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 y = a^Tx for some vector $ainmathbf{R}^n$ , where is the vector of concentrations, and the vector $a in mathbf{R}^n$ 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 ‘‘pure’’ gases in it. Denote by $x in mathbf{R}^n$ 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

for some vectors $a_i in mathbf{R}^n$ , which contain the coefficients of absorption of the gases at light frequency lambda_i . More compactly:

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 -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.