Chapter 18, Part 1. Fundamentals of Atmospheric Modeling

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1 Overhead Sldes for Chapter 18, Part 1 of Fundamentals of Atmospherc Modelng by Mark Z. Jacobson Department of Cvl & Envronmental Engneerng Stanford Unversty Stanford, CA January 30, 2002

2 Types of Equlbrum Equatons Reversble chemcal reacton dn D D + dn E E +... dn A A + dn B B +... (18.1) Dvde each dn by smallest value of dn ν D D + ν E E +... ν A A + ν B B +... (18.3) Mass conservaton k ( dn )m = 0 (18.2) Solvent Substance n whch speces dssolve n (e.g., water) Solute The dssolvng speces Soluton Combnaton of solute and solvent Solds Suspended materal not n soluton

3 Gas-Partcle Equlbrum Gas-partcle reversble reacton AB(g) AB(aq) (18.4) Gas n equlbrum wth soluton at gas-soluton nterface Examples Hydrochlorc acd HCl(g) HCl(aq) (18.5) Ntrc acd HNO 3 (g) HNO 3 (aq) Carbon doxde CO 2 (g) CO 2 (aq) Ammona NH 3 (g) NH 3 (aq) Sulfurc acd H 2 SO 4 (g) H 2 SO 4 (aq) (18.5)

4 Electrolytes, Ions, and Acds Electrolytes Substances that undergo partal or complete dssocaton nto ons n soluton Ions Charged atoms or molecules Dssocaton Molecule breaks nto smpler components, namely ons. Degree of dssocaton depends on acdty. Acdty Measure of concentraton of hydrogen ons (protons, H + ons) n soluton

5 Electrolytes, Ions, and Acds Acdty measured n terms of ph ph = log 10 [ H + ] (18.6) [H + ] = molarty of H + (moles-h + L -1 -soluton) Protons n soluton donated by acds H 2 CO 3 HCl HNO 3 H 2 SO 4 = carbonc acd = hydrochlorc acd = ntrc acd = sulfurc acd Strong acds (dssocate readly at low ph) HCl = hydrochlorc acd HNO 3 H 2 SO 4 = ntrc acd = sulfurc acd Weak acds (dssocate readly at hgher ph) H 2 CO 3 = carbonc acd

6 Electrolytes, Ions, and Acds Hydrochlorc acd dssocaton (ph above -6) HCl(aq) H + + Cl - (18.7) Sulfurc acd dssocaton (ph above -3) H 2 SO 4 (aq) H + + HSO 4 (18.8) Bsulfate dssocaton (ph above 2) HSO 4 H + + SO 2-4 (18.8) Ntrc acd dssocaton (ph above -1) HNO 3 (aq) H + + NO 3 (18.9) Carbon doxde dssocaton (ph above 6) CO 2 (aq) + H 2 O(aq) H 2 CO 3 (aq) H + + HCO 3 (18.10) Bcarbonate dssocaton (ph above 10) HCO 3 H + + CO 2-3 (18.10)

7 Bases Base Donates OH - (hydroxde on) Hydroxde ons combne wth hydrogen ons to form lqud water, ncreasng ph of soluton H 2 O(aq) H + + OH - (18.11) Ammona complexes wth water and dssocates NH 3 (aq) + H 2 O(aq) NH 4 + OH - (18.12)

8 Sold Electrolytes Suspended electrolytes not n soluton Precptaton / crystallzaton Formaton of sold electrolytes from ons Dssocaton Separaton of sold electrolytes nto ons Ammonum-contanng sold reactons NH 4 Cl(s) NH 4 + Cl - (18.15) NH 4 NO 3 (s) NH 4 + NO 3 (18.15) (NH 4 ) 2 SO 4 (s) 2NH 4 + SO 2-4 (18.15) Sodum-contanng sold reactons NaCl(s) Na + + Cl - (18.16) NaNO 3 (s) Na + + NO 3 (18.16) Na 2 SO 4 (s) 2Na + + SO 2-4 (18.16) Sold formaton from the gas phase on surfaces NH 4 Cl(s) NH 3 (g) + HCl(g) (18.17) NH 4 NO 3 (s) NH 3 (g) + HNO 3 (g) (18.17)

