Known Spectra, Uncoloured Species

We return to spectrophotometric absorbance data and Beer s law Y=CA. The matrix C of concentration profiles, as computed by the relevant function, contains the profiles of all species that are part of the model. Remember, in kinetics this relevant function is an ODE solver, e.g. Runge-Kutta, while for equilibrium investigations, such as titrations, it is the Newton-Raphson routine that computes C. The following considerations hold for both kinetic and titrimetric absorbance data Y. 

It makes no sense whatsoever to try to calculate the spectra of species that are known not to absorb, by applying the equation A=C+Y. Often C will be rank deficient and cannot be pseudo-inverted. Matlab will issue an error message that needs to be taken seriously. Even if C is not rank deficient, it is not reasonable to allow the calculation of the spectrum of a non-absorbing 

The principle is the following the base equation Y=CA can be written in a different way. The matrix Y contains the sum of all the individual contributions of the absorbing species the contribution of the j-th species is the product of the j-th column of C with the j-th row of A. 

It is possible to separate the products c-j ap into two groups. We collect all the concentration profiles and absorption spectra of the species with known spectra in the two matrices Ck and Ak. For non-absorbing species, the appropriate rows of Ak contain only zeros. Correspondingly, the matrices Cuk and A k contain the concentration profiles and spectra of the species with unknown spectra. Equation (4.84) can be arranged as  

prior to the computation of the unknown species spectra Auk, the known contribution Yk is subtracted from Y. 

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Note, that for the individual analyses, one of the two component spectra of species A or B needs to be set as colourless, i.e. non absorbing (see Known Spectra, Uncoloured Species, p.175), as the matrices Ci and C2, containing the concentration profiles, are rank deficient for this kinetic model and hence their pseudo-inverse C+ is not defined. 

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