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5Chapitre 1Introduction à la thermody-namique1.1. Le champ d’étude de la thermodynamiqueLa thermodynamique a pour objet l’étude de la matière (gaz, liquides, solides...) et lestransformations qu’elle subit. Ce sont en particulier les transferts thermiques (température)et mécaniques (par exemple compression) qui sont responsables de ces transformations detempérature, pression, volume ou changements de phase.1.1.1. HistoriqueLa première expérience réalisée en thermodynamique n’est pas récente, puisque 400 000ans avant notre ère l’Homo-erectus a appris à maîtriser le feu... Mais l’utilisation du feuest restée pendant très longtemps la seule application de la thermodynamique. Les progrèstechniques furent pourtant nombreux en mécanique ces derniers millénaires : les égyptiens ontpar exemple développé des systèmes de levage; les observations astronomiques ont motivé denombreuses recherches.Les premiers développements de la thermodynamique furent expérimentaux : l’observationd’une casserole chau¤ant sur le feu permet de constater que la vapeur est capable de pousserle couvercle pour évacuer la surpression. Il faut attendre 1629 pour voir apparaître la premièreieµmeturbine à vapeur. Le grand essor a lieu à la …n du XV III siècle et est lié aux débuts deieµmela révolution industrielle. Au XIX , l’utilisation plus rationnelle de l’énergie rend le déve-loppement de la thermodynamique primordial : il devient nécessaire de ne plus être tributairedes aléas ...
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@e^mfqob . Fkqolar‘qflk Ý i^ qebojlav* k^jfnrb
2
.+.+ Ib ‘e^jm a æqrab ab i^ qebojlavk^jfnrb I^ qebojlavk^jfnrb ^ mlro l_gbq i æqrab ab i^ j^qfåob %d^w) ifnrfabp) plifabp+++& bq ibp qo kpcloj^qflkp nr biib pr_fq+ @b plkq bk m^oqf‘rifbo ibp qo^kpcboqp qebojfnrbp %qbjmæo^qrob& ^ bq jæ‘^kfnrbp %m^o bubjmib ‘ljmobppflk& nrf plkq obpmlkp^_ibp ab ‘bp qo^kpcloj^qflkp ab qbjmæo^qrob) mobppflk) slirjb lr ‘e^kdbjbkqp ab me^pb+ .+.+.+ Efpqlofnrb I^ mobjfåob bumæofbk‘b oæ^ifpæb bk qebojlavk^jfnrb k bpq m^p oæ‘bkqb) mrfpnrb 1-- ---^kp ^s^kq klqob åob i Eljl*bob‘qrp ^ ^mmofp Ý j^ëqofpbo ib cbr+++ J^fp i rqfifp^qflk ar cbr bpq obpqæb mbka^kq qoåp ilkdqbjmp i^ pbrib ^mmif‘^qflk ab i^ qebojlavk^jfnrb+ Ibp moldoåp qb‘ekfnrbp crobkq mlroq^kq klj_obru bk jæ‘^kfnrb ‘bp abokfbop jfiiæk^fobp 7 ibp ædvmqfbkp lkq m o bubjmib aæsbilmmæ abp pvpqåjbp ab ibs^db 8 ibp l_pbos^qflkp ^pqolkljfnrbp lkq jlqfsæ ab ^ klj_obrpbp ob‘ebo‘ebp+ Ibp mobjfbop aæsbilmmbjbkqp ab i^ qebojlavk jfnrb crobkq bumæofjbkq^ru 7 i l_pbos^qflk ^ a rkb ‘^ppbolib ‘e^r ^k pro ib cbr mbojbq ab ‘lkpq^qbo nrb i^ s^mbro bpq ‘^m^_ib ab mlrppbo q ib ‘lrsbo‘ib mlro æs^‘rbo i^ promobppflk+ Fi c^rq ^qqbkaob .3/6 mlro slfo ^mm^o^ëqob i^ mobjfåob qro_fkb Ý s^mbro+ Ib do^ka bpplo ^ ifbr Ý i^ k ar XV II I i ` eme pfå‘ib bq bpq ifæ ^ru aæ_rqp ab i^ oæslirqflk fkarpqofbiib+ >r XIX i ` eme ) i rqfifp^qflk mirp o^qflkkbiib ab i ækbodfb obka ib aæsb* ilmmbjbkq ab i^ qebojlavk^jfnrb mofjloaf^i 7 fi absfbkq kæ‘bpp^fob ab kb mirp çqob qof_rq^fob abp ^iæ^p ‘ifj^qfnrbp mlro c^fob clk‘qflkkbo ibp jlrifkp Ý b^r bq Ý sbkq+ Ib aæsbilmmbjbkq abp æ‘e^kdbp ‘ljjbo‘f^ru obka kæ‘bpp^fob æd^ibjbkq ib aæsbilmmbjbkq abp qo^fkp bq sæef‘ribp Ý jlqbro bk objmi^‘bjbkq ab i^ qo^‘qflk ^kfj^ib+ Ibp qb‘ekfnrbp %bjmfofnrbp& pb plkq qlrq a _loa aæsbilmmæbp) mrfp ibp jlaåibp mevpfnrbp ^ lkq prfsf mlro bpp^vbo ab ‘ljmobkaob ibp meækljåkbp ^ k a ^jæiflobo ibp mbocloj^k‘bp) ‘ bpq Ý afob ^rdjbkqbo ibp mrfpp^k‘bp qlrq bk afjfkr^kq i^ ‘lkpljj^qflk a ækbodfb mofj^fob+ @^oklq ^ bk m^oqf‘rifbo _b^r‘lrm qo^s^fiiæ pro ibp jlqbrop bq ^ æqæ rk abp mobjfbop Ý mr ifbo ibp _^pbp _ mevpfnrbp ar clk‘qflkkbjbkq abp jlqbrop+ Lk mbrq ‘fqbo plk lrso^db ab .5/1 oæ buflkp pro i^ mrfpp^k‘b jlqof‘b ar cbr bq pro ibp j^‘efkbp molmobp Ý aæsbilmmbo ‘bqqb mrfpp^k‘b + @i^rpfrp) ?liqwj^kk) Df__p bq _fbk a ^rqobp pb plkq mbk‘eæp pro i æqrab abp æq^qp a ænrfif_ob qebojlavk^jfnrb 8 Arebj) Lkp^dbo) Mofoldfkb jæofqbkq a çqob ‘fqæp mlro ibrop qo^s^ru elop ænrfif_ob qebojlavk^jfnrb bq pro ibp meækljåkbp fooæsbopf_ibp+ .