jce-1983-393_modernization of the van deemter equation

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Modernization of the van Deemter Equation forChromatographic Zone DispersionStephen J HawkesOregon State University, Corvailis, OR 97331

In the mid-1950's, a group of Dutch chemical engineersundertook a study of zone dispersion in chromatography.Starting with the equations of Lapidus and Amundson I )they made some approximations and assumptions and arriveda t the equation 2 )

Thi s has come to be known as the "van Deemter equation."While this tit le involves some injustice to previous workers(see the review by Giddings) 3a), he naming was inevitablesince the equation was produced during the early developmentof gas chromatography and was seized upon, used, and quotedin that dramatic period of chromatographic history. The vanDeemter equation was the basis of the many improvementsin column design tha t were realized as the various parameterswere optimized. I t is now quoted in sophomore texts andphysico-chemical research papers as the definitive equation.Often i t is given in t he simplified form

11 i ~~l~tvrrutiatchnr it hay he;wne fossllii.cJ in [ h i - way.1hr euuation rt.prt.scntrd the I~r >rh l r c c ~ l d,, - . i d 111 IlrLrihut thk ensuingdecades have shown tha t chromatographicdispersion is better represented ash Blu Cu 3)

Blu C, C,)u 4)

Thi s form is as rigorously correct as current theory allows,is no more complex than van Deemter's form, and lends itselfto discussion of the geometrical parameters in the last C,u)term. It also avoids fossilizing a precise form of the Cmu termwhich is still poorly understood. The van Deemter equationomitted the C,u term because the columns of the periodcontained so much stationary liquid tha t Cmu was negligiblecompared to t he first C,u) term. However, once the equationwas widely understood and more sensitive detectors becameavailable, it became customary to use less stationary phase soas to reduce drand hence CI so that C, became relatively moreimportant . From tha t period in the early 1960's it has beennecessary to include C,u in discussions of zone dispersion.Usefulness of the Equation

Values of the parameters in eqn. (5)are seldom known withanv precision. It s main usefulness is therefore in aualitativediicussions of the effect of various parameters on column di-s~ersivitv. , and the comoromises that must be made incolumn design and in controlling temperature and flowrate.Equation 5 )can also be manipulated to give equations forresolution or analysis time. Column design, temperature, andflow rate may then be chosen to optimize the desired perfor-mance.The van Deemter equation has also been used to measure

Svmbols Usededdy diffusion r anastomosis termB coefficient of longitudinal diffusione concentrationC coefficient of mass transferd = diameter. or deothD = coeff icient of m.nlecular diffusionend = end~effectsf Giddings compressibility factor

9(pfl: )(p?lp 2 1)/8(p lp~- 2fn = funct ion ofh column dispersivity ("plate height")dames-Martin comoressihilitv factorh column capacity ratio t$t, Ku,lu,K partition coefficient, c$c,1 length of empty tube containing sample before

injectionL = column lengthp pressureq = factor describing shape of liquid phaset timelinear velocity, crn of column traversed per second= average linear velocity Llt ,J " " l l l r n ~

Subscriptsads = adsorbede columnf filmi = at in let of column1 liquid stationary phasem mobile phaseo at outlet of columnp particle

s stationary phaseGreek Svrnbols

obstruction factorviscosityX proportionality factor A A dd fraction of the mobile phase tha t is outside theporous particlesstandard deviation of dis tances of molecules of acomponent from zone centerproportionality constant C,D,ldt

DI and Dm,quite often by naively using eqn. 1). he use ofeqn. (5) makes the problems in experimental design muchclearer and shows tha t such measurements are unlikely to bereliable.The Meaning of

cessive equilihratio& in a series of "theoretical plates%ep-resenting successive segments of the column and called h the"height equivalent to a theoretical plate" or "HETP." Thiswas a logical development of Martin's previous work onmultiple liquid-liquid extraction, and the theory of distilla-tion. They used the model to show that peaks should beGaussian in the limit of a large number of plates, derived arelation between the number of plates and resolution, and

