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Electrochimica

Acta

jo u rn al h om ep age : w w w . e l s e v i e r . c o m / l o c a t e / e l e c t a c t a

A

pH

centenary

Robert

de

Levie

∗

BowdoinCollege,BrunswickME04011,USA

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received3November2013 Accepted1April2014 Availableonline18April2014 Keywords:

Acidity Ionicactivity pH

a

b

s

t

r

a

c

t

Theactivitycoefficients(andthereforetheactivities)ofsingleionicspeciesareconceptstentatively introducedbyG.N.Lewis,whichhecouldnotdefinethermodynamicallybecauseofelectroneutrality

requirements.Guggenheimsubsequentlygavetheirformal,pseudo-thermodynamicdefinitions,while

warningthattheywereimaginaryconstructswithoutphysicalsignificance.Consequently,thehydrogen ionicactivity,asapurelyconceptualbutimmeasurablequantity,cannotserveasthebasisofthepH,the globallyacceptedexperimentalmeasureofacidity.

Variousaspectsofthismismatcharedescribed,basedontheoriginalliteraturesourcesaswellas onexperimentaldatausedbytheirproponents.Anespeciallyperniciousbutapparentlywidespread misconceptionisthatthehydrogenionconcentrationcannotbedeterminedbythermodynamicmeans, despitetheextensiveworkofHarned,Robinsonandcoworkerswhoshowedotherwise.

Apathwayisindicatedtofacilitateasmoothreturntotheoriginal,thermodynamicallysounddefinition ofSørensenintermsofthehydrogenionconcentration

Wealsodescribeausefulformalismbasedontheimaginarynatureofsingleionicactivities. ©2014ElsevierLtd.Allrightsreserved.

1. Introduction

Inarecentarticle[1]commemoratingthecentenaryofthe con-ceptofpHinanofficialIUPACnewsmagazine,oneoftheco-authors ofthelatestIUPACrecommendation[2]forthedefinitionofpH wrotethatpHis“mostlikelythemostmeasuredchemical param-eterandtheonemostpeoplehearortalkabout”butlamentedthat “Infact,beyondthesimpleprocessofmeasuringpH,thereispoor understandingoftheconcept,thebasisforitsderivation,and limi-tationsofitsapplicability.”Isthisaproblemthatphysicalchemists, analyticalchemists,andelectrochemistsshouldaddressbybetter teaching?Inthiscommunication,whichreflectsatutoriallectureI presentedatthe2012ISEmeetinginPrague,Iwillarguethatthis poorpublicunderstandingisnotamatterofdeficientteaching,but ofapoorlydefined,andthereforeunteachablesubjectandis,infact, aproblemofIUPAC’sownmaking.

Thispaperwillbrieflyreviewthehistoricaldevelopmentofthe conceptofpHsince1909,whenitwasintroducedas–log[H+_],and

whyitwassubsequentlyredefinedas–logaH.Itwillconsider

dif-ferentaspectsofthisdevelopment,includingdirectquotesfrom thewritingsofitsmajorplayers,becausethosewhodevelopand advocatenewconceptstendtothinkaboutthemdeeply,and

usu-∗ Correspondingauthor.

E-mailaddress:[email protected]

allyexplainthemclearly.Inordertokeepthenotationassimpleas possible,wewillassumethatalldimensionalparametersaremade properlydimensionlesswherenecessary.

Asinanyhistoricaloverview,thefactsandcitationsmust(and will)beaccurateandobjective,buttheirinterpretationis necessar-ilysubjective.Inthiscontext,thetenorofthispaperisperhapsbest describedbyaquotefromG.N.Lewis[3]who,uponintroducingthe conceptofactivity,wroteaboutthedevelopmentofthe Guldberg-Waagemassactionlawanditsconsequences:“Asapproximations tothetruththeyhavebeenofthegreatestservice.Butnowthat theirutilityhasbeendemonstrated,theattentionofaprogressive sciencecannotrestupontheiracknowledgedtriumphs,butmust turntotheinvestigationoftheirinaccuraciesandtheirlimitations”. HerewewillapplyasimilarstandardtopH.

1.1. Theearlyhistory:Friedenthal,Sørensen&Lewis

Inthelatenineteenthcentury,GuldbergandWaage[4] formu-latedthefinal formoftheirmassactionlaw,andArrhenius[5] introducedtheideaofthepermanentpresenceofionsinelectrolyte solutions.Ostwald[6]combinedthesetoanalyzethebehaviorof weakacidsand bases, and hisextensiveworkconvinced many skepticsatthetimetoacceptthesenovelconcepts.

Friedenthal[7]introducedtheideathatthenegativeten-based logarithmofthehydrogenionconcentration(here,regardlessofthe actualspeciationofsolvatedprotons,denotedby[H+_]ingeneral,

http://dx.doi.org/10.1016/j.electacta.2014.04.006 0013-4686/©2014ElsevierLtd.Allrightsreserved.

andcHormHmorespecifically)wouldmakeausefulacidityscale.

Moreover,hetreatedacidsandbasesinthesamewayratherthan byusingtwodifferentscales,onefor[H+_{]andtheotherfor[OH}−_].

Sørensen[8]adoptedFriedenthal’sscale,gaveitthenamepH,and providedanextensivedescriptionofitsexperimental determina-tion,bybothspectroscopicandpotentiometricmeans.

Ataboutthesametime,G.N.Lewis[3]noticedthataclassof electrolytes(thenalreadycalled“strong”)didnotseemtofollow Ostwald’sdilutionlaw[6]and,byanalogytohisearlierintroduction offugacity[9],introducedmolecularactivitycoefficients,inorder toallowforthenon-idealbehaviorofsuchsolutions.Itwassoon realizedthat,indiluteelectrolytesolutions,long-distance electro-staticinteractionsweremainlyresponsibleforsuchdeviations,and Lewisthereforealsotriedtointroduceactivitiesand correspond-ingactivitycoefficientsforsingleionicspecies.However,hecould notfindasatisfactorythermodynamicdefinitionfortheactivityof singleionicspecies,incontrasttothosefortheneutralelectrolytes thatcontainedthem.Ashecommentedinhis1923bookwith Ran-dall[10]:“Indevelopingourequationswehavemadeuseofthe activitycoefficientoftheseparateions,andwehaveshownthat, forasaltlikepotassiumchloride,theactivitycoefficientisthe geo-metricmeanoftheactivitycoefficients,_␥+ofpotassiumion,and

␥−ofchlorideion.Itremainsforustoconsiderwhetherthese

sep-aratevaluescanbeexperimentallyconsidered.Thisisaproblem ofmuchdifficulty,andindeedwearefarfromanycomplete solu-tionatthepresenttime.”Andafewpageslaterinthatsamebook, theywrote[11]:“Atthepresenttimewemustconcludethatthe determinationoftheabsoluteactivityoftheionsisaninteresting problem,butonewhichisyetunsolved”.

Unfortunately,thisissuewassomewhatconfusedatthattime, becausetheionicconcentrationsofstrongelectrolyteswerenot obtaineddirectlyfromtheconcentrationsoftheelectrolytesthat contributedthem,butratherwerecomputedfromtheir conduc-tances,ontheassumption(thenalreadydisprovedbyKohlrausch [12] in hissquare-root lawfor theconductivityof strong elec-trolytes)thattheionicmobilitieswouldremainconstant.Butthere wasalsoamorefundamentalproblem,onethatLewisrecognized butcouldnotsolve.

Theconcentrationofasoluteis itsmasspervolumeof solu-tion and is then called molarity (with (symbol c), or its mass permass ofsolventand thencalledmolality(symbolm).These fundamental chemicalparameters arelinked directlyto weigh-ing,molecularmass,andtheconservationofmass.Lewisdefined activity as concentration times a correction factor, the activity coefficient. The latter is a strictly thermodynamic construction, whichthereforeexistsonlywithinathermodynamicframework. If it cannot be defined withinthat context, it simply doesnot exist or,if oneprefers the euphemism,it is a conceptwithout physicalsignificance,i.e.,an illusory,imaginaryquantity.Aswe willseebelow,thatisthecase withtheactivitycoefficientofa singleionicspecies, andistherootcauseofthedifficulty men-tioned by Camões [1], becauseIUPAC [2] now defines the pH, an experimental measure, in terms of the hydrogen ion activ-ity.

Intheir1924review,Sørensen&Linderstrøm-Lang[13]cited theaboveLewiscomments,andconcludedthat:“...itwouldseem prematureatpresenttointroducetheactivityprincipleinplace oftheconcentrationprincipleformeasurementsofhydrogenions inbiochemicalinvestigationsgenerally;...”and,alittle further-on:“Wetherefore suggest,thatinaccordancewiththepractice ofBjerrumandhiscollaborators,thetermscH,pHand␲0should

beallowedtoretaintheiroriginalsignificance, aH,paH anda␲0

beingusedtodenoterespectivelyhydrogenionactivity,exponent ofhydrogenionactivity(paH=–logaH)andthe␲0usedin

calcu-latingtheactivityofhydrogenions,...”whichtheythenrepeated intheir“ProposalsforStandardisation”as:

1)Inelectrometricmeasurementsofhydrogenions,asharp dis-tinctionshouldbemadebetweenconcentrationandactivityof thehydrogenions.

2)Instatementsofconcentrationsofhydrogenions,thetermscH,

pH and␲0 shouldbeused,retainingthesamesignificanceas

hitherto.

3)Instatingtheactivityofhydrogenions,thetermsaH,paHand

a␲0shouldbeused,indicatingrespectivelyactivity,exponentof

activity,andthe_␲0usedincalculatingtheactivityofhydrogen

ions.

ThesequotesaredifficulttoreconcilewithIUPAC’s2002use [2]ofreference[13]asitssolejustificationforusingthehydrogen activityratherthanconcentrationtodefinepH!

