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UV-Visible Spectroscopy

 

Beckman DU640 UV/Vis spectrophotometer.

 


Schematic of UV- visible spectrophotometer.

Basics of UV-Visible Spectroscopy


Introduction

 

Ultraviolet–visible spectroscopy or ultraviolet-visible spectrophotometry (UV-Vis or UV/Vis) refers to absorption spectroscopy or reflectance spectroscopy in the ultraviolet-visible spectral region. This means it uses light in the visible and adjacent (near-UV and near-infrared [NIR]) ranges. The absorption or reflectance in the visible range directly affects the perceived color of the chemicals involved. In this region of the electromagnetic spectrum, molecules undergo electronic transitions. This technique is complementary to fluorescence spectroscopy, in that fluorescence deals with transitions from the excited state to the ground state, while absorption measures transitions from the ground state to the excited state.[1]
Ultraviolet-visible (UV-vis) spectroscopy is used to obtain the absorbance spectra of a compound in solution or as a solid. What is actually being observed spectroscopically is the absorbance of light energy or electromagnetic radiation, which excites electrons from the ground state to the first singlet excited state of the compound or material. The UV-vis region of energy for the electromagnetic spectrum covers 1.5 - 6.2 eV which relates to a wavelength range of 800 - 200 nm. The Beer-Lambert Law, Equation 1, is the principle behind absorbance spectroscopy. For a single wavelength, A is absorbance (unitless, usually seen as arb. units or arbitrary units), ε is the molar absorptivity of the compound or molecule in solution (M-1cm-1), b is the path length of the cuvette or sample holder (usually 1 cm), and c is the concentration of the solution (M).
EqB1.jpg
(1)
There are three types of absorbance instruments used to collect UV-vis spectra:
  1. 1) Single beam spectrometer.
  2. 2) Double beam spectrometer.
  3. 3) Simultaneous spectrometer.
All of these instruments have a light source (usually a deuterium or tungsten lamp), a sample holder and a detector, but some have a filter for selecting one wavelength at a time. The single beam instrument (Figure 1) has a filter or a monochromator between the source and the sample to analyze one wavelength at a time. The double beam instrument (Figure 2) has a single source and a monochromator and then there is a splitter and a series of mirrors to get the beam to a reference sample and the sample to be analyzed, this allows for more accurate readings. In contrast, the simultaneous instrument (Figure 3) does not have a monochromator between the sample and the source; instead, it has a diode array detector that allows the instrument to simultaneously detect the absorbance at all wavelengths. The simultaneous instrument is usually much faster and more efficient, but all of these types of spectrometers work well.
Figure 1: Illustration of a single beam UV-vis instrument.
Figure 1 (graphics1.jpg)
Figure 2: Illustration of a double beam UV-vis instrument.
Figure 2 (graphics2.jpg)
Figure 3: Illustration of a simultaneous UV-vis instrument.

 
Figure 3 (graphics3.jpg)

 

 

 

 

Principle of ultraviolet-visible absorption

Molecules containing π-electrons or non-bonding electrons (n-electrons) can absorb the energy in the form of ultraviolet or visible light to excite these electrons to higher anti-bonding molecular orbitals.[2] The more easily excited the electrons (i.e. lower energy gap between the HOMO and the LUMO), the longer the wavelength of light it can absorb

What information can be obtained from UV-vis spectra?

UV-vis spectroscopic data can give qualitative and quantitative information of a given compound or molecule. Irrespective of whether quantitative or qualitative information is required it is important to use a reference cell to zero the instrument for the solvent the compound is in. For quantitative information on the compound, calibrating the instrument using known concentrations of the compound in question in a solution with the same solvent as the unknown sample would be required. If the information needed is just proof that a compound is in the sample being analyzed, a calibration curve will not be necessary; however, if a degradation study or reaction is being performed, and concentration of the compound in solution is required, thus a calibration curve is needed.
 
