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Environmental ModelsFugacity approach
Equilibrium
• Partition coefficients
– Definition, experimental assessment
• The fugacity approach to partition coefficients
– Fugacity definitions for a gas, a liquid, a solid
– Fugacity coefficients in the gas in the liquid phase
– Relationships between fugacity coefficients and solubility
– Calculation of partition coefficients from fugacity coefficients
– Henry ’s Law constant
– Temperature dependence of the Henry ’s Law constant
Non equilibrium: Fluxes across the air-water interface
CG
CWCP
CAAir-WaterExchange
Water-Particle Partitioning
Gas-Particle Partitioning
Dry Deposition
Wet Deposition
Vertical Fluxes
Advection
Bioaccumulation
Continental Inputs
Atmospheric Transport
Degradation
Environmental fate of organic pollutants
Chemical Fate in the Environment
Air
Soils Rivers Oceans
Biota: plankton, fish, mammals
iW
iA AW C
C K =
KAW
BCF, KOW
iS
iA AS C
C K =iW
iBiotaBW C
C BCF K ==
Air-water partitioning Air-Soil partitioningBiota-water partitioning(or bioconcentration factor)
Partition of chemicals in ideal laboratory systems
benzene HCBKAW = 0.22 0.053KOW = 135 316000
air
water octanol
eq iA, eq iO,OA C
C K =
eq iw, eq iO,ow C
C K =
eq iw, eq iA,
AW CC K =
All K are T-dependent
• Partition coefficient: ratio of concentrations at equilibrium.
If CA,eq = 1 mol m-3, then
CW,eq =
CO,,eq =4.5 10-3 mol l-1 19 10-3 mol l-1
614 10-3 mol l-1 5960000 10-3 mol l-1
NB: Octanol is a surrogatefor natural organic matter
Equilibrium partition constants and thermodynamics
iB
iA AB C
C K =
Phase A Phase B
Equilibrium partitioning between A and B
RTGiABe /AB constant K Δ−•=
)ln( Kln AB cteRT
GiAB +Δ−
=
Where ΔABGi is the free energy of transfer from A to B. It has an entropicand enthalpic controbution
Partitioning. Types of interactions
Partition of chemicals in the environment
• Experimental determination of K (KOW) at a given T
00
CiW
time
CiW,eq = CO1,eq x KOW
CiW, eq = CO2,eq x KOW
CiW,eq = CO3,eq x KOW
CiW,eq = CO4,eq x KOW
00 CiW,eq
CiO,eqKOW
Partition of chemicals in the environment
• The experimental determination of K should cover all environmental properties (T and P ranges)
Water/air: volatility
Water/octanol:hydrophobicity
Partition of chemicals in the environment
• Chemical potentials μi : not measurable, logarithmically-related to concentration
: vapor pressure of i
Phase transfer occurs from the higher to the lower μi , and stops when μi are equals in all phases.
• Fugacity f (Lewis 1901): calculable, linearly-related to concentration
C = Z f
Partition of chemicals in the environment
SiLP
⎥⎥⎦
⎤
⎢⎢⎣
⎡+= 0
i
i0ii f
f ln RTμ gas) (real μ
⎥⎦⎤
⎢⎣⎡+= S
iL
i0ii P
P ln RT μ μ
Pamol m-3
mol m-3 Pa-1
• Fugacity is a sort of partial pressure, that applies as well to compartments when no gas phase exists (organisms).
• Chemicals diffuse from compartments where their fugacity is high to the compartment where their fugacity is the lowest.
• At equilibrium, the fugacity of a chemical is equal in all phases
• The environmental partitionning of a substance is driven by its fugacitycapacity Z in all compartments: ZiA, ZiW, Zi fish.
Zi is constant for each chemical and compartment.