+.+/+ Ibp ^mmif‘^qflkp Ibp bubjmibp ab meækljåkbp bq ^mmif‘^qflkp c^fp^kq fkqbosbkfo i^ qebojlavk^jfnrb plkq qoåp s^pqbp+ I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J ^kp+
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3
@e^mfqob . Fkqolar‘qflk Ý i^ qebojlavk^jfnrb
Lk mbrq ‘fqbo m^o bubjmib 7 ibp jlqbrop qebojfnrbp %ab slfqrob) a ^sflk++++&) ibp oæcofdæo^* qbrop bq ‘ifj^qfpbrop) ib cbr %‘e^r ^db a rkb ‘^ppbolib a b^r&) ibp meækljåkbp jæqælolildfnrbp) i mfib æib‘qofnrb) ib mkbr ab _f‘v‘ibqqb %plrp mobppflk&) ib i æi^pqfnrb+++++ ^
.+/+ Ab i^ jæ‘^kfnrb Ý i^ qebojlavk^jfnrb .+/+.+ Ib ifbk jæ‘^kfnrb * qebojlavk^jfnrb Ibp pvpqåjbp æqrafæp plkq abp slirjbp V ab j^qfåob) a æ‘ebiib j^‘olp‘lmfnrb) nrf ‘lkqfbkkbkq rk qoåp do^ka klj_ob ab ‘lkpqfqr^kqp æiæjbkq^fobp+ M^o bubjmib 7 * a^kp 1 mm 3 a ^fo) fi v ^ 10 20 jliæ‘ribp 8 * a^kp 20 cl a b^r %. sboob&) fi v ^ 7 10 23 jliæ‘ribp+ Fi bpq m^o ‘lkpænrbkq elop ab nrbpqflk a æqrafbo fkafsfarbiibjbkq ‘e^‘rkb abp jliæ‘ribp+ Abru ^mmol‘ebp plkq ^ilop mlppf_ibp 7 * rkb ^mmol‘eb pq^qfpqfnrb 8 * rkb ^mmol‘eb j^‘olp‘lmfnrb+ I^ mobjfåob ^mmol‘eb) i^ mevpfnrb pq^qfpqfnrb k bpq m^p ^r moldo^jjb ab j^qe prm) Ý m^oq ibp nrbinrbp klqflkp nrf pbolkq aæsbilmmæbp a^kp ‘b ‘e^mfqob+ Biib ‘lkpfpqb Ý aæ‘ofob ibp ‘^o^‘qæofpqfnrbp abp m^oqf‘ribp m^o abp ilfp ab afpqof_rqflk pq^qfpqfnrb abp sfqbppbp lr ab i ækbodfb+ M^o bubjmib) ‘lkk^fpp^kq ib mol ^ fifqæ nr lkq ‘e^‘rkb abp m^oqf‘rib a ^slfo qbiib lr _ _ qbiib sfqbppb) lk mbrq b b‘qrbo abp jlvbkkbp pro rk do^ka klj_ob ab m^oqf‘ribp bq bk aæarfob ibro ‘ljmloqbjbkq dil_^i + Ibp klqflkp ab mobppflk lr ab qbjmæo^qrob pbolkq aæ kfbp Ý m^oqfo ab ‘bp ilfp ab afpqof_rqflk a^kp ibp m^oqfbp .+0 bq .+1 ab ‘b ‘e^mfqob+ I^ pb‘lkab ^mmol‘eb ‘lkpqfqrb i bppbkqfbi ar moldo jjb ab j^qe prm+ I^ mobppflk bq i^ ^ qbjmæo^qrob plkq fkqbomoæqæbp ‘ljjb abp do^kabrop j^‘olp‘lmfnrbp) ‘ bpq*Ý*afob ^‘‘bppf ibp _ bumæofjbkq^ibjbkq+ Rkb mobppflk mbrq pb jbprobo m^o bubjmib ‘ljjb æq^kq i^ l ‘b m^o rkfqæ c o ab proc^‘b nr ^mmifnrb rk d^w pro rkb m^olf 8 i^ qbjmæo^qrob pb jbprob ^sb‘ rk qebojljåqob+ J^fp ^s^kq ab aæ kfo ‘bp klqflkp ab mobppflk lr ab qbjmæo^qrob) fi bpq kæ‘bpp^fob ab alkkbo nrbinrbp aæ kfqflkp mlro _ b nrf pb rqfifpæ m^o i^ prfqb ubo ib sl‘^ ri^fo o^ .+/+/+ Aæ kfqflkp .+/+/+.+ I^ abkpfqæ m oqf‘ri^fob ^ Plfq V rk slirjb+ Lk klqb N ib klj_ob ab m^oqf‘ribp pb qolrs^kq a^kp V + lk ^mmbiib n V i^ abkpfqæ m^oqf‘ri^fob lr abkpfqæ jliæ‘ri^fob i^ do^kabro 7 N = n V V
A^kp i ^fo ^j_f^kq n V ∼ 10 25 m − 3 + A^kp i b^r abp l‘æ^kp n V ∼ 10 28 m − 3 Obj^onrb .+. I b^r bpq bksfolk 1000 clfp mirp abkpb nrb i ^fo Ý i^ proc^‘b ab i^ Qboob 7 a^kp i b^r ifnrfab) ibp m^oqf‘ribp pb qlr‘ebkq ibp rkbp ibp ^rqobp) ^ilop nrb a^kp i ^fo ^j_f^kq) ibp m^oqf‘ribp l‘‘rmbkq bksfolk 1 1000 ie ` me ab i bpm^‘b) bq ib obpqb bpq sfab+++
.0 j^op /--1
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Pb‘qflk .+0 I^ qeælofb ‘fkæqfnrb ar d^w m^oc^fq jlkl^qljfnrb
4
