Chapter 17 Temperature and the Kinetic Theory of Gases

Transcription

1 Chapter 7 emperature and the Kinetic heory of Gases Conceptual roblems rue or false: (a) he zeroth law of thermodynamics states that two objects in thermal equilibrium with each other must be in thermal equilibrium with a third object. (b) he ahrenheit and Celsius temperature scales differ only in the choice of the ice-point temperature. (c) he Celsius degree and the kelin are the same size. (a) alse. If two objects are in thermal equilibrium with a third, then they are in thermal equilibrium with each other. (b) alse. he ahrenheit and Celsius temperature scales also differ in the number of interals between the ice-point temperature and the steam-point temperature. (c) rue. How can you determine if two objects are in thermal equilibrium with each other when putting them into physical contact with each other would hae undesirable effects? (or example, if you put a piece of sodium in contact with water there would be a iolent chemical reaction.) Determine the Concept ut each in thermal equilibrium with a third body; that is, a thermometer. If each body is in thermal equilibrium with the third, then they are in thermal equilibrium with each other. [SS] Yesterday I woke up and it was 0 º in my bedroom, said ert to his old friend ort. hat s nothing, replied ort. y room was 5.0 ºC. Who had the colder room, ert or ort? icture the roblem We can decide which room was colder by conerting 0 to the equialent Celsius temperature. Using the ahrenheit-celsius conersion, conert 0 to the equialent Celsius temperature: 5 ( ) ( 0 ) 5 tc 9 t C so ert' s room was colder. 7

2 74 Chapter 7 4 wo identical essels contain different ideal gases at the same pressure and temperature. It follows that (a) the number of gas molecules is the same in both essels, (b) the total mass of gas is the same in both essels, (c) the aerage speed of the gas molecules is the same in both essels, (d) None of the aboe. Determine the Concept Because the essels are identical (hae the same olume) and the two ideal gases are at the same pressure and temperature, the ideal-gas law ( V Nk ) tells us that the number of gas molecules must be the same in both is correct. essels. ( a) 5 [SS] igure 7-8 shows a plot of olume ersus absolute temperature for a process that takes a fixed amount of an ideal gas from point A to point B. What happens to the pressure of the gas during this process? Determine the Concept rom the ideal-gas law, we hae nr V. In the process depicted, both the temperature and the olume increase, but the temperature increases faster than does the olume. Hence, the pressure increases. 6 igure 7-9 shows a plot of pressure ersus absolute temperature for a process that takes a sample of an ideal gas from point A to point B. What happens to the olume of the gas during this process? Determine the Concept rom the ideal-gas law, we hae V nr. In the process depicted, both the temperature and the pressure increase, but the pressure increases faster than does the temperature. Hence, the olume decreases. 7 If a essel contains equal amounts, by mass, of helium and argon, which of the following are true? (a) (b) (c) (d) he partial pressure exerted by each of the two gases on the walls of the container is the same. he aerage speed of a helium atom is the same as that of an argon atom. he number of helium atoms and argon atoms in the essel are equal. None of the aboe. Determine the Concept (a) alse. he partial pressure exerted by each gas in the mixture is the pressure it would exert if it alone occupied the container. Because the densities of helium and argon are not the same, these gases occupy different olumes and, hence, their partial pressures are not the same. (b) alse. Assuming the gasses hae been in the essel for some time, their aerage kinetic energies would be the same. Because the densities of helium and

3 emperature and the Kinetic heory of Gases 75 argon are not the same, their aerage speeds must be different. (c) alse. We know that the olumes of the two gasses are equal (they both occupy the full olume of the container) and that their temperatures are equal. Because their pressures are different, the number of atoms of each gas must be different, too. (d) Because none of the aboe is true, (d) is true. 8 By what factor must the absolute temperature of a gas be increased to double the rms speed of its molecules? Determine the Concept We can use rms R a gas to the rms speed of its molecules. to relate the temperature of Express the dependence of the rms speed of the molecules of a gas on their absolute temperature: R rms where R is the gas constant, is the molar mass, and is the absolute temperature. Because rms, the temperature must be quadrupled in order to double the rms speed of the molecules. 9 wo different gases are at the same temperature. What can be said about the aerage translational kinetic energies of the molecules? What can be said about the rms speeds of the gas molecules? Determine the Concept he aerage kinetic energies are equal. he ratio of their rms speeds is equal to the square root of the reciprocal of the ratio of their molecular masses. 0 A essel holds a mixture of helium (He) and methane (CH 4 ). he ratio of the rms speed of the He atoms to that of the CH 4 molecules is (a), (b), (c) 4, (d) 6 icture the roblem We can express the rms speeds of the helium atoms and the methane molecules using R. rms Express the rms speed of the helium atoms: rms, He R He

4 76 Chapter 7 Express the rms speed of the methane molecules: rms.ch 4 R CH 4 Diide the first of these equations by the second to obtain: Use Appendix C to find the molar masses of helium and methane: rms, He CH 4 rms.ch 4 He rms, He rms.ch 4 6g/mol 4g/mol and (b) is correct. rue or false: If the pressure of a fixed amount of gas increases, the temperature of the gas must increase. alse. Whether the pressure changes also depends on whether and how the olume changes. In an isothermal process, the pressure can increase while the olume decreases and the temperature is constant. Why might the Celsius and ahrenheit scales be more conenient than the absolute scale for ordinary, nonscientific purposes? Determine the Concept or the Celsius scale, the ice point (0 C) and the boiling point of water at atm (00 C) are more conenient than 7 K and 7 K; temperatures in roughly this range are normally encountered. On the ahrenheit scale, the temperature of warm-blooded animals is roughly 00 ; this may be a more conenient reference than approximately 00 K. hroughout most of the world, the Celsius scale is the standard for nonscientific purposes. An astronomer claims that the temperature at the center of the Sun is about 0 7 degrees. Do you think that this temperature is in kelins, degrees Celsius, or doesn t it matter? Determine the Concept Because t C K and 0 7 >> 7, it doesn t matter. 4 Imagine that you hae a fixed amount of ideal gas in a container that expands to maintain constant pressure. If you double the absolute temperature of the gas, the aerage speed of the molecules (a) remains constant, (b) doubles, (c) quadruples, (d) increases by a factor of. Determine the Concept he aerage speed of the molecules in an ideal gas depends on the square root of the kelin temperature. Because, doubling a

