Physics 3 (PHYF144) Chap 3: The Kinetic Theory of Gases - 1
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1 Physcs (PYF44) ha : he nec heory of Gases -. Molecular Moel of an Ieal Gas he goal of he olecular oel of an eal gas s o unersan he acroscoc roeres (such as ressure an eeraure ) of gas n e of s croscoc behaor (such as olecular elocy an knec energy). hs oel reresens he gas as a large nuber of olecules bouncng aroun n a close conaner wh followng assuons:. conaner wh olue conans a ery large nuber of encal olecules, each wh ass.. he olecules behae as on arcles; her sze s sall n coarson o he aerage searaon beween arcles an o he enson of he conaner.. he olecules obey ewon s laws of oon; hey oe ranoly an occasonally ake elasc collsons wh each oher an wh he walls of he conaner. 4. he olecules eer no forces on each oher ece urng he collsons; hence no graaonal oenal energy of he olecules s consere.. he conaner walls are erfecly rg an nfnely asse an o no oe. Derng he relaonsh beween olecular knec energy an he ressure* Suose ha one of he walls of he conaner s erencular o he -as, an le be he coonen of he elocy of a olecule. he coonen of he oenu of a olecule s before hs he wall. Snce wall s erfecly rg an asse, s oenu s reerse afer he h. y Snce he olecules obeys obey ewon s laws, we can aly he ulse-oenu heore o fn he aerage force on he wall by he olecule. F s he collson e. In orer for he olecule o ake anoher collson wh he sae wall, nees o rael a sance of across he conaner. ence, he collson e can also be wren as For olecules, he oal aerage force on he wall s F F equals snce all of he olecules are encal. reser, 00/0
2 Physcs (PYF44) ha : he nec heory of Gases - For sall nuber of olecules, he acual force on he wall woul ary wh e. oweer, for a ery large nuber of olecules, hese araons are soohe ou, an hence he aerage force can be consere. F, where s he aerage alue of Snce he oon of he olecules s coleely rano, y z. nowng he square of he see y z, we can wre. he oal force on wall s rewren as F. Wh hs eresson, we can fn he ressure eere on he wall. he square roo of F s calle he roo-ean-square () see of he olecules. Inerreaon: hs resul ncaes ha he ressure of a gas s rooronal o he nuber of olecules er un olue he aerage ranslaonal knec energy of he olecules Usng he eal-gas equaon n kb, he aerage ranslaonal knec energy of a sngle olecule s. kb he oal ranslaonal knec energy of olecules of gas s, oalrans k B n where kb an n. k B M, where M ( ) s olar ass n SI uns. reser, 00/0
3 Physcs (PYF44) ha : he nec heory of Gases - ale: n oygen ank conans.00 ol of oygen gas a 7.0. (a) Wha s he oal ranslaonal knec energy of he gas olecules? oalrans n (.00ol)(8.Jol )(00) 740J (b) Wha s he aerage ranslaonal knec energy of a olecule? kb (.8 0 J )(00) 6. 0 J (c) Wha s he roo-ean-square see of an oygen olecule a hs eeraure? Molar ass of oygen gas M s ( 6.0)g /ol M (8.Jol 484s.0 0 kgol )(00) oe: he SI uns of M are klogras er ole, no gras er ole. ercse:.00-ler essel conans nrogen gas a 7.0 an.00a. Fn (a) he nuber of olecules n he essel, (b) he oal ranslaonal knec energy of he gas olecules an (c) he aerage ranslaonal knec energy er olecule. nswer:.67 0 olecules,.8kj, J. Molar Secfc ea of an Ieal Gas When hea Q s ae o (or release fro) a subsance of ass, he change of eeraure of he subsance s goerne by Q c where c = secfc hea of he subsance [.e. aoun of hea requre o rase kg of ha subsance by ] If he quany of he subsance s eresse n nuber of oles n, Q s ore conenenly o be wren as Q n where s olar secfc hea. [.e. aoun of hea requre o rase ole of ha subsance by ] SI un: J ol Suose we hea u n oles of gas whle keeng s olue. he resul of ang hea o he syse s an ncrease of s eeraure n Q s he olar hea caacy a olue. reser, 00/0
