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Langmuir's Theory of Adsorption: A Centennial Review

Authors: Hans Swenson and Nicolas P. StadieThis document is the unedited author's version of a Submitted Work that was subsequently accepted for publication in Langmuir, copyright © American Chemical Society after peer review. To access the final edited and published work, see DOI: 10.1021/acs.langmuir.9b00154.

Swenson, Hans, and Nicholas P. Stadie. "Langmuir's Theory of Adsorption: A Centennial Review." Langmuir: the ACS Journal of Surfaces and Colloids 35, no. 16 (April 2019): 5409-5426. DOI:10.1021/acs.langmuir.9b00154.

Made available through Montana State University’s ScholarWorks scholarworks.montana.edu

Langmuir'sTheoryofAdsorption:aCentennialReview1‐3

HansSwensonandNicholasP.Stadie*

DepartmentofChemistry&Biochemistry,MontanaStateUniversity,Bozeman,MT,59717,USA

*E‐mail:nicholas.stadie@montana.edu

The100thanniversaryofLangmuir’stheoryofadsorptionisasignificantlandmarkfor the physical chemistry and chemical engineering communities. Despite itssimplicity, the Langmuir adsorptionmodel captures the key physics ofmolecularinteractions at interfaces and laid the foundation for further progress inunderstanding interfacial phenomena, developing new adsorbent materials, anddesigningengineeringprocesses.TheLangmuirmodelhashadanexceptionalimpacton diverse fields within the chemical sciences ranging from chemical biology tomaterialsscience ,animpactthatbecameclearerwiththedevelopmentofmodifiedadsorptiontheoriesandcontinuestoberelevanttoday.

Introduction

IrvingLangmuirisknownforwide‐rangingcontributionsinchemistryandphysics,bothinfundamentalandappliedconcepts,andisindeliblylinkedtothebirthofthescienceofinterfaces.Histhreeseminalworksonadsorptiontheory,thefoundationforthemodernunderstandingofalltypesofchemicalinterfaces,werepublishedin1916,11917,2and1918,3thelastofwhichisnowonecenturyoldandthesubjectofcelebrationinthishistoricalarticle.The“superficial”natureofthisfieldofresearchwasplayfullyacknowledgedinthepresentationspeechoftheNobelPrizeinChemistrytoLangmuirin1932.ItwasalsoacknowledgedinSöderbaum’sverynextbreaththat“inreality,surfacechemistryhascontributedgreatlytothedeepeningofourknowledgeofmatter.”4Thisremainstruetoday,withmajorworldwideresearchefforts,e.g.,inheterogeneouscatalysis,devotedtophenomenaatinterfaces.

ThehistoryandimpactofLangmuir’sthreeseminalworks,referredtoasLangmuir1916,Langmuir1917,andLangmuir1918,arebrieflyreviewedherein.Thecontextinwhichadsorptiontheorywasbirthedwillfirstbesummarized,followedbyabriefaccountofLangmuir’searlyworkthatleddirectlytothedevelopmentofhismonolayertheory.Inadditiontothesimple,idealpictureofhomogeneousmonolayeradsorptionthathascometobeknowncolloquiallyastheLangmuiradsorptionmodel,Langmuiralsodevelopedseveralmorecomplexmodelssuchasformultilayerandheterogeneousadsorptionalreadyin1918.TheingeniousearlyexperimentscarriedoutbyLangmuirandhisassistant,Mr.S.P.Sweetser,laidthegroundworkforfuturedemonstrationsofthewidespreadapplicabilityofthissimpletheoryofsite‐specificmolecularadsorption.Langmuiralsoemphasizedthenon‐universalapplicabilityofanyoneequationormodel including,ofcourse,hisownmonolayeradsorptiontheory ,3andtheneedtoaccountforawidebreadthofdifferentadsorptionmechanismsatplayoncomplicated,real‐worldsurfaces.Hence,thelastingreputationofLangmuir’sadsorptiontheoryisboththatofapowerful,simplemodelwithfar‐reachingapplicabilityandtextbookelegance,andsimultaneouslythatofamodelwithfewstrictlyadheringdemonstrativesystemsinthelaboratory.

Inthisreview,astatisticalmechanicaltreatmentofallsixofLangmuir’soriginalcasesofadsorptionisgiven,followedbyselectedcontemporaryexamplesofgas‐solidadsorbate‐adsorbentsystemsthatexhibitrepresentativebehavior.ReferencetoLangmuir’sworkhasrisensharplysince1999whenmetal‐organicframeworks MOFs wererevolutionizedtoincludematerialswithlargeinnersurfacesofcrystallinestructure;5hence,emphasisisplacedhereinonexamplesofLangmuiradsorptionontheporoussurfacesofMOFs.ThisextensionofLangmuir’s“externalsurface”theorytogasadsorptiononthe“internalsurfaces”ofporousmaterialsisoneofmanynaturalextensionsofLangmuir’soriginaltheorytointerfacesofallkinds,suchasingas‐liquidandliquid‐liquidsystems,molecular‐proteinbinding,andeveninmacromoleculeornanocrystalbinding.WeseektoputLangmuir’snowcentury‐oldtheoryintomoderncontext,withastrongemphasisonthefirst‐principlessimplicityoftheseconceptsandtheir cautious applicabilityinwide‐rangingadsorptionsystems.

HistoricalContext

EarlyAdsorptionTheory.Thephenomenonofadsorptionhasbeenknownsinceantiquity;6quantitativeinvestigationsbeganinthelate18thcentury theearliestknowninvestigationsbyScheele,Priestley,andFontanain1773‐1777,asreviewedmorecomprehensivelyelsewhere7‐8 ,andtheterm“adsorption”asdistinctfrom“absorption”wasoriginallycoinedbyEmilduBois‐

Reymond viaHeinrichKayser9 in1881.Adsorption,strictlydefined,istheincreaseindensityofafluidadsorbateataphaseboundaryorsurface.Itremainsdistinguishedfromabsorptionbyitslimitationtothesurfaceorinterfaceofthesorbent;upondiffusionbeyondtheinterfaceintothebulkofthesorbent,theadsorbatebecomestheabsorbate.Theterm“sorption”wasintroducedasreferringtotheambiguouscaseencompassingbothadsorptionandabsorptionbyJamesW.McBainin1909.10Thisterminology,inprinciple,remainstheconventiontoday.

Thehistoryofadsorptiontheoryisintimatelytiedtoboththatofintermolecularinteractionsandtheatomicstructureofsolid‐statematter,subjectsthatwerestillprematureattheturnofthe20thcentury.ItistellingthattheforcesofattractionbetweentwophasesattheirinterfacewerestillreferredtobyLangmuirin1918as“actionatadistance,”inquotationmarks.3Studiesofadsorptionwereprimarilyanexperimental asopposedtotheoretical endeavorbefore1900.WhileJ.WillardGibbsdevelopedtheearliestthermodynamictheoryof“surfacesofdiscontinuity”inthesecondpartofhislandmarkworkOntheequilibriumofheterogeneoussubstancesin1876,11hespecificallyremarkedastothequantityofmoleculesadsorbedbeing“toosmallingeneraltoadmitofdirectmeasurement.”Nevertheless,theterms“surfaceexcess”and“excessquantityofadsorption”werecoinedbyGibbsandtheconceptremainsofgreatimportancetoadsorptiontheory.Thefirstliquidsolute‐solidadsorptionequilibriawerereportedbyFriedrichStohmannandWilhelmHennebergin185812 ammoniumadsorptiononsoil andthefirstgas‐solidadsorptionequilibriawerereportedindependentlybyChappuisin187913andKayserandJoulinin18819,14 smallmoleculargasesoncharcoal ,wherethedelayforgas‐solidexperimentswasduetothemorecomplexapparatusrequired.Heatsofadsorptioncouldbemeasuredasearlyas1854byFavre15‐16andpracticaladsorptionapplicationssuchasgasseparationswerewell‐establishedbytheturnofthecentury.

PriortoLangmuir’smonolayer‐basedtheoryfirstreportedin1916,adsorptionwasunderstoodsuchthatthedensityofafluidataninterfacedecreasesgraduallytowardthebulk.ThetheoryofadsorptionwasfirstfocusedonthesolidsurfaceitselfbyHerbertFreundlichin190717,whoseempiricalequationforadsorptiondemonstratesrespectableutilityacrossrelativelywidetemperatureandpressurerangesstilltoday.However,itiswell‐knownthatFreundlich’sequationisneitherphysicallyfoundednorthermodynamicallyconsistent,preventingitsfundamentaljustification.18‐19MichaelPolanyialludedtotheconceptofacharacteristicadsorptioncurvein1914,20andfurtherlaidthefoundationofwhathasbecomeknownasPolanyi’s“potentialtheory”in1916.21Inatwistoffate,thisnon‐localizedtheorywasrejectedatthetimeforbeingincongruentwiththeemergingtheoriesofintermolecularinteractionsbasedonelectricalforces;nevertheless,itwaslaterdeveloped togetherwithFritzLondon22 intoawidelyacceptedparalleltheorytothatofLangmuir’s.23

Langmuir’stheory,onthecontrary,wasfoundedonafundamentallydiscretizedmolecularperspective,consideringtheatomicstructureofthesurfacetobecentraltoasuccessfuldescriptionofadsorption.HewasstronglyinfluencedbytheemergingpictureofcrystallinesolidmatterpaintedbyWilliamH.BraggandhissonW.LawrenceBragg,especiallyafterhearingseverallecturesgivenbytheelderBraggduringhisvisittotheeasternUnitedStatesaroundthetimeoftheoutbreakofWorldWarI 1914‐1915 .InLangmuir’swords:“theimportanceoftheworkofW.H.BraggandW.L.Bragginitsbearingonchemistryhasnot,asyet,beengenerallyrecognized.”1Thediscretenature i.e.,sequentialfilling ofwell‐definedadsorbedsurfacelayerswastheinsightfirstdevelopedbyLangmuiranddescribedextensivelyinLangmuir1916,withclearreferencetothedefinite,discretizedatomicstructureofcrystalsdiscoveredbytheBraggs.

