# Daily Archives: August 19, 2019

200 posts

## 18.84 … CALC Earth’s Atmosphere. In the troposphere the part of the atmosphere that extends from earth’s surface to an altitude of about 11 km the temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship T = T0 – ay where T0 is the temperature at the earth’s surface and T is the temperature at height y the pressure p at height y is ln a p p0 b = Mg Ra ln a T0 – ay T0 b where p0 is the pressure at the earth’s surface and M is the molar mass for air. The coefficient a is called the lapse rate of temperature. It varies with atmospheric conditions but an average value is about 0.6 C_>100 m. (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that a S 0. (c) With a = 0.6 C_>100 m calculate p for y = 8863 m and compare your answer to the result of Example 18.4. Take T0 = 288 K and p0 = 1.00 atm.

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## 18.83 … CP Dark Nebulae and the Interstellar Medium. The dark area in Fig. P18.83 that appears devoid of stars is a dark nebula a cold gas cloud in interstellar space that contains enough material to block out light from the stars behind it. A typical dark nebula is about 20 light-years in diameter and contains about 50 hydrogen atoms per cubic centimeter (monatomic hydrogen not H2) at about 20 K. (A light-year is the distance light travels in vacuum in one year and is equal to 9.46 * 1015 m.) (a) Estimate the mean free path for a hydrogen atom in a dark nebula. The radius of a hydrogen atom is 5.0 * 10-11 m. (b) Estimate the rms speed of a hydrogen atom and the mean free time (the average time between collisions for a given atom). Based on this result do you think that atomic collisions such as those leading to H2 molecule formation are very important in determining the composition of the nebula? (c) Estimate the pressure inside a dark nebula. (d) Compare the rms speed of a hydrogen atom to the escape speed at the surface of the nebula (assumed spherical). If the space around the nebula were a vacuum would such a cloud be stable or would it tend to evaporate? (e) The stability of dark nebulae is explained by the presence of the interstellar medium (ISM) an even thinner gas that permeates space and in which the dark nebulae are embedded. Show that for dark nebulae to be in equilibrium with the ISM the numbers of atoms per volume 1N>V2 and the temperatures 1T2 of dark nebulae and the ISM must be related by 1N>V2nebula 1N>V2ISM = TISM Tnebula (f) In the vicinity of the sun the ISM contains about 1 hydrogen atom per 200 cm3. Estimate the temperature of the ISM in the vicinity of the sun. Compare to the temperature of the sun’s surface about 5800 K. Would a spacecraft coasting through interstellar space burn up? Why or why not?

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## 18.82 .. DATA The statistical quantities “average value” and “root-mean-square value” can be applied to any distribution. Figure P18.82 shows the scores of a class of 150 students on a 100-point quiz. (a) Find the average score for the class. (b) Find the rms score for the class. (c) Which is higher: the average score or the rms score? Why? CHaLLenge proBLeMs

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## 18.81 … DATA The Dew Point and Clouds. The vapor pressure of water (see Exercise 18.44) decreases as the temperature decreases. The table lists the vapor pressure of water at various temperatures: Temperature 1 _C2 Vapor Pressure 1Pa2 10.0 1.23 * 103 12.0 1.40 * 103 14.0 1.60 * 103 16.0 1.81 * 103 18.0 2.06 * 103 20.0 2.34 * 103 22.0 2.65 * 103 24.0 2.99 * 103 26.0 3.36 * 103 28.0 3.78 * 103 30.0 4.25 * 103 If the amount of water vapor in the air is kept constant as the air is cooled the dew point temperature is reached at which the partial pressure and vapor pressure coincide and the vapor is saturated. If the air is cooled further the vapor condenses to liquid until the partial pressure again equals the vapor pressure at that temperature. The temperature in a room is 30.0_C. (a) A meteorologist cools a metal can by gradually adding cold water. When the can’s temperature reaches 16.0_C water droplets form on its outside surface. What is the relative humidity of the 30.0_C air in the room? On a spring day in the midwestern United States the air temperature at the surface is 28.0_C. Puffy cumulus clouds form at an altitude where the air temperature equals the dew point. If the air temperature decreases with altitude at a rate of 0.6 C_>100 m at approximately what height above the ground will clouds form if the relative humidity at the surface is (b) 35%; (c) 80%?

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## 18.80 .. DATA A steel cylinder with rigid walls is evacuated to a high degree of vacuum; you then put a small amount of helium into the cylinder. The cylinder has a pressure gauge that measures the pressure of the gas inside the cylinder. You place the cylinder in various temperature environments wait for thermal equilibrium to be established and then measure the pressure of the gas. You obtain these results: (a) Recall (Chapter 17) that absolute zero is the temperature at which the pressure of an ideal gas becomes zero. Use the data in the table to calculate the value of absolute zero in _C. Assume that the pressure of the gas is low enough for it to be treated as an ideal gas and ignore the change in volume of the cylinder as its temperature is changed. (b) Use the coefficient of volume expansion for steel in Table 17.2 to calculate the percentage change in the volume of the cylinder between the lowest and highest temperatures in the table. Is it accurate to ignore the volume change of the cylinder as the temperature changes? Justify your answer.

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## 18.79 … CP Oscillations of a Piston. A vertical cylinder of radius r contains an ideal gas and is fitted with a piston of mass m that is free to move (Fig. P18.79). The piston and the walls of the cylinder are frictionless and the entire cylinder is placed in a constant-temperature bath. The outside air pressure is p0. In equilibrium the piston sits at a height h above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance h + y above the bottom of the cylinder where y V h. (c) After the piston is displaced from equilibrium and released it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small are the oscillations simple harmonic? How can you tell?

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## 18.78 .. CALC Calculate the integral in Eq. (18.30) 1 q 0 vf 1v2 dv and compare this result to vav as given by Eq. (18.35). (Hint: Make the change of variable v2 = x and use the tabulated integral L q 0 xne-axdx = n! an+1 where n is a positive integer and a is a positive constant.)

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## 18.76 .. (a) Calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300 K. (b) Calculate the moment of inertia of an oxygen molecule 1O22 for rotation about either the y- or z-axis shown in Fig. 18.18b. Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of 1.21 * 10-10 m. The molar mass of oxygen atoms is 16.0 g>mol. (c) Find the rms angular velocity of rotation of an oxygen molecule about either the y- or z-axis shown in Fig. 18.18b. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery 110 000 rev>min2? 18.77 .. CALC (a) Explain why in a gas of N molecules the number of molecules having speeds in the finite interval v to v + _v is _N = N1 v+_v v f 1v2 dv. (b) If _v is small then f 1v2 is approximately constant over the interval and _N _ Nf 1v2_v. For oxygen gas 1O2 molar mass 32.0 g>mol2 at T = 300 K use this approximation to calculate the number of molecules with speeds within _v = 20 m>s of vmp . Express your answer as a multiple of N. (c) Repeat part (b) for speeds within _v = 20 m>s of 7vmp. (d) Repeat parts (b) and (c) for a temperature of 600 K. (e) Repeat parts (b) and (c) for a temperature of 150 K. (f) What do your results tell you about the shape of the distribution as a function of temperature? Do your conclusions agree with what is shown in Fig. 18.23?

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## 18.75 .. CALC Calculate the integral in Eq. (18.31) 1 q 0 v2 f 1v2 dv and compare this result to 1v22av as given by Eq. (18.16). (Hint: You may use the tabulated integral L q 0 x2ne-ax2 dx = 1 # 3 # 5 # # # 12n – 12 2n+1an A p a where n is a positive integer and a is a positive constant.)

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## 18.74 . Planetary Atmospheres. (a) The temperature near the top of Jupiter’s multicolored cloud layer is about 140 K. The temperature at the top of the earth’s troposphere at an altitude of about 20 km is about 220 K. Calculate the rms speed of hydrogen molecules in both these environments. Give your answers in m>s and as a fraction of the escape speed from the respective planet (see Problem 18.72). (b) Hydrogen gas 1H22 is a rare element in the earth’s atmosphere. In the atmosphere of Jupiter by contrast 89% of all molecules are H2. Explain why using your results from part (a). (c) Suppose an astronomer claims to have discovered an oxygen 1O22 atmosphere on the asteroid Ceres. How likely is this? Ceres has a mass equal to 0.014 times the mass of the moon a density of 2400 kg>m3 and a surface temperature of about 200 K.

