Anal. Chem. 1987, 59, 1692-1695
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Effect of Sample-Charge Volume on the Sensitivity of Chromatographic Analysis Michal Roth* and Josef Noviik' Institute of Analytical Chemistry, Czechoslovak Academy of Sciences, 61142 Brno, Czechoslovakia
A simple model for the dependence of the sensitivity of chromatographic determination of the mass of analyte on the sample-charge volume Is suggested and discussed. The model applies to the situation In which the mass of analyte introducedInto the column is constant and independent of the sample-charge volume, Le., the concentration (mass/voiwne) of the analyte in the sample is inversely proportional to the sample-charge volume. The model makes it possible to quantify the gain in the sensltivlty of the analyte-mass determination brought about by Concentrating the sample. Two limiting column-inlet analyte-concentration profiles are considered, viz., rectangular and exponentlal-decay concentration profiles. The theoretical results compare favorably with the outcome of a simple experimental test of the model.
In trace analysis by column chromatography, it is essential to assess the effects of the individual experimental parameters on the sensitivity of chromatographic analysis (1-3). One of these parameters is the volume of the sample charge. In analytical practice, especially if trace amounts of analytes are t o be determined, it is often necessary to apply a sampleenrichment technique prior to the chromatographic run proper. The sample-enrichment (i.e., concentrating) step may ideally be viewed as a reduction of the sample-charge volume while maintaining a constant mass of the analyte in the sample. Although the sample-enrichment techniques are applied routinely, little attention seems to be paid to quantitative evaluation of the resulting improvement in the sensitivity of analysis. In other words, if the sample is concentrated n times, what will be the corresponding increase in the sensitivity of analysis? In the present paper, a general model for the dependence of the sensitivity of analyte-mass determination on the volume of the sample charge is described and tested. The aim is to develop a model that can readily be applied with minimum computation involved. Any theoretical model is necessarily idealized; the validity of the present model is qualified by the following conditions: (i) Along the entire path of zone migration in the column, the zone-maximum analyte concentration is within the Henry-law region. (ii) The zone-maximum analyte concentration in the mobile phase a t the column outlet is within the region of dynamic linearity of the detector response. (iii) The intracolumn and extracolumn zone-spreading factors operate independently of each other. (iv) The nonzero volume of the sample charge is the only cause of the extracolumn zone spreading. (v) The peak distortion due to the fact that a record is provided of analyte concentration as a function of time a t the column outlet rather than that of analyte concentration as a function of position in the column a t a given time is negligible. 'Deceased April 6, 1986.
(vi) The peak distortion due to nonzero time constants of the detection and signal-recording system is negligible. Provided that the states of aggregation of the sample and the mobile phase are the same (gas-gas, liquid-liquid, or fluid-fluid), the treatment described below applies to any type of column chromatography.
SENSITIVITY OF ANALYTE-MASS DETERMINATION The sensitivity of chromatographic determination of the mass of analyte, i.e., the efficiency with which the chromatographic column-detector system responds to the mass of analyte introduced into the column, can be expressed as the peak-maximum response per unit mass of analyte. The sensitivity criterion so defined is given by the product of the sensitivity of detection toward a given analyte 1 and the zone-maximum analyte concentration per unit mass of the analyte in the mobile phase at the column outlet (F,",
= dR,*/dm, = (dR,*/dq,*)(dq,*/dm,) = v,,(dq,*/dm,) (1)
where m, is the mass of analyte within the zone, q,* is the maximum concentration of the analyte in the column effluent, and R,* is the detector response corresponding to qr*. Provided that the maximum concentration of analyte i in the column effluent is within the range of dynamic linearity of the detector response (cf. assumption ii above), the sensitivity of detection toward analyte i (p,J is constant, so that the sensitivity of analyte-mass determination, pml,is directly proportional to the differential quotient (dq,*/dm,). In the development of the model, it is therefore sufficient to consider the relationship between the sample-charge volume and the quotient (dqr*/drn,).
