Presentation Slides
for
Chapter 10
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
jacobson@stanford.edu
March 21, 2005
Elements of Periodic Table and Their
Lewis Symbols
Group- >
Period
1
2
I
1
II
V
VI
VII
VIII
2
0
H
He
1.008
3
1
4.003
10 0
6.941
11 1
Na
4
IV
1
Li
3
III
22.99
19 1
K
39.10
4
2
5
3
6
4
7
1
8
2
9
3
Be
B
C
N
O
F
Ne
9.012
12 2
10.81
13 3
12.01
14 4
14.01
15 1
16.00
16 2
19.00
17 3
20.18
18 0
Al
Si
P
S
Cl
Ar
24.30
20 2
26.98
31
3
28.09
32 4
30.97
33 1
32.07
34 2
35.45
35 3
39.95
36 0
Ca
Ga
Ge
As
Se
Br
Kr
40.08
69.72
72.61
74.92
Mg
78.96
79.90
83.80
Table 10.1
Lewis Structures
Molecular hydrogen
H H
H H
Molecular oxygen
O
O
O O
N
N
N
O
H
Molecular nitrogen
N
Hydroxyl radical
O H
Lewis Structures
Nitric oxide
N
O
N
O
N
O
N
O
Nitrogen dioxide
N
O
N
O
O
N
O
N
O
O
O
O
Ozone
O
O
O
O
O
O
O
O
O
O
O
Carbon monoxide
C
O
C
O
O
Lewis Structures
Carbon dioxide
O
C
O
O
C O
Sulfur dioxide
S
O
S
O
O
Methane
H
H
H C H
H
H
H
O
H
N
H C C O
O
O
H
C H
H
Peroxyacetyl nitrate (PAN)
O
O
H
C
H
O
O
C
O
O
N
O
Chemical Reactions
Elementary unimolecular photolysis reaction
NO2 + h
NO + O
(10.1)
 < 420 nm
Elementary bimolecular thermal decomposition reaction
N2O5 + M
(10.2)
NO2 + NO3 + M
Nonelementary bimolecular thermal decomposition reaction (10.3)
N2O5
M
NO2 + NO3
Chemical Reactions
Elementary bimolecular collision reaction
CH4 + OH
CH3 + H2O
Termolecular combination reaction
NO2 + NO3 + M
(10.4)
(10.5)
N2O5 + M
derived from a pair of elementary bimolecular reactions
A  B  AB *
AB * M  AB  M
Reaction Coefficients and Rates
Reaction rates with 1st-, 2nd-, 3rd-order rate coefficients (10.8)
Rate = k F A 
Rate = k SAB
Rate = k T ABC
Rates for photolysis reaction
Rate = J A 
(10.9)
Rate of Change of Reactant Conc.
Photolysis reaction, A  h  D  G
(10.10)
dA
 Rate   JA
dt
M
Nonelementary bimolecular reaction, A D + E (10.11)
dA
 Rate  k FA 
dt
M
Nonelementary termolecular reaction, A  BE (10.12)
dA dB

 Rate  k SAB
dt
dt
Rate of Change of Reactant Conc.
Elementary bimolecular reaction, A  A  E  F
(10.13)
dA
 2Rate  2kSA 2
dt
Elementary termolecular reaction, A  B  C  E  F (10.14)
dA dB dC


 -Rate  kT A BC
dt
dt
dt
Rate of Change of Concentration
Generalized reaction, Rate  kr A  B
a
b
(10.16)
aA  bB  eE + fF
Rate of change of reactant concentration
dA
 aRate  ak r A a Bb
dt
dB
 bRate  bkr Aa Bb
dt
(10.17)
Rate of change of product concentration
(10.18)
dE
 eRate  ekr A a Bb
dt
dF 
 fRate  fkr A a Bb
dt
Third Bodies
Elementary termolecular combination reaction
A  B M  E M
Rate of change of third body
(10.19)
dM
 k T ABM  k T ABM   0
dt
Number concentration of air molecules (molec. cm-3) (10.20)
pa
M  Na 
pa  pd  pv
k BT
Number concentration of nitrogen and oxygen
N 2   N 2 Nd
(10.21)
O 2    O2 Nd
Third Bodies
Example 10.1:
Calculate the number concentration of M, N2, and O2 when T =
278 K and pa = 920 hPa and the air is dry
Solution:
[M]
= 2.40 x 1019 molec. cm-3
NN2
= 1.87x 1019 molec. cm-3
NO2
= 5.02 x 1018 molec. cm-3
Rate Coefficients from Kinetic Analysis
Elementary bimolecular collision reaction
A  B D F
Expose small quantity of A to large quantity of B --> maximum
loss of B is [A]
Rate of loss of A
(10.22)
dAt
dt
 k FA t  k SAt B0
Integrate
(10.23)
At h
kS  
ln
B0 h A 0
1
Rate Coefficients from Kinetic Analysis
Unimolecular reactions
(10.24)
1 A t h
k F   ln
h
A0
Termolecular reactions
At h
1
kT  
ln
B0 C0 h A 0
(10.24)
Arrhenius Equation
Temperature-dependence of a reaction rate coefficient (10.25)
dlnkr 
dT

