REIEW & SUMMARY 59 PART Kinetic Theory of Gases The kinetic theory of gases relates the macroscoic roerties of gases (for examle, ressure and temerature) to the microscoic roerties of gas molecules (for examle, seed and kinetic energy). Avogadro s Number One mole of a substance contains N A (Avogadro s number) elementary units (usually atoms or molecules), where N A is found exerimentally to be N A 6.0 0 3 mol (Avogadro s number). (9-) One molar mass M of any substance is the mass of one mole of the substance. It is related to the mass m of the individual molecules of the substance by M mn A. (9-4) The number of moles n contained in a samle of mass M sam, consisting of N molecules, is given by n N M sam (9-, 9-3) N A M M sam. mn A Ideal Gas An ideal gas is one for which the ressure, volume, and temerature T are related by nrt (ideal gas law). (9-5) Here n is the number of moles of the gas resent and R is a constant (8.3 J/mol K) called the gas constant. The ideal gas law can also be written as NkT, (9-9) where the Boltzmann constant k is k R N A.38 0 3 J/K. (9-7) Work in an Isothermal olume Change The work done by an ideal gas during an isothermal (constant-temerature) change from volume i to volume f is W nrt ln f i (ideal gas, isothermal rocess). (9-4) Pressure, Temerature, and Molecular Seed The ressure exerted by n moles of an ideal gas, in terms of the seed of its molecules, is nmv rms 3, (9-) where v rms (v ) avg is the root-mean-square seed of the molecules of the gas.with Eq. 9-5 this gives 3RT v rms A M. (9-) Temerature and Kinetic Energy The average translational kinetic energy K avg er molecule of an ideal gas is K avg 3 kt. (9-4) Mean Free Path The mean free ath l of a gas molecule is its average ath length between collisions and is given by d N/, (9-5) where N/ is the number of molecules er unit volume and d is the molecular diameter. Maxwell Seed Distribution The Maxwell seed distribution P(v) is a function such that P(v) dv gives the fraction of molecules with seeds in the interval dv at seed v: P(v) 4 M RT 3/ v e Mv /RT. (9-7) Three measures of the distribution of seeds among the molecules of a gas are 8RT v avg (average seed), (9-3) A M RT v P A M (most robable seed), (9-35) and the rms seed defined above in Eq. 9-. Molar Secific Heats The molar secific heat C of a gas at constant volume is defined as C Q n T E int n T, (9-39, 9-4) in which Q is the energy transferred as heat to or from a samle of n moles of the gas, T is the resulting temerature change of the gas, and E int is the resulting change in the internal energy of the gas. For an ideal monatomic gas, C 3 R.5 J/mol K. (9-43) The molar secific heat C of a gas at constant ressure is defined to be C Q n T, (9-46) in which Q, n, and T are defined as above. C is also given by C C R. (9-49) For n moles of an ideal gas, E int nc T (ideal gas). (9-44) If n moles of a confined ideal gas undergo a temerature change T due to any rocess, the change in the internal energy of the gas is E int nc T (ideal gas, any rocess). (9-45) Degrees of Freedom and C We find C by using the equiartition of energy theorem, which states that every degree of freedom of a molecule (that is, every indeendent way it can store energy) has associated with it on average an energy er molecule ( RT kt er mole). If f is the number of degrees of freedom, then E int ( f/)nrt and C f R 4.6f J/mol K. (9-5) For monatomic gases f 3 (three translational degrees); for diatomic gases f 5 (three translational and two rotational degrees). Adiabatic Process When an ideal gas undergoes a slow adiabatic volume change (a change for which Q 0), g a constant (adiabatic rocess), (9-53) in which g ( C /C ) is the ratio of molar secific heats for the gas. For a free exansion, however, a constant.