Langmuirpicturedmonolayerformationtooccurbasedonaspecificbindinginteractionbetweenadsorbatemoleculesanddistinctsolidsurfacesites.Thispicturewasoriginallyfoundedontheresultsofexperimentsperformedwithhydrogenandothersmallmoleculesinthepresenceofheatedmetalfilamentsatverylowpressures.InLangmuir1918,however,itwasalreadybroadlygeneralizedtoincludecaseswheremonolayerormultilayerformationoccurs,aswellascasesrangingfromhighlyregularsurfacestocompletelyamorphoussurfacescontaininganabundanceofdifferentadsorptionsites.Thesixfundamentalcasesofadsorption alldesignedforadsorptionatbulksurfaces identifiedbyLangmuirlaidthefoundationforsurfacescience,whichhassincebeenadornedwithmodificationsandnewtheoriesdesignedtodescribecomplex,real‐wordadsorptionphenomena,includingwithinmolecularlyporousmaterials.6

IrvingLangmuir.IrvingLangmuir,bornonJanuary31st,1881,receivedhisprimaryandsecondaryeducationinBrooklyn,Paris,andPhiladelphia,followinghisfather’swork‐relatedmovements asaninsuranceexecutive .Afterthedeathofhisfatherin1899,LangmuirenteredTheSchoolofMinesatColumbiaUniversitytostudymetallurgicalengineering,because“thecoursewasstronginchemistry,ithadmorephysicsthanthechemicalcourse,andmoremathematicsthanthecourseinphysics–andIwantedallthree.”24Afterreceivinghisdegree,Langmuir,likemanyAmericanscientistsatthattime,setofftoGermanytoobtainaPhD.AttheUniversityofGöttingen,hisdesiretocouplescientificresearchwithpracticalandindustrialapplicationswouldbehonedandaffirmedbyseveralgreatprofessors,mostnotablyFelixKleinandWaltherNernst.Nernst,whoservedasLangmuir’sprimaryadvisor despiteFriedrichDolezalekbeingnamedassuch25 ,directedhisthesisprojectofexaminingthedissociativeactionofvariousgasesonthesurfaceofahotplatinumfilament.26Theseearlyexperimentswouldlaythefoundationforhislifelongstudiesofadsorptionphenomena.

Inearly1906,LangmuiracceptedafacultypositionatStevensInstituteofTechnologyinNewJersey,hopingtopursueanacademiccareerinresearch.Afterthreeyearsofintensive,time‐consumingteachingthataffordedlittletimeforresearch,LangmuirleftthispositionandjoinedthestaffoftheGeneralElectricresearchlaboratory,reunitingwithanoldcolleaguefromColumbia,ColinG.Fink.Langmuirfirstsecuredaresearchpositionforthesummerof1909,andimpressedhiscoworkersenoughtogainapermanentpositionattheresearchlab,apositionhewouldholduntilhisretirementin1950.AtGeneralElectric,Langmuir’sindependencewasconsistentlyprotectedbythelaboratory’sdirector,WillisR.Whitney.Hisfirstprojectswerelargelyacontinuationofhisthesisworkinvestigatingchemicalreactionsinthepresenceofheatedmetalfilamentswithinglassbulbs,especiallytungsteninthepresenceofhydrogenandothergases.Thereactionratescouldbeslowedandeffectivelydeconvolutedwhenperformedunderhigh‐vacuumconditions atechnicalachievementofLangmuir’swork ,culminatingintheproposalofafundamentallynewmechanismofheterogeneouschemicalreactions.27Thetwokeyfeaturesofthiswork,namelythelowpressureandinherentroleofthemetalsurface,directlyadvancedLangmuirtowardthebreakthroughtheoryofadsorptionbasedontheconceptofmolecularlayers.Therefore,despitetestinghisnewlyfoundtheoryoncasesofpurephysisorptioninLangmuir1918,theoriginalcollectionofworkonwhichthetheorywasdevelopedconsistedofoveradecadeofexperimentsobservingthebehaviorofgasesinthepresenceofheatedmetalsurfaces:purechemisorption.ItisperhapsunsurprisingthatthemonomolecularlayerthatLangmuirenvisionedremainsmuchmoreaccurateindescribingthephenomenaofchemicaladsorption wherethebindingstrengthisstrongrelativetointermolecularinteractionsinthebulkgas .

Langmuir’sTheoryofAdsorption

AKineticModel.Langmuiroriginallydescribedgasadsorptionatasolidinterfaceasbearingmechanisticsimilaritytogas‐phasecondensationataliquidsurface,whereinelasticcollisionsleadtoa“lag”orresidencetimeatthelocationofincidence.1,28Thatis,asgasmoleculesstrikethesurface,theyareheldnearthesurfacebyattractiveintermolecularforces akintocondensation forsomeshortbutfinitetimeuntilleavingintothebulkgasphaseagain akintoevaporation .Withinthispurelykineticviewofadsorption asshowninFigure1 ,andwithanewlydevisedconceptofthediscretized,surface‐orientednatureofadsorptioninhand,asimplerelationshipbetweenequilibriumsiteoccupancy, ,andthepressureofthegasphase, ,wasderived.Langmuirmadethreefundamentalassumptions:

i therateofincidenceofthemoleculesinabulkgasphaseonaunitareaofadsorbentsurface, ,isproportionaltothepressureatconstanttemperatureviathekinetictheoryofgases adaptedfromhispreviousworkonsaturatedmetalvaporsabovetheirrespectivesolids29 ,

ii therateofadsorption asopposedtometalvaporcondensation , ,dependsnotonlyontherateofincidenceofmoleculesonthesurface, ,butalsoontheprobabilityofadsorption asopposedtoelasticreflection , ,andtheprobabilityofincidenceatavacantadsorptionsite asopposedtoatanoccupiedsite , ,and

iii therateofdesorptionisequaltotherateofdesorptionatmaximumsurfacecoverage,

, ,multipliedbythefractionaloccupancyofthesurfacesitesbyadsorbedmolecules, therebyneglectinganyroleofadsorbate‐adsorbateinteractions .

Thesethreeassumptionscanalsobesummarizedbystatingthatboththegasphaseandadsorbedphasebehaveideally thereisnonotionofintermolecularinteractions andthateverybindingsiteisidentical.Combined,theyyieldthefollowingexpressionfortheequilibriumfractionaloccupancyofadsorptionsitesunderagasphaseatpressure :

, ∙ 1 Equation1

TheLangmuirconstant, ,isindependentofpressureandhenceonlydependsontemperature.Whenthetemperatureisinvariant,theisothermcanbemeasuredand experimentallydetermined,whichfromthiskineticmodelisdefinedas:

√2∙

,Equation2

TheLangmuirconstanthasunitsofinversepressure orinverseconcentrationforsoluteadsorption .Itisusefultorecognizethatthesurfaceispreciselyhalf‐occupiedatthepressurecorrespondingto ,asshowninFigure1,andsorepresentsa“characteristicinversepressureofadsorption.”AmorethoroughkineticderivationasoriginallypresentedinLangmuir1918isreproducedintheSupportingInformationusingupdatednomenclature.

Figure1. a Langmuir’skineticmodelofadsorptionacknowledgesfourprocessesoccurringbetweenagasinequilibriumwithasurface consistingofdiscretizedadsorptionsites :incidenceinc ,reflection ref ,adsorption ads ,anddesorption des .TreatingthesefundamentalprocesseswiththreesimpleapproximationsleadstothesimpleLangmuirisothermequation. b‐c ThedependenceoftheLangmuirisothermontheLangmuirbindingconstant, ,showinghyperbolicshapeinthesimpleplotoffractionaladsorptionsiteoccupancy, ,asafunctionofpressure b andsigmoidalshapeintheequivalentlog‐normalplot c .Themagnitudeofthebindingconstantisindicatedbycolor,increasingfrom0.001‐1000Pa .Thecharacteristicpressureregimecorrespondingtosignificantadsorptionuptakeoccursaround ,asshownforthecaseof 0.1Pa reddashedline .

ExperimentalResults.TheoriginalexperimentsthatinspiredLangmuir’smonolayerhypothesiswereinvestigationsofthebehaviorofhydrogenandoxygenatvacuumpressuresinsideaglasslightbulbcontainingaheatedtungstenfilament,suchasthatreportedin1912.30Langmuirfirstobservedthatthepressureofhydrogendecreaseduponexposuretotheheatedfilamentovertimeespeciallyplatinumorpalladium .Langmuirconcludedthatthehydrogendissociatedonthesurfaceoftheheatedmetal,thenescapedasamonatomicgas,andfinallychemisorbedasatomichydrogenontheinnerglasswallofthebulb.Theadsorptionofatomichydrogenonglasswasfoundtobeirreversible,althoughquantitativeresultsastotheamountadsorbedwerenotreportedatthattime.

Langmuircontinuedhisexperimentswithothergases e.g.,CH4,NH3,PH3,CO,andCO2 adsorbedonheatedmetalsurfacesandfirstoutlinedhismonolayerhypothesisin1916.1Still,equilibriumadsorptionisothermmeasurementswerenotpresentedinLangmuir1916orelsewhereatthattime.InLangmuir1917,thegas‐solidadsorptioninterfacewastemporarilyabandonedinfavoroftheliquid‐solidinterface.Thedevelopmentofmonolayerassembliesataqueousandotherinterfacesisnotwithinthefocusofthepresentreview,andtheinterestedreaderisreferredelsewhere.31‐32

Finally,inLangmuir1918,methodologicalstudiesoftheadsorptionofnumerousgasesonthreemodelmaterialsurfaceswerereported:glass amorphoussilica ,muscovite mica ,andplatinum.Langmuirperformedtheexperimentsalongthedesorptionbranch,permittingdirectobservationoftheinterplaybetweenphysisorptionandchemisorptionphenomena thechemisorbedquantitywouldnotbedesorbedatlowtemperature .Thesewerethefirstsetofquantitativeadsorptionmeasurementsofsmallmoleculargasesonsolidsurfacesofwell‐knownchemistryandgeometry.Hisexperimentswereperformedatextremelylowpressureforthetime,requiringgreatcareandnewlydevelopedvacuumequipment includingaMcLeodgaugeandLangmuir’sinvention,the“condensationpump”33,nowknownasadiffusionpump34 .

Figure2. a EquilibriumdesorptionmeasurementsofN2onmicaat90and155KasreportedinLangmuir1918,showingsingle‐siteLangmuir SSL adsorptioncharacter. b EquilibriumdesorptionmeasurementsofCOonglassat90KasreportedinLangmuir1918,showingdual‐siteLangmuir DSL adsorptioncharacter thestronglyboundspeciesattributedtochemisorptionisshownasasolidpurpleline .

Thefirstsetofexperimentsreportedin1918wereperformedona24.3gsampleofmuscovitemica ,havinganactual hand‐measured surfaceareaof0.5750m2 specificsurfacearea:0.0237m2g‐1 .LangmuirconfirmedthatN2adsorptiononmicabetween90‐155Kwaspurelyphysicalinnature i.e.,physisorption andcouldbewell‐describedbyhisequilibriumkineticmonolayermodelseeFigure2 .Modernfittinganalysisofthesedatavalidatethisconclusion,andtheresultscanbeusedtodeterminethesurfaceareaoccupiedbyasinglemoleculeofN2inacompletemonolayer:1.69nm2.Thisvalueissignificantlylarger byafactorof~10 thanthecurrentlyacceptedconventionalvalueforN2adsorptionat77K 0.162nm2 ;thisresultmaybeduetotheelevatedtemperatureofLangmuir’s1918experiments 90K .Langmuirindeedrecognizedthatthemonolayerdensitiesreportedin1918wereroughlyofthecorrectorderofmagnitude,butwerelowerthanexpectedbasedonamonolayerofliquidbyafactorofupto~9,evenat90K.Nevertheless,thisevidencestronglyfavoredLangmuir’s“monolayer”pictureofadsorptionsincethenumberofadsorbedN2moleculesdidnotexceedthatofaliquidmonolayerbutapproachedawell‐definedmaximum.

