When is beers law nonlinear

In normal laboratory instruments, the spectral bandpass is controlled by the slit width , which is adjustable by the experimenter on many instruments but not on the Spectronic 20 , which has a fixed 20 nm spectral bandpass. In this simulation you can vary the slit width of the simulated instrument from 10 nm to nm by using the Slit width control above the graph, but it can not be set below 10 nm every instrument has a minimum slit width, and therefore a minimum spectral bandpass setting; you can not set the slit width to zero because then no light would get in and the instrument would not work at all!

Note that the transmitted intensity has a triangular spectral distribution because the entrance and exit slit widths are always equal in a normal monochromator.

The peak of the slit function falls at the wavelength setting of the monochromator. You can control the wavelength setting by using the Wavelength setting control above the graph; this is equivalent to turning the wavelength knob on the spectrometer.

The other controls above the graph are for the other variables in this simulation, such as the path length of the absorption cell cm.

So that you can see how different types of absorbing species would behave, the simulation allows you to vary the maximum absorptivity of the analyte and the spectral width of the absorber that is, the width of the absorption bands that constitute the absorber's spectrum.

The last control is for the stray light. Every real monochromator passes a small amount of white light as a result of scattering off optical surfaces within the monochromator mirrors, lenses, windows, and the diffraction grating. Usually this so-called "stray light" is a very small fraction of the light intensity within the spectral bandpass, but it's important because it can lead to a significant source of deviation from Beer's Law.

In most cases the monochromator is tuned to the wavelength of maximum absorption of the analyte, in order to achieve the greatest sensitivity of analysis. But that means that stray light is less absorbed than the light within the spectral bandpass. The worst offender is stray light that is not at all absorbed by the analyte - "unabsorbed stray light", usually expressed as a percentage of the light intensity within the spectral bandpass.

In the simulation, this is set by the "Unabsorbed stray light" control. Typical monochromators have stray light rating in the 0. The stray light is always worse at wavelengths where the light source is least intense and where the detector is least sensitive. However, in this simulation, the stray light does not automatically change with wavelength.

Note: when adjusting the stray light, use the number spinner small arrows below the number rather than typing directly into cell F3. The other variables you can change either by typing or by using the number spinners. The graph on the right of the window is the analytical curve calibration curve , showing the absorbances measured for each of the standard solutions listed in the table in the top middle of the window.

You can type any set of concentrations in the concentration column of this table, up to a maximum of 10 standards. The red line in the plot sometimes obscured by the other lines represents the ideal Beer's Law absorbances, the blue dots represent the measured absorbances for each standard solution, and the blue line is the least-squares straight-line fit to the concentration-absorbance data.

Ideally, the fitted straight line blue line should go right through the middle of the blue dots. The graph below the calibration curve is the concentration prediction error. If you were to run the standards as unknowns and predict their concentrations from the straight-line fit to the calibration curve, this would be the error in prediction, expressed as a percentage of the highest concentration.

The standard deviation of those errors is a good single-number summary of those errors; it is displayed to the left. This is a more sensitive indicator of non-linearity than the R 2 value. Start the experiment with a nearly ideal case with the spectral bandpass much less than the absorption width and no stray light. Note that the ideal absorbances red line , the measured absorbances blue dots , and the least-squares fit blue line are essentially identical, even at the highest concentrations, and the R 2 is exactly 1.

You can see that in this case the absorption spectrum is almost flat over the spectral bandpass. This means that all the photons have essentially the same absorption coefficient, a fundamental requirement of Beer's Law. The concentration prediction error the graph below the calibration curve is so small it is negligible compared to other errors that are likely to be greater anyway, such as volumetric calibration accuracy and precision.

But real-world absorption measurements are never so perfect. Unabsorbed stray light limit only. Notice that the measured absorbance bends off from a straight line at the highest concentrations, but still very linear at lower concentrations. Why does the calibration curve flatten out at high concentrations? As the concentrations increases, the intensity of the transmitted light from the spectral bandpass decreases towards zero, but the unabsorbed stray light remains at the same intensity because it is unabsorbed.

