Kinetics of Complex Reactions

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1 Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet i your ow classes but commercial use is ot allowed without the permissio of the author. I a previous lesso you leared to deal with simple first ad secod order reactios by itegratig the rate laws to get expressios for the cocetratios of species as a fuctio of time. As the reactios you are iterested i become more complex, which usually meas they ivolve more steps, the itegratio of the rate laws becomes much more difficult, ad fidig a closed form solutio to the itegrals is frequetly impossible. This is where umerical solutios of the differetial equatios which describe the reactio become useful. Mathcad icludes several built-i fuctios for solvig systems of differetial equatios umerically. These methods are based o the Ruge-Katta approach, which is described i a appedix to this documet. The geeral approach is illustrated here for the case of the reactio A-->B-->C, with the first step havig rate costat k, ad the secod, k2. The solutios for the cocetratios will be obtaied usig the Mathcad fuctio Rkadapt. Use the Mathcad help facility to lear more about this fuctio. You may also wat to check the Mathcad 6. maual, Chapter 6. The geeral form of the fuctio is Rkadapt(x, x, x2, poits, D. The first term, x, is a vector cotaiig the iitial values of the cocetratios. I geeral the size of this vector will be equal to the order of the differetial equatio or the size of the system of equatios you are studyig. I the problem we have i this worksheet there are three equatios, oe for each of the compoets i the reactio system. x ad x2 are the iitial ad fial values for the idepedet variable (time i our case over which you wish to solve the set of equatios. The parameter 'poits' is the umber of poits for which you wish to solve the equatios, ad D is a vector cotaiig the first derivatives of all of the cocetratios. poits should be sufficietly large to give a good curve for the solutio but ot so large that the calculatio takes forever. The expressios to put ito the D vector come directly from the equatios that are used to defie the problem, i.e. how the cocetratio of each species varies with time. page

2 D( t, x x k. x k. x. k2 x. k2 x For the problem at had, we ca immediately write D as show to the left. The variable ames assiged to each reactio compoet is as follows: x is A, x is B ad x 2 is C. Usig the equatios foud i a text we ca see that the first row of the vector D is the first derivative of [A] with respect to time. (A, B ad C will be used as variables later i this discussio which is why x, x ad x 2 are beig used here. Note that these derivatives are writte for the formatio of each species, which gives rise to the sigs i the various terms. Note also that D is writte with a symbolic equals sig ([ctrl]. so that it will ot be processed util we defie some of the other terms. Next you defie the iitial values ad place them i the x vector. You ca choose ay values you wish; here we begi by choosig to have species A at uit cocetratio ad B ad C at zero cocetratio. The rows of the vector x correspod to the iitial cocetratios of the species i the order that their derivatives are listed i D. Later you will have the opportuity to vary the iitial cocetratios of all compoets as you explore the behavior of this series of chemical reactios. t2 if k k2,, k k2 The ext thig to do is to decide o how log the reactio will proceed. The rage of the calculatio should be large eough to allow the system to reach its fial limitig state. This documet comes with a t2 already i place. You could eter differet umbers for t2, or you could write a expressio such as the oe at the left which guaratees that the system will reach its fial state. (It makes sese to start at zero so t i the fuctio will ormally be just that. Likewise poits, the umber of poits to compute for the exercise, could be just typed i as a umber or its value ca be scaled so that it spas a time rage that is appropriate for the rate costats used i the reactio as is doe here below. This is purely a empirical approach which assures more tha poits per calculatio. Check your Mathcad maual to be sure you uderstad the 'if' statemet as used here. k poits floor if,, k2 5. k k2 5. k2 k page 2

