Ch17 ssm
- 1. Chapter 17 Temperature and the Kinetic Theory of Gases Conceptual Problems 3 • ″Yesterday I woke up and it was 20 ºF in my bedroom,″ said Mert to his old friend Mort. ″That’s nothing,″ replied Mort. ″My room was –5.0 ºC.″ Who had the colder room, Mert or Mort? Picture the Problem We can decide which room was colder by converting 20°F to the equivalent Celsius temperature. Using the Fahrenheit-Celsius conversion, convert 20°F to the equivalent Celsius temperature: ( ) ( ) C7.6 322032 9 5 F9 5 C °−= °−°=°−= tt so colder.wasroomsMert' 5 • Figure 17-18 shows a plot of volume versus 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 From the ideal-gas law, the equation of the line from the origin to the point (T,V) is .T P nR V = The slope of this line is nR/P. During the process A → B the slope of the line from the origin to (T,V) continuously decreases, so the pressure continuously increases. 15 • Suppose that you compress an ideal gas to half its original volume, while also halving its absolute temperature. During this process, the pressure of the gas (a) halves, (b) remains constant, (c) doubles, (d) quadruples. Determine the Concept From the ideal-gas law, nRTPV = , halving both the temperature and volume of the gas leaves the pressure unchanged. )(b is correct. 17 •• 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 intuitively plausible. Determine the Concept The rms speed of molecules of an ideal gas is given by M RT v 3 rms = and the speed of sound in a gas is given by M RT v γ =sound . 347
- 2. Chapter 17348 The rms speed of the molecules of an ideal-gas is given by: M RT v 3 rms = The speed of sound in a gas is given by: M RT v γ =sound Divide the first of these equations by the second and simplify to obtain: γγ 3 3 sound rms == M RT M RT v v For a monatomic gas, γ = 1.67 and: 34.1 67.1 3 sound monatomicrms, == v v For a diatomic gas, γ = 1.40 and: 46.1 40.1 3 sound diatomicrms, == v v The rms speed is always somewhat greater than the speed of sound. However, it is only the component of the molecular velocities in the direction of propagation that is relevant to this issue. In addition, in a gas the mean free path is greater than the average intermolecular distance. 21 •• Liquid nitrogen is relatively cheap, while liquid helium is relatively expensive. 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 The average molecular speed of He gas at 300 K is about 1.4 km/s, so a significant fraction of He molecules have speeds in excess of Earth’s escape velocity (11.2 km/s). Thus, they "leak" away into space. Over time, the He content of the atmosphere decreases to almost nothing. Estimation and Approximation 25 •• In Chapter 11, we found that the escape speed at the surface of a planet of radius R is gRv 2e = , where g is the acceleration due to gravity at the surface of the planet. If the rms speed of a gas is greater than about 15 to 20 percent of the escape speed of a planet, virtually all of the molecules of that gas will escape the atmosphere of the planet. (a) At what temperature is vrms for O2 equal to 15 percent of the escape speed for Earth?
- 3. Temperature and the Kinetic Theory of Gases 349 (b) At what temperature is vrms for H2 equal to 15 percent of the escape speed for Earth? (c) Temperatures in the upper atmosphere reach 1000 K. How does this help account for the low abundance of hydrogen in Earth’s atmosphere? (d) Compute the temperatures for which the rms speeds of O2 and H2 are equal to 15 percent of the escape speed at the surface of the moon, where g is about one-sixth of its value on Earth and R = 1738 km. How does this account for the absence of an atmosphere on the moon? Picture the Problem We can find the escape temperatures for the Earth and the moon by equating, in turn, 0.15ve and vrms of O2 and H2. We can compare these 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 O2 and H2. (a) Express vrms for O2: 2 2 O Orms, 3 M RT v = where R is the gas constant, T is the absolute temperature, and is the molar mass of oxygen. 2OM Equate 0.15ve and :2Orms,v M RT gR 3 215.0 earth = Solve for T to obtain: R MgR T 3 045.0 earth = (1) Substitute numerical values and evaluate T for O2: ( )( )( ) ( ) K1060.3 KJ/mol8.3143 kg/mol1032.0m106.37m/s9.81045.0 3 362 ×= ⋅ ×× = − T (b) Substitute numerical values and evaluate T for H2: ( )( )( ) ( ) K230 KJ/mol8.3143 kg/mol102.02m106.37m/s9.81045.0 362 = ⋅ ×× = − T (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.
