Presentation Slides
for
Chapter 17, Part 2
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
jacobson@stanford.edu
April 1, 2005
Solvation and Hydration
Solvation
Bonding between solvent and solute in solution
Hydration
When solvent is liquid water, solvation is hydration
Hydration of cations --> lone pairs of electrons on oxygen atom
of water attach to cations
Hydration of anions --> water molecule attaches to anion via
hydrogen bonding
Water Equation
Quantify amount of hydration with empirical water equation
Zdanovskii-Stokes-Robinson (ZSR) equation
Example with two species, x and y
m x,m m y,m

1
m x,a m y,a
(17.64)
mx,a, my,a = molalities of x and y, alone in solution at given
relative humidity
mx,m, my,m = molalities of x and y, when mixed together, at
same relative humidity
ZSR Equation
ZSR equation predictions for a sucrose (species x) - mannitol
(species y) mixture at two different water activities.
m x,m m y,m

1
m x,a m y,a
Case
1
2
mx,a
0.7751
0.9393
my,a
0.8197
1.0046
mx,m
0.6227
0.1900
my,m
0.1604
0.8014
mx,m/mx,a +
my,m/my,a
0.999
1.000
Table 17.2
Water Equation
Generalized ZSR equation
(17.64)
m k,m
 m k,a  1
k
Polynomial expression for molality of electrolyte alone in solution
at a given water activity
(17.66)
mk,a  Y0,k  Y1,k aw  Y2,k a2w  Y3,k a3w  ...
Water Equation
Water activities of several electrolytes at 298.15 K
10 20 30 40 50 60
1
Water activity
Water activity
0
0.8
0.6
NaNO3
0.4
HNO
0.2
3
H2 SO 4
HCl
0
0
10
20
30 40
Molality
50
60
Fig. 17.4a
Water activities of several electrolytes at 298.15 K
0
Water activity
Water activity
Water Equation
5 10 15 20 25 30
1
0.9
0.8
(NH ) SO
4 2
4
NH NO
4
0.7
0.6
3
NH4 Cl
0.5
0.4
0
Na2 SO 4
NaCl
5
15 20
Molality
10
25
30
Fig. 17.4b
Temp. Dependence of Water Activity
Temperature dependence of binary water activity coefficients
under ambient surface conditions is small.
Temperature dependence of water activity
(17.67)
2
m
m
 cP 
v k,a TL  L
0
ln aw T   ln aw 
 TC

* T m
m k,a 
R
 0
k,a
Polynomial for water activity at reference temperature (17.68)
12
32
0
ln aw  A0  A1mk,a  A2mk,a  A3mk,a  ...
Temp. Dependence of Water Activity
Combine (17.67), (17.68), (17.54)
(17.69-70)
12
32
ln aw T   A0  A1mk,a  A2mk,a  E3mk,a  E4m2k,a  ...

0.5l  2mv TL
El  Al 
 Ul 2  TC Vl 2 
*
R
T0

Example
mHCl= 16 m
T
= 273 K
--->
aw
= 0.09
T
= 310 K
--->
aw
= 0.11
Practical Use of Water Equation
Rearrange (17.65)
(17.71)
NC N A c

1
i, j,m 

cw 
mv
 m i, j,a 
i1 j 1

 
mi,j,a = binary molalities of species alone in solution
ci,j,m = hypothetical mol cm-3 of electrolyte pair when mixed in
solution with all other components
In a model, ion concentrations known but hypothetical electrolyte
concentrations unknown --> find hypothetical concentrations
Practical Use of Water Equation
Example 17.1:
6 mol m-3 of H+, 6 mol m-3 Na+
7 mol m-3 of Cl- , 5 mol m-3 of NO3Combine ions in a way to satisfy mole balance constraints
cH,m  cHNO3,m  cHCl,m
cNa ,m  cNaNO3,m  cNaCl,m
c Cl  ,m  c HCl,m  cNaCl,m
cNO-3 ,m  cHNO3 ,m  cNaNO3,m
Concentrations that satisfy mole balance constraints (Table 17.3)
Case
cHCl,m
cHNO3,m
cNaCl,m
cNaNO3,m
1
6
0
1
5
2
4
2
3
3
Practical Use of Water Equation
Automatic method to recombine ions into hypothetical
electrolytes
Execute the following three equations, in succession, for each
undissociated electrolyte, i,j
Electrolyte
c i,m c j,m 
ci, j,m  min 
  ,  

