the numerical integration of chemical reaction rate laws
TRANSCRIPT
The Numerical Integration of ChemicalReaction Rate Laws Using Computer
Spreadsheets
Eric L. KemerSt. Andrew's School
Middletown, DE 19709
General rate laws for chemical reactions can be solvedquantitatively using a simple numerical integration techniqueknown as Euler’s Method. When executed on a computerspreadsheet program, this method provides students with apowerful interactive tool for modeling chemical reaction kineticsand exploring the underlying dynamics of equilibrium.
†
aA + bB ´ cC + dD
†
Rr = + 1a
d A[ ]dt
= + 1b
d B[ ]dt
= - 1c
d C[ ]dt
= - 1d
d D[ ]dt
†
Rf = - 1a
d A[ ]dt
= - 1b
d B[ ]dt
= + 1c
d C[ ]dt
= + 1d
d D[ ]dt
†
Rf = k f A[ ]n B[ ]m
†
Rr = kr C[ ] r D[ ]s
†
Rnet = Rf - Rr = k f A[ ]n B[ ]m- kr C[ ]r D[ ]s
†
Q(t) =C[ ]c D[ ]d
A[ ]a B[ ]b
Forward and Reverse Reaction Rates Defined
†
limtƕ
Rnet = 0
Chemical Equilibrium
Reaction Quotient
General Reaction
Forward and Reverse Reaction Rate Laws
Net Reaction Rate Law in for a Closed System
†
kr = Ar exp DH - EA
RTÊ
Ë Á
ˆ
¯ ˜
†
kf = Af exp -EA
RTÊ
Ë Á
ˆ
¯ ˜
Arrhenius Equations for Rate Constants
†
limtƕ
Q = Keq
†
aA + bB ´ cC + dD
†
d D[ ]dt
= d ⋅ Rnet = d k f A[ ]n B[ ]m- kr C[ ]r D[ ]s( )
†
d A[ ]dt
= - a ⋅ Rnet = -a k f A[ ]n B[ ]m- kr C[ ]r D[ ]s( )
†
d B[ ]dt
= - b ⋅ Rnet = -b k f A[ ]n B[ ]m- kr C[ ]r D[ ]s( )
†
d C[ ]dt
= c ⋅ Rnet = c k f A[ ]n B[ ]m - kr C[ ]r D[ ]s( )
General rate laws for homogenous reaction in a closed system
The differential equations to be solved by numerical integration
Integrated Rate Laws: The Three Standard Cases
• Reaction rate depends only on a single reactant
• No reverse reaction considered (kr = 0)
• No connection between kinetics and equilibrium
0th Order:
†
d A[ ]dt
= - kf fi A[ ] = A[ ]0 - akf
1 stOrder:
†
d A[ ]dt
= -a ⋅ k f A[ ] fi A[ ] = A[ ]0 exp -ak f t( )
2nd Order:
†
d A[ ]dt
= -a ⋅ k f A[ ]2 fi 1A[ ]
=1A[ ]0
+ ak f t
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4 5 6 7
Time
[A]
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4 5 6 7
Time
[A]
0.000
0.500
1.000
1.500
2.000
2.500
3.000
0 1 2 3 4 5 6 7
time (s)
1/[
A]
†
A Æ products
†
d A[ ]dt
= -a k f A[ ]n B[ ]m- kr C[ ]r D[ ]s( )
†
d A[ ]dt
= - a ⋅ Rnet A[ ] , B[ ] , C[ ] , D[ ]( )Consider the rate equation:
†
A[ ] i+1 @ [A]i - a ⋅ RnetDt
†
Dt = ti +1 - ti
†
A[ ]i+1 - [A]i
ti+1 - ti
@ - a ⋅ Rnet,i A[ ]i, B[ ]i, C[ ]i, D[ ]i( )
Euler’s Method uses successive linear approximations of a rate equation over short timeintervals to numerically integrate it over an arbitrarily long time interval.
