10 34 Fall 2006 Homework 5 Due Date Wednesday Oct 11th 2006 9 AM Problem 1 Singular Value Decomposition The composition of a mixture of chemicals is usually determined by putting the mixture in a spectrometer or chromatograph The signal coming out of an analytical chemistry instrument for a mixture is typically a linear combination of the signals one would have obtained if the sample was pure For example using a spectrometer Smix n xi Si n Where S is the signal measured of a mixture with mole fractions xi and Si is the signal that would have been measured if the mixture had been pure i e if xi 1 and all other x s had been zero The spectrum S is a continuous function of the wavelength but the spectrometer only records the spectrum at N discrete set of wavelengths n Very often the composition x of a mixture changes either with time e g when we are measuring batch kinetics or as a function of some knob being turned in a chemical plant We pull samples periodically to measure these changes at K times tk and run each sample through the spectrometer obtaining a big data set S tk n which we think can be deconvoluted Sobserved tk n xi tk Si n Use SVD to write the matrix Sobserved on the Web page in the format The rows in the data file corresponds to a given time and the time is measured from 0 100 seconds The wavelength is measured from 432 482 nm both of them are measured with linear spacing S i uv T Where u is a vector with K entries v is a vector with N entries and uvT is a matrix formed by the outer product From this expansion figure out how many major light absorbing components are in the mixture Plot the normalized spectrum and timedependence of each major light absorbing species in the mixture N B When two spectral peaks are observed to move with the same time dependence this can mean a that two species coincidentally have about the same time dependence b that a single molecule has two different absorption bands or c that there are two species in rapid equilibrium with each other It is often very difficult to distinguish between the latter two possibilities An example of the last case is glucose which exists as a mixture Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY of several rapidly interconverting isomers one with a 6 membered ring one with a 5 membered ring and an open chain species Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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