- 4. Chapter 17350 (d) Express equation (1) at the surface of the moon: ( ) R MRg R MRg R MRg T moonearth moonearth6 1 moonmoon 0025.0 3 045.0 3 045.0 = = = Substitute numerical values and evaluate T for O2: ( )( )( ) K160 KJ/mol8.314 kg/mol1032.0m10738.1m/s9.810025.0 362 = ⋅ ×× = − T Substitute numerical values and evaluate T for H2: ( )( )( ) K10 KJ/mol8.314 kg/mol102.02m10738.1m/s9.810025.0 362 = ⋅ ×× = − T Because g is less on the moon, the escape speed is lower. Thus, a larger percentage of the molecules are moving at escape speed. 27 •• The escape speed for gas molecules in the atmosphere of Jupiter is 60 km/s and the surface temperature of Jupiter is typically –150ºC. Calculate the rms speeds for (a) H2, (b) O2, and (c) CO2 at this temperature. (d) Are H2, O2, and CO2 likely to be found in the atmosphere of Jupiter? Picture the Problem We can use MRTv 3rms = to calculate the rms speeds of H2, O2, and CO2 at 123 K and then compare these speeds to 20% of the escape velocity on Jupiter to decide the likelihood of finding these gases in the atmosphere of Jupiter. See Appendix C for the molar masses of H2, O2, and CO2. Express the rms speed of an atom as a function of the temperature: M RT v 3 rms = (a) Substitute numerical values and evaluate vrms for H2: ( )( ) km/s1.23 kg/mol102.02 K231KJ/mol8.3143 3Hrms, 2 = × ⋅ = − v
- 5. Temperature and the Kinetic Theory of Gases 351 (b) Evaluate vrms for O2: ( )( ) m/s310 kg/mol1032.0 K123KJ/mol8.3143 3Orms, 2 = × ⋅ = − v (c) Evaluate vrms for CO2: ( )( ) m/s264 kg/mol100.44 K123KJ/mol8.3143 3COrms, 2 = × ⋅ = − v (d) Calculate 20% of ve for Jupiter: ( ) km/s12km/s0620.0 5 1 e ==v Because ve is greater than vrms for O2, CO2, and H2, all three gasses should be found on Jupiter. 31 ••• In normal breathing conditions, approximately 5 percent of each exhaled breath is carbon dioxide. Given this information and neglecting any difference in water-vapor content, estimate the typical difference in mass between an inhaled breath and an exhaled breath. Picture the Problem One breath (one’s lung capacity) is about half a liter. The only thing that occurs in breathing is that oxygen is exchanged for carbon dioxide. Let’s estimate that of the 20% of the air that is breathed in as oxygen, ¼ is exchanged for carbon dioxide. Then 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 32 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 exhaled breath: ( ) breathOCOCO COO 222 22 Δ nMMf mmm −= −= where is the fraction of the air breathed in that is exchanged for carbon dioxide. 2COf The number of moles per breath is given by: L/mol4.22 breath breath V n = Substituting for nbreath yields: ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= L/mol4.22 Δ breath OCOCO 222 V MMfm
- 6. Chapter 17352 Substitute numerical values and evaluate Δm: ( )( ) ( ) g.010 L/mol4.22 L5.0 g/mol32g/mol4405.0Δ ≈−=m Temperature Scales 33 • The melting point of gold is 1945.4 ºF. Express this temperature in degrees Celsius. Picture the Problem We can use the Fahrenheit-Celsius conversion equation to find this temperature on the Celsius scale. Convert 1945.4°F to the equivalent Celsius temperature: ( ) ( ) C1063 324.194532 9 5 F9 5 C °= °−°=°−= tt 39 • A constant-volume gas thermometer reads 50.0 torr at the triple point of water. (a) Sketch a graph of pressure vs. absolute temperature for this thermometer. (b) What