3 The output of the Rkadapt fuctio is a matrix, which we will call Z. This matrix cotais poits+ rows, ad oe more colum tha the umber of species i the reactio scheme. The first colum of the matrix cotais the poits at which the equatios were evaluated (the times, ad other colums are the calculated cocetratios, i the order they are listed i the iitial cocetratio vector. We ca ow defie the cocetratios, ad the time, as follows: A( ( Z < > B( ( Z < 2 > C( ( Z < 3 > t( ( Z < > Note how the colums are accessed i this Mathcad documet. The <> idetifies the colum of the Z matrix. I this documet we have kept the default Mathcad umberig of colums ad rows. This scheme labels the first row as the (zero row ad the first colum as the (zero colum. Additioal rows ad colums are umbered sequetially. We are almost i a positio to plot the cocetratios as a fuctio of time. The equatios for the argumets of Rkadapt have ot really bee etered sice we used the symbolic equals sig. We eed to reeter these equatios usig the global equal sig. Furthermore, due to the way Mathcad reads equatios, we eed to eter them lower i the documet tha the values of k ad k2 (which are give just above the graph. Sice the equatios have bee give above, they are copied below ad the type of equal sig chaged. There are two other ecessary equatios. Oe defies the matrix Z as Z=Rkfixed(x,,t2,poits,D, while the other says that there is parameter which rus from to poits i icremets of. This latter oe is used to choose the row of the Z matrix to give to A( etc. The various equatios are distributed aroud the graph show below (o the ext page where we display the plot of A(, B( ad C( versus t(. page 3

4 Vary the values of the rate costats ad observe how the cocetratios chage. You may also vary the iitial cocetratios i the vector x, but the you may eed to chage the limits o the graph. Write a summary of your observatios ad use this summary to aswer the followig questios. Uder what coditios would you say that the itermediate species B reaches a steady state? What happes if oe of the rate costats is much larger tha the other? Are there limits to solvig the equatios this way? k2..39 k..34 x A( B( C( t(. sec t2 if k k2,, k k2 D( t, x k. x k. x. k2 x. k2 x,.. poits poits = 5 rows( = Z 6 Z Rkadapt( x,, t2, poits, D k poits floor if,, k2 5. k k2 5. k2 k Apply the steady-state approximatio to B, ad add that result to the above plot. What do you coclude about the validity of this approximatio for various values of k ad k2? page 4

5 Here are the symbolic expressios for the cocetratios of A, B ad C as a fuctio of time. Compare the two methods whe k = k2 (the k's are still defied from above. New k's ca be etered here just above the equatios to see the cosequece of varyig them i the itegrated rate expressios. A( A. e k. t( A. k B(. e ( k2 k A x C( A ( A( B( k. t( e k2. t( A( B( C(.5 What happes to the aalytical solutios if k=k2? t( You may be curious about how well these two methods of calculatig the cocetratios agree. You ca easily test this. The umerical method of solutio solves the equatios at a umber of poits, that umber beig based o how close the rate costats are to oe aother. That umber is called poits here, ad you ca display it by typig poits= below. Now pick ay iteger less tha or equal to poits - lets call it q - ad evaluate the expressio (A(q-A(q/A(q. This will give you a idea of how close the umerical result is to the exact result. A more rigorous approach would ivolve calculatig the sum of the squares of the deviatios over the etire rage of poits. This is left for the you to complete as a exercise. A( A( A( = % page 5

6 Exercises:. Cosider the case of A<-->B-->C, ow with three rate costats, ad derive the appropriate expressio for the vectors x ad D ad the matrix Z. You may also have to modify the expressios for the umber of poits ad the fial time. Plot the cocetratios of the species A, B ad C versus time for various values of the rate costats. Uder what coditios is the equilibrium approximatio to this mechaism valid? What about the steady state approximatio. Derive both of these approximatios ad compare them graphically to the results of the umerical solutio. 2. I her seior thesis work, Beth Teske ('94 studied multiple photo iduced eergy trasfer processes i urayl complexes. I the process, she measured the emissio decay curves as a fuctio of excitatio laser power. A emissio decay curve is a plot of the itesity of the emitted light versus time. The itesity of the emitted light is proportioal to the cocetratio of excited states. Emissio is a first order process, ad should produce a simple expoetial decay (Figure. This was what was see i may cases, but there were a umber of experimets i which curves which looked very o-expoetial were obtaied (Figure 2. Beth explaied this by a mechaism i which there were two excited states, both of which emitted i the same spectral regio (meaig the two states are very close to oe aother i eergy. These excited states (A ad B could also itercovert, as well as decayig to the groud state (C. Beth proposed that she was measurig the emissio of the sum of A ad B. Set up a kietic scheme for this mechaism (how would you write the mechaism ad test it to see if you ca fid coditios for which the sum of A ad B produces a plot similar to that show i Figure 2. page 6

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