.+/+/+/+ Pvpqåjbp cbojæp , lrsboqp Rk pvpqåjb bpq rk bkpbj_ib ab m^oqf‘ribp %‘ljjb bk jæ‘^kfnrb ‘i^ppfnrb) Ý i^ f æobk‘b a moåp nrb ‘b klj_ob ab m^oqf‘ribp bpq qoåp do^ka&+ Rk pvpqåjb bpq afq cbojæ p fi k æ‘e^kdb m^p ab m^oqf‘ribp ^sb‘ i buqæofbro+ > i fksbopb) rk pvpqåjb bpq afq lrsboq p fi æ‘e^kdb abp m^oqf‘ribp ^sb‘ i buqæofbro+ M^o bubjmib) rk pvpqåjb cbojæ mbrq*çqob i ^fo ‘lkqbkr a^kp rk mkbr bk _lk æq^q+ Lk mbrq ^ilop æqrafbo i^ s^of^qflk ab p^ qbjmæo^qrob bq ab p^ mobppflk nr^ka i ^rqljl_fifpqb olrib+ M^o ‘lkqob) pf ib mkbr sfbkq Ý pb ‘obsbo) ib pvpqåjb absfbkao^ rk pvpqåjb lrsboq %mboqb ab m^oqf‘ribp&+ .+/+/+0+ M^o^jåqobp a æq^q %lr s^of^_ibp a æq^q& Ibp s^of^_ibp a æq^q plkq ibp s^of^_ibp nrf mbojbqqbkq ab aæ kfo i æq^q a rk pvpqåjb+ M^o bubjmib) i^ qbjmæo^qrob T ) i^ mobppflk P lr i ækbodfb U ar pvpqåjb plkq abp m^o^jåqobp a æq^q+ .+/+/+1+ M^o^jåqobp fkqbkpfcp bq buqbkpfcp Ibp ^o^jåqobp a æq^q mbrsbkq pb afpqfkdrbo bk abru ‘^qædlofbp+ m Ib m^o^jåqobp fkqbkpfcp plkq abp m^o^jåqobp nrf kb aæmbkabkq m^p ar slirjb ar pvpqåjb+ Bqrafbw m^o bubjmib i ^fo nrf slrp bkqlrob 8 fplibw 1 L ab ‘bq ^fo %p^kp irf c^fob pr_fo ^r‘rkb qo^kpcloj^qflk& bq jbprobw p^ qbjmæo^qrob+ Ob‘ljjbk‘bw i^ jçjb lmæo^qflk bk mobk^kq 2 L a ^fo 7 i^ qbjmæo^qrob pbo^ i^ jçjb+ I^ qbjmæo^qrob T ) i^ mobppflk P ) i^ abkpfqæ n V m^o bubjmib plkq abp m^o^jåqobp fkqbkpfcp+ Ibp m^o^jåqobp buqbkpfcp aæmbkabkq ^r ‘lkqo^fob ar slirjb V ‘elfpf %bq plkq molmloqflk* kbip Ý V &+ I^ j^ppb a rk pvpqåjb) i ækbodfb fkqbokb U bq _fbk æsfabjjbkq ib slirjb V plkq abp m^o^jåqobp buqbkpfcp+ Pf slrp mobkbw i ^fo nrf slrp bkqlrob) i^ j^ppb ab 2 L a ^fo bpq / clfp mirp fjmloq^kqb nrb i^ j^ppb a 1 L a ^fo+ .+/+/+2+ Pvpqåjb eljldåkb Rk pvpqåjb bpq afq eljldåkb pf pbp s^of^_ibp fkqbkpfsbp plkq ‘lkpq^kqbp bk qlrq mlfkq ar pvpqåjb+ Mlro nr rk pvpqåjb plfq eljldåkb) fi c^rq alk‘ nrb p^ qbjmæo^qrob) p^ mobppflk bq p^ abkpfqæ plfbkq rkfclojbp bk qlrq mlfkq ar pvpqåjb+ .+/+/+3+ Bnrfif_ob qebojlavk^jfnrb a rk pvpqåjb Rk pvpqåjb bpq afq bk ænrfif ob qebojlavk^jfnrb pf qlrp ibp m^o^jåqobp plkq fks^* _ of^_ibp ^r ‘lrop ar qbjmp+
.+0+ I qeælofb ‘fkæqfnrb ar d^w m^oc^fq ^ jlkl^qljfnrb
I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J ^kp+
.0 j op /--1 ^
5 @e^mfqob . Fkqolar‘qflk Ý i^ qebojlavk^jfnrb _ .+0+.+ Ilfp ab afpqof rqflk abp sfqbppbp a rk d^w Ibp m^oqf‘ribp ‘lkpqfqr^kq rk d^w plkq if_obp ab pb jlrslfo a^kp k fjmloqb nrbiib afob‘qflk 8 _ ibro jlrsbjbkq %mlpfqflk bq sfqbppb& ^iæ^qlfob bq aæploalkkæ bpq ^mmbiæ jlrsbjbkq olt* kfbk + @b jlrsbjbkq ab j^o‘eb ^r e^p^oa abp m^oqf‘ribp aæmbka bk c^fq abp jriqfmibp ‘el‘p nrb pr_fppbkq ibp m^oqf‘ribp bkqob biibp ^fkpf nrb ibp ‘el‘p bkqob ibp m^oqf‘ribp bq ibp m olfp æsbk* ^ qrbiibp+ >moåp ‘e^nrb ‘el‘) i^ klrsbiib sfqbppb ^‘nrfpb m^o i^ m^oqf‘rib bpq Ý klrsb^r ^iæ^qlfob bk afob‘qflk bq klojb %bq aæmbka ab i^ ‘lk dro^qflk ar ‘el‘ nrf klrp bpq fk^‘‘bppf_ib&+ Ib klj_ob ‘lkpfaæo^_ib ab jliæ‘ribp obka fjmlppf_ib rkb ^mmol‘eb ab jæ‘^kfnrb ‘i^ppfnrb nrf ‘lkpfpqbo^fq Ý fplibo rkb m^oqf‘rib) bq ‘lkk^fpp^kq p^ mlpfqflk bq p^ sfqbppb fkfqf^ib) Ý ‘^i‘ribo plk jlrsbjbkq riqæofbro+ Fi bpq ^ilop kæ‘bpp^fob ab o^fplkkbo pro abp jlvbkkbp+ I^ sfqbppb jlvbkkb %bk klojb& afqb pm^qf^ib a rk do^ka k j_ ^oqf‘ribp Ý rk fkpq^kq l ob ab m alkkæ) ab sfqbppbp obpmb‘qfsbp v 1 ) v 2 ) +++ v i ) +++ v N bpq 7 N h v i spatiale = N 1 X v i %.