A mathematical analysis of deviations from linearity of Beer's law

Extractive Spectrophotometric Detection of Sn(II) Using 6-bromo-3-hydroxy-2-(5-methylfuran-2-yl)-4H-chromen-4-one

For the determination of tin(II) traces, an extractive spectrophotometric approach is devised. The applied method serves a powerful tool for determination of tin(II), involves the formation of yellow colored complex after the binding of 6-bromo-3-hydroxy-2-(5-methylfuran-2-yl)-4H-chromen-4-one (BHMF) and tin(II) in 1:2 stiochiometry in a slightly acidic medium (HCl). The complex shows absorbance at 434 nm with respect of the blank reagent. The outcomes of spectral investigation for complexation showed a Beer's range of 0-1.3 μg Sn mL-1, molar absorptivity, specific absorptivity and Sandell's complex sensitivity are 9.291 × 104 L mol-1 cm-1, 0.490 mL g-1 cm-1 and 0.002040 μg cm-2 at 434 nm that was stable for two days. The interferences study results showed that this method is free from interferences, when tested with metal ions including Ag, Be, Bi, Ca, Cd, Ce, Co, Hg, Mo, Re, Pt, Se,Ti, U, V, W and other common cations, anions, and complexing agents. The applied method is quite simple, highly selective, and sensitive with good re-producibility. This method has been satisfactorily by utilizing the proposed procedure, and its applicability has been tested by analyzing synthetic samples and an alloy sample of gunmetal. The procedure assumes this because of the scarcity of better methods for determining tin(II). The results are in good agreement with the certified value.

An adaptive absorption spectroscopy with adjustable moving window width for suppressing nonlinear effects in absorbance measurements

Absorption spectroscopy based on Lambert-Beer law has been widely used in material structure analysis, research in chemical reaction kinetics, and exploration of various physicochemical reaction mechanisms. However, serious nonlinearity between absorbance and measured concentration can occur in actual measurements. The idea of moving window is first introduced into the field of spectral nonlinearity in the paper. Combining with the characteristic absorption spectra of the substances to be measured, we propose an adaptive absorption spectroscopy (A-AS) with adjustable moving window parameters to effectively suppress the nonlinear effects in absorbance measurements. The validity of this method is verified by taking the differential optical absorption spectroscopy to detect SO₂ as an example. The 210-230 nm characteristic absorption band is traversed and divided by the moving window with adjustable parameters, and the estimated coefficient (k-value) of each band is calculated. On this basis, all k-values are initially and secondly screened to obtain the optimal k_best, and then the optimal concentration value is obtained by inversion. Compared with the broad-band method and narrow-band method, it shows excellent performance that the maximum error and standard deviation of A-AS is only 1.3% and 3.8 in the entire concentration range, suggesting good linearity and stability in both high and low concentration environments. Therefore, it is inferred that A-AS is universally adaptable and enables dynamic linear measurements over wide concentration range.

Simultaneous determination of additive concentration in rubber using ATR-FTIR spectroscopy

Using attenuated total reflection (ATR) Fourier transform infrared (FTIR) spectroscopy for direct quantitative analysis is highly desirable for many sample systems due to advantages such as rapid spectra collection and being completely non-destructive. However, for many complex sample matrices the feasibility of direct quantitative analysis using ATR-FTIR is uncertain. The commonly used Beer-Lambert law may not be applicable for many systems in general, besides sample related complexities such as inhomogeneity, variable optical properties, or heavily overlapping absorption bands. In this study, we consider fully formulated vulcanized rubber with carbon black or silica as the primary filler as our system of interest. We developed a method to simultaneously quantify the concentration of three different antidegradents of similar chemical structure directly on rubber samples using ATR-FTIR spectra. Results show that absorbance follows the Beer-Lambert law well for the range of antidegradent concentrations considered. Despite this, a direct application of the Beer-Lambert law to deconvolute overlapping peaks between antidegradents proved insufficient. Through the application of partial least squares (PLS) multivariate analysis, remarkable prediction accuracy of within about 0.15 wt% error for all three antidegradents was achieved for both types of rubber formulations, even with high levels of carbon black. These results show the value this method has for quantitative analysis of additives in rubber. Our investigation highlights the potential usefulness of FTIR spectroscopy in general for rapid quantitative analysis directly on samples of interest without any prior chemical separation.

Study on the origin of linear deviation with the Beer-Lambert law in absorption spectroscopy by measuring sulfur dioxide

In accordance with the Beer-Lambert law, absorbance is proportional to concentration and optical path length of the absorbers in the sample, and in a linear relationship with total column concentration (product of concentration and optical path length) at a single wavelength. However, limitation of spectral resolution will result in linear deviation with the Beer-Lambert law in actual measurement. Regarding additivity of polychromatic light intensity as the theoretical basis, this paper attributed linear deviation with the Beer-Lambert law to spectral resolution, concentration and light intensity, and verified this explanation by measuring sulfur dioxide at various total column concentrations using spectrometers with different spectral resolutions in the waveband range of 216-230 nm. It was found that linear deviation with the Beer-Lambert law was in negative correlation with spectral resolution, and in positive correlation with total column concentration, and absorbance could be considered to be linear with total column concentration (below 171.4 mg/m²) of sulfur dioxide in the wavelength range of 216-230 nm. In addition, it was also proved that linear deviation increases with decreasing light intensity at a fixed sulfur dioxide column concentration.