** iew All Solutions Here ** 530 CHAPTER 9 THE KINETIC THEORY OF GASES For four situations for an a b c d ideal gas, the table gives the energy transferred to or from the gas as heat Q and either the work W done by the gas or the work Q W W on 50 35 5 0 50 35 40 40 W on done on the gas, all in joules. Rank the four situations in terms of the temerature change of the gas, most ositive first. In the - diagram of Fig. 9-7, a the gas does 5 J of work when taken along isotherm ab and 4 J when b taken along adiabat bc. What is the change in the internal energy of the gas when it is taken along the c straight ath from a to c? 3 For a temerature increase of Fig. 9-7 Question. T, a certain amount of an ideal gas requires 30 J when heated at constant volume and 50 J when heated at constant ressure. How much work is done by the gas in the second situation? 4 The dot in Fig. 9-8a reresents the initial state of a gas, and the vertical line through the dot divides the - diagram into regions and. For the following rocesses, determine whether the work W done by the gas is ositive, negative, or zero: (a) the gas moves u along the vertical line, (b) it moves down along the vertical line, (c) it moves to anywhere in region, and (d) it moves to anywhere in region. (a) (b) (c) Fig. 9-8 Questions 4, 6, and 8. 5 A certain amount of energy is to be transferred as heat to mol of a monatomic gas (a) at constant ressure and (b) at constant volume,and to 3 mol of a diatomic gas (c) at constant 4 ressure and (d) at constant volume. Figure 9-9 shows four aths from an initial oint to four final oints on a Fig. 9-9 Question 5. - diagram. Which ath goes with which rocess? (e) Are the molecules of the diatomic gas rotating? 6 The dot in Fig. 9-8b reresents the initial state of a gas, and the isotherm through the dot divides the - diagram into regions and. For the following rocesses, determine whether the change E int in the internal energy of the gas is ositive, negative, or zero: (a) the gas moves u along the isotherm,(b) it moves down along the isotherm,(c) it moves to anywhere in region, and (d) it moves to anywhere in region. 7 (a) Rank the four aths of Fig. 9-6 according to the work done by the gas, greatest first. (b) Rank aths,, and 3 according to the change in the internal energy of the gas, most ositive first and most negative last. 8 The dot in Fig. 9-8c reresents the initial state of a gas, and the adiabat through the dot divides the - diagram into regions and. For the following rocesses, determine whether the corresonding heat Q is ositive, negative, or zero: (a) the gas moves u along the adiabat, (b) it moves down along the adiabat, (c) it moves to anywhere in region, and (d) it moves to anywhere in region. 9 An ideal diatomic gas, with molecular rotation but not oscillation, loses energy as heat Q. Is the resulting decrease in the internal energy of the gas greater if the loss occurs in a constantvolume rocess or in a constant-ressure rocess? 0 Does the temerature of an ideal gas increase, decrease, or stay the same during (a) an isothermal exansion, (b) an exansion at constant ressure, (c) an adiabatic exansion, and (d) an increase in ressure at constant volume? SSM Tutoring roblem available (at instructor s discretion) in WileyPLUS and WebAssign Worked-out solution available in Student Solutions Manual WWW Worked-out solution is at Number of dots indicates level of roblem difficulty ILW Interactive solution is at Additional information available in The Flying Circus of Physics and at flyingcircusofhysics.com htt://www.wiley.com/college/halliday sec. 9- Avogadro s Number Find the mass in kilograms of 7.50 0 4 atoms of arsenic, which has a molar mass of 74.9 g/mol. Gold has a molar mass of 97 g/mol. (a) How many moles of gold are in a.50 g samle of ure gold? (b) How many atoms are in the samle? sec. 9-3 Ideal Gases 3 SSM Oxygen gas having a volume of 000 cm 3 at 40.0 C and.0 0 5 Pa exands until its volume is 500 cm 3 and its ressure is.06 0 5 Pa. Find (a) the number of moles of oxygen resent and (b) the final temerature of the samle. 4 A quantity of ideal gas at 0.0 C and 00 kpa occuies a volume of.50 m 3. (a) How many moles of the gas are resent? (b) If the ressure is now raised to 300 kpa and the temerature is raised to 30.0 C, how much volume does the gas occuy? Assume no leaks. 5 The best laboratory vacuum has a ressure of about.00 0 8 atm, or.0 0 3 Pa. How many gas molecules are there er cubic centimeter in such a vacuum at 93 K? ** iew All Solutions Here **
** iew All Solutions Here ** PROBLEMS 53 PART 6 Water bottle in a hot car. In the American Southwest, the temerature in a closed car arked in sunlight during the summer can be high enough to burn flesh. Suose a bottle of water at a refrigerator temerature of 5.00 C is oened, then closed, and then left in a closed car with an internal temerature of 75.0 C. Neglecting the thermal exansion of the water and the bottle, find the ressure in the air ocket traed in the bottle. (The ressure can be enough to ush the bottle ca ast the threads that are intended to kee the bottle closed.) 