• 検索結果がありません。

Liquidus and Gaseous Line in Phase Diagram: University of the Ryukyus Repository

N/A
N/A
Protected

Academic year: 2021

シェア "Liquidus and Gaseous Line in Phase Diagram: University of the Ryukyus Repository"

Copied!
8
0
0

読み込み中.... (全文を見る)

全文

(1)

Title

Liquidus and Gaseous Line in Phase Diagram

Author(s)

Yogi, Tatsuhiro; Yaga, Minoru

Citation

琉球大学工学部紀要(64): 7-13

Issue Date

2002-12

URL

http://hdl.handle.net/20.500.12000/5448

(2)

BullFacultyofEngineering,Univ、oftheRyukyusNo、64,2002

LiquidusandGaseousLineinPhaseDiagram

TatsuhiroYOGI*andMinoruYAGA* Abstract

Thispaperdescribesthatnegativepair-interactionenergyaveragedoverallparticlesanditsdensity

dependencegivesvanderWaals,equationofstatethroughmean-fieldInodelingandthatliquidusand

gaseouslinefbrvapor-liquidphasearerigorouslycalculatedfrolntheequationofstate,sothatlatentheat

ofvaporizationandvaporpressureareobtained・Phasediagramisconstructedfromtheselmesanditis

shownthatthecriticaltemperaturedependsnotonlyonthesizeofmoleculeandtheaveragedmagnitude

ofinteractionenergybutalsoonthemassofmoleculeAsapropertyofsuchsystemKamerlingh-Onnes,

constantisobtajned. KeyWbrds:liquidusandgaseous Kamerlingh-Onnes,constant line,phasediagram,criticaltemperature,latentheat,vaporpressure, YbvickequationofstatefbrtheLennard-Jones6-12po- tentialderivednotonlyfromthecompressibilityfbrsev-eralisothermsbutalsofromthepressureequationisnot inagreementwithexperimentaldata・Particularlyinco‐ existentstateofliquidandvaporthisequationhasno・ solutionofexperimentalHatregion[181LiquiduSand gaseouslinesareconventionallyunderstoodbytheex‐ istenceofdoublelocalminimumofGibbsenergyina givensystem・Thepreviousworkofsuchproblemsfbr matterhavenotgivensatisfactoryexplanationfbrre sultsofexperimentsTheoccurrenceofthediscontiniu-tiesassociatedwithphasetransitionswillbefUrtherulL derstoodbythemorecorrectliquidusandgaseousline ThisleadsusmUchintensivestudyfbrexistenceofliq-uidusandgaseouslinasothatwecanconstructphase diagramnotonlywithcoexistentstatebutalsowithsep-aratephasesofvaporandliquidTherefbrethispaper willbeconcernedwithtwocurvesrepresentingliquidus andgaseouslinebelowcriticaltemperature、Wehave studiedthefirst-orderphasetransitionfbrsimplematter byusingcanonicalemsemblewithmean-fieldapproach, whichisquitesimpleandfairlypracticalandindeedshow theexistenceoftwoseparatephasesandcoexistentstate ofbothliquidandvapor・Ifweneglecttheinteraction betweenparticlesbothinliquidandvaporphases,the partitionfunctionwiththemeanfieldapproximationis availabletoobtainthedistributionofinteractionener‐ gyparparticleandthelineardependenceoftheenergy OIldensityineachphasebyMDcomputersimulation・ Thecontributionofmean-fieldtheorytothedistribu-tionanddensitynependenceofinteractionenergypar particleisacceptableoverawidetemperaturerangefbr manysubstance・ThecanonicalpartitionfUnctioninthe 1.Introduction Thephasetransitionproblemsh副ebeenalwaysat- tractedphysicistandchemistsinfUndamentalandin-dustrialapplicationfieldbothfromthepracticalview pointofseekingfbrusefUltheoryofpropertiesofmat- terandfromthefUrtherunderstandingthediscontinu-itiesassociatedwithphasetransitionsinthermodynamic fimctions・Soitisusefllltodescribesometypicalworks fbrphasetransitionbrieHyhMaリer,sworkhasbeenweU knownmthefieldofthetheoryofcondensationofgases (11~[51,howeverLeeandYapghaveshowntheinade-quacyofMayer,smethodfbrdealmgwithacondensed phase[61~[71.Theyhavealsoshownthatanexampleof two-dimentionallatticeregionsinp-Vdiagramisexactly calculatedHillhastreatedthequestionofwhetheror

