Example problems. Chapter 3: The Kinetic Theory of Gases. Homework: 13, 18, 20, 23, 25, 27 (p )

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1 Examle roblems Chater : he Kinetic heory o Gases Homework:, 8,,, 5, 7 (. 5-5)

2 9. An automobile tire has a volume o.64 x m and contains air at a gauge ressure (above atmosheric ressure) o 65 kpa when the temerature is. C. What is the gauge ressure o the air in the tires when its temerature rises to 7. C and its volume increases to.67 x m? Assume atmosheric ressure is.x 5 Pa. At state i: i i nr nr i At the inal state : nr ii i Gauge ressure: - G o

3 . A samle o an ideal gas is taken through the cyclic rocess abca shown in the igure below; at oint a, K. (a) How many moles o gas are in the samle? What are (b) the temerature o the gas at oint b, (c) the temerature o the gas at oint c, and (d) the net energy added to the gas as heat during the cycle? (a) Alying the equation o state: nr n R At oint a,.5 kn/m or 5 N/m ; m. 5 n.5 (mol) 8. aa bb nr nr. (b) 5 a At oint b, 7.5 kn/m or 75 N/m ; m. b b nr b b (K)

4 (c) see art b; c 6 K; (d) Alying the irst law o thermodynamics: E Q W W: work done by the system. For a closed cycle, E: Q W W ( )( ) b c b a W 5. 5 (J)

5 4. In the temerature range K to K, the ressure o a certain nonideal gas is related to volume and temerature by: ( 4.9 J / K) (.66 J / K ) How much work is done by the gas i its temerature is raised rom 5 K to K while the ressure is held constant? Work done by the gas is comuted by the ollowing ormula: W W i i d 4.9( ( i ) ).66( i i ) K; 5K W i (J)

6 8. he temerature and ressure in the Sun s atmoshere are.x 6 K and. Pa. Calculate the rms seed o ree electrons (mass 9.x - kg) there, assuming they are an ideal gas. v rms R M R mn A v rms (m/s)

7 . Calculate the rms seed o helium atoms at K, the molar mass o helium atoms is 4.6 g/mol. v rms R M (m/s)

8 4. At 7 K and. x - atm, the density o a gas is.4 x -5 g/cm. (a) Find v rms or the gas molecules. (b) Find the molar mass o the gas and (c) identiy the gas (hint: see able 9-). (a) Root-mean-square seed: () and (): M gas v rms R M nm ρ M nr v rms ρ () ρ n ρ () ρ.4 5 g/cm.4 kg/m. atm v rms. 494 m/s Pa

9 (b) Equation o state: M ρ n nr () () M ρ n ρr (c) M.8 kg/mol 8 g/mol From able 9., the gas is nitrogen (N )

10 5. Determine the average value o the translational kinetic energy o the molecules o an ideal gas at (a). C and (b) C. What is the translational kinetic energy er mole o an ideal gas at (c). C and (d) C? (a) he translational kinetic energy er molecule: K k K : (b) see (a): K K (c) he translational kinetic energy er mole: (d) K mole 5.65 K mole (J) (J) (J) (J) K K mole N A

11 Note: I a samle o gas has n moles (or N molecules), its total translational kinetic energy is: K n K n N total mole A K K total n K mole n N A k nr K total nr

12 Homework: 8,,, 4 (age 5)

13 8. At what requency would the wavelength o sound in air be equal to the mean ree ath o oxygen molecules at. atm ressure and. C? take the diameter o an oxygen molecule to be. x -8 cm. Mean Free Path: λ MFP k.8 J/K; 7 K; d Frequency o sound in air: v sound in air cm 4 m/s 8 : λ MFP m sound k πd. λ sound 5 Pa; vsoundin air 9. 8 m v soundin air λ MFP sound (Hz) or.68 GHz

14 . At C and 75 torr ressure, the mean ree aths or argon gas (Ar) and nitrogen (N ) are λ Ar 9.9x -6 cm and λ N 7.5x -6 cm. (a) Find the ratio o the diameter o an Ar atom to that o an N molecule. What is the mean ree ath o Ar at (b) C and 5 torr, and (c) -4 C and 75 torr? Mean Free Path: (a) he ratio d Ar to d N : (b): λ k πd d λ Ar N d λ N Ar k k λ ; λ πd πd λ λ

15 . he seeds o molecules are.,., 4.,..., km/s. What are their (a) average seed and (b) rms seed? (a) v N i i N v 6.5 (km/s) ( ) i (b) 7.(km/s) v rms v avg N N v i

16 4. wo containers are at the same temerature. he irst contains gas with ressure, molecular mass m, and rms seed v rms. he second contains gas with ressure.5, molecular mass m, and average seed v avg.v rms. Find the mass ratio m /m. RMS seed: v rms R m Average seed: v 8R πm

17 Homework: 4, 44, 46, 54, 56, 78 (. 5-55)

18 4. What is the internal energy o. mol o an ideal monatomic gas at 7 K? E nc C R.5 J mol - K - E (J) E 6.8 (kj)

19 44. One mole o an ideal diatomic gas goes rom a to c along the diagonal ath in Fig he scale o the vertical axis is set by ab 5. kpa and c. kpa, and the scale o the horizontal axis is set by bc 4. m and a. m. During the transition, (a) what is the change in internal energy o the gas, (b) how much energy is added to the gas as heat? (c) How much heat is required i the gas goes rom a to c along the indirect ath abc? a) 5 Eint nc nr 5 ( Pc c P a a ) 5J b) Q E W 5 7 J int + + c) Q E + int W ac

20 46. Under constant ressure, the temerature o. mol o an ideal monatomic gas is raised 5. K. What are (a) the work W done by the gas, (b) the energy transerred as heat Q, (c) the change E int o the gas, and (d) the change K in the average KE er atom? (a) At constant ressure: W nr (J) (b) Q E int 5 5 nc n R W (c) We use the irst law o thermodynamics: (d) For a monatomic gas: K avg Q W Eint K 95 (J) (or Eint nc nr ) k 56 (J) K avg avg k (J)

21 54. We know that or an adiabatic rocess. Evaluate constant or an adiabatic rocess involving exactly. mol o an ideal gas assing through the state having exactly.5 atm and K. Assume a diatomic gas whose molecules rotate but do not oscillate. Equation o state: nr atm C C For a diatomic gas, 5: constant nr (Pa) constant. 8.. (m ) 5.5. C + C 5 R constant.8 R + R R.8 (N m. ) (N/m (m ).4 )

22 56. Suose.L o a gas with., initially at 85 K and. atm, is suddenly comressed adiabatically to hal its initial volume. Find its inal (a) ressure and (b) temerature. (c) I the gas is then cooled to 7 K at constant ressure, what is its inal volume? ; i i i i i i i nr ' ' constant,

23 78. (a) An ideal gas initially at ressure undergoes a ree exansion until its volume is. times its initial volume. What then is the ratio o its ressure to? (b) he gas is next slowly and adiabatically comressed back to its original volume. he ressure ater comression is (.) /. Is the gas monatomic, diatomic, or olyatomic? (c) What is the ratio o the average kinetic energy er molecule in this inal state to that in the initial state? (a) ; (b) ' ' C R C C C olyatomic 6 :

24 (c) K avg k ' ' K K r avg avg ).44 (since ' ' ' ' / ' r

Physics 4C Chapter 19: The Kinetic Theory of Gases

Physics 4C Chapter 19: The Kinetic Theory of Gases Whether you think you can or think you can t, you re usually right. Henry Ford The only thing in life that is achieved without effort is failure. Source