Beer's Law – Why Absorbance Depends (Almost) Linearly on Concentration

Beer's law assumes a strictly linear dependence of the absorbance from concentration. Usually, chemical interactions and instrumental imperfection are made responsible for experimental deviations from this linearity. In this contribution we show that even in the absence of such interactions and instrumental errors, absorbance should be only approximately proportional to concentration. This can be derived from the quadratic dependence of the complex refractive index, and, by that, of the molar attenuation coefficient, from the dielectric constant and its frequency dispersion. Following dispersion theory, it is the variation of the real and the imaginary part of the dielectric function that depends linearly on concentration in the absence of interactions between the oscillators. We show that this linear correlation translates into a linear dependence of the absorbance for low concentrations or molar oscillator strengths based on an approximation provided by Lorentz in 1906. Accordingly, Beer's law can be derived from dispersion theory.     

Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration

Beer's empiric law states that absorbance is linearly proportional to the concentration. Based on electromagnetic theory, an approximately linear dependence can only be confirmed for comparably weak oscillators. For stronger oscillators the proportionality constant, the molar attenuation coefficient, is modulated by the inverse index of refraction, which is itself a function of concentration. For comparably weak oscillators, the index of refraction function depends, like absorbance, linearly on concentration. For stronger oscillators, this linearity is lost, except at wavenumbers considerably lower than the oscillator position. In these transparency regions, linearity between the change of the index of refraction and concentration is preserved to a high degree. This can be shown with help of the Kramers–Kronig relations which connect the integrated absorbance to the index of refraction change at lower wavenumbers than the corresponding band. This finding builds the foundation not only for refractive index sensing, but also for new interferometric approaches in IR spectroscopy, which allow measuring the complex index of refraction function.     

Beyond Beer's Law: Revisiting the Lorentz-Lorenz Equation

In this contribution we show how the Lorentz-Lorenz and the Clausius-Mosotti equations are related to Beer's law. Accordingly, the linear concentration dependence of absorbance is a consequence of neglecting the difference between the local and the applied electric field. Additionally, it is necessary to assume that the absorption index and the related refractive index change is small. By connecting the Lorentz-Lorenz equations with dispersion theory, it becomes obvious that the oscillators are coupled via the local field. We investigate this coupling with numerical examples and show that, as a consequence, the integrated absorbance of a single band is in general no longer linearly depending on the concentration. In practice, the deviations from Beer's law usually do not set in before the density reaches about one tenth of that of condensed matter. For solutions, the Lorentz-Lorenz equations predict a strong coupling also between the oscillators of solute and solvent. In particular, in the infrared spectral region, the absorption coefficients are prognosticated to be much higher due to this coupling compared to those in the gas phase.     

Curing kinetics of the epoxy system diglycidyl ether of bisphenol A/isophoronediamine by Fourier transform infrared spectroscopy

The curing reaction of an epoxy system consisting of a diglycidyl ether of bisphenol A (BADGE, n = 0) and isophoronediamine (IPD), was studied by Fourier transform infrared spectroscopy (FTIR). The degree of conversions and the reaction rates at different isothermal curing temperatures were calculated from the infrared spectra using a method derived from Beer's law. This method is based on the ratio of the height of the characteristic absorbance peak to reference absorbance peak. A kinetic model proposed by Sourour and Kamal has been used to fit experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

Combining linear polarization spectroscopy and the Representative Layer Theory to measure the Beer–Lambert law absorbance of highly scattering materials

Visible and Near Infrared (Vis–NIR) Spectroscopy is a powerful non destructive analytical method used to analyze major compounds in bulk materials and products and requiring no sample preparation. It is widely used in routine analysis and also in-line in industries, in-vivo with biomedical applications or in-field for agricultural and environmental applications. However, highly scattering samples subvert Beer–Lambert law's linear relationship between spectral absorbance and the concentrations. Instead of spectral pre-processing, which is commonly used by Vis–NIR spectroscopists to mitigate the scattering effect, we put forward an optical method, based on Polarized Light Spectroscopy to improve the absorbance signal measurement on highly scattering samples. This method selects part of the signal which is less impacted by scattering. The resulted signal is combined in the Absorption/Remission function defined in Dahm's Representative Layer Theory to compute an absorbance signal fulfilling Beer–Lambert's law, i.e. being linearly related to concentration of the chemicals composing the sample. The underpinning theories have been experimentally evaluated on scattering samples in liquid form and in powdered form. The method produced more accurate spectra and the Pearson's coefficient assessing the linearity between the absorbance spectra and the concentration of the added dye improved from 0.94 to 0.99 for liquid samples and 0.84–0.97 for powdered samples.     

