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new republicationof Itispublished Methuenand Manufactured the United' States - EDITED WITH BY EDBYA. INVESTIGATIONS THEMOVEMENTDEMANDEDBYTHE willbetothemolecular inatheycanbeeasily observed inaonaccountthemolecular motions thatthebediscussedwiththeso molecular however,theavailabletothe latter isinprecision,nojudgmentinthe themovementwiththe thatoneexpecttothermodynamicscanno longer be looked precisiontobodies even ofdistinguishableina exact determination ofactual atomic #en possible. Ontheother hand, had thepredictionofmovement proved tobe inwouldbeagainstthekinetic conception of I.ON USPENDEDP dissolvedin partof liquidof fromthe apartition forthesolventbutimpermeableforthesolute, a so exerted on this partition, which issufficiently arepresent inplace thedissolvedparticlesareunabletopass through thepartitionpermeabletothesolvent theclassical atleastwhenthe not interest us wouldnotexpecttofindanyforceactingonthe for according to ofthesystemappearstobeindependentofthepositionoftheofthesuspendedparticles,butonthetotalmassandqualitiesofthesus theliquidActually, for thecal thefreeenergytheenergy theboundaryshould also beconsidered thesizeandconditionofthesurfacesofdo not alter withthechangesin partition and ofthesButa different conception reachedthestandpointofthe Accordingtotheoryadifferentiated from a s whya number ofsshouldnotproducethesame osmotic pressure asthesamenumberof thattheperforman very theliquid, on THEORY themolecular movement ofthe leavingthe thepartition,exerta pressure onthemoleculesinifthereare presentinthe andtherefore inaunit and particlesarecientlyfarseparated,bea corresponding ofmagnitude signifiesthenumberofcontainedinawillbeshowninthenext paragraph thatthetheoryofactually leads towidercon osmotic pressure. ORESSUREFROMTHESTANDPOINTOFKHEORYOFHEAT arethevariables stateof Inthisparagraphthepapersoftheauthoronthe FoundationsofTh areassumedtobefamiliartothereader Anunderstanding theconclusionsreachedinthepresent paper is not dependent onaknowledgeoftheformer papers orofthisparagraph thepresent paper. completely define theconditionofthesystemample,theCoordinatesandatomsofthesystem), and thecomplete theequationsofchangeoftheseofstategiveninthe thentheentropyofthesystemis given bythe istheabsolute temperature, theenergyofthesystem, theenergyasa The integral isextended over all consistent with theconditions theprob connected with the tobeforebythe Weobtainhenceforthe THEORYBROWNIAN MOVEMENT letusconsider avolume let there be suspended particles respectively) thepor whicharethevolurne a partition theintegration limits theintegral intheexpressionsfor willbeaccordingly.Thethesolutetakenas willbe accordingtothediscussionbythe themolecularextendedtodealunit,the calculationof couldbe Accordingly,weonlyto dependsonthemagnitudeofthe inwhichallthe molecules,or arecontained. therectangularCoofthecentre thefirst particle, thesecond, oftheparticle,andallocate for thecentrestheparticlesthe Thevalueoftheintegral appearing intheexpressionfor willbesought, with thelimita thecentresgravityoftheparticleslie within a domain definedinmanner.Thethenbe independentof etc.,aswellas ofthepositionthesemipartition.But isalsochoiceoftheposthedomainsthecentresthemagnitudeof aswillbeshownimmediately.Fordomainsofthetheparticles,andthelatter givenin positionbutnot in their magnitude,andaresimilarlyallcontained ananalogous expression holds dB' Butfromthetheoryofgiven in the it equaltotheprobabilitythatatanymomentoftimethecentrestheparticlesareincluded in the themovementsofareindependentofoneanothertoadegreeoftheliquidforceontheforequalsizeofdomainstheprobabilityofofthetwosystemsthatthe Butfromthisandthelast Wehave isindependent andof integrationwe andthence hasbeen shown bythattheenceofanospressure can be deducedthemoleculartheoryof andthatas far asospressureismoleculesandsustothis HEORY THEDIFFUSION dispersed in aliquid.Westateofdynamic equilibrium, on theassumptionthata actsonthedependsontheposition,butnotthetime.will be assumed thesakeofsimplicitytheforce is everywhereinthe bethenumberofparticlesper then