Fugacity revisited The fugacity approach to environmental transport
Donald Mackay Sally Paterson Department of Chemical Engineering and Applied Chemistry Uniuersity of Toronto Toronto, Ontario, Canada
In previous articles we have described the fugacity approach to environmental modeling in general ( / ) and the simpler Level I and I1 calculations in more detail (2). Here we introduce the concepts of transfer between environmental compartments and unsteady state behavior as Level Ill and IV calculations to bring the model predictions a step closer to re-
ality. This feature article is a summary presentation of a more detailed report (3). These more advanced calculations yield more information about a chemical’s likely “behavior profile” in the environment. In addition, we suggest methods of displaying these profiles pictorially rather than numerically. To recapitulate, fugacity is a thermodynamic quantity related to chemical potential or activity that characterizes the escaping tendency from a phase. At equilibrium, fugacities (which have units of pressure) are equal. The fugacity calculations can be applied to an actual environment of defined volumes but it is more convenient to use a hypothetical or evalua-
FIGURE 1
Tank analogy d levels I-IV fugadly calculation conditions
+ Equilibrium steady-state no-flow s tem‘
+ Equilibrium steady-state flow SyStem h I II
Levs
*
Nonequiilbrium steady-state flow system Level 111
I t Before emissions
t
Approaching Level 111 steady state
t
Recovering after emissions cease
Nonequilibriumnonsteady-state flow system
Level IV
a4A
Envkm. Scl. Tedrol.. Vol. 16, No. 12,1982
t
tive environment or “unit world” consisting of volumes of homogeneous air, water, soil, bottom sediment, suspended sediment, and biota as discussed by Neely and Mackay ( 4 ) . Each phase is assigned a reasonable volume and properties, and the distribution of the compound is then calculated from a knowledge of its properties to yield a behavior profile. The behavior and concentrations in the real environment will differ from those calculated, but the dominant behavior characteristics should be mimicked. In Level I, the equilibrium partitioning of a fixed amount of a nonreacting compound is calculated using fugacity capacities that are calculated from physical chemical data and partition coefficients. The equilibrium ratios of concentrations and the relative amounts in each phase can be calculated. In Level 11, the steady-state equilibrium concentrations of the compound are calculated for fixed emissions that are balanced by reactions of various types in each phase. This provides information on the compound’s overall persistence and what proportion reacts in each phase. Figure 1 illustrates these calculation approaches by analogy with water flow into tanks in which the level in the tank is equivalent to fugacity. Level I corresponds to a fixed amount of water (chemical). Level I1 corresponds to fixed inflow (emission) balanced by the sum of various leakages (reactions). In both cases connections between the tanks are sufficient to ensure equal height of water (interphase transport is rapid). Figure 2 illustrates the unit world and Table 1 shows the properties of a hypothetical organic chemical similar to naphthalene. Figure 3 gives the Level I and I1 data in pictorial form as
0013-836X/82/0916-0654A$01.25/0 @ 1982 American Chemical Society
pie charts of mass distribution and reaction and as concentration on a log scale. Since the absolute concentrations have no significance (unless the volumes and amounts of chemical are known), the scale numbers may be eliminated and the information conveyed by only relative position on the scale. The aim of the diagram is to indicate where the substance goes, where it reacts, and what relative concentrations it adopts at a common fugacity. The most severe limitation of these calculations is that they assume that the chemical achieves equilibrium between all phases. In practice, there are transfer resistances that limit the transfer from phase to phase and thus tend to “contain” the chemical. These resistances are analogous to valves between the tanks in Figure 1. Including valves or interphase transport resistances brings us to Level 111 nonequilibrium steady state, in which each phase may have a different fugacity. Level IV introduces the concept of inflows (emissions) that change with time, causing levels in the tanks to change, i.e., conditions are nonequilihrium, unsteady state. If emissions are constant, the levels will approach the Level 111 steady-state condition at long times. The task here is to provide a simple but physically realistic method of calculating these interphase resistances. In principle, it is application to the environment of the two-resistance model concept originally devised by Whitman in 1923 for chemical Drocessing ( 5 ) . This model successfully describes diffusive transport processes that are driven by fugacity or concentration differences, i.e., in a given phase, a chemical will diffuse from regions of high to low concentration or fugacity. The diffusion process is merely a manifestation of mixing, which tends to eliminate concentration gradients. Within one phase, concentration is an adequate descriptor of diffusion “driving force” (as in Fick‘s law) but between phases, concentration fails because a chemical may diffuse from low to high concentration across a phase boundary. The correct driving force is then fugacity since diffusion always proceeds from high to low fugacity. It is usually assumed that the fugacities immediately adjacent to the phase boundary are equal in the two phases. A second method of interphase transport occurs when a quantity of material moves from one phase to another, “piggy backing” chemical with it. Examples are deposition of SO2
from air in rainfall or dustfall and sediment deposition and resuspension. These material transport processes are not driven by fugacity differences and must be treated separately.
