13

Chapter2PhysicalAdsorption

The literature pertaining to the sorption of gases by solids is now so vast that it isimpossible for any, except those who are specialists in the experimental technique,rightlytoappraisethework,whichhasbeendone,ortounderstandthemaintheoreticalproblemswhichrequireelucidation.J.E.LennardJones,19321

Adsorptionisthephenomenonmarkedbyanincreaseindensityofafluidnearthesurface,forourpurposes,ofasolid.*Inthecaseofgasadsorption,thishappenswhenmoleculesofthegasoccasiontothevicinityofthesurfaceandundergoan interactionwith it, temporarily departing from the gas phase.Molecules in this new condensedphase formedat the surface remain foraperiodof time,and then return to thegasphase.Thedurationofthisstaydependsonthenatureoftheadsorbingsurfaceandtheadsorptive gas, thenumberof gasmolecules that strike the surfaceand their kineticenergy (orcollectively, their temperature),andother factors (suchascapillary forces,surfaceheterogeneities,etc.).Adsorptionisbynatureasurfacephenomenon,governedbytheuniquepropertiesofbulkmaterialsthatexistonlyatthesurfaceduetobondingdeficiencies.*Adsorptionmayalsooccuratthesurfaceofaliquid,orevenbetweentwosolids.

14

The sorbent surface may be thought of as a twodimensional potential energy

landscape, dottedwithwells of varying depths corresponding to adsorption sites (asimplifiedrepresentation isshown inFigure2.1).Asinglegasmolecule incidentonthesurface collides in one of two fundamental ways: elastically, where no energy isexchanged, or inelastically,where the gasmoleculemay gain or lose energy. In theformer case, the molecule is likely to reflect back into the gas phase, the systemremainingunchanged. Ifthemolecule lackstheenergytoescapethesurfacepotentialwell, itbecomesadsorbed forsome timeand later returns to thegasphase. Inelasticcollisions are likelier to lead to adsorption. Shallow potential wells in this energylandscapecorrespond toweak interactions, forexamplebyvanderWaals forces,andthetrappedmoleculemaydiffusefromwelltowellacrossthesurfacebeforeacquiringthe energy to return to the gas phase. Inother cases,deeperwellsmay existwhichcorrespondtostrongerinteractions,asinchemicalbondingwhereanactivationenergyis overcome and electrons are transferred between the surface and the adsorbed

Figure2.1.Apotentialenergylandscapeforadsorptionofadiatomicmoleculeonaperiodictwodimensionalsurface.Thedepthoftheenergywellisshowninthezaxis.

z

15

molecule.Thiskindofwellishardertoescape,thechemicallyboundmoleculerequiringa much greater increase in energy to return to the gas phase. In some systems,adsorption isaccompaniedbyabsorption,wheretheadsorbedspeciespenetrates intothesolid.Thisprocess isgovernedbythelawsofdiffusion,amuchslowermechanism,andcanbereadilydifferentiatedfromadsorptionbyexperimentalmeans.

Intheabsenceofchemicaladsorption(chemisorption)andpenetrationintothebulkofthesolidphase(absorption),onlytheweakphysicaladsorption(physisorption)caseremains. The forces that bring about physisorption are predominantly the attractivedispersion forces (named so for their frequency dependent properties resemblingoptical dispersion) and shortrange repulsive forces. In addition, electrostatic(Coulombic)forcesareresponsiblefortheadsorptionofpolarmolecules,orbysurfaceswith apermanentdipole.Altogether, these forces are called vanderWaals forces,namedaftertheDutchphysicistJohannesDiderikvanderWaals.

2.1 VanderWaalsForcesAnearlyandprofoundlysimpledescriptionofmatter,theideallawcanbeelegantly

derivedbymyriadapproaches,fromkinetictheorytostatisticalmechanics.FirststatedbymileClapeyron in1834, itcombinesBoyles law (PV=constant)andCharless law(stating the linear relationship between volume and temperature) and is commonlyexpressedintermsofAvogadrosnumber,n:

Equation2.1

16

Thissimpleequationdescribesthemacroscopicstateofathreedimensionalgasof

noninteracting,volumelesspointparticles.Thegasconstant,R,isthefundamentallinkbetween microscopic energy and macroscopic temperature. While satisfactory fordescribingcommongasesat lowpressuresandhigh temperatures, it is ineffective forrealgasesoverawide temperatureandpressureregime.AbetterapproximationwasdeterminedbyvanderWaals,combining two importantobservations:2a) thevolumeexcludedbythefinitesizeofrealgasparticlesmustbesubtracted,andb)anattractiveforcebetweenmoleculeseffectsadecreasedpressure.Thesuggestionofanexcludedreal gas volume was made earlier by Bernoulli and Herapath, and confirmedexperimentally by Henri Victor Regnault, but the attractive interactions betweenmoleculeswastheimportantcontributionbyvanderWaals.Thechangeinpressuredueto intermolecular forces is taken to be proportional to the square of themoleculardensity,givingthevanderWaalsequationofstate:

Equation2.2

Atthetime,vanderWaalswasadamantthatnorepulsive forcesexistedbetweenwhathe reasonedwere hard sphere gasparticles. Interestingly, itwas JamesClerkMaxwellwhocompletedandpopularizedvanderWaals(thenobscure)workinNature3and also who later correctly supposed thatmolecules do not in fact have a hardspherenature.Nevertheless, thesumof theattractiveand repulsive forcesbetweenatomsormoleculesarenowcollectivelyreferredtoasvanderWaals forces.Forces

17

betweenanyelectricallyneutralatomsormolecules(therebyexcludingcovalent,ionic,andhydrogenbonding)conventionally fall into thiscategory.Together, these include:Keesom forces (betweenpermanentmultipoles),Debye forces (betweenapermanentmultipoleandan inducedmultipole),Londondispersion forces (between two inducedmultipoles),andthePaulirepulsiveforce.2.1.1 IntermolecularPotentials

Inhistime,therangeofvanderWaalsattractiveinteractionwaspredictedtobeofmolecular scale but the form of the potential as a function of distance, U(r), wasunknown. There was unified acceptance that the attraction potential fell off withdistance as r,with > 2 (the value for that of gravitation),but the valueofwasactivelydebated.In parallelwith the effort to determine this potential, a noteworthy advance in thegeneraldescriptionofequationofstatewasthevirialexpansionbyH.K.Onnes:4

1

Equation2.3

Perhapsmostimportantly,thisdescriptionsignifiedarealizationoftheunlikelihoodthat all gases could be accurately described by a simple closed form of equation.Additionally, the second virial coefficient, B(T), by its nature a firstneighbor term ofinteraction, sheds insight on the attractive potential between molecules. A majorbreakthrough followed, culminating in the theory now attributed to Sir John EdwardLennardJones,anEnglish theoreticalphysicist.Hisdescriptionof thepotentialenergy

18

betweentwointeractingnonpolarmoleculesused=6andarepulsivetermoforder12:5,6

Equation2.4

TheresultconstitutesabalancebetweenthelongerrangeattractivevanderWaalspotential (of order r6)with the shortrange repulsive potential arising from electronorbital overlap (of order r12), described by the Pauli exclusion principle, and is alsoreferredtoasa612potential. Itapproximatesempiricaldataforsimplesystemswithgratifying accuracy, and has the added advantage that it is computationally efficientsince r12 is easily calculated as r6 squared, an important consideration in its time.ArepresentativeplotofthispotentialisshowninFigure2.2.