will be the pressure when the thermometer measures a temperature of 300 K? (c) What ideal-gas temperature corresponds to a pressure of 678 torr? Picture the Problem We can use the equation of the graph plotted in (a) to (b) find the pressure when the temperature is 300 K and (c) the ideal-gas temperature when the pressure is 678 torr. (a) A graph of pressure (in torr) versus temperature (in kelvins) for this thermometer is shown to the right. The equation of this graph is: TP ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = K273 torr0.50 (1) 0 torr,P K,T 273 0 50.0 (b) Evaluate P when T = 300 K: ( ) ( ) torr9.54 K300 K273 torr0.50 K300 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =P
- 7. Temperature and the Kinetic Theory of Gases 353 (c) Solve equation (1) for T to obtain: PT ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = torr50.0 K273 Evaluate :( )torr678T ( ) ( ) K1070.3 torr678 torr50.0 K273 torr678 3 ×= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =T 45 ••• A thermistor is a solid-state device widely used in a variety of engineering applications. Its primary characteristic is that its electrical resistance varies greatly with temperature. Its temperature dependence is given approximately by R = R0eB/T , where R is in ohms (Ω), T is in kelvins, and R0 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 = 7360 Ω at the ice point and 153 Ω at the steam point, find R0 and B. (b) What is the resistance of the thermistor at t = 98.6ºF? (c) What is the rate of change of the resistance with temperature (dR/dT) at the ice point and the steam point? (d) At which temperature is the thermistor most sensitive? Picture the Problem We can use the temperature dependence of the resistance of the thermistor and the given data to determine R0 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 T will allow us to express the rate of change of resistance with temperature at both the ice point and the steam point temperatures. (a) Express the resistance at the ice point as a function of temperature of the ice point: K273 07360 B eR=Ω (1) Express the resistance at the steam point as a function of temperature of the steam point: K373 0153 B eR=Ω (2) Divide equation (1) by equation (2) to obtain: K373K273 10.48 153 7360 BB e − == Ω Ω
- 8. Chapter 17354 Solve for B by taking the logarithm of both sides of the equation: 1 K 373 1 273 1 1.48ln − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= B and K1094.3 K10944.3 K 373 1 273 1 1.48ln 3 3 1 ×= ×= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − B Solve equation (1) for R0 and substitute for B: ( ) ( ) Ω1091.3Ω10913.3 Ω7360 Ω7360 Ω7360 33 K273K10944.3 K273 K2730 3 −− ×− − ×=×= = == e e e R B B (b) From (a) we have: ( ) T eR K10944.33 3 Ω10913.3 ×− ×= Convert 98.6°F to kelvins to obtain: K310=T Substitute for T to obtain: ( ) ( ) kΩ31.1 Ω10913.3K310 K310K10944.33 3 = ×= ×− eR (c) Differentiate R with respect to T to obtain: ( ) 202 00 T RB eR T B T B dT d eReR dT d dT dR TB TBTB −= − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ == Evaluate dR/dT at the ice point: ( )( ) ( ) K/Ω389 K16.273 K10943.3Ω7360 2 3 pointice −= × −=⎟ ⎠ ⎞ ⎜ ⎝ ⎛ dT dR Evaluate dR/dT at the steam point: ( )( ) ( ) K/Ω33.4 K16.373 K10943.3Ω153 2 3 pointsteam −= × −=⎟ ⎠ ⎞ ⎜ ⎝ ⎛ dT dR (d) The thermistor is more sensitive (has greater sensitivity) at lower temperatures.