To make a calibration curve, at least three concentrations of the compound will be needed, but five concentrations would be most ideal for a more accurate curve. The concentrations should start at just above the estimated concentration of the unknown sample and should go down to about an order of magnitude lower than the highest concentration. The calibration solutions should be spaced relatively equally apart, and they should be made as accurately as possible using digital pipettes and volumetric flasks instead of graduated cylinders and beakers. An example of absorbance spectra of calibration solutions of Rose Bengal (4,5,6,7-tetrachloro-2',4',5',7'-tetraiodofluorescein, Figure 4) can be seen in Figure 5. To make a calibration curve, the value for the absorbances of each of the spectral curves at the highest absorbing wavelength, is plotted in a graph similar to that in Figure 6 of absorbance versus concentration. The correlation coefficient of an acceptable calibration is 0.9 or better. If the correlation coefficient is lower than that, try making the solutions again as the problem may be human error. However, if after making the solutions a few times the calibration is still poor, something may be wrong with the instrument; for example, the lamps may be going bad.
Figure 4: The molecular structure of Rose Bengal (4,5,6,7-tetrachloro-2',4',5',7'-tetraiodofluorescein).
Figure 4 (FigB4.jpg)
Figure 5: UV-vis spectra of different concentrations of Rose Bengal.
Figure 5 (FigB11.jpg)
Figure 6: Calibration curve of Rose Bengal. Equation of line: y = 0.0977x – 0.1492 (R2 = 0.996)
Figure 6 (FigB21.jpg)

Limitations of UV-visible spectroscopy

Sample

UV-vis spectroscopy works well on liquids and solutions, but if the sample is more of a suspension of solid particles in liquid, the sample will scatter the light more than absorb the light and the data will be very skewed. Most UV-vis instruments can analyze solid samples or suspensions with a diffraction apparatus (Figure 7), but this is not common. UV-vis instruments generally analyze liquids and solutions most efficiently.
Figure 7: Schematic representation of the apparatus for collecting UV-vis spectra from solid materials.
Figure 7 (Spec.jpg)

Calibration and reference

A blank reference will be needed at the very beginning of the analysis of the solvent to be used (water, hexanes, etc), and if concentration analysis needs to be performed, calibration solutions need to be made accurately (e.g., Figure 6). If the solutions are not made accurately enough, the actual concentration of the sample in question will not be accurately determined.

Choice of solvent or container

Every solvent has a UV-vis absorbance cutoff wavelength. The solvent cutoff is the wavelengthbelow which the solvent itself absorbs all of the light. So when choosing a solvent be aware of its absorbance cutoff and where the compound under investigation is thought to absorb. If they are close, chose a different solvent. Table 1 provides an example of solvent cutoffs.

SolventUV absorbance cutoff (nm)
Acetone329
Benzene278
Dimethylformamide (DMF)267
Ethanol205
Toluene285
Water180
The material the cuvette (the sample holder) is made from will also have a UV-vis absorbance cutoff. Glass will absorb all of the light higher in energy starting at about 300 nm, so if the sample absorbs in the UV, a quartz cuvette will be more practical as the absorbance cutoff is around 160 nm for quartz (Table 2).
TABLE 2: Three different types of cuvettes commonly used, with different usable wavelengths.
MaterialWavelength range (nm)
Glass380 - 780
Plastic380 - 780
Fused quartzbelow 380

Concentration of solution

To obtain reliable data, the peak of absorbance of a given compound needs to be at least three times higher in intensity than the background noise of the instrument. Obviously using higher concentrations of the compound in solution can combat this. Also, if the sample is very small and diluting it would not give an acceptable signal, there are cuvettes that hold smaller sample sizes than the 2.5 mL of a standard cuvettes. Some cuvettes are made to hold only 100 μL, which would allow for a small sample to be analyzed without having to dilute it to a larger volume, lowering the signal to noise ratio.