Volume GO Number 5 May 1983 393


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have nractical sienificance bu t are not represented in eqn. (5)and i t is a matte r of pedagogical judgment whether they arebetter discussed in terms of a more detailed and complexequation or simply as qualitative variations of eqn. 5 ) .Thefull equation would he; . ::.,

. I. _he variance c2 f me distribution of me molecules of a single substance about

th ir mean position in th column is 2d2/number of molecules.

showed that resolution varied with the square root of thecolumn length.Thi s was about the limit of usefulness of this model. I t hasgiven us the term plate height symbolized by h This is amost unfortunate inheritance because later models haveproved more tractable, easier to comprehend, and less de-ceptive.The danger of th e model is in the mental concept tha t i tconjures. If a column consisted of a series of plates, thenjoining two columns together would produce a number ofolates eaual to the sum of the d at es in the separate columns.,\n malyst using this conceln imd joining an efiit icnl n h m n11, an intffiri enl one dijappointed t o find rhnt the numberof plates is reduced and the resolution is worse.Klinkenhurg and Sjenitzer introduced a bette r model in1956 when they showed that

= cZ/Lwhere oZ s th e variance of the distribution of the moleculesof a single substance about their mean position in the columnor th in layer plate (i.e., 2d21number of molecules, in the fig-ure). an d L is the distance the zone of substance has moved.In ~uhunnhromatt~yraphy. is usunlly the column lmgth.'l'hii detinirim j more u3eful than the plate height in everyway and has rendered the plate concept obsolete^ Pedagogi-cally it is superior because u2 is directly related to the im-portant experimental phenomena it describes, viz., thespreading of the zone, the width of th e chromatographic peakand hence the resolution. Theoretical plates relate to thesephenomena only indirectly in a manner needing mathematicaltreatment. Theoretically i t is superior because i t relates di-rectly to the underlying physical processes through the Ein-stein equation 38), u2 = 2Dt, where D is the dispersioncoefficient and is the elapsed time. This has formed the basisof the random walk and the nonequilihrium treatments ofchromatography.The use of h = u2IL as a definition of h eliminates theconceptual error of additive plate numbers described above,bu t the name plate height or height equivalent to a theo-retical plate or HETP perpetuate the error. The termplate should now be dropped. For th e remainder of thispaper h will be referred to as th e column dispersivity. Un-fortunately, no such neat name has occurred to the author toreplace the popular plate number

N = L2 02In default of a better name, I have used th e name platenumber in class and indicated that th e name has no con-ceptual significance bu t is an accident of history.For the future then, the plate concept should be discardedand replaced by h = 021L or in non-uniform columns by h =du2/dL.Detailed Treatment

The remainder of this paper will he devoted t o a discussionof the reasons that eqn. (1) should nolonger be taught and thateqns. (3)-(5) should he preferred and to a discussion of th eunderlying phenomena in terms that can he transmitted t osophomores. There are some aspects of zone broadening which

The A Term, Eddy DiffusionMuch discussion and research has eventually shown thatthe A ter m is not useful in discussion of modern chromatog-

raphy. The confusion surrounding i t arises from several causesincluding conflicting definitions of the phenomena i t repre-sents, the ill-defined effect of diffusion on its magnitude, andthe fact that in one of its definitions t he effect is real a t vervhigh velocities.The term eddv diffusion is used bv chemical engineersto describe a phenomenon related t o turbulence, bu