†
A[ ] i+1is approximated concentrationat time
†
ti+1
†
[A]i is concentration at time
†
ti
Euler’s Method
t
[A]
Approximation error
†
A[ ]i+1
†
A[ ]i
†
ti+1
†
ti
Tangent line slope =
†
Rnet
Exact [A]
Comparison of Exact Solution to Euler’s Approximation for Simple First Order Rate Law
†
d A[ ]dt
= -a ⋅ k f A[ ] fi exact fi A[ ] = A[ ]0 exp -ak f t( )
†
Dt = 0.05s 40 intervals4 intervals
†
Dt = 0.5s
0.00
0.50
1.00
1.50
2.00
2.50
0 0.5 1 1.5 2 2.5
time (s)
[A]
[A], Euler
[A], exact
0.00
0.50
1.00
1.50
2.00
2.50
0 0.5 1 1.5 2 2.5
time (s)
[A]
[A], Euler
[A], exact
†
k f = 0.9 s-1
†
a =1
†
A[ ]0 = 2.0M
Outline of Spread Sheet Algorithm for Euler’s Method*
* Only [A] and [C] calculations explicitly shown
time
†
[A]
†
[B]
†
[C]
†
[D]
†
Rnet
†
Q
†
t0
†
[A]0
†
[B]0
†
[C]0
†
[D]0
†
Rnet ,0 = - k f A[ ]0n B[ ]0
m + kr C[ ]0r D[ ]0
s
†
C[ ]0c D[ ]0
d
A[ ]0a B[ ]0
b
†
t1 = t0 + Dt
†
A[ ]1 = A[ ]0 - a ⋅ Rnet ,0Dt
†
[B]1
†
C[ ]1 = C[ ]0 + c ⋅ Rnet ,0Dt
†
[D]1
†
Rnet ,1 = - k f A[ ]1n B[ ]1
m + kr C[ ]1r D[ ]1
s
†
C[ ]1c D[ ]1
d
A[ ]1a B[ ]1
b
†
t2 = t1 + Dt
†
A[ ]2 = A[ ]1 - a ⋅ Rnet ,1Dt
†
[B]2
†
C[ ]2 = C[ ]1 + c ⋅ Rnet ,1Dt
†
[D]2
†
Rnet ,2 = - k f A[ ]2n B[ ]2
m + kr C[ ]2r D[ ]2
s
†
C[ ]2c D[ ]2
d
A[ ]2a B[ ]2
b
†
t3 = t2 + Dt
†
A[ ]3 = A[ ]2 - a ⋅ Rnet ,2Dt
†
[B]3
†
C[ ]3 = C[ ]2 + c ⋅ Rnet ,2Dt
†
[D]3 etc.
to= 0 t [A] [B] [C] [D] Rnet Q∆t = 0.25 0 4.00 3.000 1 0.01 1.302 8E-04
0.25 3.674 2.6745 1.326 0.336 1.06 0.045[A]o= 4 0.5 3.409 2.4094 1.591 0.601 0.878 0.116[B]o= 3 0.75 3.19 2.1897 1.81 0.82 0.738 0.213[C]o= 1 1 3.005 2.0053 1.995 1.005 0.627 0.333[D]o= 0.01 1.25 2.849 1.8486 2.151 1.161 0.537 0.474
1.5 2.714 1.7143 2.286 1.296 0.465 0.636a= 1 1.75 2.598 1.5981 2.402 1.412 0.404 0.817b= 1 2 2.497 1.4971 2.503 1.513 0.354 1.013c= 1 2.25 2.409 1.4085 2.591 1.601 0.312 1.223d= 1 2.5 2.331 1.3307 2.669 1.679 0.275 1.445
2.75 2.262 1.2618 2.738 1.748 0.244 1.677m= 1 3 2.201 1.2007 2.799 1.809 0.218 1.917n= 1 3.25 2.146 1.1463 2.854 1.864 0.194 2.162r= 1 3.5 2.098 1.0977 2.902 1.912 0.174 2.41s= 1 3.75 2.054 1.0541 2.946 1.956 0.156 2.661
4 2.015 1.015 2.985 1.995 0.141 2.911T= 580 4.25 1.98 0.9799 3.02 2.03 0.127 3.161
4.5 1.948 0.9481 3.052 2.062 0.115 3.407∆H= -10000 4.75 1.919 0.9195 3.081 2.091 0.104 3.649
5 1.894 0.8936 3.106 2.116 0.094 3.886Ea,f = 160000 5.25 1.87 0.8701 3.13 2.14 0.085 4.116
5.5 1.849 0.8488 3.151 2.161 0.077 4.34Af= 2.79E+13 5.75 1.829 0.8294 3.171 2.181 0.07 4.556Ar= 2.79E+13 6 1.812 0.8118 3.188 2.198 0.064 4.764
6.25 1.796 0.7958 3.204 2.214 0.058 4.964kf= 0.108525 6.5 1.781 0.7813 3.219 2.229 0.053 5.155kr= 0.013643 6.75 1.768 0.768 3.232 2.242 0.048 5.337
7 1.756 0.7558 3.244 2.254 0.044 5.51
24.8 1.624 0.6241 3.376 2.386 1E-04 7.94725 1.624 0.624 3.376 2.386 1E-04 7.948
25.3 1.624 0.624 3.376 2.386 9E-05 7.94825.5 1.624 0.624 3.376 2.386 8E-05 7.949
Reagent Concentrations vs. Time
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0 5 10 15 20 25 30
Time (s)
Conce
ntr
atio
n (
M)
[A] [B][C] [D]
Reaction Quotient vs Time
0
1
2
3
4
5
6
7
8
9
0 5 10 15 20 25 30
Time (s)
Q
Sample Spreadsheet and Graphs
A B C D E F G H I
1234567891011121314151617181920212223242526272829
[A] vs. Time
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4 5
Time (s)
[A]
Reaction Rate vs. Time
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
Time (s)
React
ion
Rate
(M
/se
c)
†
Rf = k f A[ ] B[ ]
†
Rf = k f A[ ]
Comparison of Two Rate Laws
to= 0∆t = 0.05
[A]o= 2[B]o= 4[C]o= 0[D]o= 0
a= 1b= 1c= 1d= 1
m= 1n= 1r= 1s= 1
T= 600
∆H= -40000
Ea,f = 160000
Af= 2.79E+13Ar= 2.79E+13
kf= 0.327992kr= 0.000108
[A] vs. Time
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4 5
Time (s)
[A]
[A], M[A], M
Reaction Rate vs. Time
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
Time (s)
React
ion
Rate
(M
/se
c)
to= 0∆t = 0.05
[A]o= 2[B]o= 4[C]o= 0[D]o= 0
a= 1b= 1c= 1d= 1
m= 1n= 0r= 1s= 1
T= 600
∆H= -40000
Ea,f = 160000
Af= 2.79E+13Ar= 2.79E+13
kf= 0.327992kr= 0.000108
Reaction Quotient vs Time
0
10
20
30
40
50
60
0 1 2 3 4 5 6
time (s)
Q
Q (600 K)
Q (600 K)
Reaction Quotient vs Time
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50
time (s)
Q
Reagent Concentrations vs. Time
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0 10 20 30 40 50
Time (s)
Co
nce
ntr
ati
on
(M
)
[A] [B][C] [D]
Le Chatelier’s Principle: Concentration Stresses
• New equilibrium concentrations of reactants and products shifted
• Reaction quotient returns to the same equilibrium value as long astemperature remains constant.
