Presentation Slides for
Chapter 17, Part 1of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
March 31, 2005
Types of Equilibrium EquationsReversible chemical reaction (17.1)
Mass conservation (17.3)
Divide each dni by smallest value of dni (17.2)
dnDD +dnEE +... dnAA +dnBB +...
νDD+νEE +... νAA +νBB+...
ki dni( )mii∑ =0
Types of Equilibrium EquationsSolvent
Substance in which species dissolve in (e.g., water)
Solute
The dissolving species
Solution
Combination of solute and solvent
Solids
Suspended material not in solution
Gas-Particle EquilibriumGas-particle reversible reaction (17.4)
Gas in equilibrium with solution at gas-solution interface
Sulfuric acid (17.5)
ExamplesAB (g) AB (aq)
H2
SO4
(g) H2
SO4
(aq)
Nitric acid HNO3
(g) HNO3
(aq)
Hydrochloric acid
Carbon dioxide
Ammonia
HCl (g) HCl (aq)
CO2
(g) CO2
(aq)
NH3
(g) NH3
(aq)
Electrolytes, Ions, and AcidsElectrolyte
Substance that undergoes partial or complete dissociation into ions in solution
Ion
Charged atom or molecule
Dissociation
Molecule breaks into simpler components, namely ions. Degree of dissociation depends on acidity.
Acidity
Measure of concentration of hydrogen ions (H+, protons) in solution
Electrolytes, Ions, and AcidsAcidity measured in terms of pH (17.6)
Protons in solution donated by acids
pH = -log10[H+]
[H+] = molarity of H+ (mol-H+ L-1-solution)
Strong acids (dissociate readily at low pH)
HCl = hydrochloric acid
HNO3 = nitric acid
H2SO4 = sulfuric acid
Weak acids (dissociate readily at higher pH)
H2CO3 = carbonic acid
pH Scale
Fig. 10.3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Naturalrainwater
(5-5.6)
Distilledwater(7.0)
Seawater
(7.8-8.3)
Batteryacid(1.0)
Acidrain, fog(2-5.6)
More acidic More basic or alkaline
Lemonjuice(2.2)
VinegarCH3COOH(aq)
(2.8)
Apples(3.1)
Milk(6.6)
Bakingsoda
NaHCO3(aq)
(8.2)
Ammoniumhydroxide
NH4OH(aq)
(11.1)
LyeNaOH(aq)
(13.0)
Slaked limeCa(OH)2(aq)
(12.4)
pH
Electrolytes, Ions, and AcidsSulfuric acid dissociation (pH above -3) (17.7)
Nitric acid dissociation (pH above -1) (17.8)
Bisulfate dissociation (pH above 2) (17.7)
H2
SO4
(aq) H
+
+ HSO4
HSO4
H+
+ SO2-
4
HNO3
(aq) H
+
+ NO3
Electrolytes, Ions, and AcidsHydrochloric acid dissociation (pH above -6) (17.9)
Bicarbonate dissociation (pH above 10) (17.10)
Carbon dioxide dissociation (pH above 6) (17.10)
HCl (aq)H
+
+ Cl-
CO2
(aq) + H2
O(aq) H2
CO3
(aq) H
+
+ HCO3
HCO3
H+
+ CO2-
3
BasesBase
Donates OH- (hydroxide ion)
Ammonia complexes with water and dissociates (17.12)
Hydroxide ion combine with hydrogen ion to form liquid water, increasing pH of solution (17.11)
H2
O(aq) H+
+ OH-
NH3
(aq) + H2
O(aq) NH4
+ OH
-
Solid ElectrolytesSuspended electrolytes not in solution
Precipitation / crystallization
Formation of solid electrolytes from ions
Dissociation
Separation of solid electrolytes into ions
Solid ElectrolytesAmmonium-containing solid reactions (17.15)
NH4
Cl(s) NH4
+ Cl
-
NH4
NO3
(s) NH4
+ NO3
(NH4
)2
SO4
(s)2NH
4 + SO
2-
4
Solid ElectrolytesSodium-containing solid reactions (17.16)
NaCl(s)Na
+
+ Cl-
NaNO3
(s) Na
+
+ NO3
Na2
SO4
(s)
2Na
+
+ SO
2-
4
NH4
Cl(s) NH3
(g) + HCl(g)
NH4
NO3
(s) NH3
(g) + HNO3
(g)
Solid formation from the gas phase on surfaces (17.17)
Equilibrium Relation and ConstantEquilibrium coefficient relation (17.18)
{}... = Activity
Effective concentration or intensity of substance
(gas) (17.19)
(ion) (17.20)
(dissolved molecule) (17.20)
(liquid water) (17.21)
(solid) (17.22)
ai{ }kiνi
i∏ =
A{ }νA B{ }νB ...
