A Computational Introduction to Chromatographic Bandshape Analysis S. A. Kevra, D. L. Bergman, and J. T. ~ a l o ~ ' Seton Hall University, South Orange, NJ 07079
counts for the appearance of the real chromatogram shown I n this paper we describe a numerical method that has in Fikwre I land &npletely accounts for the broodenmg of been used as a n introduction to chromatographic princithe simulated chromatoprams shown in F:gure 31.Only a ples a t this university. This computer project allows the solute that is ~om~letelfunretained by thecolumn exhibstudent to develop a thorough understanding of the factors its no convective broadening in the absence of other mass that control the separation process through the statistical transport effects, for example, diffusion. moment analysis of a well-defined chromatographic bandshape, that is, one that results when a solute having fixed fractionx in the mobile phase passes through a column of Chromatographic Diagnostics q known theoretical plates. Because the actual number of Following the concept introduced by Martin and Synge theoretical plates 11is defined, this analysis also allows the (61, a chromatographic column can be envisioned a s constudent to compare different methods for assessing the sisting of a large number of identical segments, or theoretichromatographic efficiency through the determination of cal plates, in each of which equilibration is obtained. The the number of theoretical plates. number of theoretical plates within the column, 11, is a Statistical moment analysis has recently been used to measure of the efficiency of the chromatographic system; show that, in the limit of thermodynamic reversibility, the the larger the number of plates, the more efficient is the conventional plate count N = t?/02 is hut a crude approxicolumn. The conventional (7) estimate of the number of mation of the correct relationship (1,2), N = t,(t,- to)/02 theoretical plates N in a col&nn, a s determined from the where t , is the mean time required for the sample (solute) observed chromatogram (Fig. 1)is given by to elute from the column; to is the retention time of a n unretained component: and a2 is the band variance in time units. This result has been confirmed by the observation of im~rovedzoodness-of-fit with ea 2 when experimental and theoretical chromatograms a r e subjectkd to moment analysis (3, 4). In the course of analyzing a large number of simulated The results of the simulation described in this paper bandshapes the followingrelationship was discovered ( I ,2). confirm the validitv of the above observations and provide correct diagnostic Eriteria for the interpretation ofthe reversible chromatogram. In addition, those students who This paper was presented, in part, at the 1990 (Boston)Meeting of master this compu~ationaltool can readily extend it to the the American Chemical Society, paper CHED 262. simulation of other (ex., kinetically controlled) partition'Author to whom correspondence should be addressed ing mechanisms by using the finite-difference methods used herein. In elution chromatography a small, discrete sample is introduced and then carried by the mobile phase through a column containing stationary phase. When the sample molecule is in the mobile phase i t moves along the column a t the same velocity a s the mobile phase. When the molecule is in the stationary phase it makes no forward progress. The retention time is readily obtained from a statistical analysis of the (temporal) chromatogram, which is a plot of signal vs. time generated by a detector placed a t the end of the column. I t is generally held that in ideal chromatography the component zones (also called bands or peaks) would have no tendency to spread and would elute from the outlet of the column in the same volume or time interval as they were introduced a t the inlet ( 5 ) .Such a chromatogram would appear as a series of very sharp, completely separated peaks. Due to the inherent nature of the distribution of Time ( m i d the solute between the two phases. however. zones that are sharp and compact a t the inlet Figure 1. Experimental chromatogram of phenol using a SpherisorbODs II column. Mobile broaden a s they flow through the column. phase is methanol1 water (80:ZO).The displayed experimental parameters were obtained hi^ convective broadening partially ac- by moment analysis of the digitized chromatogram. ~~
~
Volume 71 Number I 2 December 1994
1023
whereh" is the correct estimate of the number of theoretical plates determined by this equation, and t, is the retention time of the unretained component. This equation was first introduced by Golay (8)who proposed its use to correct inadequacies of the conventional relationship for open-capillary gas chromatography. Fritz and Scott (9)subsequently investigated the applicability of eq 1 and another efficiency equation to liquid chromatography by carrying out a statistical analysis of the countercurrent distribution; in this study they came within a n algebraic rearrangement of obtaining Golay's original (gas chromatographic) result. Subsequent to our empirical discovery (1)Fritz and Schenk have made this necessary algebraic rearrangement and have included eq 2 in their well-known text (10) on quantitative analysis; to be consistent we have adopted their notation in this work. This equation has now been derived rigorously for a normally distributed solute in the limit of rapid distribution kinetics, and its validity has been demonstrated by comparing the goodness-of-fit achieved bv either method in the analysis of both simu1iltt.d and ~ ~ ~ e r u n e nrhrometograms~~4,. tal Either ol these esrimatci of chromato~aphicefficiency requires the statistical analysis of the ihrbmatogram to obtain the mean retention time and the variance of the temporal distribution. The observed fraction of solute in the mobile phase IT may also be obtained from the statistical analysis of the observed chromatogram (7).
