of gas at 0C. (b) If 1.000 m3 of gasoline at 0C is warmed by 20.0C, calculate its new volume. (c) Using the answer to part (b), calculate the density of gasoline at 20.0C. (d) Calculate the mass of 10.0 gal of gas at 20.0C. (e) How many extra kilograms of gasoline would you get if you bought 10.0 gal of gasoline at 0C rather than at 20.0C from a pump that is not temperature compensated?

Figure P10.27 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 30.0C. Determine the gap width when the temperature is 190C.

The concrete sections of a certain superhighway are designed to have a length of 25.0 m. The sections are poured and cured at 10.0C. What minimum spacing should the engineer leave between the sections to eliminate buckling if the concrete is to reach a temperature of 50.0C? 1 4 Macroscopic Description of an Ideal Gas

One mole of oxygen gas is at a pressure of 6.00 atm and a temperature of 27.0C. (a) If the gas is heated at constant volume until the pressure triples, what is the final temperature? (b) If the gas is heated so that both the pressure and volume are doubled, what is the final temperature?

A 20.0-L tank of carbon dioxide gas (CO2) is at a pressure of 9.50 3 105 Pa and temperature of 19.0C. (a) Calculate the temperature of the gas in Kelvin. (b) Use the ideal gas law to calculate the number of moles of gas in the tank. (c) Use the periodic table to compute the molecular weight of carbon dioxide, expressing it in grams per mole. (d) Obtain the number of grams of carbon dioxide in the tank. (e) A fire breaks out, raising the ambient temperature by 224.0 K while 82.0 g of gas leak out of the tank. Calculate the new temperature and the number of moles of gas remaining in the tank. (f) Using a technique analogous to that in Example 10.6b, find a symbolic expression for the final pressure, neglecting the change in volume of the tank. (g) Calculate the final pressure in the tank as a result of the fire and leakage.

(a) An ideal gas occupies a volume of 1.0 cm3 at 20C 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 3 10211 Pa (an extremely good vacuum) while the temperature remains constant, how many moles of gas remain in the container?

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

Gas is confined in a tank at a pressure of 11.0 atm and a temperature of 25.0C. If two-thirds of the gas is withdrawn and the temperature is raised to 75.0C, what is the new pressure of the gas remaining in the tank?

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

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 filled at atmospheric pressure and 300 K, what is its radius at liftoff?

The density of helium gas at 0C is r0 5 0.179 kg/m3. The temperature is then raised to T 5 100C, but the pressure is kept constant. Assuming the helium is an ideal gas, calculate the new density rf of the gas.

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 the temperature and the number of air molecules in the bubble remain constant during its ascent.

The ideal gas law can be recast in terms of the density of a gas. (a) Use dimensional analysis to find an expression for the density r of a gas in terms of the number of moles n, the volume V, and the molecular weight M in kilograms per mole. (b) With the expression found in part (a), show that P 5 r M RT for an ideal gas. (c) Find the density of the carbon dioxide atmosphere at the surface of Venus, where the pressure is 90.0 atm and the temperature is 7.00 3 102 K. (d) Would an evacuated steel shell of radius 1.00 m and mass 2.00 3 102 kg rise or fall in such an atmosphere? Why? 1 5 The Kinetic Theory of Gases

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

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

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

Two gases in a mixture pass through a filter at rates proportional to the gases rms speeds. (a) Find the ratio of speeds for the two isotopes of chlorine, 35Cl and 37Cl, as they pass through the air. (b) Which isotope moves faster?

At what temperature would the rms speed of helium atoms equal (a) the escape speed from Earth, 1.12 3 104 m/s and (b) the escape speed from the Moon, 2.37 3 103 m/s? (See Chapter 7 for a discussion of escape speed.) Note: The mass of a helium atom is 6.64 3 10227 kg.

A 7.00-L vessel contains 3.50 moles of ideal gas at a pressure of 1.60 3 106 Pa. Find (a) the temperature of the gas and (b) the average kinetic energy of a gas molecule in the vessel. (c) What additional information would you need if you were asked to find the average speed of a gas molecule?

Superman leaps in front of Lois Lane to save her from a volley of bullets. In a 1-minute interval, an automatic weapon fires 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 3 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, find the pressure exerted on the wall. (The mass of one N2 molecule is 4.68 3 10226 kg.) Additional Problems

Inside the wall of a house, an L-shaped section of hot-water pipe consists of three parts: a straight horizontal piece h 5 28.0 cm long, an elbow, and a straight, vertical piece , 5 134 cm long (Fig. P10.47). A stud and a second-story floorboard 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 flow is turned on, raising the temperature of the pipe from 18.0C to 46.5C.

