Chapter 3 Problems - Problems...

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Unformatted text preview: Problems _——_—__————————-—-— Kinetic Theory of Gases 3.1 Apply the kinetic theory of gases to explain Boyle’s law, Charles’ law, and Dalton’s law. 3.2 Is temperature a microscopic or macroscopic concept? Explain. 3.3 In applying the kinetic molecular theory to gases, we have assumed that the walls of the container are elastic for molecular collisions. Actually, whether these collisions are elastic or inelastic makes no difference as long as the walls are at the same temperature as the gas. Explain. 3.4 If 2.0 x 1023 argon (Ar) atoms strike 4.0 cm2 of wall per second at a 90° angle to the wall when moving with a speed of 45,000 cm 3“, what pressure (in atm) do they exert on the .wall? 3.5 A square box contains He at 25 °C. If the atoms are colliding with the walls perpendicularly (at 90°) at the rate of 4.0 x 1022 times per second, calculate the force and the pressure exerted on the wall given that the area of the wall is 100 cm2 and the speed of the atoms is 600 m s". 3.6 Calculate the average translational kinetic energy for a N2 molecule and for 1 mole of N; at 20 °C. 3.7 To what temperature must He atoms be cooled so that they have the same um,s as 02 at 25 °C? 3.8 The ems of CH4 is 846 m 3“. What is the temperature of the gas? 3.9 Calculate the value of the crms of ozone molecules in the stratosphere, where the temperature is 250 K. 3.10 At what temperature will He atoms have the same crms value as N2 molecules at 25 °C? Solve this problem without calculating the value of ems for N2. Maxwell Speed Distrbution 3.11 List the conditions used for deriving the Maxwell speed distribution. 3.12 Plot the speed distribution function for (a) He, 02, and UF6 at the same temperature, and (b) C02 at 300 K and 1000 K. 3.13 Account for the maximum in the Maxwell speed distribution curve (Figure 3.4) by plotting the following two curves on the same graph: (1) 62 versus c and (2) e‘mcz/ZkBT versus c. Use neon (Ne) at 300 K for the plot in (2). 3.14 A N2 molecule at 20 °C is released at sea level to travel upward. Assuming that the temperature is constant and that the molecule does not collide with other molecules, how far would it travel (in meters) before coming to rest? Do the same calculation for a He atom. [Hints To calculate the altitude, h, the molecule will travel, equate its kinetic energy with the potential energy, mgh, where m is the mass and g the acceleration due to gravity (9.81 m 5—2)] 3.15 The speeds of 12 particles (in cm s“) are 0.5, 1.5, 1.8, 1.8, 1.8, 1.8, 2.0, 2.5, 2.5, 3.0, 3.5, and 4.0. Find (a) the average speed, (b) the root-mean-square speed, and (c) the most probable speed of these particles. Explain your results. 3.16 At a certain temperature, the speeds of six gaseous molecules in a container are 2.0 m s“, 2.2 m s", 2.6 m s“, 2.7 m s‘l, 3.3 In 5—1, and 3.5 m 3“. Calculate the root-mean-square speed and the average speed of the molecules. These two average values are close to each other, but the root-mean-square value is always the larger of the two. Why? 69 70' Chapter 3: Kinetic Theory of Gases 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 The following diagram shows the Maxwell speed distribution curves for a certain ideal gas at two different temperatures (T1 and T2). Calculate the value of T2. “0) 0 500 1000 1500 2000 c/m-s'1 Following the procedure used in the chapter to find the value of E, derive an expression for cm. (Hint: You need to consult the Handbook of Chemistry and Physics to evaluate definite integrals.) Derive an expression for em, following the procedure described in the chapter. Calculate the values of cm, amp, and E for argon at 298 K. Calculate the value of cmp for C2H6 at 25 °C. What is the ratio of the number of molecules with a speed of 989 In S’1 to the number of molecules with this value of cm? Derive an expression for the most probable translational energy for an ideal gas. Compare your result with the average translational energy for the same gas. Considering the magnitude of molecular speeds, explain why it takes so long (on the order of minutes) to detect the odor of ammonia when someone opens a bottle of concentrated ammonia at the other end of a laboratory bench. How does the mean free path of a gas depend on (a) the temperature at constant volume, (b) the density, (c) the pressure at constant temperature, ((1) the volume at constant temperature, and (e) the size of molecules? A bag containing 20 marbles is’being shaken vigorously. Calculate the mean free path of the marbles if the volume of the bag is 850 cm3. The diameter of each marble is 1.0 cm. Calculate the mean free path and the binary number of collisions per liter per second between HI molecules at 300 K and 1.00 atm. The collision diameter of the HI molecules may be taken to be 5.10 A. Assume ideal-gas behavior. Ultra-high vacuum experiments are routinely performed at a total pressure of 1.0 x 10‘10 torr. Calculate the mean free path of N2 molecules at 350 K under these conditions. Suppose