j 
 i
Cation
ci,m  ci,m  i ci, j,m
Anion
c j,m  cj,m   jci, j,m
(17.72)
Deliquescence Relative Humidity
Deliquescence
Process by which a particle takes up liquid water, lowering its
saturation vapor pressure
Deliquescence relative humidity (DRH)
The relative humidity at which an initially-dry solid first takes
on liquid water during an increase in relative humidity. Above
the DRH, the solid may not exist.
Crystallization relative humidity (CRH)
The relative humidity at which an initially-supersaturated
aqueous electrolyte becomes crystalline upon a decrease in
relative humidity.
Deliquescence Relative Humidity
DRHs and CRHs for several electrolytes at 298 K
Electrolyte
NaCl
Na2SO4
NaHSO4
NH4Cl
(NH4)2SO4
NH4HSO4
NH4NO3
KCl
Oxalic acid
DRH(%)
75.28
84.2
52.0
77.1
79.97
40
61.83
84.26
97.3
CRH(%)
47
57-59
<5
47
37-40
<5-22
25-32
62
51.8-56.7
In a mixture, the DRH of a solid in equilibrium with the solution
is lower than the DRH of the solid alone
Table 17.4
Solid Formation
Consider the equilibrium reaction
NH 4  NO3
NH 4 NO3 s
A solid forms when
m
NH4
m
(17.73)
2


NO3
NH4  ,NO3
 KeqT
Consider the equilibrium reaction
NH 4 NO3 s
NH 3 g  HNO 3g
A solid forms when
pNH3 g,s pHNO3 g,s  Keq T 
(17.74)
Example Equilibrium Problem
Consider two equilibrium reactions
(17.75)
HCl(g)
H+ + Cl-
HSO4
2H+ + SO
4
For equilibrium concentrations, solve
equilibrium constant equations
mole balance equations
charge balance equation
water equation
with Newton-Raphson iteration
Example Equilibrium Problem
Equilibrium coefficient equations
(17.76)
m  m - 2  H ,eq Cl ,eq H ,Cl ,eq
 Keq T 
pHCl,s,eq
m
H  ,eq
m
3
SO 4 2 ,eq 2H  ,SO 4 2 ,eq
 Keq T 
2
m
 

HSO 4 ,eq H ,HSO 4  ,eq
Example Equilibrium Problem
Mole balance equations
(17.77)
CHCl(g),eq  c  CHCl(g),t h  c Cl ,eq
Cl ,t h
(17.78)
c
HSO4  ,eq
c
SO 42 ,eq
c
c

HSO4 ,th
SO4 2 ,th
Example Equilibrium Problem
Vapor pressure as a function of mole concentration
(17.79)
pHCl,s,eq  CHCl(g),s,eq R*T
Molality as a function of mole concentration
c Cl ,eq
m 
Cl ,eq c w,eq mv
Charge balance equation
(17.80)
c 
c
 2c
c +

