Teoriya kînetîk a gazan (Kinetic theory of gases)

Teoriya kînetîk a gazan modelek klasîk a sade ye ya tevgera termodînamîk a gazan. Danasîna wê rê da gelek têgehên sereke yên…

Teoriya Kînetîk a Gazan modelek klasîk a bingehîn e ji bo têgihîştina tevgera termodînamîk a gazan. Danasîna wê rê li ber damezrandina gelek têgehên termodînamîk ên bingehîn vekir. Ev teorî diyar dike ku gaz ji gelek parçikên Mîkroskopîk pêk tê ku di nav tevgera domdar û bêserûber de ne. Ev parçik niha wekî atom an molekulên ku gazê pêk tînin têne nasîn. Teoriya kînetîk têkiliya di navbera taybetmendiyên gazê yên makroskopîk (mînak, Qebare, Pesto, Germahî) û bûyerên veguhastinê (mînak, vîskozîte, gihandina germê, belavbûna girseyê) bi analîzkirina lihevketinên parçikan bi hev re û bi dîwarên Konteynerê re zelal dike.

Formûlasyona bingehîn a vê modelê gazek îdeal diyar dike. Ew lihevketinên bi tevahî elastîk wekî yekane têkiliya di navbera parçikan de diyar dike, herwiha texmîn dike ku parçik bi awayekî girîng ji dûrahiya navmolekulî ya wan a navînî piçûktir in.

Ji ber vegerandina demê ya ku di dînamîkên Mîkroskopîk de heye (ango, vegerandina Mîkroskopîk), teoriya kînetîk têkiliyek bi prensîba hevsengiya berfireh re saz dike, ku bi rêya teorema guherîn-belavbûnê (bi taybetî ji bo tevgera Brownian) û têkiliyên berevajî yên Onsager tê xuyakirin.

Ev teorî di Dîrokê de girîng bû wekî yekemîn serîlêdana eşkere ya prensîbên Mekanîka îstatîstîkî û bingeha teorîk ji bo dînamîkên gazê yên kêmkirî pêk tîne.

Dîrok

Teoriya Kînetîk a Maddeyê

Serdema Antîk

Nêzîkî 50 BZ, Fîlozofê Romî Lucretius diyar kir ku hebûnên makroskopîk ên ku xuya dikin Rawestayî ne, di bingehîn de ji atomên ku bi lez tevdigerin û bi hev re li hev dikevin di astek Mîkroskopîk de pêk tên. Ev Perspektîfa atomîstîk a Epîkûrî di sedsalên paşîn de, di dema ku têgehên Arîstotelî bi awayekî girîng serdest bûn, bala kêm dît.

Serdema Nûjen

"Germî tevger e"

Di sala 1620an de, Fîlozofê Îngilîz Francis Bacon yek ji daxuyaniyên herî destpêkê û herî bi hêz derbarê têkiliya di navbera tevgera parçikan û germê de anî ziman: "Divê neyê fikirîn ku germî tevgerê çêdike, an tevger germiyê (herçendî di hin aliyan de ev rast be jî), lê ku Cewherê germiyê ... tevger e û tiştek din nîne." Wî ev yek wekî "ne ... tevgera tevahiyê, lê ya parçikên piçûk ên laş" zelal kir. Dûv re, di sala 1623an de, Galileo Galilei di The Assayer de îdia kir ku têgihîştinên hestî yên wekî germî, Pesto, û bîhn tenê taybetmendiyên xuya ne, ku ji bûyera rastîn a tevgera parçikan derdikevin.

Di sala 1665an de, Robert Hooke, polîmatêkî Îngilîz, gotina Bacon di nav karê xwe yê Micrographia de dubare kir. Paşê, di sala 1675an de, hemdemê wî, zanyarê Anglo-Îrlandî Robert Boyle, dît ku "hêza" çakûçekî vediguhezîne tevgera parçeyên pêkhatî yên neynûkê, û ev tevger wekî cewhera germahiyê destnîşan kir. Boyle herwiha pêşniyar kir ku hemî taybetmendiyên makroskopîk, wekî reng, çêj û elastîkî, ji rêzkirin û tevgera tenê parçeyên madeyî yên nedabeşkirî derdikevin û bi bingehîn ji wan pêk tên. Di dema dersa sala 1681an de, Hooke têkiliyek rasterast di navbera germahiya tiştekî û leza parçeyên wê yên hundirîn de piştrast kir: "Germahî ... ne tiştek e ji bilî Tevgera hundirîn a Parçeyên [Laşekî]; û laşek çiqas germtir be, Parçe ewqas bi tundî tevdigerin." Di destnivîsek ku di sala 1720an de hate weşandin de, fîlozofê Îngilîz John Locke pêşniyarek pir bi heman rengî anî ziman: "Tiştê ku di hestiyariya me de germî ye, di tişt de ne tiştek e ji bilî tevgerê." Locke herwiha tevgera parçeyên hundirîn ên tiştê, ku wî jê re digot "parçeyên wê yên nedîtbar", nîqaş kir.

Di teza xwe ya sala 1744an de, Meditations on the Cause of Heat and Cold, polîmatêkê Rûs Mikhail Lomonosov serlêdanek gihîştî ya ezmûna berbelav bi kar anî da ku taybetmendiyên mîkroskopîk û kînetîk ên made û germahiyê biparêze:

Tevger divê neyê înkarkirin ji ber ku nayê dîtin. Kî dê înkar bike ku pelên daran tevdigerin dema ku ji hêla ba ve têne hejandin, tevî ku ji dûr ve nayê dîtin? Çawa ku di vê rewşê de tevger ji ber perspektîfê veşartî dimîne, ew di laşên germ de jî ji ber mezinahiyên pir piçûk ên parçeyên tevgerîn veşartî dimîne. Di her du rewşan de jî, goşeya dîtinê ew qas piçûk e ku ne tişt û ne jî tevgera wan nayê dîtin.

Lomonosov herwiha piştrast kir ku tevgera parçeyan ji bo pêvajoyên helandin, derxistin û belavbûnê girîng e. Wî mînak anîn ziman wekî helandin û belavbûna xwêyan bi çalakiya parçeyên avê li ser "molekulên xwê", helandina metalan di merkûrê de, û derxistina pîgmentên rwekê bi alkolê.

Veguhestina germahiyê bi heman rengî ji tevgera parçeyan re hate vegotin. Nêzîkî di sala 1760an de, fîzîknas û kîmyagerê Skotlendî Joseph Black destnîşan kir, "Gelekan texmîn kirine ku germî tevgera lerzokî ... ya parçeyên madeyê ye, ku ... ev tevger wan xeyal dikir ku ji laşekî bo yê din tê ragihandin."

Teoriya Kînetîk a Gazan

Di sala 1738an de, Daniel Bernoulli Hydrodynamica weşand, ku nivîseke bingehîn bû ji bo Teorîya Kînetîk a Gazan. Di nav vê weşanê de, Bernoulli pêşniyar kir ku gaz ji gelek molekulan pêk tên ku di rêgezên cûda de tevdigerin, û îdîa kir ku lêketina wan li ser Rûxarê Pesto ya gazê çêdike û Enerjiya Kînetîk a wan a navîn Germahîya gazê diyar dike. Teorîyê tavilê nehat qebûlkirin, qismî ji ber ku prensîba Parastina Xwezayê ya Enerjîyê hîn nehatibû damezrandin, û Mekanîzmaya lêketinên molekular ên bi tevahî elastîk ji Fîzîknasan re nezelal ma.