Simultaneous Spectrophotometric Estimation of Rabeprazole Sodium and Itopride HCl

Abstract

Two simple, rapid, accurate, and economical analytical methods have been developed for the simultaneous estimation of Rabeprazole Sodium and Itopride Hydrochloride in combined capsule dosage form. First method is based on the determination of Q‐value and second method is based on simultaneous equation method. Rabeprazole Sodium has absorbance maxima at 284 nm and Itopride Hydrochloride has absorbance maxima at 258 nm in methanol AR. The absorption ratio (Q‐value) was determined at 266.6 nm (Iso‐bestic point) and 284 nm (λmax of Rabeprazole Sodium). Both the drugs obey Beer's law in the concentration ranges employed for these methods. Both the methods were found to be simple, rapid, accurate, and can be adopted in routine analysis of drugs in formulations. The accuracy and reproducibility of the proposed method was statistically validated by recovery studies     

Spectrophotometric Determination of Ruthenium with 2-Amino-3-Hydroxypyridine

Ruthenium(III) complexation with 2-amino-3-hydroxpyridine is studied spectrophotometrically. The violet-red complex (λ_max 520 nm; ? 4460) contains metal and the ligand in 1 : 1 molar ratio, adheres to Beer's law from 0 to 16.5 µg/ml of ruthenium concentration and shows maximum and constancy in absorbance between pH 5.3 and 6.5. The complexation is used in the spectro-photometric determination of ruthenium in coexistence with several ions including some from platinum metals.     

Microdetermination of Trace Cobalt In Water By Direct Polyurethane From Thin-Layer Spectrophotometry

Abstract

Traces of cobalt in water sample are preconcentrated into a white thin-layer of a parallelpiped polyurethane foam, by the sorption of its thiocyanate complex from acidic medium.

The cobalt thiocyanate complex possesses a blue colour in the foam matrix which has a maximum absorption at 625 nm. the absorbance of the Dlue species is mesured directly in the foam matrix at 680 625 and 540 nm. with different media in the measuring cell. the net absorbance at 625 nm is calculated using a multiwavelength program, where the value of the blank is subtracted. Beer's law is obeyed in the concentration range 0.025-2.0 μg per dl with a detection limit of 25 μg per dl for a 100 ml sample volume. the method is specific for cobalt     

Ultraviolet Spectrophotometric Determination of Selenium by 1-Pyrrolidinecarbodithioate Method

Abstract

An ultraviolet spectrophotometric method for the determination of selenium has been developed using ammonium 1-pyrrolidinecarbo-dithioate. The selenium(IV)-1-pyrrolidinecarbodithioate complex is extracted with chloroform and the absorbance of the extract measured at 303 mμ. Conformity to Beer's law was found for 0.5 to 9 ppm of selenium.     

Spectrophotometric Determination of Sulphathiazole

Abstract

A new, rapid, simple, selective and sensitive method for the spectrophotometric determination of sulphathiazole is described. The drug forms a yellow coloured complex with Pd(II) in alkaline medium (1 M NaOH). The drug complex exhibits maximum absorbance at 410 nm, with a molar absorptivity of 1.9 × 10³ 1 mol^?1 cm^?1. Effects of molarity of NaOH, reagent concentration on the proper complex formation are discussed. The system obeys Beer's law in the concentration range of 10-160 ppm of sulphathiazole.     

Determination of Ketotifen Fumarate in Raw Material and Pharmaceutical Products Using Ion‐pair Formation

Abstract

A simple and sensitive extractive spectrophotometric method has been described for the determination of Ketotifen. The developed method is based on the formation of a colored ion‐pair complex (1∶1 drug/dye) of Ketotifen and bromocresol green in buffer pH 3 and extraction in chloroform. The extracted complex shows absorbance maxima at 423 nm. Beer's law is obeyed in the concentration range of 5.15–61.91 µg/ml. The proposed method has been applied successfully for the determination of drug in commercial tablets and syrup dosage form. No significant interference was observed from the excipients commonly used as pharmaceutical aids with the assay procedure.     