Reaction Quotient vs Time
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10
time (s)
Q Q (600 K)Q (610 K)
Le Chatelier’s Principle: Temperature Stress
• Temperature increase shifts equilibrium of exothermicreactions to the left (lowers the Keq).
• Reaction rate approximately doubles for every 10 Kincrease in temperature
to= 0∆t = 0.1
[A]o= 4[B]o= 3[C]o= 2[D]o= 1
a= 1b= 1c= 1d= 1
m= 1n= 1r= 1s= 1
T= 610
∆H= -10000
Ea,f = 160000
Af= 2.79E+13Ar= 2.79E+13
kf= 0.554907kr= 0.077247
to= 0∆t = 0.1
[A]o= 4[B]o= 3[C]o= 2[D]o= 1
a= 1b= 1c= 1d= 1
m= 1n= 1r= 1s= 1
T= 600
∆H= -10000
Ea,f = 160000
Af= 2.79E+13Ar= 2.79E+13
kf= 0.327992kr= 0.044183
Reaction Quotient vs Time
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10 12
time (s)
Q Q (600 K)Q (610 K)
Effect of Catalyst (Lowering of Activation Energy)
Reaction Quotient vs Time
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10
time (s)
Q Q (Ea,high)Q (Ea,low)
Lowering the activation energy increases reaction rate but doesnot alter equilibrium constant.
to= 0∆t = 0.08
[A]o= 4[B]o= 3[C]o= 2[D]o= 1
a= 1b= 1c= 1d= 1
m= 1n= 1r= 1s= 1
T= 600
∆H= -10000
Ea,f = 157000
Af= 2.79E+13Ar= 2.79E+13
kf= 0.598476kr= 0.080619
to= 0∆t = 0.1
[A]o= 4[B]o= 3[C]o= 2[D]o= 1
a= 1b= 1c= 1d= 1
m= 1n= 1r= 1s= 1
T= 600
∆H= -10000
Ea,f = 160000
Af= 2.79E+13Ar= 2.79E+13
kf= 0.327992kr= 0.044183
Reaction Quotient vs Time
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10 12
time (s)
Q Q (600 K)Q (610 K)
†
NO(g) + O3(g) Æ NO2(g) + O2(g)
†
A =1.204 ¥109 Lmol ⋅ s†
DH = -1.99x105 Jmol
†
k f =10.836x106†
EA =11.7x103†
T = 298 K
From CRC Handbook
Depletion of ozone by nitric oxide in the upper atmosphere proceeds by the reaction
to= 0∆t = 0.0005
[A]o= 0.000021[B]o= 0.00002[C]o= 0.000001[D]o= 0
a= 1b= 1c= 1d= 1
m= 1n= 1r= 1s= 1
T= 298
∆H= -199000
Ea,f = 11700
Af= 1.20E+09Ar= 1.20E+09
kf= 10672940kr= 1.4E-28
Reagent Concentration vs. Time
0.00
0.00
0.00
0.00
0.00
0.00
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
time (s)
Conce
ntr
atio
n (
M)
[NO]
[O3]
[NO2]
[O2]
Suggestions for Use
• As a supplement to a teacher’s class presentation.
• As a basis for guided student explorations and problem sets.
• As a tool for analyzing experimental data.
Final Comments
• Integrating rate laws (solving differential equations) is fundamental to mathematicalmodeling in science. It is not necessary that students complete advanced courses incalculus before engaging in these calculations and benefiting from the insights andunifying understandings that follow.
• Most differential equations that describe real processes cannot be solved analytically.Yet, most readily yield to numerical methods requiring only basic algebra andarithmetic. The quantitative treatment of dynamic processes need not be limited tospecial cases. The “real world” can be more readily explored.
• Numerical integration methods transfer directly to topics in physics (and visa versa).
• Introducing students to spreadsheets is generally valuable.