Modeling Stratospheric Ozone Kinetics, Part I
Contents
1. oa 09 Concentration of O2 molecules em3 Step 1 Step 2 Step 3 Step 4 O 10 O 2 10 7 O 327 107 Created August 1997 Modified May 1999 2 10 1 9998 10 1 10 210 t 1 Time sec Molecular oxygen 02 vs time 1 9999 10 7 7 0 1107 2 10 t 1 Time sec O gt 20 M O O gt M O O gt O O O O gt 20O M 9 10 T 220 Oxygen atom atoms cm3 Ozone molecules cm3 k 3 10 k 5 5 10 k 1 2210 k 6 9 10 Oxygen atom O vs time 6 10 4 10 i 2 10 0 0 1107 2 10 t 1 Time sec Ozone 03 vs time a013 2 10 a013 041510 1 10 3 5 10 2 0 1107 2 10 i 1 Time sec k 3x10 sect k 1 2 x 103 cm molecule sec k 5 5x 104 sec k 6 9 x 10 16 cm molecule sec These are stratospherically reasonable values of rate constants concentrations and temperature at an altitude of 25 km OzoneModelingPart1 mcd page 9 Authors Erica Harvey Bob Sweeney Exercise 8 Change the rate constants one at a time by a factor of 10 Make your changes by editing the equations with the triple equals signs After altering a rate constant see if the change had a discernible effect on the plots of concentrations Now return that rate constant to the default value and change another Make four general statements about trends you observe as you vary conditions Try to explain the results of thi2. ending times Here we are starting at t 0 and ending at the user specified tmax in seconds because of the units on the rate constants being used It s good practice to pay attention to the ratio of npts to tmax because the numerical method may miss interesting fluctuations if you step too quickly through the time interval The ratio is high enough if the appearance of the solutions doesn t change as you increase the ratio The last command below calls on the Stiffr Rosenbrock numerical method to solve the equations and stores the numerical solutions concentrations of each species at each time for which the equations were solved in a matrix named answers npts 1000 ETE This definition of i as the counter 7 DUn variable is useful for plotting the tmax 20000000 results in a graph below npts 5 10 tmax answers Stiffr y 0 tmax npts D J Assuming that the calculation has not been interrupted and the lightbulb is no longer flashing the answers matrix will be displayed below if you remove the extra m from the end of the word answersm The first column is time and the second third and fourth columns represent the concentrations of O O and O respectively If the matrix is so big that it blocks your reading of this document you can temporarily hide it by re inserting an extra letter at the end of the word answers This will make it an undefined variable and the matrix will disappear To get it back just rem
3. gt 20 k 6 9 x 10 16 cm3 molecule sec In Step 1 molecular oxygen is split into two oxygen atoms diradicals by hard UV light A 185 nm to 220 nm detailed absorption cross section information is given in reference 1 Solar energy is used to break the bond AH for the reaction is 119 14 kcal mol see Appendix I of reference 1 Any leftover energy is imparted as kinetic energy to the two oxygen atoms which is by definition equivalent to modestly heating the system The rate constant shown is known as a photolysis rate constant or J value It represents an effective first order rate constant that takes into account the average solar flux and average absorption cross section Step 1 O hv gt 20 k 3 x 10 sec In Step 2 an oxygen atom reacts with an oxygen molecule to form ozone In thermodynamic terms this is a spontaneous exothermic reaction at stratospheric temperatures For this reaction AH is 25 47 kcal mol and AS is 30 5 cal mol K The energy given off when the reaction occurs is given to the product molecules as kinetic energy thereby increasing the temperature of the stratosphere A third body denoted as M is required to conserve momentum and allow this reaction to occur This third body can be any atom or molecule in the stratosphere that can absorb kinetic energy If a third body is not present to carry away the released energy the ozone molecule that is formed will simply redissociate Note the concentratio
4. suggestions and comments improved the document from the beginning Mario Molina helped tremendously by directing the author to the compilation of modelling information in reference 1 and actually sending a copy of the JPL evaluation data along with words of encouragement Michelle Sprengnether and two anonymous reviewers made numerous helpful suggestions Elmore Rountree and Robin Morris helped find the on line versions of the document John Pojman helped decipher error messages given by Mathcad s differential equation solvers and Theresa Julia Zielinski provided encouragement during the preparation and refinement of the document The biggest debt of gratitude is owed to the 1997 8 and 1998 9 physical chemistry classes at Fairmont State College who plunged headfirst into unknown territory with the earlier versions of this document and figured out a great deal about Mathcad and modelling chemical kinetics Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 16 Bob Sweeney References 1 Chemical Kinetics and Photochemical Data for Use in Stratospheric Modelling Evaluation Number 12 Jet Propulsion Laboratory Pasadena 1997 JPL Publication 97 4 Copies of this document are available electronically at the following URL http remus jpl nasa gow jpl197 Hardcopies of the 274 page document can be ordered for approximately 50 through the NASA STI Bibliographic Database For information about this option g