not,,loop,,shouldbeexpectedin似一AWorp-AW

curvefbranexacttheoryoflirst-orderphasetransition usingcanonicalensemble[81AcarefUlargumenthas beendescribedbyVanHoveshowingthatnoloopisob-tainedfromcompleteevaluationofcanonicalensemble partitionfilnctionoffluid[9IThephasetransitionis tobecharacterizedbythefactthatthepartitionfilnc-tionoffinitesystemsuffersasuddenchangeatsome valueofparametersKatsurahascalculatedtheparti‐ tionfUnctionofthesystemwhichhasfinitenumberof latticepointsasitsvolumeanddescribedthepossibility ofcooperativephenomenaeveninprettysmallsystem-

s[101~[111.ManydesclMiononthelatticegasmodel

h…beenshownmtextbooks[121~[l51ThePercus‐ Recieved24June2002 *Dept・ofMechSystemsEng.,FEC、ofEng.

(3)

8 YOGI・YAGA:LiquidusandGaseousLineinPhaseDiagram presentpapergivesvanderWaals,equationofstatem eachphaseWesupposethatthermalandmechanical equilibriumareestablishedinthesystem(Le,Tl=Zh andp,=p2),therefbreliquidusandgaseouslinesarede‐ rivedfromtheequationofstateonlywithoutMaxwell,s equiarealruleandthevaporpressureareobtainedfOr severalsimplesubstanceatanygiventemperature・We

shoWthatabovetwolinesseparatethepartsofliquid,

vaporandcoexistentstateinphase.iagramandexam-inethethermodynamicpropertiesofsuchsystemThe presentpapershowsthatMaxwelFsequiarealrulebased ontheselinesissatisfiednearcriticaltemperatureOur theoryprovidesthattheisothermsstayHatincoexis‐ tentstatebetweenliquidusandgaseoslineandthatthe isothermsisafUnctiononlyoftemperatureThesefba-tureisquitedifIerentfromMayer,stheoryandLeeand Yang,stheory、Thecharacteristicfeatureofliquidusand gaseouslineexperimentanydeterminedarewellrepro-ducedbythetheory、Thermodynamicpropertiessuch aslatentheatofvaporizationaresatisfactoryinthere-gionbetweenliquidusandgaseousline、Itsholudbe emphasizedthatthepurposeofthispaperistoderive liquidusandgaseouslineinvapor-liquidcoexistentstate andtoderivethermodynamicpropertiesofsuchsystem throughmean-fieldappro錘hwewillshowthesolution fbrthediscrepancyfbrmolecularsizebetweenvander Waalsandquantummechanics・Insection21weintro-ducepra妃ticalmethodfbrliquidusandgaeouslineand constructphasediagramfbrsimplematter、Wbanalyze anddiscusstheusefUlnessofthismethodfbrresultsm sectionaTheconclusionisdescribedinsection4. expecttheproblemtorepresentsomewhatlessserious mathematicaldifIicultiesthanrigorousway、Wecalcu-lateanaveragedinteractionenergyaparticlearoundall otherparticlesbycomputersimulation(MD),soweob‐ taintheconcentratiolLdependenceofenergyineachof liquidandvaporphases,Weinvestigatethesubjectthat anaveragedinteractionenergyisofattractivetype,so wesupposethattheinteractionpotentialisofLennard-Jonestypeastypicaloneofattractivefbrce、Weconsider amonoatOmicgasorliquidwiththeinteraction