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uninterpretable. Thus, eqn. (7) gives no more informationthan the qualitative explanation that precedes i t but is de-ceptive because it seems to do so. There have been many at-tempts to provide better equations than (7) by Giddings 3b ) ,by Knox and his co-workers (8-lo), and by numerous chemicalengineers whose work is reviewed by Cluff and Hawkes (11).Knox was able to get good correlation between theory andexneriment for unretained samples, but no equation servesfor realistic samples with realistic retention times.This work will not he reviewed here because t he thesis ofthis section is that no quantita tme trea tment of this conceptshould be presented to students of analytical chemistry. I t isof relevance to chemical engineers working with high velocityfluid streams through reactor beds but it makes no significantrontribution to zone hroadenine at the velocities used in~ ~chromatography. I t was hoped &ring th e 1960's th at highvelocities would he used in liauid chromatoerauhv to obtain- -fast results, and the discoverythat h A as u- preparedthe way for such work. However, the 1970's have shown tha ti t is more productive to concentrate on short, very efficientcolumns ouerated at velocities that uroduce minimum h.The muitipath effect should he presented qualitatively withno supporting equations to deceive by oversimplification. I tshould he incorporated in the vague last term of eqn. 51,andwill account for the variation of mobile uhase mass transferC (discussed later) with velocity. It can, in fact, he discussedunder C rather than as a separate effect.The B l v Term, Longitudinal Diffusion

The van Deemter (2) expression 2yDJu or its abbreviationBju is essentially correct. There is both theoretical (3c) andexperimental 12)evidence that the obstruction fac tor y de-creases at very low velocities, but the experimental evidenceis contradictory (13). Since this term is well explained intextbooks, it will not he discussed here in further detail.The Cv Term, Mass Transfer

The mass transfer term arises because eauilihrium betweenthe stationary and mohile phases is not reached ins tanta-neously. Consequently, material in the mobile phase tends tohe swept forward ahead of the average position of the zonewhile material in the stationary phase lags behind. Similarly,material in fast flow paths is swept ahead of material in slowpaths.The more quickly a system tends toward equilibrium thesmaller this effect will he. Specifically, the effect is reducedhy fast diffusion in the mobile phase moving material fromone flow pat h to another and to the stationary phase, and byfast diffusion in a liquid stationary phase moving material tothe interface with the mohile phase. If these processes wereinfinitely fast, the material in the stat ionary phase would notlag behind that in the mobile phase and there would he nospreading from this cause.The faster the mobile phase moves the worse this effectbecomes. so the effect is urouortional to the velocity and theterm is usually written a s - ~ u : ~ h eiscussion of the kl t i p a t hterm ahove surrests that this strict uroportionality will ceaseat high velocities in packed columns bkcause the pa rt of Cuth at is due to a variety of flow paths will be reduced to theconstant term A . This deviation is probably negligible overthe range of velocities in which chromatography is actuallypracticed and should be ignored in elementary treatments.I t is usual to divide the C term into two part s representingmass transfer in the stationary and mobile phases thus

The stationary phase mass transfer C may he on an adsorbentor in a liquid stationary phase, but as simple adsorption-desorption kinetics are very fast. They make a negligiblecontribution to C,. We shall therefore discuss only masstransfer in the liquid phase, CI.

Liquid Stationary Phas e Mass Transfer, ClVan Deemter et al. (2) derived the expression

The form of thi s expression is correct hut the factor 8/n2 iswrong. Moreover, CI increases when the liquid is coated on ana d ~ o ~ b e n tupport.Van Deemter's explanation for the 8/n2was incomplete. Itstarted with the equationdcldt = a2/4) D~/d?) c, KCI

where c, and c, were the concentrations in the gaseous and theliquid phases, and K was the partition coefficient. Theequation was said to he derived from the simplified solutionof the general eauation for diffusion into a laver auuarentlvassuming th at the average concentrat ion (in the layer) doesnot differ too much from the concentration at t he surface.No reference was supplied. Diffusion into a layer is discussedat length by Crank (14) and the equations he supplies do in-volve n2,hut they donot obviously lead to the above equation.Even so, there would be no reason to distrust the 8/n2 had nottwo separate workers, Giddings ( 3 4 and Golay 15)hothderived the constant as 2/3. Giddings derived the 2/3 fromnonequilihrium theory, and Golay derived it from an electricalanalog using th e telegraphers equation. They hoth obtainedthe constant .As hoth derivations are co m~ le te nd com-. ~ ~ ~ prehensihle, the 2/3 must he accepted as a correction of vanDeemter's 81x2 (unless a full derivation of ean. 191 is everpublished). For a uniform film then

