Anal. Chem. 1981, 53, 1369-1372
of the solvent increases, dimunution of the coefficient in this sequence was observed. Next, let us consider constant c. The sums of c1 and c2 were calculated for all combinations of samples and were compared with the constant cI2of the corresponding disubstituted solutes. The results showed that the values of cI2were roughly equal to the sum of c1 and c2. In many cases, cI2was slightly larger than the sum of c1 and c2. This cannot be explained by theoretical consideration; however, it seems to be useful for the prediction of the retention behavior of the disubstituted solutes by using the data on the corresponding monosubstituted solutes, because an approximate value of cI2i s afforded simply by the addition of c1 and c2. In summary, the capacity ratio for disubstituted solutes can be expressed by the retention behaviors of the two corresponding monosubstituted solutes, the mole fraction of the stronger solvent in a binary system, and the average coefficient as log k'12 = (c1 + c2) - y(n1 + nz) log (4) In this work, a procedure simply using the mean values of constants c and n in eq 1 was employed and any selectivity due to interaction between the solute and the solvent mole-
1369
cules was disregarded. Therefore, eq 4 described above seemed not to be applicable to the precise prediction of the retention behavior for an individual sample. However, in spite of the ambiguity of this treatment, the practical procedure described in this article could be useful in order to optimize chromatographic systems using disubstituted solutes on the basis of the retentions of the two corresponded monosubstituted solutes.
LITERATURE CITED (1) Snyder, Lloyd R. "Principles of Adsorption Chromatography"; Marcel Dekker: New York, 1968. (2) Soczewinski, Edward Anal. Chem. 1969, 41, 179-182. (3) Snyder, Lloyd R. Anal. Chem. 1974, 46, 1384-1393. (4) Hara, Shoji; Fujii, Yumiko; Hirasawa. Mamiko; Miyamoto, Sayurl J . Cbromatogr. 1978, 149, 143-159. (5) Hara, Shoji; Ohsawa, Akiko J. Chromatogr. 1980, 200, 85-94. (6) b r a , Shoji; Hirasawa, Mamiko; Miyamoto, Sayuri; Ohsawa, Akiko J. Chromatogr. 1979, 169, 117-127. (7) b r a , Shoji; Nakahata, Masaaki J . Liq. Chromatogr. 1978, 1 , 43-54. (8) Hara, Shoji; Fukasaku, Noboru J. Org. Chem. 1979, 4 4 , 893-894. (9) Hara, Shoji J. Cbromatogr. 1977, 137, 41-52. (10) Hara, Shoji; Yamuchi, Noriko; Nakae, Chizuko; Sakai, Shinichiro Anal. Chem. 1980, 52, 33-38.
x,
RECEIVED for review August 11, 1980. Resubmitted January 30, 1981. Accepted April 27, 1981.
Quantitative Resolution of Severely Overlapping Chromatographic Peaks Jeffrey T. Lundeen and Richard S. Juvet, Jr." Department of Chemistry, Arizona State University, Tempe, Arizona 8528
A method is proposed to mathematically resolve severely overlapping chromatographic peaks. This method is successful even for the case in which the two peaks have identical retention as long as the peaks have different shape. Compounds with Identical chromatographic peak shape may also be analyzed if a slight dtfference exists in retention. The detector response for each component at various times is fit to a secondorder polynomial of the component'$ concentration. A series of nonlinear simultaneous equations are then solved for the concentration of each component in the mixture. Two computer simulation studies are presented.
The inability to completely separate component peaks becomes an increasingly troublesome problem as more complex systems are investigated chromatographically. The accurate, quantitative analysis of natural products, for example, in which many hundreds or thousands of components are typically present, is a challenge to the resourcefulness of the analytical chemist. The need for such analyses has led over the years to the development of high-resolution chromatographic columns, a detailed knowledge of methods for choosing the best stationary phase, and an intense study of the theory of chromatography which enables the knowledgeable chromatographer to choose the optimum experimental conditions to produce the highest separation efficiency possible for a given system. With the best of technique, however, unresolved peaks in complex mixtures commonly occur, greatly reducing the normal accuracy expected, or often completely preventing quantitative determination in the case of severely overlapping chromatographic peaks--those in which no shoulder or other indication of a mixed peak occurs.