This relationship has also been confirmed ( 4 ) by the derivations cited previously. Moment Analysis The statistical analysis of the chromatogram may be carried out by computing the zeroth, first, and second moments about the origin for that temporal distribution. The zeroth moment about the origin (equivalent to the area under the curve) is used in subsequent determinations of the first and second moments about the origin (see Table 1). The integrals that define the zeroth, first, and second moments about the origin are evaluated numerically using Simpson's rule (11). For data points evenly spaced in time, this numerical evaluation is merely the product of the summation of the points and their time spacing (dt). The results of these numerical integrations about the origin are used to compute the higher moments about the mean and thereby determine its mean and its variance. The retention time (mean) is equal to the first moment about the origin, which is equivalent to the first moment about the mean. The variance is defined by
Countercurrent Distribution Several approaches have been used to simulate the movement of bands through a chromatographic column (13-15). One approach is the Craig distribution model (16, 17) in which a specific number of equilibrations of a solute between two immiscible phases is camed out in a multistage extraction cascade. The distribution of sample molecules between the two phases is governed by a n equilibrium constant, known as the distribution coefficent K.
where C, and C , are the concentration of solute in the stationary and mobile phases. The theoretical capacity factor k, is the ratio of the moles of sample component in the stationary phase divided by the moles in the mobile phase.
where V, and V, are the volumes of stationary and mobile phases in the column, and n, and n, are the total number of moles of solute in these phases. Because the fraction of solute in the mobile phase (X) can be expressed a s a function of the total moles of solute in the stationary and mobile phases, it may be related to the capacity factor.
When X = 0, the solute does not move a t all; when X = 1, the solute moves a t the same rate as the mobile phase. Therefore, X can be used as a measure of the retention of the sample (7). Rate Theory and Plate-Model Theory Rate theory was introduced by Van Deemter and his colleagues in 1956 (18)to identify the factors that cause dispersion in a chromatographic column and to explain how Table 1. Statistical Definitions (13 zeroth, first, and second moments about the origin:
-
p;
= zeroth moment = I
VI'
f (0 dt = area under f ( 0
0
=firstmoment = t (0 dWp; 0 m
p'~= second moment = ?f (0dWp; 0
First and second moments about the mean: where pi is the second moment about the origin, and p2is the square of the first moment about the origin. Thus, any quantity such a s N' or N that depends upon the mean and the variance may be evaluated by computing the zeroth. first. and second moments about the oripin. These relarirmnhipi r121 arc also summan7ed in ihhle I . The com~ut;~tiim of higher moments relntrd to tht, skewness and kurtosis of the distribution is unnecessary for this development.
