[latexpage]
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 \(y = a^T x\) for some vector \(a \in \mathbb{R}^n\), where \(x\) is the vector of concentrations, and the vector \(a \in \mathbb{R}^n\) 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 \(n\) “pure” gases in it. Denote by \(x \in \mathbb{R}^n\) the vector of concentrations of the gases in the mixture. We illuminate the container at different frequencies \(\lambda_1, \ldots, \lambda_m\). For each experiment, we record the corresponding log-ratio \(y_i\), \(i=1, \ldots, m\), of the intensities. If the Beer-Lambert law is to be believed, then we must have
\[y_i = a_i^T x, \quad i=1, \ldots, m,\]
for some vectors \(a_i \in \mathbb{R}^n\), which contain the coefficients of absorption of the gases at light frequency \(\lambda_i\).
More compactly:
\[y = Ax,\]
where
\[A = \left( \begin{array}{c} a_1^T \\ \vdots \\ a_m^T \end{array} \right).\]
Thus, \(A_{ij}\) is the coefficient of absorption of the \(j\)-th gas at frequency \(\lambda_i\).
Since \(A_{ij}\)’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.