- 9. Temperature and the Kinetic Theory of Gases 355 The Ideal-Gas Law 47 • A 10.0-L vessel contains gas at a temperature of 0.00ºC and a pressure of 4.00 atm. How many moles of gas are in the vessel? How many molecules? Picture the Problem We can use the ideal-gas law to find the number of moles of gas in the vessel and the definition of Avogadro’s number to find the number of molecules. Apply the ideal-gas law to the gas: nRTPV = ⇒ RT PV n = Substitute numerical values and evaluate n: ( )( ) ( )( ) mol1.79mol786.1 K273Katm/molL108.206 L10.0atm4.00 2 == ⋅⋅× = − n Relate the number of molecules N in the gas in terms of the number of moles n: AnNN = Substitute numerical values and evaluate N: ( )( ) molecules101.08molmolecules/106.022mol1.786 2423 ×=×=N 55 •• After nitrogen (N2) and oxygen (O2), the most abundant molecule in Earth's atmosphere is water, H2O. However, the fraction of H2O molecules in a given volume of air varies dramatically, from practically zero percent under the driest conditions to as high as 4 percent where it is very humid. (a) At a given temperature and pressure, would air be denser when its water vapor 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 vapor molecules, and a cubic meter of air in which 4 percent of the molecules are water vapor molecules? (a) The molecular mass of N2 is 28.014 u (see Appendix C), that of O2 is 31.999 u, and that of H2O is 18.0152 u. Because H2O molecules are lighter than the predominant molecules in air, a given volume of air will be less dense when its water vapor content is higher.
- 10. Chapter 17356 (b) Express the difference in mass between a cubic meter of air containing no water vapor, and a cubic meter of air containing 4% water vapor: humiddryΔ mmm −= The masses of the dry air and humid air are related to their molar masses according to: drydry 04.0 nMm = and humidhumid 04.0 nMm = Substitute for mdry and mhumid and simplify to obtain: ( )humiddry humiddry 04.0 04.004.0Δ MMn nMnMm −= −= From the ideal-gas law we have: RT VP n atm = and so Δm can be written as ( )humiddry atm04.0 Δ MM RT VP m −= Substitute numerical values (28.811 g/mol is an 80% nitrogen and 20% oxygen weighted average) and evaluate Δm: ( )( )( ) ( )( ) ( ) g18g/mol0152.18g/mol8.8112 K300KJ/mol314.8 m00.1kPa325.10104.0 Δ 3 ≈− ⋅ =m Remarks: With 4% H2O, the remaining 96% of the air is mostly a mixture of O2 and N2. With dry air, the 4% that was H2O would be the same mix as the other 96%. Therefore, we need only consider the makeup of the 4%. 57 •• A hot-air balloon is open at the bottom. The balloon has a volume of 446 m3 is filled with air with an average temperature of 100°C. The air outside the balloon has a temperature of 20.0°C and a pressure of 1.00 atm. How large a payload (including the envelope of the balloon itself) can the balloon lift? Use 29.0 g/mol for the molar mass of air. (Neglect the volume of both the payload and the envelope of the balloon.) Picture the Problem Assume that the volume of the balloon is not changing. Then the air inside and outside the balloon must be at the same pressure of about 1.00 atm. The 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.
- 11. Temperature and the Kinetic Theory of Gases 357 Express the net force on the balloon and its contents: balloonthe insideairnet wBF −= (1) Using Archimedes principle, express the buoyant force on the balloon: gmwB fluiddisplacedfluiddisplaced == or gVB balloonoρ= where ρo is the density of the air outside the balloon. Express the weight of the air inside the balloon: gVw ballooni balloonthe insideair ρ= where ρi is the density of the air inside the balloon. Substitute in equation (1) for B and to obtain: balloonthe insideairw ( ) gV gVgVF balloonio ballooniballoononet ρρ ρρ −= −= (2) Express the densities of the air molecules in terms of their number densities, molecular mass, and Avogadro’s number: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = V N N M A ρ Using the ideal-gas law, relate the number density of air N/V to its temperature and pressure: NkTPV = and kT P V N = Substitute for V N to obtain: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = kT P N M A ρ Substitute in equation (2) and simplify to obtain: gV TTkN MP F balloon ioA net 11 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −= Assuming that the average molar mass of air is 29.0 g/mol, substitute numerical values and evaluate Fnet: ( )( ) ( )( ( )( ) ) kN.11 J/K10381.1molparticles/10022.6 m/s81.9m446 K373 1 K293 1 kPa325.101g/mol0.29 2323 23 net = ×× ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = − F