Figure2.2.TheLennardJones612potential,scaledinunitsofU0,thedepthofthewell.The

equilibriumdistancebetweentheinteractingspeciesisr=r0.

19

Atfardistances(r>>r0),themagnitudeoftheinteractionpotentialisnegligible.Ifthespeciesbecomespontaneouslyclosertogether(e.g.,astheresultofrandomcollision),itisfavorableforthemtoremainataspecificdistanceapart,namelyr=r0.Atveryshortdistance,thePaulirepulsiontermdominatesandthesystemisunfavorable.

Anumberofsimilarpotentialformsaroseshortlyafterward(usingvariousformsforthePaulirepulsiontermsuchasanexponentialorotherpowerofordernear12),andanearlytriumphofthesemodelswasaccuratelyfittingthesecondvirialcoefficientofsimplegases,suchasheliumforwhichtheequilibriumHeHedistancewascalculatedtober0=2.9in1931.Thisvalueremainsaccuratewithin2%today.

Coincidentally,LennardJonesworkwasoriginallyundertakentoattempttoexplainapuzzlingobservationmadeduringvolumetricadsorptionmeasurementsofhydrogenonnickel,7,8showingtwodistinctcharacteristicbindingpotentialsassociatedwithdifferenttemperatureregimes.Thiswouldleadtothefirstexplanationofthedifferentiablenatureofadsorptionatlowandhightemperature(nowreferredtoseparatelyasphysisorptionandchemisorption,respectively).Asaresultofitssuccess,theLennardJonesinverseseventhpowerforce(fromtheinversesixthpowerpotential)becamethebackboneofadsorptiontheory.2.1.2 DispersionForces

Ther6LennardJonespotentialwasderivedfromfirstprinciplesandfirstexplainedcorrectly by the German physicist Fritz London;9, 10 hence, the attractive force thatoccursbetweenneutral,nonpolarmoleculesiscalledtheLondondispersionforce.Itisaweak, longrange,nonspecific intermolecular interactionarising fromthe induced

20

polarizationbetweentwospecies,resultingintheformationofinstantaneouselectricalmultipole moments that attract each other. Arising by rapid, quantum inducedfluctuationsof theelectrondensitywithinamoleculeoratom (the reason theywerecoinedasdispersive forcesbyLondon), the force isstrongerbetween largerspeciesdue to those species increased polarizability. The key to the correct explanation ofdispersion forces is inquantummechanics;without theuncertaintyprinciple (and thefundamental quantummechanical property of zeropoint energy), two sphericallysymmetric specieswithnopermanentmultipole couldnot influence a forceon eachotherandwouldremainintheirclassicalrestposition.Thesubjectofdispersionforcesisimportanttomanyfields,andathoroughoverviewoftheirmoderntheorycanbefoundelsewhere.11

Thephysisorptionofnonpolarmoleculesoratomsonanonpolarsurface(aswellastheir liquefaction) occurs exclusively by dispersion forces. Dispersion forces are alsoessential forexplaining the totalattractive forcesbetweenmultipolarmolecules (e.g.,H2) forwhich typical staticmodels of intermolecular forces (e.g., Keesom or Debyeforces)accountforonlyafractionoftheactualattractiveforce.Theexistenceofnobleliquids (liquefiednoblegases) isa fundamentalverificationofdispersion forcessincethere isnootherattractive intermolecular forcebetweennoblegasatoms thatcouldotherwiseexplaintheircondensation.2.1.3 ModernTheoryofPhysicalAdsorption

Despite theircorrectexplanationover80yearsago,dispersion forcesarenotwellsimulatedby typical computationalmethods, such asdensityfunctional theorywhich

21

cannot accurately treat longrange interactions inweakly bound systems.12 The firstprinciplesmethods that are dependable are computationally intensive and are oftenforegoneforempiricalpotentialssuchasaLennardJonespotentialasdescribedintheprevious section.13, 14 For this reason, the abinitio guidance of the design ofphysisorptivematerialshasbeenmuch lessthanthatforchemisorptivematerials,andwasnotacomponentoftheworkdescribedinthisthesis.

22

2.2 GasSolidAdsorptionModels

A thermodynamic understanding of adsorption can be achieved by describing asimplified system, and a small subset of important models will be discussed. Theconstituentchemicalspeciesofthesimplestsystemareapuresolid,indexedass,andasinglecomponent adsorptive gas, indexed as a in the adsorbed phase, g in the gasphase,orxifitisambiguous.Thesystemisheldatfixedtemperatureandpressure.Westartwiththefollowingdescription:

(i) theadsorptivedensity,x,iszerowithinanduptothesurfaceofamaterial,(ii) at thematerial surface and beyond, x is an unknown function of r, the

distancefromthesurface,and(iii) atdistancefromthesurface,xisequaltothebulkgasdensity,g.

Figure2.3.Asimplifiedrepresentationofagassolidadsorptionsystem(left)andanonadsorbingreferencesystemofthesamevolume(right).Adsorptivedensity(green)isplotted

asafunctionofr.

23

Aschematicofthissystem isshown inFigure2.3 (left).The functional formofthe

densificationatthesorbentsurface,andthethicknessof theadsorption layerarenotpreciselyknown.However,theadsorbedamountcanbedefinedasthequantityexistinginmuchhigherdensitynearthesurface,andiseasilydiscernedwhencomparedtothereference case of a nonadsorptive container, shown in Figure 2.3 (right).Wemayassume that thegaspressure,Pg, isequal to the totalhydrostaticpressure,P,of thesystematequilibrium,whichisconsistentwithordinaryexperimentalconditions.2.2.1 MonolayerAdsorption

Thesimplestrepresentationofanadsorbedphaseisasanidealgas,constrainedtoatwodimensionalmonolayerwherethereisnointeractionbetweenadsorbedmolecules:

Equation2.5

Here,PaisthespreadingpressureoftheadsorptionlayerandAaisitsareaofcoverage.In the system described, the surface area for adsorption is fixed, as well as thetemperature.Ifwetakethespreadingpressureasproportionaltothatofthegasphaseinequilibriumwithit,wefindthattheamountadsorbedisalinearfunctionofpressure:

Equation2.6This relationship, called Henrys law, is the simplest description of adsorption, andapplies to systems at low relative occupancy. When occupancy increases, therelationshipbetweenthepressuresoftheadsorbedphaseandthegasphasebecomesunknown,andamoregeneraltreatmentisnecessary.

24

A firstapproximationofan imperfect twodimensionalgasmaybemadebyadaptingthevanderWaalsequationofstatetotwodimensions:

Equation2.7Forsimplicity,itisconvenienttodefineafractionalcoverageofthesurfaceavailableforadsorption,,asthenumberofadsorbedmoleculespersurfacesite,aunitlessfractionthatcanalsobeexpressedintermsofrelativesurfacearea:

Equation2.8Withthisdefinition,andamoreelegantdescriptionofthespreadingpressureofthevanderWaalstwodimensionalphase,theHilldeBoerequationisderived:

1

Equation2.9Thisequationhasbeen shown tobe accurate for certain systems,up to amaximumcoverage of = 0.5.15 Ultimately though, treatment of the adsorbed phasewithoutreferencetoitsinteractionwiththesurfaceencountersdifficulties.