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## 18.73 .. CP (a) Show that a projectile with mass m can “escape” from the surface of a planet if it is launched vertically upward with a kinetic energy greater than mgRp where g is the acceleration due to gravity at the planet’s surface and Rp is the planet’s radius. Ignore air resistance. (See Problem 18.72.) (b) If the planet in question is the earth at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass 28.0 g>mol) equal that required to escape? What about a hydrogen molecule (molar mass 2.02 g>mol?) (c) Repeat part (b) for the moon for which g = 1.63 m>s2 and Rp = 1740 km. (d) While the earth and the moon have similar average surface temperatures the moon has essentially no atmosphere. Use your results from parts (b) and (c) to explain why.

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## 18.72 . Hydrogen on the Sun. The surface of the sun has a temperature of about 5800 K and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is 1.67 * 10-27 kg.) (b) The escape speed for a particle to leave the gravitational influence of the sun is given by 12GM>R21>2 where M is the sun’s mass R its radius and G the gravitational constant (see Example 13.5 of Section 13.3). Use Appendix F to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain.

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## 18.71 .. It is possible to make crystalline solids that are only one layer of atoms thick. Such “two-dimensional” crystals can be created by depositing atoms on a very flat surface. (a) If the atoms in such a two-dimensional crystal can move only within the plane of the crystal what will be its molar heat capacity near room temperature? Give your answer as a multiple of R and in J>mol # K. (b) At very low temperatures will the molar heat capacity of a two-dimensional crystal be greater than less than or equal to the result you found in part (a)? Explain why.

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## 18.70 . (a) What is the total random translational kinetic energy of 5.00 L of hydrogen gas (molar mass 2.016 g>mol) with pressure 1.01 * 105 Pa and temperature 300 K? (Hint: Use the procedure of Problem 18.67 as a guide.) (b) If the tank containing the gas is placed on a swift jet moving at 300.0 m>s by what percentage is the total kinetic energy of the gas increased? (c) Since the kinetic energy of the gas molecules is greater when it is on the jet does this mean that its temperature has gone up? Explain.

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## 18.69 .. CP CALC The Lennard-Jones Potential. A commonly used potential-energy function for the interaction of two molecules (see Fig. 18.8) is the Lennard-Jones 6-12 potential: U1r2 = U0 c a R0 r b 12 – 2a R0 r b 6 d where r is the distance between the centers of the molecules and U0 and R0 are positive constants. The corresponding force F 1r2 is given in Eq. (14.26). (a) Graph U1r2 and F 1r2 versus r. (b) Let r1 be the value of r at which U1r2 = 0 and let r2 be the value of r at which F 1r2 = 0. Show the locations of r1 and r2 on your graphs of U1r2 and F 1r2. Which of these values represents the equilibrium separation between the molecules? (c) Find the values of r1 and r2 in terms of R0 and find the ratio r1>r2. (d) If the molecules are located a distance r2 apart [as calculated in part (c)] how much work must be done to pull them apart so that r S q?

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## 18.68 . CP (a) Compute the increase in gravitational potential energy for a nitrogen molecule (molar mass 28.0 g>mol) for an increase in elevation of 400 m near the earth’s surface. (b) At what temperature is this equal to the average kinetic energy of a nitrogen molecule? (c) Is it possible that a nitrogen molecule near sea level where T = 15.0_C could rise to an altitude of 400 m? Is it likely that it could do so without hitting any other molecules along the way? Explain.

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## 18.67 .. You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is 22.0_C. Assume that all the gas is N2 of molar mass 28.0 g>mol. (a) Find the mass of a single N2 molecule. (b) How much translational kinetic energy does an average N2 molecule have? (c) How many N2 molecules are in this balloon? (d) What is the total translational kinetic energy of all the molecules in the balloon?

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## 18.66 .. Helium gas is in a cylinder that has rigid walls. If the pressure of the gas is 2.00 atm then the root-mean-square speed of the helium atoms is vrms = 176 m>s. By how much (in atmospheres) must the pressure be increased to increase the vrms of the He atoms by 100 m>s? Ignore any change in the volume of the cylinder.

