A Laboratory Experiment on Extracolumn Band Broadening in Liquid

Charles A. Lucy, Laura L. M. Glavina, and Frederick F. Cantwell. J. Chem. Educ. , 1995, 72 (4), p 367. DOI: 10.1021/ed072p367. Publication Date: April...
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A Laboratory Experiment on Extracolumn Band Broadening in Liquid Chromatography Charles A. L U C ~ ,Laura ' L. M. Glavina, and Frederick F. Cantwell University of Alberta, Edmonton, AB, Canada T6G 2G2 peak observed. The ordinary statistical moments are deIn recent years, separation techniques, such as capillary fined by GC, capillary LC, and capillary electrophoresis, have greatly increased separation efficiencies, while simultaneously dramatically reducing t h e volume of sample required. As a result of these improvements, much greater demands are made of the instrumental components not directly involved in the separation process (e.g., injector, detector) to ensure that they do not degrade the separation where mk is the kth ordinary statistical moment; and c(t) is by broadening the peaks. This nonseparation peak broadthe concentration a t time t. enine is known as extracolumn band broadening or extraGenerally, only the zeroth and first ordinary moments column dispersion (13). This experiment introduces an advanced undergraduate or Table 1. Band-Broadening Components in a Chromatographic System (a) -praduate student to the concepts and terminology of extracolumn Chromatoaraohic Idealized Model Imoulse-Resoonse Center of Gravity Variance af band broadening. The concepts (S) Function ~ompoierit (S2) described hereTn a r e demonstrated using LC hut are equally Gaussian ~ O I rate equation applicable to GC (41, capillary Column electrophoresis ( 5 ) ,and flow injection analysis (6). Connecting tubing Golay eq. Theory Convolution of the Impulse-ReSquare wave (v/n Plug sponse Functions 2 When a sample band passes Injector mixing chamber through a chromatograph, its concentration profile (peak shape) is Detector dead volume broadened hv each component of Exponential ?-a2 diffusion chamberb the instrument. The hand-broad2Dm ening effects can he simulated mathematically in the following way A component's characteristic Detector and recorder low pass filter Exponential 7el way of modifying the peak shape is electronics described mathematically by its impulse-response function. The 'Conncn ng t ~ o n g~nless , 1 has a very large rallo of englh lo lnsloe o ameler, w!l g ve an asymmelimpulse-response function oper- r c ta lea response f ~ n con l lnal s no! G a ~ san s ~ J can I oe approx rnaled as exponenl a y moo f eo ates on the incomine - .~ e a ks h a ~ eGaussian. through a process called convolubFor a diffusionchamber: r is the inside radius of the injector or the detector flow cell: a is the inside tion to orodnce the shape of the re- radius of the inlet tube to the injector or detector. sultanipeak that exitithe compon e n t (2, 7). This convolution are used to characterize a probability density curve. The increases the peak position and zeroth moment,. mo. width. ". eives the area under the curve. The peak area is proportional to the number of moles of solute i n the band and is unaffected hv band-broadening orocStatistical Treatment of Bands esses, which alter only the location and shape of the peak. The first moment, ml, defines the center of gravity, or The chromatographic band or peak is simply a distribumean, of the distribution, which is taken a s the retention tion of concentrations over time. It can he treated as a time, tR, of the sample compound responsible for the peak. probability density curve and characterized using statisThe first moments of the response functions of each of the tics. One means of describing the size, location, and shape instrumental components of the liquid chromatograph are of a concentration distribution is by its statistical moments given in Table 1as the center of gravity (8-11).Statistical moments can be calculated for each individual impulse-response function as well as for the final Central Statistical Moments -

..

'Adnor lo wnom corrcspondcncc sno- a oe aooresseo Presenl adorcss: L n dcrsly of Calgary. Ca gay. AB Canada T2h 1 h4

Greater information about the peak shape is ohtained from higher statistical moments when these moments are Volume 72 Number 4 April 1995

367

measured using the first moment a s a reference frame. Such moments are referred to as the central statistical moments and are mathematically defined as:

The second central moment, E z , is the most useful of the higher moments because i t is the variance, a ' , of the distribution about the mean. The more traditional methods of estimating variance, such as those based on measurement of the width-at-half-height and the base-line width of the peak obtained by extrapolating tangents to the peak, work well only for Gaussian peaks (8). However, the variance obtained a s the second central moment makes no prior assumptions about the peak shape and thus is more accurate when dealing with non-Gaussian peaks (8, 12). The third and fourth central moments provide information about the horizontal and vertical asymmetry of the probability distribution curves. Center of Gravity

inj +

tvbe + tR,det

(3)

,i n units of and the variance of the observed peak time) can he taken to he the sum of the second moments of each impulse-response function.