5 emperature and the Kinetic heory of Gases 77 the temperature while maintaining constant pressure increases the aerage speed by a factor of. (d) is correct. 5 [SS] Suppose that you compress an ideal gas to half its original olume, while also haling its absolute temperature. During this process, the pressure of the gas (a) hales, (b) remains constant, (c) doubles, (d) quadruples. Determine the Concept rom the ideal-gas law, V nr, haling both the temperature and olume of the gas leaes the pressure unchanged. (b) is correct. 6 he aerage translational kinetic energy of the molecules of a gas depends on (a) the number of moles and the temperature, (b) the pressure and the temperature, (c) the pressure only, (d) the temperature only. Determine the Concept he aerage translational kinetic energy of the molecules of an ideal gas K a depends on its temperature according to K k. (d) is correct. 7 [SS] Which speed is greater, the speed of sound in a gas or the rms speed of the molecules of the gas? Justify your answer, using the appropriate formulas, and explain why your answer is intuitiely plausible. Determine the Concept he rms speed of molecules of an ideal gas is gien by R rms and the speed of sound in a gas is gien by γr sound. a he rms speed of the molecules of an ideal-gas is gien by: rms R he speed of sound in a gas is gien by: sound γr Diide the first of these equations by the second and simplify to obtain: rms sound R γr γ or a monatomic gas, γ.67 and: rms, monatomic sound.67.4

6 78 Chapter 7 or a diatomic gas, γ.40 and: rms, diatomic sound In general, the rms speed is always somewhat greater than the speed of sound. Howeer, it is only the component of the molecular elocities in the direction of propagation that is releant to this issue. In addition, In a gas the mean free path is greater than the aerage intermolecular distance. 8 Imagine that you increase the temperature of a gas while holding its olume fixed. Explain in terms of molecular motion why the pressure of the gas on the walls of its container increases. Determine the Concept he pressure is a measure of the change in momentum per second of a gas molecule on collision with the wall of the container. When the gas is heated, the aerage elocity and the aerage momentum of the molecules increase and, as a consequence, the pressure exerted by the molecules increases. 9 Imagine that you compress a gas while holding it at fixed temperature (perhaps by immersing the container in cool water). Explain in terms of molecular motion why the pressure of the gas on the walls of its container increases. Determine the Concept If the olume decreases the pressure increases because more molecules hit a unit of area of the walls in a gien time. he fact that the temperature does not change tells us the molecular speed does not change with olume. 0 Oxygen has a molar mass of g/mol and nitrogen has a molar mass of 8 g/mol. he oxygen and nitrogen molecules in a room hae (a) (b) (c) (d) (e) (f) equal aerage translational kinetic energies, but the oxygen molecules hae a larger aerage speed than the nitrogen molecules hae. equal aerage translational kinetic energies, but the oxygen molecules hae a smaller aerage speed than the nitrogen molecules hae. equal aerage translational kinetic energies and equal aerage speeds. equal aerage speeds, but the oxygen molecules hae a larger aerage translational kinetic energy than the nitrogen molecules hae. equal aerage speeds, but the oxygen molecules hae a smaller aerage translational kinetic energy than the nitrogen molecules hae. None of the aboe. icture the roblem he aerage kinetic energies of the molecules are gien by K m k Assuming that the room s temperature distribution is a ( ). a

7 emperature and the Kinetic heory of Gases 79 uniform, we can conclude that the oxygen and nitrogen molecules hae equal aerage kinetic energies. Because the oxygen molecules are more massie, they must be moing slower than the nitrogen molecules. (b) is correct. [SS] Liquid nitrogen is relatiely cheap, while liquid helium is relatiely expensie. One reason for the difference in price is that while nitrogen is the most common constituent of the atmosphere, only small traces of helium can be found in the atmosphere. Use ideas from this chapter to explain why it is that only small traces of helium can be found in the atmosphere. Determine the Concept he aerage molecular speed of He gas at 00 K is about.4 km/s, so a significant fraction of He molecules hae speeds in excess of earth s escape elocity (. km/s). hus, they "leak" away into space. Oer time, the He content of the atmosphere decreases to almost nothing. Estimation and Approximation Estimate the total number of air molecules in your classroom. icture the roblem he number of air molecules in your classroom is gien by the ideal-gas law. rom the ideal-gas law, the number of molecules N in a gien olume V at pressure and temperature is gien by: Assuming that a typical classroom is a rectangular parallelepiped, its olume is gien by: V N k where k is Boltzmann s constant. V wh Substituting for V yields: N wh k Assume atmospheric pressure, room temperature and that the room is 5 m square and 5 m high to obtain: N 0.5 ka atm ( 5 m)( 5 m)( 5 m) atm (.8 0 J/K)( 9 K) 8 0

8 70 Chapter 7 Estimate the density of dry air at sea leel on a warm summer day. icture the roblem We can use the definition of mass density, the definition of the molar mass of a gas, and the ideal-gas law to estimate the density of air at sea leel on a warm day. he density of air at sea leel is gien by: he mass of the air molecules occupying a gien olume is the product of the number of moles n and the molar mass of the molecules. Substitute for m to obtain: rom the ideal-gas law, the number of moles in a gien olume depends on the pressure and temperature according to: Substitute in the expression for ρ and simplify to obtain: Assuming atmospheric pressure, a temperature of 7 C (9 ), and 9 g/mol for the molar mass of air, substitute numerical alues and ealuate ρ: m ρ V n ρ V n ρ ρ V R V V R atm R ( 9 g/mol) atm ( 8.4 J/mol K)( 00 K). kg/m 0.5 ka 4 A stoppered test tube that has a olume of 0.0 ml has.00 ml of water at its bottom. he water has a temperature of 00ºC and is initially at a pressure of.00 atm. he test tube is held oer a flame until the water has completely boiled away. Estimate the final pressure inside the test tube. icture the roblem Assuming the steam to be an ideal gas at a temperature of 7 K, we can use the ideal-gas law to estimate the pressure inside the test tube when the water is completely boiled away. Using the ideal-gas law, relate the pressure inside the test tube to its olume and the temperature: Nk V

