surface tension - Theremarkableobservationofunmixing

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Unformatted text preview: Theremarkableobservationofunmixing Howdoesoneexplaintheremarkableobservationsofunmixingthatareknownforsome pairsofliquids.? Nearlyeveryotherifnoteveryotherexampleofmechanicalmixingthatyouencounter isaonewaystreet:youmixthingstogetherandtheystaymixed.Herearesomeexamples: Whenyoupourmilkintocoffeeandstirit,itdoesnotunmix. IfyoushakeacupwithM&Mcandies,oranythingelseforthatmatter,theyjustneverend upaslayersofuniformcolor. Theodorfromaspilledbottleofperfumeeventuallyspreadsevenlythroughoutaroom, butputtinganemptybottleintheroomneverrefillsitself. Againandagain,youcanthinkofexamplesofmixingthingsthatdonotunmix.Thenthere istheexampleofoilandwater.Youcanmakealayerofoilontopofalayerofwater,then youcanstirorshakeasvigorouslyaspossible,andstill,whenyousetthisdown,itun mixesintoalayerofoilsittingatopalayerofwater. YouhaveprobablyseenthishappeninChemistry211and216bynow.Therearesome solventsthatsimplydonotmixwithwaternomatterhowhardyoutry:ether, dichloromethane,carbontetrachloride,hexane,andsoon.Evenaftermixing(suchas shakingvigorouslyinaseparatoryfunnel,orinatesttube),thewaterandorganicsolvent form2layersbasedontheirrelativedensities(sometimesthewaterfloatsontop, sometimesitsinksbelow). Really,whenyouconsiderthisobservationrelativetoeveryotherexampleofmixingyou canthinkof,itshouldbeannoyinglymysterious. Whydoesashakenmixtureofoilandwaterspontaneouslyseparate? Andhereisaninterestingseriesofsolubilitydata:ashakenmixtureofoctanolandwater willunmix,whileamixtureofethanolandwaterwillneverunmix.Whyisthat? Methanolinwater: miscible(completelysoluble) Ethanolinwater: miscible Propanolinwater: miscible Butanolinwater: 0.11mol/100g Pentanolinwater: 0.030mol/100g Hexanolinwater: 0.0058mol/100g Heptanolinwater: 0.0008mol/100g Octanolinwater: immiscible(effectivelycompletelyinsoluble) Inordertoexplainthisphenomenon,adetourintothelandofstatisticsisnecessary. Infact,youhavealreadyrunintoprobabilityandstatisticsbefore,inChem210,when learningaboutstereochemistry.Howmanystereoisomersaretherewhenyouhave2 dissimilarstereocenters?(4becausethe2possibleoutcomes,RandS,aremultiplied, 2x2,togivethe4possiblestereoisomers,RR,SS,RS,andSR) Andhowmanystereoisomersarethereifthereare4stereocenters?(16because 2x2x2x2=16;RRRR,SRRR,RSRR,RRSR,RRRS,SSRR,SRSR,SRRSandthecorresponding enantiomersSSSS,RSSS,SRSS,SSRS,SSSR,RRSS,RSRS,andRSSR). YouhaveprobablyencounteredprobabilityandstatisticsbeforeChem210,likelywith commonexamplessuchasflippingcoinsorrollingdice. Ifyouhave4differentcoins(apenny,anickel,adimeandaquarter),whataretheodds that,upontossingthem,youendupwithallheadsoralltails?Yousolvethisexactlythe samewayasthestereocenterproblem!Thereare16possibleoutcomes,onlyoneofthem isHHHHandonlyoneofthemisTTTT.Sotheoddsare2in16,or1/8,thatyouwillendup withallheadsoralltailsifyoutoss4coins.Now,thereisnothingspecialaboutallheadsor alltailsasanoutcome.Alloftheoutcomesareequallyunique:theoddsofHHHH(1/16) areexactlythesameasgettingthepennyasHandtheotherthreeasT(1/16). Fromasetofprobableoutcomes,theindividualoutcomesarecalledmicrostates. Ifyouhave4pennies,whataretheoddsforgettingallheadsoralltails?Thisoutcomeis exactlythesameasfor4differentcoins.Consideringthe4coins,thereisthesameexact