In eqs 3 and 4 the subscripts refer to the separation column (call, the injector (inj), the connecting tubing (tube), and the detector (det). Mathematical operations, such as convolution of the impulse-response functions with one another and linear addition of centers of gravity (eq 3) and variances (eq 41, assume that the several band-broadening processes operate independently. This is generally true.'

3

4

5

Flow

6

+Flow

In

.

In ea 4., the term a? .,A is the variance described in the chromatographic plate-height equation. Under ideal analytical conditions, a Gaussian response function is associated with the separation process (Tahle 1).Thus, if an infinitesimally narrow sample plug ( a n impulse) were injected into the column, the column would convolve that impulse to produce a Gaussian peak, whose variance would equal the square of one-quarter of the base line width of the Gaussian neak. Tvnicallv. this variance would 2 ~ examole n of chromoaraohic band broadenina..orocesses which. accora ng id some a ~ l h o kdo not act moependenl y are tne eaay a H L S O ~ and !he res~slanceto mass lransfer in the moo le pnase. w n c n are cowled ,131 Tnese effectsas wet as the olner on-coL m n bandbroadening effects are incorporated in the k,,,, and of, terms in eqs. 3 and 4. 3When using moments to measure tR,col and , eq 5 would be accurate whether or not the peak eluting from the column was Gaussian or even symmetrical. That is. even if the column itself has a non-Gaussian impulse response function, using moments in equation 5 gives accurate results.

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Journal of Chemical Education

Stir Rar Class Window F g-re 1 la) Exper rnenta appara1Ls for st-dy ng extraco .mn Dana oroaoen ng See texl for aetal s (0, Cross-sectona vlew of m x ng chamber.

then be used to calculate the number of plates on the column N (i.e., the column efficiency) by the expression

where is the retention time (i.e., the peak maximum, located a t the center of the symmetrical Gaussian peak). In terms of statistical moments, tR,c,iwould he the first moment (center of gravity), and ofco1would be the second mo~nent.~ Impulse-Response Functions of the Components

Gaussian Response Function

368

2

Out

Because the shape of the overall observed peak emerging from a chromatograph can he considered to have arisen via the convolution of the impulse-response functions of the individual processes occurring in the chromatograph, the center of gravity of the overall observed peak, tR,peak.can he taken to be the sum of the first moments of each impulseresponse function, t ~peak= , t ~ , c o l+ t~

6-way Valve

1

Equations 3 and 4 show that the peak exiting the chromatograph is also affected by the injector, connecting tubing, and detector. The approximate shapes of the impulseresponse functions expected from these devices are given in Tahle 1along with expressions for their associated first and second moments ( I ) . The components mentioned in Table 1 and their impulse-response functions will be discussed below. When the impulse-response function of the extracolumn components in Table 1are convolved with the Gaussian peak from the column, the resulting peak will have a longer retention time and will he wider than the peak from the column, a s per eqs 3 and 4. As a n alternative to statistical moment analysis, if the shape of the impulse-response functions of all of the extracolumn devices is known. either individuallv or collectively, then they can he deconvolved from tLe observed peak in order to obtain the shape of the peak resulting from on-column retention and band broadening alone (11, 14). This i s verv important in size-exclusion chromatoeraphy and field-flow kactionation where the peak shape is related to the sample characteristics.

The mathematical operations of convolution and its reverse, deconvolut~on,ore cunvrniently nirrird out \.id Fast Fourier Transtbrmarion I F I T ! t 15 The Fourier transform (7) of a convolved signal i s simply the product of t h e Fourier transforms of the functions to be convolved. The functions to be convolved are the peak due to the column alone and the response function of the extracolumn components. Deconvolution i s accomplished by taking t h e Fourier transforms of both the observed peak and the extracolumn response function, dividing the former by the latter, and then taking the inverse FT of the quotient function. In this experiment the following concepts are illustrated. f.

effect of extracolumn devices on chromatographic peak an anmeciation for the maenitude of these effects

Experimental Apparatus

plastic spools for convenient handling. The third type of device connected to port 6 was a mixing chamber (Fig. lh), which had a dead volume of 3.4 mL. The chamber is wellstirred using a magnetic stirrer. Four angled baffles within the chamber further assist the mixing. Glass plates on the top and bottom of the cell allow viewing of the solution within the chamber. Two series of experiments were carried out i n which the injector volume was varied. Slider-type valves were used (CSV-2,-5,-10,-20; LDC). Because the sliders with larger volumes have larger inside diameters, they reveal the band-broadening effects due to not only a larger volume but also an injector with a larger inside diameter than its inlet tuhing. One series of these injector experiments was carried out with the six-way valves switched to port 6 in the injectorldetector mode. In the other series the six-way valves were switched to port 1(packed-column mode). Procedure