The fugacity approach to partition
(g)f (s)f (l)f iii ==
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0
100
200
300
400
500
-0.05
0
0.05
0.1
0.15
0.2
0.25
0
100
200
300
400
500
• Variation of ZiA, ZiW, fiA, fiW, CiA, CiW towards equilibrium
• Case 1: at CbenzA constant, benzene equilibrates to a water phase
CbenzA
CbenzW
ZbenzA
fbenzAin Pa
ZbenzW
fbenzW
in Pa
Time Time
Calculation of K using fugacity coefficients
ZZ Z f
Z f CC K
iWiA
iWeqiW,iAeqiA,
eqiW, eqiA,eqiAW, =×
×==
= —— = —— = 501 0.2
0.002 0.0004
mol m-3
mol m-3
At equilibrium,
=0.0004 mol m-3 Pa-1=0.002 mol m-3 Pa-1
-0.2
0
0.2
0.4
0.6
0.8
1
0
100
200
300
400
500
-0.05
0
0.05
0.1
0.15
0.2
0.25
0
100
200
300400
500
600
700
• Case 2: CbenzW constant, benzene equilibrates to the air phase
CbenzA CbenzW
ZbenzA
fbenzA
ZbenzW
fbenzW
Time Time
ZZ Z f
Z f CC K
iWiA
iWeqiW,iAeqiA,
eqiW, eqiA,
eqiAW, =××==
= —— = —— = 501 0.2
0.002 0.0004
mol m-3
mol m-3
=0.0004 mol m-3 Pa-1
At equilibrium,
=0.002 mol m-3 Pa-1
Calculation of K using fugacity coefficients
• Fugacity in the gas phase
fi (g) = φi Pi = φi xi Ptot
xi is the molar fraction in the gas phaseThe fugacity coefficient φi ≈ 1 except at low temperatures, and except if intermolecular interactions occur
• Calculation of ZiA
TotiA Tot
A iA
TotiAA iA
iAiAiA P n
n x Vn P x
1 x Vn f
C Zιι φφ ===
T Rn VP TotA Tot =
T R 1 ZiA
iφ= ZiA ≈ 10-4 mol m-3 Pa-1 for non ionic substances
Fi (g)= Pi for all non ionic substances
Fugacity and Z in the gas phase
VP n P V
n ZA Tot
TotTotA
TotiAιι φφ ==
1RT
Fugacity of gases and liquids
Fugacity in the liquid phase
• Fugacity in a liquid phase
fi (l) = ( xiW γiW fR ) = xiW γiW
xiW is the molar fraction in the liquid (water) phaseγiW is the activity coefficient. It can be viewed as the ratio of the activity (or fugacity) of i to the activity (or fugacity) that i would have in a pure solution of its own kind. L
n γi = ln γi0 (1 – xi
2) γi0 when xi ≈ 0
• Calculation of ZiW
SiLP
P Voln P n
n x Voln P x
1 x Voln f
C Z SiL W
WTotSiL iW
TotWW iW
SiL iLW
iWiWiWiW
WWiW ιι γγγ ====
P V1 Z S
iL W iW
iWγ= 1/VW! Symbol: VW corresponds to molar volume in m3 mol-1
VWater=18 10-6 m3 mol-1 m-3 mol-1
Pamol m-3 Pa-1
Fugacity and Z in the gas phase
Table 3.2 schwar. Pag 81
• Vapor pressure of a liquid and of a solidSL i,P
SL i,P
Sdsupercoole L i,P
iP
SS i,P
Fugacity in condensed phases (liquid and solid)
• Relationships between Z and solubility Si
– For a liquid at its solubility limit, thus at equilibrium
At equilibrium:
– For a solid that melts into a liquid phase, at its solubility limit, thus at equilibrium
At equilibrium:
Fugacity in the liquid phase
P (l)f SL i eqi, =
(w)f (l)f eqi, eqi, =
SL i
satiWiW eq i, P x (w)f γ=
1 xiW
satiW γ=
P Vx Z S
iL W
satiW iW =
Sdsupercoole L i
satiWiW eq i, P x (w)f γ=
P (s)f SS i eqi, =
(w)f (s)f eqi, eqi, =
F P P x
iWSdsupercoole L iiW
SS isat
iW γγ ==
W iWW
satiW
iW V1
Vx S
γ==
W iWW
satiW
iW VF
Vx S
γ==
mol m-3
– saturation of a gas that condensates into the liquid phase, at equilibrium:
At equilibrium:
1 xsatW sat
iWγ=
SL i
satiWiW eq i, P x (w)f γ=
P (g)f SL i eqi, =
(w)f (s)f eqi, eqi, =
W iWW
satiW
iW V1
Vx S
γ==
Fugacity in the liquid phase
• Exercise. Determine relevant thermodynamic properties and air-water partitioning properties of benzene, liquid at 298K from:
=12700 Pa Swater=22.8 mol m-3,R=8.314 J mol-1 K-1 VW= 18 10-6 mol m-3
Calculation of K using thermodynamic parameters
P SL i
4-10 4.04 T R1 T R
1 ==iφ
0.0018 1270022.8 P
S P VVS P V
x P V1
SL
WSL W W W
SiL W
satW