9 emperature and the Kinetic heory of Gases 7 Relate the number of particles N to the mass of water, its molar mass, and Aogadro s number N A : Relate the mass of.00 ml of water to its density: m N N m N N A m ρv.00g A (.00 0 kg/m ) 6 (.00 0 m ) Substitute for m, N A, and (8 g/mol) and ealuate N: N particles/mol (.00g) 8g/mol.46 0 particles Substitute numerical alues and ealuate : (.46 0 particles)(.8 0 J/K)( 7K) atm m 0.0 ml ml atm N/m N/m 5 [SS] In Chapter, we found that the escape speed at the surface of a planet of radius R is e gr, where g is the acceleration due to graity at the surface of the planet. If the rms speed of a gas is greater than about 5 to 0 percent of the escape speed of a planet, irtually all of the molecules of that gas will escape the atmosphere of the planet. (a) (b) (c) (d) At what temperature is rms for O equal to 5 percent of the escape speed for Earth? At what temperature is rms for H equal to 5 percent of the escape speed for Earth? emperatures in the upper atmosphere reach 000 K. How does this help account for the low abundance of hydrogen in Earth s atmosphere? Compute the temperatures for which the rms speeds of O and H are equal to 5 percent of the escape speed at the surface of the moon, where g is about one-sixth of its alue on Earth and R 78 km. How does this account for the absence of an atmosphere on the moon? icture the roblem We can find the escape temperatures for the earth and the moon by equating, in turn, 0.5 e and rms of O and H. We can compare these

10 7 Chapter 7 temperatures to explain the absence from Earth s upper atmosphere and from the surface of the moon. See Appendix C for the molar masses of O and H. (a) Express rms for O : R rms, O O where R is the gas constant, is the absolute temperature, and O is the molar mass of oxygen. Equate 0.5 e and rms, O : 0.5 gr earth R Sole for to obtain: 0.045gRearth () R Substitute numerical alues and ealuate for O : 6 ( )( m)(.0 0 kg/mol) m/s.60 0 K 8.4 J/mol ( K) (b) Substitute numerical alues and ealuate for H : 6 ( )( m)(.0 0 kg/mol) m/s ( K) 8.4 J/mol 0K (c) Because hydrogen is lighter than air it rises to the top of the atmosphere. Because the temperature is high there, a greater fraction of the molecules reach escape speed. (d) Express equation () at the surface of the moon: 0.045gmoonRmoon R 0.045( 6 gearth ) Rmoon R 0.005gearthRmoon R Substitute numerical alues and ealuate for O : m/s 6 ( )(.78 0 m)(.0 0 kg/mol) 60K 8.4 J/mol K

11 Substitute numerical alues and ealuate for H : emperature and the Kinetic heory of Gases 7 6 ( )(.78 0 m)(.0 0 kg/mol) 0K m/s 8.4 J/mol K Because g is less on the moon, the escape speed is lower. hus, a larger percentage of the molecules are moing at escape speed. 6 he escape speed for gas molecules in the atmosphere of ars is 5.0 km/s and the surface temperature of ars is typically 0ºC. Calculate the rms speeds for (a) H, (b) O, and (c) CO at this temperature. (d) Are H, O, and CO likely to be found in the atmosphere of ars? icture the roblem We can use rms R to calculate the rms speeds of H, O, and CO at 7 K and then compare these speeds to 0% of the escape elocity on ars to decide the likelihood of finding these gases in the atmosphere of ars. See Appendix C for molar masses. Express the rms speed of an atom as a function of the temperature: (a) Substitute numerical alues and ealuate rms for H : rms rms, H R ( K)( 7K) 8.4 J/mol.0 0 kg/mol.84 km/s (b) Ealuate rms for O : ( 8.4 J/mol K)( 7K) rms, O m/s kg/mol (c) Ealuate rms for CO : ( 8.4 J/mol K)( 7K) rms, CO m/s 5 esc 5 kg/mol (d) Calculate 0% of esc for ars: ( 5.0 km/s).0km/s Because is greater than rms CO and O but less than rms for H, O and CO, but not H should be present.

12 74 Chapter 7 7 [SS] he escape speed for gas molecules in the atmosphere of Jupiter is 60 km/s and the surface temperature of Jupiter is typically 50ºC. Calculate the rms speeds for (a) H, (b) O, and (c) CO at this temperature. (d) Are H, O, and CO likely to be found in the atmosphere of Jupiter? icture the roblem We can use rms R to calculate the rms speeds of H, O, and CO at K and then compare these speeds to 0% of the escape elocity on Jupiter to decide the likelihood of finding these gases in the atmosphere of Jupiter. See Appendix C for the molar masses of H, O, and CO. Express the rms speed of an atom as a function of the temperature: (a) Substitute numerical alues and ealuate rms for H : rms rms, H R ( K)( K) 8.4 J/mol.0 0 kg/mol.km/s (b) Ealuate rms for O : ( 8.4 J/mol K)( K) rms, O m/s kg/mol (c) Ealuate rms for CO : ( 8.4 J/mol K)( K) rms, CO m/s 5 esc 5 kg/mol (d) Calculate 0% of esc for Jupiter: ( 60 km/s) km/s Because e is greater than rms for O, CO and H, O, all three gasses should be found on Jupiter. 8 Estimate the aerage pressure on the front wall of a racquetball court, due to the collisions of the ball with the wall during a game. Use any reasonable numbers for the mass of the ball, its typical speed, and the dimensions of the court. Is the aerage pressure from the ball significant compared to that from the air? icture the roblem he aerage pressure exerted by the ball on the wall is the ratio of the aerage force it exerts on the wall to the area of the wall. he aerage force, in turn, is the rate at which the momentum of the ball changes during each collision with the wall. Assume that the mass of a racquetball is 00 g, that the court measures 5 m by 5 m, and that the speed of the racquetball is 0 m/s. We ll

