Environmental Models Fugacity approach · Fugacity in the liquid phase • Fugacity in a liquid...

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 )