D{ }νD E{ }νE ...=KeqT( )
A g( ){ }=pA,s
A+{ }=mA +γA+
A aq( ){ }=mAγA
H2O aq( ){ } =aw =pvpv,s
= fr
A s( ){ }=1
Equilibrium Coefficient RelationGibbs free energy (17.23)
Enthalpy
Change in Gibbs free energyMeasure of maximum amount of useful work obtained from a change in enthalpy or entropy of the system (17.24)
G* =H* −TS* =U* +paV−TS*
H* =U* +paV
dG* =d H* −TS*( ) =dU* +padV+Vdpa−TdS* −S*dT
Equilibrium Coefficient RelationChange in entropy
Change in internal energy in presence of reversible reactions (17.26)
Change in internal energy (17.25)
dS* =dQ* T
dU* =dQ* −padV=TdS* −padV
dU* =TdS* −padV+ ki dni( )μii∑
Equilibrium Coefficient RelationSubstitute (17.26) into (17.24) (17.27)
Hold temperature and pressure constant (17.28)
dG* =Vdpa −S*dT+ ki dni( )μii∑
dG* = ki dni( )μii∑
Equilibrium Coefficient RelationChemical potential (i )
Measure of intensity of a substance or the measure of the change in free energy per change in moles of a substance = partial molar free energy (17.29)
Equilibrium occurs when dG* = 0 in (17.28) (17.30)
μi =∂Gi
*
∂ni
⎛
⎝ ⎜
⎞
⎠ ⎟ T,pa
=μio T( )+R*T ln ai{ }
kiνiμii∑ =0
Equilibrium Coefficient Relation
Substitute (17.29) into (17.30) (17.31)
where
Standard molal Gibbs free energy of formation
kiνiμio T0( )
i∑ +R*T0 kiνi ln ai{ }
i∑ = kiνiΔ fGi
o
i∑ +R*T0 ln ai{ }
kiνi
i∏ =0
kiνi ln ai{ }i∑ =ln ai{ }kiνi
i∏
Δ fGio =μi
o T0( )
Equilibrium Coefficient Relation
Rearrange (17.31) (17.32)
The right side of (17.32) is the equilibrium coefficient (17.33)
ai{ }kiνi
i∏ =exp −
1
R*T0kiνiΔ fGi
o
i∑
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Keq T0( ) =exp−1
R*T0kiνiΔ fGi
o
i∑
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Temperature Dependence of Equilibrium Coefficient
Van't Hoff equation (similar to Arrhenius equation) (17.34)
Molal enthalpy of formation (J mol-1) of a substance (17.35)
= Standard molal heat capacity at constant pressure
= standard molal enthalpy of formation
dlnKeqT( )
dT=
1
R*T2 kiνiΔ fHii∑
Δ fHi ≈Δ fHio +cp,i
o T −T0( )
cp,io
Δ fHio
Temperature Dependence of Equil Const
Combine (17.34) and (17.35) and write integral (17.36)
Integrate (17.37)
dlnKeqT( )T0
T∫ =
1
R*T2 kiνi Δ fHio +cp,i
o T−T0( )[ ]i∑ dT
T0
T∫
Keq T( ) =KeqT0( )exp − kiνiΔ fHi
o
R*T0
T0T
−1⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
cp,io
R* 1−T0T
+lnT0T
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥ i
∑⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
Forms of Equilibrium EquationHenry's law
In a dilute solution, the pressure exerted by a gas at the gas-liquid interface is proportional to the molality of the dissolved gas in solution
Equilibrium coefficient relationship (17.38)
Henry's law relationship
HNO3 g( ) HNO3 aq( )
HNO3 aq( ){ }HNO3 g( ){ }
=mHNO3 aq( )γHNO3 aq( )
pHNO3 g( ),s=KeqT( )
molkg atm
Activity Coefficients ()Account for deviation from ideal behavior of a solution.