7 Suose.80 mol of an ideal gas is taken from a volume of 3.00 m 3 to a volume of.50 m 3 via an isothermal comression at 30 C. (a) How much energy is transferred as heat during the comression, and (b) is the transfer to or from the gas? 8 Comute (a) the number of moles and (b) the number of molecules in.00 cm 3 of an ideal gas at a ressure of 00 Pa and a temerature of 0 K. 9 An automobile tire has a volume of.64 0 m 3 and contains air at a gauge ressure (ressure above atmosheric ressure) of 65 kpa when the temerature is 0.00 C.What is the gauge ressure of the air in the tires when its temerature rises to 7.0 C and its volume increases to.67 0 m 3? Assume atmosheric ressure is.0 0 5 Pa. 0 A container encloses mol of an ideal gas that has molar mass M and 0.5 mol of a second ideal gas that has molar mass M 3M. What fraction of the total ressure on the container wall is attributable to the second gas? (The kinetic theory exlanation of ressure leads to the exerimentally discovered law of artial ressures for a mixture of gases that do not react chemically: The total ressure exerted by the mixture is equal to the sum of the ressures that the several gases would exert searately if each were to occuy the vessel alone.) SSM ILW WWW Air that initially occuies 0.40 m 3 at a gauge ressure of 03.0 kpa is exanded isothermally to a ressure of 0.3 kpa and then cooled at constant ressure until it reaches its initial volume. Comute the work done by the air. (Gauge ressure is the difference between the actual ressure and atmosheric ressure.) Submarine rescue. When the U. S. submarine Squalus became disabled at a deth of 80 m, a cylindrical chamber was lowered from a shi to rescue the crew. The chamber had a radius of.00 m and a height of 4.00 m, was oen at the bottom, and held two rescuers. It slid along a guide cable that a diver had attached to a hatch on the submarine. Once the chamber reached the hatch and clamed to the hull, the crew could escae into the chamber. During the descent, air was released from tanks to revent water from flooding the chamber. Assume that the interior air ressure matched the water ressure at deth h as given by 0 rgh, where 0.000 atm is the surface ressure and r 04 kg/m 3 is the density of seawater. Assume a surface temerature of 0.0 C and a submerged water temerature of 30.0 C. (a) What is the air volume in the chamber at the surface? (b) If air had not been released from the tanks, what would have been the air volume in the chamber at deth h 80.0 m? (c) How many moles of air were needed to be released to maintain the original air volume in the chamber? 3 A samle of an ideal gas is taken through the cyclic rocess abca shown in Fig. 9-0. The scale of the vertical axis is set by b 7.5 kpa and ac.5 kpa. At oint a, T 00 K. (a) How many moles of gas are in the samle? What are (b) the temerature of the gas at oint b, (c) the temerature of the gas at oint c, and (d) the net energy added to the gas as heat during the cycle? 4 In the temerature range 30 K to 330 K, the ressure of a certain nonideal gas is related to volume and temerature T by (4.9 J/K) T (0.0066 J/K ) T. How much work is done by the gas if its temerature is raised from 35 K to 35 K while the ressure is held constant? 5 Suose 0.85 mol of an ideal gas undergoes an isothermal exansion as energy is added to it as heat Q. If Fig. 9- shows the final volume f versus Q, what is the gas temerature? The scale of the vertical axis is set by fs 0.30 m 3, and the scale of the horizontal axis is set by Q s 00 J. f (m 3 ) fs 0 Q (J) Pressure (kpa) b ac Fig. 9- Problem 5. a.0 3.0 olume (m 3 ) Fig. 9-0 Problem 3. 6 An air bubble of volume 0 cm 3 is at the bottom of a lake 40 m dee, where the temerature is 4.0 C. The bubble rises to the surface, which is at a temerature of 0 C. Take the temerature of the bubble s air to be the same as that of the surrounding water. Just as the bubble reaches the surface, what is its volume? 7 Container A in Fig. 9- holds an ideal gas at a ressure of 5.0 0 5 Pa and a temerature of 300 K. It is connected by a thin tube (and a closed valve) to container B, with four times the volume of A. Container B holds the same ideal gas at a ressure of.0 0 5 Pa and a temerature of 400 K. A The valve is oened to allow the ressures to equalize, but the temerature of each container is main- Fig. 9- Problem 7. B tained. What then is the ressure? sec. 9-4 Pressure, Temerature, and RMS Seed 8 The temerature and ressure in the Sun s atmoshere are.00 0 6 K and 0.0300 Pa. Calculate the rms seed of free electrons (mass 9. 