Presentation Slides for

Chapter 17, Part 1of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

Stanford UniversityStanford, CA 94305-4020jacobson@stanford.edu

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)

SoluteThe dissolving species

SolutionCombination of solute and solvent

SolidsSuspended 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)Examples

AB (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 solutionIon

Charged atom or moleculeDissociation

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 acidHNO3 = nitric acidH2SO4 = 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 / crystallizationFormation of solid electrolytes from ions

DissociationSeparation 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

+

+ SO2-

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{ }kiνii∏ = 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 RelationSubstitute (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

oi∑ +R*T0 ln ai{ }kiνi

i∏ =0

kiνi ln ai{ }i∑ =ln ai{ }kiνii∏

Δ fGio =μio T0( )

Equilibrium Coefficient RelationRearrange (17.31) (17.32)

The right side of (17.32) is the equilibrium coefficient (17.33)

ai{ }kiνii∏ =exp − 1

R*T0kiνiΔ fGi

oi∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Keq T0( ) =exp− 1R*T0

kiνiΔ fGioi∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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,io T −T0( )

cp,ioΔ fHi

o

Temperature Dependence of Equil ConstCombine (17.34) and (17.35) and write integral (17.36)

Integrate (17.37)

dlnKeqT( )T0

T∫ = 1R*T2 kiνi Δ fHi

o +cp,io T−T0( )[ ]i∑ dTT0

T∫

Keq T( ) =KeqT0( )exp − kiνiΔ fHi

o

R*T0T0T −1⎛

⎝ ⎜ ⎞ ⎠ ⎟ +

cp,io

R* 1−T0T +ln T0T⎛ ⎝ ⎜ ⎞

⎠ ⎟ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ 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( ) mol

kg 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 CoefficientsGeometric 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 DissociationUnivalent 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 DissociationSymmetric electrolyte

Charge balance requirement

ν+=ν−

z+ν++z−ν−=0

Equilibrium Rate Expression1. (17.39)HNO3 aq( ) H++NO3−

H+{ } NO3-{ }

HNO3 aq( ){ } =m

H+γH+mNO3

- γNO3

-

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+{ }2 SO4

2−{ }Na2SO4 s( ){ } =

mNa+2 γNa+

2 mSO42−γSO4

2−

1.0

=mNa+2 m

SO42−γ

2Na+,SO42−

3=Keq T( )mol3

kg3

Equilibrium Rate Expression3. (17.43)HSO

4 H+

+ SO2-

4

H+{ }2 SO4

2-{ }H+{ } HSO4-{ }

=mH+

2 γH+2 mSO4

2-γSO42-

mH+γ

H+mHSO4-γ

HSO4-

=mH+mSO4

2-γ2H+,SO42-

3

mHSO4-

γH+,HSO4-2

=Keq T( )molkg

Equilibrium Rate Expression4. (17.44)

NH4+{ } NO3-{ }

NH3 g( ){ } HNO3 g( ){ }=mNH4

+γNH4+mNO3-

γNO3-

pNH3 g( ),spHNO3 g( ),s

=m

NH4+mNO3

- γNH4

+,NO3-

2

pNH3 g( ),spHNO3 g( ),s

=Keq T( ) mol2

kg2 atm2

NH3

(g) + HNO3

(g) NH4

+ NO3

Equilibrium Rate Expression5. (17.45)NH

3(aq) + H

2O(aq) NH

4 + OH

-

NH4+{ } OH−{ }

NH3 aq( ){ } H2O aq( ){ } =m

NH4+γ

NH4+mOH−γ

OH−

mNH3 aq( )γNH3 aq( ) fr

=m

NH4+mOH−γ

NH4+,OH−

2

mNH3 aq( )γNH3 aq( )fr

=Keq T( )molkg

Mean Binary Activity CoefficientsPitzer's method of determining binary activity coefs. (17.46)

(17.47)

lnγ12b0 =Z1Z2f γ +m12

2ν1ν2ν1+ν2

B12γ +m12

2 2 ν1ν2( )3 2

ν1+ν2C12

γ

fγ =−0.392 I12

1+1.2I12 + 21.2ln 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 =2β12

1( ) +2β122( )

4I 1−e−2I121+2I12 −2I( )⎡

⎣ ⎢ ⎤ ⎦ ⎥

I =12 m2i−1Z2i−1

2i=1

NC∑ + m2iZ2i2

i=1

NA∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Mean Binary Activity CoefficientsAlternatively, fit a polynomial expression to mean binary activity coefficient data (valid to

high molality) (17.51)

lnγ12b0 =B0 +B1m12

12 +B2m12 +B3m123 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 (b

inar

y ac

tivity

coe

ffic

ient

)

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−γH+,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

+ TLν1+ν2( )R*T0

φL +m∂φL∂m

⎛ ⎝ ⎜ ⎞

⎠ ⎟

+ TCν1+ν2( )R* φcp +m

∂φcp∂m −φcp

o⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

TL =T0T −1

TC =1+ln T0T

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −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+U3m32 +...

φcp =φcpo +V1m12 +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+F3m32 +...

Fj =Bj +GjTL +HjTC

Gj =0.5 j +2( )U jν1+ν2( )R*T0

Hj =0.5 j +2( )Vjν1+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

activ

ity c

oeff

icie

nt

Mean Mixed Activity CoefficientsBromley's method (17.58-61)

Binary activity coefficient of an electrolyte in a mixture of many electrolytes.

log10γ12m T( ) =−AγZ1Z2Im12

1+Im12 + Z1Z2Z1+Z2

W1Z1

+W2Z2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

W1=Y21 log10γ12b T( )+AγZ1Z2Im12

1+Im12⎛ ⎝ ⎜

⎞ ⎠ ⎟ +Y41 log10γ14b T( )+Aγ

Z1Z4Im12

1+Im12⎛ ⎝ ⎜

⎞ ⎠ ⎟ +...

W2 =X12 log10γ12b T( )+AγZ1Z2Im12

1+Im12⎛ ⎝ ⎜

⎞ ⎠ ⎟ +X32 log10γ32b T( )+Aγ

Z3Z2Im12

1+Im12⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟ +...

Y21= Z1+Z22

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 2 m2,m

ImX12 = Z1+Z2

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ 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=12 m1,bZ1

2 +m2,bZ22( ) =1

2 ν+m12,bZ12+ν−m12,bZ2

2( )

m1,b =ν+m12,b m2,b =ν−m12,b

m12,b = 2Imν+Z1

2+ν−Z22