Infinitely dilute solution, no deviations, = 1
Relatively dilute solutions, deviations from Coulombic (electric) forces of attraction and repulsion < 1
Concentrated solutions, deviations caused by ionic interactions, < 1 or > 1
Activity Coefficients
Geometric mean binary activity coefficient (17.40)
Rewrite (17.41)
γ±= γ+ν+γ-
ν−( )1 ν++ν−( )
γ±ν++ν− =γ+
ν+γ-ν−
Electrolyte Dissociation
Univalent electrolyte
Multivalent electrolyte
---> = 1 and = 1
---> = +1 and = -1
---> = 2 and = 1
---> = +1 and = -2
HNO3 aq( ) H++NO3−
Na2SO4 s( ) 2Na++SO42− ν+
ν+ ν−
ν−
z+
z+
z−
z−
Electrolyte Dissociation
Symmetric electrolyte
Charge balance requirement
ν+=ν−
z+ν++z−ν−=0
Equilibrium Rate Expression
1. (17.39)HNO3 aq( ) H++NO3−
H+{ } NO3
-{ }
HNO3 aq( ){ }=
mH+γ
H+mNO3- γ
NO3-
mHNO3 aq( )γHNO3 aq( )
=
mH+mNO3
- γH+,NO3
-2
mHNO3 aq( )γHNO3 aq( )
=Keq T( )mol
kg
2. (17.42)Na2
SO4
(s) 2Na+
+ SO2-
4
Na+{ }
2SO4
2−{ }
Na2SO4 s( ){ }=
mNa+2 γ
Na+2 m
SO42−γ
SO42−
1.0
=mNa+2 m
SO42−γ
2Na+,SO42−
3=Keq T( )
mol3
kg3
Equilibrium Rate Expression
3. (17.43)HSO4 H
+
+ SO
2-
4
H+{ }
2SO4
2-{ }
H+{ } HSO4
-{ }
=
mH+2 γ
H+2 m
SO42-γSO4
2-
mH+γ
H+mHSO4- γ
HSO4-
=
mH+mSO4
2-γ2H+,SO42-
3
mHSO4
- γH+,HSO4
-2
=Keq T( )mol
kg
Equilibrium Rate Expression
4. (17.44)
NH4+
{ } NO3-
{ }
NH3 g( ){ } HNO3 g( ){ }=
mNH4
+γNH4
+mNO3- γ
NO3-
pNH3 g( ),spHNO3 g( ),s
=
mNH4
+mNO3- γ
NH4+,NO3
-2
pNH3 g( ),spHNO3 g( ),s
=Keq T( )mol2
kg2 atm2
NH3
(g) + HNO3
(g) NH4
+ NO3
Equilibrium Rate Expression
5. (17.45)NH3
(aq) + H2
O(aq) NH4
+ OH
-
NH4+
{ } OH−{ }
NH3 aq( ){ } H2O aq( ){ }=
mNH4
+γNH4
+mOH−γOH−
mNH3 aq( )γNH3 aq( ) fr
=
mNH4
+mOH−γNH4
+,OH−2
mNH3 aq( )γNH3 aq( )fr
=Keq T( )mol
kg
Mean Binary Activity Coefficients
Pitzer's method of determining binary activity coefs. (17.46)
(17.47)
lnγ12b0 =Z1Z2f
γ +m122ν1ν2ν1+ν2
B12γ
+m122 2 ν1ν2( )
3 2
ν1+ν2C12
γ
fγ =−0.392I12
1+1.2I12 +2
1.2ln 1+1.2I12
( )⎡
⎣ ⎢
⎤
⎦ ⎥
Mean Binary Activity Coefficients
(17.48)
’s are Pitzer parameter’s specific to individual electrolytes
Ionic strength of solution (mol kg-1)Measure of the interionic effects resulting from attraction and repulsion among ions (17.49)
B12γ
=2β121( )
+2β12
2( )
4I1−e−2I12
1+2I12 −2I( )⎡ ⎣ ⎢
⎤ ⎦ ⎥
I =12
m2i−1Z2i−12
i=1
NC∑ + m2iZ2i
2
i=1
NA∑
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Mean Binary Activity Coefficients
Alternatively, fit a polynomial expression to mean binary activity coefficient data (valid to high molality) (17.51)
lnγ12b0 =B0 +B1m12
12+B2m12 +B3m12
3 2+...