1.2. TheDebye-Hückelmodel

Inthemeantime,Debye&Hückel[14]solvedthetheoretical problemofionicinteractionsinsufficientlydiluteelectrolyte solu-tionsbyderivingtheirequation

logfi= −z2 iA √ I 1₊Ba√I (1.2.1)

where fiis theactivitycoefficientof ionsiwithvalencyzi, the

ionicstrength Iisdefined[15]as½zi2 ci,aisthedistanceof

closestapproach of theselected,‘central’ion itoits (predomi-nantlycounter-)ions,andAandBareknownmacroscopicconstants reflectingthesolventtemperature,dielectricconstant,etc.Note thatthedefinitionofionicstrengthIisintermsofionic concen-trationsci.Likewise,theionicactivitycoefficientpredictedbythe

Debye&Hückelmodelrequirestheionicconcentrationtodefine theionicactivityai=fici.Somerelevantdetailsofthederivation

aregiveninsection1.3.

Atsufficientlylowionicstrengths,sothatthetermBa√I 1,(1.2.1)reducestotheDebye-Hückellimitinglaw, withoutany adjustableparameters:

logfi≈−z2iA

√

I (1.2.2)

Asomewhatlessrestrictiveapproximation,againforBai

√ I 1,approximates(1.2.1)to logfi≈−z2iA √ I(1−Ba√I)=−z2 iA √ I+z2 iABaI (1.2.3)

ShortlyafterDebye&Hückelpublishedtheirtheory,Hückel[16] addedanempirical termtorepresentsalting-inand salting-out effects,therebymodifyingtheexpressionforlogfito

logfi= −z2 iA √ I 1+Ba√I+bI (1.2.4)

Withitsadditionalterm,thisextendedDebye-Hückelexpression canrepresentmanymeanelectrolyteactivitiesoveraratherwide rangeofconcentrationsandionicstrengths.

TheDebye-Hückelmodeldoesnotdefinetheactivitycoefficient ofasingleionicspecies,becausetheparameteraspecifiesa dis-tanceofclosestapproachtoitsnearestions,notanionicradiusor diameter.Debye&Hückelwerequiteemphaticregardingthis lat-teraspect,andwrote[14]“DieGrö␤eami␤tdannoffenbarnicht denIonenradius,sondernstehtfüreineLänge,welcheeinen Mit-telwertbildetfürdenAbstandbisaufwelchendieumgebenden, sowohlpositiven,wienegativenIonenandaservorgehobeneIon herankommen können”,which canbetranslated as“Obviously, then,thequantityadoesnotmeasuretheionicradius.Insteadit representsalengthequaltotheaveragedistancetowhichthe (pos-itiveaswellasnegative)surroundingionscanapproachthecentral ion.”

Eachoftheexpressions(1.2.1)through(1.2.4)sufficestodefine ameanactivitycoefficientf±=



f+f−or±=√+−forthe aque-oussolutionofasinglestrong1,1-electrolyte,aswasthemainfocus ofDebye&Hückel,becauseinthatcasethedistanceaiofclosest

approachmustbethesameforbothanionandcation,andisthen indeedtheonlyrelevantdistanceparameter.Meanwhileitisuseful tokeepinmindthattheDebye-Hückelexpressionhasbeen(and canbe)testedonlyinsolutionsofelectroneutralelectrolytes,i.e., formacroscopicallymanipulablespeciesthatcan(partiallyorfully) dissociateintoelectroneutralcombinationsofanionsandcations.

The strongest experimental evidence for the validity of the Debye-Hückelapproachcomesfromitslimitingform(1.2.2).Here ishow H.S.Frank, anoutspoken criticofthefull Debye-Hückel equation,starteda 1959chapteronthis topiccoauthoredwith Thompson [17]: “There can be little doubt that the theory of DebyeandHückelgivesacompleteandcorrectaccountof activ-itycoefficientsandheatsofdilutioninionicsolutionswhichare sufficientlydilute.The finalitywithwhichit answers questions dealingwiththesepropertiesis,sotospeak,guaranteedbythe factthatitnot onlygiveslimitinglaws forlogf_± and ¯L2 which

makethesequantitieslinearfunctionsofc½_{,withslopeswhich,in}

asufficientnumberofinstances,areexperimentallyconfirmedto thehighestaccuracywithwhichexperimentscanbecarriedout, butalsospecifiestheselimitingslopesasfunctionsoftemperature, dielectricpropertiesofthesolvent,andvalencetypeofthesolute, withoutrecoursetoanyempiricaloradjustableparameters.”(¯L is therelativepartialmolarheatcontentofasoluteinthenotationof [18].)

1.3. TheDebye-Hückelmath

Wewillheresketch howDebye-Hückel derivedtheirresult, becausewewillsubsequentlyneeditinsection2.5.Inprinciple,the ionicinteractionisamany-bodyprobleminvolvingmillionsofions (becauseAvogadro’snumber,about6×1023_{moleculespermole,is}

sohuge),forwhichaclosed-formsolutioniswell-nighimpossible [19,20].Fortunately,wearenotinterestedhereinthebehaviorof individualions,butonlyintheiraverage,statisticalbehavior.Debye andHückelfoundanelegantsimplificationthatallowedthemto reachanapproximatesolutionforthelatter,bydividingthe solu-tionartificiallyintoasingle,arbitrarilychosen“central”ionand itsresulting“ionicatmosphere”,thestatisticalaverageofthe sur-roundingsofmillionsofotherions.Thereisnothingspecialabout thecentralion,andthefinalresultmustbe(andis)applicableto anionsandcationsalike,buttheabovesimplificationreducedthe numberof“particles”tobeconsideredstatisticallyfromtrillions totwo, thecentralion and itsionicatmosphere, thereby mak-ingitmathematicallytractable.Thisapproachwasanextension ofGouy’splanardiffusedoublelayermodel[21,22]toaspherical geometry,butDebye&Hückel[14]deletedtheeffectof spheric-ityduringthederivation,see(1.3.5),makingthetwotreatments formallyequivalent.

Inahomogeneouselectrolytesolution,theconcentrationprofile ofasmeared-outchargedensityaroundacentralioniisgivenby theBoltzmanndistribution

cj=c_j⊗exp



−zjF i RT



(1.3.1) wherecj⊗isthe“bulk”concentrationofionsjinthesolution,i.e.,

sufficientlyfarapartfromthecentralionsi,and iisthe

distance-dependentpotentialaroundacentralion.(Theoriginalpaperuses ␧/kinsteadofF/R,where␧istheelectronicchargeandkisPlanck’s constant.Here we have multipliedboth _␧ andk by Avogadro’s numberNinordertoavoidpossibleconfusionwiththedielectric permittivityεandchemicalrateconstantsk.)

Thechargedensityintheionicatmosphereisthen

=



j cjzjF=c_j⊗F



j zjexp



−zjF i RT



(1.3.2)

andelectroneutralityofthecentralionplusitssurroundingionic atmosphererequiresthat

∞



a

4r2dr=−ziF/N (1.3.3)

where adenotes the (statistically averaged)distanceof closest approachofionsiandj(otherwiseconsideredaspointcharges),and F/Nistheelectroniccharge.Assumingaconstantdielectric permit-tivityε,thePoisson-Boltzmannequationforsphericalsymmetry nowyields divgrad = 1 r2 d dr



r2d dr



=− ε =− F ε



j c_j⊗zjexp



−zjF i RT



(1.3.4)

Debye&Hückelexpandedtheexponentialandthentruncated theresultingseriesafteritssecondterm,therebyreducingittoa planarproblem, 1 r2 d dr

r2d dr

=−F ε



j c⊗_jzj



1−zjF i RT + 1 2

_z jF i RT

2 −...



≈−F ε



j c⊗_jzj

1−zjF i RT



=



j



c⊗_jz2 jF 2 i εRT



=F 2 i εRT



j c⊗_jz_j2=2 i (1.3.5)

when zjF i/RT  1, where the first term of the series

expan-sion,



j

c_j⊗zj,iszerobecauseofmacroscopicelectroneutrality,and

wherewehaveusedtheabbreviations

2₌ F2



j z2 jc⊗ εRT = 2F2_I εRT and I=1/2



j c⊗_jz2 j (1.3.6)

Withthefurtherabbreviationu=r iwecanrewrite(1.3.5)as

du2_/dr2_=2_{u,whichfinallyyieldsthesolution} i=ziFexp[a−r]

4εrN(1+a) (1.3.7)

wherethenecessaryintegrationisfromatoinfinity,abeingthe distanceofclosestapproachofthecentersoftheionsjintheionic atmospheretothecenterofthecentralioni.Thisresultshouldbe comparedwiththepotentialaroundanisolatedchargeziF/Ninan

infinitelylargedielectricmediumofpermittivityεwithoutother ions,

"_i = zjF

4εrN (1.3.8)

Thedifferencebetween(1.3.7)and(1.3.8),  i= i− "i = ziF 4εrN

_exp[a−r] (1+a) −1



≈− ziF 4εrN  (1+a) forr ≤a (1.3.9)

mustthereforebetheresultingloweringoftheelectricalenergyof thecentralioni,

ziF _i 2N ≈− z2 iF 2 8εN2  (1+a) (1.3.10)

due toitsinteractionwithits surroundingionicatmosphere in which,onaverage,acounterionwillbeitsclosestneighbor. Iden-tificationwiththeionicactivityterm(RT/N)lnfjthenyields

lnfi=− z2 iF 2 8εNRT  (1+a) =− z2 iA √ I (1+Ba√I) (1.3.11)

where all constants have been incorporated in A, B. Equation (1.3.11)istheDebye-Hückelresult(1.2.1).Numericalvalues for A,Bforaqueoussolutionsarelistedby,e.g.,Bates[23]for concen-trationsexpressedaseithermolarityormolality.