Utility of p-chloranilic Acid and 2,3 Dichloro-5,6-dicyano p-Benzoquinone (DDQ) for the Spectrophotometric Determination of Triamterene

Abstract

Two simple and sensitive spectrophotometric procedures are suggested for analysis of triamterene. The first procedure is based on the reaction of triamterene with p-chloranilic acid (p-CA) in methylene chloride to form a highly stable coloured product, exhibiting maximum absorbance at λ 530 nm. Beer's law is obeyed in the range of 40–220 μg.ml^?1 with a mean percentage accuracy of 99.98 ± 0.446. Limit of determination is 20 μg.ml^?1. In the second procedure, the drug is determined via charge transfer complex formation with 2,3 dichloro-5,6-dicyano p-benzoquinone (DDQ) using methylene chloride as a solvent. Here the reaction product has two well defined maxima at 460 nm and 530 nm where each has been utilized for quantitative determination. Beer's law is obeyed in concentration ranges of 25–125 μg.ml^?1 and 25–150 μg.ml^?1 with mean percentage accuracies of 99.92 ± 0.449 and 100.00 ± 0.511 for both maxima. 460 and 530 nm. respectively. Limit of determination is 12.5 μg.ml^?1 at both maxima. Optimum conditions for each procedure have been studied and the stoichiometry of both reactions was ascertained using Job's method of continuous variation. The validity of the suggested procedures was assessed by applying the standard addition technique using the drug capsules. Both procedures are statistically analyzed as compared with BP method for analysis of triamterene (non aqueous titration) revealing good accuracy and precision as indicated by t and F tests.     

Facile and Sensitive Spectrophotometric Determination of Chromium

Abstract

A simple, sensitive, and selective spectrophotometric method is developed for the determination of trace amounts of chromium(VI) in various samples. The method is based on oxidation of 2,4-dinitrophenylhydrazine (DNPH) by chromium in an acidic medium and coupling with β-naphthol to give a red-colored dye, having an absorbance maximum at 663 nm, and/or coupling with α-naphthol to give a violet-colored dye, having an absorbance maximum at 503 nm. The method obeys Beer's law in the concentration range 0.02–4.0µg mL^?1 and 0.05–9.0 µg mL^?1, respectively. The optimum reaction conditions and other important analytical parameters were established. Interference caused by various nontarget ions was also investigated. The proposed method is applied to the analysis of chromium(VI) in environmental, pharmaceutical, and steel samples. The performance of the proposed method is evaluated using the student's t and F -tests.     

Spectrophotometric Determination of Naproxen in Pure and Pharmaceutical Preparations

ABSTRACT

Naproxen reacts with 1-naphthylamine and sodium nitrite to give an orangish red colour having maximum absorbance at 460-480 nm (working wavelength 480 nm). The reaction is selective for naproxen with 0.001 mg/ml as visual limit of quantitation and provides a basis for a new spectrophotometric determination. The reaction obeys Beer's law from 0.01mg to 6.5 mg/10ml of naproxen and the relative standard deviation is 1.5%. The quantitative assessment of tolerable amount of other drugs is also studied.     

Understanding Refractive Index Changes in Homologous Series of Unbranched Organic Compounds Based on Beer's Law

Changes of the refractive index for homologous series of hydrocarbons are usually plotted versus the density. While there is a clear linear dependence for alkanes and alkenes, the linearity deteriorates for homologous series with functional groups involving heteroatoms. The slope can even become negative, e. g., for carboxylic acids. For gaining a deeper understanding and to establish a more general correlation, we reinvestigate the corresponding theories starting with the Newton-Laplace, Gladstone-Dale and the Lorentz-Lorenz rules. We revisit the concept of molar refractivity pioneered by Landolt and Brühl and show that it is closely connected with a twin of Beer's law. We conclude that the refractive index of homologues series should better be plotted versus the molar concentration of the main UV-chromophore, the C−H bond, which actually causes the refractive index changes. This new approach is not limited to alkanes and alkenes but holds for homologous series with functional groups including heteroatoms.     

