Lambert s law. Beer s law. di x / I x = -kdx (-di x = k I x dx) = - a c dx. I/I 0 = e -kl T = A = - log (T) = - log (I/I 0 )

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1 di x / I x = -kdx (-di x = k I x dx) Integrating this equation from x=0 ~ l (I x =I 0 ~I) gives ; ln I ln I 0 = -kl ln I/I 0 = -kl Expressing the number of photons absorbed by the slab as di x, and the total number of photons incident on the slab as I x, the fraction of photons absorbed by the slab is given by I 0 : Initial light intensity I : Light intensity after it passes through the sample Chapter 7 - di x / I x = -kdx (-di x = k I x dx) Integrating this equation from x=0 ~ l (I x =I 0 ~I) gives ; ln I ln I 0 = -kl ln I/I 0 = -kl T = I/I 0 = e -kl T : Transmittance the fraction of incident light at a specified wavelength that passes through a sample. For most applications, absorbance values are used since the relationship between absorbance and both concentration and path length normally is linear Transformation into a linear expression by taking the logarithm gives : = - log (T) = - log (I/I 0 ) : bsorbance ex. If = T = 0. (0%) = T = 0.0 (%) = T = 0.00 (0.%) Chapter 7 - T = I/I 0 = e -kl di x / I x = - k dx Beer s law Beer s law : the amount of light absorbed is proportional to the number of absorbing molecules(or concentration) through From lambert s law (l) = - a c dx where, k = a c higher concentration leads to a darker color = - log T = - log e -kl = kl log e l (l) Chapter 7-5 Chapter 7-6

2 Beer-Lambert law Combining the two laws gives k = a c T = I/I 0 = e -kl =e -acl From the definition of absorbance () 5 5 = - log T = - log (I/I 0 ) = - log e -acl = acl log e = εcl (a loge = const = ε) (Beer-Lambert law) where, ε = molar absorptivity (L mol - cm - ) c (mol/l) 5 Wavelength (nm) λ max Chapter 7-7 Chapter 7-8 Determination of concentration of unknown samples: Determination of concentration of unknown samples: The absorbance becomes linear with the concentration according to = log (/T) = ε c l = ε c (when l = ) - n unknown concentration of an analyte can be determined by measuring the amount of light that a sample absorbs and applying Beer's law. where, : bsorbance c : Concentration (mol/l) l : path length of light (ex. = cm) ε : molar absorption coefficient, mol - cm - (molar extinction coefficient) (molar absorptivity) c - If the absorptivity coefficient is not known, the unknown concentration can be determined using a working curve of absorbance versus concentration derived from standards. ( Calibration curve) Chapter 7-9 Chapter 7-50

3 Compound What is the significance of the molar absorptivity, ε? ε = / (c l) - In words, " ε is a measure of the amount of light absorbed per unit concentration". - Molar absorptivity is a constant for a particular substance, so if the concentration of the solution is halved so is the absorbance, which is exactly what you would expect. - Therefore, a compound with a high molar absorptivity is very effective at absorbing light (of the appropriate wavelength), and hence low concentrations of a compound with a high molar absorptivity can be easily detected. What is the significance of the molar absorptivity, ε? ε = / (c l) ex. Compound : high value of ε (ex. 00,000 L mol - cm - ) Compound B : low value of ε (ex. 0 L mol - cm - ) How much concentration is required to get an absorbance (=) (in a cm pathlength cuvette)? Compound : ε = / (xc) c = / 00,000 = mol L - Compound B : ε = / (xc) c = / 0 = 0.05 mol L - a compound with a high molar absorptivity(compound ) is very effective at absorbing light.0 Compound B c Chapter 7-5 Chapter 7-5 Limitations of the Beer-Lambert law - Limitations of Beer-Lambert law Note that the Law is not obeyed at high concentrations. deviations in absorptivity coefficients at high concentrations (>0.0M, > ~.5) - Causes of nonlinearity include bsorbance Dye conc. (mmol/l) aggregations at high concentrations deviations in absorptivity coefficients at high concentrations (>0.0M) due to electrostatic interactions between molecules in close proximity scattering of light extinction coefficient is the function of refractive index which is dependent on n ( n ) the concentration. non-ideal assumptions t Chapter 7-5 Chapter 7-5

4 ggregation & bsorption efficiency pplications of UV-visible Spectroscopy - plot of absorbance versus concentration gives according to equation: standard calibration curve. - Points that Can be joined by a straight line with slope ε x l, ifa cm cuvette is used, l = cm the slope gives the molar absorptivity. = X < X Chapter If instead of a straight line the data show that the absorbance values level off at high concentrations, then the Lambert-beers law is not obeyed at high concentrations, and equation should not be used in that concentration range. (i.e. > ) - standard calibration curve can be used to determine the concentrations Chapter 7-56 pplications of UV-visible Spectroscopy - Solvents commonly used for UVvisible spectroscopy are those with little or no absorbance in the wavelength range under investigation - In the visible region, nm, water and most organic solvents can be used without problems. - In the 0-0 nm region methanol, isopropanol, n-hexane, cyclohexane, acetonitrile and water can be used and between 90 and 0 nm, only water, n- hexane and acetonitrile are recommended. - The cuvettes used in the UV region must be made of quartz because glass absorbs UV light. - The first excited state of azobenzene is less stable, charge separated structure st excited state Ground state () () st excited state : (Charge separated structure) sextet of electrons unstable Ground state: (Uncharged structure) - The bathochormicity increases as the energy difference between the ground and st excited state decreases since Polymer Coloration Technology Chapter 7-57 Chapter 7-58

5 - donor substitution ortho or para to the azo linkage markedly stabilizes the charge separated excited state structure by donation of its lone pairs of electrons, so that a resonance form can be written () in which the outer shell of each atom has 8 electrons. st excited state () () NR Excited State E E E E = hc/ E > E, Ground state () - The energy difference between () and () is much smaller than those between () and () Bathochormic shift () NR Chapter 7-59 Ground State < Bathochromic shift : 장파장이동 Chapter 7-60 Dye Dye ther spectral properties - Molar extinction coefficient (ε max ) : molar absorptivity, parameters defining how strongly a substance absorbs light at a given wavelength, per mass density or per molar concentration ε = / (c l) (M - cm - ) - Half bandwidth ( λ / ): full width at half maximum (FWHM). - λ max - Ε max -* λ / Dye Dye *half-width (of a band) The full width of a spectral band at a height equal to half of the height at the band maximum. lso known as full width at half maximum (FWHM). The dimension of band width should be either inverse length (wavenumbers) or inverse time (frequencies) so that the values give an indication of the energies. Chapter 7-6 The full width of a spectral band at a height equal to half of the height at the band maximum. The narrower, the brighter! Chapter 7-6