1.4. Taylor’sanalysisofliquidjunctionpotentials

Theearlyelectrochemistshadintroducedasaltbridgeto sepa-ratethetwoelectrodesofanelectrochemicalcellinthehopethat, bysodoing,anychangeinthesolutioncompositionaroundone (‘indicator’)electrodewouldnotbesensedbythesecond (‘refer-ence’)electrode,andthatanyresultingliquidjunctionpotential couldeitherbemadeconstantornegligiblysmall.WhilebothLewis andSørensenhad acceptedthatapproach,itsoonbecameclear thattherewasaproblemwiththeliquidjunction,atermweuse heretodenoteanion-permeableconnectionthatisnotfully selec-tivetojustoneofthevariousionicspeciespresent.AsHarned[24] wrote:“Wearethusconfrontedwiththeinterestingcomplexity thatitisnotpossibletocomputeliquidjunctionpotentialswithout aknowledgeofindividualionicactivities,anditisnotpossibleto determineindividualionactivitieswithoutanexactknowledgeof liquidjunctionpotentials.Forthesolutionofthisdifficultproblem, itisnecessarytogooutsidethedomainofexactthermodynamics.” Taylor[25]examinedthethermodynamicbasis oftheliquid junctionmoreclosely,promptedbyathenrecentpaperbyHarned [26],andpossiblyalsobytheearlierstatementofLewis&Randall [27]that“Itistobehopedthatinthefuturewemaybesparedthe uncomfortablenecessityofguessingatthevaluesofliquid poten-tials,sinceitseemstobepossibleinnearlyallifnotallcasesto obtainthedatathatareofthermodynamicvalue,solelybymeans ofcellswhichcontainnoliquidjunctions.”

Taylor’spaper[25]startedasfollows:“ArecentpaperbyHarned onthethermodynamicbehaviorofindividualionsisrepresentative ofthepersistentattemptswhichhavebeenmadetoestablisha basisforthedeterminationofthefreeenergiesofionsbymeans ofcellswithtransference,i.e.,acellcontainingajunctionoftwo (different)electrolytes.Thepresentanalyticalstudyleadstothe conclusionthattheEMFofthecellwithtransferenceisafunction offreeenergieswhicharemolecularonly,thatitcannotpossibly bemanipulatedtoyieldionicfreeenergies,andthattheionicfree energyhasnotbeenthermodynamicallydefined.Itistobethought ofratherasapurelymathematicaldevice,whichmayindeedbe employedsafelywithconsiderablefreedom.”

Taylorexpressedtheliquidjunctionpotential“entirelyinterms of transference numbers and EMF’s of cells without transfer-ence,”sothatitcanbedescribedcompletelyintermsofdynamic (masstransport)andstatic(equilibrium)equationsforallspecies involved,“thesolutionofwhichtogetherwiththearbitrary bound-aryconditionsintimeandspacecompletelydeterminethestateof thesystem”.However,thepracticalproblemoffindingthepHof anunknownsolutionremains,andiswell-nighinsoluble.AsTaylor wrote:“InparticularthedeterminationofpHnumbersbysucha cellisnotthesimplethingitissometimesassumed,forthecell EMFdependsnotonlyontheacidactivitybutalsoontheactivity

ofeverymolecularspeciesinthecellandmobilityofeveryion.If thesearesufficientlywellknowntobeallowedfor,theacidactivity islikelytobesufficientlywellknownnottoneedmeasurement.”

The most commonapproach is tomake theliquid junction potentialassmallaspossiblethroughtheuseofdominating con-centrationsofnear-equitransferentsalts,suchasKClorNH4NO3

or,betteryet,RbCl[28] orCsCl [29],sothatanyeffects due to unknownsampleconstituentsareeffectively‘swamped’.Whilethis isadmittedlyacrudeapproach,itseemstobethebestcurrently availablemethodforpotentiometricmeasurements.Thepresent communicationwillnotaddressthisproblem.

Taylor’spaperwasquiteinfluential,becauseitleddirectlytotwo importantdevelopments:(1) Guggenheim’swork[30,31] defin-ing theionic free energyin terms of a pseudo-thermodynamic formalism(whichledtothedevelopmentof“irreversible” thermo-dynamics),and(2)studiesbyHarnedandcoworkers[32,33]onthe experimentaldeterminationofthehydrogenionconcentrations fromemfmeasurementsoncellswithoutliquidjunctions.

ItisnotnecessaryheretogointothedetailsofTaylor’spaper, whichtomyknowledgehasneverbeencontested;instead,ithas recentlybeenexpandedbyMalatesta[34].Itwillsufficetoquote Taylor[25]oncemore,becauseheis quiteexplicit:“TheEMFof thecellwithtransferenceisthusafunctionofmolecularfree ener-giessolely andisnot afunctionof ionicfreeenergies.It therefore canyieldnoinformationwhatsoeverconcerningionicfreeenergies.In factnothermodynamicinformationcanbegainedfromacellwith transferencewhichcouldnotbetterbegainedfromacellwithout transference.Conversely,withinourpresentpurviewaknowledge oftheionicenergiesisnevernecessaryforanaccountofthe ther-modynamicsofelectrolytes.Indeed,withthepossibleexceptionsof singleelectrodepotentialsandratesofreactionthereappearstobe nooccasionfortheuseofionicfreeenergiesasexperimental quan-titiesbutonlyasamathematicaldevice.”Theitalicsintheabove quotearethoseofTaylor.Notethat,atthetimeTaylorwrotethis, theexistenceof“possibleexceptions”ofsingleelectrode poten-tialshadalreadybeendisavowedbyGibbs[35,36],andforreaction ratesbytheworkofBrønsted[37]andChristiansen[38],towhich wewillreturninsections2.3through2.5.

1.5. WhyispaHextra-thermodynamic,andtherefore

immeasurable?

Thehydrogenionicactivitycannotbedefined thermodynami-cally because of macroscopic electroneutrality. The thermody-namicdefinitionofthechemicalpotential␮iofspeciesiatconstant

temperatureTandpressureP,i.e.,itspartialmolarGibbsfreeenergy G,is i=



∂G ∂ni



T,P,n_j/=i (1.5.1)

whereniisthenumberofmolesofspeciesi.However,wecannot

add,say,eitherNa+_orCl−_{ionsinsubstantial(e.g.,macroscopically}

weighable)quantitieswithoutcharge-compensatingcounterions, becauseofthecoulombic,stronglyrepulsiveforcesresultingfrom theirnetchargedensity.Thisdefinitionisthereforenotapplicable toindividualionicspecies,eventhoughitworksfineforneutral elec-trolytessuchasacids,bases,andsaltsthatcontain(ormayevenbe composedentirelyof)suchions.Andbecauseactivityisapurely thermodynamicconstruction,whenasupposedlythermodynamic quantitycannotbedefinedthermodynamically,itdoesn’texist,and hencecannotbemeasured.

Lewiswas wellaware ofthis problem, and wrote [39]: “An interestingtypeofsolutionisfurnishedbyelectrolytesdissolved inwaterorotherdissociatingsolvent.Inthiscaseitiscustomaryto assumetheexistenceofmolecularspecies,namelytheions,which

cannotbeaddedindependentlytothesolution;forexample,we havenopracticalmeansofaddingamolofsodiumionsoramolof chlorideionsalonetoasolutionofsodiumchlorideinwater.We havethereforenomeansofdeterminingthepartialmolarvolumes, orotherpartialmolarquantitiesforsuchsubstancesassodiumion andchlorideion.”

AsHarned&Owen[40]laterexplained:“Insolutionsofan elec-trolyte,electro-neutralityimposestheconditionthatthenumber ofmolsoftheindividualionicspeciescannotbevaried indepen-dently.Wemustbecareful,therefore,torefertoionicspeciesas constituentsofthesolutionratherthanascomponents,sothatthe lattertermmayretaintheprecisemeaningassignedtoitbyGibbs. Acomponentisanindependentlyvariableconstituentofasolution. Thus,inthesystemNaClandH2Otherearetwocomponentswhose

chemicalpotentialcanbemeasuredbytheapplicationof thermo-dynamicsalone.Theyare,ofcourse,NaClandH2O.Althoughthe

ionicconstituentsNa+_andCl−_{areoffundamentalimportancein}

determiningthebehaviorandpropertiesofthesystem,their con-centrationsarenotindependentvariables.Thermodynamicsdoes notpermittheevaluationofthechemicalpotentials,freeenergies, activities,etc.,oftheindividualionicspecies.Inspiteofthis lim-itationitisadvantageoustoexpressanumberofthermodynamic developmentsintermsof“hypothetical”ionicactivities,withthe strictunderstandingthatonlycertainionicactivityproducts, or ratios,haveanyrealphysicalsignificance.”

1.6. Guggenheim’sformalism

Guggenheim[30,31]developedapseudo-thermodynamic for-malismforthesingleionicactivityanditsactivitycoefficient,but explicitlywarnedthatitwas“aconceptionwhichhasno physi-calsignificance”,echoingTaylor’swarning[25]thattheionicfree energy is “apurely mathematical device”.As Guggenheim [30] explained,“...itisclearthattheinterionicenergyisstoredinthe wholeassemblyandanypartitionofitamongsttheseparatetypes ofionswouldbearbitrary.InthetheoryofDebyeandHückel,... whichtreatstheionsasrigidspheres,thisshowsitselfbythefact thatthespecificquantities,whichdistinguishsolutionsofthesame electrictype,arenotthediametersoftheindividualions,butthe distancesofclosestapproachofthevariouspairsofions.”In sec-tion1.2wealreadyquotedDebye&Hückel[14]stressingthatsame point.

Notallpotentials(orpotentialdifferences,whichitoftenmeans) aremeasurable.Gibbs,inan1899lettertoBancroft[35,36]already wroteaboutsingleelectrodepotentialsthat“...theconsideration ofthedifferenceofpotentialinelectrolyteandelectrode,involve theconsideration of quantities of which we have noapparent meansofphysicalmeasurement,whilethedifferenceofpotential inpiecesofmetalofthesamekindattachedtotheelectrodesis exactlyoneofthethingswhichwecananddomeasure.”