& i =1 Lk mbrq æd^ibjbkq aæ kfo rkb sfqbppb jlvbkkb qbjmlobiib a rkb m^oqf‘rib alkkæb pro rk qbjmp Δ t 7 h v i t =Δ1 t Z tt +Δ t v ( t ) dt %/& Qlrqbclfp) pro rk qbjmp Δ t pr§p^jjbkq ilkd mlro nr rkb m^oqf æb pr_ pp ‘rib alkk f b rk j_ ‘el‘p) bq bk pb ifjfq^kq Ý rk d^w alkq qlrqbp ibp m^oqf‘ribp plkq fabkqfnrbp) i^ do ka kl ob ab ^ sfqbppb jlvbkkb pm^qf^ib bq i^ sfqbppb jlvbkkb qbjmlobiib plkq ‘lkclkarbp+ Bk b bq) pf mbka^kq Δ t i^ m^oqf‘rib alkkæb pr_fq rk qoåp do^ka klj_ob ab ‘el‘p) biib mobkao^ pr‘‘bppfsbjbkq rk do^ka klj_ob ab sfqbppbp af æobkqbp nrf ‘loobpmlkabkq pq^qfpqfnrbjbkq ^ru afpqof_rqflkp ab sfqbppbp a rk do^ka klj_ob ab m^oqf‘ribp Ý rk fkpq^kq alkkæ+ @bqqb evmlqeåpb p ^mmbiib i evmlqeåpb bodlafnrb + I æqrab qebojlavk^jfnrb abp pvpqåjbp b b‘qræb bk j^qe prm pb ifjfqb ^ru pvpqåjbp eljldåkbp bq bk ænrfif_ob qebojlavk^jfnrb+ Bk m^oqf‘rifbo) i^ afpqof_rqflk abp sfqbppbp bpq eljldåkb %jçjbp afpqof_rqflkp pq^qfpqfnrbp abp sfqbppbp nrbi nrb plfq ib mlfkq lö i lk pb mi^‘b& bq pq^qflkk^fob %jçjb afpqof_rqflk abp sfqbppbp Ý i fkpq^kq t lr Ý i fkpq^kq t + Δ t +& Ab mirp) lk prmmlpb nr ^moåp rk ‘el‘) i^ m^oqf‘rib mbrq obm^oqfo a^kp k fjmloqb nrbiib _ afob‘qflk ab c^älk ænrfmol_^_ib 7 i^ afpqof rqflk abp sfqbppbp bpq afqb fplqolmb + Bk k) fi pbo^ kæ‘bpp^fob ^rppf mlro i^ prfqb a fkqolarfob rkb sfqbppb nr^ao^qfnrb jlvbkkb klqæb r 7 u = p h v 2 i I^ sfqbppb nr^ao^qfnrb jlvbkkb bpq i^ o^‘fkb ‘^ooæb ab i^ jlvbkkb ab i^ sfqbppb ^r ‘^ooæ+ Obj^onrb .+/ >qqbkqflk ffi Jçjb pf bk j^qe prm) i^ sfqbppb jlvbkkb bq i^ sfqbppb nr^ao^* qfnrb jlvbkkb plkq plrsbkq ‘lkclkarbp a^kp ib _rq ab pfjmif bo ibp jlaåibp molmlpæp) fi c^rq _fbk d^oabo Ý i bpmofq nrb ‘bp abru sfqbppbp plkq af æobkqbp ffi Rk mol_iåjb pfjfi^fob ^ aægÝ æqæ obk‘lkqoæ bk æib‘qolkfnrb ^sb‘ ib ‘^i‘ri a rkb qbkpflk jlvbkkb) bq a rkb qbkpflk b§‘^‘b %nrf bpq bk c^fq rkb qbkpflk nr^ao^qfnrb jlvbkkb& 7 i^ qbkpflk jlvbkkb a rk pfdk^i bpq af æobkqb ab i^ qbkpflk b§‘^‘b+ .0 j^op /--1 I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J^kp+
Pb‘qflk .+0 I^ qeælofb ‘fkæqfnrb ar d^w m^oc^fq jlkl^qljfnrb 6 Mlro rkb æqrab mirp moæ‘fpb abp ilfp ab afpqof_rqflk abp sfqbppbp) bq ab ‘^i‘ri abp sfqbppbp jlvbkkbp bq sfqbppbp nr^ao^qfnrbp jlvbkkbp) lk mlroo^ pb obmloqbo ^r m^o^do^meb .+1+ .+0+/+ Ib jlaåib pq^qfpqfnrb ar d^w m^oc^fq jlkl^qljfnrb Ib jlaåib pq^qfpqfnrb ar d^w m^oc^fq jlkl^qljfnrb obmlpb pro ibp evmlqeåpbp prfs^kqbp 7 ib d^w bpq ‘lkpqfqræ a ^qljbp jlkl^qljfnrbp %bubjmib >o) Eb&) ‘lkpfaæoæp ‘ljjb mlk‘qrbip _ bq p^kp fkqbo^‘qflk bkqob bru 8 i^ afpqof rqflkp abp sfqbppbp bpq fplqolmb + I ^fo ^j_f^kq) ‘lkpqfqræ mofk‘fm^ibjbkq ab jliæ‘ribp ab af^wlqb bq afluvdåkb) alk‘ af^ql* jfnrbp) kb obkqob m^p a^kp ib ‘^aob ar jlaåib %slfo d^w m^oc^fq mliv^qljfnrb m^o^do^meb .+2+/&+ Ib c^fq nrb i lk ‘lkpfaåob nr fi k v ^ m^p a fkqbo^‘qflk bkqob ibp ^qljbp bpq ifæ Ý i^ c^f ib _ abkpfqæ abp d^w m^oc^fqp % 999 1000 i ` eme a o i ^fo& 7 i^ qolm do^kab afpq^k‘b jlvbkkb b sfab mlr bkqob ibp jliæ‘ribp obka kædifdb^_ibp ibp fkqbo^‘qflkp æsbkqrbiibp bkqob jliæ‘ribp+ Bk k) moæ‘fplkp Ý klrsb^r nrb i æqrab pb ifjfqb Ý rk pvpqåjb b _ob qebojlavk^jfnrb k ænrfif bq eljldåkb 8 ibp jlaåibp pbolkq ^mmifnræp pro abp slirjbp pr§p^jjbkq do^kap mlro ‘lkqbkfo rk do^ka klj_ob ab m^oqf‘ribp % 1 mm 3 pr§q dækæo^ibjbkq&+ > k ab pfjmif bo i æqrab klrp ^almqbolkp rk jlaåib æiæjbkq^fob a^kp ibnrbi qlrqbp ibp sfqbppbp lkq i^ jçjb klojb %æd^ib Ý i^ sfqbppb jlvbkkb& bq ibp m^oqf‘ribp kb pb aæmi^‘bkq nrb a^kp ibp 3 afob‘qflkp loqeldlk^ibp prfs^kqbp 7 sbop + x ) − x ) + y ) − y ) + z bq − z + Bk oæ^ifqæ) ‘bqqb evmlqeåpb k bpq m^p kæ‘bpp^fob a^kp ib jlaåib ‘i^ppfnrb ar d^w m^oc^fq 7 biib mbojbq pbribjbkq ab pfjmif bo ibp ‘^i‘rip+ .+0+0+ @^i‘ri ab i^ clo‘b bubo‘æb m^o rk DMJ pro rk æiæjbkq ab m^olf+ Aæ kfqflk ‘fkæqfnrb ab i^ mobppflk I^ mobppflk) afqb mobppflk ‘fkæqfnrb a^kp ‘b jlaåib) bpq i^ clo‘b m^o rkfqæ ab proc^‘b nrb pr_fq rkb m^olf) Ý ‘^rpb abp m^oqf‘ribp nrf sfbkkbkq co^mmbo pro ‘bqqb m^olf+ paroi paroi choc choc dS= dS u z dS= dS u z