numberof(16)microstatesavailable,andonly1ofthemisallHandonly1ofthemisallT. Thistime,however,someofthemicrostatesaredegenerate,thatis,theylookthesame. Unlessthepenniesaremarked,HTHHisgoingtolookexactlylikeHHHTandTHHHand HHTH. Ashumans,webringourtendencytodivideandsorttheworldintoeverythingwedo.Soif youwereactuallytossing4pennies,youmightonlytakenoteof2differentstates:"allthe same"and"notallthesame."Eventhoughall16microstateshaveexactlythesame probabilityoftakingplace,yourhumanperceptionwillconcludethatisitmuchmorelikely forthecoinstocomeout"notallthesame"(ormixed)than"allthesame."Andthiswillbe true...butnotbecausethecoinsknoweachotherorenjoybeingmixed;itwilljustbe becausethereare14microstatesthatappeartobemixedandonly2thatappeartobe uniform. Ifyouhave6pennies,whataretheoddsthattheoutcomewillbeuniform?Ifyouhave6dice, whataretheoddsthattheoutcomewillbeuniform? YouareinChem216labandyouhaveagraduatedcylinderinwhichyoucanstack8 marbles.Youjusthappentohave8marbles:4maizeand4blue.Youputyourmarblesin anErlenmeyerflaskandyouswirlthemaround,mixingthemwell.Thenyoucarefullypour themoutintoyourgraduatedcylinder. Whataretheoddsthatyougetalayerof4blue(B)marblessittingbelowalayerof4maize (M)marbles? Onewaytosolvethatproblemistofirstconsiderallofthepossiblemicrostatesavailableif youhave8marbles.The8marblesarecompletelyindependentfromoneanother,much likeadiewith8sides.Sothetotalnumberofmicrostatesfor8marblesinarowis 8x7x6x5x4x3x2x1(or8!,whichis40,320microstates). IfIwanttoknowhowmanyofthose40,320microstatesareBBBBMMMM,thenIstartwith thebluemarblesandsaythatthenumberofpossiblebluemarblesforthefirstslotis4, then3forthenextandsoon.Sowithinthe"bluelayer"thereare4x3x2x1(4!,or24) possiblewaystoarrangethe4bluemarblestogether.Oncethebluesaretogether,thenthe maizeonesaretogether,sowithinthemaizegroup,therearealso24arrangements.Thus, foreachofthe24uniquebluearrangements,thereare24maizearrangements. So,ofthe40,320totalmicrostatesforthe8marbles,24x24(576)ofthemlooklike BBBBMMMM.Another576ofthemlookMMMMBBBB,andtheother39,168microstates lookmixedup.Remember:everysingleoneofthe40,320microstatesisunique. Hereistheresult:97%ofthetime(39,168/40,320),theresultwilllook"mixed,"and3%of thetimetheresultwilllook"unmixed."Yourbigbrainwouldregisterthisasthefollowing: ifIshakethese8marblesandpourthemout,Ijustdon'tanticipateseeing4blueand4 maizeseparatedfromeachotherintoneatgroups.Marbles"like"tobemixedup. Insummary(usingmath),thatmeansthatthepercentageoftimethatdumpingout8 marbleswilllookunmixed(BBBBMMMMorMMMMBBB)is2x[(4!x4!)/8!]x100.The formulainExcelforfactorialis=Fact(x),ifyouwanttotrythisyourself. Well,3%...maybeyouwouldexpecttoseeitnowandthen.Butwhatifitwas50marbles? Basedonthepreviousexample,youcanfigurethatthepercentageofmicrostateswith25B followedby25Mis2x[(25!x25!)/50!]x100.Anyguessesonthatone?Asthenumberof particlesincreases,theoddsofseeinganunmixedarrangementplummets!Thevaluehere is0.000000000000025%...note:everysingleoneofthe3x1064microstateshasexactlythe sameprobability,butthenumberofmixeduplookingonesjustoutnumberstheunmixed uplookingonesbyanextraordinaryamount.Whatabout100marbles? Sofar,thediscussionaboutmarblesandcoinshasbeenavoidingaveryparticularword. Haveyoufiguredthatout? Thewordisentropy. Whenyoufirsthearaboutentropy,youusuallyhearaboutitintermsof"preferencefor disorder"and"randomness."Somethingyouneverthinktoask(andbelieveme,noteacher