Or,asGuggenheim[30]putit:“Thegeneralprinciplereferred tomaybeexpressedasfollows.‘Theelectricpotentialdifference betweentwo pointsin differentmedia cannever bemeasured andhasnotyetbeendefinedinterms ofphysicalrealities; itis thereforea conception which hasnophysicalsignificance.’ The electrostaticpotentialdifferencebetweentwopointsisadmittedly definedinelectrostatics,butthisisthemathematicaltheoryofan imaginaryfluid‘electricity,’whoseequilibriumandmotionis deter-minedentirelybytheelectricfield.‘Electricity’ofthiskinddoesnot exist,onlyelectronsandionshavephysicalexistence,andthese differfundamentallyfromthehypotheticalfluidelectricityinthat theparticlesareatalltimesinmovementrelativetooneanother; theirequilibriumisthermodynamic,notstatic.”And[31]:“...we thereforehavenoknowledgeofthevalueoftheelectricpotential betweenanypairofphases,northereforeofthechemicalpotential, theactivityortheactivitycoefficientofanyindividualion.”

Guggenheim[30]definedtheionicactivityaiasafunctionof

anelectrochemical(ratherthanchemical)potential,˜i,anentity

thatcannotbethermodynamicallydefinedeither,byintroducing thepurelyformalexpressions

˜

i=˜◦_i+RTlnai+ziF

=˜◦_i+RTlnci+RTlnfi+ziF

=˜◦_i+RTlnmi+RTlni+ziF

(1.6.1)

wherethetilde∼identifiestheelectrochemical(ratherthan chem-ical) potentialand itsstandard state, thelatterdenoted bythe superscript o_{, and where c and m are the concentrations on}

a solution volume(molL−1)or solventweight (molkg−1)scale respectively.However,notonlytheelectrochemicalpotentials˜i

and˜◦_i,butalsotheionicactivityai,theionicactivitycoefficient

iorfi,andthesolutionpotential areingeneralimmeasurable

quantities.Unfortunately,Guggenheimdidnotlabelthelatteras such,butwewilldosoherebyalsoplacingtildesonai,fi,i,and

inordertoidentifythemasimmeasurable,asGuggenheimhad clearlystatedtheyare.Herewewillthereforewrite

˜

i=˜◦_i+RTln ˜ai+ziF ˜

=˜◦_i+RTlnci+RTln ˜fi+ziF ˜

=˜◦_i+RTlnmi+RTln˜i+ziF ˜

(1.6.2)

inordertoemphasizetheirpseudo-thermodynamicstatus.Note that thechemical potential hasnoterm in(nor need for)␺ becausealltermsinziF ˜ cancelforneutralelectrolytes.

AsWaserwrote[41]:“Thermodynamicsisaphenomenological theory,concerningmacroscopicquantitiessuchaspressure, tem-perature,and volume.Itis bothitsstrength andweaknessthat therelationshipsbaseduponitarecompletelyindependentofany microscopicexplanation of physical phenomena,...” or, inthe wordsofSmith[42],thermodynamics“...isessentiallya practi-calsubjectthatinterrelatesquantitiesthatcanbemeasuredinthe laboratory...”.

Unfortunately,intermsofmeasurability,electroneutrality pre-ventsusfromdeterminingionicactivitycoefficientsindependently (i.e., individually) without making additional, arbitrary model assumptions. Only those combinations that do not violate elec-troneutralitycanbemeasured.Insections2.6through2.8wewill introduceaformalismthatshowsclearlywhichcombinationsof theseimmeasurableionicactivitiesandionicactivitycoefficients canbeexpressedintermsofmeasurable,thermodynamic quanti-ties.Theelectroneutrality(orchargebalance)conditionlinksthe anionic and cationicactivity coefficients, asin ˜+˜−=±2 for a

single1,1-electrolyte,where␥±isanexampleofsuchadirectly measurablecombination.

Lewis & Randall [43] had hoped that single ionic activity coefficientsmightbeobtainablefromelectrochemicalcells with liquid junctions, ifonly one could computetheliquid junction potential,andtherebyseparatetheanionicandcathodicresponses ofthetwoelectrodes.Butbecausecellswithorwithoutliquid junc-tionsdonotrequiresingleionicactivitiesfortheirdescription,such quantitiescannotbeextractedfromthemeither,whichiswhy Tay-lor[25]consideredthesingleionicactivitya“purelymathematical device”.

Incidentally,Malatesta[44–46]showedwhathidden assump-tionsoroutrightmistakesunderliesomerecentlypublishedclaims ofhavingmeasuredunbiasedsingleionicactivitycoefficients,and Zarubin[47]derivedtheuselessnessofacommontestpurportedly validatingsuchclaims.Moreover,theliteratureisfullofearlier, sinceabandonedattempts tospecify howsingleionicactivities couldbemeasured.Buthopespringseternal.

2. WhymeasurepaH?

Ifthehydrogenionicactivityisindeedaconceptwithout physi-calsignificance,wemustaskourselveswhetherthehydrogenionic activityisausefulparametertodefineacidityinpractical, exper-imentalterms.(Debye&Hückelfirmlyestablisheditsusefulness fortheoreticalcalculations.)Toanswerthisquestion,wewillfirst posetherelatedquestion:Whatwouldwelearnifwecould some-howmeasuretheionicactivityofhydrogen?Inotherwords:Why measurepH?

TheNernstequationdefinesathermodynamicequilibrium,and thereforeshouldbewrittenintermsofactivities,asdothemass actionexpressionsforchemicalequilibria.Moreover, electrochem-icalreactionsbydefinitioninvolvechargetransfer,andtheNernst equationtherefore alwayscontainsoneor moreionicactivities, therebyofferingaseductivehopeofyieldingtheelusivehydrogen ionicactivity.Butwhatusefulpurposewouldthatserve?Belowwe willthereforeconsidertwoareasofmajoruseofpH:inchemical equilibria,andinchemicalkinetics.

2.1. WhymeasurepaHforchemicalequilibria?

AmajorareainwhichpHisusedconcernsequilibriumrelations, suchasinanalyticalchemistry,wherethepHservesasanimportant toolinacid-base,complexation,redox,andprecipitationtitrations; inphysical,inorganic,organicandbio-chemistry,whereitisused todetermineequilibriumconstantsofweakacidsandbases;in agriculture,whereitfindsapplicationsintheacidityofrainand soil;inmedicine,whereitisusedasanindicatorofbloodquality, etc.

Wecanindeedwriteequilibriumexpressionsintermsofionic activitiesbut,alternatively,wecanalwayswritetheminsteadin termsofthermodynamicallywell-definedmeanactivities.Infact, intheprefacetohis1933bookonModernthermodynamicsbythe methodsof WillardGibbs,Guggenheim[48]wrote: “Inthe treat-mentofelectrochemicalsystemsIhaveshownhowtoavoidtheuse offunctionswhicharenotthermodynamicallydefined.Amongst theseare...theindividualionicactivitycoefficientsandthe parti-tioncoefficientbetweentwosolventsofindividualionicspecies.” Hethenproceededtodoaspromised.

Guggenheim [30,31] had already illustrated this for several chemicalequilibria.Inourmodifiednotationof(1.6.2)wecanwrite foraceticacid,HAc,inequilibriumwithitsions,H+_andAc−_,either

Kao= ˜aH+˜aAc− aHAc = cH+cAc− cHAc ˜fH+˜fAc− fHAc = mH+mAc− mHAc ˜ H+˜Ac− HAc (2.1.1) or Ko a= ˜aH+˜aAc− aHAc = cH+cAc− cHAc f2 ± fHAc= mH+mAc− mHAc 2 ± HAc (2.1.2) wherethemeanactivitycoefficientsf_±,HAcand␥±,HAc(i.e.,the

geo-metricmeanoftheactivitycoefficientsoftheionsH+_andAc−_of

theweakacid), andalsotheactivitycoefficientsfHAcand ␥HAc

oftheneutralspeciesHAc,arethermodynamicallywell-defined. Numericalvaluesforf_±or_␥±asafunctionofIcanoftenbefoundin theliteratureforstrongacids,bases,andtheirsalts,andhavebeen widelytabulated;thecorrespondingdataforweakacids,basesand theirsaltsaremoredifficulttofind.AtsufficientlylowI,wecan oftenassumethatfHAc≈1and␥HAc≈1.

Likewise,fortheequilibriumbetweenNH4+andH++NH3 we

have Ko a= ˜aH+aNH3 ˜aNH+₄ =cH+cNH3 cNH+₄ ˜fH+ ˜fNH+₄ fNH3= mH+mNH3 mNH+₄ ˜ H+ ˜ NH+₄ NH3 (2.1.3)

wheretheratios ˜fH+/˜fNH+₄ and˜H+/˜NH+₄ areproperly

thermody-namicallydefined,becausetheycorrespondtothereplacementof, say,H+_{ionsbyanequivalentchargeofNH}

4+ions,whilekeeping

theanionconcentrationsunchanged,sothatthe electroneutral-ityconditionisnotviolated.Moreover,theseratiosapproach1as theionicstrengthIapproachestheregionofapplicabilityofthe Debye-Hückellimitinglaw.

Similarresultsareobtainedspectroscopicallyforconcentration ratios.ForHAcinequilibriumwithH+_+Ac−_wefind

cA cB= cH+ Ko a ˜fH+˜fAc− fHAc = cH+ Ko a f2 ±,HAc fHAc (2.1.4) which,inpractice,canoftenbesimplifiedto

cA cB≈ cH+ Kao f 2 ±,HAc (2.1.5)

FortheequilibriumbetweenNH4+ andH++NH3 welikewise

obtain cA cB= cH+ Ko a ˜fH+ ˜fNH+₄ fNH3 (2.1.6)

whichinsufficientlydilutesolutionsapproaches cA cB≈ cH+ Ko a (2.1.7) andtheequivalentrelationsintermsofmolalities.

ColorimetricpHindicatorsindeedmonitortheratiocA/cB,and

itsmodernapplicationinpHtitrationshasbeendescribed exten-sivelyby,e.g.,Polster&Lachmann[49].

Tothe bestofmy knowledge, IUPAChasnot sponsored any generaltablesofionicactivitiesorionicactivitycoefficients.(And pleasedon’tgetmewrong:Iamnotsuggestingthatitshoulddoso!) Butinonerespectithas:initspHvaluesforstandardpHbuffers.