SiL W
=====Wγ
0.22 0.00180.000404 Z
Z iWiA ==
Z benzeneA =
= benzene WZ
K benzene AW =
benzene W =γ 2437 10 18 X 22.81
VS1 x
16-
W WsatiW
===
Air-water exchange: Henry Law´s constant
• Air-water equilibrium can be characterized by two constants, the KAW and the Henry Law ’s constant, KH (or H).– KAW , also named the dimensionless constant
– Definition of the Henry Law´s constant
For a gas or a liquid, at equilibrium and
) mol l (atm C P K 1-
eqiW,eq i, H =
Pa (m3 of water) mol-1
T R P V Z
Z CC K
SiL W
iWiA
eqiW, eqiA,eqiAW,
i
iW
φγ===
(m3 of water) (m3 of air)-1
P P SiLeqi, = sat
iW eq iW, C C =
Vapor pressure at TSolubility at T
C P K sat
iW
SiL H =
Air-water exchange
• The Henry Law ’s constant is related to KAW
T RP V
n C iA
iiA ==
eqiW,
eqiA, iAW
CC K =
'HH
eqiW,i
eqiW,
eqiA, iAW K T R
K T R CP
CC K ====
Dimensionless»
T R K T R
K T R P V K H H
SiL W
eqiAW, ===ii
iW
φφγ
C P K
eqiW,eq i, H =
0iLW
Weq iW,
SiL eq iW,
H P V V
xP x
K iWWi
γγ
==
V x C
Weq iW,
eq iW, = P x )(f P (g)f SiLeqiW,Wii i γiW=== and
The fugacity approach to partition: do not forget
• Alike all K, KAW and KH are related to T
• T dependence is a critical factor of air-water partition
• T is a preminent criteria driving contaminant repartion through volatilizationand condensation.
• To understand T dependence of air-water partition, thermodynamic allows topredict relationships, and they should be validated by measurementsafterwards.
T dependence of Partitioning
2AB 1)ln()/(1
dTKln d
TH
RdTcted
RTGd iABiAB Δ
•=+Δ•−
=
SiLW H P V K iWγ=
• Temperature dependence of Henry Law ’s constant
C P K sat
iW
SiL H=
)ln( Kln AB cteRT
GiAB +Δ−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
Δ=
121AB
2AB 11 )(K)(Kln
TTRH
TT iAB
• Temperature dependence of SiLP
T RH P dT
dP 2
vapSiL
SiL Δ=
T R dTH P
dP 2 vap
SiL
SiL Δ=
A T R H - P ln vapS
iL +Δ=
T dependence of air-water exchange
• Vapor pressures of classes of contaminants , at 25oCSiLP
T dependence of air-water exchange
• Temperature dependence of γiw
T RH dT
xln d 2
excsol
satiW Δ=
B T R H xln
excsolsat
iW +Δ−=
1 xiW
satiW γ=
T dependence of air-water exchange
• Water solubilities SiW of classes of contaminants at 25oC
W iWW
satiW
iWsatiW V
1 Vx S C
γ===
T dependence of air-water exchange
B A T RH H )
V K ( ln
excsolvap
M
H −+Δ+Δ−=
• T dependence of KH
B - T R H ln
excsol
iWΔ=γ
A T R H - P ln vapS
iL +Δ=
SiLW H P V K iWγ=
Vln B A T RH H K ln M
excsolvap
H −−+Δ+Δ−=
C
T dependence of air-water exchange
• KH values of classes of contaminants at 25oC
T dependence of air-water exchange
Multimedia Environmental ModelsThe fugacity approach
iB
iA AB C
C K =
Phase A Phase B
Estimation of the fraction of chemical i in A (Fi,A)
ABBA
AiA
BiB
B
A
VVKVCVC
VV
iofmasstotalAiniofmass
11
111
1CVC
C FiBAiA
iA Ai,
+=
+=
+==
chemical i
Multimedia Environmental ModelsThe fugacity approach
iB
iA AB C
C K =
Phase A Phase B
Estimation of the fraction of chemical i in A (Fi,A) when there are k phases(air, water, aerosols, biota, sediments…)
An
k
BnnA
KiKB
A
VVK
VCVV
iofmasstotalAiniofmass
∑=
+=
+⋅⋅⋅⋅++== 11
1CVC
C FiBAiA
iA Ai,
chemical i
Phase k
Multimedia Environmental ModelsThe fugacity approach (we change C = Zf)
iB
iA
ZZ
==iB
iA AB C
C K
Phase A Phase B
Estimation of the fraction of chemical i in A (Fi,A) when there are k phases(air, water, aerosols, biota, sediments…)
An
k
BnnA
KiKB
A
VVK
VZVV
iofmasstotalAiniofmass
∑=
+=
+⋅⋅⋅⋅++== 11
1ZVZ
Z FiBAiA
iA Ai,
chemical i
Phase k
At equilibrium the fugacity (fi) in all media is the same (fA = fB =…..=fK=f )