The number of pages within the document is: 14
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William Bialek
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2019-01-31 22:13:22.791187
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1.2.BOXESANDARROWSTODIFFERENTIALEQUATIONS 43Figure1.7:Twopossiblechem- icalreactions. 1.2Boxesandarrowstodi !erentialequations WhenwedrawapicturesuchasFig1.7todescribeachemicalreaction, wecouldmeanoneoftwothings.First,wecouldsimplybestatingthe factthat,throughanunspeciÞedprocess,substance Aturnsintosubstance B,andsimilarlyinsomeotherprocess AandBcombinetomake C.For example,ifthisprocessinvolvesacatalyst,itcouldbethattherateofthe reactioncontrolledbytheavailabilityofthecatalystandthushasnothing todowiththeconcentrationof AandBmolecules.AsecondinterpretationisthatthisÞgurereallydescribesthe mechanism ofthereactions.Then A!Bmeansthattheconversionof Ainto Bisa Þrstorder reaction.Intuitively,aÞrstorderreactionisoneinwhicheach moleculemakesanindependentdecisionaboutwhethertocompletethe reaction,notdependingonencounterswithanyothermolecule.Ifthisisthe case,thenthenumberof Bmoleculeswhicharecreatedmustbeproportional tothenumberof Amoleculesthatareavailabletoreact.Itisconventional formostofchemistrytotalknotaboutthenumberofmolecules,butabout theirconcentrationornumberperunitvolume.Ifwewritetheconcentration ofspeciesias Ci,thenforoursimpleÞrstorderreactionwehave dCBdt=kCA,(1.75)wherekisthe Þrstorderrateconstant .Notethatsometimesonewrites[ A]todenotetheconcentrationof A.Itisimportanttogetusedtodi !erent notations,aslongastheyareusedconsistentlywithineachargument! CAandCBhavethesameunits,soinorderforEq(1.75)tomakesense, khastohaveunitsof1 /time,conventionally1 /s.
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