To test the hand broadening due to a certain device, the 6-way valves are switched to the appropriate port number. The experimental apparatus is designed to allow easy Except as noted, the 2-pL injection valve was used, and a and rapid switching between a variety of band-broadening M amaranth solution was iniected. After the 2.5 x devices. In the apparatus, shown in Figure l a , a chromatopeak is acquired digitally, i t is plotted and the moments graphic pump (Cheminert Model CMP-2) provides pulseare calculated usine the oroeram EXTCOL. In order to deless flow of distilled water to a slider injection valve (CSV, termine the band hroadekng due to each device, the center Laboratory Data Control (LDC), Riveria Beach, FL). In all of pravitv and the variance for the iniectorldetector peak of the experiments, except those in which the influence of m i i t in! .khtracted fnrm the ohswved Eentcr ot gl-;~vit!:; ~ n d injector volume is studied, the injection valve has a 2-pL varianre for the peak eluted from the band-brondrninedevolume. The hole through the slider has approximately the vice. same inside diameter a s that of the inlet and outlet tuhing For studies of the open glass tubes, the PTFE connecting connected to the injection valve. From the injection valve tuhing, and the mixing chamber, the flow r a t e i s 1.0 the sample flows to a six-port rotary valve (R6031 V6P, mIJmin. An iniectorldetector studv is also carried out a t LDC), which directs the flow to the chromatographic comthis flow rate to provide measuremknts that will be used to oonent to be studied. A second identical six-oort valve dicorrect for the contribution of the iniectorldetector. For r ~ogU\' dc\~~re rwts rhr rfiluent from the I ) : i ~ ~ d - l ) r ~ ~ . i d e ~ ~ ito each device, the experiment is done intriplicate. dt!tt:cror ~\Vatt!r;;1.amlnIii hlax Model 181, operating at 254 For the oacked column. the studv is done a t a flow rate nm. of 0.4-m&in and under two different conditions. I n the Caution:Amaranth is FDC Red Dye No. 2. It is an irritant first case, the pieces of tubing connecting the column to the and suspected teratogen. General precautions should he two 6-way vaives are the regular narrow-bore ones (about taken in handling. 30-cm total length of 0.3 mm i.d.1. In the second case, an additional 20-cm length of wide-bore (0.8-mm-id.) PTFE The signal from the injection of a solution of amaranth is tubing is added between the first &way valve and the displayed on a strip chart recorder (Fisher Recordall), dig~ a c k e dcolumn to ourooselv increase the maenitude of the itized by an analog-to-digital converter (LAB MASTER ~njectorldetectorh'andbroAening. In both cases, the correTM-40-PGL. Tecmar. Cleveland. OH). and stored on f l o o ~ v sponding injectorldetector study is also carried out. In disk in an I'BM-XT hicrocornp;ter. ' h e data acquisition, other words, one injectorldetector study is done with only a s well a s the subsequent data analysis, are carried out by the reeular PTFE connectine tubine. and the other is done the program EXTCOL written in ASYST (Macmillan Softwithn: extra 20-cm piece o-fwide-hre PTFE tuhing also ware Comoanv). C o ~ i e sof this oroeram can be ohtained . present. The effect of deconvolution is studied hv takine from the authors. the peak eluted from the packed column to he the observe2 The band-broadening devices studied are given below in peak and taking the corresponding injectorldetector peak order of the port number on the six-port valves. to be the extracolumn impulse-response function. The propart 1, a 30-cm-longx 2.7-mm-id.glass tube packed gram EXTCOL is used to do the deconvolutiou. with 40-wm glass beads (nonporous particles) In both series of experiments in which the injection volport 2, an empty glass tube that is otherwise identical ume was varied, the flow rate was 0.40-mIJmin i n order to to the one used in port 1 allow direct comparison with the bands ohtained in the p o r t 3, a 30-cm x 2.0-mm-i.d.empty glass tube packed-column mode. The concentration of amaranth soluport 4, a 30-em x 1.27-mm-i.d.empty glass tuhe tion injected was diluted in proportion to the increase in port 5, a 30-cm x 0.98-mm-i.d.empty glass tuhe the injector volume in order to keep constant the number of moles injected. Port 6 was used with several different devices. I t was used to measure the injectorldetector variance by connectResults and Discussion ing the tubing from the inlet and outlet six-port switching The primary objective of this experiment is for the stuvalves directly to one another. In this case the band hroadening observed is due only to the extracolumn components dents to gain a physical understanding of extracolumn (injector, switching valves, connecting tubing, and detecband-broadening phenomena. To this end the apparatus i s constructed using glass and PTFE in order to allow the tor). Port 6 was also used for measuring hand broadening student to visually ohsewe the broadening processes ocproduced by different lengths of connecting tuhing. Segcurring to the zone of red dye. Students also gain a n underments of PTFE tuhing ranging i n length from 32 to 310 cm standing of the tbeory for extracolumn effects. Thus, the were used. They were wrapped around 3.5-cm-diameter Volume 72 Number 4 April 1995