Amoreinsightfulapproachistotreatadsorptionanddesorptionaskineticprocessesdependent on an interaction potential with the surface, also seen as an energy ofadsorption.Indynamicequilibrium,theexchangeofmassbetweentheadsorptionlayerandthebulkgasphasecanbetreatedbykinetictheory;atafixedpressure,therateof

25

adsorption and desorption will be equal, resulting in an equilibrium monolayercoverage.Thisequilibriumcanbedescribedbythefollowingscheme:

The rate (where adsorption is taken to be an elementary reaction) is given by the(mathematical) product of the concentrations of the reactants, ci, and a reactionconstant,K(T):

Equation2.10

Correspondingly,forthereverse(desorption)reaction:

Equation2.11We canassume that thenumberofadsorption sites is fixed, so the concentrationofadsorbed molecules and empty adsorption sites is complementary. To satisfyequilibrium,we setEquations2.10 and2.11equal andexpress them in termsof thefractionaloccupancy,recognizedasequivalenttoca:

1

1

Equation2.12To express the temperature dependence of the reaction constants, we can use anArrheniustypeequation:

26

Equation2.13Theratiooftheadsorptionanddesorptionconstantsis:

Ifthegasphaseisassumedtobeideal,itsconcentrationisproportionaltopressure,P.Secondly, ifweassume that theenergyof theadsorbedmolecule,Ea, is the sameatevery site, and the change in energy upon adsorption,E, is independent of surfacecoverage,theresultisLangmuirsisothermequation:

1 Equation2.14

TheLangmuirisotherm,inthecontextofitsinherentassumptions,isapplicableovertheentire range of . Plots of multiple Langmuir isotherms with varying energies ofadsorptionare shown inFigure2.4 (right). In the limitof lowpressure, the LangmuirisothermisequivalenttoHenryslaw:

lim

1 1

Equation2.15Theplots inFigure2.4showthetemperatureandenergydependenceoftheLangmuirisothermequation.With increasing temperature, theHenrys law region ismarkedbymoregradualuptake,asimilartrendasfordecreasingenergyofadsorption;bothtrendsareconsistentwithexperiment(insystemsreferredtoasexhibitingtypeIisotherms).

27

Althoughitenjoysquantitativesuccessincharacterizingadsorptioninlimitedrangesofpressure for certain systems, conformity of experimental systems to the Langmuirequationisnotpredictable,anddoesnotnecessarilycoincidewithknownpropertiesofthe system (such as expected homogeneity of adsorption sites, or known adsorbateadsorbate interactions).16Nosystemhasbeen foundwhichcanbecharacterizedbyasingle Langmuir equation with satisfactory accuracy across an arbitrary range ofpressure and temperature.15 Langmuirsmodel assumes: 1) the adsorption sites areidentical,2)theenergyofadsorptionisindependentofsiteoccupancy,and3)boththetwo and threedimensional phases of the adsorptive arewell approximated as idealgases. Numerous methods have been suggested to modify the Langmuir model togeneralizeitsassumptionsforapplicationtorealsystems.

Figure2.4.Langmuirisothermsshowingthedependenceofadsorptionsiteoccupancywithpressure,varyingtemperature(left)andenergyofadsorption(right).

28

For heterogeneous surfaces with a distribution of adsorption sites of different

characteristicenergies,onepossibility is to superimposea setofLangmuirequations,eachcorrespondingtoadifferenttypeofsite.ThisgeneralizedLangmuirequationcanbewrittenas:

1

1

Equation2.16Theweightofeachcomponent isotherm,i,corresponds to the fractionof siteswiththe energy Ei. This form of Langmuirs equation is applicable across awide range ofexperimental systems, and can lend insight into the heterogeneity of the adsorbentsurfaceandtherangeofitscharacteristicbingingenergies.Itstreatmentofadsorptionsitesasbelongingtoatwodimensionalmonolayercanbemadecompletelygeneral;theadsorptionlayercanbeunderstoodasanysetofequallyaccessiblesiteswithintheadsorption volume, assuming the volume of gas remains constant as site occupancyincreases (anassumptionthat isvalidwhena>>g).TheapplicationofthisequationforhighpressureadsorptioninmicroporousmaterialsisdiscussedinSection2.4.6.2.2.2 MultilayerAdsorptionApracticalshortcomingofLangmuirsmodelisindeterminingthesurfaceareaavailableforadsorption.Typically, fittingexperimentaladsorptiondata toa Langmuirequationresults in an overestimation of the surface area.16 The tendency of experimentalisotherms toward a nonzero slope in the region beyond the knee indicates that

29

adsorptionoccurs in twodistinctphases:aprimaryphase,presumablyadsorptiononhomogeneous adsorption sites at the sorbent surface, and a secondary phasecorrespondingtoadsorptioninlayersbeyondthesurfaceoftheadsorbent.

An adaptation of Langmuirs model can be made which accounts for multiple,distinctlayersofadsorption.Acharacteristicofadsorptioninamultilayersystemisthateach layercorresponds toadifferenteffectivesurface,and thusadifferentenergyofadsorption.Thedistinctionbetweenthemultilayerclassofsystemsandageneralizedmonolayer system (such as in Equation2.16) lies in thedescriptionof the adsorbentsurface; even a perfectly homogeneous surface is susceptible to amultiple layers ofadsorption, an important consideration at temperatures and pressures near thesaturation point. Brunauer, Emmett, and Teller successfully adapted the Langmuirmodel tomultilayeradsorptionby introducinganumberofsimplifyingassumptions.17Most importantly, 1) the second, third, and ith layers have the same energy ofadsorption,notably thatof liquefaction,and2) thenumberof layersas thepressureapproaches the saturation pressure, P0, tends to infinity and the adsorbed phasebecomesaliquid.Theirequation,calledtheBETequation,canbederivedbyextendingEquation 2.12 to i layers,where the rate of evaporation and condensation betweenadjacentlayersissetequal.Theirsummationleadstoasimpleresult:

1 1

Equation2.17

30

If the Arrhenius preexponential factors A1/A2 and A2/Al (from Equation 2.13) areapproximatelyequal,atypicalassumption,16theparametercBETcanbewritten:

Equation2.18Thechange inenergyuponadsorption inanyofthe layersbeyondthesurface layer istakenastheenergyofliquefaction,andthedifferenceintheexponentinEquation2.18is referred to as the netmolar energy of adsorption. Strictlywithin BET theory, noallowanceismadeforadsorbateadsorbateinteractionsordifferentadsorptionenergiesatthesurface,assumptionsthatpreventtheBETmodelfrombeingappliedtogeneralsystemsandacrosslargepressurerangesofadsorption.

TheBETequationwasspecificallydevelopedforcharacterizingadsorption inagassolidsystemnearthesaturationpointoftheadsorptivegasforthepracticalpurposeofdeterminingthesurfaceareaofthesolidadsorbent.Despite itsnarrowsetofdefiningassumptions, itsapplicability toexperimental systems is remarkablywidespread.Asaresult, it is themostwidely applied of all adsorptionmodels, even for systemswithknowninsufficienciestomeettherequiredassumptions.Nitrogenadsorptionat77Kupto P0 = 101 kPa has become an essential characterization technique for porousmaterials,and theBETmodel iscommonlyapplied todetermine thesurfaceavailableforadsorption,calledtheBETsurfacearea.EvenformicroporousadsorbentswheretheBET assumptions are highly inadequate, this is the most common method fordeterminationofspecificsurfacearea,and itsdeficienciesaretypicallyassumedtobesimilarbetweencomparablematerials.

