F. S. Karn, R. A. Friedel, and A. G. Sharkey, Jr., Carbon, 5 , 25 (1967). J. L. Shultz and A. G. Sharkey, Jr., Carbon, 5 , 57 (1967). F. S. Karn and A. G. Sharkey, Jr., Fuel, 47, 193 (1968). F. S. Karn and J. M. Singer, Fuel, 47, 235 (1968). F. S. Karn, R . A. Friedel, and A. G. Sharkey, Jr., Fuel, 48, 297 (1969). F. S. Karn, R. A. Friedel, and A. G. Sharkey, Jr., Fuel, 5 1 , 113 (1972). A. G.Sharkey, Jr., A. F. Logar, and F. S. Karn, Fuel, 48, 95 (1969). F. S. Karn, R. A. Friedel, and A. G. Sharkey, Jr., Chem. lnd., No. 7 , 239 (1970). R. L. Hanson, N. E. Vanderborgh, and D. G. Brookins, Anal. Chem.,47,335 (1975). N . E. Vanderborgh, R. L. Hanson, and C. Brower, Anal. Chem., 47, 2277 (1975). J. P . Biscar, J. Chromatogr.,56, 348 (1971).
(22) R. L. Hanson, “Laser Pyrolysis Gas Chromatography and Plasma Stoichiometric Analysis”, Ph.D. dissertation, The University of New Mexico, August 1975. (23) W. T. Ristau, “Analytical Aspects of Laser-Induced Degradation of Organic and Inorganic Compounds”, Ph.D. dissertation, The University of New Mexico, June 1971.
RECEIVEDfor review September 23,1976. Accepted December 6, 1976. This work is taken from the Ph.D. dissertation of R.L.H., The University of New Mexico, 1975, who received support from the Coal Gasification Project a t The University of New Mexico.
Characterizing Skewed Chromatographic Band Broadening W. W. Yau E. 1. du Pont de Nemours and Company, Central Research and Development Department, Wilmington, Del. 19898
A new method of extracting column band broadening parameters from a skewed and noisy chromatographic peak is derived from an exponentially modified Gaussian-peak model. The method Is more accurate than the James-Martin method which uses a simple Gaussian-peak model and Is more precise, Le., less susceptible to baseline noise, than the common statistical moments approach. Characteristic properties of the skewed peak model are derived. These are then used to eliminate the reliance of the peak variance and skewness calculations on the 2d and 3d moments. The improved calculations depend only on the more stable, less noise influenced, zeroth and 1st moment calculations. The computatlonal algorithm of the method is slmpler and more straightforward than the curve fitting approaches.
A precise characterization of column band broadening in terms of peak shape parameters (variance and skewness) is needed for quantitative interpretation of chromatographic data. The information is used to evaluate and optimize column resolution and efficiency. Functionally, a chromatographic peak is simply a time distribution of the chromatographic height h ( t )at any retention time, t . The statistical moments of the peak are mathematically defined as: the zeroth order, the first order,
M O= MI =
Sornt
h ( t )dt
(1)
h ( t )dtlMo
(2)
and the higher order central moments,
a,,= im ( t - M I ) % @ )dtlMo
and
(8)
In principle, chromatographic peaks can be characterized by the moment calculations. However, in practice the precision of these calculations is poor. The baseline noise in actual, experimental chromatograms can greatly affect the precision of the higher order moments calculated by Equation 3. The commonly used James-Martin Method (1) uses the property of an assumed Gaussian peak model to calculate the peak variance. The method is based on Equation 9, 1 variance = - (area/peak height)2 2*
(9)