- 12. Chapter 17358 Kinetic Theory of Gases 59 • (a) One mole of argon gas is confined to a 1.0-liter container at a pressure of 10 atm. What is the rms speed of the argon atoms? (b) Compare your answer to the rms speed for helium atoms under the same conditions. Picture the Problem We can express the rms speeds of argon and helium atoms by combining andnRTPV = MRTv 3rms = to obtain an expression for vrms in terms of P, V, and M. See Appendix C for the molar masses of argon and helium. Express the rms speed of an atom as a function of the temperature and its molar mass: M RT v 3 rms = From the ideal-gas law we have: n PV RT = Substitute for RT to obtain: nM PV v 3 rms = (a) Substitute numerical values and evaluate vrms for argon atoms: ( )( )( ) ( )( ) km/s28.0 kg/mol10948.93mol1 m101.0kPa/atm101.325atm103 3 33 Arrms, = × × = − − v (b) Substitute numerical values and evaluate vrms for helium atoms: ( )( )( ) ( )( ) km/s87.0 kg/mol104.003mol1 m101.0kPa/atm101.325atm103 3 33 Herms, = × × = − − v The rms speed of argon atoms is slightly less than one third the rms speed of helium atoms. 65 •• Oxygen (O2) is confined to a cube-shaped container 15 cm on an edge at a temperature of 300 K. Compare the average kinetic energy of a molecule of the gas to the change in its gravitational potential energy if it falls 15 cm (the height of the container). Picture the Problem We can use kTK 2 3 = and ANMghmghU ==Δ to express the ratio of the average kinetic energy of a molecule of the gas to the change in its gravitational potential energy if it falls from the top of the container to the bottom. See Appendix C for the molar mass of oxygen.
- 13. Temperature and the Kinetic Theory of Gases 359 Express the average kinetic energy of a molecule of the gas as a function of its temperature: kTK 2 3 av = Letting h represent the height of the container, express the change in the potential energy of a molecule as it falls from the top of the container to the bottom: A O2 Δ N ghM mghU == Express the ratio of Kav to ΔU and simplify to obtain: ghM kTN N ghM kT U K 22 O A A O 2 3 av 2 3 Δ == Substitute numerical values and evaluate Kav/ΔU: ( )( )( ) ( )( )( ) 4 23 2323 av 109.7 m15.0m/s81.9kg/mol100.322 K300J/K10381.1molparticles/10022.63 Δ ×= × ×× = − − U K *The Distribution of Molecular Speeds 67 •• The fractional distribution function f(v) is defined in Equation 17-36. Because f(v) dv gives the fraction of molecules that have speeds in the range between v and v + dv, the integral of f(v) dv over all the possible ranges of speeds must equal 1. Given that the integral ∫ ∞ −− = 0 232 4 2 advev av π , show that , where f(v) is given by Equation 17-36.( ) 1 0 =∫ ∞ dvvf Picture the Problem We can show that f(v) is normalized by using the given integral to integrate it over all possible speeds. Express the integral of Equation 17- 36: ( ) ∫∫ ∞ − ∞ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 22 23 0 2 2 4 dvev kT m dvvf kTmv π Let kTma 2= to obtain: ( ) ∫∫ ∞ − ∞ = 0 223 0 24 dvevadvvf av π
- 14. Chapter 17360 Use the given integral to obtain: ( ) 1 4 4 2323 0 =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − ∞ ∫ aadvvf π π That is, f(v) is normalized. General Problems 73 •• The Maxwell-Boltzmann distribution applies not just to gases, but also to the molecular motions within liquids. The fact that not all molecules have the same speed helps us understand the process of evaporation. (a) Explain in terms of molecular motion why a drop of water becomes cooler as molecules evaporate from the drop’s surface. (Evaporative cooling is an important mechanism for regulating our body temperatures, and is also used to cool buildings in hot, dry locations.) (b) Use the Maxwell-Boltzmann distribution to explain why even a slight increase in temperature can greatly increase the rate at which a drop of water evaporates. Determine the Concept (a) To escape from the surface of a droplet of water, molecules