Diffusion in Dilute Solutions

8/11/2019 Diffusion in Dilute Solutions

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C H A P T E R2

Diffusionn Dilute Solutions

In thischapter,we considerhebasicaw thatunderliesiffusionand ts applicationto several impleexamples.The exampleshatwill begivenare estrictedodilute solutions.Resultsor concentratedolutionsaredeferreduntilChapter3.

This ocuson hespecial ase f dilutesolutionsmay seem trange.Surely,t wouldseemmore sensibleo treat he general aseof all solutionsand henseemathematicallyhat hedilute-solutionimitis like.Most booksuse his approach.ndeed, ecause oncentratedsolutionsarecomplex,hese ooksoftendescribe eat ransf'err fluidmechanicsirstandthen eachdiffisionby analogy.The complexityof concentratediffusionhenbecomesmathematicalancergraftedontoequations f energyandmomentum'

I haveejectedhis approachor tworeasons. irst, hemostcommondiffusionproblemsdo takeplace n dilutesolutions.For example,diffusionn living tissuealmostalwaysn-volves he ransportf smallamounts f solutesike salts.antibodies,nzymes, r steroids'

Thus manywho are nterestedn diffusionneednotworry abouthe complexitiesof con-centratedolutions;heycanwork effectivelyand contentedlywith thesimplerconceptsnthis chapter.

Second ndmore mportant,diffusionn dilutesolutionss easier o understandn phys-ical terms.A diffusionlux is therateperunit areaat whichmassmoves.A concentrationprofles simplythe variationof theconcentrationersus imeandposition. Thesedeasaremuchmore easilygrasped hanconceptsikemomentumlux,whichis themomentumperareaper ime.Thisseems articularly rueor thosewhosebackgroundsrenot n engi-neering, hosewho need oknow aboutdiftisionbutnot aboutother ranspol'thenomena.

This emphasisn dilutesolutionss found n thehistoricaldevelopmentf thebasicawsrnvolved,asdescribedinSection2.Lections2.2and2.3ofthischapterfocusontwosimplecases f {iffusion:steady-stateiffusionacross thinfilmand unsteady-stateiffusionnto

.rn nfiniteslab.This focuss a ogical choicebecausehese wocases re socommon.For

L'xample, iffusionacrosshinfilmsis basic o membraneransport, nddiffusionn slabsrr importantn the strengthf weldsand n thedecayof teeth.These wo cases re he wor.xtretnsn nature,and heybrackethe behaviorobserved xperimentally.n Sections '4.tnd2.5. thesedeasareextendedo otherexampleshatdemonstratemathematicaldeasLrsefulbr other situations.

2.1 Pioneersn Diffusion2.1.1Thomas rahamOurmodern deason diffusion areargely due to twomen,ThomasGrahamand

\dolf Fick. Grahamwas he elder.Born on December20, 1805,Grahamwas he son of

i successfulmanufacturer.At 13yearsof age he entered heUniversityof Glasgowwith:hc intentionof becominga min ister, and therehis interestn sciencewasstimulatedbyThomas homson.

l-l


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l 4 2 / Di/Jusion nDilute Solutions

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Fig .2 . l -1 .Graham'sd i f fus ion tube forgases .h isappa ra tuswasused in thebes tea r lys tudyofdiftsion. As a gas ike hydrogendifTuses utthrough theplug, the tube is lowered to ensurethat therewill be noDressure il'ference.

Graham's esearch n the ditTusionf gases,argelyconducteduring he years1828 oI833,depended tronglyon the apparatushownn Fig. 2.1- (Graham,1829, 833).This

apparatus, "diffusiontube,"consists f a straightglass ube,one endof which is closedwith a dense tuccoplug.The ube s filled withhydrogen, ndhe end s sealedwith water,as shown.Hydrogendiflises throughhe plugandout of thetube,while airdiffusesbackthroughhe plug andntothe ube.

Becausehediffusionofhydrogens fasterhan he diffusionofair, hewater eveln thistubewillrise duringhe process.Grahamsaw hat his changen waterevelwouldlead oa pressure radienthat n turnwould alterhe diffusion.To avoidhispressure radient, econtinuallyowered he ubeso hat hewaterevelstayed onstant.Hisexperimentalesultsthenconsistedof avolume-changeharacteristicf eachgasoriginallyheldin the tube.Becausehisvolume changewas characteristicf diffusion,"thediffusionor spontaneousintermixtureof twogases n contacts effctedby an nterchangef position ofinfinitelyminutevolumes,being,n the caseof eachgas, nverselyproportional othesquareoot of

thedensityof thegas"(Graham,

1833,p. 222). Gtaham's riginalexperimentwas unusual

becausehediffusionookplaceatconstant ressure, ot at constantolume Mason,1970).Grahamalsoperfbrmed mportantexperimentsn liquid diffusionusing heequipment

shownn Fig.2.l-2(Graham,850);n these xperiments,e workedwith dilutesolutions.In oneseries f experiments,e connectedwo bottleshat containedolutionsat diffrentconcentrations;e waiteil severaldaysand thenseparatedhe bottlesand analyzedheircontents.n another eries fexperiments,e placeda smallbottlecontaininga solutionofkno'uvnoncentrationn a largerar containingonly water.Afterwaitingseveral ays,heremovedhebottleandanalyzedts contents.