FIGURE 2
UnR world vdum
d-(mm3
Diffusive transport We consider two compartments in which the solute has respective concentrations, fugacities, and fugacity capacities, C1 and C2.f1 andf2. and Z I and 22 separated by an area A m2 through which the solute is diffusing at a constant rate N defined as
.
Air V = 6.0 x 100
Soil
N = D I Z ( ~-f2) I mol/h.
v = 4.5 x lo.
D is thus a transport coefficient with dimensions of mol/Pa h, which is in a very convenient form for fugacity calculations. The two-resistance model is merely the application of Fick’s first law of diffusion to each phase in series; thus N
1
Water V = 7.0 x 1W Biota V = 7.0
suspended solids V = 35.0
K I A A C I= K2AAC2 = KiAACi
where K I and K2 are phase mass transfer coefficients (m/h), and ACI and AC2 are the concentration differences (mol/m3) between the bulk of each phase and the interface. The K term is often viewed as a diffusivity divided by a “stagnant film” thickness; however, this “film” model is not physically realistic since diffusion is only partly by molecular processes, and the film thickness is unmeasurable. K is better viewed as a piston veF + y with which the solute moves
through each phase to and from the interface. If each C i s replaced by Zf and equilibrium is assumed to occur at the interface where the common interfacial fugacity i s h then N = KIAZIVI-h)
TABLE 1
Physical chemical propertl&=m u L VIIIUUJ hypothetical compound
uwthe
PhyElcal chemkal pmperuerr Molecular weiaht Aquears solublllly VWlressUe Temperatwe L-
Kow
canp.rtAlr-1,
water-2.
soil-3,
sediment-4, suspended sediment-5.
Partiuon imflkhm(Ukg) Ks2 = 26.2, K.2 = 52.4, K52 = 52.4, Ken
158.1
b
= 0.0
Du*l(lw (kghna) p3 = 1500, p, = 1500, p5 = 1500, pe = 1000
zv.lvn ( m v d Pa)
2,= l/RT= ll(8.314 X 298) = 0.4 2, = 1IH= ll(1.010.333) = 23 3 Ka2p32’/1000 = 13.1
4 = Kmp&IlOOO
26.2
Environ. Sci. Temnol., Vol. 16, No. 12. 1982
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FIGURE 3
Level II behavbr pmfik with a common lugacity 0.350
-.-.-+
--*
m
BiotaSediment, suspended solidsSoil
Emissions Reaction (moVh)
10-3
10.’