The active element of a certain laser is made of a glass rod 30.0 cm long and 1.50 cm in diameter. Assume the average coefficient of linear expansion of the glass is 9.00 3 1026 (C)21. If the temperature of the rod increases by 65.0C, what is the increase in (a) its length, (b) its diameter, and (c) its 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 20.0C, what volume does the gas occupy?

Consider an object with any one of the shapes displayed in Table 8.1. What is the percentage increase in the moment of inertia of the object when it is warmed from 0C to 100C if it is composed of (a) copper or (b) aluminum? Assume the average linear expansion coefficients shown in Table 10.1 do not vary between 0C and 100C. (c) Why are the answers for parts (a) and (b) the same for all the shapes?

A steel beam being used in the construction of a skyscraper has a length of 35.000 m when delivered on a cold day at a temperature of 15.000F. What is the length of the beam when it is being installed later on a warm day when the temperature is 90.000F?

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 245.0C, 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. P10.54). (a) If n moles of an ideal gas are in the cylinder at a temperature of T, use Newtons second law for equilibrium to show that the height h at which the piston is in equilibrium under its own weight is given by h 5 nRT mg 1 P0A where P0 is atmospheric pressure. (b) Is the pressure inside the cylinder less than, equal to, or greater than atmospheric pressure? (c) If the gas in the cylinder is warmed, how would the answer for h be affected?

A flask made of Pyrex is calibrated at 20.0C. It is filled to the 100-mL mark on the flask with 35.0C acetone. (a) What is the volume of the acetone when both it and the flask cool to 20.0C? (b) Would the temporary increase in the Pyrex flasks volume make an appreciable difference in the answer? Why or why not?

The pressure gauge on a cylinder of gas registers the gauge pressure, which is the difference between the interior and exterior pressure, P0. When the cylinder is full, the mass of the gas in it is mi at a gauge pressure of Pi. Assuming the temperature of the cylinder remains constant, use the ideal gas law and a relationship between moles and mass to show that the mass of the gas remaining in the cylinder when the gauge pressure reading is Pf is given by mf 5 mi a Pf 1 P0 Pi 1 P0 B

A liquid with a coefficient of volume expansion of b just fills a spherical flask of volume V0 at temperature Ti (Fig. P10.57). The flask is made of a material that has a coefficient of linear expansion of a. 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 DT, the liquid rises in the capillary by the amount Dh 5 (V0/A)(b 2 3a)DT. (b) For a typical system, such as a mercury thermometer, why is it a good approximation to neglect the expansion of the flask?

Before beginning a long trip on a hot day, a driver inflates an automobile tire to a gauge pressure of 1.80 atm at 300 K. At the end of the trip, the gauge pressure has increased to 2.20 atm. (a) Assuming 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 the temperature remains at the value found in part (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. P10.59a). If the temperature increases by 20.0C, what is the height y to which the spans rise when they buckle (Fig. P10.59b)?

An expandable cylinder has its top connected to a spring with force constant 2.00 3 103 N/m (Fig. P10.60). The cylinder is filled with 5.00 L of gas with the spring relaxed at a pressure of 1.00 atm and a temperature of 20.0C. (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 250C? (b) What is the pressure of the gas at 250C?

A bimetallic strip of length L 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. P10.61). Derive an expression for the angle of bending, u, 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 (Dr 5 r2 2 r1). (b) Show that the angle of bending goes to zero when DT goes to zero and also when the two average coefficients of expansion become equal. (c) What happens if the strip is cooled?

A 250-m-long bridge is improperly designed so that it cannot expand with temperature. It is made of con crete with a 5 12 3 1026 C21. (a) Assuming the maximum change in temperature at the site is expected to be 20C, find 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 DT with its ends firmly fixed is given by aY DT. (c) If the maximum stress the bridge can withstand without crumbling is 2.0 3 107 Pa, will it crumble because of this temperature increase? Youngs modulus for concrete is about 2.0 3 1010 Pa.

Following a collision in outer space, a copper disk at 850C is rotating about its axis with an angular speed of 25.0 rad/s. As the disk radiates infrared light, its temperature falls to 20.0C. 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?

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