Discrimination of two mechanisms

If the absorption coefficients in a quasi-linear photoreaction are not known, the rank of the Jacobi matrix, the eigenvalues of the reactions and the final absorbances can be calculated. Furthermore the initial values of the absorbances and their derivatives with respect to time can be determined. Any additional statements require additional information since for linear systems the following theorem is valid [15]  

Two reactions, for which Jacobi matrices result in the same rank, cannot be discriminated by spectroscopic means, if the absorption coefficients of the reaction partners are unknown 

This can be easily proved, since any system can be rearranged into another system of the same rank by linear transformation as has been demonstrated in Section 2.3.4. Since the transformation of the system of degrees of advancement to a system of absorption coefficients is done by affine representation, there is no chance that one can realise which system originally exists without additional information. 

Various additional information can be used. An optimal fact is to know all the absorption coefficients of the components participating in the reaction. This can be achieved if the reaction is interrupted at least n times and the solution analysed to determine quantitatively the concentrations of all the components. In this case concentrations a and the absorbances E are known at these times. For this reason e can be determined according to eq. (5.44). Qualitative analysis has to prove that only n components take part in the reaction. Using the absorbance coefficients for each reaction time the concentrations can be calculated according to eq. (4.9). One can proceed according to Section 5.3.1 in the following way. Because of the errors it is recommended first to use absorbances according to Section 5.3 for evaluation. Since interruption of the reaction and the determination of the concentrations 

Further helpful information can be obtained if just a single compound absorbs in a certain spectral range or if the eigenvalues can be varied. Furthermore a certain mechanism can be assumed or discarded by pure chemical knowledge. 

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