5. 0 v 0 1 10 2 10 0 1 10 2 10 t t 1 1 Time sec Time sec Exercises 12 Within the stratosphere the temperature varies from 215 K to 270 K How significant is the effect that such variations in the temperature have on the model results 13 Try using a sinusoidal function in place of the constants k and k3 to model the actual effects of sunlight as it goes through daily and or yearly intensity fluctuations You will need to alter the D and J matrices to reflect the sinusoidal natures of k and k Answers to Selected Exercises Exercise 1 Why is a third body not needed in Step 4 The energy released during the Step 4 reaction can be divided between two product molecules rather than being dumped into a single product molecule Exercise 2 The reactions above represent a chain reaction Identify the initiation propagation and termination steps Initiation Step 1 O gt 20 Propagation Step 2 M O O gt M O Step 3 O gt O O Termination Step 4 O O gt 20 Exercise 3 Add together the two chain propagation steps What is the overall reaction There is no net overall reaction all of the species cancel out The significance of this pair of propagation reactions is that photons are absorbed to drive Step 3 and then the corresponding release of energy in Step 2 is stored as increased kinetic energy of M and O Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modifie
6. 3 1 863096300790827 10 npts Exercises 9 How quickly is the photostationary state reestablished if the initial ozone concentration drops by a factor of 10 What happens to the concentrations if the atmosphere starts with only O and no O or ozone How long does the system take to reach the photostationary state in this case Hints You may need to change tmax to answer these questions To avoid errors in logarithmic graphs enter the concentrations of O and O as 0 1 molecule cm rather than zero If you make changes to the model and the given find solver has trouble finding a solution try improving your guess values by using the values of the variables near the end of the time period covered by the differential equation solver e g the final values shown in the left hand matrix equation above For instance when we set the initial ozone concentration to 7x101 to answer the first question in this exercise the solver failed when the initial values were used as guess values However simply changing the oxygen atom guess statement to pssO 5 initialO was adequate to help the solver find a solution 10 Explore the effects of changing the total concentration of atmospheric species M Note that you can either change M alone equivalent to adding or deleting species that are not involved in the chemical reactions or change the M and O concentrations in combination so that O remains approximately 20 of M equivalent to changing the pressu
7. Modeling Stratospheric Ozone Kinetics Part I Erica Harvey Bob Sweeney Department of Chemistry and Department of Chemistry Fairmont State College Fairmont State College Fairmont WV 26554 Fairmont WV 26554 eharvey mail fscwv edu Copyright 1999 by the Division of Chemical Education Inc American Chemical Society All rights reserved For classroom use by teachers one copy per student in the class may be made free of charge Write to JCE Online jceonline chem wisc edu for permission to place a document free of charge on a class Intranet Prerequisites This worksheet is appropriate for use in Junior Senior level physical chemistry classes To use the document you should have had at least a year of calculus and have completed the Mathcad tutorial In addition it is recommended that you study the chemical kinetics sections of a physical chemistry textbook as you work through this document This document requires Mathcad 6 0 or later the professional version which has specialized differential equations solvers Goal The pair of documents Modeling Stratospheric Ozone Kinetics Part I and Part II is designed to lead students into modeling the kinetics of stratospheric ozone reactions Part I focuses on the mechanics of the modeling method and considers only the Chapman cycle of reactions for stratospheric ozone In the companion document Part II students incorporate a larger set of reactions including the HO NO and ClO reacti