U=傘((急)'2-(急)`)

(4) whererfjisthedistancebetweenthei-thandj-thatoms inaphaseandびthediameterofatomsandEacou- plingconstant(themagnitudeofinteraction),indetaiL 4E=32016Band◎=3.56AfbrAr、ThevalueofEis dilIbrentfromthatusedinRahman[16IAllsampling fbrstatisticswasaveragedoverthenextlOOOOstepsafL tertheinitial4000stepscalculationfbrequilibriumwith 6t*=0.OO64asascaledtimestep)Theusualperiodic boundarycondition1inwhichthesimulatedboxissur-roundedbyimagesofitself,isappliedandweusedNose,s methodtocontrolethetemperatureofasystem[201,[211. Letdbetheaveragedenergyaparticlelnthemost probablestatefbrenergyobtainedbyMDsimulation, therepresentativeparticlesisthoughtofasmovingin someaveragedpotentialduetoalltheotherparticlesin thesystem,wethusapproximatetheenergyfUnctionin phaseasfbllows; ⅣノV

zZのが-1V①(5)

j=1J>j =1V(一k|β+α)),(6) 2.Theoreticalrramc Weconsideronecomponentclosedsystemandassume thatoneparticleinteractswithotherparticleswithat-tractivepotentialfUnctionLeMVandVbethenumber ofparticlesandvolumemsystemwiththerestrictionof N(const.)andV(variable).Wecanwritethecanonical ensemblepartitionfUnctionasfbllows; wherelElisanaveragedinteractionenergyaparticleand pisdensityofphases(seefigl) ̄ Itisshownfromeqs(5)and(6)thattheaveragedin-teractionenergyisofattractivetype、discommonly writtenbyalinearfUnctiononlyofdensityoverliquid andvaporphases、Eachmoleculeisassumedtobefree tomoveinthevolumeV-Mjo(uoisamolecularvolume), therefbreweobtainthecanonicalpartitionfUnctionas fbllows;

Q-M壼俶(苧'等iv-lwo1い`。

(7)

゜-ホルー辨立佃〆醗亘血(リ

where Ⅳ2

K=Z藍

f=1 (2)

whereβ=(んBT)~landkBisBoltzmannconstant・The

theorycanbeeasilygeneralizedtodiatomicgasorliquid; therotationalandvibrationalenergyareeasilycalculat-edincanonicalpartitionfUnction,however,theseenergy termsaredisappearedbythesettlementofthecondi‐ tionsofIl=Zhandp,=p2,whichmustbesatisfied inanequilibriumstate、Theresultshowsthelinearde pendenceofenelgyondensityandtheattractivetypeof and NN

U=ZZdが

i=1j>{ (3) Sincetheinteractionbetweenparticlesexistsinsingle

phaseaboveequation(1)cannotbeanalyticallycalcu-latedexceptsomesimplecases、Therefbreweshould

(4)

2.5 o0mmmm釦如釦印印、5 -112233445 ●●■●の。●■●

r三二一一… ̄

「.... 2 5 1 1

(、Eミロ)則◎兵』’一○二』

{①二)易□」のこの

● ● ●. ● ● ● わ ℃ 。□ ●● ・0 口● CD ひ 。● 0.5 ● 0 0020.40.60811.21.4

「h・(g/Cni3)