9 Equlbrum Relaton and Constant Equlbrum coeffcent relaton { } k ν a = { A}ν A B D { } ν B... { } ν D { E} ν E... = K eq ( T ) (18.18) {}... = Actvty Effectve concentraton or ntensty of substance { A( g) } = p A,s (gas) (18.19) { A + } = m A +γ A + (on) (18.20) { A( aq) } = m A γ A (dssolved molecule) (18.20) { H 2 O( aq) } = a w = p v = f p r (lqud water) (18.21) v,s { A( s) } =1 (sold) (18.22)

10 Equlbrum Coeffcent Relaton Gbbs free energy G * = H * TS * = U * + p a V TS * (18.23) Enthalpy H * = U * + p a V Change n Gbbs free energy Measure of maxmum amount of useful work obtaned from a change n enthalpy or entropy of the system dg * = d( H * TS * ) = du * + p a dv + Vdp a TdS * S * dt Change n entropy ds * = dq * T Change n nternal energy (18.24) du * = dq * p a dv = TdS * p a dv (18.25) Change n nternal energy n presence of reversble reactons du * = TdS * p a dv + k ( dn )µ (18.26)

11 Equlbrum Coeffcent Relaton Substtute (18.26) nto (18.24) dg * = Vdp a S * dt + k ( dn )µ (18.27) Hold temperature and pressure constant dg * = k ( dn )µ (18.28) Chemcal potental (µ ) Measure of ntensty of a substance or the measure of the change n free energy per change n moles of a substance = partal molar free energy µ = G * o = µ n ( T)+ R * T ln a T,p a { } (18.29) Equlbrum occurs when dg* = 0 n (18.28) k ν µ = 0 (18.30)

12 Equlbrum Coeffcent Relaton Substtute (18.29) nto (18.30) o k ν µ ( T0 ) + R * o T 0 k ν ln{ a } = k ν f G + R * T 0 ln { a } k ν = 0 where (18.31) k ν ln{ a } = ln a { } k ν Standard molal Gbbs free energy of formaton o f G = o µ ( T0 ) Rearrange (18.31) { a } k ν = exp 1 R * T 0 o k ν f G (18.32) The rght sde of (18.32) s the equlbrum coeffcent K eq T 0 ( ) = exp 1 R * T 0 o k ν f G (18.33)

13 Temperature Dependence of Equlbrum Constant Van't Hoff equaton (smlar to Arrhenus equaton) d ln K eq ( T) dt = 1 R * T 2 k ν f H (18.34) Molal enthalpy of formaton (J mole -1 ) of a substance o f H f H + o cp, ( ) (18.35) T T 0 o c p, o f H = Standard molal heat capacty at constant pressure = standard molal nthalpy of formaton Combne (18.34) and (18.35) and wrte ntegral T dln K eq ( T ) = T 0 Integrate K eq ( T) = K eq T 0 T T 0 1 R * T 2 o k ν f H + o [ cp, ( T T 0 )] dt (18.36) ( f Ho T )exp k ν R * 0 T 0 T 1 + c o p, R * 1 T 0 T + ln T 0 T (18.37)

14 Forms of Equlbrum Equatons Henry's law In a dlute soluton, the pressure exerted by a gas at the gaslqud nterface s proportonal to the molalty of the dssolved gas n soluton Henry's law relatonshp HNO 3 ( g) HNO 3 ( aq) Equlbrum coeffcent relatonshp { HNO 3 ( aq) } HNO 3 g { ( )} = m HNO3 aq ( ) γ HNO 3 ( aq) p HNO3 ( g),s = K eq ( T) moles kg atm (18.38)