Graham's esultswere sirnpleanddefinitive.He showedhatdiffusionn liquidswas atleasr everalhousandimesslowerhandiflisionin gases.He recognizedhat hediffusionprocess otstillsloweras heexperimentrogressed,hat"diffusionmustnecessarilyollowa diminishingrogression."ost mportant,e concludedromthe resultsn Table2.1-1that"rhequanritiesiftisedappearo be closelyn proportion . . tothequantityof saltnthe ditTusionolution"(Graham,1850,p. 6). In otherwords, he flux causedy difTusionis proportional o the concentrationiffrenceof thesalt.


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2.1 Pioneers n Dffision l 5

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Fig.2.l-2. Graham's diffusion apparatusbr liquids. The equipmentin (a) is the ancestoroffiee diffusion experiments;hat in (b) is a forerunner ofthe capillarymethod.

Table 2.1- . Graham'sesultsor liquid difrusion

Weightpercentofsodiumchloride Relativelux

r.001.993 .014.00

Source:Data rom Graham1850).

2.1.2 AdolfFick

Thenext major advance n the theory ofdiffusion camerom thework of AdolfEugen Fick. Fick was born on September, 1829, heyoungestof fve children.Hisfather,a civil engineer,wasa superintendentfbuildings.Duringhis secondarychooling,Fickwas delightedby mathematics, specially hework ofPoisson.He intended o makemathematics is career.However,anolderbrother,a professorf anatomyt the Universityof Marlburg,persuadedim to switch omedicine.

In the spring of1847, Fick went toMarlburg, where hewas occasionally utoredbyCarl Ludwig. Ludwig stronglybelieved hatmedicine,and indeed ife itself,must havea basis n mathemat ics,hysics,and chemistry.This attitudemust have beenespeciallyappealing o Fick,who saw the chance o combinehis reallove, mathematics,withhisehosen rofession.edicine.

In the fall of1849.Fick's education ontinuedn Berlin.where he did a considerableirmount fclinicalwork.In 185 he eturned o Marlburg,wherehe eceived isdegree.Hisrhesis ealtwith thevisualerrorscaused y astigmatism, gainllustratinghis determinationtocombinesc ienceandmedic ine (F ick ,852) . n the fa l lo f185,Car lLudwigbecameprofessorof anatomyn Zurich, and in the springof 1852he broughtFickalongas aprosector.udwig moved oMenna n 1855,but Fickremained nZurich until1868.

Paradoxically,he majority of Fick's scientifcccomplishmentsonotdepend n diffu-.ionstudies t all, but onhis moregeneralnvestigationsfphysiologyFick,1903).He didoutstandingworkin mechanicsparticularlyas applied o the functioningof muscles),nhydrodynamics ndhemorheology , ndn thevisual and hermalunctioningof the human

123,1


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l 6 2 / Diffusionin DiluteSolutions

body. He was an ntriguing man. However,n this discussionwe are nterested nly in hisdevelopmentf the fundamentalawsof difTusion.

In his firstdiffusionpaper,Fick (1855a)codified Graham'sexperiments hrough animpressiveombinationof qualitativeheories, asualanalogies, ndquantitativexperi-ments. Hispaper,which is refreshinglystraightforward, eserves eading oday. Fick'sintroductionof hisbasic dea s almost casual:"[T]hediffusion of thedissolvedmaterial. . . s left completelyo the nfluenceof the molecular orcesbasic o thesame aw . . . forthe spreadingf warmth n a conductorandwhich hasalreadybeenappliedwith suchgreatsuccesstothespreadingoflectricity"(Fick,855a,p.65). notherwords,diffusioncanbe described n the samemathematicalasisas Fourier's aw for heatconduction r Ohm'slaw forelectricalconduction.Thisanalogyemainsa usefulpedagogicalool.

Fickseemed nitially nervousabout his hypothesis.He buttressedt with a varietyofarguments asedon kinetictheory. Although thesearguments re nowdated, hey showphysicalnsightshatwouldbeexceptionaln medicineoday.Forexample, ick recognizedthatdiffusionis a dynamic molecularprocess.He understood he differencebetweenatrue equilibrium anda steadystate,possiblyas a resultof his studieswith musclesFick,1856).Later,Fickbecamemore confidentashe realizedhis hypothesiswasconsistentwithGraham's esulrsFick,1855b).

Using thisbasichypothesis,ickquicklydeveloped helawsof diffusion by meansof

analogieswith Fourier's work(Fourier,1822). He defineda total one-dimensionallux./1 as

r

f f i T:rfr:l]c@o ffiF,t @: Ilhr

GItlN

Jlil5

h I rnafl,

IJJ r,Nlffi [lm&iltuf ,

E pmrr r

lia

J t : A j r :- A D ?d7 .

3 r ' r / d 2 c , I A A A t \

a , : D [ * * A , ' r )

(2.1-1)

whereA is theareaacrosswhich diffusionoccurs,1 is the luxperunit a;tal.,1 s concentra-tion,andz is distance.This isthef,rstsuggestion f what s now known asFick's aw. ThequantityD, which Fickcalled the constant ependingf thenatureof the substances,"s,of course, he diffusioncoefficient.Fickalsoparalleledourier'sdevelopmento determinethe moregeneral onservationquation

(2.1-2)

When the area A is a constant, his becomes he basic equation or one-dimensionalunsteady-stateiffusion,sometimes alledFick's second aw.