369

Table 2. Effect of Column Diameter on Band Broadening in 30.0-cm-Long Open peaks produced by the passage of Glass Tubes at a Solvent Flow Rate of 1.00 mumin and Injection Volume of 2-pL the samnle throueh each of the broadening devices are acquired digitally on a computer for subColumn Diameter Center of Gravitye (s) Variancea (s2) sequent statistical moment 0.98 15.4i0.3 67.3 i 1.1 analysis or deconvolution. 184 i 2 1.27 24.6 i 0.5 If a computer is not available, 1190+10 2.00 62.2 + 0.3 the variances can also be deter2.70 m+l 3570 + 80 mined using manual methods (8, "mese are the stattstical moments corrected for the contribution from iniectoridetectorband broadening by subtract12). The 5-0method (8)underestimates the peak variances by ing the corresponding injectoridetector first (10.5 s) and second (4.96 s') moments. lo%, and the asymmetry-based Table 3. Effect of Tubing Length on Band Broadening in Nominally 0.8-mm4.d. PTFE method (12)overestimates the Tubes at a Solution Flow Rate of 1.00 mUmin and Injection Volume of 2-pL peak variances by 20%. Regardless, both methods track the true Tubing Length Center of Gravitya Variancea Average Inside FIadius4xkength variance well to illus(mm ) (cm) (S) (s2) Radius (mm) t r a t e t h e key f e a t u r e s of t h e band-broadening processes. 31.5 10.9 +0.3 21.9+1.1 0.41 8.90 Open Columns and Connecting Tubing

153.0 310.0

+

47.0 0.5 102i1

81.6 i 0.3 249 i 1

0.39 0.41

35.4 87.6

Band broadening within Open %ese are the statistical moments corrected for the contribution from the injectoridetectorby subtracting the corecolumns and connecting tubing sponding injectoridetectorfirst (10.5 s) and second (4.96 s2)moments. r e s u l t s from t h e nonuniform laminar flow profile caused by the viscous drag along the tubing walls. Injection of the amaranth dye into the open tubes results i n a parabolic flow profile that is clearly visible, especially i n the glass tubes. This visually illustrates the nonuniformity of flow. For a tube whose length is much greater than its radius, the impulse-response function for laminar flow is approximately Gaussian, with the associated time-based variance approximated by one of the terms i n the Golay equation (I).

where r is the tube radius (em); D mis the solute-diffusion coefficient (cm2/s);and F is the solvent flow rate (cm3/s). Moment analysis of peaks produced using t h e glass tubes/columns of constant length but varying inside diameter reveals that the variance (band broadening) is strongly dependent on the inside radius a s predicted by the Golay equation. The observed peaks, uncorrected for injectorldetector contributions, are shown in Figure 2. The statistical moments, corrected for injectorldetector contributions, are presented in Table 2. Plotting peak variance versus r4 yields a straight line with a slope of (1.07 f 0.03) x lo7 s2/cm4,a n intercept of zero (37 f 44 s",and a correlation coefficient of 0.999. From the slope of this plot the diffusion coefficient of amaranth in water is calculated via eq cm2/s. 6 to be D m= (2.2 f 0.1) x Moment analysis of peaks produced by the PTFE tubes, which have a nominally constant inside diameter of 0.8 mm, was used to determine the dependence of band broadening on tube length. Aplot of 0: vs. r4L was made, rather than 0: vs. L, in order to compensate for the small variation in average radius of the tubing. The plot yielded a straight line with a slope of (2.9 f 0.2) x 10%'/cm5, a n intercept of zero (-11 ? 13 s'), and a correlation coefficient of 0.997. From eq 6, this slope corresponds to a diffusion cocm2/s, efficient for amaranth of D m = (2.7 f 0.2) x which is in agreement with the value obtained from flow through the glass tubes. Evidently, the diameter of curvature of the coiled PTFE tubing (3.5 cm) is large enough that secondary flow has little effect on band broadening (16, 17). Although eq 6 is not accurate for very short tubes, i n which "end-effects" are important (16, 171, the same gen370

Journal of Chemical Education

Time (sec) Figure 2. Experimental peaks o b s e ~ e dfora 2-ILLinjection after passage through 30.0-cm open glass tubes with inside diameters of (a) 0.98 mm; (b) 1.27 mm; (c) 2.0 mrn; and (d) 2.7 mm. The peaks are uncorrected for injectorldetector broadening. Peaks b-d are offset. era1 conclusions do apply to tubes of any length. Connecting tubes that are shorter and of smaller inside diameter produce less extracolumn band broadening. This should be the guiding principle in the design of connecting tubes in a chromatograph. Detector Cell and Injector