particules en mouvement aléatoire modèle A modèle B (simplifié)
Cfd+.+.+ Ib jlaåib ‘i^ppfnrb ar d^w m^oc^fq ‘lkpfaåob nrb i^ afpqof_rqflk abp sfqbppbp bpq ^iæ^qlfob bq fplqolmb %jlaåib > ab i^ drob .+.&+ Ib jlaåib pfjmif æ nrb klrp molmlplkp f‘f ‘lkpfaåob nrb .,3 abp jliæ‘ribp pb aæmi^‘bkq a^kp i^ afob‘qflk bq ib pbkp (+ Ox ) ) .,3 prfs^kq ( − Ox ) ) .,3 obpmb‘qfsbjbkq prfs^kq ( Oy ) ) ( − Oy ) ) ( Oz ) ) ( − Oz ) %jlaåib ? ab i^ drob .+.&) bq nrb qlrqbp ibp m^oqf‘ribp lkq i^ jçjb sfqbppb æd^ib Ý i^ sfqbppb jlvbkkb klqæb v m + I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J ^kp+ .0 j^op /--1
.- @e^mfqob . Fkqolar‘qflk Ý i^ qebojlavk^jfnrb I l_gb‘qfc bpq f‘f ab ‘^i‘ribo i^ clo‘b abp ‘el‘p pro i^ m^olf bk clk‘qflk ab i^ sfqbppb abp m^oqf‘ribp mlro bk aæarfob bkprfqb i^ mobppflk+ Mlro ‘bi^) ‘lkpfaæolkp rkb p ^‘b d −→ S %slfo roc drob .+.& rq _ i‘ribo i^ s^of^qflk ab i^ nr^kqfqæ ab jlrsbjbkq molslnræb pro i^ 8 fi c^ a ^ loa ‘^ m^oqf‘rib → m^%oobrmkoæppbbrkiqæ‘belp‘r)omi^rfp d‘rljmqboibklj_obab‘el‘ppbmolarfp^ikbpqÝmb‘k^ar^pkbqa d b t pp‘reoli‘^p − proc^‘b dS ob .+.&+ I^ clo‘b pr_fb m^o ibp m^oqf‘r bpq − F → = dp − to → t dt a ^moåp ib qeæloåjb ab i^ nr^kqfqæ ab jlrsbjbkq) ^sb‘ dp − t → ot i^ s^of^qflk ab i^ nr^kqfqæ ab jlrsbjbkq qlq ib abp m^oqf‘ribp mbka^kq dt + Cfk^ibjbkq) i^ clo‘b pr_fb m^o i^ ^ m^olf bpq i lmmlpæb ab i^ clo‘b pr_fb m^o ibp m^oqf‘ribp %mofk‘fmb a ^‘qflk bq ab oæ^‘qflk&+ Ibp m^oqf‘ribp pr_fpp^kq rk ‘el‘ pro i^ m^olf plkq ibp m^oqf‘ribp pb aæmi^ä^kq prfs^kq (+ Oz ) ^sb‘ i^ sfqbppb v m pf i lk mobka ibp klq^qflkp ab i^ drob .+.+ Biibp obm^oqbkq bkprfqb a^kp i^ afob‘qflk ( − Oz ) ^sb‘ i^ sfqbppb v m + I^ s^of^qflk ab i^ nr^kqfqæ ab jlrsbjbkq ilop a rk pbri ‘el‘ bpq alk‘ d − p → = 2 ( mv m ) ( −−→ u z ) %0& Fi c^rq j^fkqbk^kq ‘ljmqbo ib klj_ob ab ‘el‘p pb molarfp^kq mbka^kq dt + Plfq n V i^ abkpfqæ m^oqf‘ri^fob+ Pbribp .,3 abp m^oqf‘ribp slkq a^kp i^ _lkkb afob‘qflk (+ Oz ) mlro ^iibo mbo‘rqbo i^ m^olf+ Ab mirp) mbka^kq dt ) ibp m^oqf‘ribp m^o‘lrobkq rkb afpq^k‘b l = v m dt 8 pbribp ibp m^oqf‘ribp pb pfqr^kq Ý rkb afpq^k‘b fkcæofbrob Ý l bq ^v^kq i^ _lkkb afob‘qflk slkq mbo‘rqbo i^ m^olf+ Fi v ^ alk‘ dN =16 n V ( ldS )=16 n V ( v m dtdS ) %1& m^oqf‘ribp mbo‘rq^kq i^ m^olf ab proc^‘b dS mbka^kq rk fkpq^kq dt + I^ s^of^qflk ab i^ nr^kqfqæ ab jlrsbjbkq qlq^ib pr_fb m^o ibp m^oqf‘ribp m^o qlrp ibp ‘el‘p ^v^kq br ifbr mbka^kq rk qbjmp dt plkq) ‘ljmqb qbkr abp obi^qflkp %0& bq %1& 7 dp − to → t = 2 ( mv m )16 n V ( v m dtdS ) ( −− u → z ) =13 n V mv 2 m dSdt ( −− u → z ) plfq rkb clo‘b − F → paroi → part = d − p → dt tot =13 n V mv 2 m dS ( −− u → z ) I^ clo‘b pr_fb m^o i^ m^olf bpq lmmlpæb 7 biib bpq alk‘ afofdæb prfs^kq u −→ z ) ‘ bpq*Ý*afob mbombkaf‘ri^fobjbkq Ý i^ m^olf bq sbop i buqæofbro+ Lo i^ mobppflk ‘fkæqfnrb bpq i^ clo‘b pr_fb m^o i^ m^olf m^o rkfqæ ab proc^‘b 7 P = FS =31 n V mv 2 m d Bk oæ^ifqæ) a^kp ib ‘^p ar jlaåib > ab i^ drob .+.) klrp ^roflkp l_qbkr ib oæpriq^q prfs^kq 7 P =13 n V mu 2 %2& ^sb‘ u i^ sfqbppb nr^ao^qfnrb jlvbkkb+ I^ pfjmif ‘^qflk ar jlaåib > bk jlaåib ? obsfbkq Ý ‘lkclkaob i^ sfqbppb jlvbkkb ^sb‘ i^ sfqbppb nr^ao^qfnrb jlvbkkb+ I^ obi^qflk %2& p æ‘ofq æd^ibjbkq) bk klq^kq V ib slirjb ab d^w ‘lkpfaæoæ bq N ib klj_ob ab m^oqf‘ribp qlq^i a^kp ‘b slirjb %^sb‘ n V = NV & 7 P V =31 Nmu 2 %3& I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J^kp+
.0 j^op /--1