31

The BET surface area of a material can be determined by fitting experimental

adsorption data to Equation 2.17 and determining the monolayer capacity, nsites.Typically,N2adsorptiondataareplotted in the formof the linearizedBETequation,arearrangementofEquation2.17:

1 1

1 1

1

1

Equation2.19TheBETvariable,BBET, isplottedasa functionofP/P0and if thedataare satisfyinglylinear,theslopeand interceptareusedtodeterminensiteswhich isproportionaltothesurfaceareaasstatedinEquation2.8(usingAsites=16.22)16.Inpractice,thelinearityoftheBETplotofanitrogen isothermat77Kmaybe limitedtoasmallrangeofpartialpressure.Anacceptedstrategyistofitthelowpressuredatauptoandincludingapointreferred toaspointB, theendof thecharacteristickneeand thestartof the linearregioninaTypeIIisotherm.15,162.2.3 PoreFillingModels

TheLangmuirmodel,anditssuccessiveadaptations,weredevelopedinreferencetoan idealizedsurfacewhichdidnothaveanyconsiderationsformicrostructure,suchasthepresenceofnarrowmicropores,which significantlychange the justifiabilityof theassumptionsofmonolayerandmultilayeradsorption.Whilenumerousadvanceshave

32

beenmade in adsorption theory since theBETmodel, anotable contributionwasbyDubininandcollaboratorstodevelopaporefillingtheoryofadsorption.Theirsuccessatovercoming the inadequacies of layer theories at explaining nitrogen adsorption inhighlymicroporousmedia,especiallyactivatedcarbons,ishighlyrelevant.TheDubininRadushkevich(DR)equation,describingthefractionalporeoccupancypore,isstatedas:

WW e

Equation2.20

The DR equation can be used for determining DR micropore volume, W0, in ananalogous way as the Langmuir or BET equation for determining surface area. ItssimilaritiestoBETanalysesarediscussedinAppendixD.

33

2.2.4 GibbsSurfaceExcess

Measurementsofadsorptionnearorabove thecriticalpoint,eitherbyvolumetric(successivegasexpansions intoanaccuratelyknownvolumecontainingtheadsorbent)orgravimetric(gasexpansionsintoanenclosedmicrobalancecontainingtheadsorbent)methods, have the simple shortcoming that they cannot directly determine theadsorbedamount.This isreadilyapparentathighpressureswhere it isobserved thatthemeasured uptake amount increases up to amaximum and then decreaseswithincreasing pressure. This is fundamentally inconsistent with the Langmuirmodel ofadsorption,whichpredictsamonotonicallyincreasingadsorptionquantityasafunctionofpressure.The reason for thediscrepancycanbe traced to the finitevolumeof theadsorbedphase,showninFigure2.3.Thefreegasvolumedisplacedbytheadsorbedphasewouldhavecontainedanamountofgasgivenbythebulkgasdensityevenintheabsence of adsorption. Therefore, this amount is necessarily excluded from themeasured adsorption amount since the gas density, measured remotely, must besubtractedfromintheentirevoidvolume,suchthatadsorptioniszerointhereferencestatewhichdoesnothaveanyadsorptionsurface.

In the landmark paper of early thermodynamics, On the Equilibrium ofHeterogeneous Substances,18 Josiah Willard Gibbs gave a simple geometricalexplanationoftheexcessquantityadsorbedattheinterfacebetweentwobulkphases,summarizedforthegassolidcaseinFigure2.5.Theabsoluteadsorbedamount,showninpurple, issubdivided intotwoconstituents:referencemoleculesshown in lightblue,existingwithintheadsorbed layerbutcorrespondingtothedensityofthebulkgasfar

34

from thesurface,andexcessmoleculesshown indarkblue, themeasuredquantityofadsorption. The Gibbs definition of excess adsorption, ne, as a function of absoluteadsorption,na,is:

, Equation2.21

Any measurement of gas density far from the surface cannot account for theexistenceof the referencemoleculesnear thesurface; thesewouldbepresent in thereference system. The excess quantity, the amount in the densified layer that is inexcessofthebulkgasdensity,istheexperimentallyaccessiblevalue.Itissimpletoshowthattheexcessuptakeisapproximatelyequaltotheabsolutequantityatlowpressures.As the bulk gas density increases, the difference between excess and absoluteadsorption increases.Astatemaybereachedathighpressure,P3 inFigure2.5,wheretheincreaseinadsorptiondensityisequaltotheincreaseinbulkgasdensity,andthustheexcessquantity reachesamaximum.Thispoint is referred toas theGibbsexcessmaximum. Beyond this pressure, the excess quantity may plateau or decline, aphenomenonreadilyapparentinthehighpressuredatapresentedinthisthesis.

35

Figure2.5.TheGibbsexcessadsorptionisplottedasafunctionofpressure(center).Amicroscopicrepresentationoftheexcess(lightblue),reference(darkblue),andabsolute(combined)quantitiesisshownatfivepressures,P1P5.Thebulkgasdensity(gray)isafunctionofthepressure,anddepictedasaregularpatternforclarity.Thevolumeofthe

adsorbedphase,unknownexperimentally,isshowninpurple(top).

P1P2P3

P4 P5

VsVadsVg

36

2.3 AdsorptionThermodynamics2.3.1 GibbsFreeEnergyAdsorptionisaspontaneousprocessandmustthereforebecharacterizedbyadecreaseinthetotalfreeenergyofthesystem.Whenagasmolecule(oratom)isadsorbedonasurface,ittransitionsfromthefreegas(withthreedegreesoftranslationalfreedom)tothe adsorbed film (with two degrees of translational freedom) and therefore losestranslational entropy. Unless the adsorbed state is characterized by a very largeadditionalentropy(perhapsfromvibrations),itfollowsthatadsorptionmustalwaysbeexothermic(Hads

37

instantaneously adjusted the pressure of the system at a location remote to theadsorbentsurfacebyaddingdngmoleculesofgas.Theadsorbedphaseandthebulkgasphase are not in equilibrium, andwill proceed to transfermatter in the direction oflower freeenergy, resulting inadsorption.Whenequilibrium is reached, thechemicalpotentialofthegasphaseandtheadsorbedphaseareequal,since:

0

Equation2.24

Determining the change in chemical potential, or the Gibbs free energy, of theadsorptive species as the system evolves toward equilibrium is essential to afundamental understanding of adsorptive systems for energy storage or otherengineering applications.We develop this understanding through the experimentallyaccessiblecomponentsoftheGibbsfreeenergy:HadsandSads.2.3.2 EntropyofAdsorptionWetakethetotalderivativeofbothsidesofEquation2.24,andrearrangetoarriveattheclassicClausiusClapeyronrelation:

Equation2.25

where:

, ,

38

Thespecificvolumeof the freegas isusuallymuchgreater than thatof theadsorbedgas,andarobust19approximationsimplifiestherelation:

Equation2.26

Werearrangetoderivethecommonequationforchangeinentropyuponadsorption:

S

Equation2.27

2.3.3 EnthalpyofAdsorptionAtequilibrium,thecorrespondingchangeinenthalpyuponadsorptionis:

H S

Equation2.28Werefertothisasthecommonenthalpyofadsorption.Tosimplifyfurther,wemustspecifytheequationofstateofthebulkgasphase.Theidealgaslawiscommonlyused,a suitable equation of state in limited temperature and pressure regimes for typicalgases(thelimitationsofwhichareinvestigatedinSection2.4.1).Foranidealgas:

H

Equation2.29

39

ThisisoftenrearrangedinthevantHoffform:

H ln 1

Equation2.30

Theisostericheatofadsorption,acommonlyreportedthermodynamicquantity,isgivenapositivevalue:*

q H ln 1

0

Equation2.31

Foradsorptionattemperaturesorpressuresoutsidetheidealgasregime,thedensityofthebulkgasphaseisnoteasilysimplified,andweusethemoregeneralrelationship:

q

Equation2.32

If theexcess adsorption,ne, is substituted for the absolutequantity in theequationsabove,theresultiscalledtheisoexcessheatofadsorption:

q

Equation2.33

*Itistypicaltoreporttheisostericenthalpyofadsorptioninthiswayaswell,andforourpurpose,increasingenthalpyreferstotheincreaseinmagnitudeoftheenthalpy,ormostspecifically,theincreaseintheheatofadsorption.

