Estimatiorr and Corrgxtion of Post-Column Dead Volume Effect in Chromatography F. A. VANDENHEUVEL Animal Research Instilute, Research Branch, Department of Agriculture, Ottawa, Canada
b Expressions relating
post-column dead volume (PCDV) effects to a few simple parameters are proposed. Applicable to both liquid elution and gas liquid column chromatography systems, they permit the estimation and correction of errors due to peak overlap in chromatograms distcrted b y PCDV effects. These expressions should b e useful in the design or correction of equipment for low PCDV distortion. They permit the setting of operating conditions under which PCDV errors will not exceed a given level or the use of operating conditions corresponding to high equipmeni efficiency under which PCDV errors (ire appreciable. Gas liquid radiocounting with ionization chambers is usec as an example of the latter application.
T
PosT-coLuMPr dead volume ( P a w ) may be defined as thc fluid-filled space lying between the exit end of the column p a d i n g and the point of collection. It includes the volume of any device placed in the path of the duct sybtem, s u ~ has: stream analyzer or n detector cell. PCDV eflect ariacs from partial remixing of the column effluent within the PCDV, with attending attenuation of concentration grac ients previously established in the colunin. Direct demonstration of PCDV effect is obtained (Figure 1) by comparing siInultaneously recorde 1chromatograms HE
'.
VOLUME
of the eluate before entering (Curve A ) and after leaving (Curve B ) a PCDV of some size. The over-all effect is to convert a neatly separated system of bands to one of much poorer definition. The observed peak and minima shifts, the lowering of peaks, and the filling-up of valleys are not indeed the most objectionable effects. As seen in Figure 2, where the PCDV chromatogram is shown as a continuous line, and the elution curves for individual components as dotted lines, overlapping of bands because of trailing results in their mutual contamination. The observed PCDV chromatogram, which is the resultant of individual elution curves, displays minima between peaks, indicated by vertical lines. hIinima positions are generally used as fraction limits for the purpose of collecting or estimating individual components. The shaded areas on the right and left of the vertical lines clearly represent impurities in such fractions. PCDV effects do not occur to a large extent in the common practice of gas liquid chromatography (GLC) where duct systems and the usual detector cells are, as a rule, of relatively small volume. They are, however, quite pronounced when the system is fitted with a radiation counting device, such as an ionization chamber, which is often chosen of large volume t o ensure satisfactory counting efficiency (1). The
effects induced by PCDV on such systems can be clearly observed by comparing the chromatogram obtained by the use of a common-type detector placed ahead of the counting chamber with the radio-chromatogram itself. The latter will show a shift of peak maxima position indicative of PCDV effect. From the example shown in Figure 2, discussed above, it is evident that this could result in counting errors. Liquid column chromatography systems, on the other hand, are affected by PCDV when the space below the plate supporting the packing is totally or partially filled with liquid, or when an analyzing cell (ultraviolet, infrared, differential refractometry) of relatively large volume is used as a monitoring device. Equations relating peak position and base width to detector volume for isolated gas chromatographic bands have been proposed (2). A method for estimating the extent of peak overlap in chromatograms distorted by PCDV effect is reported below. The proposed expressions relate PCDV effects to a few simple parameters. Their application should be useful in designing equipment for either gas liquid or liquid elution column chromatography. Furthermore, they permit the use of operating conditions corresponding to higher equipment efficiency by providing a means of estimating the extent of peak overlap ariaing under these conditions and to apply the necessary corrections. GLC radiocounting with ionization chambers is used as an illustration of this type of application.
(-
y = H K exp( - Kx)
While the total area under this curve m to m ) is given by
+
dx (11)
99.5% of this area is found between point A(V = -a) and point C(V = +a). For all practical purposes, $ ( V ) = 0 outside of the band width AC = 2a. Such function fits most symmetrical elution curves both in liquid and gas chromatograms. Substituting ( 5 ) in (4)yields 1 +; exP(-V/w)
y = A exp(-V/w)
3.