Ficknext had toprovehis hypothesishat diffusionand thermal conductioncan bedescribed y the sameequations.He wasby no means mmediatelysuccessful.First, hetried to integrateEq.2.l-2 forconstant rea,but he becamediscouragedy thenumericaleffortrequired.Second,he tried to measure hesecondderivativeexperimentally.Likemanyothers, e oundhatsecond erivativesredifficult o measure:thesecond ifferenceincreases xceptionally he efTect f[experimental]rrors."

His third effort was moresuccessful.He useda glasscylinder containingcrystallinesodiumchloride nthe bottom and a la rge volumeof watern the top, shownas he lowerapparatusn Fig. 2.1-3. Byperiodicallychanging he water n the top volume,he was ableto establish steady-stateoncentrationradientn the cylindrical cell.He foundthat hisgradientwas inear,as shown n Fig. 2.1-3. Becausehisresultcanbe predicteditherromEq. 2.1-1or from Eq.2.l-2,thiswasa triumph.


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Fig. 2 I-3. Fick's experimental esults. The crystals n the bottomof each apparatussaturate headjacentsolution,so that a fixedconcentration gradients establishedaong the narrow,lowerpartoftheapparatus.Fick'scalculationofthecurveforthefunnelwashisbestproofofFick'slaw.

Table 2.1-2.Fick's aw or diffu,sionithoutconNection

dc '- j t : D ,a zdc,- j r : D ,a rdc,- j t : D ,a r

Nole..Moregeneralquationsregivenn Table .2_.

Butthissuccesswasby no meanscomplete.Afterall, Graham'sdata or liquidsan-JlpatedFq.2.l-1. To try to strengthenhe analogywiththermalconduction,Fickused- lowerapparatushownn Fig.2.1-3.In thisapparatus,e establishedhe steady-state

- 'ncentrationroflen thesamemanneras before. Hemeasuredhisprofileand hen riedDredict heseesultsusingEq.2.1-2,inwhichthe funnelareaA availableor diffusion

,ned with the distance. When Fickcomparedhis calculationswith hisexperimental-r'.ults,he found he goodagreementhownn Fig.2.l-3. These esultswere he nitial. 'rilcationf Fick'saw .

2.1.3 Forms of Fick'sInw

useful formsof Fick'slaw indilutesolutionsare shown n Table 2.1-2. Eachr.lurtionloselyparallelshat suggestedy Fick,that s, Eq. 2.1-I. Each nvolveshe'rllephenomenologicaliffisioncoefficient.Eachwillbe combinedwith massbalances, analyzehe problemscentralo the restof thischapter.

Onemustrememberhat hese luxequationsmplyno convectionn the samedirection',' theone-dimensionaliffusion.Theyare thus specialcasesof thegeneralequationslr\en inTable3.2-1. Thislackof convectionften ndicates dilutesolution.In fct.

Forone-dimensionaliffusionnCartesian oordinates

Forradialdiffusionn cylindricalcoordinates

Forradialdiffusionn sphericalcoordinates


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t 8 2 / Diffuson in Dlute Solutions

Fig.2.2-1. Difusion acrossa thin flm. This is the simplest diffusionproblem,basic to perhaps80% of what follows. Note that the concentrationprofleis independentof the diffusioncoelhcient.

the assumption f a dilute solutions more restrictive han necessary,or there aremanyconcentrated olutionsor whichthesesimple equations an beused withoutinaccuracy.Nonetheless,or thenovice, suggest hinking of diffusionn a dilutesolution.

2.2 SteadyDiffusion Acrossa Thin FilmIn thepreviousectionwedetailed hedevelopmentf Fick's aw, hebasic elation

fordiffusion.Armedwith this aw,we cannow attack hesimplest xample:steady iffusionacross hin film. Inthis attack.we wanto findboth hediffusionlux and heconcentrationprofile.In otherwords, we wantto determineow much solutemovesacross he film andhowthe soluteconcentration hangeswithin the film.

Thisproblems very mportant. t is oneextreme fdiffusionbehavior, counterpointodiffusion n an nfinite slab. Every reader,whethercasualor diligent, should ryto masterthisproblemnow. Many will fail because ilmdiffusionis too simplemathematically.Pleasedo not dismiss his importantproblem;it is mathematicallystraightforwardutphysicallysubtle.Thinkabout t carefully.

2.2.1 The Physical Situation

Steadydiffusion acrossa thin film is illustratedschematicallyn Fig. 2.2-1. Oneachsideof the film is a well-mixed solutionof one solute,species . Both these olutionsare dilute.Thesolutediffuses rom the fixed higher concentration,ocatedat z < 0 on thelefrhand side of the film, into the fixed, essconcentrated olution, ocatedat z > / on theright-handside.