Pêşengên destpêkê yên Teorîya Kînetîk, ku beşdariyên wan ji hêla hemdemên wan ve bi giranî hatibûn paşguhkirin, di nav wan de Mikhail Lomonosov (1747), Georges-Louis Le Sage (nêzîkî 1780, di sala 1818an de hat weşandin), John Herapath (1816), û John James Waterston (1843) hebûn. Lêkolîna wan bi formûlasyona ravekirinên mekanîkî yên ji bo gravîtasyonê ve girêdayî bû.

Di sala 1856an de, August Krönig modeleke gaz-kînetîk a bingehîn pêş xist ku bi taybetî tevgera translasyonel a parçikan dihesiband. Sala paşîn, Rudolf Clausius guhertoyeke Teorîyê ya berawirdî lê tevlihevtir pêş xist, ku tevgerên molekular ên translasyonel, rotasyonel û vibrasyonel di nav de dihewand — ku ji Modela Krönig cuda bû. Di nav vê weşana heman de, Clausius têgeha rêça azad a navîn a parçikekê destnîşan kir. Di sala 1859an de, ji Karê Clausius ê li ser belavbûna molekular îlhama xwe girt, Fîzîknasê Skotlendî James Clerk Maxwell belavbûna Maxwell a Lezên molekular pêş xist, ku Rêje ya molekulên xwedî Lezek taybetî di nav rêzek diyarkirî de diyar dike. Ev yek qanûna îstatîstîkî ya destpêkê di Fîzîkê de nîşan da. Maxwell her weha argumana mekanîkî ya destpêkê pêşkêş kir ku digot lêketinên molekular dibin sedema wekhevkirina Germahîyê û, Wekî encam, pêşveçûnek ber bi hevsengiyê ve. Di gotara xwe ya sêzdeh-rûpelî ya sala 1873an de, 'Molekul', Maxwell îdîa kir: "ji me re tê gotin ku 'Atom' xalek maddî ye, ku bi 'hêzên potansiyel' ve hatiye dorpêçkirin û pêçandin û dema 'molekulên difirin' bi Berdewamî li laşek Hişk dikevin, ew dibe sedema tiştê ku jê re Pesto ya hewayê û gazên din tê gotin." Di sala 1871an de, Ludwig Boltzmann beşdariyên Maxwell berfireh kir, û belavbûna Maxwell–Boltzmann formule kir. Boltzmann her weha yekem car têkiliya logarîtmîkî di navbera Entropî û îhtîmalê de diyar kir.

Di destpêka Sedsala 20an de, gelek Fîzîknasan Atom wekî avahiyên bi tevahî teorîk dihesibandin, ne ku wekî hebûnên berbiçav. Pêşveçûnek girîng bi weşanên Albert Einstein (1905) û Marian Smoluchowski (1906) yên li ser tevgera Brownian pêk hat, ku bi serfirazî pêşbîniyên hejmarî yên rastîn ên ku ji Teorîya Kînetîk hatine wergirtin çêkirin.

Piştî ku hevkêşeya Boltzmann hate formulekirin, David Enskog (1917) û Sydney Chapman (1916) bi serê xwe çarçoveyek ji bo sepandina wê di pêşxistina hevkêşeyên veguhastinê de damezrandin. Vê çarçoveyê pêşbînîkirina taybetmendiyên veguhastinê di gazên hênik de hêsan kir û wekî Teorîya Chapman–Enskog hate nasîn. Di sedsala paşîn de, çarçove gav bi gav berfireh bû, di encamê de pêşbînîkirina taybetmendiyên veguhastinê di gazên rastîn û tîr de gengaz kir.

Hîpotez

Teorîya kinetîk, dema ku li gazên îdeal tê sepandin, xwe dispêre hîpotezên jêrîn:

Gaz ji perçeyên pir piçûk pêk tê. Mezinahiya wan a piçûk tê vê wateyê ku qebareya giştî ya molekulên gazê yên takekesî li gorî qebareya tevahî ya konteynera gazê ne girîng e. Ev pêşgotin hevwate ye bi îdîaya ku dûrahiya veqetandina navîn di navbera perçeyên gazê de ji pîvanên wan ên takekesî pir zêdetir e, û ku dema pevçûnek di navbera perçeyan û dîwarê konteynerê de li gorî navbera di navbera pevçûnên paşîn de pir kêm e.
Pîvana girîng a perçeyên niha, nêzîkatiyek statîstîkî ya pirsgirêkê bi tundî piştrast dike. Ev pêşgotina bingehîn bi gelemperî wekî sînora termodînamîk tê binavkirin.
Van perçeyên ku bi lez tevdigerin, bi berdewamî bi hev re û bi sînorên konteynerê re pevçûnên bi tevahî elastîk dikin.
Têkiliyên perçeyan, bi taybetî pevçûn, bi tenê dualî ne û tu têkiliyek nîşan nadin, ku tê vê wateyê ku têkiliyên sê-laşî an yên rêkûpêkî-bilindtir tune ne û bîra perçeyan tune ye.
Ji bilî kêliyên pevçûnê, têkiliyên navmolekulî kêm têne hesibandin, ji ber ku perçe tu hêzên din li ser hev nakin.

Wekî encam, dînamîkên tevgera perçeyan dikarin bi karanîna mekanîka klasîk werin analîz kirin, ku dibe sedema hevkêşeyên tevgerê yên dem-vegerbar.

Ji bo hêsankirinê, bi gelemperî tê texmîn kirin ku perçe xwedî girseyên yekta ne; lêbelê, çarçoveya teorîk dikare were berfireh kirin da ku belavbûnek girseyî bigire nav xwe, ku tê de her cureya girseyê bi serê xwe bandorê li taybetmendiyên gazê dike, li gorî qanûna Dalton a zextên qismî. Herwiha, gelek pêşbîniyên ku ji vî modelê hatine wergirtin, bêyî ku pevçûnên nav-perçeyî werin navandin, domdar dimînin, ku dibe sedema ku ew bi gelemperî wekî hîpotezek hêsanker di wergirtinên teorîk de werin derxistin.

Pêşketinên hemdemî, di nav de Teorîya Enskog a nûkirî û Modela Bhatnagar–Gross–Krook a berfirehkirî, yek an çend ji wan texmînên jorîn diguherînin. Van modelên pêşketî taybetmendiyên Gazên tîr û Gazên ku dereceyên azadiyê yên navxweyî nîşan didin bi duristî diyar dikin, bi tevlêkirina Qebareya parçikan, beşdariyên ji Hêzên navmolekulî û navmolekulî, zivirînên molekulî yên Kuantumkirî, bandorên sîmetrîya zivirî-lerzînî ya Kuantum, û heyecana elektronîkî. Her çend Teorîyên ku texmînên Qebareya parçikan a piçûk û lihevketinên bi tevahî elastîk sist dikin bi bandor hatine îsbat kirin, hatiye nîşandan ku sistkirina pêşşertê ji bo têkiliyên dualî û bêkorelasyon Di encamê de encamên cuda dide.

Taybetmendiyên Hêvsengiyê

Pesto û Enerjiya Kînetîk

Di nav Teorîya kînetîk a Gazan de, Pesto wekî Hêza ser yekîneya rûberê tê pênasekirin, ku ji hêla atom an molekulên Gazê yên takekesî ve tê hilberandin dema ku li Rûxara hundirîn a Konteynerê dixin û paşê jê vedigerin.

Parçikek Gazê bifikirin ku bi Lezekê, ${\textstyle v_{i}}$ , li ser rêgeha ${\hat {i}}$ di nav Qebareyek girtî de dimeşe, ku bi dirêjahiyek, $L_{i}$ , rûberek qutkirî, $A_{i}$ , û Qebareyek, $V=A_{i}L_{i}$ , tê diyar kirin. Ev parçika Gazê dê piştî demeke taybetmendî ya ku bi $t=L_{i}/v_{i}.$ tê pênasekirin, rastî Sînorekê were.

Momentum a parçikek Gazê paşê dikare wekî $p_{i}=mv_{i}=mL_{i}/t.$ were pênasekirin.

Ev formulekirin paşê bi qanûna duyemîn a Newton ve tê yekkirin, ku diyar dike Hêza ku li ser perçeyekê tê sepandin, bi rêjeya guherîna demkî ya Momentum wê re têkildar e, wekî ku tê îfadekirin: $F_{i}={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}={\frac {mL_{i}}{t^{2}}}={\frac {mv_{i}^{2}}{L_{i}}}.$ = m v i $F_{i}={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}={\frac {mL_{i}}{t^{2}}}={\frac {mv_{i}^{2}}{L_{i}}}.$ L i . {\displaystyle F_{i}={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}={\frac {mL_{i}}{t^{2}}}={\frac {mv_{i}^{2}}{L_{i}}}.} .

Niha em Pîvanek girîng, $N$ , ji perçeyên Gazê yên ku bi awayekî bêserûber di nav de Qebareyek sê-alî de hatine rêzkirin, bifikirin. Ji ber vê rêzkirina bêserûber, leza navînî ya perçeyan, ${\textstyle v}$ , li seranserê hemî aliyan yekgirtî ye, bi taybetî $v_{x}^{2}=v_{y}^{2}=v_{z}^{2}.$ = v y $N$ = v z $N$ . {\displaystyle v_{x}^{2}=v_{y}^{2}=v_{z}^{2}.} .

Wekî din, were em bifikirin ku Qebare li ser sê aliyên xwe Sîmetrî nîşan dide, bi taybetî li ser vektorên yekîneyê ${\hat {i}},{\hat {j}},{\hat {k}}$ , bi vî awayî tê wateya:

Pesto ya ku ji ber lihevketinên $N$ perçeyên Gazê bi Rûxarekê re çêdibe, bi berhevkirina beşdariyên Hêzê yên hemî perçeyan û dabeşkirina li ser Rûxara hundirîn a Qebareya girtî ve tê destnîşankirin. $P={\frac {N{\overline {F}}}{A}}={\frac {NLF}{V}}$ $\Rightarrow PV=NLF={\frac {N}{3}}mv^{2}.$ . {\displaystyle \Rightarrow PV=NLF={\frac {N}{3}}mv^{2}.}

Enerjiya Kînetîk a veguhêzbar a giştî ya gazê, ku wekî $K_{\text{t}}$ tê nîşankirin, bi fermî bi hevkêşeya jêrîn tê pênasekirin: $K_{\text{t}}={\frac {N}{2}}mv^{2},$ m v $K_{\text{t}}$ , {\displaystyle K_{\text{t}}={\frac {N}{2}}mv^{2},} . Ev pênase têkiliya jêrîn dide: $PV={\frac {2}{3}}K_{\text{t}}.$

Ev encamek girîng û ne-eşkere ya teoriya kînetîk e, ji ber ku ew têkiliyek Bingehîn di navbera Pesto, ku çavdêriyek makroskopîk e, û Enerjiya Kînetîk a veguhêzbar a molekulan, ku taybetmendiyek Mîkroskopîk e, saz dike.

Tîrbûna girseyî ya gazê, ku bi $\rho$ tê nîşankirin, wekî rêjeya girseya giştî ya parçikên gazê li ser Qebareya ku ji hêla gazê ve hatî dagirkirin tê pênasekirin: $\rho ={\frac {Nm}{V}}$ . Bi tevlêkirina vê pênaseyê, Pesto dikare bi heman awayî wekî were îfadekirin: $P={\frac {\rho v^{2}}{3}}.$ 73 . {\displaystyle P={\frac {\rho v^{2}}{3}}.}

Formulasyona relatîvîstîk a vê hevkêşeyê ev e:

Pesto $P={\frac {2\rho c^{2}}{3}}\left({\left(1-{\overline {v^{2}}}/c^{2}\right)}^{-1/2}-1\right),$ 41−v§52 $P={\frac {2\rho c^{2}}{3}}\left({\left(1-{\overline {v^{2}}}/c^{2}\right)}^{-1/2}-1\right),$ ¯/c§69 $P={\frac {2\rho c^{2}}{3}}\left({\left(1-{\overline {v^{2}}}/c^{2}\right)}^{-1/2}-1\right),$ )−82/§88 $P={\frac {2\rho c^{2}}{3}}\left({\left(1-{\overline {v^{2}}}/c^{2}\right)}^{-1/2}-1\right),$ −95),{\displaystyle P={\frac {2\rho c^{2}}{3}}\left({\left(1-{\overline {v^{2}}}/c^{2}\right)}^{-1/2}-1\right),} li vir $c$ leza Sivikê nîşan dide. Dema ku Sînorê lezên biçûk tê hesibandin, îfade hêsan dibe: $P\approx \rho {\overline {v^{2}}}/3$ ¯/160{\displaystyle P\approx \rho {\overline {v^{2}}}/3}.

Germahî û Enerjiya Kînetîk

Bi ji nû ve formulakirina encama Pesto ya berê wekî ${\textstyle PV={\frac {1}{3}}Nmv^{2}}$ 14§1617§Nmv§28 ${\textstyle PV={\frac {1}{3}}Nmv^{2}}$ {\textstyle PV={\frac {1}{3}}Nmv^{2}}, ev îfade dikare bi qanûna gazê ya îdeal re were yekkirin.

Li vir, $k_{\mathrm {B} }$ Berdewama Boltzmann temsîl dike, û $T$ germahiya mutleq destnîşan dike, wekî ku ji hêla qanûna gazê ya îdeal ve hatî pênasekirin. Ev têkilî hevkêşeyê dide: $k_{\mathrm {B} }T={\frac {1}{3}}mv^{2},$ 62§6465§mv $k_{\mathrm {B} }$ ,{\displaystyle k_{\mathrm {B} }T={\frac {1}{3}}mv^{2},} ku hêsan dibe bo îfadeyekê ji bo enerjiya kînetîk a wergerî ya navînî ji bo her Molekulê: ${\frac {1}{2}}mv^{2}={\frac {3}{2}}k_{\mathrm {B} }T.$ 96 $k_{\mathrm {B} }$ mv $k_{\mathrm {B} }$ =116 $k_{\mathrm {B} }$ kBT.{\displaystyle {\frac {1}{2}}mv^{2}={\frac {3}{2}}k_{\mathrm {B} }T.} Enerjiya kînetîk a wergerî ya giştî ya Pergalê $N$ qatên enerjiya kînetîk a yek Molekulê ye, ku wekî ${\textstyle K_{\text{t}}={\frac {1}{2}}Nmv^{2}}$ 178 $k_{\mathrm {B} }$ Nmv $k_{\mathrm {B} }$ {\textstyle K_{\text{t}}={\frac {1}{2}}Nmv^{2}} tê îfadekirin. Wekî encam, germahî $T$ rasterast bi enerjiya kînetîk a wergerî ve girêdayî ye.

Ev têkilî bêtir pêşve diçe û wiha dibe:

Hevkêşeya (3) encamek Bingehîn a Teorîya Kînetîk temsîl dike: enerjiya kînetîk a Molekulê ya navînî rasterast bi germahiya mutleq re hevrêj e, wekî ku ji hêla qanûna gazê ya îdeal ve hatî pênasekirin. Bi yekkirina hevkêşeyên () û (), tê destnîşankirin ku:

Wekî encam, berhema Pesto û Qebareya her molê hevrêjiyê bi enerjiya kînetîk a Molekulê ya wergerî ya navînî re nîşan dide.

Hevkêşeyên (1) û (4) wekî "encamên klasîk" têne binavkirin, ku dikarin ji Mekanîka îstatîstîkî jî werin derxistin.

Teorema hevpariyê diyar dike ku Enerjiya Kînetîk bi awayekî wekhev di nav hemî pileya azadiyê ya kînetîk de tê belavkirin, ku wekî D tê destnîşankirin. Mînak, Gazek yekatomî Sîmetrîya eksenî li ser her eksena fezayî nîşan dide, ku dibe sedema D = 3 pileya azadiyê, yên ku bi tevgera wergerî ya li ser her eksenê re têkildar in. Berevajî, Gazek duatomî tenê li ser yek eksenê xwedî Sîmetrîya eksenî ye, ku encam dide D = 5 pileya azadiyê, ku tevgera wergerî ya li ser sê eksenan û tevgera zivirî ya li ser du eksenan dihewîne. Gazek piratomî, wek av, li dora ti eksenê Sîmetrîya radyalî tune, bi vî awayî D = 6 pileya azadiyê dide, ku ji sê pêkhateyên wergerî û sê yên zivirî pêk tê.

Teorema parvekirina wekhev ferman dike ku Enerjiya Kînetîk bi awayekî wekhev were belavkirin, ku dibe sedem ku Enerjiya Kînetîk a giştî wekî $K=DK_{\text{t}}={\frac {D}{2}}Nmv^{2}.$ Nmv $K=DK_{\text{t}}={\frac {D}{2}}Nmv^{2}.$ .{\displaystyle K=DK_{\text{t}}={\frac {D}{2}}Nmv^{2}.} were îfadekirin.

Wekî encam, Enerjiya ku di nav Pergalê de ji bo her pileya azadiyê ya kînetîk ji bo her parçikek Gazê tê derxistin wekî ${\frac {K}{ND}}={\frac {1}{2}}k_{\text{B}}T.$ 22 ${\frac {K}{ND}}={\frac {1}{2}}k_{\text{B}}T.$ kBT.{\displaystyle {\frac {K}{ND}}={\frac {1}{2}}k_{\text{B}}T.} tê dayîn.

Ji ber vê yekê, ji bo molek Gazek îdeal a yek-atomî (D = 3), Enerjiya Kînetîk a serê kelvînê wekî $K={\frac {D}{2}}k_{\text{B}}N_{\text{A}}={\frac {3}{2}}R,$ 16kBNA=40§4243§R,{\displaystyle K={\frac {D}{2}}k_{\text{B}}N_{\text{A}}={\frac {3}{2}}R,} tê hesibandin, li cihê ku $N_{\text{A}}$ Berdewama Avogadro temsîl dike û R Berdewama Gazê ya îdeal nîşan dide.

Wekî encam, rêjeya Enerjiya Kînetîk li Germahiya mutleq ji bo Gazek îdeal a yek-atomî bi hêsanî dikare were hesibandin:

Ji bo her molekê, ev rêje 12.47 J/K ye.
Ji bo her Molekulekê, rêje 20.7 yJ/K ye, ku wekhev e bi 129 μeV/K.

Di Germahiya standard de (273.15 K), Enerjiya Kînetîk jî dikare were destnîşankirin:

Ji bo her molekê, Enerjiya Kînetîk 3406 J e.
Ji bo her Molekulekê, ew 5.65 zJ e, ku bi 35.2 meV re têkildar e.

Di Germahiyên bilind de, bi gelemperî di navbera hezaran kelvînan de, modên lerzînê çalak dibin, ku pileya azadiyê ya zêde tînin. Ev Bûyer girêdanek Germahiyê ji bo D û Enerjiya Molekulî ya giştî saz dike. Hesibandina rast a van beşdariyên sepandina Mekanîka îstatîstîkî ya Kuantum hewce dike.

Lêdanên bi Dîwarê Konteynerê re

Ji bo Gazek îdeal a di hevsengiyê de, rêjeya lêdanê bi dîwarê Konteynerê re û belavbûna leza parçikên ku li dîwarê dixin dikare ji Teorîya kînetîk a bingehîn were derxistin. Ev dîtin ji bo Analîza rêjeyên Herikîna efûzîf têne sepandin, prensîbek hêja ye di çarçoveyên wekî rêbaza belavbûna Gazê ya ku ji bo veqetandina Îzotopan tê bikar anîn.

Tê texmîn kirin ku di nav Konteynerê de, Tîrbûna hejmarê, ku wekî hejmara parçikan serê Qebareya yekîneyê tê pênasekirin, wekî $n=N/V$ tê temsîl kirin. Herwiha, tê texmîn kirin ku parçik li gorî belavbûna leza Maxwell tevdigerin:

Ji bo qadeke biçûk $dA$ li ser dîwarê Konteynerê, kerpîçek bi leza $v$ , ku bi goşeya $\theta$ ji normala qadê nêzîk dibe, dê di nav demjimêra $dt$ de li $dA$ bixe, heke Di destpêkê de ew di nav dûrahiya $v\,dt$ ji $dA$ de be. Wekî encam, hemî kerpîçên bi leza $v$ û goşeya nêzîkbûnê $\theta$ ji normalê, yên ku dikarin di nav demjimêra $dt$ de bigihîjin qada $dA$ , di nav Qebareyek silindrîkî ya xwar de ne. Ev Qebare xwedî bilindahiya $v\cos(\theta )dt$ ye û mezinahiya wê $v\cos(\theta )\,dA\,dt$ e.

Dema ku li ser hemî lezên têkildar tê entegrekirin, li gor sînorên $v>0$ 10 {\displaystyle v>0} , , ${\textstyle 0<\theta <{\frac {\pi }{2}}}$ 26 < θ < π §40 ${\displaystyle v>41§ </mfrac> </mrow> </mstyle> </mrow> <annotation encoding=$ , û , $0<\phi <2\pi$ 59§ < ϕ < §67 ${\displaystyle v>68§ <mi>π</mi> </mstyle> </mrow> <annotation encoding=$ , Pîvana encamdar hejmara lihevketinên atomî an molekularî yên ku di her yekîneya qadê de di her yekîneya demê de li ser dîwarê Konteynerê çêdibin, nîşan dide:

Di Fîzîka Valahiya Fezayê de, ev Pîvan wekî "rêjeya lêdanê" jî tê binavkirin. Girîng e ku Nota were girtin ku, ji bo hesabkirina leza navîn ${\bar {v}}$

Momentum ku ji aliyê parçikan ve ku li qadeke bêdawî biçûk $dA$ bi Lezeke $v$ û goşeyek $\theta$ li gorî normalê dixin, di navbera demeke $dt$ de, ji dîwarê Konteynerê re tê veguhestin, wiha tê hesabkirin:

Dema ku ev bi qanûna Gazê ya îdeal re tê hevberdan, ev encam dide.

Ev îfade bi qanûna Graham re hevaheng e.

Ji bo ku belavbûna Leza parçikan ku li qadeke biçûk a taybetî dixin were destnîşankirin, pêwîst e ku were zanîn ku hemî parçikên ku Lezên wan bi $(v,\theta ,\phi )$ hatine diyarkirin û li qada diferansiyel $dA$ dixin, di navbera demjimêra $dt$ de, di nav silindirekî hîpotetîkî yê xwar de ne. Bilindahiya vê silindirê $v\cos(\theta )\,dt$ ye û Qebareya wê ya têkildar jî $v\cos(\theta )\,dA\,dt$ ye. Wekî encam, belavbûna Lezê ya ku hatiye derxistin, dema ku bi belavbûna Maxwell a standard re tê berawirdkirin, Faktorekî zêde yê pirjimariyê yê $v\cos \theta$ dihewîne.

Lezên Molekular

Têkiliyên jêrîn dikarin ji formula Enerjiya Kînetîk werin derxistin: $v_{\text{p}}={\sqrt {2\cdot {\frac {k_{\mathrm {B} }T}{m}}}},$ ${\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}={\sqrt {{\frac {8}{\pi }}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},$ $v_{\text{rms}}={\sqrt {\frac {3}{2}}}v_{p}={\sqrt {{3}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},$ Di van îfadeyan de, v Lezê bi metreyan di çirkeyê de (m/s) nîşan dide, T Germahiyê bi kelvinê destnîşan dike, û m Girseya yek Molekula Gazê bi kîlogramê îfade dike. Bi taybetî, Leza herî îhtîmal, $v_{\text{p}}$ , %81.6 ji Leza Reh-navîn-çarçik, $v_{\text{rms}}$ , pêk tîne. Herwiha, Leza navîn (an jî Leza giştî), ${\bar {v}}$ , %92.1 ji Leza Reh-navîn-çarçik e, bi şertê ku belavbûna Lezan îzotropî be.

Navîn,

Average,
Navîna Aritmetîk
Arithmetic mean
Navîn
Mod (Îstatîstîk)

Rêya Navîn a Azad

Di nav Teorîya kinetîk a gazan de, rêya azad a navîn wekî dûrahiya navîn a ku ji hêla Molekulekê ve, an jî berhevokek Molekulan di nav Qebareyek diyarkirî de, berî ku ew cara yekem li hev bikevin, tê derbaskirin, tê pênasekirin. Beşa xaçê ya lihevketinê, ku wekî $\sigma$ tê nîşankirin, qebareya bandorker a ji bo lihevketinek di navbera du Molekulan de diyar dike. Li gorî pênaseyên berê, Tîrbûna hejmarê $n$ wekî hejmara Molekulan li ser Qebareya berfireh tê damezrandin, ku wekî $n=N/V$ tê îfadekirin. Tîrbûna beşa xaçê ya lihevketinê, ku wekî $n\sigma$ tê temsîlkirin, têkiliyek bi rêya azad a navîn $\ell$ re datîne, wekî ku ji hêla hevkêşeyê ve tê pênasekirin: $\ell ={\frac {1}{n\sigma {\sqrt {2}}}}$ 108 n σ §118 $\sigma$ {\displaystyle \ell ={\frac {1}{n\sigma {\sqrt {2}}}}} .

Hêjayî gotinê ye ku yekîneya ji bo beşa xaçê ya lihevketinê li ser Qebare, $n\sigma$ , berevajî dirêjahiyê ye.

Taybetmendiyên Veguhestinê

Teorîya kinetîk a gazan ne tenê pergalên di hevsengiya termodînamîk de lê di heman demê de, bi awayekî krîtîk, gazên ku ne di hevsengiya termodînamîk de ne jî, analîz dike. Ev yek serîlêdana Teorîya kinetîk ji bo Analîz kirina diyardeyên ku wekî "taybetmendiyên veguhestinê" têne binavkirin, dihewîne, ku vîskozîte, gihandina germî, belavbûna Girse, û belavbûna germî di nav de ne.

Di bingeh de, Teorîya gazê ya kinetîk bi taybetî ji bo gazên hûrkirî derbasdar e. Berfirehkirinek ji Teorîya gazê ya kinetîk ji bo ku têkelên gazê yên qelew bigire nav xwe, ku wekî Teorîya Enskog a Revîzekirî tê zanîn, di navbera salên 1983 û 1987an de ji hêla E. G. D. Cohen, J. M. Kincaid, û M. Lòpez de Haro ve hate formulekirin, li ser bingeha lêkolîna bingehîn a H. van Beijeren û M. H. Ernst.

Vîskozîte û Momentum a Kinetîk

Gelek qad encamên ku ji modelkirina gazên hênik derketine bi kar tînin, wek ku di nivîsên bingehîn ên li ser teorîya kînetîk a bingehîn de têne pêşkêş kirin. Derxistina modela kînetîk ji bo vîskozîteya qirçînê bi gelemperî bi analîzek Herikîna Couette dest pê dike, ku bi tebeqeyek gazê ya di navbera du lewheyên paralel de tê nîşankirin. Di vê sazûmanê de, lewheya jorîn di bin bandora hêzek F de bi lezek Berdewam ber bi rastê dimeşe. Di heman demê de, lewheya jêrîn Rawestayî dimîne, ku pêdivî bi hêzek wekhev û berevajî dike da ku bêlivbûna wê biparêze. Di nav tebeqeya gazê de, molekul pêkhateyek leza pêşverû, $u$ , nîşan didin, ku bi dûrahiya vertîkal $y$ ji lewheya jêrîn bi awayekî xêzî zêde dibe. Ev Herikîna ne-hevseng li ser belavkirinek hevsengiya Maxwell-Boltzmann a bingehîn a tevgerên molekular tê zêdekirin.

Di nav gazek hênik de ku di bin sazûmanek Herikîna Couette de ye, bila $u_{0}$ 10{\displaystyle u_{0}} leza pêşverû ya gazê li tebeqeyek planar a asoyî nîşan bide, bi taybetî li cihê ku $y=0$ 32{\displaystyle y=0}. Ev lez, $u_{0}$ 52{\displaystyle u_{0}}, bi awayekî asoyî ye. Hejmara molekulên ku dikevin ser elementek rûberê $dA$ li aliyekî tebeqeya gazê, ku xwedî leza $v$ û goşeyek $\theta$ li gorî normalê ne, di navbera demê $dt$ de, wiha tê îfadekirin: $nv\cos({\theta })\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{2}\sin {\theta }\,dv\,d\theta \,d\phi )$ $u_{0}$ (v $u_{0}$ sin⁡θdvdθdϕ){\displaystyle nv\cos({\theta })\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{2}\sin {\theta }\,dv\,d\theta \,d\phi )}.

Lihevketina dawî ya van Molekulan li $y=\pm \ell \cos \theta$ rû da, li vir $\ell$ rêya azad a navîn nîşan dide. Her Molekulek bi Momentumek pêşverû beşdar dibe, ku wekî $p_{x}^{\pm }=m\left(u_{0}\pm \ell \cos \theta {\frac {du}{dy}}\right),$ 80 ± ℓ cos ⁡ θ d u d y ) , {\displaystyle p_{x}^{\pm }=m\left(u_{0}\pm \ell \cos \theta {\frac {du}{dy}}\right),} tê îfadekirin, li vir nîşana erênî bi Molekulên ku ji Li ser tên re têkildar e, û nîşana neyênî jî bi yên ji jêr tên re. Girîng e ku were Nota kirin ku gradiyenta Lezê ya pêşverû $du/dy$ dibe ku li ser dûrahiyek ku bi rêya azad a navîn re wekhev e, wekî Berdewam were nêzîkkirin.

Entegrasyon li ser hemî Lezên têkildar, bi Kirdeya sînorkirinên $v>0$ 10 {\displaystyle v>0} , ${\textstyle 0<\theta <{\frac {\pi }{2}}}$ 26 < θ < π §40 ${\displaystyle v>41§ </mfrac> </mrow> </mstyle> </mrow> <annotation encoding=$ , û $0<\phi <2\pi$ 59§ < ϕ < §67 ${\displaystyle v>68§ <mi>π</mi> </mstyle> </mrow> <annotation encoding=$ , veguhastina Momentumê ya pêşverû per yekîneya demê per yekîneya Rûxarê dide, ku bi gelemperî wekî stresa qirçandinê tê zanîn: $\tau ^{\pm }={\frac {1}{4}}{\bar {v}}n\cdot m\left(u_{0}\pm {\frac {2}{3}}\ell {\frac {du}{dy}}\right)$ 101§ §102103§ v ¯ n ⋅ m ( u §132133§ ± §141 ${\displaystyle v>142§ §143<semantics>144§ </mfrac> </mrow> <mi>ℓ</mi> <mrow class=$ ) {\displaystyle \tau ^{\pm }={\frac {1}{4}}{\bar {v}}n\cdot m\left(u_{0}\pm {\frac {2}{3}}\ell {\frac {du}{dy}}\right)} .

Rêjeya net a Momentumê per Rûxarek yekîne ku li ser Rûxara xeyalî tê veguhestin, bi vî awayî wekî: $\tau =\tau ^{+}-\tau ^{-}={\frac {1}{3}}{\bar {v}}nm\cdot \ell {\frac {du}{dy}}$ 37 §3940§ v ¯ n m ⋅ ℓ d u d y {\displaystyle \tau =\tau ^{+}-\tau ^{-}={\frac {1}{3}}{\bar {v}}nm\cdot \ell {\frac {du}{dy}}} tê îfadekirin.

Bi Entegrasyonkirina hevkêşeya kinetîk a berê bi qanûna vîskozîteyê ya Newton re, $\tau =\eta {\frac {du}{dy}}$ , hevkêşeya vîskozîteya qirçînê tê bidestxistin. Ev bi gelemperî wekî $\eta _{0}$ 49 {\displaystyle \eta _{0}} tê nîşankirin dema ku li ser Gazek tenik tê sepandin: $\eta _{0}={\frac {1}{3}}{\bar {v}}nm\ell$ 72 = 79 §8182§ v ¯ n m ℓ {\displaystyle \eta _{0}={\frac {1}{3}}{\bar {v}}nm\ell }

Entegrasyona zêdetir a vê hevkêşeyê bi formula ji bo rêça azad a navîn encam dide: $\eta _{0}={\frac {1}{3{\sqrt {2}}}}{\frac {m{\bar {v}}}{\sigma }}$ 11 = 18 §2122§ §25 $\eta _{0}={\frac {1}{3{\sqrt {2}}}}{\frac {m{\bar {v}}}{\sigma }}$ m v ¯ σ {\displaystyle \eta _{0}={\frac {1}{3{\sqrt {2}}}}{\frac {m{\bar {v}}}{\sigma }}}

Dabeşkirina Maxwell-Boltzmann leza Molekulî ya navîn (hevseng) wekî jêrîn pênase dike:

Têxistina Lezê di hevkêşeya vîskozîteyê ya jorîn de formula damezrandî ya ji bo vîskozîteya qirçînê ya Gazên tenik dide, li cihê ku $\sigma$ paşê tê destnîşankirin:

Tîreya $r$ wekî tîreya beşa-têkçûnê ya pevçûnê an tîreya kinetîk tê destnîşankirin, dema ku pîvana $d$ wekî pîvana beşa-têkçûnê ya pevçûnê an pîvana kinetîk ji bo Molekulek di nav Gazek monomolekulî de tê binavkirin. Têkiliyek rasterast û giştî di navbera beşa-têkçûnê ya pevçûnê û mezinahiya Navik-hişk a Molekulek Nêzîkî gogî de tune. Ev têkilî bi veavakirina taybetî ya Enerjiya Potansiyel a Molekulê ve girêdayî ye. Ji bo Molekulek gogî ya îdeal, wekî Atomê Gazek esîl an Molekulek bi gogîbûna maqûl, potansiyela têkiliyê bi gelemperî dişibe potansiyela Lennard-Jones an potansiyela Morse. Van potansiyelan pêkhateyek neyînî heye ku li ser Molekulên din di dûrahiyên ku ji tîreya Navik-hişk derbas dibin de Hêzek balkêş pêk tîne. Wekî encam, tîreya ku bi potansiyela Lennard-Jones a sifir re têkildar e dikare wekî texmînek Nêzîkî ji bo tîreya kinetîk xizmet bike. Lê belê, bikaranîna vê texmînê pir caran dibe sedema girêdana Germahî ya nerast ji bo vîskozîteyê. Ji bo potansiyelên têkiliyê yên weha, encamên pir rasttir bi nirxandina hejmarî ya Entegralên pevçûnê yên pêwîst têne bidestxistin.

Îfadeya vîskozîteyê ya ku ji Teoriya Enskog a Revîzekirî hatiye wergirtin, di bin şertên Bêdawî helandinê de hêsan dibe bo hevkêşeya jorîn, û dikare wekî were formulekirin: $\eta =(1+\alpha _{\eta })\eta _{0}+\eta _{c}$ 13 + α η ) η §3435§ + η c {\displaystyle \eta =(1+\alpha _{\eta })\eta _{0}+\eta _{c}} Li vir, $\alpha _{\eta }$ têgehekê temsîl dike ku di Sînorê Bêdawî helandinê de nêzî sifirê dibe, û bandorên qebareya derxistî dihesibîne. Berovajî, $\eta _{c}$ têgehekê nîşan dide ku veguhestina Momentumê di navbera parçikan de li ser mesafeyek ne-sifir di dema Bûyerek lihevketinê de pîvan dike.

Ragihandina Germî û Herikîna Germê

Bi sepandina çarçoveyek mentiqî ya mîna hev, Modelê kînetîk ji bo ragihandina germî ya Gazek hênik dikare wiha were wergirtin:

Du lewheyên paralel bifikirin, ku ji aliyê tebeqeyek Gazê ve ji hev hatine veqetandin. Her du lewhe germahiyên yekreng diparêzin û li gorî tebeqeya Gazê xwedî Girseya têr in ku wekî rezervên germî werin hesibandin. Lewheya jorîn germahiyek bilindtir ji lewheya jêrîn nîşan dide. Di nav tebeqeya Gazê de, molekul xwedî Enerjiya Kînetîk a molekulî $\varepsilon$ ne ku bi dûrahiya $y$ ji lewheya jêrîn bi awayekî xêzî zêde dibe. Ev herikîna Enerjiyê ya ne-hevseng li ser belavkirina hevsengiya Maxwell-Boltzmann ya tevgerên molekulî hatiye danîn.

Bila $\varepsilon _{0}$ 11{\displaystyle \varepsilon _{0}} Enerjiya Kînetîk a molekulî ya Gazê li ser balafirek asoyî ya hîpotetîk di nav tebeqeya Gazê de temsîl bike. Pîvana molekulan ku digihîjin rûberek bêdawî biçûk $dA$ li ser Sînorek tebeqeya Gazê, xwedî lezek $v$ û goşeyek $\theta$ li gorî vektora normal, di nav zêdebûnek demê de $dt$ , wiha tê dayîn:

Her yek ji van molekulan, ku lihevketina xwe ya dawî li dûrahiya $\ell \cos \theta$ ji tebeqeya Gazê (hem li ser û hem jî li jêr) derbas kiriye, dê Enerjiya Kînetîk a molekulî ya taybetî bide.