Figure
Original chromatogram ABC of a single component and corresponding PCDV chromatogram AMKL
General Equations, Figure 3. Consider a streani of gas or liquid passing through a chamber of volume w where thorough mixing occurs, the concentration of a solute in the output stream being the same as that in the chamber. Let 1J = f ( V )
(1)
be a function of the volume V of fluid entering and leaving the chamber that expresses this concentration, and let + ( V ) describe the concentration in the input stream. For a small quantity, AV, of fluid flowing through the system, the change, Aq, in the amount of solute within the chamber-Le., in the output streamis expressed by Aq = +(V)AV - y A V
% = #4VjAV ---= w
YAJ’
w
w
AY
represents the change Ay in the concentration of the output stream. Then ay=--Y_ dV ) AV
w
It is evident that input $ ( V ) , and a fortiori, output y, attain negligible values for V O. Therefore the first term in (7) can be neglected. Further simplification in view of subsequent derivations is obtained as follows. Let x = V/a and K = a/w (8)
y =
Then,
or dV
=
HK
T~ esp(-Kx -+
adz and V / w = Kx
Substitution of A = 0 and the above expressions in (5) and (7) yields +( V ) = H exp( -4x2) (9) and hence
y = H K exp( - Kx)
JI
(2)
Thus
hence,
p - 4x2 (
~
+ Kz)dz
(10)
K2/16)s:m
While V varies from -a to +a, 2 varies from -1 to +1; $ ( V ) and y are practically equal to 0 when z = -1. Thus - m is equivalent to -1 as limit for integral in (10). The latter can be written, I
w
2
4
3
+ ( X j d X (14)
where @ ( X )is the normal distribution function. Values for the integral are found in tables (8) for any value of X Consequently they can be readily ob. 6
5
7
a
9
leads to the linear differential equation
Hence, y
=
A exp( - V / w ) J;?(V)
+ w-1 exp( - V /
w)
exp(V/u)dV
(4)
\
In this equation, $ ( V ) represents the elution curve of a single component for which a good approximation is +(TI)
=
H exp( -4V2/a2)
(5)
where H is the peak maximum [$(V)= H for V = 01, and a is the half band width. An example of this distribution function is shown in Figure 3. Note that the ordinate axis and origin are OB and 0, respectively. 1 194
ANALYTICAL CHEMISTRY
/
40.3
\
‘I,/, ,
0
I
2
Figure 4.
1
,
;
I
3
Plot of R =
40.2
1
5
I
;
, F I T 7
@(X)/s-”,(X) vs. X;
R = K/\/B plot of X M vs.
K
e#x
0.20
/ P
0.15
Figure 5. Chromatogram showing two neatly separated bands ABC (I) and
0.1
CDE (11)
e
Corresponding PCDV chrcrmatogram ( K = 2; 01 = 1 ) displays a minimiim a t m between C ' and D". PCDV chromatcgrom i s resultant of individual PCDV elution curves in dotted lines
tained for any values of z (12) and the output curve can be constructed for any K value. Such a curve, AiMKL, shown in Figure 3 for example, displays a skewed distribution with protracted trailing. I t is evident from (3) that the maximum of this curve must lie, as shown, a t the intersection with the original elution curve. Differentiating (14),,*ememberingthat dX
= 4 d . c
H
Sz/, e s p ( - K x
+ I?/lG)
A t the maximum z == w, X = and dy/dz = 0. The above expression can then be simplified and rearranged, leading to
radiochromatograms. Dividing half the base width (a) taken from the original chromatogram into the peak shift (V = zMa) observed in the radiochromatogram, yields x . ~from which K is found in the K / X Mgraph. This K value can be used, as shown below, to estimate corrections for PCDVinduced counting errors. Position of the Minimum. Figure 5 represents the original chromatogram ABCDE corresponding t o two neatly separated components the individual elution curves of which are described by (9) +(VjI =
Corresponding values for X I @ ( X ) ,and
Sfm
@ ( X ) d Xcan be found from tables.