The variance calculated by Equation 9 uses the peak area, or the zeroth order moment, and is therefore much more precise (less influenced by noise) than that calculated as the 2d moment using Equation 3. While the precisidn of the area-height method is good, the accuracy of the method is poor. First, the method does not provide for calculation of peak skewness and, second, for highly skewed peaks, the method tends to grossly underestimate the true peak variance. Thus to improve the characterization of chromatographic peaks, a more general peak shape model which permits describing skewed peaks is needed. The exponentially modified Gaussian model (2) has been chosen in this work for this purpose. The model is justified both theoretically and experimentally since it is known that the chromatographic “Gaussian” peaks are modified exponentially by extra column effects and nonequilibrium mass transfer processes ( 3 , 4 ) .The model contains only four peak shape parameters, just one more than the simple Gaussian model. The peak contour of this model is described by the following convolute integral:
(3)
where n = 2 , 3 , 4 , . . . . The statistical moments of a peak are related to the peak shape parameters as defined by Equations 4-8. peak area = Mo
(4)
average retention time = M I
(5)
peak variance = M2
(6)
peak skew =
peak excess = M4/M22 - 3
(7)
where the four essential peak shape parameters are: A , the area, 7,the time constant of the exponential modifier, tR, the peak retention time, and u the standard deviation of the Gaussian constituent. The quantity t’ in the equation is a dummy variable of integration. In Figure 1 the shape of the convoluted peak is compared to its Gaussian constituent. As shown in the figure, the peak maximum of the convoluted peak falls on the contour of the Gaussian constituent. This property of this peak shape model can be expressed mathematically as the following: ANALYTICAL CHEMISTRY, VOL. 49, NO. 3, MARCH 1977
395
where the new symbol hR is the height of the skewed peak a t the time t R , and A - is that area of the peak which elutes before the time tR, i.e., A- =
L
c
J v)
H
(19)
st"
M
w
a a
GAUSSIAN
0
I 1 I
TIME, 1
Figure 1. Peak shape models
.
where h, and t, are the peak height and peak retention time of the convoluted skewed peak, respectively. Other properties of this peak model are summarized in Equations 12-15. Ml=tR+r M2
= a2
M3
+ r2
= 2r3
(12)
(13) (14)
and Mq = 3u4
+ 6u2r2+ 9r4
(15)
In order to use Equations 12-15 it is necessary to extract the u and T value from an experimental chromatographic peak. One method for doing this has been to curve fit Equation 10 to the experimental elution peak (5).This approach uses extensive computerized iterative search calculations and on occasion leads to erroneous results because of convergence into false minima. Another way to obtain u and r is to calculate the statistical moments up to the 3d moment followed by the use of Equation 14 to solve for r and then Equation 13 to solve for u. However, since M2 and M3 are used to obtain u and r , this method has the same precision problem for the peak variance and peak skewness as the common moments method. A new and more precise method of extracting u and T is described below. This method uses additional properties of the peak model, the derivation of which is presented in the mathematical derivation section following the description of the method.
DESCRIPTION OF THE METHOD The basic formulations of the method are: = M I - tR
(16)
= (AI2 - A)/hR
(17)
T T
and
*
h ( t ) dt
t - is the average retention time of the first part of the peak, as defined by Equation 20. For mathematical convenience, the lower integration limit in Equations 19 and 20 can be changed to - m without practically affecting the numerical values of the A - and t - because invariably h i t ) will be insignificantly small a t t < 0.