COLUMN-OUTLET ANALYTE-CONCENTRATION PROFILE There are several ways to express the dependence of the column-outlet analyte concentration on time and/or volume of the mobile phase passed through the column. If the column-outlet concentration profile results from combined actions of several mutually independent effects, it is expedient ( 4 ) to use the Laplace transform or the convolution integral. The convolution integral, especially the exponentially modified Gaussian function, has often been applied to the study of the effects on the peak shape of the extracolumn zone-broadening factors (5-16). Under the above conditions (1-vi), a hypothetical zerovolume injection of a mass m, of analyte 1 into the column results in a Gaussian column-outlet concentration profile of analyte 1 , qrc( V). If the mass m, of analyte i enters the column within an inlet chromatographic zone with a concentration profile q,,J V),the resulting column-outlet concentration profile of the analyte, q Lap( V), is given by the convolution integral
where V is the volume of the mobile phase passed through the column, and V'is a dummy variable of integration. The 0003-2700/87/0359-1692$01.50/0'01987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 13, JULY 1, 1987
1693
?mi, a,p ?mi. c
0
Flgure 1. Dependence of the shape of concentration profile q,,ap(V) (eq 6) on the reduced volume of the sample charge at constant values of m,and u,: (1) a = 0,(2) a = 1, (3) a = 2, (4) a = 4, (5) a = 8.
value V = 0 corresponds to the start of introducing the sample into the column. Since both qi,J V) and qi,J V) include the mass mi of the analyte (see below), the preintegral factor l / m i is required in eq 2. Provided the intracolumn and extracolumn zone-broadening factors operate independently of each other, the variances of the three concentration profiles are related by
In the following part of the theoretical section, relationships are derived between the sensitivity of analyte-mass determination and the sample-charge volume for rectangular and exponential-decay column-inlet concentration profiles. These two particular column-inlet profiles were chosen because, from a practical viewpoint, they represent the best and worst possible alternatives of the sample introduction into the column, respectively. For each of the two profiles, the maximum concentration of the analyte in the column effluent is derived, and the corresponding sensitivity of the analyte-mass determination is expressed as a fraction of that pertaining to a hypothetical zero-volume injection of the sample. All the retention volumes mentioned below refer to the elution of the maximum concentration of the analyte in the chromatographic zone. DEVELOPMENT OF THE MODEL Rectangular Column-Inlet Concentration Profile. If the sample containing a mass mi of analyte i enters the column by plug flow from a nonmixed sample loop of a volume u, the column-inlet concentration profile of the analyte is given by for 0 < V < u qi,e(V)= mi/u =0 otherwise (4) In the case of gas chromatography, the volume u should refer to the selected reference values of temperature and pressure. The Gaussian concentration profile is described as qi,c(V) = m i / ( u c ( 2 ~ ) ” ~e) x p W - v,,J2/2u,2I
= m i / 4 Q [ ( V - V R ,~ u)/ucl - Q[(V -
V~,c)/~cll
(6) where Q(u)is the area under the normalized normal distribution curve between u and m
-Q(u) = 1 / ( 2 ~ ) ~ exp{-y2/2) / ~ 1 ~ dy
2
3
5
4
0
Flgure 2. Dependence of the sensitivity of anaiyte-mass determination on the reduced volume of the sample charge: (curve 1) rectangular inlet profile, Gaussian outlet profile: (curve 2) exponentialdecay inlet profile, Gaussian outlet profile; (curve 3) rectangular inlet profile, convoluted outlet profile; (curve 4) exponential-decay inlet profile, convoluted outlet profile. Data points show the results of the experimental test of the model.
Figure 1 shows the shapes of column-outlet concentration profiles (eq 6) corresponding to elution of a constant mass mi of analyte i contained in different sample-charge volumes. All calculations in this paper were performed with hand-held calculators and a polynomial approximation for Q(u) (17). To proceed further, it is necessary to express the maximum concentration of the analyte in the elution zone, Le., when V = VR,ap. The retention volume VR,apis given by (18, 19)
so that the maximum concentration of analyte i in the column effluent is
Qi,ap* = qL,ap(VR,ap) = mi/Uil - 2Q[~/(2ac)11
(8)
Finally, combining eq 1, 5 , and 8 yields vmi,ap/vmr,c
= qi,ap*/Qi,c* = ( 2 ~ ) ~ ” / a-( l2Q(a/2)1
(9)
where a = U / U , is the reduced volume of the sample charge, ( P ~ , , is ~ the sensitivity of the analyte-mass determination corresponding to a hypothetical zero-volume injection of a mass m, of analyte i, and is the sensitivity corresponding to plug flow of a sample containing the mass m, of the analyte into the column from a nonmixed sample loop of volume u. It is apparent from eq 9 that the sensitivity ratio ‘P~~,~,,/‘P,,,~,~ depends on the reduced volume of the sample charge only (see curve 3 in Figure 2). The sensitivity ratio can also easily be calculated on a simplifying assumption that, regardless of the sample-charge volume, the column-outlet concentration profile of the analyte remains Gaussian throughout, with a variance given by eq 3. In such a case
(5)
where VR,c is the retention volume of analyte i resulting from a hypothetical zero-volume injection of the sample. By combination of eq 2, 4, and 5, the column-outlet concentration profile of analyte i can be written as qi,ap(V)
1
pmi,ap/pmi,c
= 1/(1+ pa2)’”
(10)
where p = 1/12 for the rectangular column-inlet concentration profile. The result of this simplified treatment is shown by the dashed curve 1 in Figure 2. Exponential-Decay Column-Inlet Concentration Profile. If the sample containing a mass m, of analyte i enters the column from an ideal mixing chamber of a volume u, the column-inlet concentration profile of the analyte is given by
q,,e(V) = m,/u exp{-V/u) =0 otherwise
for V
>0 (11)
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 13, JULY 1, 1987
a simplified calculation based upon the assumption that the column-outlet concentration profile remains Gaussian throughout, regardless of the volume of the sample charge. Curve 2 in Figure 2 is described by eq 10 with p = 1.