1024
Journal of Chemical Education
p1 =first moment
p2 = second moment Relationship between moments about the mean and moments about the origin:
p2 = [1(2- p2]=
3 = variance
dispersion is related to column geometry, properties of the packing, mobile-phase flow rate, and the physical prupertie; ofthe distrihut~onsystem. rat^. theory uses the rt~sults of'platr throry to quantify this d~spersionby determining the efficiencv as a function of solvent velocitv. Even ~ - - column though the plate-&del theory originally developed by Martin and Synge (6)does not attempt to explain the causes of peak dispersion in a column, it defines efficiency as the number of theoretical nlates observed in the limit of convective control (191,that is, when other modes of mass transnort are unimoortant. The computer model of the chromatographic process (the simulation) presented in this paper uses the concepts of plate-model theory: ~
~~
Signol Observed
from Detector at Plate r = q = 10
~
~
0
..
50 100 Time - n(dt)
The column is divided into theoretical plates. Eouilihrium is established between the two phases. Axial or longitudinal diffusionof the solute is regarded as ne&ible. .. T h e partition coefficient of the salute is the same at each plate and is independent of concentration. A detector placed a t the q t h theoretical plate measures the fractional concentration of the solute a t that plate as a function of time (registered in units of theoretical transfers). To permit the entire band to pass through the deteetor, the simulation is carried out until a n insignificant amount of solute remains on the nth d a t e . This requires that the simulation be carried out'ovei plates that lie beyond the theoretical end of the column (plate q). The Computer Algorithm A computer program based on these concepts was written to generate the chromatogram that is obtained when a solute having a defined fraction ( X ) in the mobile phase passes through a column containing a known number (q) of theoretical plates. The resulting chromatogram was then subjected to moment analysis to characterize the bandshape. The variables used in this simulation are summarized in Table 2. Table 2. Basic Definitions
X= fraction of solute in the mobile phase Y = 1 - X= fraction of solute in the stationary phase
r = serial number of plates n = number of time iterations of duration dt (transfers) t = time = n(dO
f(n.r)=fractionon the !th plate beforethe nth transfer f (n,r)=fraction on the nh plate after the nfi transfer
Figure 2. Spatial (a,c) and temporal (b,d) distributionswith detectors at q = 10 and q = 100. The snapshot of the distribution on the column in panel a was taken after 20 transfers; the snapshot in c after 130. The leading edge has been detected as an unretained component (X = 1.O) in panels band d. At the beginning of a typical simulation (n = 0) all plates other than r = 0 are empty: X0,r) = 0 A unit quantity of substance is placed in plate r = 0 so that flO.0)= 1.0 A distribution coefficient is selected (e.g., X = 0.9 and Y = 0.11, and the solute contained within the r = 0 plate is partitioned accordingly. At the first transfer (n = 1) only the fraction of solute present in the mobile phase in plater = 0 goes to plate r = 1,that is, The fraction of solute in the stationary phase remains in plate r = 0:
q = serial number (0 of plate corresponding to detector f(n,q)=signal observed at detector placed at plate q during nth transfer Boundary Conditions f (n,-I) = 0
f(n.0)= Yf(n,O)
A l , O ) = (1-rnXO,O)
Afresh volume of mobile phase is added to plater = 0 from plate r = -1, but this does not alter the amount of solute contained within the r = 0 plate. Fractions of the solute present in plates r = 0 and 1 then distribute themselves between the mobile ( X ) and stationary (1 - X I phases of their respective plates. Now another transfer occurs (n = 2). The fraction
xfl1.z- 1)
f (n,n)= 0
Transport Algorithm f(n.0 = f(n+ 1 . l = ( I -4fn.r) + Xf(n,r- 1 ) (Start at r= nand ao to r = 0)