-
396
JtR
ANALYTICAL CHEMISTRY, VOL. 49, NO. 3, MARCH 1977
t- = t h(t) dt (20) A- o One may note that Equation 16 is simply a rearrangement of Equation 12, while Equations 17 and 18 are the new relationships-derived in the mathematical section later. In order to calculate u and T it is necessary to determine t~ first. This is accomplished using the following algorithm. Starting at t equal to zero, the time t is incremented and as. this tR, the two T values are signed temporarily as t ~With calculated from both Equations 16 and 17 and compared. The time t is incremented until the T value from Equation 17 becomes smaller than that from Equation 16. Then, the time is decremented by one unit and this unit is subdivided into 100 smaller units by linear extrapolation of the two data points. The calculation for r is then carried out across these 100 units until Equation 17 again gives a smaller r value than Equation 16. The value of the time t then is formally assigned to be t ~ . This algorithm is easily handled by a small computer. With tR determined, u and T values are calculated directly from Equations 16 and 18. Once the u and T values are known, the variance, skew, and excess are calculated from Equations 6 to 8 and 13 to 15. Since this calculation approach uses only the zeroth and first moment equations (calculations which can usually be done with good precision), this new method should give more precise variance, skewness, and excess values than the common moments method. The calculated values using this method should be less affected by baseline noise and the uncertainty in assignment of the peak start and the peak end in the calculations. The algorithm given above is just one way of solving Equations 16 and 18. With large random noise, this algorithm is expected to overestimate T and underestimate t ~ Im. provement of the algorithm should be possible. The validity of this method recently has been demonstrated on experimental chromatograms (6). MATHEMATICAL DERIVATIONS The precision obtained in peak characterization is inversely related to the mathematical order of the statistical moments required to specify fully the peak shape. One way to improve precision is to reduce the order of the required moment calculation. This is made possible by assuming a peak-shape model as is the case for the James-Martin method. Methods involving simpler peak-shape models are more precise but usually are less accurate. With a peak model, statistical moments can be expressed in terms of the basic peak-shape parameters of the model. These relationships are the properties of the model and are utilized in the calculation method for reducing the order of the required moment calculations. The most useful properties of the peak model are those which relate the peak-shape parameters to quantities that are precisely measurable, such as peak area, peak height, peak retention time, or even the average retention time. In the search for additional useful properties of the exponentially modified Gaussian-peak model, we decided to ex-
amine the section of the skewed peak that eluted before the time t ~where , t R is the peak retention time of the Gaussian constituent (see Figure 1).The section in question is shaded in Figure 1.First, we established the height of the skewed peak a t t R by letting t = t~ in Equation 10:
La
A hR = 7av%
exp
[-
-
51
dt'
(21)
Then we examined the area A- and the average retention time t - of the shaded section. This led us to the additional expressions for 7 and a as described in the following. Derivation of the T Expression, Equation 17. This expression was derived as a result of the examination of the area A - as defined by Equation 19. By substituting h ( t ) from Equation 10 into 19 and changing the order of integration between t and t', it was shown that: A
A- =
Jm
~
7a
d z
exp
[-:I
[
exp -
(-$-)2
-
F]
Substituting Equation 21 into 23 and rearranging terms, Equation 17 was obtained. By letting A+ = A - A-, Equation 17 becomes equivalent to: A+ - A T =
2hR
Derivation of the u Expression, Equation 18. This expression was derived as a result of the examination of the average retention time t- as defined by Equation 20. By substituting h ( t ) from Equation 10 into 20 and changing the order of integration between t and t', we obtained: A A - 7 ~ 6 X
lrn [ 1'" [- ( -4% - ")'I ] t exp
0
tR
dt
-m
One can show that the integral in the bracket of Equation 25 can be reduced to the following:
st".
. .dt]
=-1/2;;a(tR+t')erfc
2
The integral in the bracket of Equation 27 can be shown to be:
[J= . . . dt'] Substituting Equation 28 back into 27 and rearranging terms, yields Equation 29.
(A)
-u2exp
where t + is the average retention time of the other part of the peak with t > tR. With Equation 31, one can show that Equation 18 can be put in the following alternate form:
dt'] (23)
-m
(27)
which led directly to Equation 18 shown before. Since M I is the center of gravity of the entire peak, one can write:
which represents the complementary error function. Following the integration by parts, Equation 22 becomes:
[
[- t]dt'] - cr2hR/A-
or
where
t- =
X exp
[-(&)']
(26)
Substituting Equation 26 back into 25 and using the relationships established in Equations 21 and 23, yields Equation 27.