EXPERIMENTAL SECTION
I Figure 3. Dependence of the shape of concentration profile 9,,ap(V ) (eq 12) on the reduced volume of the sample charge at constant values of m,and u,: (1) a = 0,(2) a = 1, (3) a = 2, (4) a = 4, (5) a = 8.
Substitution into eq 2 from eq 5 and 11 yields after rearrangement (20)
qi,ap(V)= m i / ( u ~ l I 2 )exp(-u,2/2u2) e ~ p ( - u , Z 2 ~ ~ ~x/ u )
1:
exP(-Y21 dY
= mi/u exp(-ac2/2u2) e ~ p ( - - a , Z 2 l / ~ / u ) [l Q(z2li2)1
(12) where
Z = 1/2ll2[(V - VR,c)/uc - l / a ] In Figure 3, the shapes are shown of the column-outlet concentration profiles (eq 12) corresponding to elution of a constant mass m iof the analyte contained in different samplecharge volumes. Similarly as with the rectangular column-inlet concentration profile, it is necessary now to express the maximum concentration of the and@ in the mobile phase at the column outlet. For the exponential-decay inlet profile, however, there is no parallel to eq 7 since the relationship between V R ,and ~ ~ VR,, cannot be expressed analytically. In order t o calculate VR,ap and qi,ap*,Barber and Carr (10) fitted a second-order Gram polynomial to the points near the apex of the concentration profile. In the present paper, a different route is followed; setting the d / d V derivative of eq 12 equal to zero, it can be shown that, in the concentration maximum Z*
l / a x m exp(-y2) d y = 1/21/2 exp{-Z*2)
(13)
where
Z* = 1/2l/'(Y - 1 / a ) , Y = Since
xr
(VR,ap - VR,,)/uc
exp{-y2} d y = r1/2[1 - Q(Z*21/2)]
(14)
the reduced concentration-maximum shift, Y , can easily be calculated from eq 13 for any value of the reduced samplecharge volume, a, using the Newton-Raphson iteration method and a proper numerical technique for calculating Q(u). Substitution for the integral in eq 12 from eq 13 gives Qi,ep* = qi,ap(VR,ap) = mi/(Gc(2~)'/~)e x p ( - P / 2 ) (15) Combining eq 1, 5, and 15 yields finally pmi,ap/pmi,c
=~
XP~Y/~I
(16)
The dependence of the sensitivity ratio pmi,ap/~mi,c on the reduced volume of the sample charge is shown graphically by curve 4 in Figure 2. Curve 2 in Figure 2 shows the result of
Apparatus. A simple experimental test of the above model was carried out with a CHROM 5 gas chromatograph (Laboratory apparatus, Prague, Czechoslovakia)equipped with a flame ionization detector and modified to increase the precision of the measurement of retention data. Provided the mass flow rate of the carrier gas is kept constant and the analyte concentration in the carrier gas is very low, the response of the flame ionization detector is directly proportional to the analyte concentration in the column effluent. A glass column 2.5 m X 3 mm i.d. was packed with 5% (w/w) poly(dimethylsi1oxane)DC 200 (Packard-Becker B.V., Delft, The Netherlands) coated on ChromosorbG AW 30/60 (Carlo Erba, Milan, Italy). The column temperature was kept at 30 "C by means of a water jacket through which the water from an external thermostat (type U15C, VEB MLW Prufgerate-Werk, Medingen, GDR) was circulated. The volumetric flow rate of the nitrogen carrier gas was 5.3 cm3/min (25 "C, 0.97 bar); it was monitored continuously by a precolumn differential capillary flowmeter. The column inlet pressure was not higher than 0.09 bar above the barometric pressure. n-Pentane (p.a. grade, VEB Laborchemie,Apolda, GDR) was employed as a testing substance. Procedure. It is impossible to determine the standard deviation uc exactly. Therefore, an approximation was obtained from repeated injections of 50-pL samples of n-pentane vapor using a gas-tight syringe (type 1001, Hamilton, Bonaduz, Switzerland). The peaks obtained were nearly symmetric. The standard deviations were calculated from chart records assuming the peak shape was Gaussian. Twenty-two injections yielded the value u, = (0.795 f 0.019) cm3at 25 "C and 0.97 bar. Therefore, a 50-pL sample corresponded to a = 0.06. When the dependence of the peak height on the volume of the sample charge was tested, the injections were made from a hypodermic syringe with a stainless steel plunger. This particular type of syringe was employed in order to avoid the "memory effect" exhibited by the Teflon-coated plungers of gas-tight syringes;with pentane vapor, the effect is appreciable. A 50-pL sample