13 emperature and the Kinetic heory of Gases 75 also assume that the interal Δt between collisions with the wall is s. he aerage pressure exerted on the wall by the racquetball is gien by: Assuming head-on collisions with the wall, the change in momentum of the ball during each collision is m and the aerage force exerted by the ball on the wall is: Substituting for a yields: Substitute numerical alues and ealuate a : Express the ratio of a and atm to obtain: a a a A wh where w and h are the width and height of the wall, respectiely. Δp m a Δt Δt where Δt is the elapsed time between collisions of the ball with the wall. a a or a atm m whδt (.kg)( 0 m/s) ( 5 m)( 5 m)( s) 0.a 0.a a 6 a 0 atm 0.08 N/m and the aerage pressure from the ball is not significant compared to atmospheric pressure. 9 o a first approximation, the Sun consists of a gas of equal numbers of protons and electrons. (he masses of these particles can be found in Appendix B.) he temperature at the center of the Sun is about 0 7 K, and the density of the Sun is about 0 5 kg/m. Because the temperature is so high, the protons and electrons are separate particles (rather than being joined together to form hydrogen atoms). (a) Estimate the pressure at the center of the Sun. (b) Estimate the rms speeds of the protons and the electrons at the center of the Sun. icture the roblem We ll assume that we can model the plasma as an ideal gas, at least to a first approximation. hen we can apply the ideal gas law to an arbitrary olume of the material, say one cubic meter. (a) o the extent that the plasma acts like an ideal gas, the pressure at the center of the Sun is gien by: nr V

14 76 Chapter 7 he mass of our m olume of matter is 0 5 kg and this mass consists mostly of protons. One mole of protons has a mass of g, so in 0 5 kg, the number of moles of protons would be: Substitute numerical alues and ealuate : n m 5 0 kg 8 0 mol 0 kg protons protons protons or, counting electrons, 8 n n 0 mol protons 8 7 ( 0 mol)( 8.4 J/mol K)( 0 K) m atm a 0.5 ka atm (b) o one significant figure, the aerage speed of the protons is the same as the rms speed: Substitute numerical alues and ealuate : rms, protons rms, protons rms, protons R protons 7 ( K)( 0 K) 8.4 J/mol 0.00 kg m/s he rms speed of an electron at the center of the Sun is gien by: R rms, electrons electrons 000 R protons Substitute numerical alues and ealuate : rms, electrons rms, electrons 7 ( K)( 0 K) 8.4 J/mol 000 ( 0.00 kg) 0 7 m/s Remarks: he huge pressure we calculated in (a) is required to support the tremendous weight of the rest of the sun. he ery high speed we calculated for the plasma electrons is still an order-of-magnitude smaller than the speed of light. 0 You are designing a acuum chamber for fabricating reflectie coatings. Inside this chamber, a small sample of metal will be aporized so that its atoms trael in straight lines (the effects of graity are negligible during the brief time of flight) to a surface where they land to form a ery thin film. he sample of metal is 0 cm from the surface to which the metal atoms will adhere. How low must the pressure in the chamber be so that the metal atoms only rarely collide with air molecules before they land on the surface?

15 emperature and the Kinetic heory of Gases 77 icture the roblem We can use the expression for the mean free path of a molecule to eliminate the number of molecules per unit olume n V from the idealgas law. Assume that the air in the chamber is at room temperature (00 K) and that the aerage diameter of an air molecule is m. Apply the ideal gas law to express the pressure in terms of the number of molecules per unit olume and the temperature: Nk V n V k he mean free path of an air molecule is gien by: λ n V π d n V λπ d Substitute for n V to obtain: Substitute numerical alues and ealuate : k λπ d (.8 0 J/K)( 00 K) 0 ( 0 cm) π ( 4 0 m) 0 ma [SS] In normal breathing conditions, approximately 5 percent of each exhaled breath is carbon dioxide. Gien this information and neglecting any difference in water-apor content, estimate the typical difference in mass between an inhaled breath and an exhaled breath. icture the roblem One breath (one s lung capacity) is about half a liter. he only thing that occurs in breathing is that oxygen is exchanged for carbon dioxide. Let s estimate that of the 0% of the air that is breathed in as oxygen, ¼ is exchanged for carbon dioxide. hen the mass difference between breaths will be 5% of a breath multiplied by the molar mass difference between oxygen and carbon dioxide and by the number of moles in a breath. Because this is an estimation problem, we ll use g/mol as an approximation for the molar mass of oxygen and 44 g/mol as an approximation for the molar mass of carbon dioxide. Express the difference in mass between an inhaled breath and an Δm m ( ) exhaled breath: CO O CO breath O f where fco mco n is the fraction of the air breathed in that is exchanged for carbon dioxide. he number of moles per breath is gien by: Vbreath n breath.4 L/mol

16 78 Chapter 7 Substituting for n breath yields: Δm f Vbreath ( ) CO O CO.4 L/mol Substitute numerical alues and ealuate Δm: Δm ( 0.05)( 44 g/mol g/mol) ( 0.5 L).4 L/mol 0 4 g emperature Scales A certain ski wax is rated for use between and 7.0 ºC. What is this temperature range on the ahrenheit scale? icture the roblem We can use the fact that 5 C 9 to set up a proportion that allows us to make easy interal conersions from either the Celsius or ahrenheit scale to the other. he proportion relating a temperature range on the ahrenheit scale of a temperature range on the Celsius scale is: Δt Δt C 9 Δt 5 C 9 Δt 5 C C Substitute numerical alues and ealuate Δt : Δt 9 5 C ( 7 C ( C) ) 9 Remarks: An equialent but slightly longer solution inoles conerting the two temperatures to their ahrenheit equialents and then subtracting these temperatures. [SS] he melting point of gold is º. Express this temperature in degrees Celsius. icture the roblem We can use the ahrenheit-celsius conersion equation to find this temperature on the Celsius scale. Conert to the equialent Celsius temperature: t C t ( ) ( ) 06 C 9 4 A weather report indicates that the temperature is expected to drop by 5.0 C oer the next four hours. By how many degrees on the ahrenheit scale will the temperature drop?