However, the liquid film is never uniform. I t is held in the

Furthermore, even if the support surface were uniform i twould support only a few monolayers of uniform film and th erest would run out of the column. Giddings has allowed for thenon-uniformity of the liquid layer by replacing the Zl with ashape factor q; thus (3e),

where dc is now the debth of the non-uniform film a t its.the non-uniform film. I t normally varies from 2/ls f i r sphe;icalparticles (as in a bead of resin where df s the radius) tozjs fora uniform film. I t is possihle (3f) o conceive shapes tha t giveq outside the range 2/1: to 2/1.When the liquid is coated onsmooth glass beads and accumulates a t the contact points,then y = 2 , and if i t were held in pores tha t were wide at the

to these unlikely circumstance^When there is a variety of different shapes of pores, thenq is the mean of the individual values of q for the various typesof pore, weighted by the volume of liquid in each type of porei3e)

'Js = x s jU 121~where uj is the volume of stationary phase in pores with shapeq j and ul is the total volume of stationary phase. Thus if asmall fraction of the liquid accumulates in badly shaped blobs,the effect on h is smallK ~ ~ . v f r ,ilere i \I.I ,11 1u th er . ~ , I I I I > ~ I ~ . I I ~ . , I ~ ..\ .uri~t,.e>upp,,r t i~~:I - l : ~ l i ~ m . ~ r \iqu .~ l I~II I~ i w e s w r ~ l vN d


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diffuse through the liquid and be adsorbed a t the bottom ofit on the support surface. The liquid phase nonequilibriumis then very much increased over simple solution in the liquid.Giddings has derived a correction factor (3g) to allow for ad-sorption below a uniform film, but there is no suitable ex-pression for non-uniform films. It is evident from his deriva-tion that a multiplying factor that is a function of the ratio ofadsorbed to dissolved sample c,d,/ci must be used. For auniform film the multiplying factor is 2/ 1 c,/c,d,)3(c,dJc/cl)/(l cl/c,d,). This function approaches 3C,d$~/cl whencad, >> cl so the effect can be dramatic. The impor tant prac-tical consequence is tha t support surfaces should he no moreadsorbent than is necessary to allow the s tationary phase towet them.A final complication is tha t all the above equations are de-rived assuming that the liquid surface is in direct contact withflowing mohile phase. When it is held in the pores of particles,the sample must pass through stagnant mohile phase in thesepores before it reaches the s tationary phase. In this case, CIis increased ( I7) by a factor f lf l

mass of sample dissolved or adsorbed)- total mass dissolved or adsorbed) and in stagnant mohile phasehis correction factor is usually close t o unity and i t can heomitted from elementary discussions.Moblle Phase Mass Transfer C

Th e complexities of the mohile phase in a packed columnhave so far defied exact analvsis. By contrast, the causes ofprovements in column in all branches of chro-matography. Briefly, h increases with the square of the par-ticle diameter d;, the square of the column diameter d: theratio of d, to the diameter of the coil of a coiled column, andis affected by the geometry of the column, the uniformity ofthe vackinp, and the efficiency of the connections from thec o l n k o the injection system and the detector.Particle Diameter, d

There ar e several reasons why the particle diameter is im-~or t an t :hese are related to sta rnant mobile phase within theparr ick 2nd to the t m v patrcrn I I I I U F ~ I I the pi~rtia.l~i.A t ro.lilihriun~11c cmccn1r; iti~m r .ani l~lt . n the Iluwinymohilebhase would be the same as the concentration in thestagnant mohile phase held in the pores of the particles. In areal column th is equilibrium cannot he achieved. Moleculesin the pores lag behind those in the flow stream for a time thatdepends on the distance they must diffuse o leave the par ti-cle. This distance is proportional to the particle diameter dpand thus C, d;. For spherical particles the dependence isgiven (3h) by

any but the most advanced &dents since this term is only oneof several C terms, most of which cannot he analyzed exactly.However, the qualitative concept C, d i bas practical im-portance and should he taught at all levels.The particle must he reached by diffusion from the flowstream. Th e greater the distance between the particles, thegreater is the width of the flow stream through which thesample must diffuse. This distance is proportional to the di-

iBoften writtenC = wd; /D

where w is an ill-defined constant.