We wantto findthe soluteconcentrationrofileand hefluxacrosshis film. To do this,we first write a massbalance n a thin layer Az, locatedat somearbitrarypositionz withinthe thin film. The massbalance n this ayer s

Becausehe processs in steadystate, he accumulations zero. The diffusionrate s the

b,["

/ rate ol diffusion \/ s o l u t e \ / r a t eo d l f l u s t o n\ | . . 1 . , _ . _ _ . . . .I l : | | - | O U t o l t n e l a v e r\ a c c u m u l a t i o n /\ i n t o t h e l a y e r a t , /- \ " " ; ; ; ' , ' i ? ' ' / {N :llu-


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itr this.. \ \ i th in

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.t'iusionluxtimes he film'sareaA. Thus

0 : A( j t l .- - / r .+r.7):i ding hisequation ythe ilm's volume,AAz,and eaffanging,

(2.2-1)

(2.2-3)

( ) ) -4 \

r ? ) - 5 1

(2.2-6)

(2.2-1)

() ) _a\

(2.2-10)

o : - ( r r ' - r '/ r l ' )

r ) ) - ) t\ ( : + A r ) - z l

"henAz becomeserysmall, hisequationbecomes hedefinitionof thederivatived

0 : _ ; j t47 .

' ,nrbininghisequationwith Fick'saw ,d c t- j t : D '47 .

.' find,or a constant iffusion coefficientD,' 1 ) "

0 : D ' ; ' , '-

:r. differentialequation ssubjecto two boundaryconditions:z : 0 , c r c l o

z : I , c l: c t

:.:.rrn.ecausehis systems in steady tate,he concentrations16andc17 re ndependent' lime. Physically,his means hat the volumesof theadjacent olutionsmust be much:'rter han he volumeof the flm.

2.2.2 MuthematicalResults

Thedesired oncentration rofileand luxarenoweasilyound.First, we ntegrate- ).2-5twice o find

c t : a l b z (2 .2-8)- -;'constants andb canbe ound romEqs.2.2-6and2.2-l sohe concentrationrofiles

c l : c l o + ( c l / ' Z c1o)t

:r' li1s41 ariationwas,of course,nticipatedy the sketch nFig.2.2-1.The luxis foundby differentiatinghis profile:

dc , Dj t : - D , : ; ( c r o - c r r )07. Ir3.'rusehe system s insteady tate,he flux isa constant.

.\s mentioned arlier,his case s easymathematically.Although t is verymportant, t. ,ittenunderemphasizedecauset seems rivial. Before youconcludehis, try someofr' e\ampleshat bllowto make sureyouunderstand hat s happening.


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20 2 / Diffusionn DiluteSolutions 2 / SteadyDffisit

(a) (b) (c)

Fig.2.2-2. Concentrationprofilesacross hin membranes. In (a). the solute s moresoluble inthe membrane han in theadjacent solutions; n(b),it is less so. Both casesconespond to achemicalpotentialgradient ike that in (c).

Exampfe 2.2-l:Membranediffusion Derivehe concentrationrofileand the flux fora single solutediffisingacrossa thin membrane.As inthepreceding aseof a flm,themembrane eparateswo well-stirredolutions.Unlike he ilm,themembranes chemicallydifferent romthese olutions.

Solution Asbefore,we first writea massbalance n a thin layerAz:

0 : A( t l . -, / r:+r : )

This leadso a differentialequation denticalwithEq. 2.2-5:r )

O - C r0 : D -

az 'However, his newmassbalance s subjecto somewhat ifferentboundaryconditions:

z : 0 , c t : H C r c

z , : I , c t : H C t t

where11 s a partitioncoeffcient,he concentrationn the membrane ividedby thatn theadjacent olution.Thispartitioncoeffcient sa equilibriumproperty,o tsuse mplieshat

equilibriumexistsacross he membrane urface.Theconcentration roflehat resultsiom these elations s

c t : H ( ' t l + H l C 1 1- C , r )

whichs analogouso Eq. 2.2-9.Thisresult ooksharmless nough.However,t suggestsconcentration roflesikes hose nFig.2.2-2,whichcontainsudden iscontinuitiest theinterface. fthe solute s moresoluble nthe membranehan n the surroundingolutions,then the concentrationncreases. f the solute s lesssoluble n the membrane.hen itsconcentrationrops. Eithercaseproducesenigmas.Forexample,at the efi-hand sideofthe membranen Fig. 2.2-2(a),solutediffuses iomthe solutionat r:y6 ntothemembraneat hiphe concentration.

Thisapparentuandarys resolvedwhenwe thinkcarefullyabout he solute's iffusion.Diffusionoften can occurfiom a regionof low concentrationnto a region of highcon-centration; ndeed,his is the basisof many liquid-liquidextractions.Thus the umps inconcentrationnFig.2.2-2arenot asbizarreas hey mightappear; ather, hey aregraphical

.:cidentshat esul'rembrane.

Thistypeof dif. ,,lctly,n termsof-ross he membra,1.whichdrops ::ceresponsiblenpletelys SectiThe luxscroS:

' . r lewithFick':

. I D Hr I - r

. is parallelo E- ' r e r h i l i t r r o n ' l . '

:r'nneabilitrPsthediffusion

-:1trencesn . ,

' , mPl2.2-2:Po, :le are hans

Solution,nseron-.

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9/37

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2.2 Steudt' iJJusionAcrcss Thin Film

accidentshat esult romusinghesame caleo representoncentrationsnsideandoutsidemembrane.