FigLThedensitydependenceandattractivetypeofenergy lbrArovcrawidetemperaturerange、Othersubstanceshows thesameasthelineardependenceofenergyondensityand attractivetype. 204060801001201401601BO200220 Temp.(K) Fig.2,LiquidusandgaseouslineNe1ArandKr1whenusethere- lationd/bc=1/3,whichisindependentofthemolecularsize Thesearenotquantitativelybutqualitativelyingoodagree- mentwithexperimentaldata・Ne:dashedandsolidlines:cal-culationI□andx;experiments,Ar:dash-dootedandd顕hed lines:caIculation,xand△;experiments・Kr:dottedlines: calculation,。and+:experiments. interaction,therefbreweprogresstocalculatethecanoL icalpartitionfimctionfbrsystemdealingwiththeinter‐ actionenergybythemeanfieldtheory・Theproblemof seekingtheliquidandvaporcurvesistofindthesolu- tionwhichsatisfyZl=rhandp,=p2inthecoexistent state・Thestateequationfbrphasesare c2=1V/I'iinliquidphases,respectively,Theliquidus andgaseouslineisobtainedatI11=nandPl=p2,if thefbllowingrelationsissatislied l -bc,=dc2, (14) 1-clUo and l -bc2=。c,, (15) 1-c2Uo wheretheconstantdisdeterminedbyexPerimentsas showninlater・Therefbrewehaスノe

o1(T)一式('-,・I('6)

and

・塾(r1-式('十gQI('7)

whereノー1-./6,9=1+d/bandQ=

,=一種等

=AT(1,釜翻。+β・(筈)製)(8)

=;(」三"o

-bc),(9) whereb=β|E|mandcisnumberdenSityofphasejWV・ Aboveeqns.(8)and(9)areindeedvanderWaals,equaF tionofstate,sowederivethecriticaltemperaturefrom thelocalmaximumandmmimumvalue(denotedbyas terisk).

」(1-。..(旱)Ⅱ('0)

Cf= 3uo

。;‐上(l-c..(竿)》(u)

3uo

二('十・.富(:)L('2)

c:= 3uo

wheretan8=61/百万/(27-,),D=86/ひ0-27.IfO=O

atatemperatureineqs.(10)and(11),thencfandca coincidewitheachother,therefbreweobtainthewell knownrelationfbrcriticaltemPerature

m-鵜('31

1tisremarkedthatthecriticaltemperaturedependsnot onlyonuoandlElbutalsoonthemassofmolecule・ Wedelinethenumberdensitycl=1V/1/binvaporand 1-4(uO/b)/(92/) ThismeansthatMaxwelrsequal-arealruleisnotnecessarilyrequiredtodeterminethevaP porandliquidlinesinacoexistentstate(seefigs2~5). Thevaporpressureincoexistentstateiswritten

,(昨;MMI('8)

wherepisafilnctiononlyoftemperature(seefig6) Therectilineardiameterruleisgivenby β,+化=m(c,+c2)

ユノ

UO andtheso-caUedorderparameteris (19)

'2-'F蓋,Q

(20)

(5)

10 YOGI・YAGA:LiquidusandGaseousLineinPhaseDiagram 、25 1.4 1.2 5 1 1

〈かE○一m)do二」ロ一○二」

864 1 000

(のEC.□〉NC二』』一ロー」

0.5 0.2 0 0 204060BO100120140160180200220 Temp.(K) 50100150200250300350 Temp.(K) Fig.3.LiquidusandgaseouslinefOrN2,O2andCO2,whenuse therelationd/6。=1/3,whichisindependentofthemoIecu-larsizeAsshownin6g(2),thesearenotquantitativelybut qualitativelyingoodagreementwithexperimentaldataN2: 。零hedandsolidlines:calculation,□andx:experimentsO2: dash-dottedanddashedlines:calculation1xamd△:experi-mentsCO2:dottedlines:Ca]culation,。and+:experiments. Fig.4.LiquidusandgaseouslinefbrNe,ArandKr,wbenuse useeq.(24)as。/bDAgoodagreementwithexperimentis betterthantbatoflig.(2)quantitativelylNe:dashedandsoIid lines:calculation,□andx:experiments、Ar:。題h-dottedand dashedIines:calculation,xand△:eg(periments、Kr:dotted lines:calculation,。and+:experiments. l/E=3Uolfoneusestherelationpc=m/(3りo),then 。=9uo/8butthevalueofuowillnotgivecorrectmolec‐ ularsize,inthiscasethermodynamicpropertiesarenot ingoodagreementwithexperiments(e・glatentheatof vaporizationorvaporpressure)Thisdiscrepancycanbe solvedby2pc=m(1-./bc)んo,ifonefitsthevalueofuo withatruevalue,thenthevalueofd/bcisdetermined atacriticaltemperature,sothatthecurvesofc,(T)and c2(T),L(T)andp(T)areingoodagreementwitheach experiment、Thuswewillgivearoleoffittingparameter tothevolumeofmoleculeuo、Thisistheadvantageof ourtheory・Thevalueofuoisnotnecessarilydetermined onlywithintheframeofthermodynamics・Weexamine