This same logic shows that in an open tubular column C,d2 and Golav (15) and Giddinas (3h) have shown by dif-ferent approaches tha t for a round column

..for calculations. However, it must not he pushed too far. I tassumes a parabolic flow profile which is t rue only with per-fectly smooth walls and will be wildly inaccurate for 0.005-in.steel columns, judging from Drew and Bens' micrograph(16).The various flow vaths in a column are inevitably out ofequilibrium with each other, having different concentrationsin each flow stream. The disequilibrium is offset by diffusionbetween the flow streams, and C, is, therefore, proportionalto the sauare of the distance between them. The distance isusually few particle diameters and is assumed to he pro-portional to d,. This is again written

However, the assumption is only approximate. I t assumesthat as the particle diameter is reduced, the arrangement ofthe particles (the packing structure ) remains unaltered.This is not realistic. Even identical tubes and particles can hepacked only moderately reproducihly. Changes in particlediameter will make further changes in the structure. More-over, the very small particles now used in HPLC are packedby a different technique ( slurry packing ) from larger par-ticles with the intention of producing more uniform packing.To complicate matters further, the packing structure changesradically near the wall of the column so that to maintain auniform oackine structure as d, chances, the column diameterd , must change in direct proportion. This does not happen inoractice so C, deoends in a comvlex and poorly understood.. .manner on d , and d,. Knox's has done much t o eluci-date this prohlem 8-10) hut the resulting equations fail whenapplied to retained solutes.The packing struc ture is also strongly dependent on theuniformity of the particle diameters so that meticulouslysize-graded particles give lower C, than packings with a widerrange of particle sizes (211.Column Diameter, d,

experimentally that in a column packed with the light dia-tomaceous earth Celite there is a velocitv gradient acrossthe column, i.e., the gas velocity is faster on one side of th ecolumn than on the other and varies uniformly across thecolumn. Using this model be showed that C varied with d:and with t he square of the slope of the velocity profile. Hefurther showed tha t a great deal of previously published datawas compatible with this model. Other workers have founddifferent flow profiles. Huyten e t al. (23) found a symmetricalprofile in a 3-in. diameter column with fastest flow at the wall.This givesC d:. The parabolic flow profile in open tubularcolumns also leads to C = da as shown above. I t is unlikelythat trans-column variations are so imvortant in modernHPLC columns with the advanced technology nowused. Indeed. it is common exoerience that 'I~in. iametercolumns givebet ter ef ficiencythan '18-in. in HPLC makingthe exponent of d , negativd.Coil Diameter

When a column is coiled, the flow path on the inside of thecoil is shorter than on the outside: as both have the samepressure drop the flow is faster on the inside and gives rise totrans-column noneauilibrium. I t is for this reason that liauidchromatography coiumns are usually straight. The quanti-

396 Journal of Chemical Education


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was often used where varied from 0.3 to 0.7. This ignored theBlu term which probably was negligible. Now that liquidchromatography is most often carried ou t a t velocities closeto the velocity giving minimum h, this form is no longeruseful.Some theoreticians have extended this by adding a term t oaccount for those mass transfer terms t hat are simply pro-portional to u to give the equationh = Blu Cu Eoi13 21)