0 3 kg) there, assuming they are an ideal gas. Q s b c ** iew All Solutions Here **
** iew All Solutions Here ** 53 CHAPTER 9 THE KINETIC THEORY OF GASES 9 (a) Comute the rms seed of a nitrogen molecule at 0.0 C. The molar mass of nitrogen molecules (N ) is given in Table 9-. At what temeratures will the rms seed be (b) half that value and (c) twice that value? 0 Calculate the rms seed of helium atoms at 000 K. See Aendix F for the molar mass of helium atoms. SSM The lowest ossible temerature in outer sace is.7 K. What is the rms seed of hydrogen molecules at this temerature? (The molar mass is given in Table 9-.) Find the rms seed of argon atoms at 33 K. See Aendix F for the molar mass of argon atoms. 3 A beam of hydrogen molecules (H ) is directed toward a wall, at an angle of 55 with the normal to the wall. Each molecule in the beam has a seed of.0 km/s and a mass of 3.3 0 4 g.the beam strikes the wall over an area of.0 cm, at the rate of 0 3 molecules er second. What is the beam s ressure on the wall? 4 At 73 K and.00 0 atm, the density of a gas is.4 0 5 g/cm 3. (a) Find v rms for the gas molecules. (b) Find the molar mass of the gas and (c) identify the gas. (Hint: The gas is listed in Table 9-.) sec. 9-5 Translational Kinetic Energy 5 Determine the average value of the translational kinetic energy of the molecules of an ideal gas at (a) 0.00 C and (b) 00 C. What is the translational kinetic energy er mole of an ideal gas at (c) 0.00 C and (d) 00 C? 6 What is the average translational kinetic energy of nitrogen molecules at 600 K? 7 Water standing in the oen at 3.0 C evaorates because of the escae of some of the surface molecules.the heat of vaorization (539 cal/g) is aroximately equal to n, where is the average energy of the escaing molecules and n is the number of molecules er gram. (a) Find. (b) What is the ratio of to the average kinetic energy of H O molecules, assuming the latter is related to temerature in the same way as it is for gases? sec. 9-6 Mean Free Path 8 At what frequency would the wavelength of sound in air be equal to the mean free ath of oxygen molecules at.0 atm ressure and 0.00 C? The molecular diameter is 3.0 0 8 cm. 9 SSM The atmosheric density at an altitude of 500 km is about molecule/cm 3. (a) Assuming the molecular diameter of.0 0 8 cm, find the mean free ath redicted by Eq. 9-5. (b) Exlain whether the redicted value is meaningful. 30 The mean free ath of nitrogen molecules at 0.0 C and.0 atm is 0.80 0 5 cm. At this temerature and ressure there are.7 0 9 molecules/cm 3.What is the molecular diameter? 3 In a certain article accelerator, rotons travel around a circular ath of diameter 3.0 m in an evacuated chamber, whose residual gas is at 95 K and.00 0 6 torr ressure. (a) Calculate the number of gas molecules er cubic centimeter at this ressure. (b) What is the mean free ath of the gas molecules if the molecular diameter is.00 0 8 cm? 3 At 0 C and 750 torr ressure, the mean free aths for argon gas (Ar) and nitrogen gas (N ) are l Ar 9.9 0 6 cm and l N 7.5 0 6 cm. (a) Find the ratio of the diameter of an Ar atom to that of an N molecule. What is the mean free ath of argon at (b) 0 C and 50 torr, and (c) 40 C and 750 torr? sec. 9-7 The Distribution of Molecular Seeds 33 SSM The seeds of 0 molecules are.0, 3.0, 4.0,..., km/s.what are their (a) average seed and (b) rms seed? 34 The seeds of articles are as follows (N i reresents the number of articles that have seed v i ): N i 4 6 8 v i (cm/s).0.0 3.0 4.0 5.0 What are (a) v avg, (b) v rms, and (c) v P? 35 Ten articles are moving with the following seeds: four at 00 m/s, two at 500 m/s, and four at 600 m/s. Calculate their (a) average and (b) rms seeds. (c) Is v rms v avg? 36 It is found that the most robable seed of molecules in a gas when it has (uniform) temerature T is the same as the rms seed of the molecules in this gas when it has (uniform) temerature T. Calculate T /T. 37 SSM WWW Figure 9-3 a shows a hyothetical seed distribution for a samle of N gas articles (note that P(v) 0 for seed v v 0 ). What are the values of (a) 0 v 0 Seed v 0 av 0, (b) v avg /v 0, and (c) v rms /v 0? (d) What fraction of the articles has a seed between.5v 0 and.0v 0? Fig. 9-3 Problem 37. 