theMaxwell,sequiarealruleasanintegralofp(T)over

therangeoMandzatatemperaturewrittenby wherep1andp2aredensityofvaporandliquidstate,

respectively・Isingmodelsoflatticegashavegenerally

showntherelationofp,+p2=constI7],butthetheory

showsthateq(19)isafUnctiononlyoftemperature

Ourmethodwillgiveagoodqualitativebehaviorasan approximatetheory・Thelatentheatofvaporizationis givenbyClausius-Clapeyron,sequation dpL dTT(1/5-1/i)

志(等)念

(21)

Equation(21)yieldsthelatentheatL(T)afterusing

p(T)anddp/dTineq(18)(seeligs7and8).

Thisisnotingoodagreementwiththeexperiments,

however,ifeq(21)isconnectedtotheempiricalequation

fbrvaporpressure[l7Ithegoodagreementisobtained

asfbllows: dp

--`叫芸(…cal…1.I(22)

p then

ノ(1mⅧ-,(T)ノ(W`'

(25) ThEruleisusuallyusedtodeterminethevaporpressure ,WandlGatadesiredtemperatureandholdsonnear thecriticaltemperature,however,thecoincidenceoftwo valuesisnotgoodinthelowtemperatureneartriple pointwhenusingd=9Uo/8.Ontheotherhand,this

ruleholdsonwellinourtheoryiifa、appropriatevalue

ofuoisemployed(seefigs・10~12) KamerlinghOnnes,constantisgivenby

L-芋(等)-A

(23)

Abovelatentheatofvaporizationisshownasfig9、The

wellknownexperimentalfactthatc,(四J=c2(Z1c)and

L(、)=OrequiresQtovanishesatacriticaltempera尺

turesoweobtainthevalued K、=PCMと,(26)e2 (27) --

2ef,

whereel=1-pcuo/mande2=2el-LThisisin goodagreementwithexperimentalvaluafbrexample1 eq.(26)givesKm=O3412atr=z200AandKm=

`=」(1-梺壯等)-2‘。

2PCDD (24)

whereuoisamolecularvolume、VanderWaals,equa戸

tionofstateconventionallyyieldsthecriticalvolume

(6)