4 Physcs (PYF44) ha : he nec heory of Gases - 4 Snce he olue s ke, he work one by he gas s W 0. he frs law of heroynacs shows ha Q W Q 0 n n n (also rue for any rocess) he nernal energy of a gas s relae o he knec energy of s olecules. Le us conser he sles case of a onaoc eal gas, ha s, a gas conanng one ao er olecule, such helu or argon. he nernal energy of he gas consss only he ranslaonal knec energy: n oalrans n hs energy s a funcon of only. When hea s ae o a onaoc gas n a conaner of fe olue, all of he ae energy goes no ncreasng he ranslaonal knec energy of he aos. Q n n n (only for onaoc gas). Suose ha, whle hea s ae o he syse, he olue s allowe o change such ha he gas ressure oes no change. gan, he frs law of heroynacs shows ha Q n W n where s he olar hea caacy a ressure. he work one by he gas n hs -ressure rocess s W equaon of sae n, we can relae o as. Usng he eal-gas n. he frs law of heroynacs can be rewren as n n n oweer, he nernal energy n eens only on he eeraure an no on how he olue an/or ressure s changng. hus, he change n nernal energy urng any rocess s only eerne by he eeraure change: n n. he reous equaon can herefore be rewren as n n n ale : cylner conans.00 ol of helu gas a a eeraure of 00. ow uch hea us be ae o he gas o ncrease s eeraure o 00, f s heae a (a) olue (b) ressure? elu s onaoc, so an reser, 00/0
5 Physcs (PYF44) ha : he nec heory of Gases - (a) olue, he hea ae s Q n (.00ol)( )(8.Jol J (b) ressure, he hea ae s Q n )(00 (.00ol)( )(8.Jol. 0 J 00) )(00 00) ercse: alculae he change n nernal energy of.00 ol of helu gas when s eeraure s ncrease by.00. nswer: 74.8 J. he abac Processes for an Ieal Gas n aabac rocess s a rocess n whch no hea ransfer akes lace beween he syse an s surrounngs. Suose ha a gas s eane aabacally, hus Q = 0. onser an nfnesal change n olue. he work one by he gas urng he rocess s W. Snce he nernal energy only eens eeraure, he nernal energy change n for any rocess, aabac or no, s n n. ence, he frs law of heroynacs, n Q W, Q 0 becoes n n akng he oal fferenal of he eal-gas equaon of sae, n, we see ha n or n lnang fro hese wo equaons, we fn ha hs eresson can be rewren as 0 where = (/ ). fer negrang hs eresson we oban ln ln or reser, 00/0
6 Physcs (PYF44) ha : he nec heory of Gases - 6 Usng he eal-gas equaon n he for oban n n o elnae fro he eresson we hus For onaoc eal gas: (8.Jol ). Jol.67 ale : Durng he coresson sroke of a ceran gasolne engne, he ressure ncreases fro.00a o 0.0 a. ssung ha he rocess s aabac an ha he gas s eal, wh =.40, (a) by wha facor oes he olue change an (b) by wha facor oes he eeraure change? (c) If he coresson sars wh ol of gas a 7.0, fn he alues of Q, W, an n ha characerze he rocess..00a 0.0a.40 (a).00a a (b) Meho : Meho : n, so 0.0 a.00a (0.768). (c) = 7.0 = = 00. n = ol =. = =. = reser, 00/0
7 Physcs (PYF44) ha : he nec heory of Gases - 7 7, so n n (0.060ol)( )(8.Jol J abac rocess: Q 0. So, n W. )(.)(00) W J ercse: One ole of elu gas eans aabacally fro a ressure o a ressure. If he orgnal eeraure of he gas s 0, wha s he fnal eeraure of he gas? ( =.67) nswer: hngs o know (haer o ) 9 F c (Subsue 00 o check) 7. L L0 ; n k B ; n # of olecules, ogaro's nuber, f k B W (work one by gas) n Q Q n Q n Q W k k L L k 4 e 4 4 ne e( s oalrans ) kb (one olecule) kb n k B M n n (for any rocess) (onaoc gas) ; onsan Process n = Q - W ec Isobarc W = Q = n Isochorc W = 0 n = Q Q = n Isoheral n = 0 Q = W W = n ln ( / ) ll (,, ) no abac Q = 0 n = W = ; - = - reser, 00/0
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