Pb‘qflk .+0 I^ qeælofb ‘fkæqfnrb ar d^w m^oc^fq jlkl^qljfnrb .. Obj^onrb .+0 A ^moåp i^ obi^qflk %2& ab i^ mobppflk ‘fkæqfnrb) fi ^mm^o^ëq nrb ‘bqqb mobppflk bpq molmloqflkkbiib Ý i^ abkpfqæ m^oqf‘ri^fob ar d^w+ @bi^ pb ‘lkälfq ^fpæjbkq 7 p fi v ^ / clfp mirp ab m^oqf‘ribp) fi v ^ro^ / clfp mirp ab ‘el‘p) alk‘ i^ mobppflk alr_ib+ Ab jçjb) P bpq molmloqflkkbiib Ý i^ j^ppb abp m^oqf‘ribp 7 abp j^ppbp mirp fjmloq^kqbp cbolkq pr_fo abp clo‘bp mirp fjmloq^kqbp Ý i^ m^olf ilop ab ‘el‘p+ Bk k) i^ mobppflk bpq ^rppf molmloqflkkbiib Ý i^ sfqbppb ^r ‘^ abp m^oqf‘ribp+ ooæ
I^ mobppflk ^mm^o^ëq ‘ljjb rkb do^kabro j^‘olp‘lmfnrb 7 mlro jbprobo rkb mobppflk) fi pr§q ab jbprobo i^ clo‘b mobpp^kqb ar d^w p ^mmifnr^kq pro rkb proc^‘b S + Mlroq^kq i lofdfkb ab i^ mobppflk bpq _fbk jf‘olp‘lmfnrb 8 i^ obi^qflk P = (1 3) n V mu 2 obifb i^ mobppflk P j^‘olp‘lmfnrb %Ý d^r‘eb ab i æd^ifqæ& ^sb‘ abp do^kabrop jf‘olp‘lmfnrbp %Ý aolfqb ab i æd^ifqæ&+ Rkfqæp 7 A^kp ibp rkfqæp ar pvpqåjb fkqbok^qflk^i) i^ mobppflk p bumofjb bk M^p‘^i) klqæ P a 7 1 P a = 1 Nm − 2 J^fp ‘bqqb rkfqæ k bpq m^p ar qlrq ^a^mqæb ^ru mobppflkp ^j_f^kqbp 8 bk b bq) i^ mobppflk ^qjlpmeæofnrb Ý i^ proc^‘b ab i^ jbo bpq bk jlvbkkb ab 1 013 10 5 P a + Ib ^o lr i ^qjlpmeåob _ plkq ^ilop fkqolarfqp 7 1 bar = 10 5 P a 8 1 atm = 1 013 10 5 P a + .+0+1+ Aæ kfqflk ‘fkæqfnrb ab i^ qbjmæo^qrob I^ qbjmæo^qrob ‘fkæqfnrb a rk d^w m^oc^fq jlkl^qljfnrb bk ænrfif_ob qebojlavk^jfnrb pb aæ kfq ‘ljjb i^ jbprob ab i ækbodfb ‘fkæqfnrb jlvbkkb m^o ^qljb m^o i^ obi^qflk 7 ¿ 21 mv 2 À =32 B T k plfq bk‘lob 1 2 mu 2 =23 k B T %4& @b nrb pfdkf b nrb i^ qbjmæo^qrob bpq bk c^fq %Ý rkb ‘lkpq^kqb jriqfmif‘^qfsb 3 2 k B moåp& i ækbodfb ‘fkæqfnrb jlvbkkb abp ^qljbp+ k B bpq i^ ‘lkpq^kqb ab ?liqwj^kk 7 k B = 1 38 10 − 23 JK + I rkfqæ ab i^ qbjmæo^qrob bpq ib Hbisfk klqæb K + Obj^onrb .+1 I^ qbjmæo^qrob kb mbrq alk‘ m^p çqob kæd^qfsb) ‘^o rkb ækbodfb ‘fkæqfnrb kæd^qfsb k bpq m^p mlppf_ib+ Fi k bpq m^p klk mirp mlppf_ib a ^qqbfkaob ib wæol ^_plir % T = 0 K &) ‘b nrf pfdkf bo^fq nrb qlrp ibp ^qljbp plkq m^oc^fqbjbkq fjjl_fibp 7 i^ qbjmæo^qrob jfkfj^ib ^‘qrbiibjbkq ^qqbfkqb bpq 10 − 6 K + Obj^onrb .+2 >qqbkqflk) lk afq Hbisfk ) bq klk m^p abdoæ Hbisfk ffi Fi bpq qlrqbclfp rprbi a rqfifpbo ibp abdoæp @bipfrp %klqæp ◦ C &+ Rk æ‘^oq ab 1 ◦ C ‘loobpmlka Ý rk æ‘^oq ab 1 K 8 pbri ib wæol bpq aæ‘^iæ 7 0 ◦ C ‘loobpmlka Ý 273 15 K + I^ ‘lksbopflk bpq alk‘ 7 T ◦ C = T K − 273 15 I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J ^kp+ .0 j^op /--1
./ @e^mfqob . Fkqolar‘qflk Ý i^ qebojlavk^jfnrb .+0+2+ Bnr^qflk a æq^q a rk DMJ I^ obi^qflk %3& P V =31 Nmu 2 obifb i^ mobppflk Ý i^ sfqbppb nr^ao^qfnrb jlvbkkb+ I^ obi^qflk %4& 21 mu 2 =23 k B T obifb i^ qbjmæo^qrob Ý i^ sfqbppb nr^ao^qfnrb jlvbkkb+ Fi bpq ^ilop mlppf_ib a bk aæarfob rkb obi^qflk bkqob i^ qbjmæo^qrob bq i^ mobppflk 7 u 2 = 3 P V = 3 k B T N m m plfq P V = N k B T %5& nrf bpq i ænr^qflk a ^q ar d^w m^oc^fq jlkl^qljfnrb+ æq Obj^onrb .+3 I^ obi^qflk %5& bpq ^mmbiæb ænr^qflk a æq^q m^o‘b nr biib obifb ibp m^o^jåqobp a æq^q P ) V ) N bq T + Rkb ^rqob æ‘ofqrob ab i ænr^qflk a æq^q ar d^w m^oc^fq bpq plrsbkq rqfifpæb+ @ljjb N ) klj_ob ab m^oqf‘ribp) bpq qoåp do^ka %dækæo^ibjbkq ab i loaob ab 10 23 & bq k B qoåp mbqfq %ab i loaob ab 10 − 23 &) fi bpq mirp mo^qfnrb a rqfifpbo ibp do^kabrop R ) ‘lkpq^kqb abp d^w m^oc^fqp) bq n nr^kqfqæ ab j^qfåob) aæ kfbp ab i^ j^kfåob prfs^kqb+ Plfq a ^_loa N a i^ ‘lkpq^kqb a >sld^aol 7 N a = 6 02 10 23 %‘lkpq^kqb a >sld^aol aæ kfb a^kp ib pb‘lka^fob&+ Lk klqb n = NN a i^ nr^kqfqæ ab j^qfåob bumofjæb bk jlibp %pvj_lib mol & 8 lk klqb R = k B N a i^ ‘lkpq^kqb abp d^w m^oc^fqp %bk JK − 1 mol − 1 & 7 R = 8 314 JK − 1 mol − 1 I ænr^qflk ar d^w m^oc^fq absfbkq ^ilop 7 P V = nRT %6& Obj^ b .+4 Lk m^oib m^oclfp ab klj_ob ab jlibp ^r ifbr ab nr^kqfqæ ab j^qfåob onr mlro aæpfdkbo n ) ‘b nrf rk qbojb fjmolmob 7 ‘bi^ pbo^fq ænrfs^ibkq) m o bubjmib) Ý m^oibo ab ^ klj_ob ab jåqobp mlro aæpfdkbo rkb afpq^k‘b ffi