8. cument serves as an introduction to numerical solutions of complex rate laws and kinetic modelling using stratospheric ozone chemistry as an example Students learn how to write differential rate laws for each component to be included define initial concentrations and rate constants enter the set of differential equations as a matrix for use with a built in differential equation solver and finally generate and graph species concentration profiles over specified time intervals Specific variables that can be explored in this document include temperature total pressure initial component concentrations and rate constants The reference cited above or its successor will be a highly useful resource for advanced modelling studies based on this template The thermodynamic quantities rate constants activation parameters stratospheric concentrations and total number density of atmospheric constituents used in this document were taken to the best of the authors ability from Tables 1 and 2 and Appendices I II and IH in the NASA JPL document cited above unless otherwise noted the values apply at an altitude of 25 km and a temperature of 220 K Units are shown in the text but are not used in most of the calculations because the differential equation solver requires unitless inputs Sample answers to some of the questions posed in the text are given at the end of the document Helpful hints If Mathcad starts to show a lightbulb cursor it is performi
9. d May 1999 page 15 Bob Sweeney Exercise 4 What are the units of concentration being used Why would these units be chosen rather than molarity The units of concentration are molecules cm3 These units are useful for the very low concentrations exhibited by trace atmospheric species 1 mole liter 6 02 x 102 molecules cm Exercise 5 Explain how the cycle of reactions operates to warm the stratosphere Which two reactions are most important for the warming effect Why Steps 2 and 3 the propagation steps discussed in Exercise 3 are the most important warming steps simply because they occur over and over The ultraviolet light absorbed in Step 3 is released as heat in Step 2 increased kinetic energy of the molecules involved While Step 4 releases a great deal of energy more than Step 2 it does not occur as often as the propagation steps Exercise 6 Stop and write out the differential rate equations for the other two constituents now do 2 k 05 k M 0 05 k 03 k 0 03 The differential equations dt describing formation and loss d of each species are shown here w 27 k O 3 k M 0 0 2 k O 3 2 k 0 0 3 do rr 3 k 0 3 k M 0 0 J k 0 0 3 Acknowledgements The National Science Foundation sponsored Mathcad workshop organized by Sid Young Jeffry Madura and Andrzej Wierzbicki at the University of Southern Alabama in August of 1997 provided the impetus for the development of this document and the participants
10. d by the simultaneous equation solver with the final values from the differential equation solver we need to display the latter values in a convenient format For example to look at the last value of O generated by the differential equation solver you can type O using the period subscript then use the left bracket subscript to type npts 1 this gives the last value at tmax and the regular equals sign This has been done below for each of the components using a matrix format Thus it is very easy to do a line by line comparion of the tmax results from the differential equation solver and the photostationary state results obtained from the simultaneous equation solver If the component concentrations calculated by the two methods and displayed below are equal then a long enough tmax has been used in the differential equation solver to allow the system to reach the photostationary state In the case shown below the system has not quite reached the photostationary state This fact is also evident in the graphs that show the individual species concentrations for they are still changing with time at the end of the period for which the differential equations have been solved Differential equation solutions at tmax Photostationary state values from simultaneous equation solver O npts 1 4 354746240385475 10 pssO 4 666905102204752 10 O 2 s1 1 999844228912582 10 PSSO 2 1 999825535371536 10 a 4 1 738472797901555 10 pssO
11. e The regular equals sign is used to ask the 2 10716 ETA a A program to display the calculated value of k For the termolecular reaction incorporated into this model the temperature dependence is calculated using a two parameter equation that is different from the Arrhenius equation and unfamiliar to most undergraduate physical chemistry students The parameters given in the JPL reference for reaction 2 are the low pressure limiting rate constant k 6 0 x 10 34 cm molecule sec and n 2 3 unitless ka 6 0 10 24 ko and n are the two parameters 0 tabulated in the JPL reference n 2 3 T This equation calculates the rate constant k k o card using the temperature dependent 300 K equation given in the reference work Since temperature is defined above in units of Kelvin the 300 appearing in the k 1 22449769050059 1 10 denominator must also have units of Kelvin Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 13 Bob Sweeney Rates for photochemical reactions are also unfamiliar to many undergraduate physical chemistry students The reaction rate will depend on the solar flux and the concentration of the photoreactive constituent in addition to the match between the absorption spectrum of the photoreactive constituent and the emission spectrum of the light sunlight in this case The JPL reference is kind enough to give tables of effective unimolecular rate constants