38 Figure 9-4 gives the robability distribution for nitrogen gas. The scale of the horizontal axis is set by v s 00 m/s. What are the (a) gas temerature and (b) rms seed of the molecules? P(v) P(v) 0 v s v (m/s) Fig. 9-4 Problem 38. 39 At what temerature does the rms seed of (a) H (molecular hydrogen) and (b) O (molecular oxygen) equal the escae seed from Earth (Table 3-)? At what temerature does the rms seed of (c) H and (d) O equal the escae seed from the Moon (where the gravitational acceleration at the surface has magnitude 0.6g)? Considering the answers to arts (a) and (b), should there be much (e) hydrogen and (f) oxygen high in Earth s uer atmoshere, where the temerature is about 000 K? 40 Two containers are at the same temerature. The first contains gas with ressure, molecular mass m, and rms seed v rms. The second contains gas with ressure.0, molecular mass m, and average seed v avg.0v rms. Find the mass ratio m /m. 4 A hydrogen molecule (diameter.0 0 8 cm), traveling at the rms seed, escaes from a 4000 K furnace into a chamber contain- ** iew All Solutions Here **
** iew All Solutions Here ** PROBLEMS 533 PART ing cold argon atoms (diameter 3.0 0 8 cm) at a density of 4.0 0 9 atoms/cm 3. (a) What is the seed of the hydrogen molecule? (b) If it collides with an argon atom, what is the closest their centers can be, considering each as sherical? (c) What is the initial number of collisions er second exerienced by the hydrogen molecule? (Hint: Assume that the argon atoms are stationary. Then the mean free ath of the hydrogen molecule is given by Eq. 9-6 and not Eq. 9-5.) sec. 9-8 The Molar Secific Heats of an Ideal Gas 4 What is the internal energy of.0 mol of an ideal monatomic gas at 73 K? 43 The temerature of 3.00 mol of an ideal diatomic gas is increased by 40.0 C without the ressure of the gas changing.the molecules in the gas rotate but do not oscillate. (a) How much energy is transferred to the gas as heat? (b) What is the change in the internal energy of the gas? (c) How much work is done by the gas? (d) By how much does the rotational kinetic energy of the gas increase? 44 One mole of an ideal diatomic gas goes from a to c along the diagonal ath in Fig. 9-5. The scale of the vertical axis is set by ab a b ab 5.0 kpa and c.0 kpa, and the scale of the horizontal axis is set by bc 4.0 m 3 and a.0 m 3. During c c the transition, (a) what is the change in internal energy of the gas, and (b) how much energy is added to the a bc olume (m 3 ) gas as heat? (c) How much heat is required if the gas goes from a to c Fig. 9-5 Problem 44. along the indirect ath abc? 45 ILW The mass of a gas molecule can be comuted from its secific heat at constant volume c. (Note that this is not C.) Take c 0.075 cal/g C for argon and calculate (a) the mass of an argon atom and (b) the molar mass of argon. 46 Under constant ressure, the temerature of.00 mol of an ideal monatomic gas is raised 5.0 K. What are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change E int in the internal energy of the gas, and (d) the change K in the average kinetic energy er atom? 47 The temerature of.00 mol of an ideal monatomic gas is raised 5.0 K at constant volume.what are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change E int in the internal energy of the gas, and (d) the change K in the average kinetic energy er atom? 48 When 0.9 J was added as heat to a articular ideal gas, the volume of the gas changed from 50.0 cm 3 to 00 cm 3 while the ressure remained at.00 atm. (a) By how much did the internal energy of the gas change? If the quantity of gas resent was.00 0 3 mol, find (b) C and (c) C. 49 SSM A container holds a mixture of three nonreacting gases:.40 mol of gas with C.0 J/mol K,.50 mol of gas with C.8 J/mol K, and 3.0 mol of gas 3 with C 3 0.0 J/mol K.What is C of the mixture? Pressure (kpa) 5 ILW When.0 mol of oxygen (O ) gas is heated at constant ressure starting at 0 C, how much energy must be added to the gas as heat to double its volume? (The molecules rotate but do not oscillate.) 5 Suose.0 g of oxygen (O ) gas is heated at constant atmosheric ressure from 5.0 C to 5 C. (a) How many moles of oxygen are resent? (See Table 9- for the molar mass.) (b) How much energy is transferred to the oxygen as heat? (The molecules rotate but do not oscillate.) (c) What fraction of the heat is used to raise the internal energy of the oxygen? 