The injector and the detector, as band-broadening devices, exhibit many similarities (Table 1).In an efficiently designed injector or detector the inside diameter will be small. Also, there will be no sudden increase in inside diameter upon going from the inlet tubing to the injector or detector. An efficient injector or detector will produce a "plug" (square wave) impulse-response function. When the square wave is convolved with the Gaussian peak from the column, the resulting peak is symmetrical but wider and shorter than the Gaussian peak. An inefficient injector or detector will produce a n asymmetrical, tailing, impulse-response function. Two types of effects can cause this. Extensive convection may occur, such a s that due to eddying, that causes the

Table 4. Effect of Injector Volume on InjectorlDetector Band Broadening at a Solvent Flow Rate of 0.40 mUmina Predicted Center of Gravity (s) Injector Volume

Dimensions (length x i.d.)

Observed Variance

(mm)

Observed Center of Gravity (s)

(I4 2

6.4 x 0.61

25.3 i 0.9

5

6.4 x 0 . 9 7 ~

25.7 0.4 26.4 i 0.6

29.6 i 0.9 34.7 k 0.5 53.2 2.7

10 20

6.4 x 1.37b 9.0 x 1.57'

+

38.7 f 0.3

Plug

Predicted Variance (s2)

Mixing Diffusion Chamber chamber4

Plug

Mixing Chamber

Diffusion chamber4

(s2) 0.14 0.35

0.28

0

0.0066

0.079

0

26.4 74.2

0.042 0.17

0.50 2.00

700 5510

98.6

0.57

6.83

9720

+

0.71

0.71 1.42

121 i 2

1.31

2.61

Valve pons8 were used and the injector volume was varied. bln~ide diameter of the inlet tube to injector is 0.65 mm. 'Inside diameter of the inlet tube to injector is 0.73 mm. ' ~ a s e don experiments described above, a diffusion coefficient of 2.45 x 10" cm2/s was used

device to exhibit some mixing-chamber behavior. Another effect may arise when the inlet tubing has a much smaller inside diameter than the device, and the diameter change occurs abruptly where the tubing joins the device. In the extreme this situation can produce the very large variances associated with a diffusion chamher. Mixing Chamber Amixing chamber per se is studied, and injectors are examined under conditions in which thev exhibit some diffusion-chamber behavior. A mixing chamber is a device in which any element of entering fluid will be instantaneously mixed throughout the chamber volume. The response function for such a device exhibits a virtually instantaneous rise to a maximum value followed by a n exponential decay to zero. This exponential peak has a which is equal to the cell volume ditime constant, rCell, vided by the flow rate, as shown in Table 1. The mixing chamber used in this experiment has a volume that is much lareer than would be urudent for use in an HPLC detector. ~ L i design s allows the use of a magnetic stirring bar and the incorporation of bames (Fig. lb), both of which facilitate the nearly instantaneous mixing required to demonstrate ideal mixing-chamber behavior. The mixing chamber is also designed with viewing windows so that the student can see the exponential dilution over time. The large magnitude of the band broadening from the mixing chamber used in this experiment allows the student to treat the resultant peak a s the response function of the mixing chamber alone. (The band broadening due to the injector and detector i s so small in relation to that of the mixing chamber that they can he considered to be an impulse, even without deconvolution). To test whether the mixing-chamber response function is indeed exponential, the descending portion of the peak is linearized a s In C, = --t

1

741

+ In C,, (71

The resulting plot is linear over three half-lives with a slope ofG(5.03 0.01) x s-', anintercept of l.45+0.01, and a correlation coefficient of 1.000. From equation 7, the negative reciprocal slope gives 7-11 = 199 s (center of gravity) and = 3.96 x lo4s2(variance). Because r,ll i s equal to the chamber volume Vdivided by volumetric flow rate F = 1.00 m u m i n (Table 11, the calculated chamber volume is 3.32 mL, which is in good agreement with the measured chamber volume of 3.40 mL.