This typeof diffusioncan also be described n termsof the solute'senergyor, moreexactly, ntermsof itschemicalpotential.Thesolute's hemicalpotentiadoesnotchangeacross he membrane'snterface,because quilibriumexists here. Moreover,hispoten-tial, whichdropssmoothlywithconcentrarion,s shown n Fig. 2.2-2(c),s the drivingfbrce responsibleor the diffusion.Theexact oleof this drivingforce sdiscussedmorecompletelys Sections.4 and1.2.

The fluxacross thin membraneanbe oundby combininghe oregoingoncentrationprofilewithFick'saw :

I D H l, / t : - - ( C 1 1 y - C 1 1 . )

Thisis parallelo Eq. 2.2-10.The quantityn square rackersn this equation scalledhepermeability,and t isofien reportedexperimentally.Sometimeshis same erm iscalledthe permeabilityperunitlength. The partitioncoefficient11 s foundto varymorewidelythan he diffisioncoefficientD, so differencesn diffusionend to be less mportanthanthedifferencesn solubilitv.

Example2.2-2:Porous-membranediffusion Determinehow the resultsof thepreviousexamplearechangedf the homogeneous embranes replaced y a microporousayer.

Solution The differenceetweenhiscaseand hepreviousone s that diffusionis no longerone-dimensional;t now wigglesalong thetortuouspores hat makeup thernembrane.Rather han try to treatthis problemexactly, youcan assumean effectivediffusioncoefcientthat encompassesll ignorancef thepore'sgeometry.Allthe earlieranswers re henadopted;or example,he flux is

I D-oHfl r l - , L l t C r o - C r r )t t l

rvhereD.s i, un"*, "effective"diffusioncoefficient.Sucha quantitys a flnction notonly

of soluteand solventbutalsoofthe localseometrv.

Example2.2-3:Membrane diffusionwith fast reaction Imagine hat whilea solute:s diffusingsteadilyacrossa thin membrane, t can rapidlyand reversiblyreact withIther immobilesolutes xedwithinthe membrane.Findhow this fastreactionaffects:hesolute'slux.

Solution Theanswer ssurprising:The eactionasnoefTect. his sanexcellent:ramplebecauset requirescarefulhinking. Again,we begin by writinga massbalance.rna layerAz locatedwithinthe membrane:

/ solute \ / solute ifusionn \ / amounr roduced \\ a c c u m u l a t i o " i :\ m i n u s t h a r o u r

*\ u v c h e m i c a l r e a c r i o n

Becausehe system s in steadystate, his eads o

0 : A( rl:- ,/r. :+r;) rrAL.z

21


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22 2 / Diftusion inDilute Solutions

PorousDiaphragm

Fig. 2.2-3. A diaphragm cell for measuring diffusion coefci ents.Because he diaphragm hasa much smaller volume than the adjacent solutions. he concentrationprofilewthin thediaphragm has essentially he linear, steady-statealue.

)u0 : - . . - - r rd Z

where rl is the rate of disappearancef the mobile speciesin the membrane.A similarmassbalancebr the mmobileproduct2 gives

, ' dU - - . . / 1 f / ' 147 .

Butbecausehe product s immobile,2 is zerc,andhence11 s zero. As aresult, he massbalance or species is identicalwith Eq.2.2-3,leavinghe flux and concentrationrofleunchanged.

This result s easier o appreciaten physicalerms. After the diffusioneaches steadystate, he localconcentrations everywheren equilibriumwith the appropriate mountofthe ast eaction's roduct.Becausehese ocal concentrations onotchangewith time, heamountsof theproductdo not changeeither.Diffusion continuesnaltered.

Thiscase n whicha chemicaleactiondoesnotaffectdiffusion s unusual.For almostany other situation, hereactioncan engender ramaticallydifferent mass ransfer.f the

reactions irreversible,he flux can be increasedmany ordersof magnitude,as shown nSection 6.1. If the difTusions not steady, he apparent iffusion coefficientan be muchgreater han expected, s discussedn Example 2.3-3. However,n the casedescribed nthis example, he chemicaleactiondoesnotaffect diffusion.

Example 2,2-4: Diaphragm-cell diffusion One easyway tomeasure iffusioncoef-cients s thediaphragmcell,shown n Fig. 2.2-3. Thesecells consistof twowell-stirredvolumes eparated y a hinporousbarrieror diaphragm.n themoreaccurate xperiments,the diaphragm s ofien a sinteredlass rit; in many successful xperiments.t is ust a pieceof filterpaper(seeSection5.5). To measure diffusion coefficientwith this cell,we fillthe owercompartmentwitha solutionof known concentration nd he uppercompartmentwith solvent. After a known time, we sampleboth upper andlower compartments ndmeasure heirconcentrations.

Findan equation hat useshe known time and he measured oncentrationso calculatethe diffusion coeffi ient.

nuil

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fE rems -' , rf i " - :

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11/37

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, rriments,, . .ra plece

- : . .r i ef i l.lrrlment

.-:r-nts and

: Jrlculate

2.2/ SteadyDiJfusionAcross a ThinFilm L-')

Solution An exactsolutiono thisproblem s elaboratendunnecessary.uchasolutions known butneverused Barnes1934).The usefulapproximateolutiondependson theassumptionhat heflux across hediaphragmquicklyreachests steady-statealue(Robinsonnd Stokes,1960).This steady-statelux is approachedven houghhe concen-trationsn the upperand ower compartmentsrechangingwith time.Theapproximationsintroduced ythis assumptionwill be consideredgainater.