12. ght into stratospheric warmth Exercises Why is a third body not needed in Step 4 2 The reactions above represent a chain reaction Identify the initiation propagation and termination steps 3 Add together the two chain propagation steps What is the overall reaction 4 What are the units of concentration being used Why would these units be chosen rather than molarity 5 Explain how the cycle of reactions operates to warm the stratosphere Which two reactions are most important for the warming effect Why Now that we have written a series of reactions that occur in the stratosphere it is possible to write the expressions for the change in each of the species concentrations with time i e the differential rate expressions Each of these expressions is called a differential equation and together they form a coupled system of differential equations Mathcad is very useful for solving numerically such systems of differential equations Note that each reaction step contributing to the formation or loss of a particular constituent must be included in the differential rate law for that constituent For the oxygen atoml O reactions 1 and 3 above are formation steps and reactions 2 and 4 are destruction or loss steps The stoichiometry is included as shown below and the photon flux is included implicitly in the photolysis rate constants Note that O denotes the concentration of O Square brackets for concentration are not sho
13. in the differential equation solver often turn out to be adequate guess values to allow Mathcad to find the solutions so they are used and displayed numerically below Under the Given Mathcad is given three equations that contain three unknowns the photostationary state values for O O and O3 The first equation just states the fact that totaloxygen remains constant at the value shown above the Given The second and third equations require that dO dt and dO3 dt be equal to 0 i e that the concentrations of O and O are no longer changing with time The find function solves the system of three equations and three unknowns and returns a vector of concentrations that satisfy the conditions we imposed These are the photostationary state conditions totaloxygen initialO 2 initialO 5 3 initialO 3 totaloxygen 4 0002100001 10 Guess values pssO initialO pssO 1 10 pssO 5 initialO gt pssO 5 2 10 7 pssO 3 initialO 3 pssO 3 7 10 Given pssO 2 pssO 5 3 pssO 3 totaloxygen k pssO 2 k M pssO pssO 2 k pssO 3 2 k pssO pssO 3 0 k pssO 3 k M pssO pssO 2 k pssO pssO 3 0 pssO Here the photostationary state concentration values resulting from the find command are pssO 3 stored in a matrix pssO 9 find pssO pssO 2pssO 3 Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 11 Bob Sweeney To compare the photostationary state values generate
14. ions represented by D t y above Symbolic solutions to this system of equations would consist of mathematical functions that describe each of the concentrations as they change with time However Mathcad uses a numerical method to solve this system of differential equations rather than a symbolic method The numerical method returns the numerical values of the concentrations at user specified times rather than the mathematical functions that would allow a user to calculate the concentrations at any time Mathcad has several choices of built in functions to solve systems of differential equations Different functions are good for different types of systems The particular reaction conditions being considered in this document lend themselves to a specialized differential equation solving method that the Mathcad manual suggests for stiff systems According to Noggle reference 2 stiff equations result when a series of fast processes combine to produce a slow overall change Such conditions can arise from rate constants that have very different orders of magnitude An additional matrix that provides information about the system of equations is required as an input when the stiff solver is used This matrix J is shown below Each column in J represents a set of partial second derivatives of the functions whose first derivatives with respect to time are given in the D vector This is not as scary as it may sound The first column of the J matrix represen
15. it edu Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 17 Bob Sweeney
16. n units implied by the form of the termolecular rate constant Step 2 O O M gt 0 M k 1 2 x 10 33 cm6 molecule sec The next reaction Step 3 below requires an input of energy to occur because it is the reverse of the spontaneous reaction in Step 2 The reaction proceeds photochemically Ozone absorbs UV light in the wavelength range 210 nm to 300 nm and splits into an oxygen atom and an oxygen molecule Again any energy leftover from the bond breaking goes into increasing the kinetic energy of the products Detailed information about absorption cross sections and the atomic state of the oxygen radical is provided in reference 1 see the end of this document for references Step 3 O hv gt O O k 5 5x 107 sec Created August 1997 OzoneModelingPartl mcd Authors Erica Harvey Modified May 1999 page 3 Bob Sweeney When the chemically reactive O and O find each other a spontaneous reaction occurs to produce two oxygen molecules with lots of kinetic energy This reaction is highly exothermic AH is 93 67 kcal mol Step 4 O O gt 20 k 6 9 x 10 16 cm3 molecule sec Using this simplified set of stratospheric reactions it is possible to watch the system evolve toward a steady state condition known as a photostationary state where light is continually absorbed by the system but chemical species concentrations no longer change with time Remember though that the cycle operates continuously to change UV li
