Precise Determination of the Absorption Maximum in Wide Bands
Karl-Hugo Eriksson, Ailo Mikiver, and Walborg Thorsell National Defence Research Institute Department 5 S-172 04 Sundbyberg 4. Sweden
In connection with a report by De La Zerda et al.' about a precise determination of the absorption maxima in wide bands we want to make the following annotation. In the experimental evaluation of maximum values in wide bands of e.g. spectrometric data, there are risks for a subjective valuation. By using a mathematical method more objective values might he obtained. In this paper a method based on a logarithmic transformation of the Gaussian equation is described. By a least square fit to this transformed equation reproducible maximum values of wide hands are obtained also when several ~ e a k are s closelv located. In the expe;iments the abso;bancy at different wavelengths (wavenumbers = freauencies) of dimethvl~htalate-0.1 k n o l a r in cyc1ohexan~-was obtained, e.g. 1) from the chart of the recorder of s spectrophotometer (cf. Fig. 1) -,
2) from the digital display of a spectrophotometer by ocular oh-
servation
Figure 1. Part of a spectrum of dimemylphatalate-0.1 mmolar In cyclohexane. Experiment performed in an Acta V Spectraphotometer.The arrow indicates the maximum according to the described mathematical treatment.
3) by an automatic punch procedure followed by transfer and calculation in a computer (cf. the table)
The values thus obtained are represented in the graph (cf. Fig. 2). It is obvious from Figure 1, the table, and Figure 2 that there are difficulties in the determination of the absorption maximum. In order to facilitate the determination, a mathematical treatment of the data was performed. This is based on the assumption that the ahsorption values are distributed according to a Gaussian shaped curve. The function is = A . e-B(=-MP
(1)
where x is the calculated wavenumber. v is the molar absorbancy, A represents the maximum of M is the mean, and B is inverselv. .oro~ortional to the variance of the Gaussian . distribution. The parameters in eqn. (1) can be determined directly by a nonlinear fitting technique. The following method, however, gives simpler computations and is for that reason better suited for a mini-computer.
y;
Example of Computer Calculations
A logarithmic transformation of eqn. (1) will give a simple quadratic polynomial as follows Iny=InA-B(r-M)2=InA-BMZ+2BMx-B1-2
Molar
Wavenumber
Figure2.Partofaspectrumafdimethyiphtalate-0.1 mmolarincyclohexaneaccording to the values of lhe table. This pall is found in the square of Figure 1.The arrow Indicates the maximum according to the described mathematical treatment.
sbsorbancv
(2)
Ifz isset=Iny ko=InA-EMZ kl = 2BM kp = -B The following equation can be written from eqn. (2) The coefficients ko, k l , and ks are determined by a leastsquare fit of z on x . Since the standard deviation of y is approximately proportional t o y (and consequently the standard deviation of z is independent of z ) , an unweighted fit can be used. Only points in the surroundings of the maximum are 'De La Zerda, J., De Milleri, P., and Villaveces, J. L., J. CHEM. EDUC., 52,415 (1975).
454 1 Journal of Chemical Education
used in order t o avoid too much influence from proximate peaks. The values of A = y,,, = exp { k o - k1~14k21,M and B are then calculated. A graphical terminal or display, where the fitted curves are projected makes the fitting procedure clear. In the actual example, cf. the table, the calculated values are A = 12.81and
M = 3.642. The values of A and M are marked in Figure 2. The described method has been and is of great value in the determination of maxima in various connections with wide bands-even when close to each other-where Gaussian functions occur. A further advantage is that the method is suited for a mini-computer.
Volume 54, Number 7, July 1977 1 455