53 SSM WWW Suose 4.00 mol of an ideal diatomic gas, with molecular rotation but not oscillation, exerienced a temerature increase of 60.0 K under constant-ressure conditions. What are (a) the energy transferred as heat Q, (b) the change E int in internal energy of the gas, (c) the work W done by the gas, and (d) the change K in the total translational kinetic energy of the gas? sec. 9- The Adiabatic Exansion of an Ideal Gas 54 We know that for an adiabatic rocess g a constant. Evaluate a constant for an adiabatic rocess involving exactly.0 mol of an ideal gas assing through the state having exactly.0 atm and T 300 K. Assume a diatomic gas whose molecules rotate but do not oscillate. 55 A certain gas occuies a volume of 4.3 L at a ressure of. atm and a temerature of 30 K. It is comressed adiabatically to a volume of 0.76 L. Determine (a) the final ressure and (b) the final temerature, assuming the gas to be an ideal gas for which g.4. 56 Suose.00 L of a gas with g.30, initially at 73 K and.00 atm, is suddenly comressed adiabatically to half its initial volume. Find its final (a) ressure and (b) temerature. (c) If the gas is then cooled to 73 K at constant ressure, what is its final volume? 57 The volume of an ideal gas is adiabatically reduced from 00 L to 74.3 L. The initial ressure and temerature are.00 atm and 300 K. The final ressure is 4.00 atm. (a) Is the gas monatomic, diatomic, or olyatomic? (b) What is the final temerature? (c) How many moles are in the gas? 58 Oening chamagne. In a bottle of chamagne, the ocket of gas (rimarily carbon dioxide) between the liquid and the cork is at ressure of i 5.00 atm. When the cork is ulled from the bottle, the gas undergoes an adiabatic exansion until its ressure matches the ambient air ressure of.00 atm.assume that the ratio of the molar secific heats is. If the gas has initial temerature T i 5.00 C, what is its temerature at the end of the adiabatic exansion? 59 Figure 9-6 shows two aths that may be taken by a gas from an initial oint i to a final oint f. Path consists of an isothermal exansion (work is 50 J in magnitude), an adiabatic exansion Path i f 4 3 Path Isothermal Adiabatic sec. 9-9 Degrees of Freedom and Molar Secific Heats 50 We give 70 J as heat to a diatomic gas, which then exands at constant ressure. The gas molecules rotate but do not oscillate. By how much does the internal energy of the gas increase? Isothermal Fig. 9-6 Problem 59. ** iew All Solutions Here **
** iew All Solutions Here ** 534 CHAPTER 9 THE KINETIC THEORY OF GASES (work is 40 J in magnitude), an isothermal comression (work is 30 J in magnitude), and then an adiabatic comression (work is 5 J in magnitude). What is the change in the internal energy of the gas if the gas goes from oint i to oint f along ath? 60 Adiabatic wind. The normal airflow over the Rocky Mountains is west to east.the air loses much of its moisture content and is chilled as it climbs the western side of the mountains. When it descends on the eastern side, the increase in ressure toward lower altitudes causes the temerature to increase. The flow, then called a chinook wind, can raidly raise the air temerature at the base of the mountains. Assume that the air ressure deends on altitude y according to 0 ex ( ay), where 0.00 atm and a.6 0 4 m. Also assume that the ratio of the molar secific heats is. A arcel of air with an initial temerature of 5.00 C descends adiabatically from y 467 m to y 567 m.what is its temerature at the end of the descent? 6 A gas is to be exanded from initial state i to final state f along either ath or ath on a - diagram. Path consists of three stes: an isothermal exansion (work is 40 J in magnitude), an adiabatic exansion (work is 0 J in magnitude), and another isothermal exansion (work is 30 J in magnitude). Path consists of two stes: a ressure reduction at constant volume and an exansion at constant ressure. What is the change in the internal energy of the gas along ath? 6 An ideal diatomic gas, with rotation but no oscillation, undergoes an adiabatic comression. Its initial ressure and volume are.0 atm and 0.00 m 3. Its final ressure is.40 atm. How much work is done by the gas? 63 Figure 9-7 shows a cycle