in plater = 1goes to plate r = 2. Plate 1is left with (1 -mfil,l)
plus the quantity Xfl1,l- 1) Volume 71
Number 12 December 1994
1025
that is transferred from plate r = 0. Afresh volume of mobile phase goes into plater = 0 where
Table 3. Sample Moment Analysis of Chromatographic Bandshape Parameters
(1-rnfll,O)
of the solute remains. As the process continues the fraction on each plate (r > 0) is calculated recursively from f(r)=(l-X)Ar)+XXr-l) The recursion formula for the fraction remaining on plate r=Ois
0-9 (10 10
0.0 0.0 0.0 1.0 unretained component - X = 1.0) 0.34868 3.4868 34.868
11
0.38355
4.2190
46.409
12
0.23013
2.7616
33.139
13
0.09972
1.2964
16.853
14
0.03490
0.4886
6.841
15
0.01047
0.1571
2.356
Elution Chromatography
16
0.00279
0.0447
0.715
Elution chromatography may be modeled with the same computer algorithm by assuming that a detector is placed at a given plate r = q corresponding to the length of the column and recording the signalf(n,q) as a function of time (n(dt)).For example, by assuming that a detector is placed at r = 10 and all&ni 20 transfers to be carried out, one obtains the distributions in space (a] and time (b, for II = 10 shown in Figure 2. numerical summary of the distribution in time (the elution chromatogram) for the 10-plate column also appears in Table 3. The student may choose the detector location r = q or the fraction in the mobile phase X.Figure 2c shows the spatial distribution obtained after 130 transfers with X = 0.9; the signal that would be observed at q = 100 under these conditions is shown in Figure 2d. The fraction on each plate is calculated as the solute is transferred from plate to plate, moving past the defined detector(s). Storage requirements are minimized by using the transport algorithm shown in Table 2 and retaining only the current spatial array Fractional concentrations fln,q) at each detector are recorded as a function of time (n). Signals wn,q)l greater than some predetermined minimum (e.g., 0.000001) corresponding to the computer round-off error are stored in arrays where they are later used in statistical moment analvsis to characterize the bandshape. l to those olates conBv limitine the s ~ a t i adistribution taini"ng more-than ihe round-off minimum of Golute, the array sizes necessary to contain the entire spatial distribution have been greatly reduced. In addition, one may systematicallv reduce the number of temporal data points to minimize computation of statistical parameters for a n evenlv spaced set of data points. We have successfullv aDplied"d&a acquisition compression techniques to accomplish this. Programs have been written in BASIC and FORTRAN that allow the user to carry out each of these tasks on a PC platform, and these are available from J. T. Maloy.
17
0.00068
0.0115
0.196
18
0.00016
0.0027
0.049
f(0) = (1- m o )
This method has been used previously to obtain the theoretical spatial distributions necessary to model open-bed (e.g. thin-layer) chromatography (20).
19
0.00003
0.0006
0.012
20
0.00001
0.0001
0.002
1.1111
12.469
141.44
Sum
TIME (1 =duration of 1 theoretical transfer)
Analysis ofResults Once the elution chromatogram is obtained, moment analysis is carried out to determine the mean retention time and the variance of the peak. Table 3 shows the details of these calculations when they are carried out on the elution chromatogram that is observed by a detedor placed at r = q = 10 whenX = 0.9, as illustrated in Figure 2b. Excellent agreement is obtained between the defined value of X and the corresponding value of X", the error in this determination is less than 1%. Even for this small number of plates (N') the nonconventional estimate.of the number of theoretical plates agrees quite well (1.1% error) 1026
Journal of Chemical Education
TlME ( 1 =duration of 1 theoretical transfer) F gxe 3. S mu ated Chromatogramsfor 100- ana 400-plateco Lmns. Values of Xare assocatea wllh each cnromatogram.
