8 MeasurementofMolecularWeight andItsDistribution 8.1INTRODUCTION Asolidpolymerisamosaicofstructures.Foracrystallizablehomopolymer,for example,wecanvarytheamountandnatureofcrystallinityandtheshapeand sizeofthecrystals.Inaddition,wecanvarytheorientationofthepolymerchains inboththecrystallineandamorphousphases.Thisvariationcanbebroughtabout eitherbychangingmaterialvariablesorprocessconditions.Theformerinclude thechemicalstructure,themolecularweightanditsdistribution,theextentof chainbranching,andthebulkinessofthesidegroups.Thelatterincludethe temperatureandthedeformationrate.Itistheinterplaywithinthismultitudeof variablesthatleadstothephysicalstructurevisibleinthefinishedproduct.This structure,inturn,determinesthepropertiesofthesolidpolymer.Inthischapter, weexaminethemethodsofmeasuringthepolymer’smolecularweightandits distribution.ThesequantitiesweredefinedinChapter1,andknowledgethereof can be helpful to the process engineer in optimizing desired polymer properties. These properties include mechanical properties such as impact strength, flow properties such as viscosity, thermal properties such as the glass transition temperature, and optical properties such as clarity. There are several other reasons why we might want to measure the molecular weight. The molecular weight and its distribution determine the viscous and elastic properties of the molten polymer. This affects the processi- 340 Copyright © 2003 Marcel Dekker, Inc. bilityofthemeltandalsothebehavioroftheresultingsolidmaterial(seealso Chapter12).Tociteaspecificexample[1],aresinsuitableforextrusionmust haveahighviscosityatlowshearratessothattheextrudatemaintainsits integrity.Tobesuitableforinjectionmolding,however,thesameresinmusthave alowviscosityathighshearratessothattheinjectionpressurenotbeexcessive. Bothoftheserequirementscanbesatisfiedbyaproperadjustmentofthe molecular-weightdistribution.Moreoften,though,differentgradesofthesame polymeraremarketedfordifferentproductsthatarefabricatedviadifferent polymerprocessingoperations;theresinusedformakingpolycarbonatewater bottles,forexample,differssignificantlyinmolecularweightfromthepoly- carbonatethatgoesintocompactdisks.Differencesinmolecular-weightdistribu- tionalsoinfluencetheextentofpolymerchainentanglementandtheamountof meltelasticity,asmeasuredbyphenomenasuchasextrudateswell.Theeffectof swellshowsupduringprocessing,whereinflowresultsindifferentamountsof chainextensionandorientation,whichremainfrozenwithinthesolidifiedpart. Asaconsequence,twochemicallysimilarpolymers,processedidentically,that havethesamemolecularweightbutdifferentmolecular-weightdistributionsmay resultinproductsthatshowsignificantlydifferentshrinkages,tensileproperties, andfailureproperties[2].Forthisveryimportantreason,itisadvantageousto knowthemolecularweightandmolecular-weightdistributionofthepolymers used.Furthermore,becausepolymerscanmechanicallydegradeduringproces- singandduringuse(polymerssuchasnyloncanalsoincreaseinmolecular weight),asecondmeasurementofthemolecularweightcanrevealtheextentof chainscissionorpostcondensation.Thesemeasurementsarealsousefulin verifyingthatthevariouskineticschemespostulatedforpolymersynthesisin Chapters3–7do,indeed,producethemolecular-weightdistributionspredicted theoretically. Other situations where the molecular weight and its distribution directly influence results include phase equilibrium and crystallization kinetics. A variety of methods are available for molecular-weight determination and they are applicable in different ranges of molecular weight. Also, they