{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "N ormal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Subitem" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 3 12 1 0 2 2 270 5 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 42 "Unit 24 -- Applica tion: Chemical Reactions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu /~meade/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Industrial Mathematics I nstitute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" " " }}{PARA 256 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "ht tp://www.sc.edu/" "" }}{PARA 256 "" 0 "" {TEXT -1 19 "Columbia, SC 292 08\n" }}{PARA 256 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://ww w.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 256 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 38 "Copyright \251 2001 b y Douglas B. Meade" }}{PARA 256 "" 0 "" {TEXT -1 19 "All rights reserv ed" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 67 "-------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline of Unit 24" } }{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "24.A" 1 "" "24.A" }{TEXT -1 22 " First-Order Reactions" }}{PARA 14 "" 0 "" {HYPERLNK 17 "24.B" 1 "" " 24.B" }{TEXT -1 23 " Second-Order Reactions" }}{PARA 14 "" 0 "" {HYPERLNK 17 "24.C" 1 "" "24.C" }{TEXT -1 18 " Example: 2NO + O" } {XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 7 " -> 2NO" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initiali zation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 3 "" 0 "24.A" {TEXT -1 26 "24.A First-Order Reactions" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "An example of a first-order react ion is the conversion of " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 21 " -butyl chloride into " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 15 "-but yl alcohol:" }}{PARA 256 "" 0 "" {TEXT -1 4 " (CH" }{XPPEDIT 18 0 "``[ 3]" "6#&%!G6#\"\"$" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``[3]" "6#&%!G6#\" \"$" }{TEXT -1 18 " CCl + NaOH -> (CH" }{XPPEDIT 18 0 "``[3]" "6#&%!G6 #\"\"$" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``[3]" "6#&%!G6#\"\"$" }{TEXT -1 11 " COH + NaCl" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "A chemical reaction is first-order if the molecules of a \+ substance " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 84 " decompose into smaller molecules at a rate proportional to the amount of substance \+ " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 36 " remaining at any time. T hat is, if " }{XPPEDIT 18 0 "A(t)" "6#-%\"AG6#%\"tG" }{TEXT -1 33 " de notes the amount of substance " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 9 " at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 95 ", then the amount of substance A can be modeled with the first-order linear (and separable) ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "ode1 := diff( A(t), t ) = -k * A(t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "where the reaction rate " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 37 " is positive and has units of 1/time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The solution to this ODE, with initial condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ic1 := A(0) = A[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "can be written down on sight, but it is simpler to use Maple t o enter the result in a Maple worksheet" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sol1 := dsolve( \{ ode 1, ic1 \}, A(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "While first-order chemical reactio ns are easy to model and analyze, they are not very common." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "2 4.B" {TEXT -1 27 "24.B Second-Order Reactions" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 49 "An example of a second-order chemical reaction is" }} {PARA 256 "" 0 "" {TEXT -1 3 " CH" }{XPPEDIT 18 0 "``[3]" "6#&%!G6#\" \"$" }{TEXT -1 15 "Cl + NaOH -> CH" }{XPPEDIT 18 0 "``[3]" "6#&%!G6#\" \"$" }{TEXT -1 9 "OH + NaCl" }}{PARA 0 "" 0 "" {TEXT -1 296 "in which \+ one molecule of methyl alcohol and one molecule of sodium hydroxide co mbine to form one molecule of methyl hydroxide and one molecule of sod ium chloride. This reaction proceeds at a rate proportional to the pro duct of the remaining concentrations of methyl chloride and sodium hyd roxide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The general form for a second-order reaction is" }}{PARA 256 "" 0 "" {TEXT -1 12 " A + B -> C " }}{PARA 0 "" 0 "" {TEXT -1 38 "To model \+ a second-order reaction, let " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 41 " de note the initial amounts of chemicals " }{XPPEDIT 18 0 "A" "6#%\"AG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 35 " and den ote the amount of chemical " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 9 " at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "C(t)" "6#-%\"CG6#%\"tG" }{TEXT -1 25 ". The amount of chemical \+ " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 19 " remaining at time " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "alpha-C (t)" "6#,&%&alphaG\"\"\"-%\"CG6#%\"tG!\"\"" }{TEXT -1 12 "; likewise, \+ " }{XPPEDIT 18 0 "beta-C(t)" "6#,&%%betaG\"\"\"-%\"CG6#%\"tG!