Langmuir’ssecondsetofexperimentswereperformedonnearly200individualglassmicroscopeslides,eachdefectedslightlytopreventstackingandpermitfreeaccessofthegastotheamorphoussurface.Thetotalsamplemasswas~36.7g basedonareportedvolumeof13.6mL ,andthetotalsurfaceareawasmeasuredtobe0.1966m2 specificsurfacearea:0.00535m2g‐1 .Severalpartsoftheapparatususedintheprecedingmicaexperimentswerereplacedtopermithigheraccuracyadsorptionmeasurementsnecessitatedbytheextremelylowbindingstrengthoftheglasssurface.Impressively,totalquantitiesofadsorbedgasasminuteas20nmolontheentiresample thelowestamountdetectedisthelastpointofLangmuir’sargondesorptionisotherm,at0.1Paand90K couldbedetected.ItisunclearwhichattributeoftheglasssurfacethatLangmuirsetouttocontrastwiththemuscovite:theamorphousnatureoftheglasssurface CaseIII,describedbelow,istheoreticallydevelopedforamorphoussurfaceswithinfinitelymanydifferentsites ormerelyits

differentchemistry asSiO2 .Inanycase,themostremarkableresultswereforCOadsorptiononglassat90K;theCOcouldnotbefullydesorbedundervacuum,motivatingamulti‐sitemonolayermodel CaseII whereonetypeofsiteischaracterizedbyweakadsorption physisorption andanotherbystrongadsorption chemicaladsorption,orchemisorption ,asshowninFigure2.

Langmuir’sthirdsetofexperimentswereperformedtodifferentiatethenatureofchemicalandphysicaladsorptiononplatinummetalfoil,andtounravelsomeofthecomplexitiesoftheplatinum‐catalyzedoxidationofH2andCO.Asmallsquareof0.010mmthickfoilwasused,washedandcarefullyfolded,havingatotalmassof4.03gandatotalsurfaceareaof0.0312m2 specificsurfacearea:0.00774m2g‐1 .Langmuirfoundthatatroomtemperatureandbelow,onlyverysmallquantitiesofH2,O2,andCOcouldbeadsorbedonplatinum,farbelowthatofthecalculatedmonomolecularlayer.Aseriesofexperimentsataboveambientpressurewerethenreported,demonstratingseveralkeyfeaturesoftheplatinumsystem,namely:chemicalreaction,catalysis,andhysteresis dependingonwhatisnowknownas“activation” .Theresultinginsightsintothecatalyticreactionmechanismsatplayontheplatinumsurfacerepresentedimportantearlycontributionstoresearchinthebuddingfieldofheterogeneouscatalysis,whereLangmuir’sworkwouldhavelong‐lastingimpact.35

CorroboratingLangmuir’sTheory.Numerousinvestigationswerecarriedoutinthedecadesafter1918totestLangmuir’stheoryinothersimplegas‐solidadsorptionsystems.Ithasbeenfoundthatintheroutinecharacterizationofpracticalmaterialssuchasheterogeneouscatalystsandporousadsorbents usuallywithsubcriticaladsorptivefluidslikeN2 ,multilayeradsorptionisfarmoreoftenverifiedthansimplemonolayeradsorption.ThisledtoamoredetailedtheoryofmultilayeradsorptionchampionedbyStephenBrunauer,PaulEmmett,andEdwardTeller36 aspecificmodificationofLangmuir’sCaseVIreferredtoasBETtheory,describedbelow thatremainswell‐knowntoday.LangmuirremainedsilentduringhislifetimeontherelativeutilityofhistheoryascomparedtothatofBrunauer,Emmett,andTeller,afactthatwas perhapsrathersensitively interpretedbyBrunauertoimplyresentment.37However,itisclearthat,atleastinthecaseofgasadsorptiononexternalsolidsurfaces,nosuchdefenseoftherelativeapplicabilityofeachequationneedbemade.Langmuir’ssimplestmonolayerformulationisusuallyapplicablewhenthestrengthofinteractionbetweentheadsorbateandsolidsurfaceisfarstrongerthanthatbetweentwoadsorbatemolecules,whileBETtheory whichisitselfanextensionofLangmuir’sownmultilayertheory isapplicablewhentheinteractionsaremoresimilar.Itismerelyaquestionofwhichregimerepresentsthesystemofinterest.WenotethatLangmuir’soriginaldata e.g.,N2adsorptiononmicaandCOonglassat90KasshowninFigure2 werenotcollectedathighenoughpressuretodiscerntheaccumulationofmultipleadsorptionlayers.

Numerousadsorptionisothermequationsarenowgenerallyusedtointerpolateand/ormodelexperimentaladsorptionequilibria,typicallybasedononeofthreefundamentalunderlyingapproaches:Freundlich’sempiricalequation,Langmuir’ssurface‐basedequations,orPolanyi’spotentialtheory.Surfaceswithinporoussolids,inparticularwithinmicroporoussolids,presentashighlycomplextoaccuratelymodel,andtypicallycannotbeadequatelyaddressedwithasimpleLangmuir‐typeanalysis.Forfurtherinformation,thereaderisreferredtoseveralmoredetailedreviews6‐7,38ofthesubject.

Hill’sEquilibriumTheoryofProteinBinding.AnanalogoustheorytoLangmuir’stheoryofmonolayergasadsorptiononsolidsurfaceshadinfactalreadybeenindependentlydevelopedsignificantlyearlierthan1916,specificallytoexplaintheoxygen“dissociationcurves”associated

withhemoglobininhumanandanimalblood.Likegasadsorptiononafinitenumberofsolidsurfacesitesasafunctionofpressure,thebindingofaligand e.g.,oxygen tofunctionalsitesonafinitenumberofproteinsinsolutionwillreachavalueatequilibriumthatdependsontheconcentrationoftheligand theadsorbate .ArchibaldV.Hill,anundergraduatestudentatTrinityCollegeinCambridge,basedthistheoryondatareviewedbyBarcroftandCamisintheJournalofPhysiologyin1909,39relatingthefractionofoxygen‐saturatedproteintothepartialpressuretension ofoxygeninvariousaqueoussolutions.IthadearlierbeenobservedbyChristianBohretal.thattheshapeofsuchdissociationcurveswassigmoidal,indicatingcooperativebinding.40Hill,whowouldlatergoontoreceivetheNobelPrize inMedicinein1922 ,suggestedthatthedatacouldbeaccuratelyrepresentedbyanequationderivedfromamass‐actionframeworkforthereversiblecombinationofaligand orgeneralbindingmoiety withadiscretesetofreceptorstructures bindingsites atequilibrium.41

ThesamegenerallawsofmassactionatequilibriumunderpinboththeHillandLangmuirequations,withakeydifferenceinemphasisforHill’sdevelopment:theallowanceformultiplebindingsites andhencecooperativeadsorption onasingleprotein.ThesimplerHillequationforsystemswithonlyasinglereceptorsiteperproteinisidenticaltotheLangmuirequationforsimple,single‐sitegasadsorptiononasolidsurface.Itcanthusbederivedusingthesameequilibriumkineticmodel i.e.,thelawofmassactionforasimplecombination“reaction” asgivenabove.Asummaryofthemassactionapproach foreithersingle‐sitegasadsorptionorsingle‐receptorproteinbinding canbemadeasfollows:

Single‐SiteGasAdsorption Single‐MoleculeReceptorBinding

A ↔ A L R ↔ LR Equation3

1 LEquation4

Foroxygenbindingtomyoglobin,forexample,thissimpletreatmentisconsistentwithexperimentalresults,indicatingthatasinglereceptorbindsoxygenoneachproteinsincenoadsorbate‐adsorbatecooperationisdetected seeFigure3 .

ThemoregeneralHillequationforbindingstructureswithmultiplereceptorsitesperprotein firstreportedinitspresentformin191042 canbederivedusingasimilarmass‐actionatequilibriummodel,butforaligandbinding“reaction”involvingmultiplereceptorsonthesameprotein:

Multi‐MoleculeReceptorBinding

L R ↔ L R Equation5

Equation6

Hilloriginallypicturedsingle‐receptorproteinsthatwere“coagulated”together,causinginteractionbetweenthebindingsitesoncloselyheldproteins;thishassincegivenwaytothemodernunderstandingofmultipleligandreceptorsoneachprotein,asdeterminedbyX‐raycrystallography.43Foroxygenbindingtohemoglobin,simpleligand‐bindingtheory correspondingtoEquation4 isnotconsistentwithexperimentalresults,norwereotherempiricalequationssuch

asthatproposedbyWolfgangOstwald intheveinofFreundlich’sempiricalequationforgasadsorption .39Instead,HillshowedthatexperimentalresultswereconsistentwitheitheracombinationofEquation4andEquation6 where 2wasimposed orsimplybyEquation6alone where wasallowedtovaryasanindependentparameter .Inthelattercase,thevalueof wasinitiallydeterminedtobe~1.4,indicatingsignificantcooperationbetweentheoxygenbindingsitesonasingle,tetramerichemoglobinprotein seeFigure3 .Modernexperimentaldataconfirmahighervalueof ~2.6.44Thiscooperativebindingofoxygenbetweenthefourferroushemecomplexesinasinglehemoglobinproteinhascrucialimplicationsforoxygentransportanddeliveryinbiologicalsystems.ItisnotablethatHill’stheorydidnotallowforpartialoccupationofeachprotein e.g.,asinimposingindividualcontributionstothetotaladsorbedamountfromstatescorrespondingto 1, 2, 3, 4… infavorofasimplerequationwithfewerindependentparameters.

Figure3. a EquilibriumO2bindingonmyoglobin,asreportedbyRossi‐FanelliandAntonini45,showingsingle‐siteLangmuir SSL character. b EquilibriumO2bindingonhemoglobin,asoriginallymeasuredbyFerryandGreen46andthennormalizedbyPauling47,showingcooperativeadsorption CA character.Myoglobin proteinstructureinsetleft has1bindingsitethatbindsO2,whilehemoglobin proteinstructureinsetright has4bindingsitesthatsemi‐cooperativelybindO2,lendingaHillcoefficientof~2.6.

TheHillcoefficient, ,isrecognizedasanindicatorofcooperativeinteractions,sinceitdescribesthenumberofmoleculesboundperreceptor or,inthegas‐solidadsorptionsystem,thenumberofadsorbedmoleculespersurfacesite .Thiswasoriginallypositedtobeequalto,inthecaseofproteinbinding,theintegernumberofreceptorsperprotein;inreality,however,proteinbindingrequiresmorecomplexitytoaccuratelydescribe,andsotheHillcoefficientisonlylooselyindicativeofthecooperativitybetweenreceptorsites.Indeed,thevalueof isoftenanon‐integer.Langmuiralsoconsideredthepossibilityofcooperativeadsorptionphenomena CaseIV,detailedbelow butdidnotexplicitlyworkouttheequationforadsorptionbetween neighboringadsorbatesin1918.TheHillequationcontinuestoplayanimportantroleinbiologicalanalysisacrossawidevarietyofapplications,includingdrugdiscovery.48Theunderlyingphysicalrationaleforcooperativeeffectsinligandbinding,includingofoxygenonhemoglobin,49remainsanimportanttopicofresearchtoday.