17 emperature and the Kinetic heory of Gases 79 icture the roblem We can use the fact that 5 C 9 to set up a proportion that allows us to make easy interal conersions from either the Celsius or ahrenheit scale to the other. he proportion relating a temperature range on the ahrenheit scale of a temperature range on the Celsius scale is: Δt Δt C 9 Δt 5 C 9 Δt 5 C C Substitute numerical alues and ealuate Δt : Δt 9 5 C ( 5.0 C ) he length of the column of mercury in a thermometer is 4.00 cm when the thermometer is immersed in ice water at atm of pressure, and 4.0 cm when the thermometer is immersed in boiling water at atm of pressure. Assume that the length of the mercury column aries linearly with temperature. (a) Sketch a graph of the length of the mercury column ersus temperature (in degrees Celsius). (b) What is the length of the column at room temperature (.0ºC)? (c) If the mercury column is 5.4 cm long when the thermometer is immersed in a chemical solution, what is the temperature of the solution? icture the roblem We can use the equation of the graph plotted in (a) to (b) find the length of the mercury column at room temperature and (c) the temperature of the solution when the height of the mercury column is 5.4 cm. (a) A graph of the length of the mercury column ersus temperature (in degrees Celsius) is shown to the right. he equation of the line is: cm L 0.00 tc cm () C L, cm 0 00 t C, C (b) Ealuate L (.0 C) cm L 0.00 C 8.40 cm (.0 C) cm (c) Sole equation () for t C to obtain: t C 4.00 cm L cm 0.00 C

18 70 Chapter 7 Substitute numerical alues and t 5.4 cm : ealuate ( ) C t C ( 5.4 cm) 5.4 cm 4.00 cm cm 0.00 C 07 C 6 he temperature of the interior of the Sun is about K. What is this temperature in (a) Celsius degrees, (b) kelins, and (c) ahrenheit degrees? icture the roblem We can use the temperature conersion equations 9 t 5 tc + and tc 7.5K to conert 0 7 K to the ahrenheit and Celsius temperatures. Express the kelin temperature in terms of the Celsius temperature: (a) Sole for and ealuate t C : (b) Use the Celsius to ahrenheit conersion equation to ealuate t : t t C t C + 7.5K 7.5 K K 7.5 K 7 C 7 (.0 0 C) he boiling point of nitrogen, N, is 77.5 K. Express this temperature in degrees ahrenheit. icture the roblem While we could conert 77.5 K to a Celsius temperature and then conert the Celsius temperature to a ahrenheit temperature, an alternatie solution is to use the diagram to the right to set up a proportion for the direct conersion of the kelin temperature to its ahrenheit equialent. t K Use the diagram to set up the proportion: t 7.5 K 77.5 K 7.5 K 7.5K or t

19 emperature and the Kinetic heory of Gases 7 Soling for t yields: 95.8 t he pressure of a constant-olume gas thermometer is atm at the ice point and atm at the steam point. (a) Sketch a graph of pressure ersus Celsius temperature for this thermometer. (b) When the pressure is 0.00 atm, what is the temperature? (c) What is the pressure at ºC (the boiling point of sulfur)? icture the roblem We can use the equation of the graph plotted in (a) to (b) find the temperature when the pressure is 0.00 atm and (c) the pressure when the temperature is 444.6ºC. (a) A graph of pressure (in atm) ersus temperature (in degrees Celsius) for this thermometer is shown to the right. he equation of this graph is: atm.46 0 tc C (b) Soling for t C yields: atm t C, atm atm atm.46 0 C t, C C Substitute numerical alues and t 0.00 atm : ealuate ( ) C t C ( 0.00 atm) 0.00 atm atm atm.46 0 C 05 C Substitute numerical alues and ealuate ( C) atm C : ( C).46 0 ( C) atm.05 atm 9 [SS] A constant-olume gas thermometer reads 50.0 torr at the triple point of water. (a) Sketch a graph of pressure s. absolute temperature for this thermometer. (b) What will be the pressure when the thermometer measures a temperature of 00 K? (c) What ideal-gas temperature corresponds to a pressure of 678 torr?

20 7 Chapter 7 icture the roblem We can use the equation of the graph plotted in (a) to (b) find the pressure when the temperature is 00 K and (c) the ideal-gas temperature when the pressure is 678 torr. (a) A graph of pressure (in torr) ersus temperature (in kelins) for this thermometer is shown to the right. he equation of this graph is:, torr torr 7 K () 0 0 7, K (b) Ealuate when 00 K: 50.0 torr 7 K ( 00 K) ( 00 K) 54.9 torr (c) Sole equation () for to obtain: Ealuate ( 678 torr) : 7 K 50.0 torr 7 K 50.0 torr ( 678 torr) ( 678 torr).70 0 K 40 A constant-olume gas thermometer has a pressure of 0.0 torr when it reads a temperature of 7 K. (a) Sketch a graph of pressure s. absolute temperature for this thermometer. (b) What is its triple-point pressure? (c) What temperature corresponds to a pressure of 0.75 torr? icture the roblem We can use the equation of the graph plotted in (a) to (b) find the triple-point pressure and (c) the temperature when the pressure is 0.75 torr.

21 emperature and the Kinetic heory of Gases 7 (a) A graph of pressure ersus absolute temperature for this thermometer is shown to the right. he equation of this graph is: 0.0 torr 7 K (b) Sole for and ealuate the thermometer s triple-point (7.6 K) pressure: (), torr torr 7 K.0 torr 7, K ( 7.6 K) ( 7.6 K) (c) Sole equation () for to obtain: Ealuate ( 0.75 torr) : 7 K 0.0 torr 7 K 0.0 torr ( 0.75 torr) ( 0.75 torr).8 K 4 At what temperature do the ahrenheit and Celsius temperature scales gie the same reading? icture the roblem We can find the temperature at which the ahrenheit and Celsius scales gie the same reading by setting t t C in the temperatureconersion equation. 5 5 Set t t C in t ( t ): t ( t ) C 9 9 Sole for and ealuate t : t C t 40.0 C 40.0 Remarks: If you e not already thought of doing so, you might use your graphing calculator to plot t C ersus t and t t C (a straight line at 45 ) on the same graph. heir intersection is at ( 40, 40). 4 Sodium melts at 7 K. What is the melting point of sodium on the Celsius and ahrenheit temperature scales?