provide different amounts and kinds of information. Thus, end-group analysis and colligative property measurements yield the number-average molecular weight. Light scattering, on the other hand, furnishes the weight-average molecular weight and the size of the polymer in solution. Intrinsic viscosity supplies neither number-average molecular weight ( MM n ) nor weight-average molecular weight ( MM w ); it gives a viscosity-average molecular weight. The entire distribution can be obtained using either ultracentrifugation or size-exclusion chromatography. However, the former technique is an absolute one, whereas the latter is indirect and requires calibration. All of these methods mandate that the polymer be in solution. Other, less commonly encountered methods are described elsewhere [3]. Measurement of Molecular Weight 341 Copyright © 2003 Marcel Dekker, Inc. 8.2 END-GROUP ANALYSIS The simplest conceptual method of measuring polymer molecular weight is to count the number of molecules in a given polymer sample. The product of the sample weight and Avogadro’s number when divided by the total number of molecules gives the number-average molecular weight. This technique works best with linear molecules having two reactive end groups that each can be titrated in solution. Consequently, linear condensation polymers made by step-growth polymerization and possessing carboxyl, hydroxyl, or amine chain ends are logical candidates for end-group analysis. Nylon 66, a polyamide and one of the earliest polymers to be synthesized, contains amine and carboxyl end groups. The number of amine groups in a sample can be determined by dissolving the polymer in a phenol–water solvent [4]. Typically, ethanol and water are added to this solution and the mixture is titrated to a conductometric end point with hydrochloric acid in ethanol. Because the number of amine end groups may not equal the number of carboxyl end groups, the acid groups are counted separately by dissolving the nylon in hot benzyl alcohol, and titration is carried out with potassium hydroxide in benzyl alcohol to a phenolphthalein end point. Finally, assuming that the reaction goes to completion and that each nylon molecule has two titratable ends, it is possible to calculate MM n ¼ 2 ½NH 2 þ½COOH ð8:2:1Þ where the quantities in square brackets are concentrations of end groups in moles per gram of polymer. Results are comparable in magnitude to those obtained using osmotic pressure and vapor-pressure osmometry in the range of applic- ability of these techniques [5]. In addition to polyamides, end-group analysis has been used with poly- esters, polyurethanes, and polyethers. Besides titration, counting methods that have been employed include spectroscopic analyses and radioactive labeling. Because the number of chain ends for a given mass of sample reduces with increasing molecular weight, the method becomes less and less sensitive as the size of the polymer molecules increases. The molecular weight of most conden- sation polymers, however, is less than 50,000, and in this range, end-group analysis works fine [6]. Note also that the amount of polymer needed for end- group analysis is relatively small. Example 8.1: In order to determine the number of carboxyl end groups in a sample of polyethylene terephthalate, Pohl dissolved 0.15 g of the polymer in hot benzyl alcohol, to which some chloroform was subsequently added [7]. This solution, when titrated with 0.105 N sodium hydroxide, required 35 mLofthe 342 Chapter 8 Copyright © 2003 Marcel