15 Actvty Coeffcents Actvty coeffcents (γ ) Account for devaton from deal behavor of a soluton. Infntely dlute soluton, no devatons, γ = 1 Relatvely dlute solutons, devatons from Coulombc (electrc) forces of attracton and repulson γ < 1 Concentrated solutons, devatons caused by onc nteractons, γ < 1 or γ > 1 Geometrc mean bnary actvty coeffcent ν γ ± = γ + ν ( + γ - ) 1 ( ν + +ν ) (18.40) Rewrte γ ± ν + +ν = γ + ν + γ - ν (18.41)

16 Electrolyte Dssocaton Unvalent electrolyte HNO 3 ( aq) H + +NO 3 ---> ν + = 1 and ν = 1 ---> z + = +1 and z = -1 Multvalent electrolyte Na 2 SO 4 ( s) 2Na SO 4 ---> ν + = 2 and ν = 1 ---> z + = +1 and z = -2 Symmetrc electrolyte ν + = ν Charge balance requrement z + ν + + z ν = 0

17 Equlbrum Rate Expresson Table HNO 3 aq ( ) H + +NO 3 { H + - }{ NO 3 } { HNO 3 ( aq) } = m H + γ H + m NO 3 - γ NO 3 - m HNO 3 ( aq) γ HNO 3 ( aq) = m H + m - NO γ 2 3 H + -,NO 3 m HNO 3 ( aq) γ HNO 3 ( aq) = K eq ( T) moles kg (18.39) 2. Na 2SO 4 (s) 2Na + + SO 4 2- { Na + } 2 { SO2 4 } { Na 2 SO 4 ( s) } = m2 Na + γ 2 Na + m SO4 2 γ SO = m Na + m SO 2 γ 3 4 2Na + 2,SO 4 = K eq ( T) moles kg (18.42) 3. HSO 4 H + + SO4 2- { H + } 2 SO2-4 { } { H + }{ HSO- 4 } = m2 H + γ 2 H + m SO 2- γ 4 SO2-4 m H + γ H + m HSO 4 - γ HSO 4 - = m H + m SO 2- γ 3 4 2H +,SO2-4 m HSO - γ 2 4 H +,HSO - 4 = K eq ( T) moles kg (moles kg -1 ) (18.43)

18 Equlbrum Rate Expresson 4. NH 3 (g) + HNO 3 (g) NH 4 + NO 3 { NH+ 4 } NO- 3 NH 3 g { } { ( )} HNO 3 ( g) { } = m NH4 + γ NH 4 + m NO 3 - γ NO 3 - p NH3 ( g),s p HNO 3 ( g),s = 2 m + m NH4 NO - γ 3 NH + 4,NO - 3 p NH 3 ( g),s p HNO 3 ( g),s = K eq ( T) moles 2 kg 2 atm (18.44) 5. NH 3(aq) + H 2 O(aq) NH 4 + OH - { NH+ 4 } OH NH 3 aq { } { ( )} H 2 O( aq) { } = m NH4 + γ NH 4 + m OH γ OH m NH 3 ( aq) γ NH 3 ( aq) f r = m NH4 + m OH γ 2 NH+ 4,OH m NH 3 ( aq) γ NH 3 ( aq) f r = K eq ( T) moles kg (18.45)

19 Mean Bnary Actvty Coeffcents Ptzer's method 0 ln γ 12b ( ) 3 2 = Z 1 Z 2 f γ 2ν + m 1 ν 2 γ 12 B ν 1 +ν ν 1 ν 2 γ m12 C 2 ν 1 + ν 12 2 ( ) (18.46) f γ I 1 2 = I ln I1 (18.47) 1.2 ( 2) γ ( 1) 2β B 12 = 2β I ( ) 1 e 2I I 1 2 2I (18.48) Ionc strength of soluton (moles kg -1 ) Measure of the nteronc effects resultng from attracton and repulson among ons I = 1 N C N A 2 2 m Z m 2 Z 2 =1 =1 (18.49)