From a plot of @ ( X ) /
S_"i
ax
@ (X)
ratios against the corresponding X's (Figure 4), the value of -YMcorresponding to any K / d 8 value can be found. From XM = d8 (ZM - K / 8 ) (12), one can determine the corresponding value of z . ~and establish the ZM us. K graph shown in Figure 4. One application of this graph is in GLC radiocounting where one usually obtains a record of both original and
o
I II
IO
0.25 xI0-'
m
IO2
0 . 2 5 IO-' ~
m
lo3
0 . 2 5 ~IO-'
Y
lo4
0 . 2 5 IO-' ~
to
0.25
Plot used in the graphical determination of xm described in Appendix
(see 12, 13),
yields dy/dx =
Figure 6.
from
+(x)
I
H1exp(-4z2)
arid + ( V ) ,= H, exp(-4z2j
CC'
=
2
Kd?iexp( - K
+
The equation for the tailing section of the curve is given by exp[--K(z - 1)1 If this curve is considered in the coordinate system having D' as origin, one must write YI =
YI,~
+ +
YI = Y I J exp(-K(z 1 -3jI (18) In this expression, 6 takes into account that bands in the original chromatogram may not be exactly contiguous but may overlap (6 < 0) or be further apart (6 > 0). The expression Sa = d
(16)
respectively, when considered in their own system of coordinates, the first with BB', the second with DD' a5 ordinate axis. We will assume that AC = CE = 2a. If the PCDV curve AMC' corresponding to the first band is considered, ordinate CC', which corresponds to point C where V = a, is obtained by substituting z = 1 in (14). Thus,
yI,l =
- 2a
(19)
shows 6a as the difference between peak to peak distance d and band width in the original chromatogram. The PCDV curve corresponding to band I1 is described by (14) y11 =
H z Kd/rrexp(-Kx 2
+
K2,16)J~T(XIdX
VOL. 35, NO. 9, AUGUST 1963
(20)
1195
1
i
showing Figure 7.region Enlarged correspondiog section ofto Figure minimum 5 with hatched area representing impurities on either side of vertical MM’
L ,
=
yr
+
will reprcsc.nt C‘inD’’-i.e., the section of the observed chromatogram bptween points C‘ and 1)”. substituting (17), (lS),and (20) in (21), one obtains
I.’ =
H ,~
I
h-4*exp( - K z +
(22)
where
4
6
graphical method shown in Figure 6 and described in the Appendix. Estimation of Contamination, Figure 7 . The hatched area to the right of vertical iWM’ a t the minimum between bands I and I1 in the observed PCDV chromatogram represents the amount of the first component in fraction 11. At this point, JI‘P is the concentration of component 1 obtained in dead volume W. Since the total amount of component 2 is represented by the total area of band a
(23)
a = li2/Hi
If the above expression is differentiated using d X = dx 4 8 (13), and if the derivative is made equal to 0, one obtains an expression where z = xm, X = X, corresponding to the minimum. After simplifying and rearranging, one obtains NX)m
~-
There is no minimum when K = 1 with a comes a “shoulder” for the larger one. from pre-established r / K graph
(21)
YII
“‘-----\:.--
2
=
c
= d @ x , - K/8) + ZKS - a X, ’X’dX
K
10
8
Figure 8. Per cent impurity I t (tailing) for K values ranging from 0 to 10 with a! = 0.01, 0.1, 1, and 10, assuming 6 = 0
Thus I’
>-,
OL-
11-Le., 5 H2d\/?r,from (6)-the impurity I t expressed in per cent of band 11 is given by the following expression, combining (6), (17), (18), (22), and
5
0.1. The smaller band is submerged and beNote that r in Equation 28 i s readily obtained
1, = 50 a K ( s ,
+ 1)exp( - K z , +
J--m Note that the integral term in this expression is found in the course of zM determination as described in Figure 6. Figure 9 shows I b for various CY and K values, with 6 = 0. DISCUSSION
In the above derivations, parameter a was assumed to be the same for adja-
cent bands.