Langmuir’sClassificationsofAdsorption

Langmuiracknowledgedthatthediversityofsurfacechemistryandstructuralgeometryexhibitedbysolidmaterials,inconjunctionwiththediversityinsizeandcharacterofadsorbatespecies,manifestsinwidelydifferentphysicalmechanismsofadsorption.1Langmuiridentifiedandclassifiedsixdistinct,simplemechanismsofadsorptioninLangmuir1918,andthenderivedtheequationgoverningtheamountadsorbed, ,asafunctionofpressure, ,atconstanttemperature,,foreachcase.ThesesixclassificationsaredepictedschematicallyinFigure4.Derivationsofeachcase includingsomespecialsub‐cases,indicatedbyanasterisk follow,alongwithexamplesofrepresentativeadsorptionbehaviorreportedintherecentliterature.

AlthoughLangmuiroriginallyemployedequilibriummass‐actiontheorytoderivetheadsorptionisothermforeachcase,amorefundamentalstatisticalmechanicalderivationispresentedherein.Eachbeginsbyprescribingthepotentialenergyfieldcreatedbythesurfaceofthematerial, ,where ∈ isapointonthesolidsurface.ThepotentialenergyfieldarisesfromrelevantelectrostaticandLondondispersionforcesbetweenthesurfaceofthematerialandtheadsorbatemolecules.Usingstandardstatisticalmechanicalmethods,theadsorptionisothermthenfollowsdirectlyfromthepotentialenergyfield, ,andastipulationforhowtheadsorbedmoleculesinteractwitheachother.Thisisaguest‐hostperspectivewhereitisassumedthattheadsorbatedoesnotaffectthestructureoftheadsorbent.

Figure4.SixadsorptionclassificationswereproposedbyLangmuirin1918:single‐siteor“simple”SSL adsorption,multi‐site MSL adsorption ofwhichasub‐caseisdual‐site DSL adsorption ,generalizedmulti‐site GL adsorption,cooperativeadsorption CA ofwhichasub‐caseisquadraticadsorption QA ,dissociativeadsorption DA ,andmultilayeradsorption MLA .

CaseI:Single‐Site Simple LangmuirAdsorption SSL

Thesimplestcaseofgas‐solidadsorptionisthatonasurface coveringaunitgramofadsorbentmaterial composedof identicalelementaryadsorptionsites,eachcapableofhostingonlyasingleadsorbedmolecule,A .Inthesimplestcase,adsorbate‐adsorbateinteractionsonthesurfaceareneglected,andthebulkgasphaseistakentobeideal.Ifthepotentialenergyofanadsorbateatanybindingsiteonthesurfaceisuniformandtakentobeequalto withdimensionsofenergyper

molecule ,theadsorptionisothermequation theadsorbedquantity, ,asafunctionofbulkgaspressure, isfoundtobe:

1 Equation7

where havingdimensionsofinversepressure istheLangmuirbindingconstantequivalenttothatdefinedinEquation2.

StatisticalMechanicsDerivation.Weconsideracrystallinesurfacewithaunitcellcontainingasingle,molecularlyaccessibleadsorptionsite.Thepotentialenergyofanadsorbatemoleculeatanypoint withinasingleunitcellisdescribedas:

if ∈

∞ if ∉Equation8

where isthesetofpointscomprisingtheadsorptionsite i.e., isfor“bindingsite” ofvolume| | thenotation| |where isasetofpointsisusedtoindicatethe“volumeof ” .The

potentialenergyisinfinityoutsidetheadsorptionsite.Whenthebindingsiteisunoccupied,thepotentialenergyoftheunitcellis0.Thetotalvolumeoftheunitcell,where isthesetofpointscomprisingtheunitcell,isthen | |.

ThefundamentalassumptionsinherenttotheSSLmodelare:

i everybindingsiteisidentical,ii therearenointeractionsbetweenadsorbedmolecules,andiii therearenointeractionsinthegasphase i.e.,itisanidealgas .

Theadsorbedphaseistakentohaveaconstantvolumeandallowedtoexchangeinternalenergyandparticleswithathermal atconstant and“chemical” atconstant reservoir.Therelevantstatisticalmechanicalrepresentationisthegrandcanonicalensemble;atanygivensite,thegrandcanonicalpartitionfunction, ,isasumoverall two possiblestatesofthesite:

, , 1 1 Equation9

where istheinversetemperature , istheBoltzmannconstant, isthethermaldeBrogliewavelengthoftheadsorbatemolecule,and isthechemicalpotentialofthebulkgasphaseinequilibriumwiththeadsorbedphase.ThefirstandsecondtermsinEquation9correspondtothesingleadsorptionsitebeingemptyandoccupied,respectively.Inthisconfigurationalpartitionfunction,the“state”oftheadsorptivemoleculeisacontinuum andintegrationisperformed ,withthethermaldeBrogliewavelengthaccountingforitsunderlyingdiscretenature.

TheadsorptionsitesintheSSLmodelareindependentandhencethetotalgrandcanonicalpartitionfunctionoftheentireadsorbedphaseoverall adsorptionsites, ,isfactorizable:

, , Equation10

Finally,theexpectednumberofadsorbedmolecules, ,followsfromaderivativeofthetotalpartitionfunction:

⟨ ⟩log

, 1 Equation11

Atthermodynamicequilibrium,thebulkgasphase actingasa“chemicalreservoir” imposesitschemicalpotentialontheadsorbedphase,50whichforanidealgasisgivenby:

log Equation12

BysubstitutingEquation12intoEquation11toreplacechemicalpotentialwiththebulkgaspressure,thecommonrepresentationoftheLangmuiradsorptionisotherm Equation7 isobtained,withtheLangmuirconstantexplicitlydefinedas:

Equation13

ThisequationrelatestheLangmuirconstanttothetemperature, via ,thebindingsitevolume,,whichdeterminesthetranslationalentropyofanadsorbedmolecule,andtothepotential

energyofadsorption, .

ModernExample:XenonAdsorptiononSBMOF‐1.TheadsorptionofxenononSBMOF‐1atroomtemperaturebetween0.1‐100kPaisarepresentativeexampleofSSLadsorption.51SBMOF‐1,alsoknownasCa sdb sdb 4,4‐sulfonyldibenzoate ,iscomprisedofone‐dimensionalporeswithconstrictionsof~4.2Åindiameter;insitusingle‐crystalX‐raydiffractionstudiesandmolecularmodelsrevealthatxenonadsorbsatwell‐defined,uniformpockets bindingsites withinthesepores.ThestructureofSBMOF‐1andcontoursofthepotentialenergyofXewithinitspores viacalculationsbasedontheUniversalForceField areshowninFigure5a.Sinceeachwell‐defined,uniformpocketiscapableofholdingonlyasingleatom,theexperimentalXeadsorptionisothermexhibitsclearSSLcharacter.Themaximumbindingsiteoccupancy, ,obtainedbyfittingtheexperimentaldatatotheLangmuirmodel,iscloselymatchedtothetheoreticalmaximumnumberofXeatomsperunitgramofmaterialasestimatedbyinspectingthecrystalstructure shownasadashedblacklineinFigure5a .

Figure5. a EquilibriumadsorptionuptakeofXeonSBMOF‐1at298K,51anexampleofsingle‐siteLangmuir SSL adsorption,and b equilibriumadsorptionuptakeofCO2onMg‐MOF‐74at313K,52anexampleofdual‐siteLangmuir DSL adsorption.

CaseII:Multi‐SiteLangmuirAdsorption MSL

Wenowconsiderasurfacethatoffersmorethanonetypeofelementaryadsorptionsite,yetwhereeachsitemaystillonlyaccommodateasingleadsorbedmolecule,allbindingsitesareindependent,andadsorbate‐adsorbateinteractionsarenegligible.Thecontributionofeachtypeofsitetothetotalamountadsorbedissimplyadditive,leadingtotheMSLadsorptionisothermequation:

1 Equation14

wherethesumisoverthenumberofdifferenttypesofbindingsite, istheLangmuirconstantforsitesoftype ,andthetotalnumberofsitesissimply:

Equation15

CaseII*:Dual‐SiteLangmuirAdsorption DSL

ThesimplestcaseofMSLadsorptionisthatwherethesurfaceofferstwodistincttypesofadsorptionsite,knownasthedual‐siteLangmuir DSL model.Ifthesurfaceiscomprisedof adsorptionsitesofsitevolume , andcharacteristicbindingenergy ,for ∈ 1,2 ,theadsorptionisothermequationis:

1 1 Equation16

AkeyillustrationoftheDSLmodelisthat,asmoregasadsorbs,gasmoleculestendtofilltheadsorptionsitesonaheterogenoussurfacethatofferthelowerpotentialenergybeforeoccupyingthosewithahigherpotentialenergy.Thedisparityinbindingenergybetweenthetwositesandthetemperaturedeterminetheextenttowhichthemorefavorablesitesarefilledfirst.Atdiluteconditions,theratioofthenumberofadsorptionsitesoccupied sitesoftype1tothoseoftype2 is:

,Equation17

followingfromthederivationpresentedbelow.Ahighernumberofadsorptionsites, ,andlargerbindingsitevolume, , ,tendtoincreasethediscrepancyinoccupancybetweenthetwotypesofsite.Thediscrepancyinthepotentialenergyofadsorption, ,hasanexponentialinfluenceonthedifferencesinoccupancy,modulatedbythetemperature.Athighertemperatures,thedifferenceinpotentialenergyofadsorptionbetweenthetwotypesofsiteislessconsequentialasentropyincreasestheexplorationofbothtypesofsite;atlowertemperatures,agreaterdifferenceinoccupancyisobservedforagivendifferenceinpotentialenergies.

StatisticalMechanicalDerivation.Weconsiderasurfacecontainingtwodistincttypesofadsorptionsite.Eachsiteprovidesaspatiallyhomogeneouspotentialenergyofadsorption, ,forasinglegasmolecule,where ∈ 1,2 .Thesingle‐sitegrandcanonicalpartitionfunction, ,foreachtypeofsiteisthesameasthatderivedforSSLadsorption:

, , 1 , Equation18

Thetotalgrandcanonicalpartitionfunctionisfactorizablesinceeachadsorptionsiteisindependent,andthus:

, , Equation19

Theexpectednumberofadsorbedgasmoleculesisthen:

⟨ ⟩log

1 ,Equation20

Thisequationcanberewrittenas⟨ ⟩ ⟨ , ⟩ ⟨ , ⟩,where⟨ , ⟩istheexpectednumberofmoleculesadsorbedonthesitesoftype .Thus,thecontributionbyeachtypeofsiteispurelyadditive.UsingEquation12toreplacechemicalpotentialwiththepressureofthebulkgasphase,theDSLadsorptionisotherm Equation16 isachieved.