Dekker, Inc. alkali.Ifablanksolutionofthebenzylalcoholpluschloroformrequired5mLof thebase,howmanycarboxylendgroupswerecontainedinthepolymersample? Solution:Because30mLof0.105gramequivalentperliterofthebasereacted withthepolymer,theconcentrationofgramequivalentsofendgroupswas ð30Þð10 À6 Þð0:105Þ 0:15 ¼21Â10 À6 equivalentspergram 8.3COLLIGATIVEPROPERTIES Itiseasilyobservedthatdissolvinganonvolatilesoluteinaliquidresultsina depressionofthefreezingpoint;thatis,thetemperatureatwhichasolidphaseis formedfromsolutionislowerthanthetemperatureatwhichthepuresolvent freezes.Thisistheprincipleatworkinanicecreammakerandinsnowremoval whensaltisusedtomeltandtherebyremovesnowandicefromroads.Besides loweringthefreezingpoint,theadditionofanonvolatilesolutealsoreducesthe vaporpressureatagiventemperature,withtheconsequencethatthesolution boilsatahighertemperaturethanthepuresolventdoes.Furthermore,asolution candevelopalargeosmoticpressure(explainedlater),whichcanbemeasured withrelativeease.Thesefoureffects—depressionoffreezingpoint,elevationof boilingpoint,loweringofsolventvaporpressure,anddevelopmentofanosmotic pressure—arecalledcolligativepropertiesandtheydependonlyonthenumber concentrationofthesoluteinsolutioninthelimitofinfinitedilution.Thus, beginningwithaknownmassofsolute,aknowledgeofanyofthesecolligative propertiesrevealsthetotalnumberofmoleculesinsolution,which,inturn, allowscomputationofthenumber-averagemolecularweight.However,the relativemagnitudeoftheseeffectsissuchthatasthemolecularweightofthe soluteincreasesandthenumberofmoleculesinagivensamplemassdecreases, notallfourcolligativepropertiescanbemeasuredwithequalaccuracyorease; indeed,membraneosmometryisthemethodofchoiceformeasuringthenumber- averagemolecularweightofhighpolymers. Phaseequilibriumisthebasicprincipleusedtoobtainexpressionsforthe magnitudeofthedifferentcolligativeproperties.Itisknownfromthermo- dynamicsthatwhentwophasesareinequilibrium,thefugacity, ^ ff,ofagiven componentisthesameineachphase.Thus,if,asshowninFigure8.1,purevapor A is in equilibrium at temperature T and pressure P with a liquid mixture of A and B, where B is a nonvolatile solute, f v A ðT; PÞ¼ ^ ff L A ðT; P; x A Þð8:3:1Þ Measurement of Molecular Weight 343 Copyright © 2003 Marcel Dekker, Inc. where the superscripts v and L denote vapor phase and liquid phase, respectively, and x A is the mole fraction of A in the liquid phase. Also, a ‘‘hat’’ ( ^ )onf A signifies a component in solution as opposed to a pure component. If the mixture of A and B is sufficiently dilute, it will behave as an ideal solution, for which the following holds [8]: ^ ff L A ðT; P; x A Þ¼f L A ðT; PÞx A ð8:3:2Þ and the result is known as the Lewis and Randall rule. Consequently, f v A ðT; PÞ¼f L A ðT; PÞx A ð8:3:3Þ If at the same pressure P, pure A boils at temperature T b , then it is obvious that f v A ðT b ; PÞ¼f L A ðT b ; PÞð8:3:4Þ Dividing the left-hand side of Eq. (8.3.4) by the left-hand side of Eq. (8.3.3), taking the natural logarithm, and equating the result to the logarithm of the ratio of the corresponding right-hand sides gives the following: ln f v A ðT b ; PÞ f v A ðT; PÞ ! ¼ ln f L A ðT b ; PÞ f L A ðT; PÞ ! À ln x A ð8:3:5Þ For a pure material, d ln f dT P ¼ h 0 À h RT 2 ð8:3:6Þ where h is the specific enthalpy at temperature T and pressure P, and h 0 is the same quantity at temperature T but at a low enough pressure that the material behaves as an ideal gas. Integrating Eq. (8.3.6) from temperature T b to temperature T at constant pressure and noting that T b % T and, therefore, TT b % T 2 b , ln f ðT ; PÞ f ðT b ; PÞ ! ¼ h 0 À h R 1 T b À 1 T ! ¼ h 0 À h RT 2 b ðT À T b Þð8:3:7Þ FIGURE 8.1 Pure solvent vapor in equilibrium with a polymer solution. 