17. ng one of the calculations embedded in this document Press the escape key to interrupt the processing When you are ready you can start processing again by choosing Calculate worksheet from the Math menu Alternatively for a more drastic and long lasting solution you can go to the Math menu and turn off the checkmark beside Automatic mode When automatic mode is de selected you must push the F9 key to start EVERY calculation even seemingly trivial ones Calculation times of less than one minute are typical for calculations in this worksheet with Mathcad installed on a 120 MHz Pentium To make additional space in a Mathcad 6 0 document add blank lines by using control F9 Extra blank lines can be deleted from the document with the keystrokes control F10 Check your user manual for performing these operations with Mathcad 7 0 and Mathcad 8 0 versions of the software Created August 1997 OzoneModelingPartl mcd Authors Erica Harvey Modified May 1999 page 2 Bob Sweeney Section 1 Setting up and solving the system of differential equations We begin by considering the Chapman cycle the fundamental cycle of reactions that creates the ozone layer The Chapman cycle shown in Steps 1 4 involves only oxygen containing species each step is discussed in detail below Step 1 O hv gt 20 k 3x10 sec Step 2 M O O gt M 0 k 1 2 x 1033 cm molecule sec Step 3 O hv gt O O k 5 5x 107 sec Step 4 O O
18. o to URL http www sti nasa gov casitrs html and input either the document ID number 19970037557 or the document accession number 97N31001 into the accession number field to obtain pricing and ordering information and to read the full citation for this document 2 Noggle Joseph Physical Chemistry Using Mathcad Pile Creek Publishing Company Newark Delaware 1997 p 211 Other ozone related resources 3 Matthew Elrod s web site at Hope College provides links to environmental chemistry sites with information about stratospheric ozone chemistry He has also written several Mathcad documents that model tropospheric and stratospheric ozone kinetics and provide useful insights about the results His home page is http www chem hope edu elrod and the same address plus mathcad chapkey mcd mathcad chapt noxkey mcd smogkey mcd points to the mathcad documents 4 An ozone chemistry teaching resource directed at the general chemistry level was developed through the ChemLinks Coalition based at Beloit College and funded by NSF s systemic change initiative in chemistry The reference is T Ferrett and S Anthony Why does the ozone hole form Wiley and Sons Inc ChemConnections module Beta version published Fall 1998 To use the module contact Heather Mernitz at Beloit College Mernitzh Beloit edu or Brock Spencer the ChemLinks Project Director Spencer Beloit edu For more information on ChemLinks see http chemlinks belo
19. on cycles Performance Objectives After completing the work described in this document you should be able to 1 write differential rate expressions for species involved in a series of chemical reactions 2 set up and numerically solve systems of differential equations to show the time evolution of a chemical system 3 determine the effects of varying rate constants and initial concentrations 4 recognize and understand the Chapman cycle that controls ozone concentrations in the stratosphere Created August 1997 OzoneModelingPartl mcd Authors Erica Harvey Modified May 1999 page 1 Bob Sweeney Introduction The ozone layer is a region of the atmosphere that contains a steady state concentration of ozone resulting from a set of reactions occurring in the stratosphere Driven by a constant input of solar energy this critical set of reactions helps to maintain a temperature inversion in the stratosphere and to protect the surface of the earth from ultraviolet wavelengths of solar radiation Stratospheric ozone kinetics remains an active area of research for atmospheric chemists The basic reaction cycles are well established and a regularly updated comprehensive treatment of kinetic data is available Chemical Kinetics and Photochemical Data for Use in Stratospheric Modelling Evaluation Number 12 Jet Propulsion Laboratory Pasadena 1997 JPL Publication 97 4 available electronically at http remus jpl nasa gov jp197 This Mathcad do
