Beer-Lambert’s Law Experiment

Beer- lamberts Law ExperimentDesign Project onBeer- liters Law.Saswati RakshitAimTo learn and understand the basics and mathematical calculations of the following problem and write programs accordingly.Problem 1 Suppose an outdoor multi spectral paradigm is captured by a camera with a grade length of 1-3 microns. A part of the fervency is attenuated by the denseness of small particles in the melodic line for that spectral cranial orbit and let the scattering by the small particles for that spectral range is 0.Find the total attenuation in the spectral range utilise Simpson 1/3 , trapezoidal and Euler integration methods and comment in your findings.Scope/ApplicationBeer Lamberts impartiality relates the attenuation of light-hearted to the properties of the material finished which the light is passing. When light passes through a medium some amount of light is absorbed by the medium. For this assiduity lastingness of light reduces.Beer Lamberts practice of law states tha t the quantity of light absorbed is directly proportional to the dumbness of the substance and the path length of the light through the transmission mediumBeer Lamberts law is used to find total attenuation of light when light passes through a medium(considering scattering is zero).It is also used to find the concentration of medium in chemical analysis, medium length in some practical application 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 absorbing medium its intensity decreases exponentially as the path length of the absorbing medium increases.I = I0 e-k1 L .(i)(w here(predicate) L is the length of medium and k1 is molar extinction/absorption co-efficient for the absorbing 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 medi um increases.I = I0e-k2C ..(ii)(C concentration of medium and k2 is absorption co-efficient for the absorbing material)Combining both Beers and Lamberts law we getI = I0 e-k3CL combining eqn (i) and (ii)Where I0 = Incident light intensityI = Transmitted Light 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 go away 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 need to find the absorbance total attenuation of the incident light employ Beers Lambert Law. And then applying Simpsons 1/3, Trapezoidal and Euler Inte gration in it compare the result.System flowImplementation of the Beer Lamberts law needs a proper mathematical understanding of the Beer Lamberts law. Here I am showing 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 give birth I0 is the incident light to the medium Air of attenuation coefficient 1.64at temperature 20oC. Here path length travelled 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 now 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 economyte a new equation by integrating ranged from .001mm to .003 mm Here x is path length and c is attenuation coefficientSol ving the above eqn we get total absorbance. Thus Beer Lamberts law is successfully implemented in our problem. instantaneously 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 ConceptConsider a light incident on a medium with area A and thickness dx and concentration of molecules C. Number of molecules illuminated by light of incident intensity Ix is CAdx. contribute efficacious area CAdx. Probability of light being absorbed in thickness dx is = dx where dIx is the change 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 linear attenuation co-efficient. Here C=Absorbing co-efficient = Scattering co-efficient.The ibrightness of light decreases exponentially with depth in the m edium.So we can tell 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 decree is = f(x0) + 2) + f(xn)So, here applying the above 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, composite Simpsons rule formula is wr itten 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 near same if we take approximation i.e.0.002.CODES and OUTPUTBeer Lamberts Law embroilincludeincludeint main()float absorbtion,m,l,uprintf(nEnter spectral range)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(nnTotal absorption is %f ,absorbtion)getch()OutputSimpson 1/3rd ruleincludeincludeincludevoid main()float x10,y10,Total=0,h,tint i,n,j,k=0printf(nhow many values 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 Lamberts law can be used in image processing application where atmospheric conditi on 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 Visible 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)