344 Chapter 8 Copyright © 2003 Marcel Dekker, Inc. ApplyingEq.(8.3.7)topureAinthevaporphaseandthentopureAintheliquid phaseandintroducingtheresultsinEq.(8.3.5)gives À h 0 Àh v RT 2 b ! DT b ¼À h 0 Àh L RT 2 b ! DT b Àlnx A ð8:3:8Þ whereDT b equalsTÀT b ,theelevationinboilingpoint. Becauselnx A equalslnð1Àx B Þ,whichforsmallx B isthesameasÀx B ,we seethefollowing: DT b ¼ ðRT 2 b x B Þ Dh v ð8:3:9Þ inwhichDh v equalsh v Àh L ,themolarlatentheatofvaporizationofpuresolvent A.Fromthedefinitionofthemolefraction, x B ¼ MolesofB Totalmolesinmixture ¼ MassofB ðMol:wt:ofBÞðTotalmolesÞ Mixturevolume Mixturevolume % cv A MM n ð8:3:10Þ wherecisthemassconcentrationofB, MM n isthenumber-averagemolecular weightofB,andv A isthemolarvolumeofthesolvent.Finally,wecanderive DT b c ¼ RT 2 b v A Dh v MM n ð8:3:11Þ whichallowsustocompute MM n fromameasurementofDT b .Notethatone typicallyextrapolatestheleft-handsidetoc¼0inordertoensureidealsolution behavior.Thistechniqueofmolecular-weightmeasurementisalsoknownas ebulliometry. IfweconsiderthesituationdepictedinFigure8.2insteadofthatshownin Figure8.1,thenananalysissimilartotheonecarriedoutearlierleadstoan expression for the depression in freezing point, which is identical to Eq. (8.3.11) except that DT is now T f À T, where T f and T are the freezing points of the pure solvent and the solution, respectively. Also, Dh becomes the molar latent heat of fusion of the pure solvent, and T b is replaced by T f . This measurement is known as cryosc opy. For an ideal solution, the vapor pressure p A of the solvent in solution is given by Raoult’s law as follows [8]: p A ¼ x A P A ð8:3:12Þ Measurement of Molecular Weight 345 Copyright © 2003 Marcel Dekker, Inc. where P A is the vapor pressure of the pure solvent at temperature T. This lowering in vapor pressure is utilized for the measurement of molecular weight in the technique known as vapor-pressure osmometry. Figure 8.3 shows a schematic diagram of a vapor-pressure osmometer. Two thermistor probes are positioned inside a temperature-controlled cell that is saturated with solvent vapor. If syringes are used to introduce drops of pure solvent on the thermistor probes, then at thermal equilibrium, the temperature of the two probes is the same and an equal amount of solvent vaporizes and condenses at each probe. If, however, one of the solvent drops is replaced by a drop of solution, there is an initial imbalance in the amount of solvent condensing and vaporizing at that probe. Because of the lowering in vapor pressure, less solvent vaporizes than condenses, which leads to a rise in temperature due to the additional heat of vaporization. When equilibrium is reestablished, the temperature T at this probe is higher than the temperature T S at the other probe which is in contact with the drop of pure solvent. Under these conditions, the vapor pressure of pure solvent at temperature T S equals the vapor pressure of the solvent in solution at temperature T, and the situation is analogous to ebullio- metry. Therefore, we can use Eq. (8.3.11) again if we define DT as T À T S . The temperature difference itself is measured as a difference in electrical resistance by FIGURE 8.2 Pure solid solvent in equilibrium with a polymer solution. FIGURE 8.3 Schematic drawing of a vapor-pressure osmometer. 