EmpiricalGasLaws

• EarlyexperimentsconductedtounderstandtherelationshipbetweenP,VandT(andnumberofmolesn)

• Resultswerebasedpurelyonobservation– Notheoreticalunderstandingofwhatwastakingplace

• Systematicvariationofonevariablewhilemeasuringanother,keepingtheremainingvariablesfixed

Boyle’sLaw(1662)

• Foraclosedsystem(i.e.nogascanenterorleave)undergoinganisothermalprocess(constantT),thereisaninverserelationshipbetweenthepressureandvolumeofagas(regardlessoftheidentityofthegas)

Vα 1/P• Toturnthisintoanequality,introduceaconstantofproportionality(a)à V=a/P,orPV=a

• SincetheproductofPVisequaltoaconstant,itmustUNIVERSALLY(forallvaluesofPandV)beequaltothesameconstant

P1V1 =P2V2

Boyle’sLaw• Thereisaninverse relationship

betweenPandV(isothermal, closedsystem)

https://en.wikipedia.org/wiki/Boyle%27s_law

Charles’Law(1787)

• Foraclosedsystemundergoinganisobaricprocess(constantP),thereisadirectrelationshipbetweenthevolumeandtemperatureofagas.

Vα T

• Proceedinginasimilarfashionasbefore,wecansaythatV=bT(theconstantneednotthesameasthatofBoyle’sLaw!)andthereforeV/T=b

Charles’Law(continued)

• ThisequationpresumesaplotofVvsTwouldgiveastraightlinewithaslopeofbanday-interceptof0(i.e.equationgoesthrough theorigin).

• Howevertheexperimentaldatadidhaveanintercept!• LordKelvin(1848)decided toextrapolatethedatatoseewhereitwouldcrossthe

x-axis(T)– Regardlessofthenatureofthegas,thedataalwayswouldyieldthesamevalue:-273.15oC

– Thereforetomakeitgothrough zero,justadd273.15toeachpoint!• ThisestablishedtheKelvin(absolute) scale

Absolutetemperature(Kelvin)

• Theestablishmentofanabsolutetemperaturescalewasbasedonexperiments

http://chemed.chem.purdue.edu/genchem/history/charleslaw.html

Gay-Lussac’sLaw(1809)

• Foraclosedsystemundergoinganisochoricprocess(constantV),thereisadirectrelationshipbetweenthepressureandtemperatureofagas

• Pα T• OrproceedingasforCharles’Law,P/T=c

• Inthisrelation,wealsomustuseabsolutetemperature

http://www.meritnation.com/ask-answer/question/what-is-gay-lussacs-law-need-help/science/1780970

Avogadro’sLaw(1811)

• Foranopensystem(massisallowedtobetransferredinorout),thevolumeofgasisdirectlyproportionaltotheamountofgaspresent(givenisothermalandisobaricconditions)

Vα n

• MathematicallythiscanbewrittenasV=dn(yetanotherconstant)

Avogadro’sLaw(continued)

• SincetheTandPmustbeconstant,itwouldbeusefultodefineareferencestate(TandP)sothatgasescanbecomparedtoeachother

• STP(standardtemperatureandpressure)– T=0oCandP=1atm

• Standardstate– T=25oCandP=1bar

CombinedGasLaw(?)

• Itwouldappearthatalltherelationshipscanbecombinedintooneequation.Forexample,volumeisseentobeinverselyproportionaltopressure,directlyproportionaltothetemperatureanddirectlyproportionaltothenumberofmolesofgas.Therefore

Vα nT/P• Or• Thiswilleventuallyleadtotheideal(perfect)gaslaw,PV=nRT,aswell.