u = &(t+
A+AA2
- t-) -
(32)
DISCUSSION Peak broadening affects the accuracy of calibration and molecular weight calculations in gel permeation chromatographic analysis (7). In a method developed by us, we have considered the correction for a Gaussian (a) type of band broadening (8).The work to extend our method to include the T correction is forthcoming. The ability to isolate the individual u and 7 contributions to band broadening is helpful for identification of the causes of chromatographic dispersion. It can be utilized in studies of basic dispersion mechanisms and in the practical development of chromatographic columns and instrumentation. Processes which lead to symmetrical (a) dispersion, for example, include eddy and longitudinal diffusion. Extracolumn effects, channeling in the column packing, and slow lateral diffusion tend to cause skewed ( 7 ) type dispersions. The skewness of a peak increases with increasing T , as exemplified by the additional T occurring because of the finite response time of a recorder. In this case, the added 7 increases the peak (t,) and the average retention time ( M I )of the peak but does not change the t~ value. The increase in skewness can also be caused by changes in operating conditions, such as increasing flowrate or solvent viscosity, or decreasing column temperature. In these cases, the center of gravity of the peak becomes the invariant, while both t , and t~ move to shorter retention time. Theoretical curves to simulate this latter skewing mechanism would have the same first moment, M I . They can be generated by replacing t~ by ( M I - T ) in Equation 10 and keeping MI constant. (For the case of changing flowrate, it is understandable that tR, t,, and M I should be treated in the retention volume bases). ANALYTICAL CHEMISTRY, VOL. 49, NO. 3, MARCH 1977
397
ACKNOWLEDGMENT The author is grateful to D. D. Bly and J. J. Kirkland for helpful discussions in aspects of this work. LITERATURE CITED (1) A. T. James and A. J. P. Martin, Analyst(London), 77, 915 (1952). (2) E. Grushka, Anal. Chem., 44, 1733 (1972). (3)J. C.Sternberg, "Advances in Chromatography," Vol. 2, J. C.Giddings and R. A. Kelier, Ed., Marcel Dekker, New York, N.Y., 1966.
(4) H. M. Gladney, B. F. Dowden. and J. D. Swalen, Anal. Chem., 41, 883 li9fi91 \
(5) A. H. Anderson, T. C.Gibb, and A. B. Littlewood, J. Chromatogr. Sci., 8,640 (1970). (6) R. E. Pauls and L. B. Rogers, Anal. Chem., 49, in press. (7) A. E. Hamielec, J. Appl. Polym. Sci., 14, 1519 (1970). (8)W. W. Yau, H. J. Stoklosa, and D. D. Biy, J. Appl. Polym. Sci., in press.
for review October 7, lg76*Accepted November 23, 1976.
Immiscible Solvent Extraction Scheme for Biodegradation Testing of Polyethoxylate Nonionic Surfactants D. H. J. Anthony and I?.S. Tobin* Analytical Methods Research Section, Canada Centre for Inland Waters, P.O. Box 5050, Burlington, Ontario, Canada, L 7R #A6
An Immiscible solvent extraction scheme for the Isolation of polyethoxylate nonionic surfactants and their polyglycol biodegradation products is described. Surfactant is extracted from biodegradation cultures with benzene. Surfactant and polyglycol material are extracted by salting-out into chloroform. The two species are recovered separately In a sequential extraction procedure.
The ease of destruction of surfactant properties by microorganisms has been a major concern in the development of methods for evaluating nonionic surfactant biodegradability. Many extraction procedures used in nonionic surfactant biodegradation experiments have therefore been designed to selectively isolate the parent surfactant material (1-12). An expanding knowledge of the subtlety of environmental impact demands that biodegradability testing must include observation of the appearance of transient or recalcitrant products as well as the disappearance of a parent compound. For nonionic surfactants, some of the techniques mentioned above are inadequate in this respect because of their selectivity for the parent surfactant; others are time consuming or poorly described quantitatively or qualitatively. A technique described previously in a brief manner (13) has been improved and the results are presented here. This technique allows selective isolation of surfactant and of polyglycol biodegradation products (4,8,13,14) with high recoveries and good precision offering a useful alternative to isolation techniques previously used in nonionic surfactant biodegradation testing. The biodegradation of a nonionic surfactant is examined using this procedure.