40

It is common to include the ideal gas assumption in the isoexcessmethod since theassumptionsarevalidinsimilarregimesoftemperatureandpressure,giving:

q ln 1

Equation2.34

For caseswhere the adsorptive is in a regime far from ideality, it is sometimesnecessary touseabetterapproximation for the change inmolar volume specified inEquation 2.26. For example, gaseous methane at high pressures and nearambienttemperatures (nearcritical) has amolar volume approaching that of liquidmethane.Therefore,amoregeneralequationmustbeused,andherewesuggesttoapproximatethemolarvolumeoftheadsorbedphaseasthatoftheadsorptiveliquidat0.1MPa,vliq:

Equation2.35

Thisgivesthefollowingequationsfortheisostericandisoexcessheatofadsorption:

q

Equation2.36

q

Equation2.37

41

Insummary:

Assumptions: IsostericMethod IsoexcessMethod( )

IdealGas

S ln 1

q ln1

Eq.2.31

S ln 1

q ln 1

Eq.2.34

NonidealGas

S

q

1Eq.2.32

S

q

1Eq.2.33

NonidealGas

S

q

Eq.2.36

S

q

Eq.2.37

TheHenrys law value of the isosteric enthalpy of adsorption, H0, is calculated byextrapolationoftheadsorptionenthalpytozerouptake:

H lim HEquation2.38

42

A common technique fordetermining theenthalpyof adsorption in the ideal gas

approximationistoplotlnPasafunctionofT1,avantHoffplot,andtofindtheslopeofthelinealongeachisostere(correspondingtoafixedvalueofn).IfthedataarelinearinacertainrangeofT 1, theenthalpyofadsorption isdetermined tobe temperatureindependentinthatrange,correspondingtotheaveragetemperature:

2 1 1

Equation2.39

Iftheslope isnonlinear,asubsetofthedatabetweentwotemperatures,T1andT2, isfoundwheretheslopeisapproximatelylinear.Adifferenceof10Kisconsideredtobeacceptableformostpurposes.15Insidethiswindow,theaveragetemperatureis:

2 1 1

Equation2.40

Witha seriesofwindows,a temperaturedependenceof thedatacanbedeterminedwithintherangeoftemperaturescollected.

43

2.4 ThermodynamicCalculationsfromExperimentalData

Simplicityandaccuracyaredesired insolvingthethermodynamicrelationsderivedinSection2.3forcalculatingtheentropyandenthalpyofadsorptionfromexperimentaldata.The limitationsof the twomost commonlyapplied simplifications, the idealgasandisoexcessassumptions,arediscussedinSections2.4.12.

The primary obstacles to a completely assumptionless derivation of thethermodynamicquantitiesof interestaretwofold.Most importantly,boththe isostericand isoexcess treatments require tabulations of the equilibrium pressure, or ln P, atfixedvaluesofuptake,naorne.Experimentally, it ispossibletocontrolPandmeasurethe amount adsorbed, but exceedingly difficult to fix the adsorbed quantity andmeasure the equilibrium pressure. Therefore, a fitting equation is often used tointerpolatethemeasuredvalues.The interpolationofadsorptiondata isverysensitivetothefittingmethodchosen,andsmalldeviationsfromthetruevaluecausesignificanterrors in thermodynamiccalculations.2022Sections2.4.36discuss theuseof thedatawith andwithout a fitting equation, and compare the results across three types ofmodelindependent fitting equations in an effort to determine the most accuratemethodologyforcalculatingthermodynamicquantitiesofadsorption.

Secondly,thedeterminationoftheabsolutequantityofadsorptionrequiresamodelwhich defines the volume of the adsorbed phase as a function of pressure andtemperatureinthesystem.Numerousmethodshavebeensuggestedtodefineit,andasuccessfulmodelispresentedinSection2.5.

44

Thedatausedforthecomparisonsismethaneadsorptiononsuperactivatedcarbon

MSC30, a standard carbon material whose properties are thoroughly discussed inChapters46.Itisawellcharacterizedmaterialwithtraditionalsorbentpropertiesandalargesurfacearea(givingalargesignaltonoiseratioinmeasurementsofitsadsorptionuptake).Thedatasetconsistsof13isothermsbetween238521K,andspanspressuresbetween0.059MPa,asshowninFigure2.6.

Figure2.6.EquilibriumadsorptionisothermsofmethaneonMSC30between238521K,the

testdataforthermodynamiccalculationsofadsorption.

45

2.4.1 IdealGasAssumption

Theidealgaslawisdefinedbytwomainapproximations:thatthegasmoleculesdonotinteractorhaveinherentvolume.Inthelimitofzeropressure,allgasestendtowardideality since they are dilute enough that interactions are improbable and the totalvolumeofthesystem isapproximatelyunchangedby includingthemolecules in it.Forhydrogen, the ideal gas law holds approximately true for a large pressure andtemperatureregimearoundambient;forexample,theerrorindensityisonly6%evenat 10 MPa and 298 K (see Figure 2.7a). However, at low temperatures and highpressures that are desired inmany adsorption applications, nonideality of the gasphase isstrikinglyapparent formostcommongases.Figure2.7bshowsthedensityofmethane as a function of pressure at various temperatures of interest for storageapplications. The very significant nonideality of both hydrogen and methane isapparent at room temperature and elevated pressures. In addition, each gas showssignificant nonideality in their respective low temperature regimes of interest forstorageapplications,evenat lowpressures.This isan issue formostadsorptivegasessincethelatentheatofphysicaladsorptionisveryclosetothatofliquefaction.

ThevanderWaalsequationofstatepredictsadensity forhydrogenandmethanethatisgenerallyanacceptableapproximationofthetruedensityatpressuresupto10MPa,asshowninFigure2.8.Forhydrogenat77K,theerrorindensityislessthan0.4%between 05 MPa, an acceptable figure for isosteric heat calculations, and muchimprovedcomparedtothe4.5%errorintheidealgasdensity.

46

Ultimately, realgasesexhibitproperties thatcannotbeaccuratelymodeled inour

pressureandtemperatureregimeof interestbyanysimplemeans.Themostaccuratepure fluidmodel available at this time is the32termmodifiedBenedictWebbRubin(mBWR) equation of state.23 For this work, we refer to the mBWR model, asimplementedbytheREFPROPstandardreferencedatabase,24astherealgasequationofstate.

Figure2.7.Acomparisonofidealgasdensity(dottedlines)to(a)hydrogenand(b)methaneatvarioustemperatures,andpressuresupto100MPa.

Figure2.8.Acomparisonoftheidealgas,vanderWaals(vdW)gas,andreal(mBWR)gasdensityofmethaneandhydrogenat298Kbetween0100MPa.