1.4

li

lll

{矩。ご呵・二」.一.二一

(。}「)|ロ①二一Eの一画】

02

潔一

50100150.200250300350 Temp(K) □■ Fig.5.LiquidusandgaseouslinefbrN2,O2andCO2,when useeq.(24)asd/bc,whichisdependentonamolecularsize Asshownmfig.(4),thesearequalitativelyingoodag塵ement withexperimentaldataN2:dashedandsoIidlines:calcu-lation,□andx:experimentsO2:dash-dottedanddashed lines:calculation,xand△:experiments・CO2:dottedlines: calcuIation,can。+:eD(periments. 6080100120140160 Temp.(K) Fig.7.TheIatentheatofvaporizationofN2,O2andAr,whenuse d/bc=l/3.Dashedandsolid1ines:calcuaItionfmmeq.(21). Boldlines:drawnfromeq(23)△,xand*:experime肋ts. 叩釦如釦、知的釦・叩〃卯0 544332211 {g「)|ロ⑩二一こ①一日 7 6 5 {ユニ).⑭巴旦 4 3 2 6080100120140160180 TemplK) Fig.8.Themolecula炉sizedependenceoflatentheatfbrN2,O2 andArdrawnbyeqs(21)and(24).Dashedandsolidlines: r=2130AIbrAr,r=2.270AfbrN2andr=z140AfbrO2・Tl1e peculiarbeh誠ior1appearsinlowtemperatureregion,which intendstodiverge・Boldlines:r=2.lO5AlbrAr,r=2.222A lbrN2andr=z092AfbrO2. 1 、 BOgO100110120130140150160 Temp.(K) Fig.6.ThevaporpressurecurvesofAr.(a)Experiments.(b) whenuseeq(24)withr=2.105A.(c)whenused/bc=1/3. Whenuser=2.105Aineq.(24),theagr℃ementisbetterthan thatofd/bc=1/3. vapor-liquidphasearerigorouslycalculatedfromthee- quationofstateWefbundtheexistenceoftwophas-esofliquidandvaporandcoexistentstatethroughthe canonicalpartitionfUnctionwithmean-fieldapproach・ Thevapor,liquidandflatregionphasesarecontinuous-lyconnectedinpllasediagram,thuswecanidentifythe pvTdiagram、ThePercus-Yevickequationofstatefbr theLennard-Jones6-12potentialderivednotonlyfrom thecompressibilityfbrseveralisothermsbutalsofrom thepressurefUnctionisnotinagreementwithexperi-mentaldata・Particularlyincoexistentstateofliquid andvaporthisequationhasnosolutionofexperimen-talHatregion[181,[l9IThermodynamicpropertiesgiven byeq.(21)arelesssatisfactoryinthelatentheatofva‐ O2548atr=2300Awitheq.(24)fbrAr,ontheother hand,ifoneemploystherelationofd=9uo/8,then Km=O375Equation(26)alsogivesfairlygoodvalue fbranothersubstanceandwecolljecturethattheinterval ofKmandpcuo/mareO<Km<O5andO<pcuo/m< 0.5,respectively, WbcanconstructtheisothermsinpvTdiagramfiPom eqs.(8),(9)and(18)(seefigl3) 3.ResultandDiscussion NegativepaiPinter錘tionenergya凡reragedoverallpa炉 ticlesanditsdensitydependencegivesvanderWaals, equationofstateandthatliquidusandgaseouslinefbr

(7)

12 YOGI・YAGA:LiquidusandGaseousLineinPhaseDiagram 19876543210 ■●■■■①●のひ 000000000 ~回逗C (g「}一口①二一g一口] シ・② 2 「 ̄ Ne 6080100120140160 Temp.(K) Fig.9.The]atentheatofvaporizationofN2,O2andArdrawn byeqs.(23)and(24).r=z105AfbrAr,r=2.222AfOrN2 andr=2.O92AfbrO21o,+andx:experiments・Boldlines: drawnfromeq(23)Thesecurvesareingoodagreementwith experiments. 050100150200250300350 Temp.(K) Fig.11.TheMaxwelI,sequiarealrulewitheq.(24)andineq.(25), apartfromthefactorβ・Thisruleissatis6edinNe、N2,CO, Ar,KrandCO2well,whenr=2.126A(Ar),1.755A(Ne),2.282 A(Kr),2.295A(CO),2.302A(CO2)and2.258A(N2),respec-tivclylfthesuitablevalueofuDisemployed1thenagood agreementwithexperimentsisobtainedOthersubstances showsgoodagreementwitbrespectiveexperimentslwhenthe adequatevaluesofuoareused 1 98765432l0 q■■■●■■■■ 000000000 Ⅲ

02 berequiredfbrgoodagreementwithexperimentaldata preciselyexaminedasshowninfig8.

