presentation slides for chapter 17, part 1 of fundamentals of atmospheric modeling 2 nd edition

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)

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