must have enough kinetic energy to overcome the attractive forces from their neighbors. Therefore the molecules that escape will be those that are moving faster, leaving the slower molecules behind. The slower molecules have less kinetic energy, so the temperature of the droplet, which is proportional to the average kinetic energy per molecule, decreases. (b) As long as the temperature isn’t too high, the molecules that evaporate from a surface will be only those with the most extreme speeds, at the high-energy ″tail″ of the Maxwell-Boltzmann distribution. Within this part of the distribution, increasing the temperature only slightly can greatly increase the percentage of molecules with speeds above a certain threshold. For example, suppose that we set an initial threshold at E = 5kT1, then imagine increasing the temperature by 10% so T2 = 1.1T1. At the threshold, the ratio of the new energy distribution to the old one is ( ) ( ) ( ) 366.11.1 51.1 5 2 3 2 1 1 2 12 2 3 ==⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − − +− eeee T T TF TF kT E kT E an increase of almost 37%. 75 •• In attempting to create liquid hydrogen for fuel, one of the proposals is to convert plain old water (H2O) into H2 and O2 gases by electrolysis. How many moles of each of these gases result from the electrolysis of 2.0 L of water? Picture the Problem We can use the molar mass of water to find the number of moles in 2.0 L of water. Because there are two hydrogen atoms in each molecule
- 15. Temperature and the Kinetic Theory of Gases 361 of water, there must be as many hydrogen molecules in the gas formed by electrolysis as there were molecules of water and, because there is one oxygen atom in each molecule of water, there must be half as many oxygen molecules in the gas formed by electrolysis as there were molecules of water. See Appendix C for the molar masses of hydrogen and oxygen. Express the electrolysis of water into H2 and O2: 222 O2 1 HOH nnn +→ Express the number of moles in 2.0 L of water: mol110 mol g 18.02 L g 1000L2.0 OH2 = × =n Because there are two hydrogen atoms for each water molecule: mol1102H =n Because there is one oxygen atom for each water molecule: ( ) mol55mol1102 1 OH2 1 O 22 === nn 79 •• Current experiments in atomic trapping and cooling can create low- density gases of rubidium and other atoms with temperatures in the nanokelvin (10–9 K) range. These atoms are trapped and cooled using magnetic fields and lasers in ultrahigh vacuum chambers. One method that is used to measure the temperature of a trapped gas is to turn the trap off and measure the time it takes for molecules of the gas to fall a given distance. Consider a gas of rubidium atoms at a temperature of 120 nK. Calculate how long it would take an atom traveling at the rms speed of the gas to fall a distance of 10.0 cm if (a) it were initially moving directly downward and (b) if it were initially moving directly upward. Assume that the atom doesn’t collide with any others along its trajectory. Picture the Problem Choose a coordinate system in which downward is the positive direction. We can use a constant-acceleration equation to relate the fall distance to the initial velocity of the molecule, the acceleration due to gravity, the fall time, and mkTv 3rms = to find the initial velocity of the rubidium molecules.
- 16. Chapter 17362 (a) Using a constant-acceleration equation, relate the fall distance to the initial velocity of a molecule, the acceleration due to gravity, and the fall time: 2 2 1 0 gttvy += (1) Relate the rms speed of rubidium atoms to their temperature and mass: Rb rms 3 m kT v = Substitute numerical values and evaluate vrms: ( )( ) ( )( ) m/s10918.5 kg/u106606.1u47.85 nK120J/K10381.13 3 27 23 rms − − − ×= × × =v Letting vrms = v0, substitute in equation (1) to obtain: ( ) ( ) 22 2 13 m/s81.9m/s10918.5m100.0 tt +×= − Use the quadratic formula or your graphing calculator to solve this equation for its positive root: ms142s14218.0 ==t (b) If the atom is initially moving upward: m/s10918.5 3 0rms − ×−== vv Substitute in equation (1) to obtain: ( ) ( ) 22 2 13 m/s81.9m/s10918.5m100.0 tt +×−= − Use the quadratic formula or your graphing calculator to solve this equation for its positive root: ms431=t