of pentane vapor was diluted with air to yield the volume required. Before an injection, the sample was allowed to equilibrate in the syringe for a few minutes. The maximum volume of the sample charge was 2.8 cm3 (a = 3.6). Although a glass syringe with a metallic plunger was employed, it was impossible to maintain a constant mass of pentane in the sample throughout the procedure. Therefore, the peak areas were evaluated by use of a CI 100 computing integrator (Laboratory apparatus, Prague, Czechoslovakia), and the peak heights were normalized to a unit peak area. The results of the test are represented by the points in Figure 2. As can be expected, all experimental points fall into the region between the curves 3 and 4. When the volume of the sample charge is increased, the experimental points converge toward the curve derived for a rectangular column-inlet concentration profile. At the same time, however, the peaks showed increasing amounts of tailing. This apparent contradiction results from the fact that a syringe was employed to introduce the samples into the column and that the instrument was operated at a constant flow rate of the carrier gas. In such a case, an increased volume of the sample charge is bound to cause an increased pressure surge on injection. Therefore, when the volume of the sample charge is increased, the effective value gradually becomes smaller than the apparent one.
RESULTS AND DISCUSSION The model has been derived showing the dependence of the sensitivity of analyte-mass determination on the volume of the sample charge for two limiting column-inlet concentration profiles of the analyte. The word "limiting" means that the two column-inlet profiles represent the best and worst possible cases of the sample introduction into the column. Since the final eq 9 and 16 are written in terms of dimensionless (reduced) quantities, they apply generally to any column chro-
Anal. Chem. 1987, 59, 1695-1700
matography system as long as the qualifying conditions (see introductory section) are satisfied. In other words, in any system of column chromatography and for any volume of the sample charge, the sensitivity of analyte-mass determination should range between the “extreme” values given by eq 9 and 16 (or by curves 3 and 4 in Figure 2). Although the reducing parameters u, and v,,,,,~ cannot be determined exactly, fair approximations to these may be obtained from a chromatogram of the smallest possible sample charge. Once the volume uc is known, the increase in the sensitivity of analyte-mass determination on concentrating the sample may readily be calculated. T o do so, eq 9 or eq 13 and 16 are applied to the volumes of charges of the concentrated and original samples, respectively. Naturally, the volumes are concerned containing the same amounts of analyte. For the model to be applicable directly, the state of aggregation of both concentrated and original samples should be the same as that of the mobile phase. The increase required is then given by the ratio of the two results. The choice between the two column-inlet profiles depends upon the performance characteristics of the particular sample-introduction device employed. In principle, relationships analogous to eq 9 and 16 may be derived for any other column-inlet concentration profile. It may be shown is that for large values of a, the sensitivity ratio pmi,ap/pmi,c nearly equal to (2a)’i2/a regardless of the column-inlet concentration profile. The use of eq 9,13, and 16 may be managed even with a programmable pocket calculator. Finally, a brief evaluation should be given of the relative significance of qualifying conditions (i-vi) (see introductory section). Obviously, the most severe limitations to the practical applicability of the model are those imposed by conditions iv and vi. The adherence of the behavior of a real instrument to the model described improves on decreasing the dead volume of the chromatographic system and on de-
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creasing the time constants of the detection and registration system. For very slightly retained compounds and/or for very quick analyses, condition v may impose significant restriction to the applicability of the model.
