BurstingMoneyBins
Theiceandwaterstructure
FrancoBagnoli,
Dept.ofPhysicsandAstronomyandCenterfortheStudyofComplexDynamics,Universityof
Florence,Italy
ViaG.Sansone,150019SestoFiorentino(FI)Italy
[email protected]
IntheclassiccomicsbyCarlBarks,“TheBigBinonKillmotorHill”[1],UncleScrooge,tryingto
defendhismoneybinfromtheBeagleBoys,followsasuggestionbyDonaldDuck,andfillsthe
binwithwater.Unfortunately,thatnightisgoingbethecoldestoneinthehistoryof
Ducksburg.Thewaterfreezes,burstingthe``ten-footwalls''ofthemoneybin,andfinallythe
giganticcubeoficeanddollarsslipsdownthehilluptotheBeagleBoyslot.
Thatwaterexpandswhenfreezingisawell-knownfact,anditisatthebasisofanexperiment
thatisofteninvoluntaryperformedwithbeerbottlesinfreezers.Butwhydoesthewater
behavethisway?And,moredifficult,howcanoneillustratethisphenomenoninsimple
terms?
Firstofall,wehavetorememberthatthetemperatureisrelatedtothekineticenergyof
molecules,whichtendtostayintheconfigurationofminimalenergy.Ingeneral,ifoneadopts
asimpleballmodelforatoms,theconfigurationofminimalenergyismorecompactthana
moreenergetic(andthusdisordered)configuration.Butthisisnotthecaseforwater.
FIG.1:ASchematicrepresentationofthewatermolecule
Indeed,watermoleculesresembletheheadofMickeyMouse(seeFig.1),thetwohydrogen
atomsbeingtheears.Itisdifficulttodrawthecrystalstructureofwater,sinceitis
intrinsicallythree-dimensional,butonecanmakeuseofthe``MercedesBenz''model[2].
FIG.2:WatermoleculesintheMercedes-Benzmodel
Inthismodel,thewatermoleculesarerepresentedasthefamousMercedes-Benzsymbol(see
Fig.2),andtheminimumenergyisobtainedwhentwospikesjoin(anunorientedversionof
thehydrogenbond).ItisnotdifficulttomakesomecardboardMercedes-Benzmoleculesand
letpeopleplaywiththem.
FIG.3:WaterandiceintheMercedes-Benzmodel
Bymeansofthismodelitisnoweasytoillustratethattheminimumenergyconfiguration
containsalotof``void''(seeright-handsideofFig.3),andthisexplainsthelowerdensityof
icewithrespecttowater.Itmayalsoservetoillustratethephenomenonofregelation
(meltingunderpressureandfreezingagainwhenthepressureisreduced).
However,inordertohaveastableicestructure,wefirsthavetoreducethekineticenergyof
watermolecules,andthenremovealsothedifferenceinenergyfromtheVanderWaalstothe
hydrogenbonds,i.e.,wehavetoextractheat.Inmacroscopicterms,thefirstiscalledtheheat
capacityofwater(specificheat:C=4.187kJ/kgK),andthesecondthelatentheatofmelting(I
=334kJ/kg).Ifonewantstofurtherlowerthetemperatureofice,onehasalsotoextractthe
energycorrespondingtothespecificheatofice(2.108kJ/kgK).
InCarlBarks’comics,UncleScroogeoftenstatesthathismoneybinis“threecubicacresbig”,
whichisobviouslynonsense,becausetheacreisasurfaceunit.Ifweassumethateachfaceof
thecubeisanacre(around4,000m2),wehaveasideofabout60m,andavolumeVofabout
200,000m3.Iwasunabletofindthedensityofarandompackingofcoins(itseemsthatonly
spheresandM&M’shavebeenstudied[3]),soIarrangedasmallexperimentwith174ten
centscoins.Theresultingpackingdensityisabout0.55,muchlessthanspheres(0.64),but
therearepronouncedfinite-sizeeffects1.ThetotalamountofwateristhusW=
0.45×2×105×103=9×107kg.Assumingthatinitiallythemoneybin(andthewaterinside)isat
T=10oC,thetotalamountofheattoberemovedtoobtainiceat0oCisW×(C×T+I)=
3.4×1013Jifwedisregardtheheatcapacityofcoins–thatofcopper/brassisamere0.4kJ/kg
K–andthatofthewalls,about0.8kJ/kgK.Ifthefreezingoccursin,say,5hours(disregarding
thetimerequiredfortheheatconduction),thismeansacoolingcapacityof113GW!Indeed,
tofreezewaterisnotaneasytask,anditisacommonpracticetoputcansfullofwaternear
fruittreesinordertoprotectthemfromfrost.
Finally,weclosewithachallenge:whatisthetemperatureatthebottomofthesea?2
References
[1]http://coa.inducks.org/story.php?c=W+WDC+135-02(1951).Availableonlineon
http://it.paperpedia.wikia.com/wiki/File:Big_bin_on_killmotor_hill.pdf
[2]K.A.T.Silverstein,A.D.J.Haymet,K.A.Dill,ASimpleModelofWaterandtheHydrophobic
Effect,J.Am.Chem.Soc.120,3166-3175(1998).doi:10.1021/ja973029k
[3]A.Donev,I.Cisse,D.Sachs,E.A.Varano,F.H.Stillinger,R.Connelly,S.Torquato,P.M.
Chaikin,ImprovingtheDensityofJammedDisorderedPackingsUsingEllipsoidsScience303,
990-993(2004),doi:10.1126/science.1093010
1
3
ThisallowsustocomputehowrichScroogeMcDuckis.Thevolumeofa1dollarcoinisabout1.1cm ,and
3
consideringa0.55packingdensityitoccupiesanaveragevolumeof2cm .DividingVbythisnumberweobtainabout
11
10 dollars,onehundredthousandmillionsoronehundredbillions.
2
Letusfirstanalysethecaseoffreshwater(adeeplake).Themaximumdensityofwater,accordingtoourmodel,is
o
somewhatabovezero(inrealwateritisabout4 C).Belowthistemperature,ice-likestructures(ruledbyhydrogen
bonds)arecommon,loweringthedensitytowardsthatofice.AbovethistemperaturetheVanWaalsinteractions
dominateandthewaterbehavesmorelikea``standard''liquid.Inthepresenceofsaltthingschangeabit,the
o
freezingpointlowersandsodoesthemaximumwaterdensity.Sothebottomoftheoceanisabout3 C.