22 74 Chapter 7 icture the roblem We can use the Celsius-to-absolute conersion equation to find 7 K on the Celsius scale and the Celsius-to-ahrenheit conersion equation to find the ahrenheit temperature corresponding to 7 K. Express the absolute temperature as a function of the Celsius temperature: Sole for and ealuate t C : t C t C + 7.5K 7.5 K 7K 7.5 K 98 C Use the Celsius-to-ahrenheit conersion equation to find t : t t 9 5 C ( 97.9 ) + 4 he boiling point of oxygen at.00 atm is 90. K. What is the boiling point of oxygen at.00 atm on the Celsius and ahrenheit scales? icture the roblem We can use the Celsius-to-absolute conersion equation to find 90. K on the Celsius scale and the Celsius-to-ahrenheit conersion equation to find the ahrenheit temperature corresponding to 90. K. Express the absolute temperature as a function of the Celsius temperature: Sole for and ealuate t C : t C t C + 7.5K 7.5K 90.K 7.5K 8 C Use the Celsius-to-ahrenheit conersion equation to find t : t t 9 5 C ( 8 ) + 44 On the Réaumur temperature scale, the melting point of ice is 0ºR and the boiling point of water is 80ºR. Derie expressions for conerting temperatures on the Réaumur scale to the Celsius and ahrenheit scales. icture the roblem We can use the following diagram to set up proportions that will allow us to conert temperatures on the Réaumur scale to Celsius and ahrenheit temperatures.

23 emperature and the Kinetic heory of Gases C R 80 t C t t R 0 0 Referring to the diagram, set up a proportion to conert temperatures on the Réaumur scale to Celsius temperatures: Simplify to obtain: Referring to the diagram, set up a proportion to conert temperatures on the Réaumur scale to ahrenheit temperatures: tc 0 C tr 0 R 00 C 0 C 80 R 0 R t C t R 5 t C 4 tr t tr 0 R 80 R 0 R Simplify to obtain: t t R t t 4 R + 45 [SS] A thermistor is a solid-state deice widely used in a ariety of engineering applications. Its primary characteristic is that its electrical resistance aries greatly with temperature. Its temperature dependence is gien approximately by R R 0 e B/, where R is in ohms (Ω), is in kelins, and R 0 and B are constants that can be determined by measuring R at calibration points such as the ice point and the steam point. (a) If R 760 Ω at the ice point and 5 Ω at the steam point, find R 0 and B. (b) What is the resistance of the thermistor at t 98.6 º? (c) What is the rate of change of the resistance with temperature (dr/d) at the ice point and the steam point? (d) At which temperature is the thermistor most sensitie? icture the roblem We can use the temperature dependence of the resistance of the thermistor and the gien data to determine R 0 and B. Once we know these quantities, we can use the temperature-dependence equation to find the resistance at any temperature in the calibration range. Differentiation of R with respect to will allow us to express the rate of change of resistance with temperature at both the ice point and the steam point temperatures.

24 76 Chapter 7 (a) Express the resistance at the ice point as a function of temperature of the ice point: Express the resistance at the steam point as a function of temperature of the steam point: Diide equation () by equation () to obtain: Sole for B by taking the logarithm of both sides of the equation: B 7 K Ω R0e () B 7K Ω R0e () Ω 48.0 e 5Ω 760 B 7 K B 7 K ln 48. B 7 and ln 48. B K K 7 K K Sole equation () for R 0 and substitute for B: R 0 760Ω B 7K e ( 760Ω).9 0 e.9 0 ( 760Ω) K 7K Ω Ω e B 7K (b) rom (a) we hae: K ( ) R.9 0 Ω e Conert 98.6 to kelins to obtain: 0K Substitute for to obtain: R( 0 K) (.9 0 Ω).kΩ e K 0 K (c) Differentiate R with respect to to obtain: dr d d d B ( R e ) 0 B R 0e B R e 0 B RB d d B

25 emperature and the Kinetic heory of Gases 77 Ealuate dr/d at the ice point: dr ( 760Ω)(.94 0 K) d ice point 89Ω / K ( 7.6 K) Ealuate dr/d at the steam point: dr ( 5Ω)(.94 0 K) d steam point ( 7.6 K) 4.Ω / K (d) he thermistor is more sensitie (has greater sensitiity) at lower temperatures. he Ideal-Gas Law 46 An ideal gas in a cylinder fitted with a piston (igure 7-0) is held at fixed pressure. If the temperature of the gas increases from 50 to 00 C, by what factor does the olume change? icture the roblem Let the subscript 50 refer to the gas at 50 C and the subscript 00 to the gas at 00 C. We can apply the ideal-gas law for a fixed amount of gas to find the ratio of the final and initial olumes. Apply the ideal-gas law for a fixed 00V00 50V50 amount of gas: or, because 00 50, V V Substitute numerical alues and ealuate V 00 /V 50 : V V 50 ( ) ( ) K K 00 or a 5% increase in olume [SS] A 0.0-L essel contains gas at a temperature of 0.00ºC and a pressure of 4.00 atm. How many moles of gas are in the essel? How many molecules? icture the roblem We can use the ideal-gas law to find the number of moles of gas in the essel and the definition of Aogadro s number to find the number of molecules.