20. ove the extra letter you added Note In Mathcad8 set the Format Result answersm Matrix Display style to Automatic before removing the m in answersm shown here to the left Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 7 Bob Sweeney The next set of commands breaks the answers matrix apart into columns vectors The first column the series of time values for which concentrations have been calculated will become the data used on the x axis of a plot of concentrations versus time To get the symbol lt gt use control 6 or go to the matrices palette The second third and fourth columns are stored as the concentrations of O O and O respectively Mathcad will let us plot all three concentrations on the y axis at the same time T lt 0 gt t answers lt 1 gt O answers lt 2 gt O 2 answers lt 3 gt O 3 answers Finally the moment of truth has arrived The plots below show the concentrations of all species as a function of time individually and grouped on a single plot Double clicking on the axes allows you to change from logarithmic to linear plots A single click on the axes will bring up the upper and lower limits which can be modified to rescale a plot Directly above the plots are the global definitions of the variables in the model including rate constants for each reaction and initial concentrations of each of the chemical species To access the triple equal
21. quals sign at the bottom of the document right beside a plot of concentrations versus time The triple equals sign is a way of letting the user specify variables after the point in the document where the variables first have been used in equations Mathcad initially scans through the document and sets all of the variables that are globally defined then works stepwise from the beginning to the end of the document Since the variables called initialO initialO and initialO are globally defined assigned numerical values below Mathcad doesn t object when we use them here Definitions initialO yE initialO gt Yo is the initial conc of O ae y is the initial conc of O initialO 3 E y is the initial conc of O Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 5 Bob Sweeney The next step is to make an equation vector D whose elements describe the rates of change of each species The elements of the vector are just the right hand sides of the differential equations that you wrote above with Mathcad friendly y s in place of O O and O Since the y vector contains the initial concentrations of each component the equation vector D actually represents the initial rate of change of each of the components with time 2ky ky M yyy kyy kyYo D t y kyi Ky M ygyt ky 2 kyyoy kyy t ky M yyy kyyo We are soon going to ask Mathcad to solve the system of coupled differential equat
22. re by changing the volume of the air mass 11 What is the effect of increasing or decreasing the number of points and the time scale Discuss the tradeoffs involved with changing npts and tmax At what values of npts and tmax do you consider the results to be reliable Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 12 Bob Sweeney Section 3 A closer look at the rate constants The temperature of the stratosphere varies considerably with altitude and the rate constants for the reactions involved vary considerably with temperature For bimolecular reactions the JPL source actually reports Arrhenius factors rather than the rate constants themselves Substitution of the activation energy and pre exponential terms into the Arrhenius equation gives the rate constant as a function of temperature For example the activation energy for the reaction corresponding to k4 is given as 2060 the value given is actually E R and therefore has units of K and the preexponential factor is 8 0x10 2 cm3 molecule sec Once these variables are defined and the Arrhenius equation is entered the value of k will be automatically updated as the variable T temperature in Kelvins is changed For convenience the temperature is globally defined just above the plots at the end of this section E 4 2060 K 12 A 4 8 0 10 4 Note that E really represents E R for k 5A ge the reaction in step 4 of our kinetic schem
23. s experiment using your knowledge of chain reaction kinetics Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 10 Bob Sweeney Section 2 Photostationary state PSS the final solution The following section shows the use of simultaneous equations to solve for the photostationary state abbreviated pss concentrations of the atmospheric constituents i e the long term steady state concentrations that will persist as long as light continues to be absorbed This section is useful when you are entering lots of different initial values and trying to see the effects on your model A quick comparison of the calculated concentration values at the end of the chosen differential equation solver time period with the values calculated below tells you whether or not the system has truly stopped changing i e reached the long term steady state The totaloxygen variable defined below gives the total number of O atoms either loose or bonded in molecules per cubic centimeter of atmosphere Mass balance requires that this number must remain constant The initial concentrations used below to calculate totaloxygen are defined by the triple equals signs above and will be updated each time you change variables above The values for the photostationary state variables entered under the Guess values subheading are needed to help Mathcad solve the set of simultaneous equations The values we set for the initial values
