In effect, this iteration method apparently seeks out those peak heights where the adsorption terms are negligible and the solubility isotherms are linear, thus yielding infinite dilution bulk solution retention volumes. The resulting KL values are listed in Table 11. They each represent unique solutions to the sets of equations solved. Interestingly, the hj values found for the amine were the values at the peak maxima, as would be expected. In view of the simplicity of this method, the observed agreement with the other approaches is encouraging. CONCLUSIONS
The relative advantages and disadvantages of the three approaches are summarized in Table 111. The new approach measures up very well against the other two. However, all three are capable of giving consistent KL results for solutes which exhibit multiple sorption mechanisms. The implications of this study and, largely, the earlier studies of Conder
and Purnell (2-5, 14, 16, 17) are evident. In particular, bulk solution thermodynamic quantities for testing theories of nonelectrolytic mixtures (15) or for determining hydrogenbond association constants (5, 6 ) can now be obtained with greater confidence for systems which were previously considered to be too troublesome and error prone. Also, meaningful bulk solution retention parameters can be confidently assigned for such compounds as alcohols. Finally, analytical and preparative scaie GLC separations of mixtures where one or more of the compounds is a multiple sorber can be approached with a better understanding of the effect of sample size and column loading on peak shape and retention.
RECEIVED for review September 13,1971. Accepted November 30, 1971. This work was supported by a grant from the National Science Foundation. (16) J. R. Conder and J. H. Purnell, Trans. Faraday Soc., 64, 3100 (1968). (17) Zbid.,65,824(1969).
Short Cut Fused Peak Resolution Method for Chromatograms S. M. Roberts IBM, Data Processing Division, Palo Alto, Calv. 94304 This paper describes a simple, fast, and effective computation method for resolving fused peaks of chromatograms. If the underlying curves are Gaussian, the method has a sound theoretical basis and can produce in practice fits comparable to least squares approximations. If the underlying curves are not Gaussian, the user can still apply the method to the mathematical model he deems appropriate. Numerical results are given and compared with the results of the least squares method.
where fi(tc) = the Gaussian function for the jth peak at time
Aj to,j
wj
INAN EARLIER PAPER the author and his coworkers described the practical application of the least squares method to chromatograms ( I ) . As part of an effort to accelerate the curve fitting process, this paper describes a “short cut” computation method of resolving fused peaks of chromatograms. If the underlying curves are Gaussian, the method has a sound theoretical basis and can produce in practice fits comparable to least squares approximations (2-4). The principal advantages of the method are its speed of computation, its requirements for only two data points per peak, its simplicity, its ease of programming, and its satisfactory fits. Its prime disadvantage is the validity of the assumption of the Gaussian curves as the mathematical model. The technique is sufficiently general, however, that other models may be employed at the user’s option. THEORY
We assume that the trace of each peak of the chromatogram is described by a Gaussian curve (1) S. M. Roberts, D. H. Wilkinson, and L. R. Walker, ANAL. CHEM.,42, 886-893 (1970). (2) N. R. Draper and H. Smith, “Applied Regression Analysis,” Wiley, New York, N.Y., 1966, Chapter 10. (3) F. B. Hildebrand, “Introduction to Numerical Analysis,” McGraw-Hill, New York, N.Y., 1956, Chapter 10. (4) E. L. Stiefel, “An Introduction to Numerical Mathematics,” Academic Press, New Yoxk, N.Y., 1963,Chapter 4. 502
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ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
ti
maximum amplitude of the jth Gaussian peak = time at which the amplitude of the jth peak occurs = “width” of the jth peak = one standard deviation =
We assume that the measured data can be represented as a sum of Gaussian curves S
Y(ti) =
A j exp j=l
(ti
2
to,j)’
wj2
1
, i = 1 , 2 , . . , 2 s (2)
where Y(ti) = measured data point (corrected for base-line drift) at the time ti = total number of Gaussian curves s
If we assume that the t o , jpoints are fixed and known as observed in the trace, then each peak is characterized by two parameters A , and wj. This means that with two measurements (corrected for base-line drift) per peak that A , and wj can be determined. Once the parameters are known, it is a simple matter to integrate the Gaussian curves and obtain the relative areas under each peak. Some investigators feel that t o , jshould not be considered fixed but should be considered as variables. In this case t o , j is handled in the same manner as the parameters A , and wj. The solution method consists first of linearizing the transcendental Equations 2, and then solving iteratively for the parameters A, and wj, j = 1,2,, , ,s. Each Gaussian function in Equation 2 is approximated by a Taylor series up to and including first order terms.
The process converges numerically when
where €1 and where
= kth estimate of w j , j =
- A j ( k ) )