Weconcludethat,forthermodynamicequilibria,ionicactivities maybeconvenientbutarenotrequired;andionicconcentrations plusmeanactivitycoefficientsorothermeasurableproductsand ratiosofioniccationoranionactivitycoefficients(or concentration-basedKa-values)suffice.ThestatementbyBates[50]that“ThepsH

hasvirtuallynomeaning.ThepaHvaluehasinitselfnosignificance intermsofphysicalreality,yetitsrôleinchemicalequilibriais simplyandunequivocallydefined.Forthesereasons,thegeneral adoptionoftheactivityscaleseemswarranted.”isthereforetruly baffling,becausethatrôle(foraquantitythathas“nosignificance intermsofphysicalreality”)isentirelyunnecessary.

2.2. WhymeasurepaHforchemicalkinetics?

Inthe1907paper[3]inwhichLewisintroducedactivities,he carefully addressed the questionof reaction kinetics, explicitly restrictingittomolecularspecies:“Wewillnowconsiderthose pro-cessesinwhichthemolecularspeciesreactwitheachothertoform newspecies,anditwillbeshownthattheactivityofagivenspecies isnotonlyameasureofthetendencyofthespeciestoescapeinto someotherphase,butisalsoaperfectmeasureofthetendencyof thespeciestotakepartofanychemicalreaction.Inotherwords,the activityisanexactmeasureofthatwhichhasbeenrathervaguely calledthe“activemass”ofasubstance.”And,inthesummaryofthat samepaper:“Ithasbeenshownthataquantitynamedtheactivity, andcloselyrelatedtothefugacityoftheprecedingpaper,maybe sodefinedthatitservesasanidealmeasureofthetendencyofa givenmolecularspeciestoescapefromtheconditioninwhichit is.”Inotherwords:Lewisemphasizedmolecularspecies,anddid notmentionionicactivities.In thisrespect,Lewis’analogywith thefugacitywasapt:neutralspeciescanhavemeasurablevapor

pressuresbut,absentbalancingcounter-ioniccharges,individual ionicspeciescannot.

Unfortunately,Lewissubsequently wasnotalwaysascareful withhisoccasional,off-handsuggestionsthattheratesofchemical reactionsinvolvingionsshouldalsobewrittenintermsof activi-tiesratherthanconcentrations.Inafootnoteinhis1923bookwith Randall[51],e.g.,hewrote:“Wedonotmentionthemethodwhich dependsupontherateofcatalysisofareactionasdeterminedbythe amountofsomeionpresent.Evenifthismethodcouldbebroughtto greaterrefinement,itwouldgiveactivitiesratherthan concentra-tions.Atleastthisisthecaseintheneighborhoodofanequilibrium wherethereactionratesaredefinitelyrelatedtotheequilibrium constant.”

Thiswasaneasytraptofallinto.Achemicalequilibriumsuch asbetweenA+BandC+Dcanbeunderstooddynamicallyasthe equalityofthetworeactionratesA+B→C+DandA+B←C+D, andifthefirstratewouldbeproportionalto kaAaB,andthesecond

to ៭kaCaD,wherethek’sarethecorrespondingforwardandreverse

rateconstants,thentheirequalityatequilibriumwouldindeedlead toko_a= ៭k/k =aCaD/aAaB.

Inanycorrectformulationofachemicalequilibrium,thesum ofthechargesoneachsideoftheequilibriumexpressionmustbe thesame,sothatnetelectroneutralityismaintained.However,to useequilibriumexpressionstodrawconclusionsaboutthe math-ematicalformoftheindividualrateexpressions,especiallywhen ionsareinvolved,wasonebridgetoofar,andoftenturnedoutnot toagreewithexperimentalobservations.

Infact,itwasknownmuchearlierthatchemicalreactionrates notnecessarilyreflectequilibriumexpressions,andoftendon’t.In 1850Wilhelmy[52]alreadyreportedthattherateofsucrose inver-sionwasfirstorderinstrongacid,eventhoughneithertheacidnor hydrogenionsentertheequilibriumexpression,asiscommonwith catalysts.AndHarcourt[53]wrotein1867thattherateof oxida-tionofiodidebyhydrogenperoxideisfirstorderwithrespectto bothH2O2andI−,whereastheequilibriumexpressionH2O2+2HI

2H2O+I2wouldhavesuggestedasecondorderiniodide.

Ingeneral,

(1)reaction kineticstend toreflect theslowest (so-called rate-determining)process(es)inapossiblyquitecomplicatedchain ofevents,whereasequilibriumrepresentstheratioofthe over-allreactionsestablishingthatequilibrium;and,moregenerally, (2)knowingaratio,suchastheexpressionforanequilibrium con-stant,is in generalinsufficientto specify itsnumerator and denominatorseparately,justasknowingthemeanactivityofa saltdoesnotspecifytheindividualactivitiesofitsconstituent ionicactivities,butonlytheirproduct.Theneedforactivities ratherthanconcentrationsinchemicalrateexpressions there-forecannotbesettledapriori,asLewistriedtodo,butmustbe establishedexperimentally.

We will first consider the logical application of the idea expressedintheabove-quotedfootnote[51]toreactionkinetics. Lettherateofachemicalreactionbetweenmolecular(i.e.,neutral) speciesAandBbedeterminedbyarateconstanttimesfunctionsof AandB.Atinfinitedilution,suchfunctionsaretheconcentrations ofAandB,andtherateexpressions[54]are

dcA/dt=dcB/dt=−kcAcB (2.2.1)

oritsequivalentintermsofmolalitiesm.Inmoreconcentrated solutions(2.2.1)shouldthenbereplacedby

dcA/dt=dcB/dt=−k˜aA˜aB (2.2.2)

becauseafixedrateconstantcannotanticipateeffectscausedby, e.g.,changesintheionicstrengthofthesolution,whiletheterms

dcA/dtanddcB/dtarepurelyaccountingdevicessafeguardingthe

conservationofmass,andthereforewillnotbeaffectedby ener-getics.

Butwhen Aand/or Bareions, the principlethat “the activ-ity... isalsoa perfect measureof thetendencyof thespecies totakepartofanychemicalreaction”cannotbeappliedinthis simpleform,becausetheirsingleionicactivitiescannotbedefined thermodynamically,andhencehavenophysicalmeaningsandare immeasurable.Or,toputitdifferently,singleionicspecies can-not‘escape’assuch,butonlyincombinationwithanequivalent chargeofcounterions,i.e.,aselectroneutralspecies.Inthatcase wecanonlymeasuretheso-calledmeanionicactivitiesand activ-itycoefficients,alreadyencounteredinsection2.1,whichingeneral aredefinedas c++− ± =c++c−−, f±++−= ˜f++˜f−− (2.2.3) or m++− ± =m++m−−, ±++−=˜++˜−− (2.2.4)

when+and–arethestoichiometriccoefficientsoftheneutralsalt

M−X+ofthecationMofvalencyzMandtheanionXofvalency

–zX,andwheretheelectroneutralityconditionreads+z++–z–=0.

Fortherateofformationofa1,1-saltABfromitsions,A+_+B–_→AB,

wethenfind

dcA/dt=dcB/dt=−kcAcBf±2 (2.2.5)

or

dmA/dt=dmB/dt=−kmAmB_±2 (2.2.6)

andcorrespondinglymorecomplicatedexpressionsforsaltswith differentstoichiometries and/orvalencies. Butstrict thermody-namicsdoesn’tapplytosingleionicspecies,i.e.,inrateexpressions forwhich+z++–z– /= 0,andonlyexperimentcandecidewhatis

thecorrectformalismtouseforthecorrespondingreactionrate.As Kortümwrote,introducingBrønsted’stheoryinhisinfluential Trea-tiseonElectrochemistry[55]:“Therateofachemicalreactionis proportionaltotheconcentrationofthereactants,accordingtothe familiarlawsofkinetics.Iftherate-determiningprocessinvolves ions,departuresfromthislawareobserved.Ithasalsobeenfound thatsimplesubstitutionofactivitiesforconcentrationsdoesnot,as mightbeexpected,accountfortheexperimentalresults.”In sec-tion2.3wewillseethatsucha“simplesubstitutionofactivitiesfor concentrations”indeedfailsexperimentaltests,andinsections2.4 and2.5whythisisso.

2.3. Brønsted’sobservations

ThismatterwasclearedupbytheextensivestudiesofBrønsted [37] and,following thepublicationof theDebye-Hückel theory [14],bytheeleganttheoreticalworkofChristiansen[38].Brønsted scannedtheliteratureavailable atthat timefor casesin which reaction ratescould becompared in the absenceand presence of external interioniceffects. He noticedthat strong anomalies (called “primary” salt effects) appear even at highdilutions in reactionrates betweenionic species,but not in those between non-electrolytesorbetweenelectrolytesandnon-electrolytes.

Brønstedfoundthat,ingeneral,reactionratesinitiallyincreased withtheadditionofinertelectrolyteforreactionsbetweenanions andanions,orbetweencationsandcations,i.e.,whenthevalencies ofthetworeagentswereofthesamesign(eventhoughtheywould electrostaticallyrepeleachother),butdecreasedforthosebetween reagentswithvalenciesofoppositesigns,i.e.,between(mutually attracting)anionsandcations.Andthereactionrateswererather indifferenttochangesinionicstrengthwhenoneorbothofthe reagentswasuncharged.

Thisisdefinitelynotwhatonewouldexpectfromthe Debye-Hückel limiting law(1.2.2) ifthe rateexpression werewritten intermsofionicactivities,becausethenareactionrate expres-sionsuchasdcA/dt=dcB/dt=–k ˜aA˜aB=−kcAcB˜fA˜fBwouldalways

yieldaninitialdecreaseoflnkapp with√Iaslongasatleastoneof

thereagentsisanion.In general,however,suchbehavior isnot observed.

