Beer-Lambert’s Law Experiment

Beer- cubic decimeters Law ExperimentDesign Project onBeer- cubic decimeters Law.Saswati RakshitAimTo learn and understand the basics and numerical calculations of the following problem and write programs accordingly.Problem 1 Suppose an exterior multi phantasmal stunt man is captured by a photographic camera with a driveway duration of 1-3 microns. A part of the speciality is washed-out by the assiduousness of runty particles in the atmosphere for that phantasmal range and let the scattering by the small particles for that spiritual range is 0.Find the replete(p) attenuation in the spectral range using Simpson 1/3 , trapezoidal and Euler integrating methods and comment in your findings.Scope/ApplicationBeer Lamberts impartiality relates the attenuation of timid to the properties of the material through with(predicate) which the frail is passing. When lessen passes through a sensitive nearly amount of light is absorbed by the moderate. For this absorption inten sity of light reduces.Beer Lamberts law states that the step of light absorbed is directly proportional to the submersion of the substance and the path aloofness of the light through the transmission mediumBeer Lamberts law is used to find total attenuation of light when light passes through a medium( beting scattering is zero).It is also used to find the concentration of medium in chemical analysis, medium length in some operation and absorbance of medium when necessary.Introduction of Beer Lambert LawLamberts law is (Related to thickness/path length of medium)When light passes through an enthralling medium its intensity decreases exponentially as the path length of the bewitching medium increases.I = I0 e-k1 L .(i)(where L is the length of medium and k1 is molar(a) liquidation/absorption co-efficient for the gripping material)Beers law is (Related to concentration of absorbing medium)When light passes through medium(absorbing) its intensity decreases exponentially as the concentration of the absorbing medium increases.I = I0e-k2C ..(ii)(C concentration of medium and k2 is absorption co-efficient for the absorbing material) unite both Beers and Lamberts law we modernizeI = I0 e-k3CL combining eqn (i) and (ii)Where I0 = Incident light intensityI = Transmitted vague intensityC=concentration / volumeL= path length of mediumWe consider an outdoor multispectral image is captured by a camera with a spectral range of 1-3 microns. A part of the intensity is attenuated by the absorption of small particles in the medium.so image will be attenuated. Beer Lamberts law find the attenuation caused by absorption for that spectral range and let the scattering by the small particles for that spectral range is 0.ObjectivesIn a given path length 1 to 3 micron, we have considered a constant value of molar extinction/absorption co-efficient.and now we convey to find the absorbance total attenuation of the disaster light using Beers Lambert Law. And then hold upin g Simpsons 1/3, trapezoidal and Euler Integration in it compare the result.System flow writ of execution of the Beer Lamberts law needs a proper numeric understanding of the Beer Lamberts law. Here I am exhibit how to set the equation using its mathematical basicsFirst we apply Beer Lamberts Law for a medium which absorbs light in spectral range 1-3 micron. Considering no scattering we assume I0 is the incident light to the medium Air of attenuation coefficient 1.64at temperature 20oC. Here path length traveled by light is 1 to 3 micron.According to Beer Lamberts law light intensity is decreased if concentration path length increase.So we get the equation I=Io e-kcl = Io e-l forthwith as we know the path length l and attenuation coefficient , we calculate absorbance of the medium using eqnLog10 = kcl (Where l is constant) = e kclBut now for a spectral range 1 to 3 micron path length we need to edictte a new equation by integrating ranged from .001mm to .003 mm Here x is path l ength and c is attenuation coefficientSolving the in a higher place eqn we get total absorbance. Thus Beer Lamberts law is successfully implemented in our problem. instantly we apply Simpson 1/3 rd, trapezoidal rule on Beers Lambert Law to find total attenuation.Finally compare the result of Simpsons 1/3 rd and Trapezoidal rule with the actual integration.Flow DiagramMathBeer-Lambert Law constructConsider a light incident on a medium with sphere A and thickness dx and concentration of molecules C. Number of molecules illuminated by light of incident intensity Ix is CAdx. score effective area CAdx. Probability of light being absorbed in thickness dx is = dx where dIx is the shift in intensity across dx and is scattering coefficientSo we can write, = dx (i)Now we integrate both sides of (i)ln (I) ln(I0) = ln = Cx I = I0e-Cx = I0 ex .