This can be true when
-.--,
_ 1
-
__ - ~
-
r
-
-
-
-
Ib
I
*
i
(25). It
=
loo€ -- eap{--I((z, Ly
+ l)\
(27)
where c =
exp( - K 4-
(24)
where C
=
=;:J
K exp( - 2 K )
ad8
J
dE(1 - K / 8 )
@(X)dX
(25)
with Hi = R; exp(6K) = aeaK
(26)
Constant C can be calculated for any combination of K and fi values. One can then find X,, and therefore zm from X, = (zm - IO and the bands get farther apart. Examination of Figure 7 shows pronounced contamination due to PCDV trailing effect for K values below 5, and the more so for low CY values. Conclusions regarding what limit to adopt for the dead volume will dcpend on what is considered an acceptahle level of contamination. In making this appraisal, it should be borne in mind that ideal conditions assumed to txist in the present treatment, such azi perfect mixing and originally symmetiical bands, may not be met in practice. Kor will original bands always be nt?atly separated (6 < 0). Departure from these conditions may result in an appreciable increase of the PCDV effect. Taking this into consideration, one must conclude that the practically complete suppression of PCDV effect wi 1 not be assured unless K is greater than 15 or 20. In other words, the dead volume should not exceed to the volume of the smallest band of interest expected from the column system used. While one can conceivably endeavor to work under such conditions as will increase band volume, or band separation (6 > O)-i.e., longer columns and lower temperature-it would seem more practical to design equipment with negligible PCDV effecm Liquid elution chromatographic column and monitoring cell systems meeting the latter requirement have been proposed (4, 6). Although mos, of the recent GLC chromatographs nave PCDV of relatively small volume, a further reduction of the latter would, in some cases, improve the resolution of peaks having the shortest emergence times. In continuous GLC radiocounting with ionization chambers, one is limited in this direction by the necessity of using large chambers for the sake of counting efficiency. Increasing band volumes, and therefore K valueq, by sweeping the chamber with a diluting gas, will I?
offset PCDV effects to a certain extent. The effectiveness of this procedure can be judged from the shift in peak position observed by comparing the original and the radiochromatograms. It is clear from the graph in Figure 4 that a situation where this shift is less than 5Oj, of the band width will nor entail extensive correction of the count (K > 10). Comparing Figure 7 with Figure 8 shows I t contamination to be much more extensive than its I* counterpart under the same conditions. This is not necessarily true in GLC radiocounting with large chambers, because of posd!e large differences in specific radioactivity leading to low CY values, and because of the low K values prevailing in this technique. Correction of PCDV-Radiocounting Errors. As pointed out above, PCDV effects can be avoided by working under conditions leading to a wide spread of bands in the original chromatogram. Understandably, the larger the counting chamber: the longer the elution time required to bring about this situation and the smaller the number of analyses obtainable in a day's work from an expensive piece of equipment. On the other hand, since PCDV errors can be readily estimated by the use of appropriate graphs, it would seem that a satisfactory compromise could be reached with faster elution rates. In GLC radiocounting with ionization chambers, two chromatograms are usually recorded simultaneously. One of these is the original chromatogram obtained with an ordinary-type detector positioned ahead of the counting chamber. The other one is the radiochromatogram itself. The latter is effectively a PCDV chromatogram having been further modified by relative activity effects on peak heights. Determination of impurities I t and I b , as shown by (27), (28), and (291, requires values for CY, K , and .z, These are easily obtained from the two chromatograms. As pointed out in the discussion, parameter CY is the ratio of peak heights in the radiochromatogram. If this is expressed by a, = h,/hl, hl is the peak height of the contaminated band in the case of I f , and that of the contaminating band in the case of I b . Parameter K is obtained from the zm us. K graph in Figure 4. Here zm = V,/a where V , is the peak shiftLe., the observed displacement of corresponding peak maxima in thc radiochromatogram. Measuring a, the halfbase width in the original chromatogram, may be impractical in situations where, for instance, bands overlap to some extent. However, if a0.6 represents the half-width a t half-height, then a = a0.5/0.417 = 2.4 ao.5