KineticMolecularTheory

• Providesatheoreticalexplanationforthebehaviorofgases• KMTissimplyamodel– itisnotaperfectdescriptionofreality– Goodenough!(ifnotwewillfixitlater)

PostulatesoftheKMT

• Agasiscomposedofparticles(moleculesoratoms)thatareperfectspheres

• Gasparticlesareinconstant,randommotion• Gasparticlesmoveinstraightlines(i.e.notaccelerating)• Gasparticlesareveryfarapart– Vgas <<Vcontainer (mostofagasisemptyspace)

• Thetemperature isproportionaltotheaveragekineticenergyofthemotion

MorepostulatesoftheKMT

• Gasparticlesmoveindependently ofeachother– Thepositionandmomentumofoneparticlearenotaffectedbytheposition/momentumofanother.

– Therearenoforcesofattractionorrepulsionbetweenparticles• Eventuallygasparticleswillcollide.– Collisionswithotherparticleswillbeperfectlyelastic– Collisionswiththewallsofthecontainerwillresultinpressure.

ApplicationsoftheIdealGasLaw

• Determinationofthemolarmassofagas• SincePV=nRTwecansaythatn=PV/RT• Wecanalsosaythatthenumberofmolesisgivenbyn=m/M• SettingtheseexpressionsequaltoeachotheryieldsPV/RT=m/M,whichcanberearrangedtosolveforthemolarmass:

ApplicationsoftheIdealGasLaw

• Determinationofthedensityofagas• Let’susethefactthatρ =m/VandrearrangetheidealgaslawtoyieldV=nRT/P

• Combiningthiswithn=m/Manddoingsomealgebragives

Whatisamole?

• Smallfurryanimal

• Facialblemish

Whatisamole?

• Amolerepresentthenumberof“particles”(elementaryentities)presentinasample.

• Avogadro’sNumber(NA)– 6.022X1023 /mol

• Themolecanalsoberelatedtomass– n=m/M– n=#ofmoles– m=mass–M=molarmass(orformulamassoratomicmass)

Molecularvelocities

• TheKMTcanbeusedtocalculatetheroot-mean-squarevelocity

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 500 1000 1500 2000 2500

Freq

uency

Velocity(m/s)

Maxwell-BoltzmannDistributionofCO2 atVariousTemperatures

Realgases

• Thereareanumberofreasonswhytheidealgaslawmightbreakdown–Molecularforces• Attractiveandrepulsiveforcesdoexist,andmaybesubstantial(i.e.polarmolecules)

– Conditions• Underextremecasesof“high”pressureand/or“low”temperaturegasesstarttobehavemorelikecondensed phases(liquids andsolids)andintermolecular forcescannotbeignored

vanderWaalsequation

• Firstaccountforthefinitevolumethatagasmoleculeoccupies– Và V-nb

• Sincethiswilldecreasethevolume“available”forthemoleculestocollide, theyshouldcollidemoreoften.Thiswillmeanthatthepressureshould increase.

• Mathematicallyacollisionisdefinedastwoparticlesinthesameplaceatthesametime.Wecanthinkofthe“particledensity”asbeingn/V.Thusthenumber ofcollisionshouldbeproportional to(n/V)(n/V) =n2/V2

– Pà P+an2/V2

vanderWaalsequation

• Puttingallofthistogetherleadsto

or

wherev=V/n.visknownasthespecificvolume,ormolarvolume.

Itisimportanttonotethattheconstants(aandb)aredifferentforeachsubstance.

vanderWaalsequationSubstance a(L2atm/mol) b (L/mol)

He 0.0341 0.02370

Ar 1.34 0.0322

H2 0.244 0.0266

O2 1.36 0.0318

CO2 3.59 0.0427

CCl4 20.4 0.1383

• Typicallycorrectionsare“small”butcanimproveagreement– v=22.4L/molforidealgasatSTPsob/vis<<1

• Correctionstendtobelargerforlargermolecules,aswellasforpolarmolecules

Acomparisonofthethreemainphasesofmatter

• Gases,liquidsandsolidsdifferfromeachotherintherelativemagnitudesofinter- andintramolecularforces

