Fugacity Models. I. Model Framework

Fugacity Models

A primary objective in environmental chemistry is to forcast the concentrations of pollutants in the environments with respect to space and time variables. Our knowledge of the behavior of

pollutants can be used to model the space and time domains of pollants once emissions are known or estimated. Fugacity models are distribution based models incoporating all environmenal compartments, and are based on steady-state fluxes of pollutans across compartment interfaces.

I. Model Framework

A. Fugacity defined as chemical activity of a gas, and expresses

escaping tendancy from a compartment. Fugacity is linearly related to concentration at dilute concentrations via:

Ci,j = Zi,jfi,j (1)

Ci,j = concentration of ith pollutant in jth compartment (mol/m3)

Zi,j = fugacity capacity of ith pollutant in jth compartment (mol/m3-Pa)

fi,j= fugacity of ith pollutant in jth compartment (Pa)

Fugacity may be expressed for any environmental compartment or phase, and represents equivalent gas phase fugacity.

B. Equilibria: partitioning process between two phases @ equilib fA,1 = fA,2

CA,1/ZA,1 = CA,2/ZA,2

K12 = CA,2/CA,1 = ZA,2/ZA,1

A

1

A

Z constants are proportionality constants which are often referred to as half-partition coefficients.

C. Fugacity Capacities

Fundamental fugacity equation used for derivation of Z constants is in the form of Raoult’s Law

fi,j = Ci,jvjγi,jfR (2)

fi,j = fugacity if ith pollutant in jth compartment (Pa)

Ci,j = concentration of ith pollutant in jth compartment (mol/m3)

vj = molar volume of jth compartment (mol/m3); derived from phase

density

γi,j = activity coefficient of ith pollutant in jth compartment

fR = reference fugacity (Pa)

By convention, the reference fugacity is the liquid state vapor pressure of the organic pollutant (i.e., PoL)

1. Z for air PV = nRT n V P RT = ∴ C f 1 RT =   (f = P at dilute conc.)

(

)

2. Z for water

@ equilib fAir = fWater for pollutant

fAir = fWater = CwvwγwfR Z 1 v f P S K w w R L o w,liq 1 H 1 = =      = − − γ

ZWater = 1/KH mol-m3/Pa

3. Z for sediment, suspended particulates, colloids, and biota @ equilib fAir = fWater = fSediment/particulates/colloids/biota for pollutant

CwvwγwfR = CsvsγsfR (C = Zf) Zwfwvwγw = Csvsγs C Z f v v fZ K f K K s w w w w s s w d ' s d H = γ = = γ ρ Z K K mol m Pa s s d H 3 = − ρ alluvial sediments Z K K mol m Pa p p d H 3 = − ρ

particulates; fluvial sediments

Z K K mol m Pa c c d H 3 = − ρ colloids

Z K K mol m Pa b b d H 3 = − ρ biota

4. Typical phase densities

Phase ρ, Kg/m3 Air 1.19 Water 1,000 Sediments 1,500 Colloids 1,000 Biota 1,000

II. Strategy in Setting Up Fugacity Model

1. Define unit world (UW); determine all relevant UW dimensions and properties.

2. Calculate phase volumes for all relavant compartments and sub-compartments (m3)

3. Choose fugacity model level (I-IV)

4. Obtain or estimate physical properties, e.g., KH, Koc, Kdoc, Kb, etc.

5. Calculate Z constants for all relevant compartments and subc-ompartments

6. Calculate mass transport coefficients for all relevant interface boundaries (for level III and IV models only)

8. Obtain and/or calculate all relevant degradation rate constants for pollutant (levels II-IV) (1/hr).

9. Set up worksheet.

10. Calculate prevailing fugacity (levels I and II); for level III, phase fugacity must solved through all equns simultaneously).

11. Calculate environmental concentrations and mole percentages. III. Fugacity Models

A. Level I Fugacity Model

1. One time input of pollutant of known quantity, MT

2. No reaction or advective gain or loss. 3. System at equilibrium

4. Fugacity is prevailing (i.e., equal in all phases) B. Level I calculations

Mi,j = Ci,jV j = fZi,jVj (3)

Mi,j = mol of ith pollutant in jth compartment

Ci,j = concentration of ith pollutant in jth compartment (mol/m3)

V j = volume of jth compartment (m3)

f = prevailing fugacity (Pa)

(

)

Once f calculated, phase concentrations estimated as: C = Zf C. Level II fugacity model