  1. Applications

    An example of a UV/Vis readout
    UV/Vis spectroscopy is routinely used in analytical chemistry for the quantitative determination of different analytes, such as transition metal ions, highly conjugated organic compounds, and biological macromolecules. Spectroscopic analysis is commonly carried out in solutions but solids and gases may also be studied.
    • Solutions of transition metal ions can be colored (i.e., absorb visible light) because d electrons within the metal atoms can be excited from one electronic state to another. The colour of metal ion solutions is strongly affected by the presence of other species, such as certain anions or ligands. For instance, the colour of a dilute solution of copper sulfate is a very light blue; adding ammonia intensifies the colour and changes the wavelength of maximum absorption (λmax).
    • Organic compounds, especially those with a high degree of conjugation, also absorb light in the UV or visible regions of the electromagnetic spectrum. The solvents for these determinations are often water for water-soluble compounds, or ethanol for organic-soluble compounds. (Organic solvents may have significant UV absorption; not all solvents are suitable for use in UV spectroscopy. Ethanol absorbs very weakly at most wavelengths.) Solvent polarity and pH can affect the absorption spectrum of an organic compound. Tyrosine, for example, increases in absorption maxima and molar extinction coefficient when pH increases from 6 to 13 or when solvent polarity decreases.
    • While charge transfer complexes also give rise to colours, the colours are often too intense to be used for quantitative measurement.
    The Beer-Lambert law states that the absorbance of a solution is directly proportional to the concentration of the absorbing species in the solution and the path length.[3] Thus, for a fixed path length, UV/Vis spectroscopy can be used to determine the concentration of the absorber in a solution. It is necessary to know how quickly the absorbance changes with concentration. This can be taken from references (tables of molar extinction coefficients), or more accurately, determined from a calibration curve.
    A UV/Vis spectrophotometer may be used as a detector for HPLC. The presence of an analyte gives a response assumed to be proportional to the concentration. For accurate results, the instrument's response to the analyte in the unknown should be compared with the response to a standard; this is very similar to the use of calibration curves. The response (e.g., peak height) for a particular concentration is known as the response factor.
    The wavelengths of absorption peaks can be correlated with the types of bonds in a given molecule and are valuable in determining the functional groups within a molecule. The Woodward-Fieser rules, for instance, are a set of empirical observations used to predict λmax, the wavelength of the most intense UV/Vis absorption, for conjugated organic compounds such as dienes and ketones. The spectrum alone is not, however, a specific test for any given sample. The nature of the solvent, the pH of the solution, temperature, high electrolyte concentrations, and the presence of interfering substances can influence the absorption spectrum. Experimental variations such as the slit width (effective bandwidth) of the spectrophotometer will also alter the spectrum. To apply UV/Vis spectroscopy to analysis, these variables must be controlled or accounted for in order to identify the substances present.[4]
    UV-Vis spectroscopy is also used in the semiconductor industry to measure the thickness and optical properties of thin films on a wafer. UV-Vis spectrometers are used to measure the reflectance of light, and can be analyzed via the Forouhi-Bloomer dispersion equations to determine the Index of Refraction (n) and the Extinction Coefficient (k) of a given film across the measured spectral range.

    Beer–Lambert law

    Main article: Beer–Lambert law
    The method is most often used in a quantitative way to determine concentrations of an absorbing species in solution, using the Beer-Lambert law:
    A=\log _{{10}}(I_{0}/I)=\varepsilon cL,
    where A is the measured absorbance, in Absorbance Units (AU), I_0 is the intensity of the incident light at a given wavelength, I is the transmitted intensity, L the path length through the sample, and c the concentration of the absorbing species. For each species and wavelength, ε is a constant known as the molar absorptivity or extinction coefficient. This constant is a fundamental molecular property in a given solvent, at a particular temperature and pressure, and has units of 1/M*cm or often AU/M*cm.
    The absorbance and extinction ε are sometimes defined in terms of the natural logarithm instead of the base-10 logarithm.
    The Beer-Lambert Law is useful for characterizing many compounds but does not hold as a universal relationship for the concentration and absorption of all substances. A 2nd order polynomial relationship between absorption and concentration is sometimes encountered for very large, complex molecules such as organic dyes (Xylenol Orange or Neutral Red, for example).

    Practical considerations

    The Beer-Lambert law has implicit assumptions that must be met experimentally for it to apply otherwise there is a possibility of deviations from the law to be observed.[5] For instance, the chemical makeup and physical environment of the sample can alter its extinction coefficient. The chemical and physical conditions of a test sample therefore must match reference measurements for conclusions to be valid.

    Spectral bandwidth

    Monochromaticity of light incident on the sample cell, which is the width of the triangle at one half of the peak intensity. A given spectrometer has a spectral bandwidth that characterizes how monochromatic the light is. It is important to have monochromatic source of radiation for analysis of the sample.[5] If this bandwidth is comparable to the width of the absorption features, then the measured extinction coefficient will be altered. In most reference measurements, the instrument bandwidth is kept below the width of the spectral lines. When a new material is being measured, it may be necessary to test and verify if the bandwidth is sufficiently narrow. Reducing the spectral bandwidth will reduce the energy passed to the detector and will, therefore, require a longer measurement time to achieve the same signal to noise ratio.