47

2.4.2 IsoexcessAssumption

For thermodynamic or other calculations from experimental adsorption data atsufficiently low pressures, such as in determining BET surface area, the measured(excess)quantityofadsorption(fromEquation2.21)sufficesforapproximatingthetotaladsorptionamount.However,thisassumptionquicklybecomesinvalidevenatrelativelymodest pressures and especially at low temperatureswhere the gas density is highcomparedtothedensityoftheadsorbedlayer.20,21

Nevertheless, the simplest approach to thermodynamic calculations usingexperimentalisothermdataistoproceedwiththemeasuredexcessuptakequantity,ne,in place of the absolute adsorption quantity, na. Since the volume of adsorption isfundamentally unknown, amodel is required to determine the absolute adsorptionamount,a task that isbeyond the scopeofmanyadsorption studies.This shortcut isvalid in the low coverage limit since, from Equation 2.21, if the pressure of the gasapproacheszero:

lim lim 0 Equation2.41

ThisapproachcanthereforebeeffectiveforapproximatingtheHenryslawvalueof the isosteric enthalpy of adsorption. However, it leads to significant errors indetermining thedependenceof theenthalpyonuptake (discussed further in Section2.4.35)andanyisoexcessenthalpyofadsorptionvaluescalculatedatcoverageshigherthanne/nmax=0.5shouldonlybeacceptedwithgreatcaution.

48

2.4.3 CalculationsWithoutFitting

Controlling the pressure of experimentalmeasurements is possible, but it provesvery difficult to perform experiments at specific fixed quantities of uptake. Bycoincidence, such as in a largedata set like theoneused in this comparison, itmayhappen that numerous data points lie at similar values of excess adsorption, andanalysisof the isostericenthalpyofadsorptionmaybeperformedwithout fitting.ThevantHoffplotofthesecoincidentallyalignedpointsisshowninFigure2.9.

Figure2.9.ThevantHoffplotofthe(unfitted)methaneuptakedataatparticularvaluesofexcessadsorptiononMSC30.Thedashedcoloredlines,frombluetoorange,indicatethe

temperaturesoftheisothermsfrom238521K,respectively.

49

Sincetheexperimentalquantityisexcessadsorption,theisoexcessmethodmustbe

used; the resulting isoexcess enthalpy of adsorption is shown in Figure 2.10. Thismethodisseverelylimitedsinceitreliesonthestatisticalprobabilityoftwodatapointslyingon thesame isostere in thevantHoffplot.This likelihood isdramatically lessathigh uptake where only low temperature data are available, and requires intenseexperimental effort to be accurate. The fixed values of ne are necessarilyapproximated (in thiscase,0.05mmolg1)and theabsoluteadsorbedamount isnotused, resulting in a significant, unphysical divergence at high values of uptake.Nonetheless, without any fitting equation or model, we achieve an acceptabledetermination of the Henrys law value of the enthalpy ofmethane adsorption onMSC30(asanaverageovertheentiretemperaturerange):15kJmol1atTavg=310K.

Figure2.10.TheaverageisoexcessenthalpyofadsorptioncalculatedovertheentiretemperaturerangeofdatashowninFigure2.9,238521K.

50

2.4.4 LinearInterpolation

A primitive method for fitting the experimental adsorption data is by linearinterpolation. Thismethod does not require any fitting equation and is the simplestapproach to determining values of P at specific fixed values of ne to be used in theisoexcessmethod.TheMSC30datasetfittedby linear interpolationandtheresultingvantHoffplotisshowninFigure2.11.Thedataareoftenapparentlylinearforanalysisacrosstheentirerangeofne,butarelimitedtothevaluesofnewheretherearemultiplemeasured points (favoring themeasurements at low uptake and low temperature).Additionally,athighvaluesofnethe isoexcessapproximationbecomes invalid,ornena,anderrorsbecomelargerasneincreases.

Tocalculatetheenthalpyofadsorption,the isoexcessmethod isusedbecausetheabsolute adsorption is unknown; the results are shown in Figure 2.12. The averageenthalpyofadsorptionovertheentiredataset(whereTavg=310K)isplottedatthetop.The temperature dependent results are plotted at top, middle, and bottom, usingwindowsofsizes3,5,and7temperatures,respectively.Thecolorsfrombluetoorangerepresent the average temperatures of each window, from low to high. Twoobservationsmaybemade:thereisnosignificanttrendofthetemperaturedependenceofthedata,andincreasingthewindowsizehastheeffectofdecreasingoutliersbutnotelucidatinganyfurtherinsightintothethermodynamicsofadsorptioninthissystem.Asbefore,theenthalpyofadsorptiondivergesathighvaluesofuptakewheretheisoexcessapproximationisinvalid(ne>10)andcannotberegardedasaccurate.

51

Figure2.11.Linearinterpolationoftheexperimental(excess)uptakedataofmethaneonMSC30(top),andthecorrespondingvantHoffplotshowingtherelativelinearityofthedataevenusingthisprimitivefittingtechnique(bottom).Thecoloredlines,frombluetoorange,

indicatethetemperaturesofthemeasurementsfrom238521K,respectively.

52

Figure2.12.IsoexcessenthalpyofadsorptionofmethaneonMSC30,usinglinearinterpolationfitsand3(top),5(middle),and7(bottom)temperaturewindows.A

calculationoveralldata(asingle13temperaturewindow)isalsoshown(dashedblack).

53

2.4.5 VirialTypeFittingEquation

Numerousequationshavebeen suggested to fitexperimentaladsorptiondata forthermodynamic calculations. The most common is to fit the data with a modelindependentvirialtypeequation25andproceedwiththeGibbsexcessquantities,ne,asdirectlyfitted.Theadvantagestothismethodarethatthenumberoffittingparametersiseasilyadjustedtosuitthedata.Thefittingequationhastheform:

ln 1

Equation2.42

The parameters ai and bi are temperature independent, and optimized by a leastsquaresfittingalgorithm.TheexperimentaldataofmethaneadsorptiononMSC30areshown inFigure2.13, fittedusing first, third,and fifthorder terms (with4,8,and12independentparameters,respectively).Itcanbeobservedthatthedataareverypoorlyfitted beyond the Gibbs excess maximum and cannot be used. In addition, as thenumberoffittingparametersisincreased,awellknownlimitationofpolynomialfittingoccurs:acurvatureisintroducedbetweendatapoints.However,theisoexcessmethodis never employed beyond the Gibbs excess maximum, and an analogous plot ofoptimized fits of only selected data is shown in Figure 2.14. The data are wellapproximated even for i = 1,but some improvement is gained for i = 3. There isnosignificant improvementbyaddingparametersto i=5.Thedataarewellfittedat lowtemperaturesupto~4MPaandatallpressuresforisothermsatandabove340K.

The enthalpy of adsorption is calculated by the isoexcess method, using:

54

H ln 1

Equation2.43

Thefunctionalformoftheenthalpyisthatofanithorderpolynomial.Therefore,ifonlyfirstorderparametersareused,theenthalpywillbeastraightline(asshowninFigure2.15).Thisisacceptableforsomepurposes,butallowslittlepotentialinsighttothetruedependenceoftheenthalpyonuptake.Afitusinghigherordertermsmaybepreferredfor that reason.Nevertheless, in this comparison, third order terms do not lend anysignificantcontribution to theanalysis (seeFigure2.16).TheHenrys lawvalueof theenthalpycalculatedinEquation2.43is:

H lim

Equation2.44

ThefitsofselectdatagivevaluesoftheHenrys lawenthalpy intherangeof13.517.5kJmol1.Sincetheparametersareconstantwithtemperature,itisnotpossibletodeterminethetemperaturedependenceofthe isoexcessenthalpywithasingle fittingequation, and it is often assumed to be negligible. If amoving windowmethod isemployed,a temperaturedependence isaccessible,butdoesnotyieldany significanttrends in this case (forwindowsof3 temperatures,with i =1or i=3)as shown inFigures 2.1516. All calculations show an increasing isoexcess enthalpy of adsorptionexceptforsmallwindowsofisothermsusingonlyfirstorderfits,andoverall,thisfittingmethodologyisunsatisfactoryforanalyzingmethaneadsorptiononMSC30.