undergone by.00 mol of an ideal monatomic gas. The temeratures are T 300 K, T 600 K, and Adiabatic T 3 455 K. For :, what are (a) heat Q, (b) the change in internal energy E int, and (c) the work done 3 W? For : 3, what are (d) Q, (e) E int, and (f) W? For 3 :, what are (g) Q, (h) E int, and (i) W? For the full olume cycle, what are (j) Q, (k) E int, and (l) Fig. 9-7 Problem 63. W? The initial ressure at oint is.00 atm (.03 0 5 Pa). What are the (m) volume and (n) ressure at oint and the (o) volume and () ressure at oint 3? Additional Problems 64 Calculate the work done by an external agent during an isothermal comression of.00 mol of oxygen from a volume of.4 L at 0 C and.00 atm to a volume of 6.8 L. 65 An ideal gas undergoes an adiabatic comression from.0 atm,.0 0 6 L, T 0.0 C to.0 0 5 atm,.0 0 3 L. (a) Is the gas monatomic, diatomic, or olyatomic? (b) What is its final temerature? (c) How many moles of gas are resent? What is the total translational kinetic energy er mole (d) before and (e) after the comression? (f) What is the ratio of the squares of the rms seeds before and after the comression? 66 An ideal gas consists of.50 mol of diatomic molecules that rotate but do not oscillate. The molecular diameter is 50 m. The gas is exanded at a constant ressure of.50 0 5 Pa, with a transfer of 00 J as heat. What is the change in the mean free ath of the molecules? 4 3 Pressure 67 An ideal monatomic gas initially has a temerature of 330 K and a ressure of 6.00 atm. It is to exand from volume 500 cm 3 to volume 500 cm 3. If the exansion is isothermal, what are (a) the final ressure and (b) the work done by the gas? If, instead, the exansion is adiabatic, what are (c) the final ressure and (d) the work done by the gas? 68 In an interstellar gas cloud at 50.0 K, the ressure is.00 0 8 Pa.Assuming that the molecular diameters of the gases in the cloud are all 0.0 nm, what is their mean free ath? 69 SSM The enveloe and basket of a hot-air balloon have a combined weight of.45 kn, and the enveloe has a caacity (volume) of.8 0 3 m 3.When it is fully inflated, what should be the temerature of the enclosed air to give the balloon a lifting caacity (force) of.67 kn (in addition to the balloon s weight)? Assume that the surrounding air, at 0.0 C, has a weight er unit volume of.9 N/m 3 and a molecular mass of 0.08 kg/mol,and is at a ressure of.0 atm. 70 An ideal gas, at initial temerature T and initial volume.0 m 3, is exanded adiabatically to a volume of 4.0 m 3, then exanded isothermally to a volume of 0 m 3, and then comressed adiabatically back to T.What is its final volume? 7 SSM The temerature of.00 mol of an ideal monatomic gas is raised 5.0 K in an adiabatic rocess. What are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change E int in internal energy of the gas, and (d) the change K in the average kinetic energy er atom? 7 At what temerature do atoms of helium gas have the same rms seed as molecules of hydrogen gas at 0.0 C? (The molar masses are given in Table 9-.) 73 SSM At what frequency do molecules (diameter 90 m) collide in (an ideal) oxygen gas (O ) at temerature 400 K and ressure.00 atm? 74 (a) What is the number of molecules er cubic meter in air at 0 C and at a ressure of.0 atm (.0 0 5 Pa)? (b) What is the mass of.0 m 3 of this air? Assume that 75% of the molecules are nitrogen (N ) and 5% are oxygen (O ). 75 The temerature of 3.00 mol of a gas with C 6.00 cal/mol K is to be raised 50.0 K. If the rocess is at constant volume, what are (a) the energy transferred as heat Q, (b) the work W done by the gas, (c) the change E int in internal energy of the gas, and (d) the change K in the total translational kinetic energy? If the rocess is at constant ressure, what are (e) Q,(f) W, (g) E int, and (h) K? If the rocess is adiabatic, what are (i) Q,(j) W, (k) E int, and (l) K? 76 During a comression at a constant ressure of 50 Pa, the volume of an ideal gas decreases from 0.80 m 3 to 0.0 m 3. The initial temerature is 360 K, and the gas loses 0 J as heat. What are (a) the change in the internal energy of the gas and (b) the final temerature of the gas? 77 SSM Figure 9-8 shows a hyothetical seed distribution P(v) 0 Seed v 0 Fig. 9-8 Problem 77. ** iew All Solutions Here **
** iew All Solutions Here ** PROBLEMS 535 PART for articles of a certain gas: P(v) Cv for 0 v v 0 and P(v) 0 for v v 0. Find (a) an exression for C in terms of v 0, (b) the average seed of the articles,and (c) their rms seed. 78 (a) An ideal gas initially at ressure 0 undergoes a free exansion until its volume is 3.00 times its initial volume. What then is the ratio of its ressure to 0? (b) The gas is next slowly and adiabatically comressed back to its original volume. The ressure after comression is (3.00) /3 0. Is the gas monatomic, diatomic, or olyatomic? (c) What is the ratio of the average kinetic energy er molecule in this final state to that in the initial state? 