Phase

Volume Shape Compressibility Fluidity

Gas Indefinite Indefinite High HighLiquid

Definite Indefinite Low High

Solid Definite Definite Low Low

Phasediagrams

• Graphicalrepresentationofthestatesofmatterasafunctionoftemperatureandpressure

http://serc.carleton.edu/research_education/equilibria/phaserule.html

CriticalPoints

• CriticalTemperature(Tc)– highesttemperaturethatliquidandgascanexistasdistinctphases– Aliquidcanbeproducedbysimplyincreasingthepressureofthegas

• CriticalPressure(Pc)– highestpressurethatliquidandgascanexistasdistinctphases– Aliquidcanbeproducedbysimplydecreasingthetemperatureofthegas

• Beyondthecriticalpoint,supercriticalfluidexists

CriticalPoint– inpictures

• Phaseboundarydisappears,sothetwophasesareindistinguishable

http://www.rsc.org/Education/EiC/issues/2008November/SupercriticalProcessing.asp

TriplePoint• Foraonecomponentsystem,thereexistsaunique temperatureandpressurewhereallthreephasescoexistatequilibrium

• Itisaphysicalpropertyofthesubstanceandcan’tbevaried!

• Ex.H2OT=0.0098oCandP=4.58mmHg

http://www.maniacworld.com/water-at-triple-point.html

Solids

• Mostdifficultphasetomodelbecauseparticles(ions,atoms,molecules)areinveryclosecontact– Strongestintermolecularforces

• Amorphous – disorganizedclusterswithnolong-rangeorder• Crystalline – highlyorderedlattice-likeassemblies

TypesofCrystallineSolids

• Molecular– Nonpolar– Polar– H-bonded

• Networkcovalent• Ionic• Metallic

Crystallattices

• 14different types• Lengths(a,b,c)• Angles(α,β,γ)• Differenttypesofsymmetry• Cubichasthegreatestdegreeofsymmetry(a=b=c,α=β=γ=90o)

http://blogs.agu.org/georneys/2011/09/15/geology-word-of-the-week-p-is-for-pleochroism/

CubicArrangements

• Simplecubiccell– Hasitsconstituentsonlyattheedges(corners)ofacube

• Body-centeredcubic(bcc)– Hasanadditional constituentatthecenterofthecube

• Face-centeredcubic(fcc)– Hasanadditional constituentatthecenterofeachfaceofthecube

http://martine.people.cofc.edu/111Lectweek14.htm

TypesofsolutionsSolute Solvent SolutionPhase ExamplesGas Gas Gas Air,naturalgasGas Liquid Liquid Club soda(CO2 inH2O),artificialblood (O2 in

perfluorodecalin)Liquid Liquid Liquid VodkaSolid Liquid Liquid SalineGas Solid Solid H2/PdSolid Solid Solid 14-karatgold(AginAu)

Energetics ofsolutionformation

• 1)Puresolventà separatedsolventmolecules– ΔH1>0becauseintermolecularforcesarebeingbroken

• 2)Puresolventà separatedsolutemolecules– ΔH2>0becauseintermolecularforcesarebeingbroken

• 3)Separatedsolventandsolutemoleculesà solution– ΔH3<0becauseintermolecularforcesarebeingformed

ΔHsolution=ΔH1+ΔH2+ΔH3

Whatitmeanstobeideal

• Forcondensedphases,weknowthatthereareintermolecularforces,whichmaybefairlysignificant.Consideringthecaseofjusttwodifferent typesofmoleculesinasolution(AandB),therearereallythree typesofinteractions:A-A,A-BandB-B.

• Ifthesolutionisideal,thenthemagnitudesoftheseinteractionsareallequal– i.e.itdoesn’treallymatterwhoyourneighboris!