ThereactionratesBrønstedconsideredweremostlyofthetype A+B_→product(s),butthedependenciesoftheirinitiallogarithmic reactionratesonaddedinertelectrolyteappearedtodependonthe productzAzBofthevalenciesofthereagentsAandB,ratherthan

onthesumoftheirsquareszA2+zB2,aswouldbeexpectedfrom

thecombinationof(2.2.2)withtheDebye-Hückeltheory. Brønstedinterpretedthischemicallytoimplythatthereaction involvedtheequilibriumformationofareactionintermediate,an “activatedcomplex”or“transitionstate”,asinA+BAB#_→

prod-uct(s)(where the superscript # _{indicatesits transitorynature),}

combinedwiththeadditional,rathercounterintuitiveassumptions that,inallsuchcases,theobserveddecompositionrate(an ener-geticallydownhillreaction)ofthattransitionstateAB# _wouldbe

therate-determiningstep and,paradoxically, wouldbedirectly proportionaltotheconcentrationratherthantotheactivityofthe transitionstateAB#_{.Notethatitwasthelatterassumption,viz.that}

therateofdecompositionofAB#_{wouldbedirectlyproportionalto}

itsconcentrationratherthanitsactivity,whichwasasillogicalin thisframeworkasitwascrucialtoBrønsted’sinterpretation.

Withtheaboveassumptions,theeffectofaddedinertsaltwould thenderivefromthethermodynamicpre-equilibriumbetweenthe reagentsAandBandtheintermediateAB#_as

K#₌ aAB#

˜aA˜aB=

c_AB#f_AB# cA˜fAcB˜fB

and kapp=kK#

so that the assumption of the down-hill yet rate-determining decompositionofAB#_wouldyield

˜fA˜fB

f_AB#cAcB=−kappcAcBK

#˜fA˜fB

f_AB#cAcB=−kappcAcB

(2.3.1) inwhichcasetheDebye-Hückellimitinglaw(1.2.2),theformof whichMilner[19]andBrønsted[56]hadalreadyanticipated cor-rectly,yields

logkapp=logkK#−(z2A+zB2−z2AB)A

√ I=logkK#_+2z AzBA√I (2.3.2) becausezAB=zA+zBorzAB2 =(zA+zB)2=z2A+z 2 B+2zAzB,sothat −(z2 A+z 2 B−zAB2 )=+2zAzB (2.3.3)

inquantitativeagreement withtheliteratureeffects onreaction rateswhichBrønstedhadfound.

On the other hand, starting either from dcA/dt=dcB/dt=

−k ˜aA˜aB=−kcAcB˜fA˜fBorfromdcA/dt=dcB/dt=−ka_AB#=−kK# cAcB˜fA˜fBwouldyieldaproportionalityofthelogarithmofthe

appar-entrateconstantkappwith–(zA2+zB2)A√I,whichinitiallywould

alwaysdecreasetheobservedratewithincreasingionicstrength I as long as at least one of thereagents is ionic. Such behav-ior Brønsted clearly didnot find,except when zB=–zA so that

zAB=zA+zB=0,inwhichcasethetwomodelspredictidentical(and

thereforeexperimentallyindistinguishable)results. 2.4. Christiansen’smodel

ShortlyaftertheDebye-Hückeltheory[14]appearedinprint, Christiansen[38]showedthatinvokinganintermediarytransition

stateandmakingitsassociated,counter-intuitiveassumptionsisin factcompletelyunnecessary,byderivingBrønsted’sresultdirectly from theDebye-Hückel law,as subsequently confirmed explic-itlybyDebye[57].Theinfluenceofaddedinertionsontheionic strength I modifies thedistance of closest approach aof ionic reagents,andtherebyaffectstheirreactionrate,yieldingprecisely thesameresultasthatreportedbyBrønsted,aswillbederivedin section2.5.Ingeneral,thisresultisdifferentfromthatobtainedby thethoughtlessapplicationofeq.(2.2.2)oritsequivalentstoionic reactants.Equationssuchas(2.2.2)arenolongerfoundinrecent treatisesonchemicalkinetics,butunfortunatelystillpoppedupin arecentelectrochemicaltextbook[58].

Inthiscontextitisusefultoconsiderthenarrowregionofionic strengthsinwhichtheDebye-Hückellimitinglawapplies,because onlyinthisregiondoesthattheorypredictspecificionicactivity coefficientswithoutaneedforadjustableparameters,suchasa distanceofclosestapproach.Ifweassumethat,inthis limiting-lawregion,theDebye-Hückelmodelapplies,thentheChristiansen interpretationmustholdthereaswell,whichwouldseemtomake Brønsted’sexplanationnotmerelysuperfluousbuterroneous,asit wouldleadtodouble-countingtheeffect.

In the Debye-Hückel limiting law region, the Brøn-sted–Christiansen model (Brønsted’s for collecting the experimentalevidenceandfindingitsregularities,Christiansen’s for his insightful theoretical interpretation thereof) hasindeed beenverysuccessfulinrepresentingexperimentaldata.An exten-sivereviewbyLaMer[59]gavemanymoreexamples,suchasthe rate-determiningreactionsand theirapproximate experimental slopes d ln kapp/d√ I) upon theaddition of inertsalt at ionic

strengthsbelow0.1molkg−1,aslistedinTable2.4.1.

Moreover,theBrønsted-Christiansenmodelwasspectacularly successfulinidentifyingthetruenatureofsolvatedelectronsin water,whenHart[60]generatedtransientreactivespeciesin aque-ous solutions bybombarding themwithhigh-energy electrons. TheirreactionrateswithelectronscavengerssuchasH+_,N

2O,H2O2

andNO2−couldthenbefollowedspectrometrically.

Intheselatterreactions,therewasnoprimarysalteffectwith unchargedN2OorH2O2,whereastheeffectswithNO2−andH+were

equalinmagnitudebutofoppositesigns,inquantitativeagreement withtheBrønsted-ChristiansenfactorzA zB,butagainnot

inter-pretableintermsofreagentactivities,wherethatfactorwouldhave been–(zA2+zB2)andthereforealwaysofthesamesignexceptwhen

bothzAandzBwouldbezero.Hart’sexperimentswithanionic,

neu-tral,andcationicquenchersunambiguouslyindicatedachargeof–1 forthespeciesgeneratedbytheelectronbombardment,supporting itsidentificationasasolvatedelectronratherthanas,e.g.,an acti-vatedhydrogenatom‘instatunascendi’,aconceptstillinuseat thattime.Infact,thetransientspectrumHartobservedinwateris quitesimilartothepermanentspectrumofsolvatedelectronsin, e.g.,dryliquidammonia.

Table 2.4.2 comparesthe predictionsof the Lewis“activity” modelwiththoseoftheBrønsted-Christiansenmodelasa func-tionoftheparameterszAandzB.OnlywhenzA+zB=0dothetwo

Table2.4.1

SeveralreactionsquotedbyLaMer[59],andtheirslopesinaplotoflogkvs.√I,see Fig.2ofref.59.

2[Co(NH3)5Br]2++Hg2++2H2O→(to[Co(NH3)5H2O]3+ andHgBr2)

initialslope+4 CH2BrCOO–+S2O32−→(toCH2S2O3COO2–andBr–) initialslope+2 S2O82–+2I–→(toSO42–andI2) initialslope+2 [NO2NCOOC2H5]–+OH−→(toN2O,CO32–andC2H5OH) initialslope+1 C12H22O11+OH–→(inversionofsucroseto

glucose+fructose)

initialslope0 H2O2+H++Br–→(toH2OandBr2) initialslope–1 [Co(NH3)5Br]2++OH–→to[Co(NH3)5OH]2+andBr–) initialslope–2

Table2.4.2

ThepredictedfactorsoftheBrønsted-Christiansen(bold)and“activity”(italics)models,+2(ZAZB)and–



Z_A2+Z_B2



respectively,fortheinitialcoefficientd(logkapp)/d(A√I) atlowionicstrengthsoftherateofA+B→asafunctionofthevalenciesofAandB.Thetwomodelspredictdifferentresultsexceptinthecasesshownwithgraybackgrounds (onthe‘main’diagonaloftheTable),inwhichcasethetwomodelsleadtoidentical,experimentallyindistinguishablepredictions.

zA= –4 –3 –2 –1 0 1 2 3 4 zB= 4 –32–32 –24–25 –16–20 –8–17 0–16 8–17 16–20 24–25 32–32 3 –24–25 –18–18 –12–13 –6–10 0–9 6–10 12–13 18–18 24–25 2 –16–20 –12–13 –8–8 –4–5 0–4 4–5 8–8 12–13 16–20 1 –8–17 –6–10 –4–5 –2–2 0–1 2–2 4–5 6–10 8–17 0 0–16 0–9 0–4 0–1 0–0 0–1 0–4 0–9 0–16 –1 8–17 6–10 4–5 2–2 0–1 –2–2 –4–5 –6–10 –8–17 –2 16–20 12–13 8–8 4–5 0–4 –4–5 –8–8 –12–13 –16–20 –3 24–25 18–18 12–13 6–10 0–9 –6–10 –12–13 –18–18 –24–25 –4 32–32 24–25 16–20 8–17 0–16 –8–17 –16–20 –24–25 –32–32

modelspredictthesameresults,andthereforecannotbeusedto differentiatebetweenthem.Inallothercases,theexperimental dataclearlyfavortheBrønsted-Christiansenmodel.