26 78 Chapter 7 Apply the ideal-gas law to the gas: Substitute numerical alues and ealuate n: V V nr n R ( 4.00 atm)( 0.0L) ( L atm/mol K)( 7K) n.786 mol.79 mol Relate the number of molecules N in the gas in terms of the number of moles n: N nn A Substitute numerical alues and ealuate N: N 4 (.786 mol)( molecules/mol).08 0 molecules 48 A pressure as low as torr can be achieed using an oil diffusion pump. How many molecules are there in.00 cm of a gas at this pressure if its temperature is 00 K? icture the roblem We can use the ideal-gas law to relate the number of molecules in the gas to its pressure, olume, and temperature. Sole the ideal-gas law for the number of molecules in a gas as a function of its pressure, olume, and temperature: V N k Substitute numerical alues and ealuate N: N 8 6 (.00 0 torr)(. a/torr)(.00 0 m ) (.8 0 J/K)( 00K) You copy the following paragraph from a artian physics textbook: snorf of an ideal gas occupies a olume of.5 zaks. At a temperature of glips, the gas has a pressure of.5 klads. At a temperature of 0 glips, the same gas now has a pressure of 8.7 klads. Determine the temperature of absolute zero in glips. icture the roblem Because the gas is ideal, its pressure is directly proportional to its temperature. Hence, a graph of ersus will be linear and the linear equation relating and can be soled for the temperature corresponding to zero pressure. We ll assume that the data was taken at constant olume.

27 emperature and the Kinetic heory of Gases 79 A graph of, in klads, as function of, in glips, is shown to the right. he equation of this graph is:, klads.5.8 klads klads glips , glips When 0: Sole for 0 to obtain:.8 klads klads glips glips 50 A motorist inflates the tires of her car to a gauge pressure of 80 ka on a day when the temperature is 8.0ºC. When she arries at her destination, the tire pressure has increased to 45 ka. What is the temperature of the tires if we assume that (a) the tires do not expand or (b) that the tires expand so the olume of the enclosed air increases by 7 percent? icture the roblem Let the subscript refer to the tires when their gauge pressure is 80 ka and the subscript to conditions when their gauge pressure is 45 ka. Assume that the air in the tires behaes as an ideal gas. hen, we can apply the ideal-gas law for a fixed amount of gas to relate the temperatures to the pressures and olumes of the tires. (a) Apply the ideal-gas law for a fixed amount of gas to the air in the tires: Sole for : V V () where the temperatures and pressures are absolute. because V V. Substitute numerical alues to obtain: ( 65K) 45ka + 0 ka 80 ka + 0 ka 6.K 5 C

28 740 Chapter 7 (b) Use equation () with V.07 V. Sole for : V 07 V (. ) Substitute numerical alues and ealuate : (.07)( 6.K) 76 C 49.K 5 A room is 6.0 m by 5.0 m by.0 m. (a) If the air pressure in the room is.0 atm and the temperature is 00 K, find the number of moles of air in the room. (b) If the temperature increases by 5.0 K and the pressure remains constant, how many moles of air leae the room? icture the roblem We can apply the ideal-gas law to find the number of moles of air in the room as a function of the temperature. (a) Use the ideal-gas law to relate the number of moles of air in the room to the pressure, olume, and temperature of the air: Substitute numerical alues and ealuate n: V n () R n ( 0.5ka)( 6.0m)( 5.0m)(.0m) ( 8.4 J/mol K)( 00K).66 0 mol.7 0 mol (b) Letting n represent the number of moles in the room when the temperature rises by 5 K, express the number of moles of air that leae the room: Apply the ideal-gas law to obtain: Diide equation () by equation () to obtain: Substitute for n to obtain: Δ n n n' V n' () R' n' and n' n n ' ' Δ n n n n ' '

29 Substitute numerical alues and emperature and the Kinetic heory of Gases 74 Δ n (.66 0 mol) ealuate Δn: 60mol 00K 05K 5 Image that 0.0 g of liquid helium, initially at 4.0 K, eaporate into an empty balloon that is kept at.00 atm pressure. What is the olume of the balloon at (a) 5.0 K and (b) 9 K? icture the roblem Let the subscript refer to helium gas at 4. K and the subscript to the gas at 9 K. We can apply the ideal-gas law to find the olume of the gas at 4. K and a fixed amount of gas to find its olume at 9 K. See Appendix C for the molar mass of helium. (a) Apply the ideal-gas law to the helium gas to express its olume: nr m R V mr Substitute numerical alues and ealuate V : ( 0.0 g)( L atm/mol K)( 5.0K) ( 4.00 g/mol)(.00atm) V 5.5 L 5. L (b) Apply the ideal-gas law for a fixed amount of gas and sole for the olume of the helium gas at 9 K: Substitute numerical alues and ealuate V : V V and, because, V V V 9K 5.0K ( 5.5L) 60.L 5 A closed container with a olume of 6.00 L holds 0.0 g of liquid helium at 5.0 K and enough air to fill the rest of its olume at a pressure of.00 atm. he helium then eaporates and the container warms to room temperature (9 K). What is the final pressure inside the container? icture the roblem We can apply the law of partial pressures to find the final pressure inside the container. See Appendix C for the molar mass of helium and of air.

30 74 Chapter 7 he final pressure inside the container is the sum of the partial pressures of helium gas and air: + () final He gas air he pressure exerted by the air molecules at room temperature is gien by the ideal-gas law: air nairr V ρairr air mairr V air ρairvr V air he pressure exerted by the helium molecules at room temperature is also gien by the ideal-gas law: nher V He gas mher V He Substituting in equation () yields: final mher ρairr + V He m He ρair + V He air air R Substitute numerical alues and ealuate : final 0.0 g g 0 m L mol L a atm 0.5 ka kg.9 m J g 8.8 mol K mol.atm ( 9 K) 54 An automobile tire is filled to a gauge pressure of 00 ka when its temperature is 0ºC. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.) After the car has been drien at high speeds, the tire temperature increases to 50ºC. (a) Assuming that the olume of the tire does not change and that air behaes as an ideal gas, find the gauge pressure of the air in the tire. (b) Calculate the gauge pressure if the tire expands so the olume of the enclosed air increases by 0 percent. icture the roblem Let the subscript refer to the tire when its temperature is 0 C and the subscript to conditions when its temperature is 50 C. We can apply the ideal-gas law for a fixed amount of gas to relate the temperatures to the pressures of the air in the tire.