+

An Ideal Diffusion Chamber In a n ideal diffusion chamber the stream of solution from the narrower uustream tubine -.uasses throueh the solution in the larger-bore downstream component without experiencing any convective mixing. In the larger-bore component the transport of solute between the narrow flowing stream and the surrounding stationary solution occurs exclusively by diffusion. The behavior in a n ideal diffusion chamber stands in marked contrast to that in an ideal mixmg chamher where conwctio~iin the component i:,lnstantaneous and cumplerr. Realistically, neither ol these deal models is likely to describe mass-transport behavior when a narrow stream flows into a larger-bore component. I t is likely that there will be some, but not complete, convective eddying that leads to centers of gravity and variances that fall somewhere between those predicted by the mixingchamber and diffusion-chamber models. Influence of the Injection Volume The influence of the injection volume on band broadening was demonstrated using injection valves with 2-, 5-, 10- and 20-pL volumes. The dimensions of the holes in the sliders are presented in the second column of Table 4. The dimensions of the inlet tubes are given i n the footnote to Table 4. Presented in the third and fourth columns of the table are the observed centers of gravity and variances for the injectorldetector (no column) experiment. These observed quantities include not only the injector contribution but also-the contributions from the deteitor and the tubing that connects them. The centers of gravity and variances observed for the 2-WLand 5-pL injections are nearly the same, suggesting that the band broadening observed for the 2-pL injector is essentially equal to that produced exclusively by the detector and connecting tubing. Thus, the contribution of each of the four injectors alone can be approximated by subtracting 25.3 s from the observed centers of gravity and 29.6 s2 from the observed variances. After making these subtractions the injedoronly values are

.

0 s and 0 s2for the 2-pL injector 0.4 s and 5 s2for the 5pL injector 1 sand 24 s2 for the 10-pL injector 13 sand 91 s2for the 20-pL inject01

These values may be compared with the theoretical values of the centers of a a v i t y and variances ~ r e d i c t e dbv the equations in Table c f o r &g, mixing-chamber, and diffusion-chamber behavior. These theoretical values are presented in columns 5-10 in Table 4. Volume 72 Number 4 April 1995

371

Comparison with Idealized Injector Models Comparison of the injector-only centers of gravity and variances with the theoretical ones for the 5-, lo-, and 20pL injectors reveals that the injection profiles produced by these injectors do not conform exclusively to any one of the three idealized injector models. It is evident from a comparison of the variances t h a t t h e band broadening i s greater than predicted by plug or mixing-chamber behavior, and that the contribution from the "diffusion-chamberlike" processes becomes greater a s the difference between the inside diameters of the injector and the inlet tubing increases. In other words, convection is less complete in the larger injectors. For the 20-pL injector this contribution is quite significant and reveals clearly the importance of avoiding sudden increases in diameter anvwhere in the chromatographic system, whether i t be in the injector, detector, or elsewhere. Nevertheless, i n some instances a wider-bore is necessary. In these cases, proper design is essential. For example. the dimensions of the detector flow cell used here (1.0 cm k 0.067 cm i.d.1 could result in 0.4 to 390 s2 of hroadening depending on whether the cell behaves as a connecting tube or a diffusion chamber. Additional Sources An additional source of detector band broadening not covered in this experiment i s the finite response time of the detector/recorder electronics. The response function of the electronics can he modeled a s that of a simple low-pass filter circuit (I, 15). This circuit will have ankxponeAtial response function with a time constant, re,, as shown in Table 1.

Figure 3. Experimental peaks o b s e ~ e dfor a 2-pL injection after passage through (a) 30.0-cm long by 2.7-mm-i.d. open glass tube at 1.0 mumin; (b)30.0-cm long by 2.7-mm-i.d.packed tube at 0.4 mumin; and (c) same as b except for a 20-pL injection. The horizontal axis is plotted as time divided by the center of gravity of the respective peak. The peaks are uncorrected for injectorldetector broadening. 0.31 a

Case

Packed Column A comoarison of the hand hroadenine hv a - exoerienced . nonsorhed sample component as it passes through a n open tube and through an identical tuhe packed with particles should he made a t the same interstitial linear velocity of solvent. In this experiment a 2.70-mm-id. x 30.0-cm-long empty glass tube is compared with a n identical tuhe that is packed with 40-pm diameter spherical, nonporous glass particles. As a n approximation, the interparticle void space in such a tuhe would be about 40% of the emutv-tube volume (18).Therefore, operating the packed tube i b a flow rate of 0.40 m u m i n will produce approximately the same linear velocity as that produced in the empty tuhe a t 1.00 mllmin. Tailing of the Peak The observed peaks from the empty and packed tubes, uncorrected for~injectorldetectoreffects, are shown a s curves a and h i n Firmre 3. The horizontal axis is time divided by the first moment of the respective peak to eliminate any remaining differences in the linear velocity. Despite appearances, the peak area and center of gravity (scaled by t ~of)peaks a and b in Figure 3 are identical. For the peak a from the unpacked column, much of the area is in the tail, thus shifting the center of gravity onto the tail of the peak. Likewise the variance is dramatically enhanced by the tailing of the peak, as can he seen quantitatively by comparing the variance of the peak from the packed tube, in Table 5, with that of the empty tuhe, in Table 2. This comparison reveals t h a t the peak from the packed tuhe is much narrower than the peak from the empty tuhe. The reality of this difference is reinforced by observing the zones of amaranth broaden a s they pass through the two glass columns. The packing reduces the variance primarily