2

HSO 4 ,eq
SO 4 ,eq
Cl ,eq
H ,eq
Example Equilibrium Problem
Water equation
(17.81)
 c

c
c
1  H  ,Cl - ,m
H  ,HSO 4  ,m
2H  ,SO 4 2 ,m 
c w,eq 




mv m
m
m





2

H
,Cl
,a
H
,HSO
,a
2H
,SO
,a


4
4
Hypothetical mole concentration constraints
(17.82)
c 
 c  - c 
 2c 

H ,eq
H ,Cl ,m
H ,HSO 4 ,m
2H ,SO 4 2  ,m
c 
c  Cl ,eq
H ,Cl ,m
c
HSO 4  ,eq
c
SO 42  ,eq
c 
H ,HSO 4 ,m
c 
2
2H ,SO 4
,m
Mass-Flux Iterative Method
Solve each equation iteratively and iterate over all equations
Initialize species concentrations so that charge is conserved
No intelligent first guess required
Solution mass and charge conserving and always converges
Example solution for one equilibrium equation
Equilibrium equation and coefficient relation
DD   EE  ...
AA  B B  ...
AA BB ...
D
 E  Keq T 
D E ...
Mass-Flux Iterative Method
1) Calculate smallest ratio of mole concentration to moles in
denominator and numerator, respectively
(17.83)
C D,0 C E,0 
Qd  min 
,



 D
E 
c A,0 c B,0 
Qn  min 
,



 A
B 
2) Initialize two parameters
z1  0.5(Qd  Qn)
x1  Qd  z 1
Mass-Flux Iterative Method
Add mass flux factor (x) to mole concentrations
(17.84)
c A,l1  c A,l   A xl
c B,l1  cB,l   B x l
CD,l1  CD,l   D x l
CE,l 1  CE,l  E x l
3) Compare ratio of activities to equilibrium coefficient (17.85)

F

 
A m B  A
B
m A,l1
B,l 1 AB,l1
D
E
pD,l1 pE,l1
1
Keq T 
Mass-Flux Iterative Method
4) Cut z in half
zl1  0.5zl
5) Check convergence
 1

F   1

 1
(17.86)

xl 1  z l 1

xl 1  z l 1

convergence
Return to (17.84) until convergence occurs
Analytical Equilibrium Iteration Method
Solve most equations analytically but iterate over all equations
Reactions of the form DA
Solve the equilibrium equation
(17.87)
c A,c
c A,0  x fin

 Kr
c D,c cD,0  x fin
Solution for change in concentration
(17.88)
c D,0 K r  c A,0
x fin 
1  Kr
Final concentrations
cA,c  cA,0  x fin
cD,c  cD,0  x fin
Analytical Equilibrium Iteration Method
Reactions of the form D+EA+B
Solve the equilibrium equation




(17.89)
c A,0  x fin c B,0  x fin
c A,cc B,c

 Kr
c D,cc E,c
c D,0  x fin c E,0  x fin
Solution for change in concentration
(17.90)


c A,0  c B,0  c D,0 Kr  cE,0 Kr
cA,0  cB,0  cD,0Kr  c E,0Kr 
41  Kr c A,0 c B,0  c D,0c E,0 
2

x fin 
21 Kr 
Analytical Equilibrium Iteration Method
Final concentrations
cA,c  cA,0  x fin
cB,c  cB,0  x fin
cD,c  cD,0  x fin
cE,c  cE,0 x fin
Analytical Equilibrium Iteration Method
Reactions of the form D(s)2A+B
Check if solid can form
(17.91)
cA,0  2cD,0 cB,0  2cD,0  Kr
2
If so, solve the equilibrium equation

(17.92)


2
2
c A,cc B,c  c A,0  2x fin c B,0  x fin  Kr
Analytical Equilibrium Iteration Method
Iterative Newton-Raphson procedure
(17.93)
fn x   x 3fin,n  qx 2fin,n  rx fin,n  s  0
f x   3x 2fin,n  2qx fin,n  r
q  c A,0  c B,0
r  c A,0 c B,0  0.25c 2A,0
s  c 2A,0 c B,0  K r
fn x 
x fin,n1  x fin,n 
fnx 
Analytical Equilibrium Iteration Method
Final concentrations
cA,c  cA,0  2x fin
cB,c  cB,0  x fin
cD,c  cD,0  x fin
Equilibrium Solver Results
g m-3 -3