Bi pêkanîna entegrasyonê li ser hemî lezên têkildar, ku bi sînorên $v>0$ 10 {\displaystyle v>0} ji bo lezê, ${\textstyle 0<\theta <{\frac {\pi }{2}}}$ 26 < θ < π §40 ${\displaystyle v>41§ </mfrac> </mrow> </mstyle> </mrow> <annotation encoding=$ ji bo goşeya polar, û $0<\phi <2\pi$ 59§ < ϕ < §67 ${\displaystyle v>68§ <mi>π</mi> </mstyle> </mrow> <annotation encoding=$ ji bo goşeya azîmûtal ve girêdayî ye, veguhastina enerjiyê ya her yekîneya demê li ser her yekîneya rûxarê dide, ku bi gelemperî wekî herikîna germê tê zanîn: $q_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}n\cdot \left(\varepsilon _{0}\pm {\frac {2}{3}}mc_{v}\ell {\frac {dT}{dy}}\right)$ 107§ §108109§ v ¯ n ⋅ ( ε §137138§ ± §146 ${\displaystyle v>147§ §148<semantics>149§ </mfrac> </mrow> <mi>m</mi> <msub> <mi>c</mi> <mrow class=$ ℓ d T d y ) {\displaystyle q_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}n\cdot \left(\varepsilon _{0}\pm {\frac {2}{3}}mc_{v}\ell {\frac {dT}{dy}}\right)} .

Girîng e ku were zanîn ku veguhastina enerjiyê, wekî ku berê hatibû derxistin, di arasteya $-y$ de pêk tê, ku îşareta neyînî ya di hevkêşeya berê de rave dike. Wekî encam, herikîna germê ya netî ya li ser rûxara hîpotetîk wiha tê îfadekirin: $q=q_{y}^{+}-q_{y}^{-}=-{\frac {1}{3}}{\bar {v}}nmc_{v}\ell \,{\frac {dT}{dy}}$ 64 §6667§ v ¯ n m c v ℓ d T d y {\displaystyle q=q_{y}^{+}-q_{y}^{-}=-{\frac {1}{3}}{\bar {v}}nmc_{v}\ell \,{\frac {dT}{dy}}} .

Entegrasyona hevkêşeya kinetîk a jorîn bi qanûna Fourier re, $q=-\kappa \,{\frac {dT}{dy}}$ , hevkêşeya Bingehîn ji bo ragihandina termal dide, ku bi gelemperî wekî $\kappa _{0}$ 53 {\displaystyle \kappa _{0}} ji bo gazek hûrkirî tê temsîlkirin: $\kappa _{0}={\frac {1}{3}}{\bar {v}}nmc_{v}\ell$ 76 = 84 §8687§ v ¯ n m c v ℓ {\displaystyle \kappa _{0}={\frac {1}{3}}{\bar {v}}nmc_{v}\ell } .

Mîna tedawiya vîskozîteyê, Teorîya Enskog a Revîzekirî îfadeyekê ji bo gihandina germî peyda dike ku di bin şertên bêdawî yên diluasyonê de ber bi hevkêşeya jorîn ve diçe. Ev îfade wiha tê formulekirin: $\kappa =\alpha _{\kappa }\kappa _{0}+\kappa _{c}$ 25 + κ c {\displaystyle \kappa =\alpha _{\kappa }\kappa _{0}+\kappa _{c}} . Li vir, $\alpha _{\kappa }$ faktorekî temsîl dike ku di bin diluasyona bêdawî de nêzî yekî dibe, bandora qebareya derxistî dihewîne. Berovajî, $\kappa _{c}$ veguhestina Enerjiyê ya ku di dema bûyerên pevçûnê de li ser dûrahiyek sînordar di navbera parçikan de çêdibe, diyar dike.

Hevkêşeya Dîfûzyonê û Herikîna Dîfûzyonê

Bi berfirehkirina çarçoveya mentiqî ya berê, mimkun e ku modelek kînetîk ji bo dîfûzyona girseyê ya gazek diluandî were formulekirin.

Ji bo ronîkirinê, senaryoyek dîfûzyona rewşa-domdar bifikirin ku di navbera du herêmên cuda yên gazek yekbûyî de çêdibe, ku ji hêla tebeqeyek navîn a heman gazê ve têne veqetandin û ji hêla rûxarên bi tevahî rût û paralel ve têne sînorkirin. Dema ku her du herêm tîrbûnên hejmarî yên yekreng diparêzin, herêma jorîn li gorî herêma jêrîn tîrbûnek hejmarî ya mezintir nîşan dide. Di bin şertên rewşa-domdar de, tîrbûna hejmarî di her xalek diyarkirî de li gorî demê neguher dimîne. Lêbelê, di nav tebeqeya veqetîner de, tîrbûna hejmarî $n$ wekî fonksiyonek dûrahiya $y$ ji plakaya jêrîn zêdebûnek yekreng nîşan dide. Ev Herikîna molekulî ya ne-hevseng li ser dabeşkirina hevsengiya Maxwell–Boltzmann a tevgerên molekulî yên bingehîn tê zêdekirin.

$n_{0}$ 10 {\displaystyle n_{0}} wekî tîrbûna hejmarî ya gazê bifikirin ku li ser rûxarekî asoyî yê hîpotetîk di nav tebeqeyê de ye.

Van molekulan pevçûna xwe ya dawîn li dûrahiyek $\ell \cos \theta$ ji tebeqeya gazê, hem li ser û hem jî li jêr wê, jiyan. Li van cihan, tîrbûna hejmarî ya herêmî ji hêla $n^{\pm }=\left(n_{0}\pm \ell \cos \theta \,{\frac {dn}{dy}}\right)$ 50 ± ℓ cos ⁡ θ d n d y ) {\displaystyle n^{\pm }=\left(n_{0}\pm \ell \cos \theta \,{\frac {dn}{dy}}\right)} ve tê dayîn.

Li gorî nîşankirina berê, nîşana erênî bi molekulên ku ji li ser tên re têkildar e, dema ku nîşana neyînî yên ku ji jêr tên destnîşan dike. Girîng e ku were nota kirin ku gradyenta tîrbûna hejmarê $dn/dy$ dikare li ser dûrahiyek ku bi rêça azad a navîn re hevwate ye, wekî Berdewam were nêzîk kirin.

Bi entegrekirina li ser hemî lezên têkildar, di bin sînorkirinên $v>0$ 10 {\displaystyle v>0} , ${\textstyle 0<\theta <{\frac {\pi }{2}}}$ 26 < θ < π §40 ${\displaystyle v>41§ </mfrac> </mrow> </mstyle> </mrow> <annotation encoding=$ , û $0<\phi <2\pi$ 59§ < ϕ < §67 ${\displaystyle v>68§ <mi>π</mi> </mstyle> </mrow> <annotation encoding=$ , veguhastina molekulî ya her yekîneya demê li ser her yekîneya rûxarê, ku wekî herika belavbûnê jî tê binavkirin, wekî jêrîn tê destnîşankirin: $J_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}\cdot \left(n_{0}\pm {\frac {2}{3}}\ell \,{\frac {dn}{dy}}\right)$ 107§ §108109§ v ¯ ⋅ ( n §134135§ ± §143 ${\displaystyle v>144§ §145<semantics>146§ </mfrac> </mrow> <mi>ℓ</mi> <mspace width=$ ) {\displaystyle J_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}\cdot \left(n_{0}\pm {\frac {2}{3}}\ell \,{\frac {dn}{dy}}\right)} .