WenowreturntotheequilibriumformalismA+BC+Das theratioofequalforwardandbackwardreactionrates.Interms oftheBrønsted-Christiansenmodelwehave,intheDebye-Hückel limitinglawregion,−→k cAcBe+2zAzBA

√

I₌←_k−_c

CcDe+2zCzDA

√

I _and,

forindividual ionicactivitycoefficients, ln ˜fi =–zi2 A√I or ˜fi=

e−Zi2A√I.Wethenhave Ko_{= ៭k/k=(c} CcD/cAcB)e+2(zCzD−zAzB)A √ I =cCcD×10−(z 2 C+z 2 D)A √ I /cAcB×10−(z 2 A+z 2 B)A √ I =cCcD˜fC˜fD/cAcB˜fA˜fB=aCaD/aAaB (2.4.1)

because electroneutrality requires that zA+zB=zC+zD hence

(zA+zB)2=(zC+zD)2,whichyields

−(z2

C+z2D)+(z2A+z2B)=+2zCzD−2zAzB (2.4.2)

Thesameappliestomorecomplicatedequilibriumexpressions, becausethesealwaysrequireachargebalancebetweenreagents and products. Note that there is therefore no conflict between rateexpressionsintermsofconcentrationsplustheirappropriate Brønsted-Christiansencorrectiontermsforinterionicinteractions, andthecorrespondingequilibriumconstantswrittenexplicitlyin terms of activities.The charge balance conditionregarding the reagentsintheexpressionzA+zB=zC+zD forabalanced

chemi-calequilibriumequationisclearlymuchlessrestrictivethanthe requirementsthatbothzA+zBandzC+zDareeachzerointhe

indi-vidualkineticexpressions. Insummary:

(1)ForequilibriumpHproblems,ionicactivitiesarenotneeded,and meansaltactivitiesormeasurableactivityratiossuffice. (2)Forreactionratesinvolvingions,usingindividualionicactivity

coefficientsintherateexpressionsoftenleadsto demonstra-blyincorrectpredictions.Atlowionicstrengthsamuchsimpler descriptioncan bebased onionic concentrationsby consid-ering primary salt effects separately, and this turns out to beinfullaccordwiththeexperimentalevidence.Ingeneral, the Brønsted-Christiansen model predicts a different initial dependence of the logarithm of the apparentrate constant onthesquarerootoftheionicstrengththandoesthe“ionic activity”model,andexperimentaldatasupportthe Brønsted-Christiansenmodel.

(3)IntherangeofvalidityoftheDebye-Hückellimitinglaw,the Brønsted-ChristiansenrateexpressionsdcA/dt=dcB/dt=– kcAcB

e+2 zA zB A√I _anddc

C/dt=dcD/dt=– ៭kcCcD e+2 zA zB A

√_I

lead tothethermodynamicequilibriumequationKo_=a

CaD/aAaBin

terms oftheactivities ofreagents andproducts, as a direct

consequenceoftheelectroneutralityconditionzA+zB=zC+zD

fortheequilibriumA+BC+D.However,equilibrium equa-tionscannotbeusedinreverseto“give activitiesratherthan concentrations”,assurmisedbyLewis&Randall[51].

(4)Sinceneitherchemicalequilibrianorchemicalkineticsrequire ionicactivities,itisnotclearwhatbenefitonecanevenhopeto derivefromcontinuedeffortstomeasuretheimmeasurable. 2.5. Christiansen’smath

Brønsted’smodel [37] assumed thepresence of a transition state,andtherebybecametheprecursorof(andmodelfor)Eyring’s “absolute”ratetheory[61,62].However,longbeforeEyring’swork, Christiansen[38]hadalreadyderivedthesameresultbycarefully consideringtheencounterprobabilitiesofreagentions,thereby demonstratingthatBrønsted’smodelassumptionwasnotrequired. Because Christiansen’s explanation was rather terse, and was buriedinmuchothermaterial,itishereoutlinedmoreexplicitly.

Christiansenassumedthat,whenAandBareuncharged,their encounterprobabilityissimplyproportionaltotheproductoftheir volumedensities,i.e.,totheproductoftheirmolarconcentrations, andthattheirreactionratesshouldsimplyreflectthis.Whenboth AandBareions,theirmutualelectrostaticattractionorrepulsion mustalsobetakenintoaccount,butbecausetheirvalenciesare invariant,thiseffectcanbeabsorbedintotheresultingrate con-stant.However,whentheionicstrengthisvaried,e.g.,bychanging theconcentrationsofAand/orB,orbyaddingotherwiseinert elec-trolyte,theirionicatmospheresaremodified,andthereforetheir distancesofclosestapproacha.Christiansenfoundthatthese mod-ificationssufficetoaccountquantitativelyfortheentireprimarysalt effectobservedbyBrønsted.

Inordertoexplaintheprimarysalteffect(ofotherwise non-participating,“inert” electrolyte)inionicreactions, Christiansen assumedthatthe(statistical)reactiondistancecanbeequatedto thedistanceofclosestapproachaintheDebye-Hückeltheory,and usedtheBoltzmanndistributiontocomputetheequilibrium con-centrationsoftheionsiandjatthatdistanceas

ci=ci⊗exp



−ziF j RT



and cj=cj⊗exp



−zjF i RT



(2.5.1)

Becausethosepotentials containtheconstantterms that areunaffectedbythepresenceorabsenceofasalteffect,see(1.3.8), theseterms canbeincorporatedintothereactionrateconstant. Consequently,Christiansonreplaced(2.5.1)forthispurposeby

ci=ci⊗exp



−ziF _j RT



and cj=cj⊗exp



−zjF _i RT



(2.5.2)

Here, 

jisgivenby(1.3.10)which,uponsubstitutioninto(2.5.2)

whilepayingcarefulattentiontotheindicesiandj,yields ci=ci⊗exp



−ziF  j RT



=c⊗_i exp



zizjF2 8εNRT



 (1+a)= zizjA √ I (1+Ba√I) (2.5.3) Likewiseweobtain cj=c⊗_j exp



−zjF _i RT



=c_j⊗exp



zizjF2 8εNRT



 (1+a) = zizjA √ I (1+Ba√I) (2.5.4) sothatthegeneral,Debye-Hückel-compatibleresultreads lnkapp=lnk+ 2zizjA

√ I

(1+Ba√I) (2.5.5)

or,inthelimitingcasewhereBa√I«1, lnkapp≈lnk+2zizjA

√

I (2.5.6)

ThisisindeedBrønsted’sresult,butderivedherepurelyonthe basisoftheDebye-Hückeltheory,withoutadditionalmodel assump-tionsand,inthewordsofDebye[57],“withoutthehelpofany intermediary.”Thesemodelassumptionswerethereforeentirely unnecessary.

2.6. Aformalismfordealingwithsingleionicactivities

Guggenheim[30,31]gaveionicactivitiesaformaldescription, whilewarninghisreadersthatthesewerehypotheticalquantities, withoutphysicalreality,andthatonlysomeveryspecific combi-nations of them would bephysically meaningful and therefore experimentally accessible. Here we will call such hypothetical quantitiesextra-thermodynamic,basedontheclassical understand-ingofthermodynamicsasthestudyofmathematicalrelationships betweenmacroscopicallymeasurablequantities,while disregard-ing the microscopic, atomic nature of matter, which of course includesionization.

Guggenheimuseda separatesymbol,¯ (or,as isnow more customary, )˜ for hiselectrochemicalpotentialin orderto dis-tinguishit explicitly fromthechemical potential,but he did not use any notational markers to express the special status of his ionic activities and activity coefficients. This has led to an unfortunate confusion between thermodynamic and extra-thermodynamicquantitiesandproperties.Butsuchconfusion(1) canbeavoidedbyalsolabelinghissingleionicactivitiesand activ-itycoefficients,andhissingleelectrodepotentials,see(1.6.2),and (2)byexploitingthecommonmathematicalcustomofusingthe imaginarynumberj=√(–1)toidentifyusefulyetnon-real num-bers.AsNahin[63]wroteinhisdelightfulAnimaginarytale:the storyof√–1,“Associatingtheappearanceofimaginarynumbers withthephysicallyimpossibleisaroutineconcepttothemodern engineerorphysicist...”

Inmathematics,certaincombinationsofimaginarynumbersare real,suchastheirproduct,e.g.,√(−5)×√(−3)=j√5×j√3=−√15 becausej2_{=−1,ortheirratio,asin}√_(−5)/√_(−3)=j√_5/j√₃₌√_(5/3)

becauseinthelattercasethej’scancel.Usingasimilarapproach, wefirstpostulateasimpleformalismforstrong1,1-electrolytes suchasNaCl.We will usetheactivityformalism,i.e., eitheras ˜ai=ci˜fiwhenionicconcentrationsciareexpressedinunitsofmols

per unit volumeof solution (typically:mol L−1 or M) and ˜fi is

thecorrespondingionicactivitycoefficient,oras ˜ai=mi˜iwhen

miismeasuredinmolsperunitweightofthesolvent(molkg−1)

with˜iasitscorrespondingionicactivitycoefficient.Inorderto

avoidunnecessarynotationalduplicationwewillprimarilyfocus

ontheactivity ˜ai.Itshouldbeunderstoodthatourparametersdo

notpertaintoindividualions,butrepresentthestatisticalagesof macroscopicamountsofsingleionicspecies,evenifnotisolatable assuch.

ForNaCl,thecationsNa+_andanionsCl−_{willherebedenoted}

bythesubscripts+and−respectively,asintheirvalenciesz+=+1

andz₋=−1,stoichiometricfactors␯+=1and␯−=1,andintheir

activitiesandactivitycoefficients.Wenowwrite ˜a+=ja+instead ofGuggenheim’sa+,and ˜a−=−ja−insteadofGuggenheim’sa−,

for theionicactivitiesaofthesemonovalentions,and we will insist that only non-negative real combinations of such modi-fiedquantitiesrepresentmeasurable,thermodynamicactivities.As Guggenheimpointedout,forasingle,strong1,1-electrolytethereis onlyonesimplecombinationthatisphysicallymeaningful,viz.the productofitssingleionactivitiesoritspositivesquareroot,and thatproductintheabove-modifiednotationwouldbeja+times

−ja₋=a+a−becausej×−j=+1.Thisproductalreadyhasaspecial

thermodynamicsymbol,a±2,thesquareofthemeanactivitya±. Moreover,inamixtureoftwostrong1,1-electrolytes,sayNaCl+KI, therearetwoactivityratios(ortheirinverses)thatarealso physi-callymeaningful,viz.thecationactivityratioaNa/aKandtheanion

activityratioaCl/aI,andthesetworatiosalsoyieldnon-negativereal

valuesintheabovenotationbecausetheiraccompanyingj-terms cancelinthoseratios.