If chromatographic peaks are partially resolved, matheatical resolution has been an alternative to "brute-force'' separation attempts. Some methods of mathematical resolution now in common use may lead to enormous errors, however, and must be used with caution. Several means have been proposed to apportion the area under the curve of a mixture into its constituent components. These methods include geometric methods, curve fitting, principal component analysis, and the solution of simultaneous equations. When the degree of peak overlap is only moderate, geometric methods such as perpendicular drop a t a valley, triangulation, and extrapolation under a shoulder have been used. These are the methods most commonly used in commercial integrating systems. When the degree of overlap increases or when a large difference in peak size exists, these methods give progressively worse results or fail completely. Curve fitting methods have been proposed assuming the peak to be a predefined shape, often Gaussian (1-3). It is well-known, however, that chromatographic peaks are not truly Gaussian in shape ( 4 3 ) ,and exponentially modified Gaussian ( 4 , 7 ) ,bi-Gaussian (5, a), Poisson (5), Gram-Charlier (9),and a combination of Gaussian, exponential, and hyperbolic tangent (6) have been applied. Grushka and co-workers ( 1 0 , I I ) have shown that compounds with varying functional groups have different peak shapes. Moreover, as will be shown in the present work, peak shape changes with sample size. Therefore any curve fitting procedure which assumes that all components have the Same peak shape (Gaussian or otherwise) or that the peak shape remains constant at different concentrations is subject to error. Principal component analysis as well as factor analysis have also been used in studying mixtures of overlapping components (12). Principal component analysis assumes that the I
0003-2700/81/0353-1369$01.25/0 0 1981 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
a 0 c
a
n
u:
a VI
a
C
m
a
Volume
Flgure 1. Computer-simubted overlapping chromatographic peaks and shoulderless composite peak. Response shown for composite peak and indivMual peaks at volume i .
curves from each of the components are the eigen vectors of a response matrix, composed of the curves of various mixtures of the two components. These eigen vectors when linearly combined will fit the curve of the mixture. The method does not assume a peak shape. It does, however, assume constant peak shape. Moreover, Macnaughtan et al. (12) found that a precision chromatograph, in which the flow rate and column temperature were precisely controlled, was required. Mori (13) proposed the use of linear simultaneous equations in which the detector response at a particular time is assumed to vary linearly with concentration. Gourlia and Bordet (14) have used a method of simultaneous equations according to the proposal of Goldberg (15)for the analysis of overlapping components. These methods assume the detector response at any time varies linearly with concentration, which is shown in this paper not to be the case except near the peak maximum. None of these methods are applicable when there is no resolution between the components of the mixture. Baudisch and co-workers (16) observed that peak fronts and half-widths do not change significantly with sample size, but the peak maxima and peak shape do vary with sample size. Thus, assuming a constant peak shape or a response at a particular time that is directly proportional to concentration is inaccurate over much of the peak. The present study is based on relating the response at a particular time to a second degree polynomial in concentration. The response from the mixture may then be expressed as a series of nonlinear simultaneous equations which may be solved to give the concentration of the components of the mixture with a high degree of accuracy even for the case in which the retention is identical for two components or for the case in which peak shapes are identical but a slight difference exists in retention.
THEORY Figure 1 shows a computer-simulated chromatogram of a two-component mixture in which the individual component peaks are so close in retention that only a single composite peak is produced and that peak shows no shoulder. Such a system obviously cannot be analyzed by the perpendicular drop method or by the tangential approach since in these cases a valley between the peaks or, at least, a well-developed shoulder must be present on the composite peak.
- -___ Volume
FWe 2. Analysis of 1 , 2 , 3 , and 4 % ndodecane in hexane (hexane peak not shown). Column was 12% Carbowax 20M on 80/80mesh Chromosorb WAW at 100 O C . Injection port temperature was 260 OC.
A basic assumption of the method being proposed is that the response from the mixture is equal to the sum of the responses from each component. Inherent in this statement is the assumption that the components do not interact with each other at the low concentrations of the solutes in the liquid phase. Experimental data to be published elsewhere show that this is generally a correct assumption. A data envelope containing peaks of interest is divided into evenly spaced volume increments. Volume increments are used rather than time increments since account must be made of variations in the flow rate of the mobile phase and variations in the column temperature during the period of analysis, both of which would alter the peak position. Either the flow rate and column temperature must be held constant so that the peak position does not vary or else these parameters must be accurately measured and the peak position adjusted to the position it would have had had these parameters not changed. Accurate, continuous measurement of flow rate and column temperature thus becomes an essential part of the successful use of the proposed method. The computer-assisted measurement of these parameters for liquid chromatography and for isothermal and programmed temperature gas chromatography is the subject of articles to be published elsewhere. If RT, is the total response from the mixture at volume i and R1,and RB are the responses at volume i from component 1 and 2, respectively, then It is important to realize that generally peak shape changes with sample size as shown in Figure 2, a series of chromatograms of n-dodecane on a Carbowax 20M column at 100 O C . If the response a t a particular volume is plotted against concentration, calibration curves similar to those shown in Figure 3 are obtained. Since peaks generally increase in asymmetry as concentration increases and a relatively sharp front with tailing is common, the leading edge of the peak does not change in proportion to sample size and calibration curves measured before the peak maximum are concave downward. Calibration curves obtained near the peak maximum are more linear. Tailing increases with sample size, and calibration curves after the peak maximum are concave upward. Asym-
ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
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The response at any particular time for the mixture is related to the amounts of each component in the mixture by the previously evaluated constants. Each equation will have some error associated with it, and it is desired to minimize these errors. Lam et al. (17),proposed several different error equations that can be minimized. In the present study we choose to minimize the sum of the squares of these errors S = C(RTi - aix2 - b,x - c,y2 - d,y)2 (9) i
To minimize this equation, its partial deviations with respect to 1: and y are set equal to zero 6S/6X = k ( x , y ) = -2xCaiRTi - Cb;RTi + 2x3Ca? + 3x2Caibi x C b ? + i
i
i
2xy2Caici + y2Cbici + 2 x y C a i d i 1
6S/6Y = g k y ) = -2yCCiR~i- E d i & i
i
I
+
i
i
+ yCbidi = 0
(10)
I
+ 2y3Cc? + 3 ~ ~i C c .+dy. Ci d i 2 + i 1
i
1
2x2yCaici + x2Caidi + 2xyCbici + x C b & = 0 (11) i
i
Flgure 3. Calibration curves for various regions of a typical asymmetic chromatographic peak: (A) near peak maximum; (B) tailing portion of peak; (C) leading portion of peak.