Mean Binary Activity Coefficients
Fig. 17.2
Comparison of measured (Hammer and Wu) and calculated (Pitzer) activity coefficient data
-3
-2
-1
0
1
2
3
4
5
0 1 2 3 4 5 6
Pitzer
Hammer
and Wu
HNO
3
NH
4
NO
3
HCl
ln(binary activity coefficient)
m
1/2
ln (
bina
ry a
ctiv
ity
coef
fici
ent)
Mean Binary Activity CoefficientsEquilibrium coefficient expression for hydrochloric acid
(17.50)
Equilibrium coefficient expression for nitric acid
mH+mCl−γH+,Cl−2
pHCl(g),s=1.97×106
mH+mNO3−γ
H+,NO3−
2
pHNO3 g( ),s=2.51×106
Temp Dependence of Mean Binary Activity Coefficient
Temperature dependent equation (17.52)
Temperature-dependent parameters (17.53)
lnγ12b T( )=lnγ12b0
+TL
ν1+ν2( )R*T0
φL +m∂φL∂m
⎛ ⎝ ⎜
⎞ ⎠ ⎟
+TC
ν1+ν2( )R* φcp +m
∂φcp∂m
−φcpo⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
TL =T0T
−1
TC =1+lnT0T
⎛ ⎝ ⎜
⎞ ⎠ ⎟ −
T0T
Temp Dep of Mean Binary Activity CoefPolynomial for relative apparent molal enthalpy (17.54)
Polynomial for apparent molal heat capacity
= binary activity coefficient at temperature T
L = relative apparent molal enthalpy (J mol-1)
= apparent molal heat capacity (J mol-1 K-1)
= apparent molal heat capacity at infinite dilution
φL =U1m12+U2m+U3m
32 +...
φcp =φcpo +V1m
12 +V2m+V3m32 +...
γ12b T( )
φcpφcp
o
Temp Dep of Mean Binary Activity CoefCombine (17.51) - (17.54) --> (17.55)
Coefficients for equation (17.56-7)
lnγ12b T( )=F0+F1m12 +F2m+F3m
32 +...
Fj =Bj +GjTL +HjTC
Gj =0.5 j +2( )U j
ν1+ν2( )R*T0
Hj =0.5 j +2( )Vj
ν1+ν2( )R*
F0 = B0 j = 1...
Sulfate and Bisulfate
Fig. 17.3
Binary activity coefficients of sulfate and bisulfate, each alone in solution. Results valid for 0 - 40 m.
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
0 1 2 3 4 5 6 7 8
201 K
273 K
298 K
328 K
Binary activity coefficient
m
1/2
H
+
/ HSO
4
-
2H
+
/ SO
4
2-
Bin
ary
acti
vity
coe
ffic
ient
Mean Mixed Activity CoefficientsBromley's method (17.58-61)
Binary activity coefficient of an electrolyte in a mixture of many electrolytes.
log10γ12m T( ) =−AγZ1Z2Im
12
1+Im12 +
Z1Z2Z1+Z2
W1Z1
+W2Z2
⎛
⎝ ⎜
⎞
⎠ ⎟
W1=Y21 log10γ12b T( )+AγZ1Z2Im
12
1+Im12
⎛
⎝ ⎜
⎞
⎠ ⎟ +Y41 log10γ14b T( )+Aγ
Z1Z4Im12
1+Im12
⎛
⎝ ⎜
⎞
⎠ ⎟ +...
W2 =X12 log10γ12b T( )+AγZ1Z2Im
12
1+Im12
⎛
⎝ ⎜
⎞
⎠ ⎟ +X32 log10γ32b T( )+Aγ
Z3Z2Im12
1+Im12
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ +...
Y21=Z1+Z2
2⎛ ⎝ ⎜ ⎞
⎠ ⎟ 2 m2,m
ImX12 =
Z1+Z22
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 2 m1,m
Im
Mean Mixed Activity CoefficientsMolalities of binary electrolyte found from (17.62)
Molalities of cation, anion alone in solution
Molality of binary electrolyte giving ionic strength of mixture (17.63)
Im=12m1,bZ1
2 +m2,bZ22
( ) =12
ν+m12,bZ12+ν−m12,bZ2
2( )
m1,b =ν+m12,b m2,b =ν−m12,b
m12,b =2Im
ν+Z12+ν−Z2
2