LITERATURE CITED Kaimanovskii, V. I.; Zhukhovitskii, A. A. J . Chromatogr. 1965, 18, 243-252. Karger, B. L.; Martin, M.; Guiochon, G Anal. Chem. 1974, 4 6 , 1640-1647. Guiochon, G.; Colin, H. “Analytical Techniques in Environmental Chemistry”;Proceedings of fhe 2nd International Congress, Barcelo na, Spain, November 1981; Aibaiges, J., Ed.; Pergamon: OxfordNew York-Sydney, 1981; pp 169-176. Sternberg, J. C. Advances in Chromatography;Giddings, J. C., Keiier, R. A., Eds.; Marcel Dekker: New York, 1966; Voi. 2, pp 205-270. McWiiiiam, I . G.; Boiton, H. C. Anal. Chem. 1969, 4 1 , 1755-1762. Giadney, H. M.; Dowden, B. F.; Swaien, J. D. Anal. Chem. 1969, 4 7 , 883-888. Anderson, A. H.; Gibb, T. C.; Littiewood, A. B. J . Chromatogr. Sci. 1970, 8, 840-646. Grushka, E. Anal. Chem. 1972, 4 4 , 1733-1738. Pauis, R. E.; Rogers, L. B. Anal. Chem. 1977, 4 9 , 625-628. Barber, W. E.; Carr, P. W. Anal. Chem. 1961, 53, 1939-1942. Foiey, J. P.; Dorsey, J. G. Anal. Chem. 1963, 55, 730-737. Foley, J. P.;Dorsey, J. G. J . Chromatogr. Sci. 1984, 22, 40-46. Anderson, D. J.; Waiters, R. R. J . Chromatogr. Sci. 1984, 22, 353-359. Deiiey, R. Chromafographia 1984, 18, 374-382. Hanggi, D.; Carr, P. W. Anal. Chem. 1985, 57,2395-2397. Deiiey, R. Anal. Cbem. 1986, 5 8 , 2344-2346. Handbook of Mathematical Functions; Abramowitz, M., Stegun, I. A., Eds.; National Bureau of Standards: Washington, DC, 1964; Applied Mathematics Series No. 55, p 932. van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Cbem. Eng. Sci. 1956, 5,271-289. Porter, P. E.; Deai, C. H.; Stross, F. H. J . Am. Chem. SOC.1956, 78, 2999-3006. Gradshtein, I. S.; Ryzhik, I. M. fablitsy Integralov, Summ, Ryadov i Proizvedenii (fables of Integrals, Summations, Series, and Products), 4th ed.; Fizmatgiz: Moscow, 1982; p 321, formula 3.322.
RECEIVED for review October 10, 1986. Accepted March 3, 1987.
Identification of Mutagenic Methylbenz[ a ]anthracene and Methylchrysene Isomers in Natural Samples by Liquid Chromatography and Shpol’skii Spectrometry Philippe Garrigues,*’ Marie-Pierre Marniesse,’ Stephen A. Wise: Jacqueline Bellocq,’ and Marc Ewald’ Groupe d’OcCanographie Physico-chimique, LA 348 C N R S , Universitd de Bordeaux I , 33405 Talence Cedex, France, and Organic Analytical Research Division, National Bureau of Standards, Gaithersburg, Maryland 20899
Chromatographic extracts of natural samples (rock and air partlculate matter) have been examlned by high-resolution Shpoi’skll spectrometry (HRS) at 15 K In n-alkane polycrystalline frozen solutlons for the ldentificatlon of the 12 methylbenr[a ]anthracenes (MBA) and the six methylchrysenes (MC). This Is the flrst report on the unamblguous identlflcation of each MBA Isomer in real samples which wlll provide a better understanding of carcinogenic potency and further quantlflcation of these compounds in tetraaromatic fractlons.
Polycyclic aromatic hydrocarbons (PAH) and their alkylated derivatives are well recognized as ubiquitous contamiUniversitB de Bordeaux. 2National Bureau of Standards.
nants of the environment. The major analytical problem in the determination of PAH in complex natural mixtures is the separation and the identification of individual components in the presence of the numerous other isomeric parent and alkyl-substituted PAH. Since the biological activity of aromatic compounds is isomer specific, the identification of each compound in an alkylated aromatic series is a vital part of understanding the carcinogenic activity of PAH mixtures. Methylbenz[a]anthracenes (MBA) and methylchrysenes (MC) are among the most biologically active alkylated aromatic series found in man’s environment (Figure 1) (1-6). There are 12 possible isomers in the MBA series which vary significantly with respect to carcinogenicity(Figure 1). 7-MEiA has been recognized as the most tumorigenic compound, followed by 6-, 8-, and 12-MBA, which are of equal carcinogenicity, while 9- and 11-MBA are the next most carcinogenic compounds. The low tumorigenicity of the 1-,2-, 3-, and 4-MBA has been generally cited in support of the bay region
0003-2700/87/0359-1695$01.50/0 0 1987 American Chemical Society