in theconditionofdynamic is such a tionoftheenergy vanishes foran ofthesuspendedsubstance.We O. THEORY beassumedthattheliquidhas perpendicular.tothe isboundedbythe have, andand required condition ofequilibriumisthere The last equation equilibrium with the brought about byusedtofindthecoefficientofdiffusionthesuspendedsubstance.Wecanuponthedynamic equilibrium condition considered here asa superposition twoprocessesproceedinginopposite directions, namely thesuspendedundertheinfluenceoftheforce onsingle suspended particle. processofdiffusion,belookeduponasaresultoftheirregularmovementoftheproducedby thesuspended(radiusofthe theliquidhasacoefficientofviscosity thenthe partstothesingleparticlesa andthere unit area perunitoftime particles. thecoefficientof thesuspended themassparticle,astheresultofarea in aunitof grams particles. Lectures BROWNIAN there must be dynamic equilibrium, we Wecan calculate the fromthetwofoundforthedynamic equilibrium. Weget N Thecoefficientdiffusionofthesuspendedsubtherefore depends (except foruniversal onlyonthecoefficientofviscosityoftheliquidandonthethesuspended NRREGULARMFPUSPENDEDIN LIQUIDRFOD turnnowtoa theirregularmovement,andgive rise totheinvestigated in themustbeexecutesamovement themovement all other particles the oneandthesame time must beconsideredaslongaswethinkoftimeasnottoosmall.Wewillintroduceatime in our diswhichistobe very small compared with theobservedintervaloftime,ofsuch a thatthebyaparticle in intervalsoftime aretobeconsideredasthere are altogether suspendedparticlesina liquid. Inan interval of thesingle hasa different value (positive or lawwill hold the' theparticlesexperience in the a displacement which willbeexpressedby an theform from zero for very small values of thecondition investigatenowthecoefficientofdependson tothecasewhenthe oftheparticles per volumeis onlyon Puttingforthe particlesperunit calculatethedistri theparticlesata fromthedistributionatthetime Fromthe thefunctionobtainedthenumberoftheparticleswhicharelocatedat two planes perpendicular tothex Weget isverysmall,wecanputFurther,we inpowers Wecanbringundertheintegralsince only very small values anythingtothelatter.We SMALLPARTICLES therightsidethes vanish third, fifth, etc., terms, every withtheandputting andtakingthefirstand therighthandside,wegetfromthewell diffusion,and considerationcanbetothismethodofdevelopment. thesingleparticlesareall system.Butsincethe thesingleare mutually independent. We nowreferthemotionofeachparticletoa origin coincides atthe withthepositionoftheparticlesinquestion with nowgivesthenumberofthe ordinatehasincreased between the andthetime byaquantity Inthisalsothe theequationFurther,we oand oand must evidently Theproblem,accords with theproblemofthediffusionfroma sibilitiesofexchangebetweenthediffusingpar thesolution Theprobabledistributionoftheplacements in a thereforethesameasthatwhichwasto be expected.Butsignificanthowthetheexponential are related tothediffusion.We nowcalculate with thehelp OFSMALL equationthedisplacement inthedirectionoftheXwhicha average,or rootofthe thesquares displacementsinthedirection theX Themeanthesquare thetime.easilybethatthethemean thetotaldisplacementsoftheparticleshasthe FORMULA EANDISPLACEMENTOFSUSPENDEDP NETHODETERMININGEALSIZE wefound for the liquid in theFurther,we themean value theparticlesinthe theXaxisin Byeliminating weobtainshowshow dependson Wewillcalculatehowgreat is for one second, if is taken equal to in accordance with thekinetic theory ofwaterat astheliquid andthediameteroftheparticles mm.Weget cm. Themeandisplacementinonewouldbe, theotherhand,therelationfound can beusedforthedeterminationof Weobtain behopedenquirermayshortlyinsolvingtheproblem suggested whichisso withthe Heat. (Received, THETHEORYTHEBROWNIAN OONaftertheappearanceofmypaper themovementsofparticlessus liquids demanded by theoryof informedmethathe inthefirst (ofLyons)hadbeen convinced bythatthesocausedby themoleculesofthe Not only thequalitative properties ofthemotion,butalsotheorderofmagnitudeofthepathsdescribedbythewiththe thetheory.notattemptherea theslendermaterialatmydisposal with the d.