An empirical investigation of deviations from the Beer-Lambert law in optical estimation of lactate

The linear relationship between optical absorbance and the concentration of analytes-as postulated by the Beer-Lambert law-is one of the fundamental assumptions that much of the optical spectroscopy literature is explicitly or implicitly based upon. The common use of linear regression models such as principal component regression and partial least squares exemplifies how the linearity assumption is upheld in practical applications. However, the literature also establishes that deviations from the Beer-Lambert law can be expected when (a) the light source is far from monochromatic, (b) the concentrations of analytes are very high and (c) the medium is highly scattering. The lack of a quantitative understanding of when such nonlinearities can become predominant, along with the mainstream use of nonlinear machine learning models in different fields, have given rise to the use of methods such as random forests, support vector regression, and neural networks in spectroscopic applications. This raises the question that, given the small number of samples and the high number of variables in many spectroscopic datasets, are nonlinear effects significant enough to justify the additional model complexity? In the present study, we empirically investigate this question in relation to lactate, an important biomarker. Particularly, to analyze the effects of scattering matrices, three datasets were generated by varying the concentration of lactate in phosphate buffer solution, human serum, and sheep blood. Additionally, the fourth dataset pertained to invivo, transcutaneous spectra obtained from healthy volunteers in an exercise study. Linear and nonlinear models were fitted to each dataset and measures of model performance were compared to attest the assumption of linearity. To isolate the effects of high concentrations, the phosphate buffer solution dataset was augmented with six samples with very high concentrations of lactate between (100-600 mmol/L). Subsequently, three partly overlapping datasets were extracted with lactate concentrations varying between 0-11, 0-20 and 0-600 mmol/L. Similarly, the performance of linear and nonlinear models were compared in each dataset. This analysis did not provide any evidence of substantial nonlinearities due high concentrations. However, the results suggest that nonlinearities may be present in scattering media, justifying the use of complex, nonlinear models.

Modelling and Optimization of Processing Factors of Pumpkin Seeds Oil Extraction under Uniaxial Loading

In the present study, a Box–Behnken design of response surface methodology (RSM) was employed to optimize the processing factors (force: 100, 150, and 200 kN; speed: 3, 5, and 7 mm/min; and temperature: 40, 60, and 80 °C) for extracting pumpkin seeds oil under uniaxial compression. The design generated 15 experiments including twelve combinations of factors and three replicates at the center point. The responses: oil yield (%), oil expression efficiency (%), and energy (J) were calculated, and the regression models determined were statistically analyzed and validated. The optimum factors combination: 200 kN, 4 mm/min and 80 °C predicted the oil yield of 20.48%, oil expression efficiency of 60.90%, and energy of 848.04 J. The relaxation time of 12 min at the optimum factors increased the oil efficiency to 64.53%. The lower oil point force was determined to be 57.32 kN for estimating the maximum oil output. The tangent curve and generalized Maxwell models adequately (R2 = 0.996) described the compression and relaxation processes of pumpkin seeds oil extraction. Peroxide value increased with temperatures. The study provides detailed information useful for processing different bulk oilseeds under uniaxial loading for optimizing the mechanical oil pressing in large-scale oil production.

Beer's Law-Why Integrated Absorbance Depends Linearly on Concentration

As derived by Max Planck in 1903 from dispersion theory, Beer's law has a fundamental limitation. The concentration dependence of absorbance can deviate from linearity, even in the absence of any interactions or instrumental nonlinearities. Integrated absorbance, not peak absorbance, depends linearly on concentration. The numerical integration of the absorbance leads to maximum deviations from linearity of less than 0.1 %. This deviation is a consequence of a sum rule that was derived from the Kramers-Kronig relations at a time when the fundamental limitation of Beer's law was no longer mentioned in the literature. This sum rule also links concentration to (classical) oscillator strengths and thereby enables the use of dispersion analysis to determine the concentration directly from transmittance and reflectance measurements. Thus, concentration analysis of complex samples, such as layered and/or anisotropic materials, in which Beer's law cannot be applied, can be achieved using dispersion analysis.

Beer's law derived from electromagnetic theory

We present a rigorous derivation of Beer's law from electromagnetic theory which allows to establish the connection of wave optics and Maxwell's equations with quantitative UV/VIS and infrared spectroscopy. As a result of this derivation, the attenuation constant is shown to be a function of the index of refraction. For small concentrations and an index of refraction not very different from unity, however, the empirical law is retrieved. Only for very strong absorptions with attenuation constants beyond 10³ l/(mol·cm) and high concentrations, noticeable deviations from the empirical law and its linear concentration dependence are predicted to occur. The derivation is presented in a form, which is not only similar to that leading to the Clausius-Mossotti equation, but also shares the same starting point. Accordingly, it can easily be included in the curricula of physical chemistry.

Aggregation of slipped-cofacial phthalocyanine J-type dimers: Spectroscopic and AFM study

Direct metallation of 2-hydroxyphthalocyanine J-type slipped-cofacial dimeric ligand by Mg, Zn, Cu, Ni and Co salts has been carried out to obtain corresponding metal complexes selectively without any noticeable dissociation or polymerization of the starting ligand. Integrated analysis of aggregation properties in the synthesized series has been conducted with the involvement of AFM microscopy, UV/Vis spectroscopy and theoretical assessment. As a result, a nonlinear relationship between absorption and concentration was found, with aggregation beginning to appear at concentrations above 3.3 × 10^-5 mol L^-1 with predominant formation of trimers from the dimeric molecules in THF solutions.