Understanding Beer-Lambert Law at Low Concentrations: A Biologist's Dilemma

[Thiago Augusto]

Dear All,

I am PhD student in nanomaterials, and a biologist trying to find the way to understand Beer-Lambert Law. Considering that only at low concentrations the relation between concentration and absorption is linear, I diluted a high concentration sample, dilution factor 10, resulting an adjustment curve, R2 = 0.9592, Equation, x = (y + 0.2118)/0.0071. How should I consider low concentrations samples, considering values below 0.2118 in absorbance (not negative values)?

 

[Charles Link]

I would suggest you make the line go through the origin and perhaps make the best linear fit using just points with lower concentrations. $\\$ Additional idea: The straight line that you have obtained is actually the result of what appears to be a less than linear response for higher concentrations. Your data is most likely quite accurate=Instead of the equation being $y=Ax$, it contains higher order terms which could be modeled as $y=Ax-Bx^2$ where $B$ is a small positive constant and $-Bx^2$ is the approximate correction term. I believe you could do a least squares fit or something similar for the curve $y=Ax-Bx^2$ to determine the constants $A$ and $B$. For small $x$, the equation $y=Ax-Bx^2$ becomes $y=Ax$ to a very good approximation.

 

[Thiago Augusto]

I would suggest you make the line go through the origin and perhaps make the best linear fit using just points with lower concentrations. \\

Thank you for your reply. If I apply low concentrations values, which fits better my predicted values, the higher concentrations will be underestimated.

Instead of the equation being y=Axy=Ax y=Ax , it contains higher order terms which could be modeled as y=Ax−Bx2y=Ax−Bx2 y=Ax-Bx^2 where BB B is a small positive constant and −Bx2−Bx2 -Bx^2 is the approximate correction factor.

I did not get your additional idea yet, since the calibration curve must be linear. Moreover, my Supervisor suggested me to categorize the low values, for instance, absorbances below 0.2118 should be presented as <10 microliters per mililiter solution. It is an addaptive resolution to go throw.

 

[Charles Link]

Thank you for your reply. If I apply low concentrations values, which fits better my predicted values, the higher concentrations will be underestimated.

I believe the slope of the graph at low concentrations is actually higher than for larger concentrations=which means that your line $y=Ax$ will overestimate the value of $y$ for high concentrations. The mathematics here is actually very much what one would expect. Your data set looks to be very good.