Beer-Lambert’s law proves a direct correlation between the absorbance (A) of a molecule to the concentration (c) and the path length (b) of the sample as has been observed in the article for the Derivation of Beer Lambert Law. This relationship is a linear for the most part. However, under certain circumstances the Beer Lambert relationship breaks down and gives a non-linear relationship. These deviations from the Beer Lambert law can be classified into three categories:
- Real Deviations - These are fundamental deviations due to the limitations of the law itself.
- Chemical Deviations- These are deviations observed due to specific chemical species of the sample which is being analyzed.
- Instrument Deviations - These are deviations which occur due to how the absorbance measurements are made.
I shall attempt to clarify each of these with suitable examples so as to improve your understanding of spectroscopy.
Real Limitation and Deviation of Beer-Lambert Law
Beer law and Lambert law is capable of describing absorption behavior of solutions containing relatively low amounts of solutes dissolved in it (<10mM). When the concentration of the analyte in the solution is high (>10mM), the analyte begins to behave differently due to interactions with the solvent and other solute molecules and at times even due to hydrogen bonding interactions.
- At high concentrations, solute molecules can cause different charge distribution on their neighboring species in the solution. Since UV-visible absorption is an electronic phenomenon, high concentrations would possibly result in a shift in the absorption wavelength of the analyte. At times, even electrolyte concentrations (such as those present in buffers) play an important role in altering the charge distributions and affecting UV-visible absorbance. Some large ions or molecules show deviations even at very low concentrations. For e.g. methylene blue absorptivity at 436 nm fails to observe Beer Lambert law even at concentrations as low as 10μM.
- It is also possible that the concentration is so high, that the molecules create a screen for other molecules thereby shadowing them from the incident light.
- High analyte concentrations can also possibly alter the refractive index (η) of the solution which in turn could affect the absorbance obtained. If the addition of solute causes a significant change in the refractive index of the solution a correction to the Beer Lambert formula can be placed as:
A = εbc (η2 + 2)2
This correction is normally not required below concentrations of 10mM.
Chemical Deviations and Limitations to Beer-Lambert Law
Chemical deviations occur due to chemical phenomenon involving the analyte molecules due to association, dissociation and interaction with the solvent to produce a product with different absorption characteristics. For example, phenol red undergoes a resonance transformation when moving from the acidic form (yellow) to the basic form (red). Due to this resonance, the electron distribution of the bonds of molecule changes with the pH of the solvent in which it is dissolved. Since UV-visible spectroscopy is an electron-related phenomenon, the absorption spectrum of the sample changes with the change in pH of the solvent.
- Acid and Base forms of phenol red along with their UV spectra at different pH demonstrates chemical deviations of Beer-Lambert law in UV-Visible spectroscopy
Instrumental Deviations and Limitations to Beer-Lambert Law
- Acid and Base forms of phenol red along with their UV spectra at different pH demonstrates chemical deviations of Beer-Lambert law in UV-Visible spectroscopy
A] Due to Polychromatic Radiation (Also the reason why absorbance measurements are taken at the wavelength of maximum absorbance λmax)
Beer-Lambert law is strictly followed when a monochromatic source of radiation exists. In practice, however, it is common to use a polychromatic source of radiation with continuous distribution of wavelengths along with a filter or a grating unit (monochromators) to create a monochromatic beam from this source. For example (see figure below), consider a molecule having molar absorptivities ε’ and ε” at wavelengths λ’ and λ”. The absorbance (Am) for such a species can be calculated as:
Equation to calculate absorbance of a sample with polychromatic light source.
When the molar absorptivities are the same at both wavelengths (i.e. ε’ = ε”) , the relationship between absorbance and concentration follows Beer-Lambert law to obtain a straight line. However, as the difference between ε’ and ε” increases, the deviations from linearity also increases.
Why absorption measurements are taken at wavelength of maximum absorbance λmax?
If the band of wavelength selected on the spectrometer is such that the molar absorptivities of the analyte is essentially constant, deviations from Beer-Lambert law are minimal. However, if a band is chosen such that the molar absorptivity of the analyte at these wavelengths changes a lot, the absorbance of the analyte will not follow Beer-Lambert law. It is observed (as demonstrated in the figure below) that the deviations in absorbance over wavelengths is minimal when the wavelength observed is at the λmax. Due to this reason absorption measurements are taken at wavelengths.
Figure A: Shows the difference in deviations in absorbance when values are obtained at maximum wavelength of absorbance (band A) vs other wavelengths of absorbance (band B). Figure B: shows the deviations in Beer-Lambert law due to observations made at wavelengths other than lambda max.