Girîng e ku were nota kirin ku veguhastina molekulî ya ku berê hate vegotin di aliyê $-y$ de pêk tê, ku nîşana neyînî ya di hevkêşeyê de rave dike. Wekî encam, herika belavbûnê ya netî ya li ser rûxara hîpotetîk wekî jêrîn tê îfadekirin: $J=J_{y}^{+}-J_{y}^{-}=-{\frac {1}{3}}{\bar {v}}\ell {\frac {dn}{dy}}$ 64 §6667§ v ¯ ℓ d n d y {\displaystyle J=J_{y}^{+}-J_{y}^{-}=-{\frac {1}{3}}{\bar {v}}\ell {\frac {dn}{dy}}} .

Hevkêşeya kinetîk a jorîn, dema ku bi qanûna yekem a belavbûnê ya Fick re tê hev kirin, $J=-D{\frac {dn}{dy}}$ , hevkêşeya ji bo belavbûna Girse dide. Ev bi gelemperî wekî $D_{0}$ 49 {\displaystyle D_{0}} ji bo Gazek hûrkirî tê nîşandan: $D_{0}={\frac {1}{3}}{\bar {v}}\ell$ 71 = 79 §8182§ v ¯ ℓ {\displaystyle D_{0}={\frac {1}{3}}{\bar {v}}\ell } .

Îfadeya heman rengî ya ku ji Teorîya Enskog a Revîzekirî hatiye wergirtin, dikare wekî $D=\alpha _{D}D_{0}$ 22 {\displaystyle D=\alpha _{D}D_{0}} were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were 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were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were were

Detailed Balance

Fluctuation and Dissipation

According to the kinetic theory of gases, the microscopic reversibility inherent in the detailed dynamics of gas particles mandates that the system adheres to the principle of detailed balance. More precisely, the fluctuation-dissipation theorem is applicable to Brownian motion (or diffusion) and the drag force, culminating in the Einstein–Smoluchowski equation: $D=\mu \,k_{\text{B}}T,$ , where:

D denotes the mass diffusivity;
μ represents the "mobility," defined as the ratio of the particle's terminal drift velocity to an applied force, expressed as μ = v_d/F.
k_B represents the Boltzmann constant.
T denotes the absolute temperature.

The mobility, expressed as μ = v_d/F, is derivable from the gas's viscosity. Consequently, the Einstein–Smoluchowski equation establishes a correlation between the mass diffusivity and the gas viscosity.

Onsager Reciprocal Relations

The mathematical congruence observed among the expressions for shear viscosity, thermal conductivity, and the diffusion coefficient in an ideal (dilute) gas is not fortuitous. Instead, it directly stems from the Onsager reciprocal relations, which describe the detailed balance of reversible particle dynamics. This principle applies specifically to the convection (i.e., matter flow driven by temperature gradients and heat flow induced by pressure gradients) and advection (i.e., matter flow resulting from particle velocity and momentum transfer due to pressure gradients) within an ideal (dilute) gas.

Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) Hierarchy of Equations

Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations
Boltzmann Equation
Chapman–Enskog Theory
Collision Theory
Critical Temperature
Gas Laws
Heat
Interatomic Potential
Magnetohydrodynamics
Maxwell–Boltzmann Distribution
Mixmaster Universe
Thermodynamics
Vicsek Model
Vlasov Equation

References

Citations

Cited Sources

Chapman, Sydney, û Cowling, Thomas George (1939/1970), Teorîya Matematîkî ya Gazên Ne-yekreng: Hesabek ji Teorîya Kînetîk a Vîskozîte, Ragihandina Termal û Belavbûnê di Gazan de, (çapa yekem 1939, çapa duyem 1952), çapa sêyem 1970 bi hevkariya D. Burnett hat amadekirin, Cambridge University Press, London.

Sydney Chapman û Thomas George Cowling (1939/1970), Teorîya Matematîkî ya Gazên Ne-yekreng: Hesabek ji Teorîya Kînetîk a Vîskozîte, Ragihandina Termal û Belavbûnê di Gazan de, (çapa yekem 1939, çapa duyem 1952), çapa sêyem 1970 bi hevkariya D. Burnett hat amadekirin, Cambridge University Press, London.
Hirschfelder, Joseph Oakland, Curtiss, Charles Francis, û Bird, Robert Byron (1964), Teorîya Molekulî ya Gaz û Şilemeniyan, çapa nûkirî (Wiley-Interscience), ISBN 978-0471400653.
Liboff, Richard Lawrence (2003), Teorîya Kînetîk: Danasînên Klasîk, Kuantum û Relatîvîstîk, çapa sêyem (Springer), ISBN 978-0-387-21775-8.
Rahimi, Behnam, û Struchtrup, Henning (2016), "Modelkirina Makroskopîk û Kînetîk a Gazên Polîatomîk ên Kêmkirî", Kovara Mekanîka Şilemeniyan, 806, pp. 437–505, DOI 10.1017/jfm.2016.604.

Kîmya Fîzîkî: Gaz.

KÎMYA FÎZÎKÎ – Gaz.
Teorîyên Destpêkê yên Gazan.
Termodînamîk, beşa pirtûkek dersê ya serhêl.
Germahî û Pesto ya Gazek Îdeal: Hevkêşeya Rewşê.
Danasînek ji teorîya molekulî ya kînetîk a gazan, ku ji hêla Lijneya Dibistanê ya Navçeya Kanada Jorîn ve hatî peyda kirin.
Anîmasyonek Java ji Zanîngeha Arkansas, ku teorîya kînetîk nîşan dide.
Nexşeyek herikînê ji HyperPhysics, ku têgehên cûrbecûr yên teorîya kînetîk girêdide.
Appletên Java yên înteraktîf ku ji bo xwendekarên lîseyê hatine sêwirandin da ku bandora faktorên cûrbecûr li ser rêjeyên reaksiyona kîmyayî lêkolîn bikin û fam bikin.
Amûrek pêşandanê ku tevliheviya termal di gazan de nîşan dide.

Teoriya kînetîk a gazan (Kinetic theory of gases)

Teoriya kînetîk a gazan (Kinetic theory of gases)

Dîrok

Teoriya Kînetîk a Maddeyê

Serdema Antîk

Serdema Nûjen

"Germî tevger e"

Teoriya Kînetîk a Gazan

Hîpotez

Taybetmendiyên Hêvsengiyê

Pesto û Enerjiya Kînetîk

Germahî û Enerjiya Kînetîk

Lêdanên bi Dîwarê Konteynerê re

Lezên Molekular

Rêya Navîn a Azad

Taybetmendiyên Veguhestinê

Vîskozîte û Momentum a Kinetîk

Ragihandina Germî û Herikîna Germê

Hevkêşeya Dîfûzyonê û Herikîna Dîfûzyonê

Detailed Balance

Fluctuation and Dissipation

Onsager Reciprocal Relations

Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) Hierarchy of Equations

References

Citations

Cited Sources

Kîmya Fîzîkî: Gaz.

Teoriya kînetîk a gazan çi ye?

Etîketên babetê

Lêgerînên gelemperî li ser vê babetê

Arşîva Neverok: Zanist û Zanîn