EXPERIMENTAL Reagents. Chemical structures of the surfactants used in this study are provided in Table I; sources of these materials are listed below. Dobanol 25-9: Shell Research Laboratories, Egham, Surrey, England. lOEO (NPEOlo): Chemische Werke Huls, AkNonylphenol tiengesellschaft, Marl Kries, Recklinghausen, Germany. Lissapol NX: IC1 Ltd., Billingham, Teeside, England. Pluronic F68, L61, P65: BASF Wyandotte Corp., Wyandotte, Mich. Brij 52, 56, 58, and G3780A: Atlas Chemical Industries (Canada) Ltd., Brantford, Ontario, Canada. Varonic K202, L202, T202: Ashland Chemical Co., Columbus, Ohio. Alkylbenzene sulphonate, tetrapropylene derived (TBS): Associ-
+
396
ANALYTICAL CHEMISTRY, VOL. 49, NO..3, MARCH 1977
ation of American Soap and Glycerine Products Inc., New York, N.Y. Polyethylene glycols (PEG'S) were GLC liquid phase grade. Low molecular weight glycols, polypropylene glycols (PPG's), cationic surfactants, and analytical reagents were reagent grade materials. Pesticide grade solvents were employed. Analytical Method. The BIAS method (2-4) was chosen as the most precise routine method for the determination of trace amounts of long chain (>6 EO units) polyethoxylate nonionic surfactants and polyglycols. Analyses were performed as described by the O.E.C.D. ( 1 5 ) using a platinum-calomel combination electrode and a Radiometer semi-automated titration system to determine Bi3+ cation. Recovery factors ( R ) were evaluated by comparison of the BIAS response of extracted solute (1.0ml of 500 pg ml-I stock solution diluted to 100 ml and extracted) with that of a reference sample of unextracted solute (1.0 ml of 500 pg ml-' stock solution) after correction for blank values. Brij 52 and PPG-400 stock solutions were prepared in 25% aqueous methanol; all other stock solutions were aqueous. The precision of determination of solutes in reference samples by the BIAS method was typically S,,I = 0.03 (n = 3) when evaluated as described by Dean and Dixon (16). Development of Extraction Techniques. The extraction techniques described briefly by Tobin et al. (13)formed the starting point for this investigation. Extraction variables studied included solvent type, salting-out behavior, p H effects, relative solvent volume, and agitation time. Related aspects investigated included sample pretreatment, coextraction of interferences, and concentration of organic extracts. The techniques were developed using Dobanol 25-9 and PEG-400 as model substrates, then applied to other classes of nonionic surfactants and polyglycols. Biodegradation of Dobanol25-9. A 3-1. shake-flask biodegradation experiment using Dobanol25-9 as the carbon source was carried out as prescribed by the O.E.C.D. (15).Samples of appropriate volume were extracted by the recovery techniques developed and analyzed by the BIAS method.
RESULTS Standardized Extraction Technique. The final extraction procedures developed are described below. Extraction Method A. Adjust pH of sample, at room temperature, to the range of p H 7.0 f 0.5 with dilute NaOH or HCl. Transfer 100 ml of pH-adjusted sample to a 250-ml separatory funnel, add 0.5 g NaCl and shake to dissolve. Extract the sample with one 40-ml portion then three 20-ml portions of C&, mixing the layers vigoriously for 30 s each time. Carry-over of NaCl with the organic layer is minimized by allowing the phases to separate adequately. The vessel used to measure the volume of the sample is rinsed each time with the fresh C6H6 portion. Discard aqueous layer and rinse the