Concentration (g m) )
Concentration (
Aerosol composition versus NaCl concentration when the relative
humidity was 90%. Other initial conditions were H2SO4(aq) = 10
g m-3, HCl(g) = 0 g m-3, NH3(g) = 10 g m-3, HNO3(g) = 30 g
m-3, and T = 298 K.
30
NO3-
25
20
H2O(aq) x 0.1
15
SO 42-
10
NH4+
5
Cl -
0
0
5
10
15
20
25
NaCl concentration (g m -3)
30
Fig. 17.4
Equilibrium Solver Results
g m-3
) -3)
Concentration (g m
Concentration (
Aerosol composition versus relative humidity. Initial conditions
were H2SO4(aq) = 10 g m-3, HCl(g) = 0 g m-3, NH3(g) = 10 g
m-3, HNO3(g) = 30 g m-3, and T = 298 K.
25
20
NH4NO3(s)
NO3-
15
(NH4)2SO 4(s)
10
SO 42NH4+
5
H2O(aq) x 0.1
0
0
20
40
60
80
Relative humidity (percent)
100
Fig. 17.5
Dissolutional Growth
Saturation vapor pressure of gas q over particle size i
pq,s,i 
(17.95)
m q,i
Hq
Saturation vapor pressure as function of gas mole concentration
(17.96)
pq,s,i  Cq,s,i R* T
Molality as function of particle mole concentration
m q,i 
c q,i
mvc w,i
(17.97)
Dissolutional Growth
Substitute (17.95) and (17.97) into (17.96)
C q,s,i 
pq,s,i
m q,i
(17.98)
c q,i
c q,i
 *


*

R T
R TH q m vcw ,i R TH q Hq,i
*
where
(17.99)
Hq,i
  mvc w,i R* TH q
Dissolutional Growth
Condensational growth equations
dc q,i,t
dt

 k q,i,t h Cq,t  Sq,i,t
 h Cq,s,i,t h
(16.67)

(16.68)
dCq,t
dt

NB
 h Cq,s,i,t 
kq,i,t hCq,t  Sq,i,t
i1
Dissolutional Growth
Substitute (17.98)
--> Dissolutional growth equations
dc q,i,t
dt
(17.100)

cq,i,t 
 k q,i,t h 

q,i,t h
Cq,t  S

H

q,i,t h 

(17.101)
dCq,t
dt
N B 

cq,i,t 

   k q,i,t h 

q,i,t h
Cq,t  S

H


q,i,t h 


i1 
Analytical Predictor of Dissolution
Integrate (17.100) for final aerosol concentration
c q,i,t 
Hq,i,t
 h C q,t
Sq,i,t
 h
(17.102)

H q,i,t
 h Cq,t   hSq,i,t
 h k q,i,t h 
 


c q,i,t h  S 
 exp 


H

q,i,t h 
q,i,t h



Mole balance equation
Cq,t 
(17.103)
NB
NB
i1
i 1
 cq,i,t  Cq,t h   cq,i,t h
Substitute (17.102) into (17.103)
NB 
(17.104)

 hSq,i,t

 h k q,i,t h 


C q,t h 


cq,i,t h 1  exp 


Hq,i,t
  h






i1 

C q,t 
N B  H 
 hSq,i,t
 h k q,i,t h 
 q,i,t h 

1
1  exp 







Hq,i,t
  h
h 




 Si,q,t

i1 


Growth During Dissociation
Growth equation for hydrochloric acid

(17.105)
dc Cl,i,t
 k HCl,i,t h CHCl,t  SHCl,i,t

h C HCl,s,i,t
dt
Total dissolved chlorine

(17.106)
c Cl,i,t  c HClaq,i,t  c Cl -,i,t
Find saturation mole concentration from equilibrium expressions
(17.107)
HClHCl(aq)
(17.108)
HCl(aq)H++Cl-
Growth During Dissociation
Equilibrium coefficient relations
m HClaq ,i
mol
 H HCl
pHCl,s,i
kg atm
m H + ,i m Cl - ,i  2 + i,H Cl
m HClaq ,i
(17.107)
(17.108)
mol
 KHCl
kg
Equilibrium coefficient relations in terms of mole concentration
(17.109)
cCl,i
CHCl,s,i 
KHCl,i