etry is an indicator of nonlinear isotherms. Let x j be the concentration of component 1 in standard solution j and R , , be the response of component 1 at volume i and concentration xi. The calibration curves of Figure 3 may then be represented by a second degree polynomial
R 111. . = a.x.2 1 I + b.x. 1 I
(2)
There is no constant term included in the polynomial because when there is no sample present, there should be no response above base line. The best least-squares estimates for ai and bi are
and
i
I
The problem of solving a set of equations similar to eq 8 has now been reduced to solving two nonlinear equations, eq 10 and 11,in two unknowns. These equations can be solved by Newton's method (18). Expanding k ( x , y ) and g(x,y) in Taylor series about some initial estimate of the concentration of each component, xo and yo, and neglecting the higher order terms gives k ( x , y ) = k(X0,YO) + ( x - xo)6lt/dx(xo,Yo) + 0, - Yo)6k/6y(xo,Yo) = 0 (12)
g ( x , y ) = g(X0,Yo) + ( x - xo)&/~x(xo,yo)+ 0,- yo)~g/dy(xo,yo)= 0 (13) where k(xo,yo),6k/6r(xo,yo),6k/6y(ro,yo),and the corresponding g terms are the function and its partial derivatives evaluated at the point (xo,yo),respectively. Solving eq 12 and 13 for ( x - xo) and (y - yo) yields x-xo= g(xo,Yo)6~/6Y(~o,Yo) - ~(~o,Yo)~g/~Y(~o,Yo) (14) 6k/bX (~o,Yo)~g/~Y(~o ,6Ygo/)~ x ( x o , y o )/6Y(XO,YO) 6~ and
Y-Yo= k ( ~ o , Y o ) ~ g / ~ ~ (-~ g(xo,yo)bk o , Y o ) /MXO,YO) For each volume, i, a t which calibration data have been obtained, a set of u and b values are computer derived to fit the response. At these same volume increments calibration curves are also made for component 2 and fit to the relationship R21]
=
+ 'YJ
(5)
where y, is the concentration of component 2 in standard solution j and R2,]is the response for this concentration at volume i. Equations similar to eq 3 and 4 can be derived for c, and d,, respectively. In the mixture, the response of each component at any volume can be related to its concentration by
Rll = alx2 + b,x
6k /6x(xo,Yo)6g/6Y (X0,YO)- ~ g / ~ x ( x , , y o ) /6Y 6 k (X0,YO)
(15)
Because the higher powers of the Taylor series were ignored, eq 14 and 15 will not give the values of x and y that satisfy eq 10 and 11 initially. However, by setting up iteration equations the values of x and y may be progressively improved. Xi+l = g(xi,yi)6k/6y(xi,Yi)- k(xi,yi)&/6y(xi,yi) xi + 6k / 6 x (xi,yi)6g/JY(Xi,yi)- 6g/ 6~ (Xi,yi)Gk/ 6 y (Xityi) (16)
Yi+l = k(xi,yi)Jg/Jx(xi,yi)- g(xi,yi)6k/6y(xi,yi) y i + 6 k / h x (xi,yi)6g/6y(xi,yi)- Jg/dX (xi,yi)bk/ay(xi,yi)
(6)
(17)
and
R2, = c y 2 + d y
(7) where 3c and y are the concentrations of components 1 and 2 in the mixture, respectively. Substituting eq 6 and 7 into eq 1 yields RT, =
+ blx + c y 2 + d y
U,X~
(8)
The iteration process can be judged complete when bi+l
- Xi1
< t and
bi+1 -
yi
Yil
< t
Xi
or
Ixi+l - xi( < and Iyi+l - yil