Table 4. Moment Analysis Results for Digital Simulation of Chromatographic Bandshapes
nonconventional method shows good agreement (errors less than 1%)with the defined number of theoretical plates. The conventional estimate always exceeds the number of plates that are actually present on the column. However, the greatest relative errors occur with highly mobile solutes. (When X = 0.9 the error in the conventional estimate is 892%.) Also displayed in Table 4 are the computed values for the solute fraction in the mobile phase (X') and the corresponding computed capacity factor, k'. These quantities were determined by eqs 7 and 3. Excellent agreement is obtained between the defined value of X and the corresponding value ofX' in all cases. Conclusion This computer project shows the student how to use moment analysis to deduce chromatographic effkiency and retention characteristics from digital chromatoma~hic data. All of the fundemental chrornatographlc concepts are d ~ f i n r dnaorouslv in thls a p ~ r o a c hThe product of thls effort is a working model o f t h e c h r ~ m a ~ o g r a ~ process hic that can be used to address the basic issues of retention and resolution present in any separation. In its present form it is left to the student to develop the software to do this. As a n outgrowth of this model development the validity of our unconventional estimate of number of theoretical plates has been demonstrated for partition chromatography. This is of some theoretical importance because the spatial distribution in countercurrent chromatography is binomial in nature. Because a binomial distribution may be expressed a s a skewed Guassian distribution, this theoretical development indicates that eq 2 may be generally valid and not limited to solutes that move down the column in normally distributed bands. This is especially important because the conventional method for estimating the number of theoretical plates grossly overestimates the chromatographic efficiency for highly mobile solutes. The e n k t of this error can ho assessed by considering the followinr relation41io that mav be obtained b\ combining eqs 1-3.~
X = defined fraction in mobile phase X = computed value of X K = capacity factor corresponding to X N = conventional estimate of 11 N = correct estimate of 11 (this work)
with the defined value of 11, whereas N, the conventional estimate, does not agree a t all (828% error). the change in retention time that is F i ~ ~ r3tillustrates ! observed from 100- and 400-plate columns whenX is varied from 0.9 to 0.1. As the solute becomes less mobile a transition to longer retention times and increasing peak widths is observed. Changes in bandshape are also evident: When the solute is mobile, the chromatogram is skewed; a s the solute mobility decreases, the chromatogram achieves the appearance of a Guassian distribution. The improvement in resolution that is possible with columns having a greaternumber of theoretical plates is evident from the relative narrowing of the peak widths observed in going from 11 = 100 to 11 = 400. For example, it is obvious from Figure 3 that the peak corresponding to X = 0.7 cannot be separated on a 100-plate column from theX= 0.6 peak on the basis of retention characteristics alone; the separation is feasible, however, on the 400-plate column. The results of the moment analysis carried out on each of the chromatoerams shown in Fimre 3 are summarized in Table 4. withuthe values for theretention time and the variance determined by this analysis, the number of theoretical plates can be calculated conventionally (eq 1) or nonconventionally (eq 2). In all cases, N' calculated by the
N ' = N ( l -X)
(8)
I n other words, the conventional estimate of the chromatographic efficiency can be corrected by multiplying the conventional plate count by (1-X).This implies that the relative error i n the conventional estimate is equal to 1/(1-X).Although this may be insignificant for less mobile solutes (see Table 4), it is quite significant for highly mobile solutes asxapproaches 1.0. Indeed, much of the astronomical efficiencv attributed to ca~illarvchromatoma~hv .. . r21.22, and capillary zone e l w t n ~ p h ~ ~ r e,191 s i s may be the result ofan imdied division In zero that is inherent withln the conventio~alestimate of [he plate count for these highmobility methods, just as it would be in open-capillary gas was probably inchromatography. Golay's original work (8) tended to address this issue. The model that has been developed assumes that a fixed fraction of solute is contained within the mobile phase a t each theoretical plate. One way to obtain this condition is equilibrium partitioning on each plate. The efflciency equation t h a t we have developed may be used, therefore, to determine the number of theoretical plates in the limit of thermodynamic reversibility. Thus, eq 2 is to the assessment of chromatogra~hic efficiency what the .. . Nernst equation is to the assessment of electrode kmetlcs: It provides the hiah rate-llmitlna cond~tionfor the process. TI& observationis essential forthe study of contemporary problems in chromatography.