In thispseudosteady tate,he flux across hediaphragms that given for membranediffusion:

I D H I/ r : | , - l t C r . r o * . r - C lp p . r )t t l

Here, hequantity H includes hefractionof the diaphragm'sarea hat is availableorJiffusion.We nextwrite an overallmassbalanceon the adjacentompartments:

dCt . lu*" ,Vtower# - -Ajt

d t

dCr.upp.,Vu o o c r - - - . : : l A j t

" d

.vhereA is the diaphragm'sarea. If thesemassbalances redividedbYVru*",andyuppcr,

-espectively,and the equationsare subtracted,ne can combine he resultwith thefluxtquation o obtain

d ^;;{Cr tu*.,

Ct.upp",) Df(Ct,opp., Cl.lo*.,)

r whichA H l I 1 \R - - l - r - lY - / \ v t " * t ' ' v ' , , " )

. a geometricalonstant haracteristicf theparticulardiaphragmcellbeingused. This-.l-ferential quations subjecto the obviousnitialcondition

/ : 0, Cl.ror.. Ct.upp., C.to*.. Cf.uno.t- rheuppercomparrments initiallyfilled withsolvent, hents initialsoluteconcentration:llbe zero.Integrating hedifferentialequation ubjecto thisconditiongives hedesired esult:

Cl.lo*". - Cl.upp., ^ - f tDr: (C.,n*.. C?.uoo"'

I / cP ., , . ' t t \D : - l n | ' ' c r - L l u P P c r I

pt \ C' . 'u*", Cr.upper/

,-an measurethe time r and the various concentrationsdirectly. We can also determineSeometricfactor B by calibrationof the cellwith aspecies whose diffusioncoeffcientro.uvn.Then we can determinethe diffusioncoefficientsof unknown solutes.


12/37

) 1 2 / Diffusionin DiluteSolutions

Thereare womajorways nwhichthis analysisan bequestioned. irst, hediffusioncoefficientusedhere s an effctivevaluealteredby the ortuosityn thediaphragm.Theo-reticiansoccasionallysserthatdifferentsoluteswill havedifferentortuosities,o hat hediffusioncoefficientsmeasured illapplyonlyto thatparticulardiaphragmell andwill notbe generallyusable.Experimentalistsavecheerfullygnoredheseassertionsy writing

| / c 9 . - c P \D - - r n 1 ' i h ' u c r ' l u P P e rB ' t \C ' . ' "*" ,

- Ct . ,p r, ,/

wherep' is a newcalibrationonstanthat ncludesany orluosityefTects. o ar, heexper-imentalistsavegottenawaywiththis:Diffusioncoefficientsmeasured ith thediaphragmcelldo agreewiththosemeasured y other methods.

The secondmajor questionbouthis analysisomes romthecombinationf thesteady-state luxequationwithan unsteady-stateassbalance.You mayfindthis combinationobe oneofthoseareaswheresuperfcialnspections reassuring,ut wherecarefuleflectionis disquieting.havebeen emptedo skipover hispoint,but havedecided hat hadbetternot. Heregoes:

The adjacentcompartments re muchlargerthan the diaphragm tselfbecause hey

containmuchmorematerial.Theirconcentrationshangeslowly,ponderously,s a resul tof the transferof a lot of solute. Incontrast, he diaphragmtselfcontains elativelylittlematerial. Changesn its concentrationrofileoccur quickly. Thus,even f thisprofileis initiallyverydiffrentrom steadystate, t willapproacha steadysratebeforethe concentrationsn theadjacentompartmentsanchangemuch.As a result,he profleacrosshe diaphragmwillalwaysbe close o itssteady alue,even houghhe compartmentconcentrationsre imedependent.

These deas anbe placedon a morequantitativeasisby comparinghe relaxationimeof thediaphragm,21o, withthat of the comparrmenrs,l(Dp) Theanalysisusedherewillbe accuratewhen(Mills,Woolf,andWatts, 1968)

Stead,tDiffusionAt

cl o

Smoldiffusion

----

c o e f f i c i e n t

t r i o r a - J C -

the diffu'ior. -

: r f i lm. [ r tc : i r

i -(.t l _

ci --

' j . . l t .t h e . - . .

- - D .

trr,t.l,/ :" :vai"p,.ogn'1 + I )| /( p Dr1, ".r \ Vt,,*., Vuppr,

Thistypeof "pseudosteady-statepproximation"s commonand willbe found o underliemostmass ransferoefficients.

Example2.2-5:concentration-dependentdiffusion In all the exampleshus far,wehaveassumedhat hediffusioncoefficients constant.However.n somecases his is nottrue;the diffusioncoeffcientan suddenlydrop froma highvalue o a much lowerone.Suchchanges anoccur or waterdifTusioncrosslmsand n detergentolutions.

Findthe fluxacrossa thin film inwhichdiffisionvariessharply.To keep he problemsimple,assume hat belowsomecriticalconcentration1.,diffusion s tast,butabove hisconcentrationt is suddenlymuchslower.