wantsyoutoaskit)is"so,howdothemarblesknowtheyaremixedup?" "Entropy"iswhatwecallthestatisticalfactthat,ingeneral,thenumberof"mixedup" microstateswilloutnumbersogreatlythenumberof"unmixed"microstates.Thefirst peoplewhothoughtaboutthisusedtermslike"preferencefordisorder"and"randomness" becausetheyhadnotquiteworkedoutamodelforit,yet. Gobacktothe50marbles:thereisnointrinsicreasontoexpectthatanyoneofthe3x1064 microstatesisanybetterthananyother.Infact,theyareallthesame.However, 99.999999999999975%ofthemLOOKmixedup,andyourbigbrainjustcannotkeeptrack ofthedifferencebetweenonemixeduparrangementrelativetoanotherone.Theonlyones youwouldreallynoticearetheonesthatarenotmixedup.Andthereisnotarealistic chanceofseeingthathappeninamillionmillionlifetimesofshakingthatflaskof50 marbles! So:herewego.Youhaveaflaskof50marbles...25maizeand25blue...andyoupourthem outintoacylinder.Younever(Imean,neverever)seealayerof25maizeplusalayerof25 blue.Whyisthat?Youcansay:becausetheentropyformixedupmaizeandblueishigher thanunmixedmaizeandblue(or,theentropyforunmixedmaizeandblueisreallytruly low),whichtellsyounothing.Oryoucansaythattheprobabilityofgettingamixedup lookingmicrostateisoverwhelming.Thisisthesamethingassayingthereishigher entropyassociatedwiththeconditionofbeingmixedup.Themarbles,asitturnsout,do nothavetoknowanythingabouteachother.It'sallabouttheodds. OK,nowbacktothosepeskymolecules. DoyouseenowwhyIthinkyououghttobereally,reallycuriousaboutunmixinginthe caseofoilandwater...ordichloromethaneandwater? Ifyouhad18mLofwater(18gr,1mol,6x1023molecules)and18mLofdichloromethane (13.85g,0.16mol,1x1023molecules),andthemoleculeswerelikemarbles,thenwhatare theoddsofseeingtheunmixedstate(nevermindhavingitactuallyunmix!)?Itisnot evenworthtryingtogetanExcelspreadsheettodealwiththesesortsofnumbers! So:whyisitbetterforwateranddichloromethanetobeunmixed? Youknowthatwaterexistsasahighlyhydrogenbondednetwork.Withinthebulkof water,anygivenwatermoleculeissurroundedbymanyotherwatermolecules,soitcan takeonmanydifferentorientationsandstillremainhydrogenbonded.Consequently, watermoleculeswithinthebulkofwaterhavemanydifferentmicrostatesavailableto thembasedonorientation. Twothingsareconsistentwiththisideathatwatermoleculesinthebulkofwaterhavelots ofmicrostatesbasedonorientation.First,whenwaterisheated,itstaystogetherasa liquidforalongtime.Asacollectionofmoleculeswarmsup,themoleculesareundergoing greatermotion.Thatmotioncanbevibrational,asthebondswiggleandstretch.That motioncanberotational,asthemoleculesspininplace.Orthatmotioncanbe translational,wheretheymovefromonelocationtoanother(suchasgoingfromasolidto aliquid,oraliquidtoagas).Waterhasahighheatcapacity.Thatis,watermoleculescan takeonlotsofmotionandnotflyapart,partlybecauseevenifawatermoleculestartsto rotate(spin)itcanstillendupassociatingwithneighborsonallsidesbyhydrogen bonding. Onthesurfaceofwater,itisanotherstoryaltogether.Ifthewatersurfaceisopentotheair, thenthereisnothingthereforthewatermoleculestohydrogenbondwith.Asaresult,the surfaceofwaterisconstructedfromamuchmoretightlyboundsetofwatermolecules,all hangingontotheirneighborsmuchmorestrongly,andwithfarfewermicrostates availablebecausemicrostatesthatpointthebondsoutintotheairarefarlessstable.So waterendsupwithalargedegreeofsurfacetension.Andwatermoleculesatthesurface