See if you can devise a rule that will predict the plateau absorbance for a given stray light percent. Calculate the weight concentration. Experimental measurements are usually made in terms of transmittance T , which is defined as:. The relation between A and T is:.

Issue 20, From the journal: Physical Chemistry Chemical Physics. Alexander Yu. Pushkarev , a Larisa G. Tomilova ab and Nikolay S. Zefirov ab. You have access to this article.

Please wait while we load your content Something went wrong. Try again? Cited by. All isomerization reactions were followed for several hours, until a photostationary state was reached. The measurements were repeated at different illumination intensities regulated with the power supply and at different dye concentrations.

This is important because the concentration of molecules in the trans state at every instant was determined from the absorbance at nm. Absorbance was measured at several concentrations. After this point, aggregation effects start playing a role and the basic Lambert-Beer law is no longer valid, undermining the theoretical relationship given by the equation For our detailed dynamic experiments, a very important issue was the viscosity of the solution.

In fact, at high illumination intensity we have encountered an unexpected problem. Figure 9 shows that the transmission of light through a low-viscosity dye solution in pure toluene displays a characteristic oscillatory behaviour. Detailed analysis of this phenomenon is under further investigation. The periodic instability was reproducible in all low-viscosity experiments.

In order to avoid this difficulty, the dye solutions were prepared in a mixture of toluene and polystyrene of high molecular weight. Adding polystyrene increases the viscosity of the solution by over 2 orders of magnitude, and in this way prevents fluid motion in the cuvette on the time scales of our measurements.

Polystyrenetoluene solutions were prepared at a fixed weight ratio. Adding polystyrene to toluene increases the Rayleigh scattering of the solution, but we felt that we could safely do that because on one hand the absorption dynamics is not affected we used the same concentration for all the measurements and for the reference spectrum , and on the other we measured the transmittance of the toluene-polystyrene solution, which is almost equal to the pure toluene solution at nm.

A traditional description of the kinetics of isomerisation would predict an exponential decay of the absorption over time, but from figure 10 we see a strong deviation. Two data sets at higher intensity show the transmitted I x, t reach saturated values. In this case we are confident of the fit because we have to match both the slope and the amplitude of the curve. We found that one particular output of experimental recording, the absorbance plateau value photo-bleached at long times, was extremely sensitive to the reading of reference intensity I 0.

The latter measurement could be affected by various stray factors and in a few cases we had to rescale the raw absorbance readings with a proper reference value. Three values of irradiation intensity are labelled on the plot. At higher irradiation intensities we achieve the saturation and the steady-state value I x corresponding to the solution of equation 6.

The change of curvature, notable in figures 8 and 10 , is not so clear here even at the highest I 0. However, in the comparative analysis of data we now take a different approach. It is clear that the theory is in excellent agreement with the data. The integration can then be carried out analytically, giving. The fits of the data for I t are again in good agreement with the full theory. This is in fact the rate originally seen in the kinetic equation 2.

Therefore, if we instead fit the family of experimental curves in figure 12 and several other data sets we measured by the simple exponential growth of the absorbance, we can have an independent measure of the relaxation rates obtained by this fit. The inset in figure 12 plots these rates for all the I 0 values we have studied. A clear linear relation between the relaxation rate and I 0 allows us to independently determine the molecular constant:.

The consequences of this nonlinear behaviour have in the last year raised an interested in some research groups who studied the azobenzene-based actuators.

In fact, the dynamic photobleaching is the reason why heavily doped cantilevers, where the penetration depth is very small, can still bend if irradiated with sufficiently intense beams.

Because the contraction of cantilevers is due to the force generated by the differential contraction of the top and bottom layers, if the light was propagating exponentially in the medium the bending would be impossible, because the thin layer where the light propagates is too small to generate enough force. A non-exponential propagation of light due to photobleaching, instead, can explain this effect.

Subsequent work by Van Oosten, Corbett et al. Lee et al. Because absorption spectroscopy is so widely used in biology, we want to show the effect of dynamic photobleaching on a biological molecule, and we chose chlorophyll, an important substance in biology and in everyday life. Chlorophyll has a very recognisable absorption spectrum, which shows two clear peaks, one in the blue and the other in the red region which procures its green colour of the electromagnetic spectrum.