ModernExample:CO2AdsorptiononMg‐MOF‐74.ArepresentativeexampleofDSLadsorptionisthatofCO2onMg‐MOF‐74at313K,asshowninFigure5b.52TheMgformofMOF‐74,alsoknownasMg2 dobdc dobdc 2,5‐dioxido‐1,4‐benzenedicarboxylate ,iscomprisedofone‐dimensional,hexagonalchannelsboundedbyexposedsquarepyramidalMg2 cationsthatareknowntobesitesofstrongadsorptionformanysmallgasmolecules.Theinflectionintheadsorptionisothermat~105PaisaclearindicationofDSLcharacter,andisduetothelargedisparityinbindingenergybetweentheadsorptionsitesneartheexposedMg2 metalcentersandelsewhereonthesurface~42and~24kJmol ,respectively .FittinganalysisperformedhereinyieldstwodistinctLangmuirconstants: 2.49 10 Pa and 1.28 10 Pa .IncontrastwithCOadsorptiononglassasreportedinLangmuir1918,bothdistinctadsorptionsitesinMg‐MOF‐74areofphysisorptivetype,allowingtheentireadsorptionisothermtobereversibleat313K.

CaseIII:GeneralizedLangmuirAdsorption GL

WhileCaseIIisdesignedtohandlemultipletypesofdistinctadsorptionsites,anamorphousmaterialmaycompriseanintractablenumberofdifferentadsorptionsiteswithdifferingaffinitiestowardtheadsorbate.Langmuirsuggestedtotreatthedistributionofsiteaffinitiesinsuchanamorphousmaterialasacontinuum.3Neglectingadsorbate‐adsorbateattractions,theadsorptionisothermfollowsfromthedistributionofbindingenergiesamongtheadsorptionsites.

Let bethenumberofelementaryspacesontheentiresurfacewithacharacteristicbindingenergybetween and .Weneglecttoemployaunitcell,permittingthetreatmentofeithercrystallineoramorphoussurfaces.Necessarily, ,thetotalnumberofelementaryspacesofferedbythesurface.Whentheroleofadsorbate‐adsorbateattractionscanbeneglected,theadsorptionisothermissimplythecontinuumanalogytotheMSLmodel:

1 Equation21

where istheLangmuirconstantofasiteofferingenergyofadsorption .

Langmuirnotedthatanimpracticallyintimateknowledgeofthesurfacestructurewouldbeneededtowrite explicitly.Nevertheless,Langmuirspeculatedthatconclusionsaboutthecharacterof

couldinfactbedrawnfromtheexperimentaladsorptionisotherm,3whichwaslatervalidatedbySips.18AstrongassumptioninherenttoEquation21isthat isthesameforalladsorptionsites,53butSipsnotedthat can,inprinciple,alsobewrittenasafunctionof .19AsimplecaseofEquation21iswherethedistributionisaDiracdeltafunctioncenteredat ,orinotherwordswhere .Inthiscase,alladsorptionsitesareidenticalandtheSSLmodelisrecovered asinEquation7 .

CaseIII*:UniformLangmuirorUnilanAdsorption UL

AnespeciallyusefulcaseofEquation21isknownastheUnilan for“uniformLangmuir” ,orULmodel.54Inthiscase,allofthebindingsitesareuniqueandthebindingenergiesaretakentobeuniformlydistributedbetween .19Thisisexpressedas:

if ∈ ,

0 otherwise

Equation22

Byimposingthisdistribution,Equation21resultsintheUnilanequation:

1 Equation23

where isrelatedtotherangeofadsorptionenergiesas:

2 Equation24

IntheUnilanformalism, istheLangmuirconstantofahypotheticalmaterialwithuniformadsorptionsitesexhibitingtheaveragebindingenergy,givenas:

Equation25

StatisticalMechanicalDerivation.ThestatisticalmechanicalderivationoftheUnilanequationisequivalenttothatforCaseII MSL exceptwithaverylargenumberofdifferentsingle‐sitepartitionfunctionsequaltothenumberoftotalsites .EachpartitionfunctionhasthesameformasEquation18.Thetotalgrandcanonicalpartitionfunctionissimplytheproduct:

, , … Equation26

TheUnilanequation Equation23 followsasusualbydeterminingtheexpectednumberofadsorbedmoleculesatconstanttemperature,volume,andchemicalpotential.18

ThegeneralizationofLangmuir’sSSLadsorptionmodeltoaccountfordifferenttypesofadsorptionsites MSLadsorption permitsitsapplicationtomaterialswithheterogeneoussurfaces.However,theextremecase,acompletelyamorphoussurfacewhereeverybindingsiteisdifferent,wouldrequiremorethan characteristicparameterstobemodelledinthisway,anunreasonablylargenumber.TheULequationisanexampleofapracticalsolutiontothisproblem,allowingforawiderangeofbindingenergiesbutenforcingauniformdistributionofthem,andtherebyminimizingthenumberofcharacteristicparametersofthesystem.TheULmodelisjustifiedforanysystem

exhibitingadistributionofbindingsitessolongasadsorbate‐adsorbateinteractionsremaininsignificantandthebulkgasphaseisideal.

ModernExample:H2AdsorptiononZTC.Atypicalcharacteristicofgasadsorptiononamorphoussurfacesisagentleslopingsigmoidaluptakeprofilewhenplottedonlog‐normalaxesasafunctionofpressure.ThisasdemonstratedbyH2adsorptiononzeolite‐templatedcarbon ZTC 55at77K,asshowninFigure6a.56Theexperimentaldatapointsareeachuniquelycoloredtorepresentdifferentcharacteristicbindingsites.TheamorphousnatureofthemicroporoussurfaceofZTCleadstoamultitudeofdifferentbindingenergies;theaverageLangmuirconstantis 2.9 10 Pa .ItisclearthatnooneSSLequationwouldfitthesedataoveranappreciablerangeofpressure.TheSSLisothermforahypotheticalmaterialwithidenticalbindingsitesofthesameaveragebindingenergyasinthedistributiononthesurfaceofZTC correspondingto isshownasapurplelineandissignificantlysteeper.AULmodelfitsthedatawell shownasablackline ,usingonlythreeindependentparameters,owingtothefactthatH2hasweakintermolecularinteractionsandthatZTChasnoshort‐rangeorderedmolecularstructure.57Owingtoitssimplefunctionalformyetcomplexphysicalrepresentation,theULmodelisalsosuccessfulinengineeringapplicationssuchasthedescriptionofH2adsorptiononMOF‐5acrossawiderangeoftemperatureandpressure.58

Figure6. a EquilibriumadsorptionuptakeofH2onzeolite‐templatedcarbon ZTC at77K,56anexampleofgeneralizedLangmuir GL adsorption. b EquilibriumadsorptionuptakeofN2simulatedonIRMOF‐16at77K,59anexampleofmultilayeradsorption MLA withinaporousmaterial.

CaseIV:CooperativeAdsorption CA

Wereturntothecasewhereeachbindingsiteonasurfaceisidentical,butnowwhereeachsiteispermittedtohostmultipleadsorbedmolecules.Theadsorptionofthefirstmoleculeonagivensiteaffectstheenvironment potentialenergyofadsorptionandvolumeavailablefortranslationalentropy ofthesecondadsorbingmolecule,andsoonasmoremoleculesadsorbonthesamesite.Inotherwords,adsorptioniscooperativesincethepresenceofotheradsorbatesonthesamesiteaffectstheenergeticsoffurtheradsorption.Thegeneralcasewhere neighboringmoleculesareallowedtointeractiscomplex,andtheinterestedreaderisreferredelsewhere.53

CaseIV*:QuadraticAdsorption QA

ThesimplestCAmodelisthatofasurfacewhereeachbindingsitecanaccommodateuptotwoadsorbedspecies,andwhereanadsorbate‐adsorbateintermolecularinteractiontakesplacewhenbothsitesareoccupied.Inthisparticularcase,thequadraticadsorption QA isothermisobtained:

2 21 2 Equation27

where istheLangmuirbindingconstantcorrespondingtoeachhalf‐site identicaltoEquation13 , isthenumberofadsorptionsites eachbindinguptotwomolecules ,and isafactordescribingthemagnitudeofadsorbate‐adsorbateinteractions whentwomoleculesareboundonasinglesite .Fornon‐interactingmolecules, 1andEquation27reducestothesimpleLangmuiradsorptionisotherm Equation7 withamaximumadsorptionoccupancyof2 .Inthenon‐trivialcase, 1forattractiveadsorbate‐adsorbateinteractionsand 1forrepulsiveadsorbate‐adsorbateinteractions.Inthelattercase,adsorptionofthesecondmoleculeislessfavoredthanthatofthefirst,forexampleduetosterichindrance.

StatisticalMechanicalDerivation.Thepotentialenergyofeachadsorptionsitedependsonitsmicrostate;eachsitecanaccommodateuptotwomoleculesandsohasthreedistinctmicrostatesempty,singlyoccupied,ordoublyoccupied .Welet , ∈ 0,1 determinethestateofeachhalfofanadsorptionsite,where 0correspondstothehalf‐site asemptyand 1tothehalf‐siteasoccupied.Thepotentialenergyofanindividualadsorptionsite, ,isthen:

, Equation28

where isthespatiallyuniformpotentialenergyexperiencedbyasinglemoleculeadsorbedoneitherpartoftheadsorptionsiteduetoitsinteractionwiththesurface,and isthepotentialenergyassociatedwiththeinteractionbetweentwoadsorbedmoleculeswhenthesiteisdoublyoccupied.Thegrandcanonicalpartitionfunctionforthesingleadsorptionsiteisthenfoundbysummingoverallofthemicrostates atotaloffourmicrostates,withtwodegeneratesinglyoccupiedstates :

, , , ,

1 2Equation 29

The termarisesfromtheconfigurationalpartofthepartitionfunctionwheretheBoltzmannfactorisintegratedoverallparticlecoordinates; isthefreevolumeofferedbyeachadsorptionhalf‐siteinthepair.Thefirst,second,andthirdtermsinthesumcorrespondtothethreedistinguishablemicrostatesidentifiedabove.For independentadsorptionsites upto pairsofadsorbedmolecules ,thetotalgrandcanonicalpartitionfunctionisfactorizable:

, , Equation30

Theexpectednumberofadsorbedmoleculesisthen:

⟨ ⟩log

1 2 Equation31

ByreplacingthechemicalpotentialwiththepressureoftheidealgasviaEquation12,theresultgivenabove Equation27 isachievedwith asintheSSLmodeland .Thusif 0 forattractiveinteractions ,then 1,andif 0 forrepulsiveinteractions ,then 1,asexpected.

CaseIV*:HillEquation

Wheneachadsorptionsiteisrequiredtobeeitheremptyoroccupiedbythemaximumnumberofoccupants,theHillequation Equation6 isobtained.Ifthemaximumnumberofoccupantspersiteistwo,theformalderivationfollowsthatofthemoregeneralQAmodelabove,exceptwithonlytwotermsinthesingle‐sitepartitionfunction:

, , 1 Equation32

Thetotalgrandcanonicalpartitionfunctionremainsfactorizable:

, , Equation33

TheHillequationfor 2followsfromtheusualderivative:

⟨ ⟩log

1 Equation34

ByreplacingthechemicalpotentialwiththepressureoftheidealgasphaseviaEquation12,thefollowingadsorptionisotherm equivalenttoEquation6 isachieved:

⟨ ⟩ 21 Equation35

Themagnitudeoftheinteraction, ,isdefinedasaboveandtheLangmuirconstantisdefinedasinEquation13.TheHillequationisanover‐simplifiedapproximationformostadsorptionsystemswhereanynumberofmoleculesbetweenzeroandthemaximumnumberofoccupantscanoccupyagivensite ,butremainsacommonlyemployedmodel,nevertheless.