.+0+3+ Bkbodfb fkqbokb a rk DMJ I ækbodfb fkqbokb klqæb U a rk pvpqåjb bpq i^ pljjb ab qlrqbp ibp ækbodfbp jf‘olp‘lmfnrbp ar pvpqåjb 7 ibp ækbodfbp mlqbkqfbiibp a fkqbo^‘qflk jliæ‘rib*jliæ‘rib) ibp ækbodfbp a fkqbo^‘* qflk ^qljb*^qljb ^r pbfk ab ‘e^nrb jliæ‘rib ^fkpf nrb ibp ækbodfbp ‘fkæqfnrbp a ^dfq^qflk qebojfnrb+ Lo) m^o evmlqeåpb a^kp ib jlaåib ar d^w m^oc^fq) ibp ækbodfbp a fkqbo^‘qflkp bkqob abru m^oqf‘ribp plkq kriibp %slfo m^o^do^meb .+0+/&+ Ab mirp) pf ib d^w bpq jlkl^qljfnrb) fi .0 j^op /--1 I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J^kp+
Pb‘qflk .+1 Klqflk ab mevpfnrb pq^qfpqfnrb ab ?liqwj^kk %E+M+& .0 k bufpqb m^p a fkqbo^‘qflk ^qljb*^qljb ^r pbfk a rkb m^oqf‘rib+ I ækbodfb fkqbokb U a rk pvp* qåjb ab N m^oqf‘ribp bq alk‘ pfjmibjbkq i^ pljjb abp ækbodfbp ‘fkæqfnrbp ab ‘e^‘rkb abp ^ f‘ribp 7 m oq N U = X 21 mv i 2 i =1 = N ¿ 21 mv 2 À 1 µ 2 mu 2 ¶ = N plfq U = 3 N k T = 3 n 2 B 2 RT %.-& Obj^onrb .+5 T bpq rk m^o^jåqob a æq^q fkqbkpfc bq n rk m^o^jåqob a æq q buqbkpfc 7 U bpq ^ alk‘ rk m^o^jåqob a æq^q buqbkpfc+ Obj onrb .+6 > ‘e^nrb qbjmæo^qrob T ‘loobpmlka rkb ækbodfb fkqbokb U + I ækbodfb fk* ^ qbokb a rk d^w m^oc^fq jlkl^qljfnrb kb aæmbka m^p ab i^ mobppflk+
.+1+ Klqflk ab mevpfnrb pq^qfpqfnrb ab ?liqwj^kk %E+M+& @b m^o^do^meb bpq qo fqæ bk ‘lrop ^ .+2+ Ibp ^rqobp rfabp bq me^pbp ‘lkabkpæbp .+2+.+ Aæ kfqflkp .+2+.+.+ I^ mobppflk Mi^älkp rkb m^olf pæm^o^qof‘b bkqob ib d^w bq ib sfab+ I^ mobppflk bpq aæ kfb ‘ljjb æq^kq i^ clo‘b m^o rkfqæ ab proc^‘b p ^mmifnr^kq pro ‘bqqb m^olf bq ‘oææb m^o ib d^w+ Obj^onrb .+.->qqbkqflk ffi Ab j^kfåob dækæo^ib) i^ mobppflk k bpq m^p ifæb rkfnrbjbkq ^ru ‘el‘p abp m^oqf‘ribp pro i^ m^olf) ‘ljjb ‘ bpq i^ ‘^p mlro ib d^w m^oc^fq+ @ljjb klr ^iilkp p ib slfo a^kp i^ prfqb ab ‘b ‘e^mfqob) i^ mobppflk bpq i^ pljjb ab i^ mobppflk ‘fkæqfnrb %arb ^ ru ‘el‘p& Ý i^nrbiib lk ^glrqb i^ mobppflk jliæ‘ri^fob %arb Ý i fkqbo^‘qflk abp m^oqf‘ribp bkqob biibp) fkqbo^‘qflk nrf ^ ^rppf rkb fk rbk‘b fkafob‘qb pro i^ clo‘b ab mlrppæb p ^mmifnr^kq pro i^ m^olf %slfo m^o bubjmib m^o^do^meb .+2+0+. &+
I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J ^kp+
.0 j^op /--1
@e^mfqob . Fkqolar‘qflk Ý i^ qebojlavk^jfnrb
.1 .+2+.+/+ I^ qbjmæo^qrob Mlro ib d^w m^oc^fq) i^ qbjmæo^qrob bpq molmloqflkkbiib Ý i ækbodfb ‘fkæqfnrb jf‘olp‘lmfnrb jlvbkkb a rkb m^oqf‘rib+ @ bpq ib ‘^p æd^ibjbkq mlro ibp ^rqobp rfabp+ .+2+.+0+ I ækbodfb fkqbokb I ækbodfb fkqbokb) klqæb U ) bpq i^ pljjb ab qlrqbp ibp ækbodfbp jf‘olp‘lmfnrbp ^r pbfk ar pvpqåjb 7 * ibp ækbodfbp ‘fkæqfnrbp a ^dfq^qflk qebojfnrb 8 * ibp ækbodfbp a fkqbo^‘qflk jliæ‘rib*jliæ‘rib 8 * ibp ækbodfbp a fkqbo^‘qflk ^qljb*^qljb r pbfk ab ‘e^nrb jliæ‘rib+ ^ @ljjb fi ^ æqæ sr moæ‘æabjjbkq) rk d^w m^oc^fq jlkl^qljfnrb kb ‘lkqfbkq ^r‘rkb clo‘b a fkqbo^‘qflk fkqbokb 7 i ækbodfb fkqbokb U kb aæmbka alk‘ nrb ab i^ qbjmæo^qrob+ M^o ‘lkqob) mlro rk rfab nrbi‘lknrb) fi c^rq ^glrqbo ibp ækbodfbp a fkqbo^‘qflk bkqob ibp af æobkqbp m^oqf* ‘ribp 8 i ækbodfb fkqbokb U aæmbka alk‘ ^rppf ar slirjb V ar pvpqåjb+ Bk b bq) pf ib slirjb a rk pvpqåjb V bpq jlaf æ) i^ afpq^k‘b jlvbkkb bkqob ibp m^oqf‘ribp bpq jlaf æb) ‘b nrf ‘e^kdb alk‘ ibp ækbodfbp a fkqbo^‘qflk) alk‘ i ækbodfb fkqbokb U + .+2+.+1+ I^ ‘^m^‘fqæ ‘^ilof nrb Ý slirjb ‘lkpq^kq Bk ‘lk‘irpflk ab ‘b nrf sfbkq a çqob sr 7 * mlro rk d^w m^oc^fq jlkl^qljfnrb 7 U ( T ) 8 * mlro rk rfab nrbi‘lknrb 7 U ( T V ) + I^ ‘^m^‘fqæ ‘^ilofnrb Ý slirjb ‘lkpq^kq ) klqæb C V ) bpq ^ilop aæ kfb 7 C µ ∂∂UT ¶ V %..