University Physics AI No. 11 Kinetic Theory

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1 Unersty hyscs AI No. 11 Knetc heory Class Number Name I.Choose the Correct Answer 1. Whch type o deal gas wll hae the largest alue or C -C? ( D (A Monatomc (B Datomc (C olyatomc (D he alue wll be the same or all. he derence between the molar specc heat at constant pressure and that at constant olume s equal to the unersal gas constant: C -C R.. What would be the most lkely alue or C, the molar heat capacty at constant temperature? ( D (A 0 (B 0< C < C (C C <C <C (D C Usng the denton o the molar specc heat: heat capacty s C. C molar 1 dq, so when constant, the molar n d. Whch o the ollowng speeds ddes the molecules n a gas n thermal equlbrum so that hal hae speeds aster, and hal hae speeds slower? ( B (A p (B a (C rms (D Non o the aboe. Accordng to the physcal meanng o p,, and gas so that hal hae speeds aster, and hal hae speeds slower. a rms, non o them ddes the molecules n a 4. Whch o the ollowng speeds corresponds to a molecule wth the aerage knetc energy? ( C (A p (B a (C rms (D Non o the aboe he aerage knetc energy s 1 ae m, and or rms, we hae 1 ae m rms, hen the answer s (C.

2 II. Flg the Blanks 1. he number o partcles n a cubc mllmeter o a gas at temperature 7 K and 1.00 atm pressure s.9 1. o get a eeg or the order o magntude o ths number, the age o the unerse n seconds assumng t s 15 bllon years old s s. he number o partcles s Q nr N nn R A N A N nn A he age o the unerse n seconds s (s 1. A sample o oxygen gas (O s at temperature 00 K and 1.00 atm pressure. One molecule, wth a speed equal to the rms speed, makes a head-on elastc collson wth your nose. Ouch! he magntude o the mpulse mparted to your schnozzle s Kg m/s. he speed o an oxygen molecule s R / ( 1/ rms ( 48.59(m/s M Usng the Impulse-Momentum heorem, the magntude o the mpulse mparted to the schnozzle s I m M N A (Kg m/s. When helum atoms hae an rms speed equal to the escape speed rom the surace o the Earth ( escape 11. km/s, the temperature s.01 4 K. Accordng to the problem R rms ( M 1/ rmsm ( So the temperature s.01 (K R he rms speed o hydrogen gas (H at temperature 00 K n the atmosphere s 1.9 km/s. Compare t wth the escape speed rom the Earth (11. km/s. Snce hydrogen s the least masse gas, hydrogen molecules wll hae the hghest rms speeds at a gen temperature. How can ths calculaton explan why there s essentally no hydrogen gas n the atmosphere o the Earth?

3 he rms speed o hydrogen gas s rms R ( 1/ ( M 1/ 1.9 (m/s he rms speed o hydrogen gas s less than the escape speed rom the Earth (11. km/s. Snce hydrogen s the least masse gas, hydrogen molecules hae the hghest rms speeds at a gen temperature. Hydrogen hae hghest escape rate n the atmosphere o the Earth, so there s ery ew hydrogen gas n the atmosphere o the Earth. 5. he aerage knetc energy o a partcle n a gas at temperature s gen by Equaton 14.1: ae k, or 1 m k. he specal theory o relatty states that there s an upper lmt on the speed o an partcle: the speed o lght c.00 8 m/s. For a gas o hydrogen atoms, the mmedately precedng equaton mples an upper lmt on the temperature. he absolute temperature such that or a gas o hydrogen atoms s equal to the square o the speed o lght s 4.5 K. In act, there s no upper lmt on the temperature; so the classcal expresson or the knetc energy cannot be ald or speeds approachng the speed o lght. Accordng to the problem c, so m k mc k 8 ( (K III. Ge the Solutons o the Followng roblems 1. Consder helum gas at temperature 00 K near the surace o the Earth. (a Calculate the aerage knetc energy o one o the helum atoms. (b Calculate the gratatonal potental energy o a sngle helum atom near the surace o the Earth. Choose the zero o gratatonal potental energy to be nntely ar away rom the Earth. (c What s the absolute alue o the rato o the gratatonal potental energy o the helum atom to ts aerage knetc energy? Is t justable to neglect the gratatonal potental energy n theory? Why or why not? (a he aerage knetc energy o one o the helum atoms s ae k (J (b he gratatonal potental energy o a sngle helum atom near the surace o the Earth s E gra mgrearth (J