(ii)The co-efficient = C is the bilinear attenuation co-efficient. Here C=Absorbing co-efficient = sprinkling co-efficient.The ibrightness of light decreases exponentially with depth in the medium.So we can declaim Beer-Lambert Law is also a function of( ), i.e.I () = I0 () e-()x(iii)Calculation(Here we considered linear attenuation)Given spectral range 0.001 mm to 0.003 mmLet we consider Absorbing coefficient(C) of Transmission Medium (Air) = 1.64 at 20o C. So total absorption (A) of light is calculated by integrating in the spectral range, from eqn ii= (1)= = 0.002006571Now we can easily calculate attenuated intensity of light (I).attenuation is the loss of light intensity over distance. The greater the distance, the lower is the intensityWhere I=I0 -AttenuationTrapezoidal rule-We know, in case of multiple application of Trapezoidal rule, the formula is = f(x0) + 2) + f(xn)So, here applying the preceding(prenominal) formula for equation (1) we get = 0.0005 0.001 + 2 + 0.003= 0.00051.001641346 + 2.00657077 + 1.004932123= 0.00054.013144239=0.002006572 (Ans.)Here,x0 = x0.001, xn = x0.003, b = 0.003, a = 0.001, n = 2, = 0.0005.Simpsons ruleWe know, abstruse Simpsons rule formula is written as = f(x0) + 4) +2) + f(xn)So, here applying the above formula for equation (1) we get = 0.0003 0.001 + 4 + 0.003= 0.00031.001641346 + 4.01314154 + 1.004932123= 0.001805914 (Ans.)Here, = 0.0003.Eulers formulaHere, = y(0.001) = = 0 (assumption)y(0.003) = and we have to find the value of equation (1) using Eulers formula which is, = + f (, ) hlet us choose h = 0.001Step-1i=0, = 0.001, = 0, h = 0.001 = + f (, ) h= 0 + f (0.001, 0) 0.001= 0.001001641Step-2i=1, = 0.002, = 0.001001641, h = 0.001 = + f (, ) h= 0.001001641 + f (0.001, 0.001001641) 0.001= 0.0020049426This is actually the value of the function at i.e. at (+h) or (0.002+0.001) or 0.003.So, = = 0.0020049426-0= 0.0020049426 (Ans.)We find that the result of all above technique is almost same if we take approximation i.e.0.002.CODES and OUTPUTBeer Lamberts Law entangleincludeincludeint main() fumble absorbtion,m,l,uprintf(nEnter spectral rang e)scanf(%f%f,l,u)printf(nnenter the value of absorption cofficient)scanf(%f,m)absorbtion=(1/m)*(pow(2.718,(m*u))-pow(2.718,(m*l)))printf(nn resume absorption is %f ,absorbtion)getch()OutputSimpson 1/3rd ruleincludeincludeincludevoid main()float x10,y10, check=0,h,tint i,n,j,k=0printf(nhow many set you will enter )scanf(%d,n)for(i=0 i printf(nn x%d ,i)scanf(%f,xi)printf(nn f(x%d) ,i)scanf(%f,yi) h=x1-x0n=n-1Total = Total + y0for(i=1i if(k==0) Total = Total + 4 * yik=1 else Total = Total + 2 * yik=0 Total = Total + yiTotal = Total * (h/3)printf(nn I = %f , Total)getch()Trapezoidal ruleincludeincludeincludeint main()float x10,y10, Total =0,hint i,n,j,k=0float fact(int)printf(nhow many values of ranges you will be enter )scanf(%d,n)for(i=0 iprintf(nn x%d ,i)scanf(%f,xi)printf(nn f(x%d) ,i)scanf(%f,yi) h=x1-x0n=n-1for(i=0iif(k==0) Total = Total + yik=1 elseTotal = Total + 2 * yiTotal = Total + yiTotal = Total * (h/2)printf(nn I = %f , Total)getch()Future Work ScopeThis Beer Lambert s law can be used in image processing application where atmospheric condition is poor to find the attenuation of light and image by absorption of light.Implementing Euler Method.ReferencesWeisstein, Eric W. Simpsons Rule. From MathWorldA Wolfram Web Resource. http//mathworld.wolfram.com/SimpsonsRule.html. (Accessed on 26.04.2015)Basics of Trapezoidal and Simpson Rules, www.math.umd.edu/jmr/141/Simpson.pdf.Lal, A. K., Simpsons Rule, 2007, http//numericalmethods.eng.usf.edu.(Accessed on 20.04.2015)http//numericalmethods.eng.usf.edu. (Accessed on 19.03.2015)Garrett, P., Absorption and Transmission of light and the Beer-Lambert Law, Lecture 21, 2006, www.physics.uoguelph.ca/pgarrett/Teaching.html. (Accessed on 26.04.2015)Mudakavi, J. R., Modern Instrumental Methods of Analysis, Lecture 07, Ultraviolet and manifest Spectrophotometry 3 Theoretical Aspects, http//nptel.ac.in/courses/103108100/7(Accessed on 26.04.2015).www.chemwiki.ucdavis.edu. (Accessed on 19.03.2015)