Absorption spectrometry: Using measurements at different light frequencies.
Return to the absorption spectrometry setup described here.
The Beer-Lambert law postulates that the logarithm of the ratio of the light intensities is a linear function of the concentrations of each gas in the mix. The log-ratio of intensities is thus of the form 
for some vector 
, where 
is the vector of concentrations, and the vector contains the coefficients of absorption of each gas. This vector is actually also a function of the frequency of the light we illuminate the container with.
Now consider a container having a mixture of 
“pure” gases in it. Denote by 
the vector of concentrations of the gases in the mixture. We illuminate the container at different frequencies 
. For each experiment, we record the corresponding log-ratio 
, 
, of the intensities. If the Beer-Lambert law is to be believed, then we must have
![\[y_i = a_i^T x, \quad i=1, \ldots, m,\]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-95502016af98e75af2b19889265896c7_l3.png "Rendered by QuickLaTeX.com")
for some vectors 
, which contain the coefficients of absorption of the gases at light frequency 
.
More compactly:
![\[y = Ax,\]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-80d1ae1580de8ee5781c2c8354db08db_l3.png "Rendered by QuickLaTeX.com")
where
![\[A = \left( \begin{array}{c} a_1^T \\ \vdots \\ a_m^T \end{array} \right).\]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-ade3ba0aa0d5837583caa08c5ddfeca9_l3.png "Rendered by QuickLaTeX.com")
Thus, 
is the coefficient of absorption of the 
-th gas at frequency .
Since ‘s correspond to “pure” gases, they can be measured in the laboratory. We can then use the above model to infer the concentration of the gases in a mixture, given some observed light intensity log-ratio.