B] Due to Presence of Stray Radiation
Stray radiation or scattered radiation is defined as radiation from the instrument that is outside the nominal wavelength band selected. Usually the wavelength of the stray radiation is very different from the wavelength band selected. It is known that radiation exiting from a monochromator is often contaminated with minute quantities of scattered or stray radiation. Usually, this radiation is due to reflection and scattering by the surfaces of lenses, mirrors, gratings, filters and windows. If the analyte absorbs at the wavelength of the stray radiation, a deviation from Beer-Lambert law is observed similar to the deviation due to polychromatic radiation.
C] Due to Mismatched Cells or Cuvettes
If the cells holding the analyte and the blank solutions are having different path-lengths, or unequal optical characteristics, it is obvious that there would be a deviation observed in Beer-Lambert law. In such cases when a plot of absorbance versus concentration is made, the curve will have an intercept k and the equation will be defined as:
A = εbc + k
In today’s instrument this problem is generally not observed, however if it is present, appropriate linear regression to quantify this deviation must be made.
Books on Analytical Chemistry and Spectroscopy
Check out these good books for analytical chemistry and spectroscopy
References
- Analytical Chemistry: An Introduction (Saunders Golden Sunburst Series) 7th Ed., by Douglas A. Skoog, Donald M. West, F. James Holler. 1999.
- Practical Pharmaceutical Chemistry 4th Ed. Edited by A. H. Beckett and J. B. Stenlake. 1988.
- Fundamentals of Analytical Chemistry 8th Ed., by Douglas A. Skoog, Donald M. West, F. James Holler, Stanley R. Crouch. 2003.
- Brian Lamp, Lecture notes Chemistry 322. UV-Vis Techniques. (accessed on 13th May, 2012).
It is completely explained. Kudos
2.It is also possible that the concentration is so high, that the molecules create a screen for other molecules thereby shadowing them from the incident light.
Might want to elaborate on this part if you get a chance. I could think of FRET as a type of example.
Nice article.
Hi Rob,
FRET is off course slightly different as it uses fluorescence but the idea of local concentrations is well understood for people who understand FRET. Thanks for the compliment. I hope to write similar articles (possibly with animation) for fluorescence in the future.
I agree with Rob, It would be good with an elaboration of that part; e.g. how high an concentration (or how long the path length trough the sample) can be, before measurable deviations appear. Of course this would be instrument dependent, but a rule of thumb or a formula would be great.
I suppose it would be also dependant on the absorptivity of the molecule. This could make for someone’s thesis if the work is already not done
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Akul,
You are promoting a fundamental misconception about the nonlinearity of Beer’s Law at high optical depths. While there are many instrumental and chemical reasons for nonlinearity (polychromatic light, stray light, varying pathlengths, chemical equilibria, bulk refractive index changes and perturbations of wavefunctions at high concentrations), absent these, the Beer-Lambert-Bouger law holds to infinite optical depths. Better said, if you have a dilute number density of absorbers (low number density means that equilibria, wavefunction perturbations and bulk refractive index changes are all negligible), but increase your optical depth by increasing pathlength, Beer’s Law will hold to infinite pathlength- i.e. infinite optical depth. Of course, the fraction of light absorbed is not linear at high optical depths because of molecules lying in the shadow of other molecules. Absorbance, however, IS linear with pathlength forever. I suggest Daniels, J. Chem Ed., “A Random Number Model for Beer’s Law- Atom Shadowing”, 76, 1, 1999. pp. 138-141. The author explicitly addresses this misconception on page 140.
Hi Jonathan,
Thank you for the valuable feedback. The article addresses practical deviations observed in the law. Theoretically the law should be linear however practical deviations are observed. I do not see what fundamental misconception I am promoting. Although the mathematics in the paper may prove that theoretically it may be possible to obtain linearity, practically it is impossible to obtain linearity at high concentrations (although some systems nowadays are beginning to show good linearity even up to an absorbance of 3). The paper you mention only deals with theory and has no practical verification of the phenomenon. Secondly the paper shows linearity in the range of absorbance values 0 to 0.3 and proves only conditions of low concentration. In the words of the author “At low analyte concentrations, where the probability of one analyte atom or molecule lying behind another is low, an absorption curve is linear. However, as analyte concentrations increase, the probability of analyte units lying in the shadows of others increases, and the absorption curve bends towards the abscissa.” The article above clearly states “… the concentration is so high, that the molecules create a screen for other molecules thereby shadowing them from the incident light” which is in agreement with the paper. So I still do not see what fundamental misconception I am promoting. Could you please highlight the sentence which you feel is wrong.