    Wavelength error

    In liquids, the extinction coefficient usually changes slowly with wavelength. A peak of the absorbance curve (a wavelength where the absorbance reaches a maximum) is where the rate of change in absorbance with wavelength is smallest.[5] Measurements are usually made at a peak to minimize errors produced by errors in wavelength in the instrument, that is errors due to having a different extinction coefficient than assumed.

    Stray light


    Another important factor is the purity of the light used. The most important factor affecting this is the stray light level of the monochromator [5]
    The detector used is broadband; it responds to all the light that reaches it. If a significant amount of the light passed through the sample contains wavelengths that have much lower extinction coefficients than the nominal one, the instrument will report an incorrectly low absorbance. Any instrument will reach a point where an increase in sample concentration will not result in an increase in the reported absorbance, because the detector is simply responding to the stray light. In practice the concentration of the sample or the optical path length must be adjusted to place the unknown absorbance within a range that is valid for the instrument. Sometimes an empirical calibration function is developed, using known concentrations of the sample, to allow measurements into the region where the instrument is becoming non-linear.
    As a rough guide, an instrument with a single monochromator would typically have a stray light level corresponding to about 3 Absorbance Units (AU), which would make measurements above about 2 AU problematic. A more complex instrument with a double monochromator would have a stray light level corresponding to about 6 AU, which would therefore allow measuring a much wider absorbance range.

    Deviations from the Beer–Lambert law

    At sufficiently high concentrations, the absorption bands will saturate and show absorption flattening. The absorption peak appears to flatten because close to 100% of the light is already being absorbed. The concentration at which this occurs depends on the particular compound being measured. One test that can be used to test for this effect is to vary the path length of the measurement. In the Beer-Lambert law, varying concentration and path length has an equivalent effect—diluting a solution by a factor of 10 has the same effect as shortening the path length by a factor of 10. If cells of different path lengths are available, testing if this relationship holds true is one way to judge if absorption flattening is occurring.
    Solutions that are not homogeneous can show deviations from the Beer-Lambert law because of the phenomenon of absorption flattening. This can happen, for instance, where the absorbing substance is located within suspended particles (see Beer's law revisited, Berberan-Santos, J. Chem. Educ. 67 (1990) 757, and Absorption flattening in the optical spectra of liposome-entrapped substances, Wittung, Kajanus, Kubista, Malmström, FEBS Lett 352 (1994) 37). The deviations will be most noticeable under conditions of low concentration and high absorbance. The last reference describes a way to correct for this deviation.
    Some solutions like copper(II)chloride in water changes colour at a certain concentration because of changed conditions around the coloured ion (the divalent copper ion). For copper(II)chloride it means a shift from blue to green,[6] which would mean that monochromatic measurements would deviate from the Beer-Lambert law.

    Measurement uncertainty sources

    The above factors contribute to the measurement uncertainty of the results obtained with UV/Vis spectrophotometry. If UV/Vis spectrophotometry is used in quantitative chemical analysis then the results are additionally affected by uncertainty sources arising from the nature of the compounds and/or solutions that are measured. These include spectral interferences caused by absorption band overlap, fading of the color of the absorbing species (caused by decomposition or reaction) and possible composition mismatch between the sample and the calibration solution.[7]

