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First_Order_UV_SVD_v7

# First_Order_UV_SVD_v7 - Irreversible One-step First Order...

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Irreversible One-step First Order Reaction: Generation and SVD Analysis of Noisy UV-Vis Absorbance Data Dr. Kalju Kahn UC Santa Barbara, 2004-2010 This Notebook illustrates how to generate and then deconvolute multi-wavelength absorbance data containing random noise for the irreversible process A fi k B . The main goal is to illustrate how to determine the number of species that contribute to absorbance change via singular value decomposition (SVD). The value of isosbestic point for two-component system is illus- trated. First, we generate concentration profiles for A and B for 30 time points with 0.2 second spacing. The concentrations follow first-order kinetics. Data Generation In[1]:= Remove @ "Global` * " D onestep = DSolve @ 8 concA ¢ @ time D == - kconcA @ time D , concA @ 0 D == A0, concB ¢ @ time D == kconcA @ time D , concB @ 0 D == 0 < , 8 concA @ time D , concB @ time D< , time D Flatten; Concentration Profiles In[3]:= cA = concA @ time D . onestep Simplify cB = concB @ time D . onestep Simplify Out[3]= A0 ª - ktime Out[4]= A0 - A0 ª - ktime In[5]:= timeval = Table @ i, 8 i, 30 <D 5 N; cAtime = cA . 8 A0 fi 1.0, k fi 0.693, time fi timeval < ; cBtime = cB . 8 A0 fi 1.0, k fi 0.693, time fi timeval < ; cAdata = Transpose @8 timeval, cAtime <D ; cBdata = Transpose @8 timeval, cBtime <D ;

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In[10]:= Needs @ "PlotLegends`" D ListPlot @8 cAdata, cBdata < , AxesLabel fi 8 "Time", "Conc" < , PlotLegend fi 8 " @ A D ", " @ B D " < , LegendPosition fi 8 0.2, - 0.10 < , LegendSize fi 8 0.35, 0.35 < , PlotLabel fi "Concentration Profiles", LabelStyle fi 8 Medium, FontFamily fi "Helvetica" < , Background fi ColorData @ "Atoms", "He" DD Out[11]= 1 2 3 4 5 6 Time 0.2 0.4 0.6 0.8 1.0 Conc Concentration Profiles @ B D @ A D In[12]:= datatable = Transpose @8 timeval, cAtime, cBtime <D ; PaddedForm @ TableForm @ datatable, TableHeadings fi 8 None, 8 "Time", " @ A D ", " @ B D " <<D , 8 3, 3 <D ; Spectra of Species Now we generate spectra for A and B. The spectrum of A is modeled as a Gaussian curve with a maximum at 280 nm, and the spectrum of B is modeled as a Gaussian curve with a maximum at 400 nm. The Mathematica function NormalDistribu- tion[ Μ , Σ ] generates the Gaussian distribution with mean Μ and standard deviation Σ . In[14]:= ndist280 = NormalDistribution @ 280, 30 D ; ndist400 = NormalDistribution @ 400, 35 D ; The function PDF[dist, x] calculates the probability density function of this distribution. We multiply the probability values with coefficients (100 and 90 below) to get reasonable units. You can think of these values as molar absorptivities in Μ M - 1 cm - 1 - units. In[16]:= pdf280 = 100PDF @ ndist280, x D pdf400 = 90PDF @ ndist400, x D Out[16]= 5 3 ª - H - 280 + x L 2 1800 2 Π Out[17]= 9 7 ª - H - 400 + x L 2 2450 2 Π 2 First_Order_UV_SVD_v7.nb
In[18]:= Plot A8 pdf280, pdf400 < , 8 x, 150, 500 < , PlotRange fi All, PlotLabel fi "Molar Absorptivity of A and B", AxesLabel fi 9 " Λ , nm", " Ε , Μ M - 1 cm - 1 " = , Background fi ColorData @ "Atoms", "He" DE Out[18]= 200 250 300 350 400 450 500 Λ , nm 0.2 0.4 0.6 0.8 1.0 1.2 Ε , Μ M - 1 cm - 1 Molar Absorptivity of A and B We next convert these continuous functions into two discrete data sets (named peakA, and peakB) with range from 200 to 500 nm and spacing of 1 nm. The real spectra below 200 nm are rather crowded because of several Σ fi Σ * transitions. Furthermore, this spectral region is difficult to observe and we hide it in our plots. To facilitate matrix inversion during the spectral deconvolu-

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First_Order_UV_SVD_v7 - Irreversible One-step First Order...

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