Solution This problemis best idealizedas two filmsthat are stuck together(Fi$.2.2-4).The interfacebetweenhese lmsoccurswhentheconcentrationqualsc1..

l l

= t )

D


13/37

. Solutions

:ic diffision, rg l l . TheO-J. . \(l that the. .Lndwi l l no t' h1rvriting

.ii. heexpef-:- . : iaphragm

I thesteady-::brnationo:r. il eflection.:: hadbetter

' iailuse hey

.. , J \ a result: : -r e la t ive ly.: ' . n if this-:-,tr' etbre

: : te ro t le- ' r t .Lrtment

.ri - Lrll tlllle. - . iJ here

.rnderlie

-. . dr. wO. : . : : - . .s no t

. . \ r one.

'.:J rroblem,. -:' r\e this

. - \ t l rge the r" 3 . lua lstc .

[),isiorrAcross Thin Film

L0 rg ed i f u s i o nc o e f f i c i e n t

z = Q z = z c z = l

::g. 1.2-.1.Concentration-dependentiffusion acrossa thin fiim. Above the concentration1, ,::e drffusioncoeffcient s small; belowthis critical value, t is larger.

'- j: tm. a steady-state assbalanceeads o the sameequation:

, d j tdz

- -;.ult.the fluxj1 is a constant verywheren the film. However, n the effhand fi lm- -': .oncentrationroduces small diffusioncoefficient:

dc,/ t - D ,a?.

. ::.ult iseasily ntegrated:

t . : f ,I i ta, - -D I dct

. t t t J , , u

- : the esul t

D, / r - ( c r o - c r . )z( .

--' risht-handilm,the concentrations small, and he diffusion coefficient s large:

t . .l r - D ?az .

D: , - ( c l r _ c u )L _ a a


14/37

/(r 2 / Difrusionn DiluteSolutions

The unknownposition2,.canbe found byrecognizinghat he fluxfilms:

It' : o r-r-t'l ( c 1 9 - c 1 , . )

Thefluxbecomes

is the sameacross oth

D(c ' ro c r, . ) D(c r,- c t t ). I l -

If thecriticalconcentrationquals he average f c1sandc11,henthe apparent ifTusioncoefficientwill bethe arithmeticaverage f the twodiffusioncoefficients.

In passing,we shouldecognizehat the concentrationrofile shownn Fig'2.2-4im'plicitlygivestheratioofhediffisioncoefflcients.hefluxacrossthefilmisconstantandis proportionalo the concentrationradient.Becausehe gradients largeron the eft, thedifTusionoeffcients smaller.Becausehe gradient s smalleron theright,thediffusioncoefficients larger. To testyourunderstandingf thispoint,you shouldconsiderwhatthe concentrationrofle will looklikeif the diffusioncoeffcientsuddenlydecreases stheconcentrationrops. Suchconsi


15/37

t n

. , n

: l l -

-, : ' lJ: |lr '

. ' t]

: : . 1 t' :\

' L . 1 1

. ifc'

- r l l

- , | a

. . , . I

- i I' , t l,

J . - l

: ' ,\o

2.2 Stead,-DiJJusionc'ross Thin FiLm 2l

I P 1 i P ? i

p 1 p ?

Loyer

P1,qos

Gosouts idethe body

P2, t i s sue

z=O

z= A z=lA+ lAFig. 2.2-5.Gas difiusion acrossskin. Thegas pressures hown are those n equilibriurn withthe actualconcentrations. n the specifccase consideredhere,gas 2 is morepermeable nlayer B,and gas I is morepermeable n layer A.The resultingtotal pressurecan have majorphysiologiceffcts.

Theseprofiles,which are shown n Fig. 2.2-5,mply why rashesorm inthe skin.Inparticular,hese raphsllustrate he ransportfgas I fromthe suffoundingsntothe issueandthe simultaneous iffusion ofgas 2 across he skin in the oppositedirection. GasIis morepermeablen layer A than n layer B;as a result, tspressure nd concentrationgradientsall lesssharply n layer Athan n layer B. The reverses true brgas2; it is morepermeablen layerB than n A.

Thesedifferentpermeabilitiesead o a totalpressurehat will have a maximum at thernterfaceetween he twoskin layers. This totalpressure, hown by the dotted ine inFig. 2.2-5,may exceed he surroundingpressure utside he skin and within the body. Ifit doesso,gasbubbleswill formaround heinterfacebetween he two skinlayers.Thesebubbles roducehemedicallyobserved ymptoms.Thus his conditions a consequencefunequal ifTusionor,moreexactly,unequalpermeabilities)cross iffrent ayersof skin.

The examples nthis sectionshow that diffusion across hin films can be diffcult tounderstand. hedifficulty doesnotderiverom mathematicalomplexity; he calculationrs easyand essentiallynchanged.The simplicity of the mathematicss the reasonwhyJrffusionacross hin filmstends o be discussed uperfcially n mathematically rientedbooks. Thedifculty in thin-filmdiffusion comes rom adapting he samemathematicsttr widely varyingsituationswith different chemicalandphysicaleffects. This is what isJifcultto understandbout hinfilmdiffusion. It is an understandinghatyoumustgainbefbreyoucan docreativeworkon hardermass ransferproblems.