that helium’atoms in a sealed container all start with the same speed, 2.74 x 104 cm s“. The atoms are then allowed to collide with one another until the Maxwell distribution is established. What is the temperature of the gas at equilibrium? Assume that there is no heat exchange between the gas and its surroundings. Compare the collision number and the mean free path for air molecules at (a) sea level (T = 300 K and density = 1.2 g L‘l) and (b) in the stratosphere (T = 250 K and density = 5.0 x 10‘3 g L‘l). The molar mass of air may be taken as 29.0 g, and the collision diameter is 3.72 A. Calculate the values of Z1 and Zn for mercury (Hg) vapor at 40°C, both at P = 1.0 atm and at P = 0.10 atm. How do these two quantities depend on pressure? ‘ a - lt Problems Gas Viscosity 3.31 Account for the difference between the dependence of viscosity on temperature for a liquid and a gas. 3.32 Calculate the values of the average speed and collision diameter for ethylene at 288 K. The viscosity of ethylene is 99.8 x 10‘7 N s m‘2 at the same temperature. 3.33 The viscosity of sulfur dioxide at 210°C and 1.0 atm pressure is 1.25 x 10‘5 N s m‘z. Calculate the collision diameter of the S02 molecule and the mean free path at the given temperature and pressure. Gas Diffusion and Effusion 3.34 Derive Equation 3.23 from Equation 3.14. 3.35 An inflammable gas is generated in marsh lands and sewage by a certain anaerobic bacterium. A pure sample of this gas was found to efl‘use through an orifice in 12.6 min. Under identical conditions of temperature and pressure, oxygen takes 17.8 min to effuse through the same orifice. Calculate the molar mass of the gas, and suggest what this gas might be. 3.36 Nickel forms a gaseous compound of the formula Ni(CO)x. What is the value of x given the fact that under the same conditions of temperature and pressure, methane (CH4) effuses 3.3 times .faster than the compound? 3.37 In 2.00 min, 29.7 mL of He effuse through a small hole. Under the same conditions of temperature and pressure, 10.0 mL of a mixture of CO and CO2 effuse through the hole in the same amount of time. Calculate the percent composition by volume of the mixture. 3.38 Uranium-235 can be separated from uranium-238 by the effusion process involving UF6. Assuming a 50:50 mixture at the start, what is the percentage of enrichment after a single stage of separation? 3.39 An equimolar mixture of H2 and D2 effuses through an orifice at a certain temperature. Calculate the composition (in mole fractions) of the gas that passes through the orifice. The molar mass of deuterium is 2.014 g mol“l. 3.40 The rate (reg) at which molecules c0nfined to a volume V effuse through an orifice of area A is given by -(1/4)nNAEA/ V, where n is the number of moles of the gas. An automobile tire of volume 30.0 L and pressure 1,500 torr is punctured as it runs over a sharp nail. (a) Calculate the effusion rate if the diameter of the hole is 1.0 mm. (b) How long would it take to lose half of the air in the tire through effusion? Assume a constant effusion rate and constant volume. The molar mass of air is 29.0 g, and the temperature is 32.0°C. Equipartition of Energy 3.41 Calculate the various degrees of freedom for the following molecules: (a) Xe, (b) HCl, (c) CS2, (d) C2H2, (e) C6H6, and (f) a hemoglobin molecule containing 9272 atoms. 3.42 Explain the equipartition of energy theorem. Why does it fail for diatomic and polyatomic molecules? 3.43 A quantity of 500 joules of energy is delivered to one mole of each of the following gases at 298 K and the same fixed volume: Ar, CH4, H2. Which gas will have the highest rise in temperature? 3.44 Calculate the mean kinetic energy (Ems) in joules of the following molecules at 350 K: (a) He, (b) CO2, and (c) UF6. Explain your results. 3.45 A sample of neon gas is heated from 300 to 390 K. Calculate the percent increase in its kinetic energy. 3.46 Calculate the value of CV for H2, C02, and S02, assuming that only translational and rotational motions contribute to the heat capacities. Compare your results with the values listed in Table 3.3. Explain the differences. :- 71 Chapter 3: Kinetic Theory of Gases 3.47 One mole of ammon_ia initially at 5 °C is placed in contact with 3 moles of helium initially at 90 °C. Given that C V for ammonia is 3R, if the process is carried out at constant total volume, what is the final temperature of the gases? 3.48 The typical energy differences between successive rotational, vibrational, and electronic energy levels are 5.0 x 10‘22 J, 0.50 x 10‘19 J, and 1.0 x 10‘17 J, respectively. Calculate the ratios of the numbers of molecules in the two adjacent energy levels (higher to lower) in each case at 298 K. 3.49 The first excited electronic energy level of the helium atom is 3.13 x 10“18 J above the ground level. Estimate the temperature at which the electronic motion will begin to make a significant contribution to the heat capacity. That is, at what temperature will the ratio of the population of the first excited state to the ground state be 5.0%? 