55

Figure2.13.First(top),third(middle),andfifthorder(bottom)virialequationfitsofmethaneadsorptionuptakeonMSC30between238521K,fittingthecompletedataset

includingtheuptakebeyondtheGibbsexcessmaximum.

56

Figure2.14.First(top),third(middle),andfifthorder(bottom)virialequationfitsofmethaneadsorptionuptakeonMSC30between238521K,fittingonlythedatabelowthe

Gibbsexcessmaximum.

57

Figure2.15.Isoexcessenthalpyofadsorption,usingfirstordervirialequationfitsand3temperaturewindows.Alsoshownistheresultforafirstordervirialequationfitusingthe

entiredataset,a13temperature(13T)singlewindow.ThecalculatedenthalpiesarecomparedtothoseforageneralizedLangmuirfit(giveninSection2.5).

Figure2.16.Isoexcessenthalpyofadsorption,usingthirdordervirialequationfitsand3temperaturewindows.Alsoshownistheresultforathirdordervirialequationfitusingthe

entiredataset,a13temperature(13T)singlewindow.ThecalculatedenthalpiesarecomparedtothoseforageneralizedLangmuirfit(giveninSection2.5).

58

2.4.6 GeneralizedLangmuirFittingEquation

A major limitation of the virialtype equation is that it does not accuratelyinterpolatethedataathighadsorptionuptakeinthenearorsupercriticalregimewherethemaximuminGibbsexcessisaprominentfeatureofthedata.AnotherapproachistoincorporatetheGibbsdefinition(Equation2.21)ofadsorptionintothefittingequation,whichcanbedoneinamodelindependentway.

An effective strategy is to choose a functional form for na that ismonotonicallyincreasing with pressure, consistent with the physical nature of adsorption.20 TheLangmuir isotherm is one example, although others have been suggested (e.g.,LangmuirFreundlich,21 Unilan,22 and Toth26 equations). If an arbitrary number ofLangmuir isotherms are superpositioned, referred to as a generalizedLangmuirequation,thenumberofindependentfittingparameterscanbeeasilytunedtosuitthedata. Langmuirwas the first to generalize his equation for a heterogeneous surfaceconsistingofnumerousdifferentadsorptionsitesofdifferentcharacteristicenergies.27ImplementingthegeneralizedLangmuirmodel(Equation2.16)inamodelindependentway,absoluteadsorptiontakestheform:

, 1

1Equation2.45

The volumeof adsorption in theGibbs equation alsohas apressuredependencethat is fundamentallyunknown,butwhich is generally accepted tobemonotonically

59

increasing in most systems. It too can be approximated by a generalizedLangmuirequation,simplifyingthefinalequationandkeepingthenumberofparameterslow:

, 1

Equation2.46

Theexcessadsorptiondataarethenfittedto:

, 1 1

,

, , 1

Equation2.47

TheKiareequivalenttotheequilibriumconstantsofadsorptionintheclassicalLangmuirmodel,but arenot required tohavephysicalmeaning for thispurpose. They canbetakenas constantwithpressure,buthavingadependenceon temperature similar toEquation2.13:

Equation2.48

Thefittingparameters(cn,cV,i,Ci,andEi)areconstants.Iftheidealgaslawappliestothepressureandtemperatureregimeofinterest,theequationsimplifies:

,

1

Equation2.49

60

TheexperimentaldatafittedbyadoubleLangmuirisotherm(i=2)withtheidealgas

assumptionareshowninFigure2.17.Forallbutthehighestpressures,thedataarewellapproximatedbythismethod.Thecurvatureoftheinterpolationbetweenpoints,evennearorat theGibbsexcessmaximum, is representativeof thephysicalnatureof thesystem, and the extrapolation to high pressures shows much improved behaviorcompared to theprevious fittingmethods.A limitationof the idealgasassumption isthattheexcessuptakeisothermscannotcrossathighpressure,thelowesttemperaturedatadecreasingproportionallywithpressure.This isnotconsistentwithexperimentalresultswhere it frequentlyoccurs that low temperaturedata falls significantlybelowhighertemperaturedataatthesamepressure.Thisphenomenonisentirelyduetononidealgasinteractionsfromanonlinearchangeingasdensity,andcanbeaccountedforbyusing themoregeneral formof theexcessadsorption (Equation2.47).Data fittedusing 1, 2, and 3 superimposed Langmuir isotherms (with 4, 7, and 10 independentparameters,respectively)andusingthemBWRgasdensityareshowninFigure2.18.

Figure2.17.DoubleLangmuirfitofmethaneuptakeonMSC30usingtheidealgaslaw.

61

Figure2.18.Single(top),double(middle),andtriple(bottom)LangmuirequationfitsofmethaneadsorptionuptakeonMSC30between238521K.

62

ThedoubleLangmuir fit is satisfactory for thermodynamic calculations,and is the

preferredmethod inthisstudy.ThetripleLangmuir fit isnotsignificantly improvedtojustifytheadditionof3independentfittingparameters.Theextrapolationofthedatatohighpressuresshowsbehaviorconsistentwithotherexperimentalresultsofhydrogenandmethaneadsorptionatlowtemperature.

2.5 GeneralizedLangmuirHighPressureAdsorptionModelTo understand the true thermodynamic quantities of adsorption from

experimentally measured adsorption data, a model is necessary to determine theabsolute adsorption amount as a function of pressure. The necessary variable thatremainsunknown is thevolumeof theadsorption layerandnumerousmethodshavebeensuggestedtoestimateit.Typicalmethodsincludefixingthevolumeofadsorptionasthetotalporevolumeofthesorbentmaterial,20usingavolumeproportionaltothesurface area (assuming fixed thickness),28 or deriving the volume by assuming theadsorbed layer is at liquid density.29 Some approaches are specific to graphitelikecarbonmaterials,suchastheOnoKondomodel.30,31Themostgeneralapproach istolet the adsorption volume be an independent parameter of the fitting equation ofchoice,equivalenttotheparametercVinthemodelindependenttreatmentabove.