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## 18.65 .. A sealed box contains a monatomic ideal gas. The number of gas atoms per unit volume is 5.00 * 1020 atoms>cm3 and the average translational kinetic energy of each atom is 1.80 * 10-23 J. (a) What is the gas pressure? (b) If the gas is neon (molar mass 20.18 g>mol) what is vrms for the gas atoms?

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## 18.64 . The size of an oxygen molecule is about 2.0 * 10-10 m. Make a rough estimate of the pressure at which the finite volume of the molecules should cause noticeable deviations from idealgas behavior at ordinary temperatures 1T = 300 K2.

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## 18.63 .. You have two identical containers one containing gas A and the other gas B. The masses of these molecules are mA = 3.34 * 10-27 kg and mB = 5.34 * 10-26 kg. Both gases are under the same pressure and are at 10.0_C. (a) Which molecules (A or B) have greater translational kinetic energy per molecule and rms speeds? (b) Now you want to raise the temperature of only one of these containers so that both gases will have the same rms speed. For which gas should you raise the temperature? (c) At what temperature will you accomplish your goal? (d) Once you have accomplished your goal which molecules (A or B) now have greater average translational kinetic energy per molecule?

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## 18.62 .. bio A person at rest inhales 0.50 L of air with each breath at a pressure of 1.00 atm and a temperature of 20.0_C. The inhaled air is 21.0% oxygen. (a) How many oxygen molecules does this person inhale with each breath? (b) Suppose this person is now resting at an elevation of 2000 m but the temperature is still 20.0_C. Assuming that the oxygen percentage and volume per inhalation are the same as stated above how many oxygen molecules does this person now inhale with each breath? (c) Given that the body still requires the same number of oxygen molecules per second as at sea level to maintain its functions explain why some people report “shortness of breath” at high elevations.

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## 18.61 .. bio How Many Atoms Are You? Estimate the number of atoms in the body of a 50@kg physics student. Note that the human body is mostly water which has molar mass 18.0 g>mol and that each water molecule contains three atoms.

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## 18.59 .. CP A large tank of water has a hose connected to it (Fig. P18.59). The tank is sealed at the top and has compressed air between the water surface and the top. When the water height h has the value 3.50 m the absolute pressure p of the compressed air is 4.20 * 105 Pa. Assume that the air above the water expands at constant temperature and take the atmospheric pressure to be 1.00 * 105 Pa. (a) What is the speed with which water flows out of the hose when h = 3.50 m? (b) As water flows out of the tank h decreases. Calculate the speed of flow for h = 3.00 m and for h = 2.00 m. (c) At what value of h does the flow stop? 18.60 .. CP A light plastic sphere with mass m = 9.00 g and density r = 4.00 kg>m3 is suspended in air by thread of negligible mass. (a) What is the tension T in the thread if the air is at 5.00oC and p = 1.00 atm? The molar mass of air is 28.8 g>mol. (b) How much does the tension in the thread change if the temperature of the gas is increased to 35.0oC? Ignore the change in volume of the plastic sphere when the temperature is changed.

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## 18.58 .. A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 0.300 atm at 20.0_C. The round part of the tank has a radius of 10.0 cm and the gas is supporting a piston that can move up and down in the cylinder without friction. There is a vacuum above the piston. (a) What is the mass of this piston? (b) How tall is the column of gas that is supporting the piston?

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## 18.57 .. CP A balloon of volume 750 m3 is to be filled with hydrogen at atmospheric pressure 11.01 * 105 Pa2. (a) If the hydrogen is stored in cylinders with volumes of 1.90 m3 at a gauge pressure of 1.20 * 106 Pa how many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if both the gas in the balloon and the surrounding air are at 15.0_C? The molar mass of hydrogen 1H22 is 2.02 g>mol. The density of air at 15.0_C and atmospheric pressure is 1.23 kg>m3. See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass 4.00 g>mol) instead of hydrogen again at 15.0_C?

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## 18.56 .. A flask with a volume of 1.50 L provided with a stopcock contains ethane gas 1C2H62 at 300 K and atmospheric pressure 11.013 * 105 Pa2. The molar mass of ethane is 30.1 g>mol. The system is warmed to a temperature of 550 K with the stopcock open to the atmosphere. The stopcock is then closed and the flask is cooled to its original temperature. (a) What is the final pressure of the ethane in the flask? (b) How many grams of ethane remain in the flask?

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