24. s sign for this global definition feature use the Evaluation and Boolean palette or the keystroke shift tilde For your convenience in playing around and changing variables the chemical equations to which the rate constants correspond have been reproduced below the graphs along with atmospherically reasonable values for the variables Note that these have been entered with a control equals sign to produce a bold equal sign so that they do not interact with the calculations and can serve as a constant reference point The concentration of M is the total concentration of all stratospheric constituents This number is essentially independent of the nature of the constituents and the elapsed reaction time so it is simply entered as the total number density at an altitude of 25 km We assumed that the ratio of N to O was the same as in the troposphere and calculated the concentration of O as 20 of the total number density at an altitude of 25 km The other constituent concentrations were read from plots of trace constituent concentrations versus altitude in Appendix III of reference 1 Rate constants were calculated as described in Section 3 of this template Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 8 Bob Sweeney Global definitions of model variables M 9 10 7 initialO 10 initialO 5 2 10 7 initialO 3 7 00 10 All Concentrations vs Time Concentrations molecules em3
25. that incorporate average solar fluxes and spectral information These are treated as temperature independent constants in the present document Note that the k s defined with triple equals signs beside the graphs in Section I are the ones used by the differential equation solver in that section The numerical integration steps and concentration plots are reproduced below so that you are able to visualize the effects of altering the k s by adjusting T in this section 2kry ky M yyy kyy kg YoY D t y kyi Ky M ygyt ky 2 kyyoy kyy t ky M yyy kyYo 0 kyMy key 2 kM yo Bas Eo J t y 10 k My 2 k y 0 kyMy k y ky MY ky Ky Yo kK kyMy ki 2 kyy answers Stiffr y 0 tmax npts D J The temperature can be changed 2 lt 0 gt i t answers right here for convenience Be sure lt I gt to use the Kelvin units O answers O 2 answers T 220 K lt 3 gt O 3 answers All Concentrations vs Time 18 z Oxygen atom O vs time 17 6 10 Concentrations molecules em3 Li Li P Oxygen radical molecules cm3 4 7 0 i107 2 10 t t 1 Time sec Time sec Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 14 Bob Sweeney a Molecular oxygen O2 vs time Ozone 03 vs time e713 5 2 10 2 10 amp a 5 g E 3 13 5 3 1 5 10 E O2 E 5 03 1 9999 10 2 ig E 5 2 1 10 E g z fo S 1 9998 10 7 5 1
26. ts the partial derivatives with respect to t time of the functions in the D vector Remember that the D vector itself represents a bunch of first derivatives of concentration functions with respect to time dO dt is the second element for example Therefore the first column of the J matrix represents the second derivative functions with respect to time 820 5t represents the second element of the first column for example The second column of the J matrix represents the partial derivatives with respect tO y of the functions in the D vector i e the second element of the second column could be represented as 620 dtdy The third column represents the partial derivatives with respect to y of the functions in the D vector and the fourth column represents the partial derivatives with respect to y of the functions in the D vector Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 6 Bob Sweeney Exercise 7 Write out the partial derivatives with respect to time yg y and y for each of the elements in the D vector and verify that your answers agree with the entries in the J matrix here immediately below 0 kyMy k y 2 2k ky My k ky K t y 0 kyMy 2kyy k k My k 2 k y 0 kyMy kyy k M y k kyyy To use the stiff differential equation solver in Mathcad the user is required to specify the number of points at which the equations will be solved and the starting and
27. wn because Mathcad uses them for a different purpose d o eS g k M 0 0 2 k O 37 k 0 0 3 Created August 1997 OzoneModelingPart1 mcd Authors Erica Harvey Modified May 1999 page 4 Bob Sweeney Exercise 6 Stop and write out the differential rate equations for the other two constituents now Use the calculus palette to find the d dt symbol and use the control equals sign The convention throughout this document is to use the period subscript for chemical formulas and the left bracket subscript for rate constants For convenience the reaction cycle under consideration has been reproduced below Add and delete extra lined from your Mathcad document as directed in the manual for your version of Mathcad Step 1 O hv gt 20 k 3x10 sec Step 2 M O O gt M O k 1 2 x 1033 cm molecule sec Step 3 O hv gt O O k 5 5x 104 sec Step 4 O O gt 20 k 6 9 x 10 16 cm molecule sec For Mathcad s sake it is necessary to change from the chemically intuitive formulation of rate equations above to a vector formulation The first step is to define a vector y which is filled with elements y arranged in a vertical stack The matrix palette is used to define a 3 row 1 column matrix The elements yo y y correspond to the initial concentrations of O O and O3 respectively For convenience in making changes later the actual numerical values for the initial concentrations will be globally defined triple e
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