346 Chapter 8 Copyright © 2003 Marcel Dekker, Inc. makingthetwothermistorsbethetwoarmsofaWheatstonebridge.Commercial instrumentscan,atbest,measureDTdowntoabout5Â10 À5 C.Becauseofheat lossesandsolutionnonidealities,themeasuredDTdoesnotequalthevalue calculatedbasedonEq.(8.3.11).Itisnecessarytocalibratetheinstrumentusing amaterialofknownmolecularweight.Therangeofcommercialvaporpressure osmometersisfrom40to50,000g=mol,withthelowerlimitbeingsetbysolute volatility[6]. Forpolymermolecularweightsof100,000andgreater,thetemperature differencespredictedbyEq.(8.3.11)foradilutepolymersolutioninatypical organicsolventareabout10 À5 C(seeTable8.1).Suchsmallchangesin temperatureareverydifficulttomeasurewithanydegreeofprecision.Conse- quently,whenworkingwithhigh-molecular-weightpolymers,weturntoother techniquesofmolecular-weightdetermination,especiallyosmoticpressure. Whenapolymersolutionisseparatedfromthepuresolventbya semipermeablemembranethatallowspassageofthesolventbutnotthesolute, then(asshowninFig.8.4)thetendencytoequalizeconcentrationsresultsinflux of the solvent across the membrane and into the solution. As mass transfer proceeds, a pressure head builds up on the solution side, tending to slow down and ultimately stop the flow of solvent through the membrane. At equilibrium, the liquid levels in the two compartments differ by h units; the difference in pressure p is known as the osmotic pressure of the solution. Note that if additional pressure is applied to the solution, solvent can be made to flow back to the solvent side from the solution side; this is known as reverse osmosis. As the following analysis demonstrates, osmotic pressure can be employed to measure the number- average molecular weight of a polymeric solute. If we designate solvent properties by subscript 1 and solute properties by subscript 2, then the following relations hold at thermodynamic equilibrium, using the condition of phase equilibrium: f 1 ðT; PÞ¼ ^ ff 1 ðT; P þ p; x 1 Þ%x 1 f 1 ðT; P þpÞð8:3:13Þ TABLE 8.1 Colligative Properties of Polystyrene-in-Toluene Solutions at a Mass Concentration of 0.01 g=cm 3 MM n DT b ðKÞ DT f ðKÞ p (cm of solvent at 25 C) 50,000 7:8  10 À4 8:5  10 À4 5.8 100,000 3:9  10 À4 4:25  10 À4 2.9 500,000 7:8  10 À5 8:5  10 À5 0.58 5,000,000 7:8  10 À6 8:5  10 À6 0.058 Measurement of Molecular Weight 347 Copyright © 2003 Marcel Dekker, Inc. where the second equality follows from the Lewis and Randall rule. Conse- quently, ln f 1 ðT; PÞ f 1 ðT; P þ pÞ ¼ ln x 1 ð8:3:14Þ For a pure material, however, laws of thermodynamics give d ln f dP T ¼ v 1 RT ð8:3:15Þ where v 1 is the molar volume of the solvent. Integrating Eq. (8.3.15) between P and P þ p at constant temperature yields ln f ðT ; P þpÞ f ðT ; PÞ ! ¼ v 1 p RT ð8:3:16Þ Comparing Eqs. (8.3.14) and (8.3.16) reveals Àln x 1 ¼ v 1 p RT ð8:3:17Þ Because it is possible to write the left-hand side as x 2 for dilute solutions, a further use of Eq. (8.3.10) converts Eq. (8.3.17) to p c ¼ RT MM n ð8:3:18Þ where c is the mass concentration of the solute. Again, we typically extrapolate p=c to c ¼ 0 to ensure that ideal solution behavior is obtained and Eq. (8.3.18) FIGURE 8.4 Osmosis through a semipermeable membrane. 348 Chapter 8 Copyright © 2003 Marcel Dekker, Inc. holds.Expectedvaluesoftheosmoticpressurefordilutesolutionsofpolystyrene intoluenearelistedinTable8.1. Atypicalplotofexperimentaldataforaqueoussolutionsofpolyethylene oxideat20 CisshowninFigure8.5[9];thesedataareextremelyeasytoobtain eventhough2daysarerequiredforequilibriumtobereached.Itisseenthatthe plothasanonzeroslope,andsignificanterrorcanoccurifextrapolationtozero concentrationisnotcarriedout.Thisnonzeroslopecanbetheoreticallypredicted ifrealsolutiontheoryisusedinsteadofassumingidealsolutionbehavior.For instance,ifweemploytheFlory–Hugginstheory(consideredindetailinChapter 9)andequatethefugacities(or,equivalently,thechemicalpotentials)ofthe solventonbothsidesofthemembrane,theuseofEq.