& V = I^ ‘^m^‘fqæ ‘^ilof nrb Ý slirjb ‘lkpq^kq bpq i ækbodfb Ý ^mmloqbo ^r pvpqåjb mlro ^rd* jbkqbo p^ qbjmæo^qrob a rk Hbisfk) nr^ka i bumæofbk‘b bpq b b‘qræb Ý slirjb ‘lkpq^kq+ Obj^onrb .+.. I fkaf‘b V bpq fkafpmbkp^ ib a^kp i^ aæ kfqflk %..&+ Bk b bq) i^ klq^qflk _ ( ∂U∂T ) p^kp fkaf‘b kb sbrq ofbk afob ffi Rkb aæofsæb m^oqfbiib %lr aæofsæb olkab & pfdkf b nrb i lk aæofsb U m^o o^mmloq Ý T bk d^oa^kq ibp ^rqobp do^kabrop ‘lkpq^kqbp) bk m^oqf‘rifbo f‘f bk d^oa^kq V bq P ‘lkpq^kqbp+ Lo Ý ‘^rpb ab i ænr^qflk a æq^q nrf bufpqb mlro qlrq pvpqåjb %‘ljjb i ænr^qflk ar d^w m^oc^fq m^o bubjmib&) ibp m^o^jåqobp P ) T bq V plkq obifæp+ Aæofsbo U m^o o^mmloq Ý T pfdkf b nrb i lk ‘^i‘rib i^ s^of^qflk ab U nr^ka lk c^fq s^ofbo T 8 lo fi bpq fjmlppf_ibp ab c^fob s^ofbo T bk d^oa^kq P bq V ‘lkpq^kqp Ý i^ clfp) Ý ‘^rpb ab i ænr^qflk a æq^q+ Fi c^rq ^ilop moæ‘fpbo pf ‘ bpq P lr pf ‘ bpq V nrb i lk d^oab ‘lkpq^kq nr^ka lk aæofsb m^o o^mmloq Ý T 7 µ ∂∂UT ¶ V 6 = µ ∂∂TU ¶ P %./& Ib qbojb ab d^r‘eb ab i bumobppflk %./& pfdkf b nrb i lk jbprob i^ s^of^qflk ab U nr^ka T s^ofb bk d^oa^kq V ‘lkpq^kq 7 P s^ofb ^ilop fjmif‘fqbjbkq %‘^o P = f ( T V ) &+ Ib pb‘lka jbj_ob ab i bumobppflk %./& pfdkf b nrb i lk æqrafb i^ s^of^qflk ab U nr^ka T s^ofb bk d^oa^kq P ‘lkpq^kq %bq alk‘ V s ofb fjmif‘fqbjbkq&+ Bumæofjbkq^ibjbkq) mlro æs^irbo ib mobjfbo ^ qbojb) fi c^rao^fq ‘e^r bo ib pvpqåjb a^kp rk oæ‘fmfbkq ^ru m^olfp ofdfabp %alk‘ V = cte 7 m^o bubjmib a^kp rk ^rql‘rfpbro& 8 i^ mobppflk ^rdjbkqbo^fq ^ilop mol_^_ibjbkq+ Mlro æs^irbo ib pb‘lka qbojb) fi c^rao^fq ‘e^r bo ib pvpqåjb Ý mobppflk ‘lkpq^kqb) alk‘ m^o bubjmib bk ib .0 j^op /--1 I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J^kp+
Pb‘qflk .+2 Ibp ^rqobp rfabp bq me^pbp ‘lkabkpæbp .2 mi^ä^kq a^kp rkb bk‘bfkqb æi^pqfnrb mi^‘æb a^kp i ^qjlpmeåob ab mobppflk ‘lkpq^kqb 7 fi bpq mlppf_ib nrb ib slirjb ^rdjbkqb ^ilop ilop ar ‘e^r ^db+ A^kp ib ‘^p ar d^w m^oc^fq jlkl^qljfnrb) ‘ljjb U = (3 2) nRT ) ^ilop 7 C V ( GP M ) = µ ∂∂TU ¶ V =23 nR Lk mbrq æd^ibjbkq aæ kfo rk C V j^ppfnrb klqæ c v nrf bpq ib C V m^o rkfqæ ab j^ppb 7 C V c v = m bq rk C V jli^fob ) nrf bpq rk C V m^o jlib klqæ C V m 7 C V m = C V n Rkfqæp 7 * C V bpq bk Glrib m^o Hbisfk % JK − 1 & 8 * c v bpq bk JK − 1 kg − 1 8 * C V m bpq bk JK − 1 mol − 1 %jçjb rkfqæ nrb R &+ A^kp ib ‘^p ar d^w m^oc^fq jlkl^qljfnrb 7 C V m ( GP M ) = (3 2) R .+2+/+ Ibp d^w m^oc^fqp mliv^qljfnrbp Ibp evmlqeåpbp ar d^w m^oc^fq mliv^qljfnrb plkq ibp prfs^kqbp 7 fi k v ^ m^p a fkqbo^‘qflk jliæ‘rib*jliæ‘rib 8 ibp jliæ‘ribp plkq ‘lkpfaæoæbp mlk‘qrbiibp bq i^ afpqof rqflk abp sfqbppbp bpq _ fplqolmb+ M^o ‘lkqob) fi v ^ abru ^qljbp m^o jliæ‘rib 7 fi c^rq alk‘ qbkfo ‘ljmqb ab i fkqbo^‘qflk ^qljb*^qljb ^r pbfk ab i^ jliæ‘rib+ Obmobklkp ^ilop ibp afsbop oæpriq^qp ar d^w m^oc^fq jlkl^qljfnrb mlro ibp dækæo^ifpbo ^r d^w m^oc^fq mliv^qljfnrb+ I^ mobppflk ‘fkæqfnrb %arb ^ru ‘el‘p pro i^ m^olf& bpq qlrglrop 7 = 1 n mu 2 P 3 V ^ qbj ^ ‘fkæqfnrb bpq qlrglrop aæ kfb m^o 7 I mæo qrob 21 mu 2 =32 k B T Fi c^rq pfjmibjbkq obj^onrbo nrb m bpq i^ j^ppb a rkb jliæ‘rib %bq klk m^p a rk ^qljb&+ I^ ilf ar d^w m^oc^fq jlkl^qljfnrb pb dækæo^ifpb alk‘ m^oc^fqbjbkq Ý qlrp ibp d^w m^o* c^fqp ) jçjb mliv^qljfnrbp 7 P V = nRT M^o ‘lkqob) a^kp i ækbodfb fkqbokb) fi c^rq ^glrqbo abp clojbp a ækbodfb klrsbiibp m^o o^mmloq ^r d^w m^oc^fq jlkl^qljfnrb arb Ý i^ moæpbk‘b abp ^qljbp a^kp ‘e^nrb jliæ‘rib %ækbodfbp ‘fkæqfnrbp ab olq^qflk ab i^ jliæ‘rib pro biib jçjb bq ækbodfbp a fkqbo^‘qflkp bkqob ibp ^qljbp&+ Lk ^ajbq nrb i lk ^ qlrglrop rkb obi^qflk ab i^ clojb U = nC V m T I+ Jbkdrv) Iv‘æb Jlkqbpnrfbr Ib J ^kp+
.0 j^op /--1
