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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.
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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
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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
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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
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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
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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.
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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
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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
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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.
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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.
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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.
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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
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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:
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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).
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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).
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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
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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:
, ,
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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
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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.
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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51
Figure2.11.Linearinterpolationoftheexperimental(excess)uptakedataofmethaneonMSC30(top),andthecorrespondingvantHoffplotshowingtherelativelinearityofthedataevenusingthisprimitivefittingtechnique(bottom).Thecoloredlines,frombluetoorange,
indicatethetemperaturesofthemeasurementsfrom238521K,respectively.
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52
Figure2.12.IsoexcessenthalpyofadsorptionofmethaneonMSC30,usinglinearinterpolationfitsand3(top),5(middle),and7(bottom)temperaturewindows.A
calculationoveralldata(asingle13temperaturewindow)isalsoshown(dashedblack).
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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:
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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.
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55
Figure2.13.First(top),third(middle),andfifthorder(bottom)virialequationfitsofmethaneadsorptionuptakeonMSC30between238521K,fittingthecompletedataset
includingtheuptakebeyondtheGibbsexcessmaximum.
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56
Figure2.14.First(top),third(middle),andfifthorder(bottom)virialequationfitsofmethaneadsorptionuptakeonMSC30between238521K,fittingonlythedatabelowthe
Gibbsexcessmaximum.
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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).
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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
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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
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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.
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61
Figure2.18.Single(top),double(middle),andtriple(bottom)LangmuirequationfitsofmethaneadsorptionuptakeonMSC30between238521K.
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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
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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
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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).
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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
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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).
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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).
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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.
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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
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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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