(9.3.30)alongwiththe knowndependenceofthechemicalpotentialontemperatureleadstothe followingresult(seeChapter9): p¼À RT v 1 lnf 1 þf 2 1À 1 m þw 1 f 2 2 ! ð8:3:19Þ wheref 1 andf 2 arethevolumefractionsofthetwocomponents,respectively,m istheratioofthemolarvolumeofthesolutetothemolarvolumeofthesolvent, andw 1 istheinteractionparameter. FIGURE8.5Osmoticpressureofaqueouspolyethyleneoxidesolutionsat20 C.(From Ref. 9.) Measurement of Molecular Weight 349 Copyright © 2003 Marcel Dekker, Inc. [...]... 78.15 81.6 84.4 86.7 88 .9 5.065 5.113 5.161 5.2 09 5.256 5.303 5.3 49 5. 395 5.440 5.485 5.530 5.574 5.618 5.662 5.705 5.7 89 5.87 90 .7 92 .2 93 .7 94 .8 95 .8 96 .6 97 .3 97 .9 98.4 98 .7 99 .1 99 .3 99 .5 99 .7 99 .8 99 .9 100.0 8.16 Use the data given in Problem 8.15 to plot the mole fraction distribution and the weight fraction distribution as a function of the logarithm of the degree of polymerization Copyright... 2.865 2 .92 9 2 .99 2 3.056 3.1 19 3.181 3.243 3.305 3.366 3.427 3.488 3.548 3.607 3.667 3.725 3.784 3.842 3 .90 0 3 .95 7 0.0 0.005 0.020 0.052 0.105 0.185 0.343 0.475 0.706 0 .99 9 1.38 1.88 2.51 3.30 4.28 5.46 6.87 8.56 10.50 12.7 4.014 4.070 4.126 4.182 4.237 4. 292 4.346 4.440 4.454 4.507 4.560 4.612 4.664 4.715 4.766 4.817 4.868 4 .91 8 4 .96 7 5.016 15.2 18.1 21.5 25.2 29. 3 33.7 38.5 43.4 48.5 53.5 58.3 62 .9 67.3... 1614–1616, 195 4 Smith, J M., H C Van Ness, and M Abbott, Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 199 6 Cherutich, C K., Osmotic Pressure of Polymer Solutions in the Presence of Fluid Deformation, M.S thesis, Chemical Engineering, State University of New York, Buffalo, 199 0 Young, R J., and P A Lovell, Introduction to Polymers, 2nd ed., Chapman & Hall, London, 199 1... Press, New York, 196 9 Hiemenz, P C., Polymer Chemistry, Marcel Dekker, New York, 198 4 Zimm, B H., Apparatus and Methods for Measurement and Interpretation of the Angular Variation of Light Scattering; Preliminary Results on Polystyrene Solutions, J Chem Phys., 16, 1 099 –1116, 194 8 Kratochvil, P., Advances in Classical Light Scattering from Polymer Solutions, Pure Appl Chem., 54, 3 79 393 , 198 2 Hiemenz, P... of Colloid and Surface Chemistry, Marcel Dekker, New York, 197 7 Budd, P M., and S Chakrabarti, Ultracentrifugal Studies of the Degradation of a Fracturing Fluid Polymer: Hydroxypropyl Guar, J Appl Polym Sci., 42, 2 191 –2 196 , 199 1 Bird, R B., W E Stewart, and E N Lightfoot, Transport Phenomena, 2nd ed., Wiley, New York, 2002 Einstein, A., Eine neuve bestimmung der molekuldimension, Ann Phys., 19, 2 89 ... Phys., 19, 2 89 306, 190 6 Walia, P S., Influence of Polymeric Additives on the Melting and Crystallization Behavior of Nylon 6,6, Ph.D thesis, Chemical Engineering, West Virginia University, Morgantown, 199 8 Kurata, M., and W H Stockmayer, Intrinsic Viscosities and Unperturbed Dimensions of Long Chain Molecules, Fortschr Hochpolym.