ModernExamples.Cooperativeadsorptionisrareinthemodernadsorptionliterature,andcasesofpureQAbehaviorinagas‐solidsystemareunknowntotheauthors.SomerepresentativeexamplesofCA‐likebehavioraregivenbyCO2adsorptionontheUSO‐2‐MseriesofMOFsat298K,especiallyinthenickelform M Ni .60‐61AsecondexampleisthatofethaneonvariousMOF‐74‐typematerials,especiallyMn‐andCo‐MOF‐74,wherethemaximumoccupancycorrespondsroughlytotwoethanemoleculesperformulaunit.62‐63TheadsorptionofCO2onthealuminumformofMIL‐91alsoexhibitsaCA‐likeinflection.64Lastly,thepresenceofdynamicmoietieswithinaporouscrystalmayleadtophenomenaresemblingCAbehavior.65

CaseV:DissociativeAdsorption DA

Inthepreviouscases,itwasassumedthattheadsorbatedoesnotundergoanysortofchemicalchangeuponadsorptionordesorption.InCaseV,Langmuirconsideredthepossibilityofadsorptionbeingatwo‐foldprocess:residenceatthesurfaceadsorptionsiteincombinationwithmoleculardissociation aresultofchemicalbonding .Thistypeofbehaviorcommonlyoccursatthesurfaceofacatalyst,forexample,whereadiatomicmolecule A adsorbsatabindingsiteandsubsequentlydissociatesintotwoatoms A ,whicharethenpermittedtodiffuseonthesurface freelyexploring

theavailableadsorptionsitesasanatomicspecies .Thereverseprocessrequirestwoneighboringatomsonthesurfacetoreassociateintothediatomicmolecule A A → A andleavethesurfacei.e.,undergodesorption .Inthisspecificcaseofhomonucleardiatomic/monatomicdissociativeadsorption DA ,theadsorptionisothermoftheatomicspecies,A,asafunctionofthepressureofthebulkgaseousmolecularspecies,A ,is:

1 √Equation36

StatisticalMechanicalDerivation.Atthermodynamicequilibriumofthesimplestdissociationreactiongivenabove A ↔ A A ,thechemicalpotentialsmustsatisfythefollowingrelationship:

2 Equation37

Thatis,thebulkgasphaseofthemolecularspecies,A ,imposesachemicalpotentialontheadsorbedatomicspecies,A,atthermodynamicequilibrium.ThismodeldoesnotpermitanydiatomicA toremainadsorbedonthesurface;dissociationandadsorptionorreassociationanddesorptionareconcomitantinstableequilibrium.Let bethespatiallyhomogeneouspotentialenergyexperiencedbyanatomicspeciesAadsorbedonthesurfaceand bethefreevolumeofthebindingsite thataccommodatesasingleatom .Then,thesingle‐sitegrandcanonicalpartitionfunctionis:

, , 1 Equation38

ThechemicalpotentialanddeBrogliewavelengthbothcorrespondtotheatomicspecies,A,sinceitistherelevantspeciesintheadsorbedphase.Thetotalgrandcanonicalpartitionfunctionforasurfacecontaining independentadsorptionsitesisthen:

, , Equation39

Theexpectednumberofmonatomicadsorbedspecies,A,isthen:

⟨ ⟩log

, 1 Equation40

UsingEquation37andEquation12torelatethebulkgaspressuretothechemicalpotentialoftheatomicadsorbedspecies,A,theDAisothermisobtained Equation36 .TheLangmuirbindingconstantisexplicitlydefinedas:

Equation41

withdimensionsofpressuretothepowerof½,wherethepressurereferstothediatomicgas.

ModernExamples.Typicalexamplesofdissociativeadsorption DA areforsimplediatomicmoleculeslikeH2onspecificsurfacessuchaspalladium,platinum,tungsten,andsilicon e.g.,Honsilicon 111 7 7at950K66 .Athoroughreviewofhydrogenonsiliconisgivenelsewhere.67TheauthorsarenotawareofanystrictexamplesofdissociativeadsorptiondirectlyonMOFsurfaceswithoutthepresenceofmetalnanoparticles,asinhydrogenspillover68 ,whichhasbeenattributedtothehighactivationbarriertodissociation.69

CaseVI:MultilayerAdsorption MLA

Lastly,Langmuiraddressedthecaseofmultilayeradsorptionuponafinitesetofunderlyingadsorptionsitesonasurface.Inthesimplestsuchcase,eachadsorptionsiteistakentobeidenticalandindependent;afterthefirstmoleculeisadsorbedonagivensite,anadditionalmoleculeispermittedtoadsorbaboveit.Inthisway,despitethatthenumberofadsorptionsitesislimited,thenumberofadsorbedmoleculesisunlimited assumingalargechemicalreservoirisheldincontactwiththesurfaceatconstantchemicalpotential anddivergestoinfinityatthesaturationpressureinthecaseofsubcriticaladsorption .Langmuirdidnotexplicitlywriteaclosed‐formadsorptionisothermforCaseVI,butdidremarkthatsuchanisothermwouldbegreatlysimplifiedifthesecondlayerandallabovewereconsideredunderidenticalconditions oreven,slightlymorecomplexly,thethirdlayerandallabove .3

CaseVI*:Brunauer‐Emmett‐Teller BET MultilayerAdsorption

Themostwell‐knownmultilayeradsorptionisothermistheBrunauer‐Emmett‐Teller BET model;itstipulatesthatadsorptioninthefirstlayeroccurswithaconstantbindingenergyandthatadsorptioninthesecond,third,andallhigherlayersoccurswithabindingenergyequivalenttotheenergyofcondensation liquefaction ofthebulkgasphase.36Langmuir’smoregeneralCaseVIpredatesBETtheorybytwodecadesbutisinfactinclusiveoftheBETstipulation.Hence,inthisreviewwehighlighttheBETequationasaspecificexampleofCaseVIsinceitisthemostwidelyusedadsorptionisothermequation,oftenemployedtocalculatethespecificsurfaceareaofporousandnonporousmaterials.IntheBETmodel,eachbindingsitecanaccommodatean“adsorptioncolumn”consistingofasingleadsorbedmoleculeatthebaseanduptoaninfinitenumberofadsorbedmoleculesaboveit.Adsorbate‐adsorbateinteractionswithineachcolumnareignored,aswellasinteractionsbetweenadsorptionsites.Forasurfacecontaining adsorptionsites,theBETmultilayeradsorptionisothermis:

1 1 Equation42

Asusual, istheLangmuirbindingconstantdescribingtheinteractionbetweentheadsorbateinthefirstmonolayerandthesolidsurface; isananalogousbindingconstantdescribingtheaffinityofanadsorbateinoneofthehigherlayerstowardtheadsorbatemoleculebelowit takentobeequivalenttothatofbulkliquefaction .

StatisticalMechanicalDerivation.AsfortheQAmodel,thepotentialenergyofadsorptionmustfirstbeexpressedasafunctionofeachmicrostateofagivenadsorptionsite.Asbefore,let bethepotentialenergyofamoleculeadsorbedinthefirstlayer,directlyinteractingwiththeadsorbentsurface.Inallsubsequentlayers,thepotentialenergyassociatedwithadsorptionis ,aconsequenceofinteractingwiththegasmoleculeinthelayerbelow.Thereisaninfinitesetofmicrostatesforeachadsorptionsite:empty,singlyoccupied,doublyoccupied,andsoon,uptoinfinitelyoccupied.With asthenumberofmoleculesadsorbedonanadsorptionsite,thepotentialenergyofasinglecolumnofmoleculesasafunctionofitsmicrostateis:

0 if 0

1 if ∈ 1, 2, …Equation43

Thegrandcanonicalpartitionfunctionofasingleadsorptionsite column isthenasumoverallpossiblemicrostatesincludingtheemptystate ∈ 0,1, 2, … :

, , 1 ⋯ Equation44

Thecanonicalpartitionfunctionsofasinglemoleculeadsorbedonthefirstlayerandinsubsequentlayersare,respectively:

Equation45

Equation46

ByfactoringoutthecontributionineachterminEquation43fromthemoleculeinthefirstlayer,werecognizethegeometricseries:

, , 1 1 ⋯ 11 Equation47

Thetotalgrandcanonicalpartitionfunctionforasurfacecontaining independentadsorptionsitesisfactorizable:

, , Equation48

Thetotalexpectednumberofadsorbedmoleculesisthen:

⟨ ⟩log

, 1 1 Equation49

ReplacingchemicalpotentialinEquation49withthepressureofthebulkgasphaseviaEquation12,wearriveattheBETadsorptionisotherm Equation42 with definedasinEquation13and:

, Equation50

ModernExample:N2AdsorptiononIRMOF‐16.ArepresentativeexampleofBETadsorptionwithinaporousmaterialisN2adsorptiononthesurfaceoflarge‐poreMOFssuchasIRMOF‐12,‐14,and‐16,wherethelatterisshowninFigure6b.59,70The~2nmwideporesofIRMOF‐16canaccommodatemultilayeradsorptionoveralimitedpressurerangeupto~20kPawhencross‐poreinteractionsbecomesignificant.Thiseffectisobviousathigherpressureswherethemeasuredadsorptionuptakedepartsdramaticallyfromtheexpecteduptake,firsttowardhigheruptakeasporefillingcompletes exhibitingstrongintermolecularinteractionsbetweenmultilayersonbothsidesofthepore andthenshowingabruptfillingandnofurtheradsorption.SimilarMLAisothermshavebeenmeasuredforN2adsorptiononotherlarge‐poreMOFssuchasNU‐100,71NU‐1300,72andNU‐109andNU‐11073.

Belowthe“rougheningtemperature,”layer‐by‐layeradsorptiononcrystallinesurfacesisdistinctlyobservedtooccur,resultingfromsignificantlydifferentbindinginteractionscharacteristicofeachlayer e.g.,Xeadsorptiononpalladium 100 between60‐70K74 .SuchanadsorptionsystemisaccuratelydescribedbyamorecomplexMLAequationthanthatoftheBETmodel,withindividualLangmuirconstantsassignedtoeachlayer.

ApplicationsandExtensionsofLangmuir’sTheory

TheLangmuirtheoryofadsorptionisusefulformodelinganddesigningengineeringprocesses,bothqualitativelyandquantitatively.SinceeachLangmuirmodelisfoundedonreasonablephysicalassumptions,thermodynamicconsistencyisalsoinherenttoeachadsorptionisotherm,andthecalculationofthermodynamicquantitiessuchastheenthalpyofadsorptionisdirectandanalytical.ThemostcommonmaterialpropertiescalculatedviaLangmuir’stheoryarethespecificsurfaceareaandtheisostericenthalpyofadsorption.Twocommonengineeringprocessesofinteresttogas‐solidadsorptionsystemsarestorageandseparation.