ザ急

シoL 4.Conclusion Thecomputersimulationshowstheattractiveinterac- tionasaveragedenergyovcrallparticlesandthedensitv-dependenceofpairzinteractionenergyineachphase,Wb showedalternativemethodderivingtheequationofs-tatebothofliquidandvaporphasesthroughmeanfield modelingwithattractivetypefbrceTherehavebeen someattemptswithmodels,asseeninprevioussection, ourtheorywillbeasimpleandoneofthepracticalap-proximationmethods,Theliquidusandgaseouslinein phasediagramarederivedandisingoodagreementwith experimentaldatafbrseveralsimplesubstance,Wecan identifytheliquid,vaporandtransitionregionsinthe pvTdiagramandtheisothermsthusobtainedareHat inthetransitionregionandriseveryrapidlywithde- creasingvolumeintheliquidphase、Thevaporpres- sureiscalculatedandissomewhathigherthanexper-imentalvaluefbrsimplesubstanceinhighertemper鉦 ture,howeverinlowtemperaturethepressureisfairly ingoodagreementwithexperimentfbrvaporpressure・ Itshouldberemarkedthatacriticaltemperatureiscon-cemedwithstructureofmolecule,massofmoleculeand theaveragedmagnitudeofinteractionenergyaparticle inthephase,Thelatentheatofvaporizationobtained ineq(21)iscomparedwithexperimentalvalueandthe agreementwithexperimentisnotgood(abouthalfof experimentaldataLTheheatofvaporizationofeq.(23) ismgoodagreementwithexperimant,Thediscrepancy お旬乞 。□・・・耐 L

Iik

gG

i2

■■■■■■●■ロ

● 6080100120140160 T(K) Fig.10TheMaxweII,sequiarealruleineq.(25)with。/DC=1/3, apartftOmthefa鰹toraThisruleholdsonnearthecritical temperaturewell1howeverthisruledoesnotholdoninlow temperaturereglonThecurvもsdenotedbyasteriskmeans theleft-handsideofeq.(25)andtheothertheright-handside ofeq.(25). porization,whichissmallerthanthevaluecalculatedby empiricalfbrmula1howeverthisresultsfromthesecond lawofthermodynamicSIfthevalueofuoisconven-tionallyemployedbypc=m/(3Uo)invanderWaals, equationofstate,thenoneobtainsd/bc=1/3.Howev‐ erthisvaluedoesnotcoincidewiththevalueobtained fromtherelationd/bc=1-2pcuo/mwithappropnate valueofuo、Thepeculiarbehaviorinthelatentheatof vaporizationandMaxwell'sequiarealruledisappearin thelowtemperatureneartriplepoint,iftheappropriate valueofuoisemployed(seefigs、9andll).Theappropri‐ atevalueofdensityalongliquidandgaseouslineshould

(8)

1.4 icvalues.Althoughourtheoryshouldberestrictedto asystemwhosepotentialisofattractivetypethrough themean-fieldtheoryibycomparisonofourresultswith experimentaldata,weconcludethatsuchmean-fieldap-proachworklnoresatisfactorilythanthoseofprevious work[11~[31,[61~[81,[101,1181.Iftheinteractionbetween particlesisofrepuIsivetypeasshownmeqs(5)and(6), wewillpredictthenegativecriticaltemperaturefrom eq.(13)Thisremarkableresultcorlもspondstothesit-uationofferromagnetismwithCurietemperatureand antifbrromagnetismwithNeeltemperatureinthesecond orderphasetransitionofmagnetism,Therefbrewewill showtheclassificationofphasetransitionbywhetherthe interactionbetweenparticlesinHuidsystemisattractive orrepulsive. 1.2 1