372

Journal of Chemical Education

F gdre 4 El rn naton of exlracoLrnn oano oroaoenmg oy deconvo .!,on la) pea* from nfecloraeleclor oroaden ng fb, pea* ooserved for 30.0-cm-longby 2.7-mm-i.d.packedglass tube; and (c) peak due to effects of packed tube alone obtained from deconvolution of peak a with peakb. All injection volumes were Z p L . Peaks a and c are offset. See tevt and Table 5 for details of cases.

by greatly diminishing the magnitude of the nonuniform flow profile of the solvent along the tube. Efficiency ofthe Column In chromatographic studies, one often wishes to determine both the degree to which the sample has been retained by the column and the efficiency of the column. If the peak has been delayed, broadened, and distorted by extracolumn hand hroadening in the injector, connecting tuhina, detector cell, and detectorlrecorder electronics, i t is

Table 5. Deconvolution of Peaks Obtained for 2-WLInjections at 0.40 mumin from the 30.0-cm-Lona bv 2.70-mm4.d. Packed Tube with ADDrO.. priate ~njector/~etec~or'lmpulse-~esponse Functions

give the centers of gravity ( t ~ . ~ band J the variances (o~I,,~)of the eluted peaks. (The discrepancy between the observed center of gravity a n d variance for t h e packed column with a 2yLinjector in Tahle Center of Gravity (s) Variance (sZ) 6 with the values for the same packed colCase I umn in case 1 of Table 5 (also done with a lnjectorIDetector 23.7 f 0.3 2 7 3 i 1.1 2-pL injector) is due to slight variations in t h e experimental conditions (e.g., flow Packed Tube plus InjectorIDetector 159i2 139f 2 rate): The injector volume experiments Packed Tube Alone 135f 2 112f 3 were carried out six months after the other (by difference of moments) experiments described in this paper.) 136f 2 105f 1 Packed Tube Alone (by deconvolution) As expected, both the center of gravity Case Za and variance increase a s the volume inInjectorIDetector 42.2 f 1.O 112f2 jected increases. The variance, in particular, experiences a dramatic increase: The Packed Tube plus InjectorIDetector 179f I 239 f 5 observed variance is nearly twice as much Packed Tube Alone 137+2 127+7 for a 20-pL injection as for a 2-pL injection. (by difference of moments) The two elution peaks are shown in Figure Packed Tube Alone (by deconvolution) 136f 2 111 + 5 3 where i t is evident that the greater injectorldetector contributions of the 20-pL in'In case 2, 20-crn of 0.8-mm-i.d.PTFE tubing was insetled aHer the first 6-way valve to increase the jection has caused curve c ( 2 0 y L injection) extracolumn band broadening. to become shorter and more tailed than Table 6. Effect of lnjector Volume on Band Broadening of a 30.0-cm-Long curve b (2-pL injection). Packed Columna Packed Column Alone Injector Volume Observed Observed tn.obs - h.10 0 2 0 b s -210 Presented i n columns 4 and 5 of Tahle 6 Center of Gravity, Variance, (S) (sZ) (PL) are the differences obtained when the cenk 0 b s (S) 0 2 0 b s (s2) ters of gravity and variances for the injec96f3 126f2 153i 1 126f2 toridetector alone, from Table 4, are sub92f2 148i2 127f 1 122f2 tracted from the values oft^,,^, in column 2 101 f 7 15511 154f4 129i-2 and sob: in column 3, in Table 6 . These dif101 f 7 129 +2168fl 222 f 5 ferences are the retention times and variances produced by t h e packed column 'Flow Rate is 0.40 mUmin. Valve ports-1 were used, and the injector volume was varied. Subscripts alone. within experimental these col"Obs"and " I D refer to the 'observed' and "lnjectorIOeteclo? alone. umn-alone retention times (126 f 3 s) and variances (98 i 4 s? are constant, as expected. necessary to remove these extracolumn effects from the observed peak. a n d o ~ , , , , ~from eqs 3 and 4, In other words, if tR,peak rather than tR,eoi and of,,^, are used for the determination of column efficiency via eq 5, then the calculated value of N will be in error. If the first and second moments are known for all of the extracolumn devices in the chromatograph, either for each device independently or for all of the extracolumn devices collectively, they may he suhtracted from the first and second moments of the overall peak to obtain the center of gravity ( t ~ , ~ and , , ) the variance ($, ) resulting from on-column retention and band broadening alone. For instance, for the data given in Table 5 the variance of the packed column alone was 112 s2,whereas the overall observed peak variance was 139 s2.The additional 20% of band broadening was due to the extracolumn devices: injector, connecting tubing, and detector. I n the case where the additional F'TFE tubing is present, the variance of the packed column alone was 127 s2, whereas the overall observed peak variance was 239 s2.Here, the presence of the additional tubing increased the extracolumn band broadening fivefold. cB,