The origins of this equation are rather obscure. The earliestuse seems to be by Done, Kennedy, and Knox in 1972 (33).They give the last term s Eun and suggest that n = 0.33 is thebest value for velocities such that ud,lD, = 10 to 100. Thisis much higher than the minimum of the h l u plot that repre-sents velocities tha t are now usual in modern liquid chroma-tography, so higher values than can now he expected. Thevalue of n = 0.33 comes from an earlier paper by Knox andParcher 9),ut reference to the paper shows that i t refers toa form of the coupling equation

which requires some simplification to reduce to eqn. (20). In

engineerrug da ta involving both laminar and turbulent flow.This was first used in chromatoeraohv hv Huber 35).. ..In rquation similar I,, t:qn ,221was deriwd ihmreticull?fur unrttained ch i c - ; IVElmwith and [.in ~ . r l i thilt reducesto the form

h = Alu Bl 1 Cu- I3 ) Du2I3 EuThis was shown to provide a be tter fit to experimental datafor unretained solutes in packings of glass beads than otherpublished equations. While this was a real step toward un-derstanding zone dispersion, it is improbable that it willrepresent the real world of retained solutes. Moreover, it wasbased on Pfeffer and Happel's model 40 )which showed thatthe mass transfer coefficient is proportional to u 3 a t valuesof the Peclet number ud,/D, (known in chromatographicnomenclature as the reduced plate height) greater than 30 anda porosity of 0.4. Values of ud,lD, are seldom that high inchromatographic practice, although they would occur a t th every high velocities at which simple versions of the vanDeemter eauation break down.It would-be unwise to introduce most students to semi-empirical equations such as these. There mav be occasion in

group in optimization of thin-layer chromatography 36).Otherwise, eqns. (3) and 4) are to be preferred, perhapssupplemented with a qualitative description of the deviationsto be expected a t higher velocities and the uncertain effect ofthese deviations at lower velocities. Unfortunately, they havebegun to appear in textbooks without any explanation of theirorigin or their very limited usefulness.Limitations of the van Deemter Equation

Whatever form of the van Deemter equation is used, i t isstrictly applicable only to infinitely long columns. The ap-proximations involved in its derivation become progressivelymore important as the plate number N defined by

N = L2/02 Llhbecomes smaller. Giddings ( 3 j ) has defined this conditionapproximately as

mso tha t the equation is certainly inapplicable when N < 100and douhtful when N < 1000.Moreover, the flow pattern in the regions near the two endsof the column is different from that in the main bodv of th ecolumn. This has been theoretically elucidated by Gill andSankasubramanian 129 for emutv tubes, hut there is noguid:u~re, i n h i , I1 theauthtx iidwarv, to suggest what lel~ytho ia vurkrd cAlmn hus ~~nrc:r)rcicmtarirclow pro:ilea. O n thchasis of the work on empty tubes, bowever,-it would seemunwise to auulv the van Deemter equation to columns that areless than 5 0times as long as they are wide.These limitations do not affect the qualitative usefulnessof the dispersion equation, hut unfortunately, a number ofworkers have attempted t o use it to derive kinetic constantsusing columns that were too short to justify its use. It istherefore important t o make students aware of these limita-tions and prevent future errors of this kind.Literature Cited

11) Lapidus, L., and Amundson, N. R., J Pkya. Ckem., , , 984 (1%2).12) an Deemler, J. d., Zuiderweg, F. J.. and Klinkenberg, A., Ch em En Sci , 271

I821 Schett10r.P. D.,Eikelberger .M.,snd Giddings,J. C., Anal. Chern.,39,146 (19671.1331 Done. J. N Kennedv.G .JLand Knur. J. H.. GaqChromatoeraohv 1972. S C Pcrrv

136) Huber,J. F K. , J Chiornat g. S c i . 7.85 (1968).I361 Guiochun, G..Brersol le ,F.,and Siouffi, A , J Chiornotag. Sci., 17,368 (1979).I371 Msrfin,A.d. P .,and Syn8e.R. L. M., Bioihem. 11.35,1358(1941) .138) Einstein. A,. Ann der Phyaik, 17.549 (1905).(39) Horunth, C., andLin. H. .J Chromatog., 126.401 (1976).(40) P r e f f e r , ~ . ,nd J . , A I c h . E J . , 10,605 (1964).

398 Journal of Chemical Education

jce-1983-393_modernization of the van deemter equation

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