If Beer's Law appears to apply to a high concentration (>0.01M) of a solute, is it valid to use for concentration calculation?

Internet consensus seems to define "high concentration" for Beer's Law as >0.01M. If one has a solution at higher concentrations than that, but the absorbance vs. concentration is still linear, is it valid to use Beer's Law to calculate concentration from absorbance data (assuming that the concentration is less than the highest point for which one has experimental data)?

• I have never encountered this "internet consensus". There are a number of underlying assumptions for Beer's law to be valid: I have seen references to as many as 13 conditions. And you would not want the analyte to form dimers, as concentration increased, because this would change the molar absorptivity. But there is nothing intrinsically problematic about, say, 0.1 M analyte. You appear to have a handle on this, so best of success! – Ed V Jul 18 at 23:26
• @EdV, do you remember the reference which listed 13 conditions? Since you are an expert on the limit of detection, we should start a post on this concept. Please share your thoughts, in your experience which is the "best" way to determine LOD. – M. Farooq Jul 19 at 0:28
• @M.Farooq I just e-mailed two papers stating 11 conditions (not 13: my mistake). As for detection limits, take a look at my short list of references on the topic: stats.stackexchange.com/a/410372/247352 . This includes my book, Joel Tellinghuisen's Analytical Chemistry paper (published last month), and lots more. What I recommend on detection limits is in my book and multiple papers on the topic, and at my book's associated web site (URL at the bottom of my e-mail). Frankly, I prefer not to get into that at this Q & A site: it is way too complicated. – Ed V Jul 19 at 0:53
• Thank you Ed. Most people take LOD as a fixed definition. Nice to see that feature in Analytical Chemistry. Will have a look. The acknowledgement is interesting "I have profited greatly from frequent exchanges with EdV on matters concerning detection limits. I cannot thank him enough." – M. Farooq Jul 19 at 0:58
• @M.Farooq We actually exchanged 185 e-mails, over about 4.5 months, and more than 150 files: sims, spreadsheets, pics, papers, etc. It was an enjoyable intellectual adventure. – Ed V Jul 19 at 1:11

3 Answers

Internet consensus seems to define "high concentration" for Beer's Law as >0.01M

Keep in mind that Beer's law is an approximation. Look at the more advanced version. Beer's law-advanced version. The statement that a concentration of 0.01 M is the upper limit of Beer's law is incorrect due to several reasons. It is the absorbance value which we should worry about rather than the concentration. One can have a solution which is 0.01 M yet its extinction coefficient is not that large at that particular wavelength, we would still be fine.

This trick is used in atomic absorption spectroscopy, where you choose wavelength, which is not a resonance line of that particular atom. As a result, one can use relatively high concentration without getting a non-linear curve.

Secondly, chemists tend to infatuate little bit with linear calibrations. It is okay to have a non-linear curve. Quadratic equations can be used to fit the empirical curve and one can get accurate analytical results.

• ... moreover, what a chemist would call a nonlinear curve may still be a linear from a statistics/data analysis/curve fitting point of view: as a chemist I understand linear calibration to refer to a function that is linear in the concentration. As chemometrician/data analyst, I can also understand linear model refering to a calibration function that is linear in the coefficient: in curve fitting/statistical modeling the coefficient is what needs to be calculated, and the fitting methods that can be used depend on this type of linearity, nonlinear terms in concentration are "harmless" here. – cbeleites Jul 19 at 16:59
• Looking at the actual transmission points to a 2nd way of adjusting: choosing appropriate optical path length. OTOH, concentration will actually matter if at too high concentration new chemical species appear which have different absorption characteristics (dimerization, changes in solvation). – cbeleites Jul 19 at 17:03
• @cbeleites Your point about linearity in coefficients is one that should be far better known! And the distinction between y - a = bx and y = a + bx, in terms of mathematical linearity, also should be better known. – Ed V Jul 19 at 22:31
• Dear EdV, Could you elaborate more on linear in coefficients and your distinction b/w y-a=bx vs. y=a+bx. What is a test for that? – M. Farooq Jul 20 at 0:51
• @M.Farooq So, exactly as cbeleites said, "fitting methods that can be used depend on this type of linearity", i.e., whether the equation is linear or nonlinear in the coefficients. This paper, which I will e-mail to you right after this comment is posted, explains things very well: J. Tellinghuisen, "Statistical Error Propagation", J. Phys. Chem. A 105, (2001) 3917-3921. See the bottom of the first page and left half of the second page, in particular. – Ed V Jul 20 at 12:04

The concentration is abitrary; take copper sulfate (CuSO4), with a molar extinction coefficient of ~50 dm3 mol-1 cm-1; its absorbance will be relatively low (A $$\approx$$ 0.5 at 0.01 M in 1 cm cell, so T $$\approx$$ 32 %).

Compare this with KMnO4, $$\epsilon \approx 220$$ dm3 mol-1 cm-1. At 0.01 M, A $$\approx$$ 2.2 (T $$\approx$$ 0.6%).

Even more striking are organic dyes with extinction coefficients upwards of $$50\,000$$ dm3 mol-1 cm-1 and suddenly you're at a point where a concentration of 0.01 is sufficiently high that only one photon in 10500 is actually getting through to your detector!

It's the absorbance that counts; a general rule of thumb is to configure your concentrations so that you keep your absorbance in the range 0.1-0.6 as this corresponds to 80% to 25% transmission, well within the dynamic range of most instruments. If you have a shiny instrument, you may be able to work outside this range.

As for linearity, this is an obsession of physical scientists. The linearity of BL is only an approximation. If you have done an investigation with your analyte and looked at how absorbance varies with concentration, and you are satisified with its approximation to linearity, then run with it. If however you notice a significant curve to your Abs/Conc data, then you may wish to revisit the BL law and its logarithmic derivation.

HTH.

Concentration alone is not a criterion here, and I would refrain even more from stipulating one threshold concentration applicable for all samples. True, from perspective of the sample (possible unwanted association of the analyte's molecule) and linearity of the signal, you are in a better situation if your analyte is not too concentrated.

From the perspective of the spectrometer's detector, the old-school approach is to record the absorption bands around $$\mathrm{Abs} \approx 1$$. For a cell of $$\pu{1cm}$$, given $$\mathrm{Abs} = \varepsilon \cdot c \cdot l$$, once you know about your strongest absoption $$\varepsilon_{\mathrm{max}} = 10^n$$ in your region of interest, you then would record at a concentration not higher than $$c \approx 10^{-n}$$.

Do not forget that the simultaneous presence of (very) strong and weak absorption bands does not hinder you to split the range to record into multiple sections -- each recorded at a different (diluted) concentration. Instead of $$\mathrm{Abs}$$ on the ordinate, taking into account the sample's concentration, you would trace $$\log \varepsilon$$ as function of wavelength ($$\mathrm{nm}$$), or proportional to energy (e.g., $$\pu{cm^{-1}}$$) on the abcissa.