havefewermicrostatesavailabletothemthaninthebulk...whichisanotherwayofsaying thatthemoleculesatthesurfacehavealowerentropythatwatermoleculesinthebulkof water. Asaresult,itismuchmorelikelytoseewaterwithaminimumsurfacearea...because therearesimplythatmanymoremicrostatesavailableforwaterwhenitisinthebulk. Ifyouincreasethesurfaceofwaterthatisnotincontactwithotherwatermolecules,then thenumberofavailablemicrostatesgoesdown(likeseeingallheadswhenyoutossthe coins),andthissituationislesslikelytooccursowaterisalwaysseenwiththemost minimumamountofsurfacearea. Now:thinkaboutsomebulkwater,andtrytoimaginewhathappenswhenyouinserta hydrophobicmolecule,suchashexane,intoit.Thehexanecannothydrogenbondwiththe water,sothewatermoleculessurroundingthehexanemoleculeareactuallyforminganew surfaceandatthatsurface,thenumberofmicrostatesavailableforthewatermolecules goesdowndramatically.Thatis,whileinsertingthehexaneintothecollectionofwater moleculeshasafavorableincreaseinentropy(thehexaneandwaterparticlesaremore mixed),theeffectonthewateritselfittodecreaseitsentropy(becauseforeveryone hexanemoleculethatmixesin,abunchofwatermoleculeshavetogetmoreordered,that is,theyloseavailablemicrostatesbecauseofthenewsurfaceareathatforms). Reviewofthemixing/unmixingofwaterandhexane: Waterminimizesitssurfaceareabecausethisrepresentsthemaximumnumberof microstateswatermoleculesinthebulkhavemoremicrostatesavailablethanwater moleculesatasurface.Surfacewater,then,haslowerentropythatbulkwater. Nowyouaddhydrophobichexanemoleculestothis. Fromthepointofviewofmixingparticles(likemaizeandbluemarbles),thehexane moleculesandthewatermoleculesoughttostaymixed.However,whenyouplaceahexane moleculeintothebulkofwater,thissmallgaininentropyiscounteredbyalargelossin entropybecauseyouhavecreatedanewsurface,surroundingthehexanemolecule,inside thebulkofthewater.Asaresult,mixingthehexanemoleculeswiththewatermolecules actuallyrepresentsanetdecreaseinentropyforthesystem! Foreveryhexanemoleculethatgetsmixedwiththewatermolecules,thewatermolecules themselvesactuallygetalittleunmixed(thenewsurfacethatsurroundsthehexane).Asa result,thesystemdrivestowardthestatewiththegreatestnumberofmicrostates,thatis, thestatewiththehighestentropy(orthegreateststateofmixing). Quitecounterintuitively,thestateofhighestmixing(entropy)iswhenthehexaneandthe waterareinseparatelayers.Byhavingtheunmixedlookinglayerofhexanefloatingon thewater,theamountoflowerentropysurfaceareaforthewaterisminimized.Sowhile thesituationdoesnotlook(toyou)asbeingverygoodintermsofmixing(forthehexane pluswater),itturnsoutthemaximumamountofmixingwithinthewateritselfhastaken place.So:byunmixing,thehexanepluswatersystemisactuallyachievingthegreaterstate ofoverallmixing(entropy),thattakingplacewithinthewateritself. Thetendencyforhydrophobicmoleculestoaggregateawayfromwateriscalledthe hydrophobiceffect. Takealookbackatthesolubilitydata:ashakenmixtureofoctanolandwaterwillunmix, whileamixtureofethanolandwaterwillneverunmix.Youneedtobeabletoexplainboth observations! Methanolinwater: miscible(completelysoluble) Ethanolinwater: miscible Propanolinwater: miscible Butanolinwater: 0.11mol/100g Pentanolinwater: 0.030mol/100g Hexanolinwater: 0.0058mol/100g Heptanolinwater: 0.0008mol/100g Octanolinwater: immiscible(effectivelycompletelyinsoluble) ...
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