Er
R*T 2
Activation energy
Smallest amount of energy required for reacting species to
form an activated complex or transition state before products
are formed
Integrate (10.25)
(10.26)
lnk r  lnAr 
Er
*
RT
Arrhenius Equation
Measure kr vs. T --> plot to find Ar and Er
Useful form of Arrhenius equation
 Er 
Cr 
k r  Ar exp  *   Ar exp  
 T 
 R T 
(10.27)
Fig. 10.1
Temperature Dependence
Er ≈0 --> collisional prefactor strongly depends on T
(10.28)
300 Br
C r 
k r  Ar   exp  
 T 
 T 
Br found by fitting equation to data
Example 10.2:
NO + O3
k1  1.80  10
NO2 + O2
12
exp1370 T   1.81 10
14
cm3 molec.-1 s-1 at T = 298 K
O + O2 + M
O3 + M
34
2.8
k 2  5.63  10
300 T 
34
 5.74  10
cm6 molec.-2 s-1 at T = 298 K
Pressure-Dependence of Reactions
Rate of pressure-dependent reaction
kr 
k,T k 0,T [M]
k ,T  k 0,T [M]
(10.30)
 
2 1

k 0, T [M ] 
 
 
1 lo g1 0
k ,T 


 


Fc
Low pressure limit rate coefficient x M
(10.31)
k0,T [M]  lim kr
[M]0
High pressure limit rate coefficient
k,T  lim kr
[M]
(10.32)
Pressure-Dependence of Reactions
Example 10.3:
M
OH + NO2
HNO3
pa
T
[M]
Fc
= 140 hPa
= 216 K
= 4.69 x 1018 molec. cm-3
= 0.43
--> k0,T
--> k0,T [M]
--> k∞,T
--> kr
= 2.60 x 10-30(300/T)2.9 cm6 molec.-2 s-1
= 2.47 x 10-11
= 8.16 x 10-11 cm3 molec.-1 s-1
= 1.11 x 10-11 cm3 molec.-1 s-1
PAN Rate Coefficient
10 -4
10 -5
275 K
10 -6
1
10
100
Pressure (hPa)
1000
-1
PAN rate coefficient (s-1) )
298 K
PAN rate coefficient (s
10 -3
-1
) -1)
PAN rate coefficient (s
PAN rate coefficient as a function of pressure and temperature,
respectively
10 -1
10 -3
10 -5
10 -7
10 -9
10-11
10-13
10-15
10-17
1013 hPa
200
250
300
Temperature (K)
Fig. 10.2
Photolysis Coefficient
J q, p 
(10.35)

0 4I p, ba,g,q,,T Yq, p,,T d
Photoprocesses
NO3 + h
(10.36)
NO2 + O
410 nm < < 670 nm
NO + O2
590 nm < < 630 nm
Photolysis Rate Coefficients
Pressure (hPa)
300
400
500
600
700
800
900
1000
-5
a
0.0001
0.001
0.01
0.1
HCHO
b
O
3
b
HCHO
HONO
a
b
O
NO
3
3
a
NO
2
10
-5
1
0.0001
0.001
0.01
-1
Photolysis rate coefficient (s )
NO
3
0.1
1
Pressure (hPa)
Pressure (hPa)
10
Fig. 10.3
Set of Reactions
(10.37)
NO + O3
NO2 + O2
(10.38)
O + O2 + M
O3 + M
(10.39)
NO2 + h
NO + O
(10.40)
NO2 + O
NO + O2
Reaction Rates
(10.37)
Rate1  k1NO O 3
(10.38)
Rate 2  k2 OO2 M
(10.39)
Rate 3  JNO 2
(10.40)
Rate 4  k3NO 2 O
ODEs For Set of Reactions
(10.41)
dNO
 Pc  Lc  Rate 3  Rate 4  Rate1  J NO2  k3 NO 2 O  k1 NOO3 
dt
dNO 2
dt
(10.42)
 Pc  Lc  Rate1  Rate3  Rate 4  k1NOO3  JNO 2  k 3NO 2 O 
(10.43)
dO
 Pc  L c  Rate3  Rate 2  Rate 4  J NO2  k2 OO2 M  k3 NO2 O
dt
(10.44)
dO3 
 Pc  Lc  Rate 2  Rate1  k2 OO 2 M  k1NO O 3 
dt
Production and Loss for Reaction Set
Production term for NO
(10.45)
Pc  J NO 2 k3NO 2O
Loss term for NO
(10.45)
Lc  k1NOO 3 
E-folding Lifetime
E-folding lifetime
Time a species takes to have its concentration reduced to 1/e its
original value
Overall lifetime (s)
(10.46)
A 
1
 A1

1
1
 A2
 ...
1
 An
E-folding Lifetime
Unimolecular reaction
A  products
First derivative
dA dt  kF A
E-folding lifetime occurs when
E-folding lifetime
A  1 kF h
 e
A 0 e
1
 A1  h 
kF
(10.47)
(10.48)
E-folding Lifetime
Bimolecular reaction
A  B  products
Loss rate of species A
dA
 kSA B0
dt
E-folding lifetime
(10.49)
1
 A2 
kSB0
E-folding Lifetime
Termolecular reaction
A  B  C  products
E-folding lifetime
(10.50)
 A3 
1
k T B0 C 0
Stiff System of Equations
Example 10.4:
CH4 + OH --> CH3 + H2O
k
= 6.2 x 10-15 cm3 molec-1 s-1 at 298K
[OH.] = 5.0 x 105 molec. cm-3
1
--->  CH 4 
. = 10.2 years
k OH
 
O(1D) + N2 --> O + N2
k
= 2.6 x 10-11 cm3 molec-1 s-1 at 298K
[N2] = 1.9 x 1019 molec. cm-3
 
1
--->  O 1 D 
= 2 x 10-9 seconds
k N2 
---> stiff set of equations
Half-Lifetime
Unimolecular reaction
Half-lifetime occurs when
(10.51)
A  1 k F h
 e
A 0 2
Solve for half-lifetime
0.693
 1 2 A1  h 
kF
(10.52)
Bimolecular reaction
(10.53)
0.693
 1 2 A2 
k SB 0
Termolecular reaction
0.693
 1 2 A3 
k T B 0 C 0
(10.53)