SurfaceAreaMeasurements.Specificsurfaceareaisanimportantcharacteristicofporousmaterials,especiallyfordeterminingtheapplicabilityofamaterialforengineeringapplicationssuchasstorageorseparation.ThetwomostwidelyusedequationsaretheBETandSSLmodels,thelatterofwhichwasfirstusedbyLangmuirhimselfin19183.Whileneithermodelisstrictlyappropriateforthedescriptionofgasadsorptionwithinnarrowmicropores,ithasbeenshownthattheBETmodelisoftensuitablefortheestimationofthetruesurfaceareaofmicroporousandmesoporousmaterialsincludingMOFsandzeolites despitethatthenumberofmultilayersisconstrained whenproperconsistencycriteria75areemployedindeterminingtherangeofpartialpressureoverwhichtofitthedata.59,76‐77

TheBETformalismprevalentintheliteratureiswrittenasfollows:

/1 / 1 1 / Equation51

where isthesaturationpressureoftheadsorbate,definedas:

Equation52

and istheBETconstant,definedas:

Equation53

Acommonlinearizedrearrangement,whereBissimplythe“BETvariable”,is:

1 1/ Equation54

Brunauer,Emmett,andTellerdescribedthreeregionsalongtheBETisotherminEquation54:aconcaveregionatlowpressure,aconvexregionathighpressure,andalinearregionatintermediatepressure.Theyoriginallyspecifiedtherelativepressure / rangeof0.05‐0.3asthelinearregionfromwhichthenumberofsurfacesites, ,canbedependablyextracted.Inthecaseofhighsurfacearea,microporousmaterialssuchaszeolitesandMOFs,however,thisrangehasproventobeinadequateasauniversalstandardandanowwidelyacceptedsetofself‐consistencycriteriahavebeenproposed.75Oncethelinearrangeisdeterminedand isextracted,theBETsurfacearea, ,iscalculatedas:

∙ Equation55

where isthesurfaceareaofasinglebindingsite.Theconventionalcross‐sectionalareaofaN2moleculeforBETsurfaceareacalculationis0.162nm2,whichallowsresearchersacrossdifferentlaboratoriestohaveastandardformaterialscomparison.Forexample,inthecaseofIRMOF‐16,theBETsurfacearea monolayercapacity,shownasadashedblacklineinFigure6b determinedhereincorrespondsto61.8mmolg or6030m g .Thisisconsistentwiththegeometricalsurfaceareaofthecrystal ~6000m g asdeterminedbyaMonteCarlointegrationtechnique59 .ItshouldbenotedthatchallengesstillpersistinusingtheBETmodeltoaccuratelyestimatethemonolayercapacityofporousmaterials,especiallythosewithadiversityofporesizes.77

EnthalpyofAdsorption.Anotherimportantcharacteristicfordeterminingtheapplicabilityofaporousmaterialtostorageandseparationprocessesistheadsorptionbindingstrength.Whilethebindingenergy orenergies,inthecaseofamulti‐siteadsorptionmodel couldinprinciplebedirectlyextractedfromanyofLangmuir’ssimplemodelsviatherelationshipinEquation13,amoregeneral lessmodel‐dependent approachistocalculatethedifferenceinenthalpybetweentheadsorbedphaseandthebulkgasphase,∆ ,viatheClapeyronequationoravariantthereof.78‐79Whenthebulkgasphaseisideal,therearrangedClausius‐Clapeyronequationisemployed:

∆ ∆ Equation56

Thisrequiresthemeasurementofmultipleisothermsatdifferenttemperatures,andinvestigationofthetemperature‐pressurerelationshipalongalineofconstantadsorbedamount anisostere .Theresultingthermodynamicquantityisreferredtoastheisostericenthalpy oroften“isostericheat” ofadsorption;forSSLadsorptionunderabulkgasphasethatisideal,itisequaltothenegativeofthebindingenergy i.e.,∆ .TheSSL,DSL,andULmodelsareroutinelyemployedtointerpolatemeasureddata,animportantstepinthecalculationoftheisostericenthalpyofadsorption.58,80Whenasufficientquantityofdataaremeasured,theenthalpyofadsorptionmeasuredbythesorptionisostericmethodcomparesreasonablyaccuratelytocalorimetricmeasurements.81Itshouldbenotedthatthereremainsdebateastothevalidityoftheisostericmethodinthehighpressure non‐idealgas regimewherethemeasuredquantityofadsorption the“Gibbssurfaceexcess” departsfromtheactualquantity;undersuchconditions,aso‐called“isoexcess”methodhasbeenproposedwhichdoesnotrelyontheuseofanyadsorptionmodel.82‐83

DeliverableGasStorage.Thekeymaterialdesignmetricforadsorptivestorageisthedeliverablecapacity:theamountadsorbedatthestoragepressure, ,minusthatremainingintheadsorbedphasewhenthepressureisbelowtheusefulthreshold, e.g.,whenthestoragevesselcannolongersupplyafuelcellorcombustionengine .Forexample,theUSDepartmentofEnergy DOE targetfornaturalgasstorageon‐boardmobilevehiclessets 6.5MPaand 0.5MPafordeliveryatroomtemperature.InanadsorptionsystemcharacterizedbySSLtheory Equation7 ,thedeliverablecapacity, ,foranisothermalpressureswingbetween and is:

1 1Equation57

Foragivenstorageanddeliverypressure,thedeliverablecapacitydependsimportantlyonthematerialproperties, and .Whenthenumberofadsorptionsites, ,isfixed i.e.,maximized ,theoptimalLangmuirconstanttomaximize is:

1Equation58

Ifthebindingenergyistooweak,correspondingto ,thelowstoragecapacityatthestoragepressure, ,limitsthedeliverableamountofadsorbedgas.Ontheotherhand,ifthebindingenergyistoostrong,correspondingto ,asignificantquantityofadsorbateremainsboundattheminimumdischargepressure, ,limitingthedeliverablecapacityaswell.TheseeffectsareoptimallybalancedforamaterialthatexhibitsaLangmuirconstantgivenby .Then,byassumingatypicalentropyofadsorptionbasedonthedifferenceinentropybetweenthebulkgasandtheadsorbedphases,thisanalysiscanbeusedtoestimatetheoptimalenthalpyofadsorptiontomaximizethedeliverablestoragecapacity.84SeveralMOFshavebeenidentifiedbyasimilarmethodasrecord‐holdingmaterialsforbothmethane85andhydrogen86storageatambientconditions,andthesearchcontinuesforhighercapacitymaterialsusingadesign‐drivenapproach.

Mixed Competitive Adsorption.Anotherchiefapplicationofporousmaterialsistoseparategaseousmixturesthroughselectiveadsorption.Hence,itisusefultoconsiderthecaseofmixedgasadsorptionwheretwodifferentadsorbatespeciescompeteforthesamesetofadsorptionsites.Considerasurfacecomprising adsorptionsitesexposedtoanidealgasmixture A B .IftheAandBmoleculesarecomparableinsizesothateachadsorptionsiteaccommodatesonlyasingleAorBmolecule,theadsorptionisotherm where isthetotalnumberofAandBmolecules is:

1Equation59

where and aretheLangmuirconstantscorrespondingtothebindingofspeciesAandB,andand arethepartialpressuresofspeciesinthemixedgasphase oftotalpressure .87The

statisticalmechanicalderivationuncoversthedependenceoftheLangmuirconstantonthevolumeofthebindingsite, , ,andbindingenergy, ,wherebothcharacteristicsvarydependingonthespecies occupyingtheadsorptionsite,aswellasthetemperature.Thereby,theratioofthenumberofadsorptionsitesoccupiedbyspeciesAtospeciesBatdiluteconditionsis:

,Equation60

whichdependsontheratioofpartialpressures,ontheratiooffreevolumeavailabletoeachadsorbate anentropiceffect ,andexponentiallyonthedifferenceinbindingenergies.AsintheDSLmodel,lowertemperatureservestoamplifytheoccupancyratiodependenceonthedifferenceinpotentialenergiesofadsorptionofthetwospecies.ArepresentativeexampleofthedirectapplicationofEquation59tomeasurementsofmixed‐adsorptionequilibriaisfoundintheadsorptionofbinarymixturesofCO2,H2,andN2onactivatedcarbon.88

Themeasurementofmixed‐adsorptionequilibriarequiresanadditionalstepoverpureadsorptionmeasurements:varyingthecompositionofthebulk gaseousorsupercritical fluidandthenobtainingthecorrespondingcompositionoftheadsorbedphase,whichistime‐intensiveandtechnicallycomplicated.89Tocircumventtheselimitations,adsorptionselectivityisveryoftenestimatedbycomparingthepuregasadsorptionequilibriaofeachofthecomponentsofthedesiredmixtureandapplyingasimplemodel.Idealadsorbedsolutiontheory IAST 90isasimplemodelfordescribingthemixedadsorptionstatebasedonthepurecomponentadsorptionequilibria,analogoustoRaoult’slawforidealvapor/liquidsolutions.Themainprincipleliesintheconceptof

aconstantspreadingpressureoftheadsorbedphaseatequilibrium,whereinalladsorbatespeciesareassumedtobenon‐interacting.Fromthepure‐stateadsorptionisotherms, ○ ○, ,thespreadingpressure, ,isestimatedforeachcomponentasafunctionofthepressureofthepurebulkphaseaboveit, ○;inturn,thisrelationshipisextendedtotheco‐adsorptionstatebyassumingthatthe individual adsorbedphaseofeachcomponentwillbehavethesameasafunctionofthepartialpressureofthatcomponentinthemixedstate.Thebinarymixed‐statetotaladsorptionisotherm, , , ,foragivengasphasecompositiondefinedby , canthereforebeestimatedinadditiontotheindividual mixed‐state adsorptionisothermsofeachcomponent,

TheresultsofIASTanalysisareremarkablyaccuratewhencomparedtoexperimentalmixed‐adsorptionequilibria91‐92andalsowhenIASTisappliedtoGrandCanonicalMonteCarlo GCMC pure‐adsorptionsimulationsandcomparedtomixed‐adsorptionGCMCsimulations93‐94.However,theoverallaccuracyofthemethodisdependentonthesimilarityofthecomponentsoftheadsorbedphaseaswellasthebindingstrengthtowardandhomogeneityoftheadsorbentsurface;correctionscanbeappliedforsystemswheretheco‐adsorbedphaseissignificantlynon‐ideal,referredtoasRAST.95‐96TheanalysisofnumerousadsorptionpairsandmixtureshavebeencarriedoutusingIAST,demonstratingextraordinaryprospectsforusingMOFs,97zeolites,andotherdesignableporousmaterialsassolid‐phaseseparationmedia:e.g.,asincarboncapture,98‐99hydrogenrecycling,100hydrocarbonseparations,62,101andharmfulgasseparationsfromair,102amongmanyotherapplications.