    Ultraviolet-visible spectrophotometer

    The instrument used in ultraviolet-visible spectroscopy is called a UV/Vis spectrophotometer. It measures the intensity of light passing through a sample (I), and compares it to the intensity of light before it passes through the sample (I_o). The ratio I/I_o is called the transmittance, and is usually expressed as a percentage (%T). The absorbance, A, is based on the transmittance:
    A=-\log(\%T/100\%)
    The UV-visible spectrophotometer can also be configured to measure reflectance. In this case, the spectrophotometer measures the intensity of light reflected from a sample (I), and compares it to the intensity of light reflected from a reference material (I_o) (such as a white tile). The ratio I/I_o is called the reflectance, and is usually expressed as a percentage (%R).
    The basic parts of a spectrophotometer are a light source, a holder for the sample, a diffraction grating in a monochromator or a prism to separate the different wavelengths of light, and a detector. The radiation source is often a Tungsten filament (300-2500 nm), a deuterium arc lamp, which is continuous over the ultraviolet region (190-400 nm), Xenon arc lamp, which is continuous from 160-2,000 nm; or more recently, light emitting diodes (LED)[1] for the visible wavelengths. The detector is typically a photomultiplier tube, a photodiode, a photodiode array or a charge-coupled device (CCD). Single photodiode detectors and photomultiplier tubes are used with scanning monochromators, which filter the light so that only light of a single wavelength reaches the detector at one time. The scanning monochromator moves the diffraction grating to "step-through" each wavelength so that its intensity may be measured as a function of wavelength. Fixed monochromators are used with CCDs and photodiode arrays. As both of these devices consist of many detectors grouped into one or two dimensional arrays, they are able to collect light of different wavelengths on different pixels or groups of pixels simultaneously.
    Schematic of UV- visible spectrophotometer.
    A spectrophotometer can be either single beam or double beam. In a single beam instrument (such as the Spectronic 20), all of the light passes through the sample cell. I_o must be measured by removing the sample. This was the earliest design and is still in common use in both teaching and industrial labs.
    In a double-beam instrument, the light is split into two beams before it reaches the sample. One beam is used as the reference; the other beam passes through the sample. The reference beam intensity is taken as 100% Transmission (or 0 Absorbance), and the measurement displayed is the ratio of the two beam intensities. Some double-beam instruments have two detectors (photodiodes), and the sample and reference beam are measured at the same time. In other instruments, the two beams pass through a beam chopper, which blocks one beam at a time. The detector alternates between measuring the sample beam and the reference beam in synchronism with the chopper. There may also be one or more dark intervals in the chopper cycle. In this case, the measured beam intensities may be corrected by subtracting the intensity measured in the dark interval before the ratio is taken.
    Samples for UV/Vis spectrophotometry are most often liquids, although the absorbance of gases and even of solids can also be measured. Samples are typically placed in a transparent cell, known as a cuvette. Cuvettes are typically rectangular in shape, commonly with an internal width of 1 cm. (This width becomes the path length, L, in the Beer-Lambert law.) Test tubes can also be used as cuvettes in some instruments. The type of sample container used must allow radiation to pass over the spectral region of interest. The most widely applicable cuvettes are made of high quality fused silica or quartz glass because these are transparent throughout the UV, visible and near infrared regions. Glass and plastic cuvettes are also common, although glass and most plastics absorb in the UV, which limits their usefulness to visible wavelengths.[1]
    Specialized instruments have also been made. These include attaching spectrophotometers to telescopes to measure the spectra of astronomical features. UV-visible microspectrophotometers consist of a UV-visible microscope integrated with a UV-visible spectrophotometer.
    A complete spectrum of the absorption at all wavelengths of interest can often be produced directly by a more sophisticated spectrophotometer. In simpler instruments the absorption is determined one wavelength at a time and then compiled into a spectrum by the operator. By removing the concentration dependence, the extinction coefficient (ε) can be determined as a function of wavelength.

    Microspectrophotometry

    UV-visible spectroscopy of microscopic samples is done by integrating an optical microscope with UV-visible optics, white light sources, a monochromator, and a sensitive detector such as a charge-coupled device (CCD) or photomultiplier tube (PMT). As only a single optical path is available, these are single beam instruments. Modern instruments are capable of measuring UV-visible spectra in both reflectance and transmission of micron-scale sampling areas. The advantages of using such instruments is that they are able to measure microscopic samples but are also able to measure the spectra of larger samples with high spatial resolution. As such, they are used in the forensic laboratory to analyze the dyes and pigments in individual textile fibers,[8] microscopic paint chips [9] and the color of glass fragments. They are also used in materials science and biological research and for determining the energy content of coal and petroleum source rock by measuring the vitrinite reflectance. Microspectrophotometers are used in the semiconductor and micro-optics industries for monitoring the thickness of thin films after they have been deposited. In the semiconductor industry, they are used because the critical dimensions of circuitry is microscopic. A typical test of a semiconductor wafer would entail the acquisition of spectra from many points on a patterned or unpatterned wafer. The thickness of the deposited films may be calculated from the interference pattern of the spectra. In addition, ultraviolet-visible spectrophotometry can be used to determine the thickness, along with the refractive index and extinction coefficient of thin films as described in Refractive index and extinction coefficient of thin film materials. A map of the film thickness across the entire wafer can then be generated and used for quality control purposes.[10]