-f 10-5
Water-
Emissions -.-.--.+
1.o n
10-6 0x105
10-7
10-8
Air-
Mass distribution t(r9
Reaction distribution
Eliminatingfi yields
f = fugacity (Pa) C = concentration(molimJ) m = amount (mol) k = reactionrate constant (h-‘)
hence 0 1 2 = l / ( r l r2). Since the resistances are in series the total resistance is the sum of the two phase resistances, and D is an overall conductivity or the reciprocal of the total resistance.
less obvious. A large Z implies that the solute accumulates at a large concentration at a given fugacity; thus, when the piston velocity is applied to this large concentration a large flux results. More cars can pass along a road if they move fast (large K ) , the road is wider or multilane ( A ) , and if they are packed or follow more closely ( Z ) . For environmental calculations it is very useful to know the relative magnitudes of these resistances since a small resistance may be ignored if it is in series with a large resistance. Figure 4 illustrates the two-resistance concept in a hypothetical steady-state case. The concentration profile is discontinuous at the interfaces while the fugacity profile is continuous. For example, for water (1) to air (2) transfer, K I is Kw, ZI is 1 / H , K Z is K A , and Z2 is IIRT; thus r I is H/ KwA, rz is RT/KAA, and thus
Physical signitieance of terms It is important to grasp the physical significance of these terms. A large K (fast piston velocity) obviously results in a low resistance and a high D. Large area also contributes to faster transfer by a large D. The role of the Z terms is
N = Ifi -fz)l(HjKwA + RT/KAA) = ( c w - RTCA/H)A/(l/Kw RT/HKA). Since RTCA is the air-phase solute partial pressure PA, this reduces to the familiar Whitman two-resistance model, which is often stated using an
N = Vi - ~ Z ) I ( V K I A Z I +IIKzAZz) = Vi - f z P i z . is defined as a combination of A , K , and Z terms. It may be conceptually easier to grasp the significance of this equation if expressed as resistance terms (denoted by r ) in which by analogy with Ohm’s law, N is current, f i s voltage, and r is ohms. The ( 1 / K A Z ) terms are resistances r; thus, N can be written as 012
N = VI - f i ) / r ~= K -fz)/b = Vi - f M r i
+ rz);
+
0MA
Environ.Sci. Technol.. Vol. 16. NO. 12, 1982
+
10-10
c-
+
Concentrations (movm3)
overall mass transfer coefficient K AW . N = KAwA(CW- PA/H) where I/KAw= I/Kw+ RTjHKA. If molecular or eddy diffusion can beapplied over a path length y ( m ) ,K can be replaced by Bjy where B is diffusivity (mz/h) The resistance is then y/ABZ.
In summary, for calculating the flux between two phases, values of A , K , and Z are obtained; the individual phase resistances are estimated, added, and then applied to the fugacity differences in the form of the interphase transport coefficient D.
Material transport When a volume of one phase physically moves to another, carrying solute with it, the flux N is simply SC where S is the volume flow rate (m3/h). Replacing C by Zfshows that N is SZfor Df,and D is thus SZ;fand Z apply to the source phase. The process is, of course, “one-way” in contrast to diffusion, which is “two-way” or reversible. Indeed diffusion may be viewed as the net difference between the two processesflD12 andfzDlz, which are proceeding in opposition. When ma-
‘terial transport is included (for example, wet or dry deposition or sediment movement to or from water), the D term is added but only in one direction. In the interest of simplicity this process is not considered in the examples given later, but its inclusion is straightforward. Estimating transport coefficient (0) Environmental compartments tend to fall into two categories. First are those in which exchange is between layers such as soil, the atmosphere, water, and sediment. The interphase area is simply the phase volume divided by the mean depth or thickness. Second, there are phases consisting of particles that are totally enveloped in a phase, such as fish or suspended particles. The area of such phases of effective diameter d is r d 2 and the volume is rd3/6; hence the “effective thickness” or volume/area ratio is d/6. This characteristic thickness can be used to estimate the area from the dispersed phase volume. Clearly, small particles have very high area to volume ratios, and transfer will usually be rapid. For phases that are not in contact, such as sediment and air, A is zero and there is no flux. The two mass transfer coefficients that apply to each interphase transport process can be estimated experimentally by measuring N, A, and concentration or fugacities. It is convenient to select two transferring solutes of quite different Z values such that in one case KlZl >> K2Z2; hencerl