• ΔHsolution =0,ΔVsolution =0

Waystomeasureconcentration

• Relative/qualitativeterms– Diluteorconcentrated

• Solubility– usuallybasedongofsolute/100mLofwater– Unsaturated– undersolubilitylimit– Saturated– atsolubilitylimit– Supersaturated– oversolubilitylimit

Waystomeasureconcentration

• %bymass

– Totalmass=massofsolute+massofsolvent

• %byvolume

– Totalvolume≈volumeofsolute+volumeofsolvent

Waystomeasureconcentration

• Molarity–Mostcommonunitofconcentrationinchemistry

–M=n/V• Molality

– m =n/m– Betterbecausewithmolarity youareunsureoftheactualvolumeofliquidbeingadded

Waystomeasureconcentration

• Normality– Usedalmostexclusivelyforacidsandbases– N=M*E,whereMisthemolarity andEisthe#ofequivalents(#ofH+ thatwilldissociateinanacid,or#ofOH- thatwilldissociateinabase)

– Usefulfortitrations• Molefraction

– Usefulforcolligative properties, chemicalprocesses

Raoult’sLaw

• Consider asolution madeupofsolventA(largepurplespheres) andsoluteB(smallgreenspheres).

• TherateatwhichAleavesthesurface(vaporization)isproportional tohowmanyyouhaveonthesurface,whichisproportional tothemolefraction:r=kxA

• TherateatwhichAcomesback(condensation) isproportional totheconcentrationofthegas,whichisproportional tothepartialpressure:r=k’PA

• Sincethesetworatesmustbethesame:.

Forapureliquid xA=1sok/k’=PA*. Thismeansthat

Idealsolutions

• Anidealsolution isonewhereRaoult’s Lawisobeyed.

• SinceP=PA+PB,andthevaporpressureforaliquid isthesameasthatforagas,foranidealsolutionwecansaythat

• Furthermore, sincexA+xB=1,or

Exampleofanidealsolution

• Solutionstendtobehaveideallywhenthesolvent(A)andsolute(B)are“similar”toeachotherintermsofmolecularstructure,polarity,intermolecularforces,etc.

• Ex.Benzeneandtoluene(methylbenzene)

Colligativeproperties

• Thepresenceofasolutecanaffectthepropertiesofasolution• Thiseffect is(primarily)duetotheamountofsolute(i.e.concentration)butnotnecessarilyonthenatureofthesolute

• Threemaincolligativeproperties:– Boilingpointelevation– Freezingpointdepression– Osmoticpressure

Freezingpointdepressionandboilingpointelevation

• Consider solutions whereonly thesolvent isvolatile,andthesoluteonly dissolves intheliquidphaseofthesolvent

• ΔTf =- iKfm• ΔTb =iKbmi =van’t Hofffactor(i =1foranon-

electrolyte,≈1foraweakelectrolyte,or#ofparticlesforastrongelectrolyte)

Kf =freezing-pointdepression(cryoscopic)constant

Kb =boiling-pointelevation(ebullioscopic)constant

m =molality

https://www.boundless.com/chemistry/solutions/colligative-properties-nonelectrolyte-solutions/freezing-point-depression/

Osmoticpressure

• Osmosis - netflowofsolventmolecules throughasemipermeable membrane– Solventmoleculesgofromasolutionoflowerconcentrationtoasolutionofhigherconcentration(soluteisnotabletopassthrough)

• Osmoticpressure(π)=pressurerequiredtostoposmosis

• π =MRT,whereM=molarity

http://chemistry.tutorvista.com/physical-chemistry/osmotic-pressure.html

Whatitmeanstobenon-ideal

• Intermolecularforcesbetweensoluteandsolventmoleculesarestronger thanotherintermolecularforces– ΔH3>ΔH1+ΔH2

– ΔHsolution <0,ΔVsolution <0• Intermolecularforcesbetweensoluteandsolventmoleculesareweaker thanotherintermolecularforces– ΔH3<ΔH1+ΔH2

– ΔHsolution >0,ΔVsolution >0– Ifforcesaremuchweaker,thenasolutionmaynotformatall!