4 (c he absolute alue o the rato s E gra 1 ae 7 It s justable to neglect the gratatonal potental energy n theory. As the aerage knetc energy s a statstcal alue or the whole system regardless the gratatonal potental energy.. A well-nsulated 4.00 lter box contans a partton ddng the box nto two equal olumes as shown n Fgure 1. Intally,.00 g o molecular hydrogen gas (H at 00 K s conned to the let-hand sde o the partton, and the other hal s a acuum. (a What s the rms speed o the partcles n the gas? (b What s the ntal pressure o the gas? (c he partton s remoed or broken suddenly, so that the gas now s contaned throughout the entre box. Assume that the gas s deal. Does the temperature o the gas change? What s the change n the nternal energy o the system? (d When the gas reaches equlbrum, what s the nal pressure? (a he rms speed o the partcles n the gas s Insulaton Gas hn partton acuum Fg.1 rms R ( 1/ ( M 1/ (b Usng the equaton o state or an deal gas: nr So the ntal pressure o the gas s 1.9 m M R (m/s m R (a M (c he temperature s not changed. he nternal energy o the system s not changed too. (d Because t s so ast that the gas reaches equlbrum. So t s an adabatc process. For 00K and 1 1, So the nal pressure s (a. A gas ntally at temperature, pressure and olume has ts pressure reduced to a nal alue a one o the ollowng types o processes: (1 sochorc; ( sothermal; ( adabatc. (a Sketch each process schematcally on a - dagram. (b In whch process s the work done by the gas zero? (c In whch process s the work done by the gas greatest?

5 (d Show that the rato o the absolute magntude o the heat transer to the gas durng the sothermal process to that durng the sochorc process s Qsothermal R Qsochorc c (e In whch process s the absolute magntude o the change n the nternal energy o the gas the greatest? Explan your reasonng. (a Isochorc process Isothermal process Adabatc process (b See the area under the cure o the graph, the work done by the gas s zero n sochorc process. (c See the area under the cure o the graph, the work done by the gas s greatest n sothermal process. (d For sothermal process: So the heat transer s For sochorc process: Q sothermal nr nr So the heat transer s Q nc Q sochorc ( Qsochorc nc hus the rato o the absolute magntude o the heat transer to the gas durng the sothermal process to that durng the sochorc process s

6 Q Q sothermal sochorc nr nc ( C R ( R C (e For sothermal process: U 1 0. Usng the rst law o thermodynamcs, or an adabatc process: Q 0 U For sochorc process: W 0 U W nc ( Q nc ( Q > U > U So the absolute magntude o the change n the nternal energy o the gas s the greatest n adabatc process. 4. An deal gas experences an adabatc compresson rom 1kpa,.7m, -.0 C to 1450kpa, 1.m. (a Calculate the alue o γ. (b Fnd the nal temperature. (c How many moles o gas are present? (d What s the total translatonal knetc energy per mole beore and ater the compresson? (e Calculate the rato o the rms speed beore to that ater the compresson. (a For adabatc process:, γ 1450 γ.7 γ 1450 Hence ( ( γ 1 1., γ (b Usng the equaton o state or an deal gas: nr, we hae (7 So the nal temperature s 77.(K 1.7 (c Usng the equaton o state or an deal gas: nr, we hae

7 1.7 n 7.97 (mole R 8.15 (7 (dhe total translatonal knetc energy per mole beore and ater the compresson s beore ater n R (J n R (J (e he rms speed s R rms ( M 1/ Hence the rato o the rms speed beore to that ater the compresson s rms rms ( 1/ 7 ( 77. 1/ 0.81

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