\"\"" } {TEXT -1 37 " is the remaining amount of chemical " }{XPPEDIT 18 0 "B " "6#%\"BG" }{TEXT -1 38 ". Hence, the amount of C is modeled by" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ode2 := diff( C(t), t ) = k * ( alpha - C(t) ) * ( beta - C(t) ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 98 " i s a reaction constant. This first-order ODE is nonlinear (but separabl e). With initial condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ic2 := C(0) = C[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "the amount of chemical " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 15 " is found to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sol2 := dsolve( \{ ode2, ic2 \}, C( t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "provided " }{XPPEDIT 18 0 "alpha<>beta" "6#0%&al phaG%%betaG" }{TEXT -1 5 ". If " }{XPPEDIT 18 0 "alpha=beta" "6#/%&alp haG%%betaG" }{TEXT -1 22 ", the solution becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify( d solve( \{ eval( ode2, beta=alpha ), ic2 \}, C(t) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Note that it is not possible to obtain this solution simply by \+ substituting " }{XPPEDIT 18 0 "beta=alpha" "6#/%%betaG%&alphaG" } {TEXT -1 26 " into the general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eval( sol2, beta=alpha );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 32 "but is obtained in the limit as " }{XPPEDIT 18 0 " beta" "6#%%betaG" }{TEXT -1 12 " approaches " }{XPPEDIT 18 0 "alpha" " 6#%&alphaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "map( limit, sol2, beta=alpha );" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " 24.C" {TEXT -1 21 "24.C Example: 2NO + O" }{XPPEDIT 18 0 "``[2]" "6#&% !G6#\"\"#" }{TEXT -1 7 " -> 2NO" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"# " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Two molecules of nitrous oxide combine with one molecule of oxygen to form two mole cules of nitrogen dioxide:" }}{PARA 256 "" 0 "" {TEXT -1 8 " 2NO + O" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 7 " -> 2NO" } {XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 61 "At room temperature this reaction can be modeled with t he IVP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "ode3 := diff( NO2(t), t ) = k * ( alpha - NO2(t) )^2 \+ * ( beta - NO2(t)/2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "i c3 := NO2(0) = A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "NO2(t)" "6# -%$NO2G6#%\"tG" }{TEXT -1 50 " is the concentration of nitrogen dioxid e at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 49 " is the initial concentration \+ of nitrogen oxide, " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 45 " \+ is the initial concentration of oxygen, and " }{XPPEDIT 18 0 "A" "6#% \"AG" }{TEXT -1 50 " is the initial concentration of nitrogen dioxide. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The s olution to this IVP with a separable ODE is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sol3 := dsolve( \+ \{ ode3, ic3 \}, NO2(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "This \"explicit\" solution is of almost no use." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Note, however, that the qualitative behavior of solut ions can be determined directly from the ODE. There are two equilibria . The equilibrium at " }{XPPEDIT 18 0 "NO2=2*beta" "6#/%$NO2G*&\"\"#\" \"\"%%betaGF'" }{TEXT -1 47 " is stable and, because of the quadratic \+ term, " }{XPPEDIT 18 0 "NO2=alpha" "6#/%$NO2G%&alphaG" }{TEXT -1 16 " \+ is semi-stable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Assuming numerical values for the parameters are availabl e, a numerical solution is easy to obtain." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "At a temperature of 25C, the ra te constant is 7130 liter" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" } {TEXT -1 6 "/(mole" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 52 " second). Assume the initial concentration of NO is " }{XPPEDIT 18 0 "alpha=0.003" "6#/%&alphaG-%&FloatG6$\"\"$!\"$" }{TEXT -1 43 " mole/li ter, the initial concentration of O" }{XPPEDIT 18 0 "``[2]" "6#&%!G6# \"\"#" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "beta=0.0021" "6#/%%betaG-%&F loatG6$\"#@!\"%" }{TEXT -1 47 " mole/liter and the initial concentrati on of NO" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "A=0" "6#/%\"AG\"\"!" }{TEXT -1 12 " mole/liter." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "param3 := \{ k=7.13*10^3, alpha=0.003, beta=0.002, A=0 \};" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "A plot of the numerical solution to this IVP for the fir st hour (3600 seconds) of the reaction is obtained with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "sol3n \+ := dsolve( eval( \{ ode3, ic3 \}, param3 ), NO2(t), numeric ):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "odeplot( sol3n, [t,NO2(t)], 0..3600 , view=[0..3600,0..0.003] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Note that the concentrat ion is increasing to the equilibrium solution " }{XPPEDIT 18 0 "NO2=0. 003" "6#/%$NO2G-%&FloatG6$\"\"$!\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Slightly more informat ion can be obtained by superimposing the solution curve on the directi on field for this problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "DEplot( eval(ode3,param3), NO2(t), \+ t=0..3600, [[0,0]], NO2=0..0.01, stepsize=25 );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "Somet imes the slopes are so small that the direction field is not very usef ul. In such cases, it can be useful to plot the right-hand side of the (autonomous) ODE as a function of the dependent variable." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "plot( eval(rhs(ode3),param3 union \+ \{NO2(t)=NO2\}), NO2=0.001..0.005, labels=[NO2,\"NO2 ' \"] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "From this plot it is seen that the rate of change of the \+ concentration of NO" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 41 " is positive when the concentration of NO" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 61 " is in [0,0.003) or (0.003,0.004) and th e concentration of NO" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 53 " decreases only when the concentration exceeds 0.004." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE P owertool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }