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Beer's Law - Linear RegressionSlope Spectroscopy® is fundamentally a UV-Vis technique based on the Beer-Lambert law, which describes the linear relationship observed when light and matter interact. C Technologies, Inc. created variable pathlength technology to take advantage of the pathlength term of the governing equation, which is often neglected by being held constant. The consequence of holding pathlength constant generally means that analysts end up diluting and changing the very sample they want to measure. Our goal was to change the measurement system, not the sample, by creating a technology that dynamically varies the pathlength to adapt to the sample being measured.
The SoloVPE Solution is essentially a Linear Range Finder that identifies the Beer-Lambert law compliant region of the Section Plot (Absorbance vs. Pathlength Plot) by making absorbance measurements at various pathlengths as shown in the left-hand (green) region of the displayed graph. The system only relies on the linear region for quantifying things like concentration based upon the proportionality between the slope of the linear curve and Extinction coefficient of the sample.
There are many reasons why Section Data may reveal non-linearities as are shown in the middle (yellow) region of the displayed graph. But the SoloVPE system, through its search algorithm, works to identify when this occurs and avoids relying on the non-linear data for quantification. This is significantly different than traditional Absolute Absorbance methods which typically rely on single point measurements and unlike Slope Spectroscopy results, provide no context nor confirmation that the reported results are based on data from the linear range.
The right-hand (grey) region of the graph shows where the Beer-Lambert law has broken down. This most frequently occurs due to limitations of the light detecting capabilities of the instrument. It is always important for users of the system to understand the tool and the technique to be able to interpret the data-rich results produced by the SoloVPE Solution. However, the adaptability of the technology allows it to successfully measure the widest range of samples and concentrations with minimal user intervention and without costly and error-prone dilution and background correction steps.
Slope SpectroscopySlope Spectroscopy® is fundamentally a UV-Vis technique based on the Beer-Lambert law, powered by C Technologies’ variable pathlength technology. It enables scientists to measure samples directly on the system without the need for costly and error-prone dilutions, sample preparation steps, or even background correction in most instances. It leverages the power of pathlength by making absorbance measurements at various pathlengths unlike the fixed pathlength measurements of traditional spectroscopic methods. The linear range of a traditional spectrophotometer is frequently limited by the fixed pathlength cuvettes that are used. The SoloVPE dynamically adjusts the pathlength between 0.005 mm and 15.000 mm to identify the linear data region and thereby extends the power and capabilities of its spectroscopic engine by orders of magnitude.
Using a special light-guiding component called a Fibrette™, the pathlength is defined by the distance between the tip of the Fibrette and the inside bottom of the sample vessel. The pathlength is dynamically adjusted through precise movements of the Fibrette up and down in the sample. The pathlength changes are directed by motion control hardware and software integrated into the system. Absorbance measurements made at multiple pathlengths are used to identify the linear region of the data which can then be used for quantifying concentration, drug to antibody ratios, and more.
AbsorbanceThe Beer-Lambert law or Beer’s law states that absorbance of a solution is directly proportional to the concentration of the absorbing species in the solution and the pathlength. This has been the basis for UV-Vis spectroscopy since the inception of the spectrophotometer. Traditionally scientists will dispense the solution into a 10mm cuvette and run a scan of the solution to determine the peak absorbance across a specified wavelength range. If this absolute absorbance value is within the linear range of the spectrophotometer, the concentration of the solution can be determined using the calculated extinction coefficient. However, in cases where this absolute absorbance value is outside the linear range of the instrument, the scientist is then required to alter the concentration as the pathlength remains fixed.
Absorption vs. WavelengthSpectra and Section Data
Traditional UV-Vis methods rely on spectral data or discrete absorbance values. Spectra, as shown in the graph, are absorbance vs. wavelength plots that reveal information about the sample such as the wavelength location(s) of highly absorptive peaks and magnitude of the absorptivity. A typical SoloVPE survey dataset (shown here) displays a collection of UV-Vis spectra, with each spectrum having been collected at a unique pathlength. As predicted by the Beer-Lambert law, the absorbance varies consistently with the differences in pathlength. These curves help scientists identify the wavelength(s) at which to develop their quantitative methods.
Variable pathlength techniques leverage section data, which are absorbance vs. pathlength plots. Section datasets are useful for identifying the Beer-Lambert-law-compliant region of the collected data and drive the ability of the SoloVPE system to make rapid determinations of concentration and more. The Slope Spectroscopy® equation, which is derived from the Beer-Lambert law, makes this possible since the linear slope of a section plot is directly proportional to the sample concentration based upon the extinction coefficient of the sample.
Beer's LawThe derivation of Slope Spectroscopy:
A = the measure Absorbance
c = the sample concentration
l = the pathlength
m = the slope of the regression line
α = the wavelength dependent molar absorption coefficient
Beer’s law is expressed as: A= α* l * c
The Slope Spectroscopy equation is: m = A / l
Therefore: c = m / αFast and accurate determinations can be made when the absorbance versus pathlength regression has a region of strong correlation across multiple data points (>5). The subsequent R2 value close to one confirms that the absorbance values are changing proportionally with pathlength in accordance with Beer’s law.