Z08

0.6 0.4 02 0 6080100120140160 TemP(K) Fig.12.TheMaxwelrsequiarealrulefbrO2ineq.(24)and(25), apartfmmthefactorβ、Thesecurv巳sdonotsatisfiedtherule, evenwhenr=2.129A・Thisruleisfi1lnl1⑨dneartbecritica】 temperature1howeverdoesnotholdoninlowtemperature・ solidline:theIeft-handsideofeq(25)。:theright-handside ofeq.(25). Refbrences l11JEMayer,JChemPhys、5,67(1937) [21JEMayerandPhGAckermann,J・ChemPhys、 5,74(1937) [31J・EMayerandS.F・Harrison,JChemPhys、 6,87,101(1938) [4]BKahnandGEUhlenbechk1Physica5,399(1938) [51M.BornandK・Fnchs,ProcRoySoc・A166,391(1938). [61TDLeeandON;Yang,Phys・Rev、87,No藝3,404(1952) [71TDLeeandON・Yang,PhysPev87,3,410(1952). l81T、HillJ・ChemPhys23,812(1955). (91L・VanHove,Physica,XV,951(1949) [l01SKatsura,Pro9.TheonPhys、11,No.4-5,476(1954) [111SKatsura,JChemPhys23,812(1955) [121T,Hill,SmtiStfmlMecノmndcs,(McGmwRill,NewYbrk, 1956). [131T・Hill,A〃mtmductiontoSmtfstlcqlllhermodWamjcs, (Addison-WesleyiMassachusetts,1952) [141JE・MayerandMGMayer,StotiStiCQlMechQrmjCS,(Wi-leyiNewYbrk,1940). [15ID、terHarr,StqtdsticqlMechqnics,(ButterworthHeine-mann,1995) [161ARahman,PhysRev、136,No.2A,A405(1964). [l71MWZemansky,Hmto"dThermodWamics,(McGmw-HilLIbkyo,1968) [181R○.Watts,J,ChemPhys、48,50(1968). [l91JKPercusandG.』・Yevick,Phys,Rev、110,1(1958). [201SNose,MoLPhys.,52,255(1984) [211SNose,JChemPhys.,81,511(1984) ごこ)△

Wcm3/moIe)

Fig.13.TheisothermsfbrArinpvTdiagrams、Onecaniden-ti【ytheHatregionofcoexistenceofvaporandliquidOther substancerevcaIsthesamebehaviorasAr. betweenourresultsandexperiments(orempiricalfbr‐ mula)willindicatenottoresultsfromneglectingthein-teractionbetweenparticlesmbothphasesbuttoresult fromthesecondlawofthermodynamics、Thethermody- namicpropertiesareshowntobesensitivetothemolec-ularsizeinthiSpaper;iftheequationpc=m/(3U0)is usedtodeterminethevolumeofmolecule1thisvaluedoes notcoincidewithothertheory(e9.,quantummechanics etc),ontheotherhand,ourtheoryhasanadvantage thatonecanemploycorrectvaluesofcriticaldensity andmolecularsizeineq.(28)Liquidusandgaseousline, thelatentheatofvaporizationandMaxwell,sequiareal ruleshouklbesatisfiedbyusingsuitablevaluesofuo,

参照

関連したドキュメント

61歳一一70St,71歳一80歳,81歳一90歳ノ年齢別 ノ8組二分チ,更二男女別二分類シ限局性緻密

チ   モ   一   ル 三並 三六・七% 一〇丹ゑヅ蹄合殉一︑=一九一︑三二四入五・二%三五 パ ラ ジ ト 一  〃

顧  粒 減少︑飛散︑清失︒ 同上︑核消失後門モ粟酒スルアリ︑抵抗力強シ︒

一門 報一 生口鍬  卵q 山砕・ 学割  u60 雑Z(  ヨ 

Amount of Remuneration, etc. The Company does not pay to Directors who concurrently serve as Executive Officer the remuneration paid to Directors. Therefore, “Number of Persons”

⇒ The CR was fully inserted and the CR index tube was stored in CRD guide tube at the time of the accident, so it is assumed that the cylindrical structure is CR guide tube and

(以下、福島第一北放水口付近)と、福島第一敷地沖合 15km 及び福島第二 敷地沖合

栄養成分表示 1食(○g)当たり エネルギー ○kcal たんぱく質 ○g 脂質 ○g 炭水化物 ○g 食塩相当量 ○g カルシウム ○mg. 鉄