If it is irradiated by UV light or very strong visible light it undergoes a photo-chemical bleaching which degrades the molecule irreversibly and leads it to precipitate from solution, as many studies reported [ Mirchin et al.

We wish to observe a dynamical reversible bleaching due to the absorption of light, rather than this chemical degradation process. In the previous section, the theoretical model was verified in the case of azobenzene, a molecule with a very long lived excited state.

Because the kinetics of transition could be followed by a spectrometer, it was also possible to model it with the kinetics law equation Fluorescent molecules have an excited state with a characteristic lifetime of a few nanoseconds, which is still much slower than the typical time of excitation. These characteristic times, though, are too short to be followed with conventional spectroscopes, and the transition kinetics cannot be followed as in the previous case.

The model, however, also makes predictions also about the transmittance at the photostationary state, which differs from the LB law transmittance. To clarify, in figure 10 the Beer limit would be the transmittance at time zero, and the stationary state the transmittance at long times. It was important, for our experiments, to rule out all possible mechanisms leading to failure of LB law. As it was previously discussed, LB law has many limitations. It fails at high concentration of dyes, when they start to interact with each other and form aggregates; it fails if the stray light is high and the apparent absorption seems to reach a saturation level; it can fail at high intensity of the incident light if nonlinear effects like multiple photon absorption, or saturable absorption occur [ Abitam et al.

In order to rule out all these possible effects, we place ourselves in the most favourable experimental conditions: low concentration of dye and low illumination intensity. According to the model, the behaviour at the stationary state is described by equation The important thing to observe is that the absorbance or, equivalently, the transmittance also depends on the intensity of the incident light I 0.

In order to experimentally verify this dependence, five different solutions of chlorophyll at known concentrations were measured at various light intensities. In this section all the absorbances will be reported in base logarithmic form. Figure 13 shows the outcome of measurements of chlorophyll absorption of the same solution using different incident intensities. The result was striking: the change in the measured absorbance was very substantially affected by this parameter. Some interesting consequences of this effect are shown in figure 14 and The values correspond to the steady-state absorption at the peak wavelength.

Indeed it is possible to see a strong dependence on the incident light intensity which is enhanced at high solute concentrations. Figure 14 shows the dependence of the absorbance on the intensity at various concentrations. Equation 16 cannot be explicitly solved for A, but only for I 0. Figure 15 shows the same data in the classical absorbance-concentration plot, for different intensities. It is important to remark that the experimental points can be satisfactorily fitted with a straight line in all cases as the LB law says but the line slopes are very different.

Therefore the absorption coefficient may have different values if it is measured with a different light source. The exchangeability of results between different laboratories is thus in question. We obtained analogous results with Nile Blue, a simpler chromophore. We decided to test this dye, described in the Material and Methods section, because it has an absorption spectrum similar to chlorophyll in the red region, but it is a simpler and well studied molecule.

This also proves that the results are general, and that aggregation phenomena which may occur in chlorophyll solutions giving rise to scattering phenomena from still intact chloroplasts are ruled out as a possible cause for the observed behaviour. All of the experiments were repeated several times and the behaviour was reproducible. Moreover, the intensity of light was increased and decreased alternatively to exclude the hypothesis of a chemical permanent photobleaching as a reason for absorption decrease.

Due to the phenomenon of reversible dynamic photobleaching, a simple absorption experiment like the one described in the introduction is in practice impossible. Absorption of chlorophyll as a function of the intensity of incident light. One can see an increase of absorbance at low intensities. The values are reported for five different concentrations.

This comparison is done in order to show that the range of intensity of our set-up is the same as a more conventional one. Absorption as a function of concentration for the different values of incident light. The LB limit was extrapolated from the ideal limit of zero intensity. In light of this, can we use the theoretical model to find a new method to determine concentrations using absorption spectroscopy, removing this dependence on the incident light intensity?

We took a series of concentrations of Nile Blue solutions and the corresponding absorbance measured at different intensities of incident light. Therefore the deviation from the literature value is still consistent with our findings.