since from (6) it can be shown that for +(V) = H / 2 , V / a = 0.417. The value of 2, is obtained from zm = - D ' M ' / a where D'M' (Figure 7) is the distance from the minimum between bands I and I1 to a perpendicular drawn t)hrough the original peak maximum in band 11. Following the facilr determination of the above values from the recorded data, corrections I f and I* can be obtained, practically without calculation, from preconstructed plots and nomographs representing expressions (27), (2S), and (29). APPENDIX
Procedure for the Determination of z,,, (Figure 6). In expression ( 2 4 , both z, and X, are always negative, while XI in expression (25) is negative for K > 8. Since tables of the normal curve of error [SI described by
give only @(X)and @ ( X ) d X for positive values of X , the following expressions are used to derive those needed in the present computation @(-X)
f"
J
=
@(-X)dX = 1
@(X)
-
( a)
r"
+ (S)dX
J-a
-a
'
(b)
Values calculated in this way have been used to construct curves I to V in Figure 6. Each curve represents a plot of B@(X) os. 8 J q X ) d X where e = lo", with n a digit from 0 to 4; the insert at the lower right corner of Figure 6 shows the correspondence of curves to e values. Determination of zm requires prior calculation of constant C, using expression (25), for given values of /3 and K . From (25),
c
log,, K / d K - 0.838K
Log,, C =
log a
+
- 0.434K
Thus Log,, C can be obtained from a nomograph constructed by lumping the three first terms into a single K-dependent value. The calculated C and given K values are substituted in the following expression
PQ choosing
e
=
OC
+ K/10 4s
(as defined above)
(c) BO
that
PQ < 0.25. The value used for e
corresponds to one of the curves in Figure 6. PQ and ec from expression (e) are then plotted in Figure 6, PQ on ordinate VOL. 35, NO. 9, AUGUST 1963
1197
VQ, and SC as O N . Line N P intersects the curve corresponding t o the 0 used; the ordinate of the point of intersection gives
or found on a nomograph corresponding to this expression.
m(-xm) = eqx,) +(-xm)= ST/^ is ob-x, > 0 is then found in
I am very grateful to Jean R. Proctor, of the Statistical Research Service, Department of Agriculture, for gracious assistance in checking my results.
ST =
from tained; tables and z,,,can be calculated (24) from .Y,
=
&x,
ACKNOWLEDGMENT
LITERATURE CITED
(1) Cacace, F., Inam-C1-Hay, 131, 732 (1960).
- K/8)
Science
( 2 ) Johnson, H. W., Stross, F. H., AKAI.. CHEM.31, 357 (1959). (3) “Tables for Statisticians and Bio-
metricians,” Karl Pearson, ed., Part I, Table 11, Cambridge University Press, 1948.
( 4 ) Vandenheuvel, F. A . , S~POS, J. ANAL.CHEM.33, 286 (1961).
c.,
( 5 ) Vandenheuvel, F. A., Sipos, J. C., J . Chromatog. 10, 131 (1963).
RECEIVED for review December 5, 1962. Accepted May 23, 1963. Contribution Yo. 115, Animal Research Institute.
Hydrocarbon Gas Analysis Using Differential Chemical Absorption and Flame Ionization Detectors W. B. INNES, W. E. BAMBRICK, and A. J. ANDREATCH American Cyanamid
Co., Stamford,
Conn.
b Parallel chemical absorbers combined with dual flame ionization detectors are used to determine the olefinic, paraffinic, and aromatic constituents in hydrocarbon gas mixtures. Design of the analyzer and a description of its various modes of operation are given. Applications illustrating its use include chromatographic analysis of gases in which the olefinic and paraffinic components are separately recorded, continuous analysis of auto exhaust for olefinic and paraffinic content during cyclic operation, studies of the effect of engine spark timing on the paraffinic and olefinic content of exhaust gases, and determination of olefinic, paraffinic, and aromatic content of gasolines.