2 * 
 KHCl mv cw,i R T 
K
HCl,i = HHCl 1
2

+
c

H ,i

i,H + Cl - 




(17.110)
Dissolution of Acids/Bases
Substitute saturation mole concentration into growth equation
(17.111)

dc Cl,i,t
cCl,i,t 
 k HCl,i,t h CHCl,t  SHCl,i,t


h
dt
K


HCl,i,t h 
Mole balance equation
CHCl,t 
(17.112)
NB
NB
i 1
i1
cCl,i,t  CHCl,t h   cCl,i,th
Dissolution for Dissociating Species
Integrate (17.111) for final aerosol concentration
(17.113)
Substitute (17.113) into (17.112)
C HCl,t 
(17.114)
KHCl,i,th

CHCl,t
c Cl,i,t 
SCl
 - ,i,t h

 hk HCl,i,t h SHCl,i,t
KHCl,i,t


h CHCl,t 
h 
 cCl,i,t h 
 exp 

SHCl,i,t

K

h
 
HCl,i,t h

N B 


 hk HCl,i,t  h S

HCl,i,t h 
C HCl,t h   c Cl,i,t h 1 exp 

K 


HCl,i,t h






i1
N B 




 K 
hk
S

HCl,i,t h
HCl,i,t h HCl,i,t h 
1   
1  exp 

SHCl,i,t
K HCl,i,t




h

h





i1
Solve for Ammonia/Ammonium
Charge balance equation
(17.115)
cNH4 +,i,t  cH+,i,t  c,i,t  0
where
(17.116)
c ,i,t  cNO 3- ,i,t  c Cl - ,i,t cHSO 4- ,i,t  2cSO 4 2- ,i,t   zc q,i,t h
q
Mole balance equation
CHCl,t  CNH 3,t 
(17.117)
NB
 cNH3 aq,i,t  cNH4 +,i,t 
i1
 CNH3 ,t h 
NB
cNH3 aq,i,th  cNH4 +,i,th  Ctot
i 1
Solve for Ammonia/Ammonium
Equilibrium expressions
NH3(g)NH3(aq)
NH3(aq)+H+NH4+
Equilibrium coefficient expressions
m NH 3 aq ,i
mol
 H NH3
pNH
kg atm
(17.118)
(17.119)
(17.118)
3
(17.119)
m NH  ,i  i,NH 
kg
4
4
 KNH 3
m NH 3 aq ,i m H + ,i  i,H +
mol
Solve for Ammonia/Ammonium
NH4+/H+ activity coefficient relationship
(17.120)
2

 i,NH 4   i,NH4   i,NO3 
i,NH 4 NO3


 i,H +
 i,H +  i,NO 3
2 +

i,H NO3
2
 i,NH 4   i,Cl -  i,NH4  Cl 

 i,H +  i,Cl 2 + i,H Cl
Equilibrium coefficient relations in terms of mole concentration
(17.121,2)
c NH3 aq ,i
mol
*
 HNH
 3 ,i
H


H
R
NH 3,i
NH3 Tm vc w,i
C
mol
NH3
c NH4 + ,i
 i,H +
cm3
1
 KNH
 3 ,i
K
NH3 ,i = KNH 3
+
c NH3 aq ,i c H ,i
mol
mv cw,i  i,NH4 
Solve for Ammonia/Ammonium
Ion concentration in each size bin
(17.124)
c ,i,t CNH ,t HNH
 ,i,t h K NH
 ,i,t h
3
3
3
c NH4 + ,i,t 
CNH3 ,t HNH
 3 ,i,t h K
NH3 ,i,t h  1
Substitute into mole-balance equation
(17.125)
CNH3 ,t H 