The generalizedLangmuir equation20 (as shown in Section 2.4.6) and numerousothers(e.g.,LangmuirFreundlich,21Unilan,22andToth26equations)havebeenshowntobe suitable fitting equations fordetermining absolute adsorption from excessuptakeisothermssincetheyaremonotonicallyincreasingandcontainarelativelysmallnumber

63

offittingparameterstoachieveasatisfactoryfittotheexperimentaldata.Weconsiderthe following fitting equation forGibbs excess adsorption as a function of pressure,identicalinformtothemodelindependentequationabove(Equation2.47):

, , 1

1

Equation2.50

Theminimum number of independent parameters is desired, andwe find that i = 2yieldssatisfyingresultsacrossanumberofmaterialsinsupercriticaladsorptionstudiesofbothmethaneandhydrogen(wheretheregimebeyondtheexcessmaximumiswellcharacterized),givingthereducedequation:

, , 1 1

1

Equation2.51

We have found that the assumption that the total adsorption volume scalesproportionallywithsiteoccupancyisrobust,andthisfittingequationhasbeenshowntobesuccessfulforbothcarbonaceous(seeChapter6)andMOFmaterials.20WerefertothismethodasthedoubleLangmuirmethod,andiftheabsolutequantityofadsorptionis held constant, it yields the true isosteric quantities of adsorption. The absolutequantity,fromtheGibbsdefinition,is:

, 1 1

1

Equation2.52

64

Thefractionalsiteoccupancy,alsocalledthesurfacecoverage,is:

, 1 1

1

Equation2.53

LeastsquaresfitsofmethaneuptakeonMSC30tothedoubleLangmuirequationareshown in Figure 2.19, the fitted excess adsorption (left) and calculated absoluteadsorption (right) at all temperaturesmeasured. The goodness of fit is satisfactoryacrosstheentirerangeoftemperatureandpressure,witharesidualsumofsquareslessthan0.04mmolg1perdatapoint.TheoptimalfittingparametersforMSC30aregiveninTable5.2,andtheirrelationtothematerialspropertiesisdiscussedinSection5.2.3.

To derive the isosteric enthalpy from the generalized Langmuir equation, the

derivativeofpressurewithrespecttotemperatureisdecomposedasfollows:

Figure2.19.DoubleLangmuirequationfitsofmethaneadsorptionuptakeonMSC30between238521K,showingcalculatedexcessuptake(left)andcalculatedabsoluteuptake(right)assolidlines,andthemeasuredexcessuptakedataasfilleddiamonds(leftandright).

65

P

K

Equation2.54

FromEquations2.5053,therespectivecomponentsofthederivativearegivenby:

1 1

1

1 1

1

1 1

1

1 1

1 Y

K

12

K

12

12

Thesearecombinedtofindtheisostericenthalpyofadsorption,inEquations2.3137:

P XYZ

Equation2.55

TheisostericenthalpyofadsorptionofmethaneonMSC30isshowninFigure2.20,using the ideal gas law to approximate the density in the gas phase. The results areconsistentwith reported results fornumerous sorbent systemsand showaphysicallyinsightfuldependenceof the isostericenthalpyonbothuptakeand temperature.TheHenrys lawvalue isbetween14.515.5kJmol1, consistentwith the isoexcess results

66

calculatedwithouta fittingequation (thebestapproximateof theHenrys lawvalue),andtheenthalpydeclineswithuptaketo14kJmol1.Duetothe idealgasdependenceofthedensitywithpressure,theenthalpyreachesaplateauathighvaluesofuptake.

When the real gas data is used, the calculation of isosteric enthalpy changessignificantlyathighpressures,asshown inFigure2.21.Nonidealityofmethane inthegasphaseissubstantialundertheseconditionsandmustbetakenintoaccountforthemost accurate description of adsorption thermodynamics in MSC30, and was alsonecessarywhen extended to other systems. The tendency of the isosteric heat to aconstantvalueathighuptake32iscommonlyreportedasevidenceofpropercalculationprocedures20,21;howeveraplateauwasnotobservedwhenidealgasassumptionswere

Figure2.20.Isostericenthalpyofadsorption,usingadoubleLangmuirfitwiththeidealgasassumption(Equation2.31).

67

Figure2.21.Isostericenthalpyofadsorption,usingdoubleLangmuirfits:(left)withtheidealgasassumption(Equation2.31),and(right)withtherealgasdensity(Equation2.32),both

employingthetypicalmolarvolumeassumption.

Figure2.22.Isostericenthalpyofadsorption,usingdoubleLangmuirfitsandrealgasdensity:(left)withthetypicalmolarvolumeassumption(Equation2.32),and(right)withthe

suggestedliquidmethaneapproximation(Equation2.36).

68

omitted from the calculations in this study.We suggest that a general exception bemadeforadsorption inthesignificantlynonidealgasregimewherethere isnoreasontosuggestthattheisostericenthalpyofadsorptionwouldpersisttoaplateauvalue.

Secondly, the change in molar volume on adsorption must also be carefullyconsideredathighpressureforcertainadsorptives,wherethemolarvolumeinthegasphaseapproachesthatofitsliquid.Thisisnotthecaseacrossawidetemperatureandpressureregime forhydrogen, forexample,but ishighly relevant tomethaneevenattemperaturesnearambient.FormethaneadsorptiononMSC30inthisstudy,theusualapproximation, treating theadsorbedphasevolumeasnegligiblecompared to thatofthegasphase,holds inthe lowpressure limitbutbecomes invalidbeyond1MPa.Thedifferenceinisostericheatcalculatedwithorwithouttheapproximationis>1%beyond1MPa, as shown in Figure 2.22. To approximate themolar volume of the adsorbedphase,wesuggesttousethatof liquidmethane(seeEquations2.3637),afixedvaluethat can be easily determined and which is seen as a reasonable approximation innumerousgassolidadsorptionsystems.Specifically formethane,weuseva=vliq=38mLmol1,thevalueforpuremethaneat111.5Kand0.1MPa.ForourdataofmethaneonMSC30, the difference between themolar volume of the gas and the adsorbedphasesbecomessignificantatalltemperaturessincevais530%themagnitudeofvgat10MPa.Thevariationof the liquidmolarvolumewith temperatureandpressurewasconsidered;thedensityalongthevaporizationlineis~20%,andsocanbeconsideredanegligible complicationwithin the error of the proposed assumption. A fixedmolarvolumeoftheadsorbedphasewasusedthroughoutalltemperaturesandpressures.

69

2.6 Conclusions

In summary, the method used to determine the uptake and temperaturedependenceofthe isosteric(or isoexcess)enthalpyofadsorptionofmethaneonMSC30 had a significant effect on the results. The virialtype fitting method has theadvantage thatwith few parameters, one can fit experimental isotherm data over alarge rangeofPandT inmany systems.Thebest fits toEquation2.42are found forsystemswith aweakly temperature dependent isosteric heat. For high temperatures(near the critical temperature and above) andup tomodestpressures, thisequationoften sufficeswith only 24 parameters (i = 01).However, application of this fittingproceduretomoderatelyhigh (definedasnearcritical)pressuresor lowtemperatures(where a Gibbs excess maximum is encountered) lends substantial error to theinterpolated results even with the addition of many parameters. In every caseimplementing the isoexcess assumption, the enthalpydiverged athighuptakewheretheassumptionisfundamentallyinvalid.Itissimpletoshowthatsincene

70

AdoubleLangmuirtypeequationwithabuiltindefinitionofGibbssurfaceexcesswasasuitablefittingequationofmethaneuptakeonMSC30inthemodelindependentcase,andcouldalsobeusedasamodeltodeterminetheabsolutequantityofadsorptionandtoperformatrueisostericenthalpyanalysis.Inboththeisoexcessandabsoluteresults,the enthalpy of adsorption showed physically justifiable characteristics, and gave aHenryslawvalueclosesttothatforamodelfreeanalysisofthelowpressuredata.Allsimplifyingapproximationswithinthederivationoftheisostericenthalpyofadsorptionwerefoundtobeextremelylimitedinvalidityformethaneadsorptionwithinapressureand temperature range close to ambient. Therefore, real gas equation of state datamustbeusedandasimpleapproximationof the finitemolarvolumeof theadsorbedphasewasproposed.

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