3.50 Consider 1 mole each of gaseous He and N2 at the same temperature and pressure. State which gas (if any) has the greater value for: (a) c", (b) cms, (c) Em“, (d) Z1, (e) Z”, (1') density, (g) mean free path, and (h) viscosity. 3.51 The root-mean-square velocity of a certain gaseous oxide is 493 m s‘1 at 20 °C. What is the molecular formula of the compound? 3.52 At_ 298 K, the CV of S02 is greater than that of C02. At very high temperatures (>1000 K), the C y of C02 is greater than that of 802. Explain. 3.53 Calculate the total translational kinetic energy of the air molecules in a spherical balloon of ' radius 43.0 cm at 24°C and 1.2 atm. Is this enough energy to heat 200 mL of water from 20°C to 90 °C for a cup of tea? The density of water is 1.0 g cm‘3, and its specific heat is 4.184 J g-1 °c-1. Additional Problems 3.54 The following apparatus can be used to measure atomic and molecular speed. A beam of metal atoms is directed at a rotating cylinder in a vacuum. A small opening in thecylinder allows the atoms to strike a target area. Because the cylinder is rotating, atoms traveling at different speeds will strike the target at different positions. In time, a layer of the metal will deposit on the target area, and the variation in its thickness is found to correspond to Maxwell’s speed distribution. In one experiment, it is found that at 850 °C, some bismuth (Bi) atoms struck the target at a point 2.80 cm from the spot directly opposite the slit. The diameter of the cylinder is 15.0 cm, and it is rotating at 130 revolutions per second. (a) Calculate the speed (m s") at which the target is moving. (Hint: The circumference of a circle is given by 2m, where r is the radius.) (b) Calculate the time (in seconds) it takes for the target to travel 2.80 cm. (c) Determine the speed of the Bi atoms. Compare your result in (c) with the can, value for Bi at 850 °C. Comment on the difference. Rotating cylinder / Bi atoms /Target Slit/ 3.55 From your knowledge of heat capacity, explain why hot, humid air is more uncomfortable than hot, dry air and cold, damp air is more uncomfortable than cold, dry air. 3.56 The escape velocity, v, from Earth’s gravitational field is given by (2GM/r) , where G is the universal gravitational constant (6.67 x 10‘11 m3 lrg‘l 3‘2), M is the mass of Earth a n 1/2 Problems i. (6.0 x 1024 kg), and r is the distance from the center of Earth to the object, in meters. Compare ‘ the average speeds of He and N2 molecules in the thermosphere (altitude about 100 km, T = 250 K). Which of the two molecules will have a greater tendency to escape? The radius of Earth is 6.4 x 106 m. 3.57 Suppose you are traveling in a space vehicle on a journey to the moon. The atmosphere in the i vehicle consists of 20% oxygen and 80% helium by volume. Before takeoff, someone noticed a small leakage that, if left unchecked, would lead to a continual loss of gas at a rate of 0.050 atm , day’1 through effusion. If the temperature in the space vehicle is maintained at 22 °C and the . volume of the vehicle is 1.5 X 104 L, calculate the amounts of helium and oxygen in grams that must be stored on a 10-day journey to allow for the leakage. (Hint: First calculate the quantity of gas lost each day, using PV = nRT. Note that the rate of effusion is proportional to the pressure of the gas. Assume that effusion does not affect the pressure or the mean free path of the gas in the space vehicle.) 3.58 Calculate the ratio of the number of O3 molecules with a speed of 1300 m s‘1 at 360 K to the number with that speed at 293 K. 3.59 Calculate the collision frequency for 1.0 mole of krypton (Kr) at equilibrium at 300 K and 1.0 atm pressure. Which of the following alterations increases the collision frequency more: (a) doubling the temperature at constant pressure or (b) doubling the pressure at constant temperature? (Hint: Use the collision diameter in Table 3.1.) 3.60 Apply your knowledge of the kinetic theory of gases to the following situations. (a) Two flasks of volumes V1 and V2 (where V2 > V1) contain the same number of helium atoms at the same temperature. (i) Compare the root-mean-square (rms) speeds and average kinetic energies of the helium (He) atoms in the flasks. (ii) Compare the frequency and the force with which the He atoms collide with the walls of their containers. (b) Equal numbers of He atoms are placed \ in two flasks of the same volume at temperatures T1 and T2 (where T2 > T1). (i) Compare the rms speeds of the atoms in the two flasks. (ii) Compare the frequency and the force with which the He atoms collide with the walls of their containers. (c) Equal numbers of He and neon (Ne) 1 atoms are placed in two flasks of the same volume. The temperature of both gases is 74°C. Comment on the validity of the following statements: (i) The rms speed of He is equal to that of Ne. (ii) The average kinetic energies of the two gases are equal. (iii) The rms speed of each He atom is 1.47 x 103 m s". 1 ‘1 ...
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Chapter 3 Problems - Problems...

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