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Volume 71 Number 12 December 1994
1027
For example, one might use this approach to study other chromatoeraohic mechanisms (i.e.. adsomtion (20). (i.e., lon: gradient elution? aAd other transport gitudinal diffusion) with well-established finite difference techniques (23).One such study is already well under way (24). Because these reoresent nonesuilibrium situations, there is no reason to Gelieve that the equilibrium bandshaoe analvsis oresented herein will be eenerallv valid. the results may still be interpreted on t&e basis of the same statistical (moment) analysis, and the results may be compared with those equations that have been shown to be valid in the limit of thermodynamic reversibility. Because there is no preconceived functional form of chromatographic bandshape, any observed deviation from the eouilibrium bandshaoe is a result of the oartitionine process and not an artificially adjustable skewing parameter (25). It is. therefore. auite likelv that useful mechanistic (rate) infknation co;ld be obtained from this sort of analysis of the bandshape.
ow ever,
~
~
-
~A
Literature Cited 1. h m b n y a , A: Maloy, J. T. Resented a t the Pittsburgh Conference on Anslytied Chemistty and Applied Spectmscopy.Atlantic City, NJ, 1986: paper 946. 2. Onwubuya. A,: Maloy. J. T Presented at Eastern Analytical Symposium. New York, 1987; paper 24.
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Journal of Chemical Education
3. Onwubuya, A,: Malox J. T Presented a t the Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy. New Orleans. 1988: paper 057. 4. h w b u y a , A. Ph.D. Theeie, Seton Hall University, 1991. 5 . Pecsok, R.L.:Shields, L. D.:Cairns. T.: MeWillism, I. G.Madprn Mdh& ofChomim l Andysis; Wiley: New York, 1976. 6. Martin, A. J. P: Synge, R. L. M. Bimhem Jour. 1941,35,135&1868. 7. Karger B. L.; Snyder, L. R.; Hamath, C. An Infmiucfion to Szpamfion Sc&nce; Wiley: New York. 1973. 8. Golay, M. J. E. as dted by E m e , L. S. Own Tl~bulorCalumns in Gzs Chromofogmphy; Plenum Pleas: New York, 1965; p 54. 9. Fritz, J. S.;Scott. D. M. J Chromatogr 1983,271, 193-212. 10. Fritz. J. S.; Schenk. G. H. Quonfitotiw h o l y t i m l Chemrstry, 5th. ed.; AUyn and Bacon: Newfon. MA, 1981. 11. Nor& A C. Comoutolionol Chomistrv: An lnfmduction to Numarial M ~ f h m b : wile$ New ~ork:1986. 12. Hod, P G. Introduction to Mothemofiml Statistics; Wiley: New York, 1962. 13. Czoch, M.; Guimhon, G h l . Chem I s W , 62,189-200. 14. Eble. J. E.: Grab. R. L.: Antle. P E.: Snvder. L. R. J. C h m m t o m 1987.405.5146. . . 15. EI ~ a l 1 ~ h . M z ..; ' G & ~ ~ ~G , h o i c i e m . ' ~ w6~3,, 8 5 ~ 6 1 T 16. Craig, L. C. J. B i d Chom. 1M4,155,519534. 17. Laitinen, H. A ; Hamis, W E. Chemical Analysis, Mffiraw-Hill: New York, 1975. 18. Van Oeemter, J. J.: Zuidemeg, F J.; hlinkenberg,A Chem. Eng. Sci. ISSB,5,271. 19. Smtt, R. P W Liquid Chmmdogruphy Column Theory; Wiley: NewYork, 1992. 20. h a . S. A ; B e d l a , B. M.; Maloy. J.T Resented a t t h e 182nd National ACS meeting. New York, 1981: paperANYL 14. 21. Delinger. S. L.; Da+s,J.M.Anol Chem. 1992.64,1947-1959. 22. Jennings. W. Gus Chmmatogmphy ldth Capillary Columns: Academic Press: New V. ". ?. k.., . 14110 .... 23. Malox J. T. In Labomtory lhehnipuee in Elchoonolyticol Chemistry; &singer, P. T;Heineman, W R., Eds.; Mareel Dekker: New York, 1985: Chapter 16. 24. Maloy. J. T.: B e w a n , D. Presented a t the Plttaburgh Conference on Analytical Chemistryand Applied Speetmsmpy. Atlanta. 1993: paper 76. 25. Folev. J.P.: Dorsev. J.G.Anol. Cham. 1983,55. 73%737: and references contained
