Chapter 19: The Kinetic Theory of Gases Questions and Example Problems
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1 Chapter 9: The Kinetic Theory of Gases Questions and Example Problems N M V f N M Vo sam n pv nrt Nk T W nrt ln B A molar nmv RT k T rms B p v K k T λ rms avg B V M m πd N/V Q nc T Q nc T C C + R E nc V V P P p V int γ C γ γ / C pv constant TV constant P V Questions a. Why does the boiling temperature of a liquid increase with pressure? b. When two gases are mixed, if they are to be in thermal equilibrium they must have the same average molecular speed. Is the statement correct? Why or why not? c. If you fill a saucer with water at room temperature, the water, under normal conditions, will evaporate completely. It is easy to believe that some of the more energetic molecules can escape from the water surface, but how can all of them eventually escape? Many of then in fact the vast majority do not have enough energy to do so. d. Suppose that you want to heat a gas so that its temperature will be as high as possible. Would you heat it under conditions of constant pressure or constant volume? Why? e. The plunger of a bicycle tire pump is pushed down rapidly with the end of the pump sealed so that no air escapes. And there is little time for heat to flow through the cylinder wall. Explain why the cylinder of the pump becomes warm to the touch. Ignore friction and assume that air behaves as in ideal gas. Example 9. Water has a molar mass of 8 g/mol. a. How many water molecules are there in your body? (Assume that you are nearly all water.) b. How many drops of water are there in all the oceans of the world? (The mass of the world s oceans is about 0 kg) c. Which of these two numbers from (a) and (b) is the larger? V T Example 9. A quantity of ideal gas at 0 o C and 00 kpa occupies a volume of.5 m. a. How many moles of the gas are present? b. If the pressure is now raised to 00 kpa and the temperature is raised to 0 o C, how much volume does the gas occupy? c. Interpret these results using a pv- and energy bar diagrams. Example 9. A sample of ideal gas expands from an initial pressure and volume of atm and.0 L to a final volume of.0 L. The initial temperature of the gas is 00 K. What are the final pressure and temperature of the gas and how much work is done by the gas during the expansion, if the expansion is isothermal? 9.
2 Example 9.4 Determine the rms speed of a single molecule of an ideal gas (a) H and N at room temperature (0 o C), (b) H in outer space, and (c) free electrons in the sun atmosphere ( 0 6 K)? Example 9.5 One mole of an ideal diatomic gas goes from to along two different paths as shown. a. Rank the (i) change in internal energy and (ii) heat transfers for transitions and. Explain your reasoning. b. During the transition along, what is the (iii) change in internal energy of the gas and (iv) heating done on the gas? c. Repeat part (b) for transition. Example 9.6 The figure shows a thermodynamics process followed by 0.00 moles of helium gas (He ). a. Draw an accurate energy bar diagram for the cycle that clearly shows appropriate height changes. Rank all of the workings, heatings, and changes in internal energy. b. Determine the pressure, temperature, and volume of the gas at points,, and. Put your results in a table for easy reading. c. Determine the workings, heatings, and changes in internal energies for each path? d. As checks, the following must be true of your numerical values. (i) Is ΣΔE int 0 and (ii) Q net W net? (iii) Do the numerical values from part (c) agree with your rankings from part (a)? 9.
3 Example 9.7 The figure shows a thermodynamics process followed by 0 mg of helium. e. Determine the pressure, temperature, and volume of the gas at points,, and. Put your results in a table for easy reading. f. How much work is done on the gas during each of the three segments? g. How much heat energy is transferred to or from the gas during each of the three segments? I need to figure out the signs and draw energy bar diagrams. a. Helium gas is a monatomic gas and the given number of moles is mhe 0 0 g n 0.00 mol n MHe 4 g/mol The underlying structure is the st law but the details are given by the IGL. In other words, to get the (p, V, T) at anyone state, is use the IGL equations with the IGL process to setup the equations. Let s setup a table and fill-in the values to make sure I get them all. State-: nrt pv nrt p V n 0.00 mol T K V 0 cm 0 m Transition is Pa.0 0 Pa atm 0 isochoric (constant volume process) so that the IGL implies T T p V constant T T 5T 00 K p p p T 406K p /p 5 Transition is isothermal so T constant p V p V V p V 5V 5 0 m I now summarize everything into my table p T 406K p /p 5 p(0 5 Pa) T(K) V(m ) b. To determine the work done through each transition, we do the following: Transition is isochoric: W 0 Transition is isothermic: 9.