2.7. Extensiontoionsofarbitraryvalence

We cangeneralizetheabove,ad-hocstatementstothe gen-eralstrongbinaryelectrolyteCv+Av−whereCrepresentsacation, Adenotesananion,and␯+and␯−aretheirstoichiometric num-bers.

As our model salt we will use ferric sulfate, Fe2(SO4)3,

where +=2, z+=3, −=3 and z−=−2. In this case we replace the cation and anion activities ã+ and ã− by jz+ _a

+ and jz− a−

respectively, so that their activity product (˜a+)v+ (˜a−)v− must be replaced by (jz+_a+)v+_(jz−_a−)v−_=j(z+v++z−v−)_(a

+)v+(a−)v−=

(a+)v+(a−)v−=(a+)v++v−=(a±)v, because of the electroneutrality conditionz+

v

++z−

v

−=0andthenotationaldefinition

v

++

v

−≡

v

.

Notethat,indeed,jz−_=j−1_{=−jwhenz−=−1intheearlier} exam-pleofNaCl.The traditionalthermodynamicformalism ignoresthe specific requirement of electroneutrality for parameters involving single ionic species, whereas the notation proposed here enforces it.

Likewise, in electrolyte mixtures, such as that of Fe2(SO4)3

plus NaNO3, we can in principle (e.g., by using ion-selective

electrodes for these ions, insofar as these respond thermody-namically)measurethecationratioãFe/(ãNa)3andtheanionratio

ãSO4/(ãNO3)2,ortheirinverses.Thisissobecause,intheproposed

notation,theywillagainbenon-negativerealnumbers,sincewe have zFe=3 and zNa=1, so that ãFe/(ãNa)3=(j3aFe)/(j1aNa)3=j3−3

aFe/(aNa)3=aFe/(aNa)3, and likewise zSO4=−2 and

zNO3=−1, hence ãSO4/(ãNO3)2=(j−2aSO4)/(j−1aNO3)2=j−2+2

aSO4/(aNO3)2=aSO4/(aNO3)2. Here the requirement is that

zFe␯Fe=zNa␯Na and zSO4␯SO4=zNO3␯NO3, so that the exchange

ofoneFe3+_{ionforthreeNa}+_{ions,orthatofoneSO}

42− fortwo

NO3−,isindeedelectroneutral.Allsuchcombinationsthatviolate

electroneutrality are imaginary in this notation, thereby exposing theirtrueextra-thermodynamic,immeasurablenature.

Because we can always write activities as the product of a concentration and an activity coefficient, and concentrations are real, non-negative quantities, the complex notation must beassociated withtheiraccompanyingactivitycoefficients. For the electrolyte Cv+Av− we then find, as expected, that the imaginary terms vanish when we consider theirelectroneutral combination

±≡+˜++−˜− =+˜◦++−˜◦−+(+z++−z−)RTlnj++z+RTln(a+) +−z−RTln(a−)+(+z++−z−)F ˜ =+++−−++z+RTln(a+)+−z−RTln(a−) =◦±+RTln(a±) (2.7.1) where

v

+z++

v

−z−=0,

v

≡

v

++

v

−,

v

a±≡

v

+˜a++

v

−˜a−,

v

˜◦±≡

v

+˜++

v

−˜−. (2.7.2)

2.8. Extensiontochemicalequilibriaandchemicalkinetics

Wecanapplythisformalismtochemicalequilibria,inwhich case we seethat equilibrium constants based onwell-balanced chemical equations aremeasurable quantities. Thismust beso becausewecanexpresssuchequilibriumconstantsas

K=



p (˜ap)p



r (˜ar)r =



p (jzp_ap)p



r (jzr_ar)r = j˙p zpvp



p ap p j˙r zrvr



r ar r =



p ap p



r ar r (2.8.1)

whererreferstothereagents,andptotheproducts,fromwhich we immediatelysee the effect of thenecessary charge balance



p

zpp=



r

zrrbetweenreagentsandproducts.

However,suchcancellationoftheimaginarynumbersdoesnot necessarilyoccurinequationsforindividualreactionrates.Instead, whenweapplythisnotationtoexpressionsforsuchratesinterms of activitiessuchas dc/dt=k



r

˜avr

r rather than concentrations

dc/dt=k



r

cvr

r wefindthattheirratesareimaginaryaslongas



r

zrr /=0.

In theabove examples,toparaphrase Hadamard, the short-estpathbetweentworeal,directlymeasurableconceptspasses throughthecomplex domain.Byassigningimaginaryvalues to immeasurablequantities,theproposednotationcanbeusedasa convenientrealitycheck.Iftheabovenotation,orsomeequivalent thereof,hadbeenincommonuseduringthepasthalfcenturyto distinguishrealfrom“hypothetical”quantities, wemightnotso readilyhave associatedaverypractical quantitydesigned tobe directlymeasurable,suchaspH,withthesingleionicactivity ˜a of hydrogenions,anextra-thermodynamicquantitythatlacks“any realphysicalsignificance”

3. MeasuringpH

DefiningpHisamatteroffindingtheoptimalquantity repre-sentingsolutionacidity.Itboilsdowntothreequestions:which parameterdoweneedtoknow,whichonecanwecompute,and whichonecanwemeasure.Therearenoconflictsbetweenthe parameterweneedtoknowandthatwecancompute,sinceany quantityfirmlybasedinchemicaltheorywillservebothpurposes. Whetherwecanmeasureit,andbywhatmeans,isadifferent mat-ter.HerewewillfirsttakeabrieflookattheopinionofBates,the mainarchitectofthecurrentIUPACpHrecommendation.

Forastrong1,1-electrolytesuchasHCl,theseriouscandidates toconsiderherearepmH,pm˜aH=pmH˜H,andpmH±=pmH˜+˜−,

ortheir molarequivalents.On a molal scale, Sørensen’s defini-tion [8]used pmH, which is a thermodynamically well-defined

quantity. It is hard to argue against the concept of ionic con-centrationswithinthecontextofthepHofelectrolytesolutions, becausemostofourbasicunderstandingofsuchsolutionsrelies onmassandchargebalanceequations,bothaccountingdevices, and therefore written strictly in terms of ionic concentrations ratherthanactivities.Likewise,theionicstrengthintroducedby Lewis & Randall [15], and used by Debye & Hückel [14], is a purelyconcentration-basedquantity. Moreover,evenBates[64] initially defines asconcentration-based solutionproperties Van Slyke’sbuffervalue [65] ˇ=dH_F/dlog[H+_{] orHenderson’s [66]}

numericallysimplerequivalent,thebufferstrengthB=dH_{F/d ln}

[H+_]=[H+_]{dH_F/d[H+_{]}=ˇ/ln(10),here(generallyaswellas}

com-pactly)definedintermsoftheprotonfunctionH_{F[67].Inshort,}

Batesdidn’targuethatthereisanythingfundamentallywrongwith pH=pcHorpmH,whichhesometimesdenotedaspcH.

Wehavealreadycommentedontheallureof ˜aH,eventhough

thatquantityisnowgenerallyrealizedtobeimmeasurable. More-over,Bates[68]commentedthat“Itisnowwellrecognizedthatthe activityofasingleionicspeciesplaysnorealpartinthe develop-mentofthee.m.f.ofagalvaniccell,whetherornotthatcellisofthe typewithaliquidjunction.Itisprobablethatthesameistrueof otherphenomenainfluencedbyhydrogenion.”Whenoneis look-ingforanexperimentalparameterquantifyingthechemicaleffects ofhydrogenions,asdeterminedthroughemfmeasurements,one wouldthinkthatbothitsimmeasurability,aswellasitsirrelevance topotentiometry,wouldeachbeabsolutedisqualifiers.

The third candidate, pmH±=p(mH



˜

H˜X), which Bates

baptized ptH, involves a counterion X (assuming a single 1,1-electrolyte), which would make it a function of two solution constituents.Moreover,␥±cannotbedefinedunambiguouslyin

electrolytemixturescontainingseveralanionsX.Aswehaveseen insection2.1,mHandpmH_±2=pmH˜H˜Cloccurofteninthe

ther-modynamicexpressions forsolutions of single electrolytes,but mH±doesnot.Batesclaimed[69]“thatthereisnorealdifference

betweenptHandpaH,becausesomeofthecommonconventions

identifythesingleionicactivitycoefficientwiththemeanactivity coefficientofanaverageuni-univalentelectrolyte.”Maybeso,but theBates-Guggenheimapproximation[70]doesnot.

Batesthenconsideredthepracticalquestionofmeasurability, asseenthroughthelensofpotentiometry.Sørensenusedan

equa-tionoftheformpH=(Emeas–Eref.el.)/k,whereEmeasisthemeasured

cellpotential,Eref.el.thepotentialassignedtoanexternalreference

electrodeconnectedtothemeasurementcellbyaliquidjunction, andk=RTln(10)/F=0.059157V.Bates[71]calledthisquantitypsH, andcommentedasfollows:“ThefactthatthepsH(Sørensen)scale hasvirtually nofundamental meaningdoesnotpreventitfrom beingusefulforreproduciblecomparisonsofacidity.ThepaHvalue

hasinitselfnomeaningintermsofphysicalreality,yetitsrolein chemicalequilibriacanbesimplyandunequivocally,although con-ventionally,defined.Thisunitpossesses,therefore,allthevirtuesof psHand,inaddition,thepossibility,underidealexperimental con-ditions,ofalimitedamountoftheoreticalinterpretation.”Atthe endofsection2.1wealreadycommentedonthefactthatonecan indeeddefinearoleforsingleionicactivities,butthatsuchactivities areunnecessaryindescribingchemicalequilibria,andcanbe out-rightmisleadinginchemicalkinetics.Moreover,thepHasdefined byBatesandIUPAChastwopracticalproblems:notonlyisp˜aH

immeasurable,butsoistheliquidjunctionpotential.

BatesalsobrieflylookedatpmH±,HCl=p ˜aH˜Cl,a

thermody-namicallywell-defined quantitywhich hedenoted bypwH.He observed[50] that,for amonobasic weakacidat abufferratio of1(i.e.,mHA=mA),pwHisapproximatelyconstant,andthat“a