31 emperature and the Kinetic heory of Gases 74 (a) Apply the ideal-gas law for a fixed amount of gas and sole for pressure at the higher temperature: V () and V because V V Substitute numerical alues to obtain: (b) Sole equation () for : Because V.0 V : K 9K ka and ( 00 ka + 0ka) ka 0ka, gauge V V V.0V.0 ka Substitute numerical alues and K.0 9K ealuate and,gauge : ( )( ) ( 00 ka 0ka ) + 0 ka and 0 ka 0ka, gauge 0ka 55 [SS] After nitrogen (N ) and oxygen (O ), the most abundant molecule in Earth's atmosphere is water, H O. Howeer, the fraction of H O molecules in a gien olume of air aries dramatically, from practically zero percent under the driest conditions to as high as 4 percent where it is ery humid. (a) At a gien temperature and pressure, would air be denser when its water apor content is large or small? (b) What is the difference in mass, at room temperature and atmospheric pressure, between a cubic meter of air with no water apor molecules, and a cubic meter of air in which 4 percent of the molecules are water apor molecules? icture the roblem (a) At a gien temperature and pressure, the ideal-gas law tells us that the total number of molecules per unit olume, N/V, is constant. he denser gas will, therefore, be the one in which the aerage mass per molecule is greater. (b) We can apply the ideal-gas law and use the relationship between the masses of the dry and humid air and their molar masses to find the difference in

32 744 Chapter 7 mass, at room temperature and atmospheric pressure, between a cubic meter of air containing no water apor, and a cubic meter of air containing 4% water apor. See Appendix C for the molar masses of N, O, and H O. (a) he molecular mass of N is 8.04 amu (see Appendix C), that of O is.999 amu, and that of H O is 8.05 amu. Because H O molecules are lighter than the predominant molecules in air, a gien olume of air will be less when its water apor content is lower. (b) Express the difference in mass between a cubic meter of air containing no water apor, and a cubic meter of air containing 4% water apor: he masses of the dry air and humid air are related to their molar masses according to: Δm m dry dry. 04 m m 0 n humid dry and m 0. n humid 04 humid Substitute for m dry and m humid and Δm 0.04n simplify to obtain: 0.04n( ) dry dry 0.04n humid humid rom the ideal-gas law we hae: atmv n R and so Δm can be written as 0.04atmV Δm dry R ( ) humid Substitute numerical alues (8.8 g/mol is an 80% nitrogen and 0% oxygen weighted aerage) and ealuate Δm: Δm ( 0.04)( 0.5 ka)(.00 m ) ( 8.4 J/mol K)( 00 K) ( 8.8g/mol 8.05 g/mol) 8 g 56 A scuba dier is 40 m below the surface of a lake, where the temperature is 5.0ºC. He releases an air bubble that has a olume of 5 cm. he bubble rises to the surface, where the temperature is 5ºC. Assume that the air in the bubble is always in thermal equilibrium with the surrounding water, and assume that there is no exchange of molecules between the bubble and the surrounding water. What is the olume of the bubble right before it breaks the surface? Hint: Remember that the pressure also changes.

33 emperature and the Kinetic heory of Gases 745 icture the roblem Let the subscript refer to the conditions at the bottom of the lake and the subscript to the surface of the lake and apply the ideal-gas law for a fixed amount of gas. Apply the ideal-gas law for a fixed V V V V amount of gas: he pressure at the bottom of the lake is the sum of the pressure at its surface (atmospheric) and the pressure due to the depth of the lake: atm + ρgh Substituting for yields: V ( + ρgh) Substitute numerical alues and ealuate V : V V atm ( 5 cm )( 98 K) [ 0.5 ka + (.00 0 kg/m )( 9.8 m/s )( 40 m) ] 78 cm ( 78 K)( 0.5 ka) 57 [SS] A hot-air balloon is open at the bottom. he balloon has a olume of 446 m is filled with air with an aerage temperature of 00 C. he air outside the balloon has a temperature of 0.0 C and a pressure of.00 atm. How large a payload (including the enelope of the balloon itself) can the balloon lift? Use 9.0 g/mol for the molar mass of air. (Neglect the olume of both the payload and the enelope of the balloon.) icture the roblem Assume that the olume of the balloon is not changing. hen the air inside and outside the balloon must be at the same pressure of about.00 atm. he contents of the balloon are the air molecules inside it. We can use Archimedes principle to express the buoyant force on the balloon and we can find the weight of the air molecules inside the balloon. You ll need to determine the molar mass of air. See Appendix C for the molar masses of oxygen and nitrogen. Express the net force on the balloon and its contents: Using Archimedes principle, express the buoyant force on the balloon: B () net w air inside the balloon B wdisplaced fluid mdisplaced fluidg or B ρ o Vballoong where ρ o is the density of the air outside the balloon.

34 746 Chapter 7 Express the weight of the air inside the balloon: wair inside the balloon ρivballoong where ρ i is the density of the air inside the balloon. Substitute in equation () for B and ρ V g ρ V w air inside the balloon to obtain: ( ) V g net o ρ o balloon ρ i i balloon balloon g () Express the densities of the air molecules in terms of their number densities, molecular mass, and Aogadro s number: Using the ideal-gas law, relate the number density of air N/V to its temperature and pressure: Substitute for V N to obtain: ρ NA N V V Nk and ρ NA k N V k Substitute in equation () and simplify to obtain: net N Ak o N Ak o V i balloon g ( π d )g 6 i Assuming that the aerage molar mass of air is 8.8 g/mol, substitute numerical alues and ealuate net : net [ ]( 9.8m/s ) 97 K 48K 6 ( particles/mol)(.8 0 J/K) ( 8.8g/mol)( 0.5 ka) π ( 5.0m).0 kn 58 A helium balloon is used to lift a load of 0 N. he weight of the enelope of the balloon is 50.0 N and the olume of the helium when the balloon is fully inflated is.0 m. he temperature of the air is 0ºC and the atmospheric pressure is.00 atm. he balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is 0.0 N. Neglect any effects due to the changes of temperature as the altitude changes. (a) How many moles of helium gas are contained in the balloon? (b) At what altitude will the balloon be fully inflated? (c) Does the balloon eer reach the altitude at which it is fully inflated? (d) If the answer to (c) is Yes, what is the maximum altitude attained by the balloon?