NH3 ,i,t h
NB


 ,i,t h KNH
 ,i,t h  Ctot  0
CNH3 ,t 
 c,i,t CNH 3 ,t HNH
3
3


i 1 C
H

K


1
NH
,t
NH
,i,t
h
NH
,i,t
h


3
3
3

Solve for Ammonia/Ammonium
Iterate for ammonia gas concentration


CNH3 ,t,n1  CNH3 ,t,n 
fnCNH ,t,n 
3
(17.126)
fn CNH3 ,t,n
where

(17.128)

fn CNH 3,t,n 
c ,i,t H 


NH3 ,i,t h K 
NH3 ,i,t h
 3,i,t h 
HNH

CNH3 ,t,n H 
NH3 ,i,t h K 
NH3 ,i,t h  1
N B 

2
1   
 c ,i,t CNH ,t,n HNH
 ,i,t h KNH
 ,i,t h

3
3
3
i1 

2
 C

H

K

1
NH
,t,n
NH
,i,t
h
NH
,i,t
h

3
3
3





Simulations of Growth/Dissociation
dN (No. cm-3) / dlog10 Dp
dM (g m-3) / dlog10 Dp
Initial distributions for simulation
Fig. 17.7
Aerosol concentrations, summed over all sizes, during
nonequilibrium growth plus internal aerosol equilibrium at
RH=90 percent when h=5 s.
g m-3

Summed concentration (g m-3))
Summed concentration (
Simulations of Growth/Dissociation
30
(b) h=5s
25
NO
-
3
20
H O x 0.1
2
15
S(VI)
10
Cl
5
NH
-
+
4
+
Na
0
0
2
4
6
8
Time from start (h)
10
12
Simulations of Growth/Dissociation
-3
 gmm-3) )
Summed concentration (g
Same as previous slide, but h=300 s
30
(c) h=300s
25
NO
-
3
20
H O x 0.1
2
15
S(VI)
10
Cl
5
NH
-
+
4
+
Na
0
0
2
4
6
8
Time from start (h)
10
12
Nonequilibrium Growth of Solids
Gas-solid equilibrium reactions
NH4NO3(s)NH4(g)+HNO3(g)
NH4Cl(s)NH4(g)+HCl(g)
Solids can form when
(17.129)
(17.130)
(17.131)
pNH3 pHNO3  KNH4NO3
(17.132)
pNH3 pHCl  KNH4 Cl
Nonequilibrium Growth of Solids
Gas-solid equilibrium coefficient relation
(17.133)
 
*
CNH3 ,s,t CHNO 3 ,s,t  KNH 4 NO3 R T
2
(17.134)
 
*
CNH3 ,s,t CHCl,s,t  KNH4Cl R T
2
Nonequilibrium Growth of Solids
Growth equations for gases that form solids (solids formed during
operator-split equilibrium calculation)
dc NO3 ,i,t
dt

 k HNO3 ,i,th CHNO3 ,t  S
HNO3 ,i,th CHNO3,s,t
dc Cl  ,i,t
 k HCl,i,t h CHCl,t  SHCl,i,t

h CHCl,s,t
dt



Simulations of Solid Growth
g m-3 -3)

Summed concentration (g m )
Time-dependent aerosol concentrations, summed over all sizes,
during nonequilibrium growth plus internal aerosol equilibrium at
RH=10 percent when h=5 s.
NH NO (s)
4
(NH ) SO (s)
3
4 2
4
10
NaCl(s)
(a) h=5s
Na SO (s)
2
1
NaNO (s)
4
NH Cl(s)
4
3
0
2
4
6
8
Time from start (h)
10
12
Fig. 17.8
Simulations of Solid Growth
g m-3-3)

Summed concentration (g m )
Same as previous slide, but h=300 s
NH NO (s)
4
(NH ) SO (s)
3
4 2
4
10
NaCl(s)
(b) h=300s
Na SO (s)
2
1
NaNO (s)
4
NH Cl(s)
4
3
0
2
4
6
8
Time from start (h)
10
12
Fig. 17.8