-Forsch., 3, 196 –312, 196 3 Brandrup, J., and E H Immergut, Polymer Handbook,... Halliday, Physics Part I, Wiley, New York, 196 6 Berne, B J., and R Pecora, Dynamic Light Scattering, Wiley, New York, 197 6 Pecora, R (ed.), Dynamic Light Scattering, Plenum, New York, 198 5 Oster, G., The Scattering of Light and Its Application to Chemistry, Chem Rev., 43, 3 19 365, 194 8 Tanford, C., Physical Chemistry of Macromolecules, Wiley, New York, 196 1 Kerker, M., The Scattering of Light and Other... number of particles are dropped into the tube, then, in the absence of particle–particle interactions, the mass flux of spheres at any cross section is given by Flux ¼ vc ð8:5:5Þ where c is the local mass concentration of spheres As time proceeds, spheres build up at the bottom of the tube, and the tendency to equalize concentrations causes a diffusive flux of spheres upward in the tube The magnitude of. .. ½ZM =NA is the volume of a polymer molecule multiplied by 2.5, whereas cNA =M is obviously the number of polymer molecules per unit volume As a consequence, ½Zc, which is the product of these two quantities, represents the volume fraction of polymer multiplied by 2.5 If this number is small compared to unity, the polymer solution is considered to be dilute; if it is of the order of unity, the solution... 197 4 Thomas, D P., and R S Hagan, The Influence of Molecular Weight Distribution on Melt Viscosity, Melt Elasticity, Processing Behavior, and Properties of Polystyrene, Polym Eng Sci., 9, 164–171, 196 9 Ezrin, M (ed.), Polymer Molecular Weight Methods, Advances in Chemistry Series No 125, American Chemical Society, Washington, DC, 197 3 Waltz, J E., and G B Taylor, Determination of Molecular Weight of . Inc. whereRistheuniversalgasconstant.Itcanbeseenthatameasurementofthe terminalvelocitymakesitpossibletocomputethemolecularweightiftheother quantitiesinEq.(8.5.4)areknown. If,insteadofasingleparticle,alargenumberofparticlesaredroppedinto thetube,then,intheabsenceofparticle–particleinteractions,themassfluxof spheresatanycrosssectionisgivenby Flux¼vcð8:5:5Þ wherecisthelocalmassconcentrationofspheres. Astimeproceeds,spheresbuildupatthebottomofthetube,andthe tendencytoequalizeconcentrationscausesadiffusivefluxofspheresupwardin thetube.ThemagnitudeofthefluxisgivenbyFick’slaw(seeChapter13)as follows: Flux¼D dc dz ð8:5:6Þ whereDisthesamediffusioncoefficientappearinginEq.(8.5.2)andzisthe distancemeasuredalongthetubeaxis.Forasteadystatetobereachedinthe sphereconcentration,thefluxesgivenbyEqs.(8.5.5)and(8.5.6)havetobeequal inmagnitude.Equatingthesetwoquantitiesandreplacingtheterminalvelocity byanexpressionobtainedwiththehelpofEq.(8.5.4)gives dc dz ¼ cMg RT 1À r r s . on the geometric shape of the scattering particle; it is from the deviation of the data from the predictions of Eq. (8.4.3) that we estimate the radius of gyration of the polymer molecule. 352. ! DT b Àlnx A ð8:3:8Þ whereDT b equalsTÀT b ,theelevationinboilingpoint. Becauselnx A equalslnð1Àx B Þ,whichforsmallx B isthesameasÀx B ,we seethefollowing: DT b ¼ ðRT 2 b x B Þ Dh v ð8:3 :9 inwhichDh v equalsh v Àh L ,themolarlatentheatofvaporizationofpuresolvent A.Fromthedefinitionofthemolefraction, x B ¼ MolesofB Totalmolesinmixture ¼ MassofB ðMol:wt:ofBÞðTotalmolesÞ Mixturevolume Mixturevolume % cv A MM n ð8:3:10Þ wherecisthemassconcentrationofB, MM n isthenumber-averagemolecular weightofB,andv A isthemolarvolumeofthesolvent.Finally,wecanderive DT b c ¼ RT 2 b v A Dh v MM n ð8:3:11Þ whichallowsustocompute MM n fromameasurementofDT b .Notethatone typicallyextrapolatestheleft-handsidetoc¼0inordertoensureidealsolution behavior.Thistechniqueofmolecular-weightmeasurementisalsoknownas ebulliometry. IfweconsiderthesituationdepictedinFigure8.2insteadofthatshownin Figure8.1,thenananalysissimilartotheonecarriedoutearlierleadstoan expression