Determination of Vanadium with Chlorosubstituted Hydroxamic Acids for Photometric and Atomic Absorption Spectrophotometric Determination

Abstract

The eight newly synthesized chlorosubstituted hydroxamic acids are described for the extraction and spectrophotometric determination of vanadium. The sensitive and selective reagent, N-m-Chlorophenylpalmito hydroxamic acid, (m-CPPHA), which gives violet coloured vanadium complex was extracted with chloroform from 6M HCl. The violet coloured complex thus obtained has a maximum absorbance at 520 nm and molar absorptivity 4.9 ± 10³1mol^?1cm^?1. The Beer's law obeyed in the region 0.50-12.0ppm. Effects of acidity, reagent concentration, diverse ions have also been investigated. A comparison has been made with atomic absorption spectrophotometric method. Vanadium has been determined in the environment, e.g. plant, soil, rock, etc.     

Spectrophotometry Determination of Amitriptyline-HCl in Pure and Pharmaceutical Preparations.

ABSTRACT

Amitriptyline-HCl reacts with ammonium molybdate in concentrated sulphuric acid after heating for 20 min. at 100°C to give a green colour having maximum absorbance at 660 nm. The reaction is selective for amitriptyline with 0.01 mg/10ml as visual limit of identification and provides a basis for a new spectrophotometric determination. The reaction obeys Beer's law from 0.01 mg to 1.4 mg/10ml of amitriptyline-HCl and the relative standard deviation is 0.35%. The quantitative assessment of tolerable amount of other drags is also studied.     

Extractive Spectrophotometric Methods for Determination of Clarithromycin in Pharmaceutical Formulations Using Bromothymol Blue and Cresol Red

Two simple and sensitive extractive spectrophotometric methods have been described for the analysis of clarithromycin in pure form and in pharmaceutical formulations. The methods involved formation of yellow colored chloroform extractable ion‐association complexes of clarithromycin with bromothymol blue (BTB) and cresol red (CR) in buffered aqueous solution at pH 4. The extracted complexes showed maximum absorbance at 410 and 415 nm for BTB and CR, respectively. Beer's law is obeyed in the concentration ranges 0.1–20 μg mL^?1 and 2.0–20 μg mL^?1 of clarithromycin with molar absorptivity of 2.01 × 10⁴ and 4.378 × 10³ for BTB and CR, respectively. The composition ratio of the ion‐association complex was clarithromycin: BTB and CR = 1:1 as established by Job's method. The methods have been applied to the determination of drug in commercial formulations. The results of analysis were validated statistically and through recovery studies.     

Atomic Absorption Spectrophotometric Determination of Copper (I) after Adsorption of its 4′-(p-Methoxyphenyl)-2, 2′:6′, 2″-Terpyridine Tetraphenylborate Ion-Associated Complex on Microcrystalline Naphthalene

A new atomic absorption spectrophotometric method has been developed for the trace determination of copper after adsorption of its 4′-(p-methoxyphenyl)-2, 2′:6′,2″-terpyridine tetraphenylborate ion-associated complex on microcrystalline naphthalene. This complex is adsorbed on microcrystalline naphthalene in the pH range 0.5-11.0 by shaking for a few seconds. The solid mass so formed is separated by filtration, dissolved in dimethylformamide and the absorbance is measured at 324.7 nm. Beer's law is obeyed in the concentration range 5.0-60.0 μg of copper in 10 ml of the final dimethylformamide solution. The replicate determinations of a sample containing 30 μg of copper gave a mean absorbance of 0.171 with a standard deviation of 0.0019 and a relative standard deviation of 1.11%. The method has been applied to the determination of copper in certain standard reference materials.     

Spectrophotometric Determination of Cephalosporins in Pure Form and in Pharmaceutical Preparations

Abstract

A spectrophotometric determination is described for cephalosporins, offering adequate sensitivity and good precision. The procedure applies successfully to a wide variety of cephalosporins, also in pharmaceutical preparations: cephalothin, cefacetrile, cephapirin, cefotaxime, ceftizoxime, cephaloridine, cefazolin, cefamandole nafate, cephalexin, cefadroxil, cefoxitin and cefuroxime. The method employs a reaction with ammonium molybdate in sulphuric acid medium. The antibiotic is heated at 91.5°C for 15 min and the absorbance of the coloured product is measured at 670 nm against a reagent blank treated similarly. Beer's law is obeyed up to 125 to 150 μg of cephalosporin in the 5-ml final solution. The effect of reagent concentration and reaction conditions are discussed.