The Contribution of Injector Volume a n d Design The contribution of injector volume and design to band broadening of peaks eluting from the packed column was also studied. The same 2-, 5-, lo-, and 20-pL slider injectors were used as in the injectorldetector study described above. The results are shown in Table 6. Columns 2 and 3

Deconuolution The effect of the extracolumn band broadening can also be removed from an observed peak using deconvolution. This is illustrated for case 1and case 2 described in Table 5. The impulse-response function due to all of the extracolumn band-broadening devices is determined by removing the column from the system and connecting the injector directly to the detector. The impulse-response function due to the extracolumn devices in case 1 (Table 5) is shown as peak a in Figure 4 (case 1).Peak b in case 1of the figure is the overall observed peak eluting from the system with the packed tube in place, and peak c is the peak resulting from deconvolution of peak b (observed peak) with peak a (extracolumn peak). Peak c is due to passage of the amaranth zone through the ~ a c k e d column alone and is free of contri1)utioni f r ~ mthc cxtrvcolunin d e w t s . The de~~onvolution Ii.is removed a~vroximatelv25'; of the \.nrinncr from tht, observed peak'(ib) and has decreased the retention time by 27 s. The peak from case 2 in which additional F'TFE tubing is present to increase the extracolumn band broadening, was also deconvolved with its corresponding "injector/detector" response function. The lower plot of Figure 4 shows impulse-response functions of the extracolumn components (peak a), the column (peak c), and the observed peak (peak b) for case 2, and the associated statistical moments are eiven in Table 5. Here too. the resulting deconvolved ~ e a k w,is f r w of c m t r i h t i ~ r ~ if;i l m the cxirvcolumn romponc~its.as ran be >een .n Table 5 bv the agreement of thew results with the deconvolved values obtained in case 1 Volume 72 Number 4 April 1995

373

5. Otsuka, K,Tersbe, S. J. Chmmntogr 1989,480.91-94,

above. I n both cases, the moments of the deconvolved peaks agreed with the values obtained by simply subtracting the appropriate "injector/detector" moments from the observed packed-column values (Table 5 ) .

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Acknowledgment

This work was supported by the Natural Sciences and ~ ~ ~~~~~~~~h i council ~ ~of canada ~ and f thei univer. ~ sity of Alberta. Literature Cited Sternberg. J. C. InAdvaneer in Chrornoiography; Giddings, J. C.; Keller, R. A., E d s ; Marcel Dekker: New York. 1966: Vol. 2. Chapter 6. 2. H u e , K. P: Jonker. R.J.;Rozing, 0. J Chromalogr 1384,285,253-265. 3. Freebairn. K. W.: Knor, J . H. Chmmoiogrophin 1984. 1 9 , 3 7 4 7 . 4. Berngard,A. K.: Colrnsjo, A. L. J High R e d Chromatog. 1 9 8 0 13,689-693. 1.

374

Journal of Chemical Education

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and refer.

en-8 therein.

~

8. Bidlingmeyer. B.A.: Warren, F V , JrAnoi. Chem. 1984.56, 1583A-1596A. 9. Anderson, D. J.: Walters, R. R. J. Chmmolmi Scr. 1984,232,353-359. la. Ruzida, J.; Hansen, E. H. Flow Injection Analysis; John Wiley: New York, 1981: Chapter 3. n. Wright,N. A.:Villalanti, D. C.; Burke, M. FAnal. Chem 1982,54,1735-1738. 12. F k J. E;Domw J . G-Anal. Chsm 1983.55.73(1-737. 13. Giddings, J . C. Dynamics ~JChromoiogmphy;Marcel Dekker: New York, 1965: chapter 2. 14. Sehiire, M. R.;Barman, B. N.; Giddings, J. C.Ano1. Chem. 1989,61,273%2743. 15. ~ ~ i ~E.hO. a~h~ ~ F.S S ~F O U ~ W ~ ~P ~ ~ ~ ~ - HE~ I fI : ~ cliffs. ~ NJ, ~ 1974: Chapter 9. 16. Shsnksr, A,: Lenhoff. A. M. J Chmmnrogr 1991,556.23%248. 17. T W m , R A n d Chim.Aela 1930. 114.71-89. 18. Giddings, J. C. Dynamics dChrnmofogmph.y, Marcel Dekker: New York. 1965: Chapter 5.

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