T
determination of the olefin hydrocarbon content is important in assessing the smog potential of automotive exhaust gases. Numerous workers (7, 16, 22, 23) have shown that the paraffinic and C r C s aromatic hydrocarbons are relatively inactive and can almost be disregarded as photochemical smog precursors. Altshuller has indicated (1-2) that the Cs+ aromatics need to be considered. The hexane-sensitized, nondispersive infrared hydrocarbon analyzer has a high response to parafins and a low response to unsaturates (3, 6, 13). Its sensitivity for the analysis of treated exhaust gas samples is marginal. Spectral interferences are encountered from NOz, HzO, and COz. Its use for measuring the mean smog potential of treated exhaust can therefore be questioned. The bromocoulometric (8) method for the determination of olefins in gaseous samples is based upon the measurement of the amount of bromine HE
1 198
ANALYTICAL CHEMISTRY
reacting with the double bonds in the olefin molecules. Although this method is applicable to auto exhaust gases, the speed of response is slow, the response varies for different olefins, and NOz interferes. Flame ionization detectors (3-5) have been used to determine the total hydrocarbon content of exhaust gases. They have sufficient sensitivity to measure the low concentrations found in treated exhaust, they are insensitive to inorganic gases, including CO, CO,, HzO, and oxides of nitrogen, and the response is linear up to high hydrocarbon concentrations. However, the olefins comprise only about 25% of the hydrocarbons emitted by automobiles. Therefore, this approach has limited value in asJessing smog potential. The methods described herein combine flame ionization detectors with selective chemical absorbers. Various hydrocarbon classes such as olefins plus acetylenes and paraffins plus aromatics are determined on a continuous, intermittent, or chromatographic basis.
By electrically subtracting the two signals, the resultant output can be used as a measure of olefin plus acetylene concentration. The 10-mv. outputs are electrically coupled (Figure 2) so that the output of each analyzer and the differential output can be recorded. ,4variable-gain potentiometer is placed in each output so that the electrical signals can be varied to give exact compensation. The outputs are recorded on a dual 10-mv. recorder, although individual recorders have also been employed Optimum performance of the analyzer for exhaust gas analysis requires: insensitivity to small change in flow of hydrogen or auxiliary air; stability of flame toward small changes in sample gas flow; linear response up to 5 mole yo hydrocarbon concentration; and insensitivity to small changes of the oxygen concentration in the sample gas. The following flow ranges to each analyzer appear to meet these requirements :
EXPERIMENTAL
Air and hydrogen flow (Figure 1) is controlled by pressure regulators a and b and capillary restrictors 2, 4, 5, and 6. Flow rates are measured with pressure gauges i and j . Sample gas flow is controlled by varying the forepressure (liquid bubbler level). Water is used in the bubbler, but high density 5:OjO saturated salt solutions-e.g., HgBrl 34% BaBrz with a specific gravity of 3.0-or mercury would be suitable for higher pressures. Sample flow is observed with a low-pressure gauge, k (0 to 30 inches of HzO), prior to restrictors 1 and 3, and may be controlled by needle valves A and B. Both time synchronization and equal instrument re3ponse for a given carbon concentration are required before vari-
Apparatus Design. Two flame ionization detectors (Carad Corp., Palo *klto, Calif.) are used because it is normally desired t o measure the concentrations of both olefins and total hydrocarbons. The analyzer is illustrated in Figure 1. (A detailed list of components is available from the authors.) For the analysis of olefinic hydrocarbons, the untreated gaseous sample is fed to one detector, while a gaseous sample which has been scrubbed through a HgS04-H2S04absorber is fed to a second detector. Thus the first detector has a signal proportional t o t,otal hydrocarbons, while the second corresponds to the saturate plus aromatic hydrocarbon content of the sample.
Hydrogen flow rate Sample flow rate Hydrogen-sample gas ratio Auxiliary air flow rate
+
30-70 cc./minute 20-50 cc./minute 1.0-1.5 500-1000 cc./minute