20 Mean Bnary Actvty Coeffcents Table Ptzer parameters for three electrolytes. ( 1) β 12 ( 2) β 12 γ C 12 HCl HNO NH 4 NO Polynomal ft from data (vald to hgh molalty) 0 ln γ 12b = B 0 + B 1 m 12 + B2 m 12 + B 3 m (18.51)

21 Mean Bnary Actvty Coeffcents Fg Comparson of measured (Hammer and Wu) and calculated (Ptzer) actvty coeffcent data ln(bnary actvty coeffcent) Ptzer Hammer and Wu NH 4 NO 3 HCl HNO m 1/2 Equlbrum coeffcent expresson for hydrochlorc acd m H + m Cl 2 γ H +,Cl p HCl(g),s = (18.50) Equlbrum coeffcent expresson for ntrc acd m H + m 2 NO3 γ H +,NO 3 p HNO 3 ( g),s =

22 Temperature Dependence of Mean Bnary Actvty Coeffcents Temperature dependent equaton 0 ln γ 12b ( T )= ln γ 12b (18.52) T + L ( ν 1 + ν 2 )R * φ L + m φ L T + C T 0 m ( ν 1 + ν 2 )R * φ c p + m φ cp m φ o c p Temperature-dependent parameters T L = T 0 T 1 (18.53) T C = 1+ ln T 0 T 0 T T Polynomal for relatve apparent molal enthalpy φ L = U 1 m U 2 m+ U 3 m (18.54) Polynomal for apparent molal heat capacty o φ c p = φ c p +V 1 m V 2 m +V 3 m γ 12b ( T) = bnary actvty coef. at temperature T φ L = relatve apparent molal enthalpy (J mole -1 ) φ c p = apparent molal heat capacty (J mole -1 K -1 ) o φ c = apparent molal heat capacty at nfnte dluton p

23 Temperature Dependence of Mean Bnary Actvty Coeffcents Combne (18.51) - (18.54) --> ln γ 12b ( T )= F 0 + F 1 m 12 + F 2 m + F 3 m (18.55) Coeffcents F 0 = B 0, F j = B j + G j T L + H j T C j = 1... (18.56) G j = 0.5( j +2 )U j ( ν 1 + ν 2 )R * (18.57) T 0 0.5( j +2 )V j H j = ( ν 1 + ν 2 )R *

24 Sulfate and Bsulfate Fg Bnary actvty coeffcents of sulfate and bsulfate, each alone n soluton. Results vald for 0-40 m. Bnary actvty coeffcent H + - / HSO H + 2- / SO m 1/2 201 K 273 K 298 K 328 K

25 Mean Mxed Actvty Coeffcents Bromley's method Bnary actvty coeffcent of an electrolyte n a mxture Z log 10 γ 12m ( T) = A 1 Z 2 I1 2 m γ I + Z 1 Z 2 m W 1 + W 2 Z 1 + Z 2 Z 1 Z (18.58) 2 W 1 = Y 21 log 10 γ 12b T ( ) + A γ Z 1 Z 2 I m I m 1 2 Z +Y 41 log 10 γ 14b ( T)+ A 1 Z 4 I1 2 m γ I m (18.59) W 2 = X 12 log 10 γ 12b T Y 21 = Z 1 + Z X 32 log 10 γ 32b T 1+ I m (18.60) X I 12 = Z 1 + Z 2 2 m 1,m (18.61) m 2 I m ( ) + A γ Z 1 Z 2 I m m 2,m Molaltes of bnary electrolyte found from I m = 1 2 m 2 1,bZ ( m2,b Z 2 ) = 1 2 ν 2 +m 12,b Z ( ν m 12,b Z 2 ) (18.62) Molaltes of caton, anon alone n soluton m 1,b = ν + m 12,b m 2,b = ν m 12,b Molalty of bnary electrolyte gvng onc strength of mxture m 12,b = 2I m ν + Z ν Z 2 2 (18.63) ( ) + A γ Z 3 Z 2 I m I m 1 2 +

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