79 SSM An ideal gas undergoes isothermal comression from an initial volume of 4.00 m 3 to a final volume of 3.00 m 3. There is 3.50 mol of the gas, and its temerature is 0.0 C. (a) How much work is done by the gas? (b) How much energy is transferred as heat between the gas and its environment? 80 Oxygen (O ) gas at 73 K and.0 atm is confined to a cubical container 0 cm on a side. Calculate U g /K avg, where U g is the change in the gravitational otential energy of an oxygen molecule falling the height of the box and K avg is the molecule s average translational kinetic energy. 8 An ideal gas is taken through a comlete cycle in three stes: adiabatic exansion with work equal to 5 J, isothermal contraction at 35 K, and increase in ressure at constant volume. (a) Draw a - diagram for the three stes. (b) How much energy is transferred as heat in ste 3, and (c) is it transferred to or from the gas? 8 (a) What is the volume occuied by.00 mol of an ideal gas at standard conditions that is,.00 atm (.0 0 5 Pa) and 73 K? (b) Show that the number of molecules er cubic centimeter (the Loschmidt number) at standard conditions is.69 0 9. 83 SSM A samle of ideal gas exands from an initial ressure and volume of 3 atm and.0 L to a final volume of 4.0 L. The initial temerature is 300 K. If the gas is monatomic and the exansion isothermal, what are the (a) final ressure f, (b) final temerature T f, and (c) work W done by the gas? If the gas is monatomic and the exansion adiabatic, what are (d) f, (e) T f, and (f) W? If the gas is diatomic and the exansion adiabatic, what are (g) f, (h) T f, and (i) W? 84 An ideal gas with 3.00 mol is initially in state with ressure 0.0 atm and volume 500 cm 3. First it is taken to state with ressure.50 and volume.00. Then it is taken to state 3 with ressure 3.00 and volume 3 0.500. What is the temerature of the gas in (a) state and (b) state? (c) What is the net change in internal energy from state to state 3? 85 A steel tank contains 300 g of ammonia gas (NH 3 ) at a ressure of.35 0 6 Pa and a temerature of 77 C. (a) What is the volume of the tank in liters? (b) Later the temerature is C and the ressure is 8.7 0 5 Pa. How many grams of gas have leaked out of the tank? 86 In an industrial rocess the volume of 5.0 mol of a monatomic ideal gas is reduced at a uniform rate from 0.66 m 3 to 0.308 m 3 in.00 h while its temerature is increased at a uniform rate from 7.0 C to 450 C. Throughout the rocess, the gas asses through thermodynamic equilibrium states. What are (a) the cumulative work done on the gas, (b) the cumulative energy absorbed by the gas as heat, and (c) the molar secific heat for the rocess? (Hint: To evaluate the integral for the work, you might use a bx bx dx A Bx B an indefinite integral.) Suose the rocess is relaced with a twoste rocess that reaches the same final state. In ste, the gas volume is reduced at constant temerature, and in ste the temerature is increased at constant volume. For this rocess, what are (d) the cumulative work done on the gas, (e) the cumulative energy absorbed by the gas as heat, and (f) the molar secific heat for the rocess? 87 Figure 9-9 shows a cycle consisting of five aths: AB is isothermal at 300 K, BC is adiabatic with work 5.0 J, CD is at a constant ressure of 5 atm, DE is isothermal, and EA is adiabatic with a change in internal energy of 8.0 J. What is the change in internal energy of the gas along ath CD? A E ab ba B ln(a Bx), Fig. 9-9 Problem 87. 88 An ideal gas initially at 300 K is comressed at a constant ressure of 5 N/m from a volume of 3.0 m 3 to a volume of.8 m 3. In the rocess, 75 J is lost by the gas as heat. What are (a) the change in internal energy of the gas and (b) the final temerature of the gas? D B C ** iew All Solutions Here ** www.meetyourbrain.com Ask questions get answers, homework hel