    Additional applications

    UV/Vis can be applied to determine the kinetics or rate constant of a chemical reaction. The reaction, occurring in solution, must present color or brightness shifts from reactants to products in order to use UV/Vis for this application.[2] For example, the molecule mercury dithizonate is a yellow-orange color in diluted solution (1*10^-5 M), and turns blue when subjected with particular wavelengths of visible light (and UV) via a conformational change, but this reaction is reversible back into the yellow "ground state".[11]
    The rate constant of a particular reaction can be determined by measuring the UV/Vis absorbance spectrum at specific time intervals. Using mercury dithizonate again as an example, one can shine light on the sample to turn the solution blue, then run a UV/Vis test every 10 seconds (variable) to see the levels of absorbed and reflected wavelengths change over time in accordance with the solution turning back to yellow from the excited blue energy state. From these measurements, the concentration of the two species can be calculated.[12] The mercury dithizonate reaction from one conformation to another is first order and would have the integral first order rate law : ln[A](time t)=−kt+ln[A](initial). Therefore graphing the natural log (ln) of the concentration [A] versus time will graph a line with slope -k, or negative the rate constant. Different rate orders have different integrated rate laws depending on the mechanism of the reaction.
    An equilibrium constant can also be calculated with UV/Vis spectroscopy. After determining optimal wavelengths for all species involved in equilibria, a reaction can be run to equilibrium, and the concentration of species determined from spectroscopy at various known wavelengths. The equilibrium constant can be calculated as K(eq) = [Products] / [Reactants].
  2. References


  3. Skoog, Douglas A.; Holler, F. James; Crouch, Stanley R. (2007). Principles of Instrumental Analysis (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 169–173. ISBN 9780495012016.

  4. Metha, Akul (13 Dec 2011). "Principle". PharmaXChange.info.

  5. Metha, Akul (22 Apr 2012). "Derivation of Beer-Lambert Law". PharmaXChange.info.

  6. Misra, Prabhakar; Dubinskii, Mark, eds. (2002). Ultraviolet Spectroscopy and UV Lasers. New York: Marcel Dekker. ISBN 0-8247-0668-4.

  7. Metha, Akul (14 May 2012). "Limitations and Deviations of Beer-Lambert Law". PharmaXChange.info.

  8. Ansell, S.; Tromp, R. H.; Neilson, G. W. (1995). "The solute and aquaion structure in a concentrated aqueous solution of copper(II) chloride". J. Phys.: Condens. Matter 7 (8): 1513–1524. doi:10.1088/0953-8984/7/8/002.

  9. Sooväli, L.; Rõõm, E.-I.; Kütt, A. et al. (2006). "Uncertainty sources in UV-Vis spectrophotometric measurement". Accred. Qual. Assur 11: 246–255. doi:10.1007/s00769-006-0124-x.

  10. Forensic Fiber Examination Guidelines, Scientific Working Group-Materials, 1999, http://www.swgmat.org/fiber.htm

  11. Standard Guide for Microspectrophotometry and Color Measurement in Forensic Paint Analysis, Scientific Working Group-Materials, 1999, http://www.swgmat.org/paint.htm

  12. "Spectroscopic thin film thickness measurement system for semiconductor industries", Horie, M.; Fujiwara, N.; Kokubo, M.; Kondo, N., Proceedings of Instrumentation and Measurement Technology Conference, Hamamatsu, Japan, 1994,(ISBN 0-7803-1880-3).

  13. Sertova (June 2000). "Photochromism of mercury(II) dithizonate in solution". Journal of Photochemistry and Photobiology A: Chemistry 134 (3): 163–168. doi:10.1016/s1010-6030(00)00267-7. Retrieved 2014-11-11.

  14. UC Davis. "The Rate Law". ChemWiki. Retrieved 2014-11-11.

 

Bibliography

  • D. A. Skoog, F. J. Holler, S. R. Crouch, Principles of Instrumental Analysis, 6th Ed., Thomson Brooks/Cole (2007).
  • J. P. Sibilia, Materials Characterization and Chemical Analysis, 2nd Ed., Wiley-VCH, New York (1996).
  • D. C. Harris, Quantitative Chemical Analysis, 7th Ed., Freeman, New York(2007).



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