16/37

28 2 / Diffusionn DiluteSolutions

2.3 UnsteadyDiffusionn a SemiinfinitelabWe now turnto a discussion f diffusionin a semiinfniteslab. We considera

volume ofsolutionhatstartsat an nterfaceandextends verylongway'Sucha solutioncan bea gas, iquid,or solid.We wantto find howtheconcentrationaries n thissolutionas a resultof a concentrationhangeat its interface.ln mathematicalerms,we want tofncl heconcentrationnd lux as unctionsof positionand ime'

Thistypeof mass ransfers oftencallediee diffusion(Gosting, 1956)simplybecausethis s brieferhan"unsteadydiffusionn a semiinfiniteslab."At firstglance,hissituationmayseem arebecause o solutioncanextendan nfinitedistance.Theprevious hin-filmexamplemademoresense ecausewe can hinkof manymore hinfilmsthansemiinfiniteslabs.Thus wemightconcludehat hissemiinfinitease s notcommon.Thatconclusionwouldbe a serious rrol.

The importantcaseof an infiniteslab s commonbecause nydiffusionproblemwillbehave s f theslab s infinitelyhickat shorlenoughimes.For example,maginehatoneof thethinmembranes iscussedn theprevioussectionseparateswo identicalsolutions,so that it initiallycontainsa soluteat constant oncentration.Everythings quiescent, tequilibrium.Suddenlyhe concentrationn the leffhandinterfaceof the membranesrased,as shown n Fig.2.3-1.Just after his suddenncrease,he concentrationear his

leftinterf'aceises apidlyon itsway [oa newsteadystate. nthese irstfewseconds,he

concentrationt therightinterfaceemainsunaltered,gnorantof the turmoilon theleft'The leftmightas wellbe infinitelyfar away;the membrane,or these irstfew seconds,mightas *"ll b.infinitelythick.Of course, t larger imes, he systemwillslitherntothesteidy-stateimit in Fig.2.3-l(c). But in those irstseconds,he membranedoesbehavelike a semiinfnitelab.

This examplepoints to animportantcorollary,whichstates hat cases nvolvinganinfiniteslab an


17/37

- 4 . l

. i - L r l l

. i 1 ' r l l' ' : I r )

. . , 1 l

: .Lnl. i l g

. . ' r l l

r l

. . ' i s, . : h is

J -a i t ., - I- . , u .

" j a n. . : 1 e s .

' j . i b' . 1 . t O

- Th e. i f l AS

::10ugh: sndent

: r l n we

1 .3 -

' ) .3-2)

l l l' n eI l .

2.3 UnsteadyDffision in a SemiinfiniteSlab 29

Concen ro t ion rof i le ino membroneol equi l ib r ium

Concent rofon profle s l igh yof e r heconcent rof ion nhe le f i s ro ised

I ncreose

L i m i i n gc o n c e n l r o l i o np r o f e o f l o r g e m e

Fig. 2.3- . Unste ady- versus steady-state iffusion. At smal l times, difTusionwill occur onlynear the lefrhand side of the membrane. As a result.at thesesmall times. the diffusion willbethe same as f the membrane wasinfinitely t hick. At large times, the results become those nthe thin fim.

Fig.2.3-2. Free diffusion. ln this case, heconcentration at thelefta higher constant value.Diflsion occurs n the region to the rightFig.2.2-1 are basic to most diffusionproblems.

is suddenly ncreased oThis case and that in


18/37

30 2 / Dilt'usion n Dilute Solutions

We divide by AA: to find

Dcr ( j t l ,+ t , - , r r.: - l - la t \ ( : + a : ) - : /

We then et A: go to zero and use he definitionof the derivative

^ ^ 1d C r d ' C t- D :At ':2

cr _ _ /r

3 t 3 :(2.3-4)

Combining this equationwithFick's law,and assuminghat the diffusion coefficientsindependentl 'concentralion.e get

(2.3-3)

(2.3-s)

(2.3-0)

tr' : "".

'rromr*':IIlr :".

q L U " *

Thisequations sometimes alledFick's second aw, and it is oftenreferred o as oneexampleof a"diffusionequation."n this case,t is subject othe following conditions:

/ : 0 , a l l , c l : c r e/ > 0 . z : 0 . c l : c t o

i : @ , c l : c l x

(2.3-6)(2.3-1)(2.3-8)

Notethatbothcl6Ddct0aretakenasconstants.heconcentrationcl-isconstantbecauseit is so fr fiom the interfceas o be unaffcted y events here; heconcentration 1e skept constant y addingmaterialat the nterface.

2.3.2 Mathematical Solution

Thesolutionof thisproblems easiest sing hemethodof "combinationof vari-ables."This method s easy o fbllow, but it must have been difficultto invent. Fourier,Graham,andFick failed n the attempt; t requiredBoltzman'sortured maginationBoltz-man,1894).

The trick to solving hisproblems to definea new variable

'/4Dt(2.-r-9)

The differentia l equation can then be w

d t , / ( \ d l , , / ; t . 1 :. I l : D , - ; t . ,t i 1 \ t / d ( - \ a z lor

, , ,d 'c r d r : t | 1 r - nLt< ' d

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Diffusion in Dilute Solutions