1. Constant emission of known rate (mol/hr) 2. Reaction and advection may occur

3. System is at steady-state wrt to pollutant 4. Fugacity is prevailing

D. Level II calculations

1. Model reactions as first-order processes: hydrolysis, photolysis, redox, and biolysis

Rate = k[P]

k = first order rate constant (1/hr)

[P] = concentration of pollutant (mol/m3)

Associations exist between reaction pathway and phase; example reactions shown below:

Reaction type phase

hydrolysis water>biota>air

photolysis air>water>particulates

biolysis sediments>particulates>water redox sediments>particulates>water

With first order assumption, reaction rate equations for each phase of given reaction type (k1, k2, etc.) can be summed within a jth

compartment as

Rate = k1[P]j + k2[P]j + k3[P]j + ...

Rate constants within a single phase or compartment can be factored out Rate = (k1 + k2 + k3 + ...) [P]j

Rate = Ki[P]j (5)

Ki= overall first order reaction rate constant for ith pollutant (1/hr)

[P]j = concentration of pollutant in jth compartment (mol/m 3

)

At steady state, the loss of pollutant due to reaction in each phase is balanced by chemical emission rate (I); thus, I can be calculated as: I = ΣVjCi,jKi (mol/hr)

(6)

I = input rate of pollutant in unit world (mol/hr)

2. Advection also introduces and removes pollutant from unit world; air and water are major compartments for advection

a. G = horizontal movement of phase (also defined as bulk flow) (m3/hr)

Amount of pollutant introduced to unit world by advective inflow is:

b. Advective loss modeled using first-order rate constant K G V a j j = (8)

Ka= first order advection rate constant (1/hr)

Gj= advection rate of jth compartment (m3/hr)

Vj = volume of jth compartment (m3)

Loss of pollutant through reaction and advection is overall loss rate = ΣVjCi,j(Ki + Ka)

d. prevailing fugacity calculated by setting up mass balance eqn combining all input and loss terms:

I + ΣGjCi = ΣVjCi,j(Ki + Ka) = fΣVjZi,j(Ki + Ka) (9) f I G C V Z (K K ) j i, j j i, j a i = + + Σ Σ (10)

In the above eqn, the numerator represents all input terms (emission and advective inputs) and the denominator represents all out put terms

(reaction and advective loss).

Level I and II models assume an instantaneous distribution of pollutants upon emission or advection into the system. There exist no barriers to mass transfer from one phase to another.

E. Level 3 calculations

1. Fugacity is equal within a compartment (air, water , and sediments) for all defined subcompartments, but not equal between compartments. 2. System is at steady-state

3. D values are used to determine steady-state fluxes.

4. Pollutant transfer between compartments occurs by bulk and diffusive processes

bulk process - one-way transfer which is associated with transport of component from one compartment to another: examples include rain out of pollutant from air to water; particle deposition of material from water column to alluvial sediments.

diffusive process - two-way transfer which is associated with

molecular motion of pollutant across interface from one compartment to another: examples include dry gas deposition/volatilization from water; sorption/desorption from sediments into water.

Bulk processes: N = GC = GZf = Df where D = GZ Diffusive processes: N = D12f1 - D21f2

N = pollutant flux in mol/hr

D12, D21 = transport coefficients for all transfer from compartment 1 to 2

and 2 to one, respectively (mol/hr-Pa)

f1, f2 = fugacities of pollutant in compartment 1 and 2, repspectively

Zk = Σk φj,kZj,k (11)

Zk = fugacity capacity of pollutant in kth compartment

φj,k = volume fraction of jth sub-compartment in kth compartment (=

Vj/Vk)

Zj,k = fugacity capacity of pollutant in jth subcompartment in kth

compartment

6. Fugacities - steady state mass balance equations can be set up for each compartmen via the equation

NEi + DAifBi - fi[ΣDij + DAi + DRi] + Σj Djifj] = 0 (12)

Fugacities in feach compartment can be solved from eqn 12 algebraically.

References

Mackay, D. and S Paterson (1981). Calculating fugacity. Environ. Sci. Technol. 15, 1006-1014. Mackay, D. and S. Paterson (1982). Fugacity revisited. Environ. Sci. Technol. 16, 645A-660A. Mackay, D. and S. Paterson (1991). Evaluating the multimedia fate of organic chemicals: a level III fugacity model. Environ. Sci. Technol. 25, 427-436.

Mackay, D. (1991). Multimedia Environmental Models: The Fugacity Approach. Lewis Publishers, Inc., Chelsea, MI.