Flexible,Gas‐ResponsiveMaterials.Itisusuallyassumedthattheadsorptionofgasmoleculesdoesnotchangethethermodynamicpropertiesorstructureoftheadsorbentitself.Suchaguest‐hostperspectiveisasimplificationthatisnotapplicableinporousmaterialsthatundergostructuralchangesinducedbytheadsorptionofgasmolecules:so‐called“flexible,”“soft,”or“breathing”materials.103‐105ThestructuralchangesinflexibleMOFsoftenhaveaprofoundinfluenceonadsorptionandcanleadtocounterintuitivepropertiessuchasnegativegasadsorptionsteps.106‐107TheLangmuirmodelservesasaplatformontowhichfurthercomplexity,suchasflexibility,canbeintegrated.

Gate‐openingmaterialsexistinacollapsed,nonporousstateinvacuumthatpersistsevenwhentheyareexposedtolowpressures lowchemicalpotential ofthegaseousadsorbate.108Asthechemicalpotentialofthegaseousreservoirisincreased,atsomeconditionthematerialabruptlyexpandstoaporousstateandanadsorptionstepoccurs.Aflagshipexampleofgate‐openingbehaviorisintheadsorptionofCH4onCo bdp wherebdp2‐ 1,4‐benzenedipyrazolate .109Atroomtemperature,gateopeningoccursconcomitantwithanabruptstepat1.6MPaontheadsorptionbranchoftheCH4uptakeisotherm;thereversetransitionisdetectedat0.7MPaonthedesorptionbranch.Importantly,thisstepoccursbetweenthedesiredpressureswingrangeinvolvedinvehicularadsorbednaturalgasstorage,therebyimpartingCo bdp withanextremelyhighdeliverablecapacityofCH4: 197LSTPL‐1 ~10mmolg‐1 .Moreover,thestructuraltransitioninCo bdp isendothermic,offsettingaportionoftheheatofadsorptionandmitigatingstorageinefficienciesuponrefilling.

Breathingmaterials,likegate‐openingmaterials,exhibittwostructuralformsthatarestableunderdifferentconditions:anarrow‐poreandlarge‐porestate.Thelarge‐porestateofabreathingmaterialisstableinvacuumanduptolowpressures lowchemicalpotential ofthegaseousreservoir .Atanintermediatetransitionpressure,thelarge‐porestatecollapsestothenarrow‐

porestate,affordingastrongeradsorptioninteraction.Atahighertransitionpressure,theframeworkexpandsagaintothelarge‐porestate,accommodatingmoreadsorbedmoleculeswithaweakercharacteristicbindingenergy.AflagshipexampleofbreathingbehaviorisinCO2adsorptiononM‐MIL‐53 M Cr,Al .110‐111AtroomtemperatureonAl‐MIL‐53,thefirstabruptstepiscenteredat24 15kPaandthesecondat425 75kPawherethepressurerangecorrespondstothewidthofthehysteresisloop.111

Ageneralstatisticalmechanicaltreatmentofgate‐openingandbreathingingas‐solidadsorptionsystemshasbeendeveloped.112Insteadofemployingthegrandcanonicalensembleasintheabovesixderivations wheretheadsorbedphaseisheldatconstant , ,and ,theosmoticensembleispreferred atconstant , ,and .Inthistreatment,thechemicalpotentialimposedbythebulkgasphasecanbedecoupledfromthemechanicalpressuresothattheeffectsofapplyingmechanicalforceonthesolidadsorbentcanbestudied.SuchexperimentshavebeenperformedwithCo bdp ,illustratingthattheapplicationofmechanicalforcecanservetocollapsetheporousstateofagate‐openingmaterialandexpeltheadsorbedgas.Thetwo“rigidhost”adsorptionsystemsforthetwodistinctstatesofthegate‐openingadsorbentcouldbeaccuratelymodeledbySSLequations.109

Structuraltransitionsofadsorbentmaterialsinthepresenceofgaseousadsorbatemayalsooccurwithoutanyexpansionorcontractionoftheunitcell.ArepresentativeexampleofsuchbehavioristhatofXeadsorptiononNi SiF6 pyz 2 pyz pyrazine ,apillaredsquaregridMOFwithrotatablepyrazinestruts.113TheXeadsorptionisothermatroomtemperatureexhibitsapronouncedinflectionat~50kPa;insituX‐raydiffractionmeasurements,densityfunctionaltheorycalculations,andclassicalMonteCarlomolecularsimulationsindicatethattheinflectionisaresultoftherotationalstatesofthepyrazineligandsorganizingtoachieveamorefavorablegas‐hostinteraction.AsimplestatisticalmechanicalmodelofporousMOFswithrotatingligandswassubsequentlydevelopedwhichcouldaccuratelyreproducetheadsorptioninflectionobservedinexperiment.65ThemodelisanextensionoftheSSLmodelwithadditionalcomplexitytoaccountforthetwostatesoftherotatingligands,whichmodifiestheadsorptionbindinginteraction.Studiesofgate‐opening,breathing,androtatingligandbehaviorareleadingtoimportantadsorbentmaterialdesignprinciplesaimedatexploitingthesigmoidal S‐shaped isothermtomaximizedeliverableadsorptioncapacityandseparationselectivity.

Conclusions

ThelegacyofIrvingLangmuiriswidespread,frombiologytoatmosphericscience,butnofieldismorecloselyassociatedwithhisnamethanadsorption:thescienceofinterfaces.Inthecenturysincethepublicationofhissimpletheoryofgas‐solidadsorption,therehasneverbeenatimewhenhisworkwasmorehungrilyconsumed i.e.,referencedinthescientificliterature thanitisnow.DespitethatLangmuir’ssixadsorptionclassificationswereinitiallydevelopedtotreatgasadsorptiononplanar,externalsurfaces withtheexceptionofCaseIIIwhereLangmuirremarksaboutapplicabilitytoporousmaterials ,recentlydiscoveredclassesofnanostructuredandporousmaterialsnowdominatethelandscapeofgas‐solidadsorptionstudies.Mostnotably,metal‐organicframeworks MOFs exhibitwidelytunablecrystallineenvironmentsforbindingadsorbates e.g.,stronglybindingopen‐metalsitesandweaklybindingorganiclinkerandpocketsites ,inadditiontointernalporositieswithunprecedentedsurfaceareasforadsorption e.g., 6000m2g‐1 .Porousframeworkcarbon‐basedmaterialssuchaszeolite‐templatedcarbon ZTC 55andmicroporoussilicates zeolites 114alsopresentunusualandhighlyinterestingenvironmentsforgas,liquid,andionadsorption.Thistypeofadsorptioninconfined“nanospaces”,whereapotentialenergyfieldfor

adsorptioniscreatedinathree‐dimensionalspace,bringsanewlevelofcomplexityandrichnesstostudiesofadsorption.ThelimitationsofLangmuir’stheoryintreatingcomplexadsorbentshave,ofcourse,longbeennoted.115Itisoftenappropriatetoemployapotentialtheoryofadsorptioninsuchsystems,116‐118oramorecomplexmodelsuchasonebasedondensityfunctionaltheory119‐120ormolecularsimulations121;nevertheless,theLangmuirmodelstillservesasarobust,simpleplatformontowhichfurthercomplexitycaneasilybeintegratedtomeettheneedsofbothroutineandunusualproblemsinadsorptionscience.Thefundamentaltheoriesofmonolayerandmultilayeradsorption,bothattributedtoLangmuir’spivotal1916‐1918works,remainwidelyusedtoday.

AssociatedContent

Nomenclature

ThenomenclatureusedinLangmuir1918hasnotbeenadoptedinthisreviewinfavorofmoremodernnomenclatureandstatisticalmechanicalconventions,asshownbelow.Thephaseoftheadsorptivemolecule“A”isindicatedbyeither“g”forgasor“a”foradsorbed.Theuseoftheterm“gas”orphase“g”isinclusiveofsupercriticalfluids.

Symbol Quantity TypicalUnits Langmuir’sSymbol3

surfacerate m s ,

probability dimensionless

fractionalsiteoccupancy dimensionless

‐‐ relativeadsorptionlifetime m s

number dimensionless

molenumber mol

temperature K

pressure Pa

volume m

molecularmass kg

gasconstant J K mol

Langmuirbindingconstant Pa

Langmuirscalingconstant mmol g

exponentialprefactor varies ‐‐

weightingconstant dimensionless

Hillinteractionconstant dimensionless ‐‐

BETvariable g mmol ‐‐

BETconstant dimensionless ‐‐

numberadsorbed dimensionless

Langmuirscalingconstant g

adsorptionsitebindingenergy J

chemicalpotential J

, single‐sitepartitionfunction dimensionless

, totalpartitionfunction dimensionless

thermaldeBrogliewavelength m

spreadingpressure N m

SupportingInformation

TheSupportingInformationisavailablefreeofchargeviatheInternetathttp://pubs.acs.org.ExperimentalNotesandKineticandStatisticalMechanicalDerivationsofLangmuir’sTheory.

I. AuthorInformation

CorrespondingAuthors

*Email:nicholas.stadie@montana.edu

NicholasP.Stadie:0000‐0002‐1139‐7846

Theauthorsdeclarenocompetingfinancialinterest.

Acknowledgements

WethankCorySimonandArniSturlusonforinvaluablecontributionstothestatisticalmechanicalderivationsandacknowledgeseveralimportanthistorical6‐8andbiographical24,122‐125referencesusedextensivelyinthepreparationofthismanuscript.

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Biographies

HansSwensongraduatedfromhighschoolinBozeman,Montana,andiscurrentlyanundergraduatestudentatMontanaStateUniversity,withaspirationsofpursuingaPhDinchemistry.Hisresearchhasbeenfocusedonmeasuringtheadsorptionofheliumonmicroporousmaterialstoelucidatetheeffectsofnarrowporesizeonerrorsinvoidvolumedetermination.

NicholasP.StadiereceivedaB.S.inChemistryfromArizonaStateUniversitywhereheperformedresearchonzeoliticimidazolateframeworks ZIFs underProf.MichaelO’Keeffe.AttheCaliforniaInstituteofTechnology,heundertookstudiesofhighsurfaceareacarbonaceousframeworkmaterialsandtheirgasadsorptionpropertiesunderProf.BrentFultz,receivingaPh.D.inMaterialsSciencein2013.Hisattentionturnedtoreactive,unstableporousphasesofhydrogen‐richmaterialssuchasmagnesiumborohydrideunderProf.AndreasZüttelattheSwissFederalLaboratoriesforMaterialsScience&Technology Empa .AfterasecondpostdoctoralpositionatETHZürichapplyingporouscarbonmaterialsasexoticbatteryelectrodesinthelaboratoryofProf.MaksymKovalenko,hereturnedtoNorthAmericain2017asanassistantprofessoratMontanaStateUniversity.Hisresearcheffortscombinesyntheticsolid‐statechemistrywithexperimentalstudiesofgas‐surfaceinteractions,especiallyunderhigh‐pressureconditions.

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Montana Cooperative State Grazing District Directory …dnrc.mt.gov/divisions/cardd/conservation-districts/montana-grass... · Montana Cooperative State Grazing District Directory

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