4 W V p V ln 5 0 Pa 0 m 5 ln 85 J W Transition is isobaric: W 5 p (V V ).0 0 ( 5) J W 5 V c. To determine the heating through each transition, we do the following: Transition is isochoric heating: Q E nc T nr(t T ) (p V p V ) int V 5 5 Transition is isothermic heating: Q W 85 J Q Pa m 609 J Q Transition is isobaric heating: 5 Q nc T nr(t T ) (p V p V ) P J 0 J Q Example 9.7 In a bottle of champagne, the pocket of gas (primarily carbon dioxide) between the liquid and the cork is at pressure of p 5.00 atm and temperature of 5 o C. When the cork is pulled from the bottle, the gas undergoes an adiabatic expansion until its pressure matches the ambient air pressure of.00 atm. a. Draw a pv-diagram for the situation and what is the ratio of the molar specific heats? Explain. b. What is its temperature at the end of the adiabatic expansion? Let p, V, and T represent the pressure, volume, and temperature of the initial state of the gas, and let p, V, and T be the pressure, volume, and temperature of the final state. Since the process is adiabatic p V γ p V γ. Combining with the ideal gas law, pv nrt, we obtain γ γ γ γ γ γ γ γ pv p (T / p ) p T constant p T p T With γ 4/ which gives ( γ)/ γ /4, the temperature at the end of the adiabatic expansion is γ / 4 p γ 5.00 atm p.00 atm T T (78 K) 86 K 87 C T. 9.4
5 Example A The temperature of.00 mol of a gas with C V 6.00 cal/mol K is to be raised 50.0 K. a. If the process is at constant volume, draw a pv-diagram of the situation and determine Q, W, E int, and K (total translational kinetic energy) of the gas? b. Repeat (a) if instead the process is at constant pressure. c. Repeat (a) if instead the process is adiabatic. I would expect that because the internal energy and the translation kinetic energy are temperature dependent, these values will be the same for all three processes. Furthermore, since the question specifically states translational kinetic energy, it means monatomic: C V R/. The given quantities are T 50 K, n.0 mol, and converting joules to calories in the ideal gas constant value gives R.0 cal/mol K. a. Constant volume process: since the work done by the gas is W 0, and the change in the internal energy is Eint ncv T cal Eint The first law gives Eint Q W Q 900 cal The change in the total translational kinetic energy is K nr T cal K b. Constant pressure process: since the change in internal energy is the same as above ( E int 900 cal), the work done at constant pressure is W p V nr T cal W pv nrt The first law gives Q Eint + W cal Q The change in the translational kinetic energy is K nr T cal K c. Adiabiatic process: once again the change in internal energy is the same as above ( E int 900 cal) but by definition of adiabatic Q 0. The first law leads to Eint Q W W 900 cal The change in the translational kinetic energy is K nr T cal K Example B An ideal gas undergoes an adiabatic compression from (p, V, T) (.0 atm, L, 0.0 o C) to (p, V) ( atm,.0 0 L). (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? a. We use p V γ p V γ to compute γ: 5 ln( p p) ln(.0atm.0 0 atm) 5 γ monatomic 6 ln V V ln.0 0 L.0 0 L b. Using the gas law in ratio form, the final temperature is 9.5
6 ( 5 )(.0 0 atm.0 0 L ) ( ) pv pv.0atm.0 0 L 4 T T 7K.7 0 K 6 c. The number of moles of gas present is 5 (.0 0 Pa)(.0 0 cm ) pv RT 8. J/mol K 7K 4 n mol ( ) d. The total translational energy per mole before the compression is K RT 8. J/mol K 7K.4 0 J and after the compression, 4 5 K RT ( 8. J/mol K)(.7 0 K).4 0 J 9.6
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