## Nx cxvx

Substitution of the next-to-last equation into the last one gives nx = fZxVx

When we sum the nx values over all phases x of interest, we must obtain the total number of moles. Thus ntotai =/2ZxVx

Rearrangement of this equation allows us to calculate the value of the system fugacity:

/=ntotal/XZxVx

### An Example of a Fugacity Calculation

As an example of how fugacity calculations are carried out in practice, consider the distribution of 1 mole of DDT among three phases: air, water, and sediment in a model compartment of Earth (Figure 1). As discussed later, we take the volume of air to be 1010 m3, the water volume to be 7 X 106 m3, and the volume of accessible sediment to be 2 X 104 m3. The values of the Zx constants for DDT, in units of mol/atm m3, are determined from experimental data to be for the air phase, 40.3

for the water phase, 3.92 X 104

for the sediment phase, 2.25 X 109

In the evaluations of Zx values from experimental data, a temperature of 25°C is usually assumed for simplicity. The Zx values for sediment (and biota) are assumed to be proportional to the octanol-water partition coefficients Kow discussed earlier in the chapter.

After substitution of the values for Zx and Vx, the value of the fugacity in this case is

/= 1.0/(40.3 x 1010 + 3.92 x 104 X 7 X 106 + 2.25 x 109 X 2 x 104)

The concentration of the chemical can now be computed for each phase:

(continued on p. 464)

1 km

1 km

1 km

1 km

Sediment (includes biota) 2 104 m

FIGURE 1 Model world parameters used in fugacity calculations.

The Environmental Distribution of Pollutants (continued)

BOX 10-3

DDT concentration in air = 2.19 X 10"14 X

40.3 = 8.8 X 10~"13 mol/m3 DDT concentration in water = 2.19 X 10~14 X

3.92 X 104 = 8.6 X 10-10 mol/m3 DDT concentration in sediment = 2.19 X 10~14 X 2.25 X 1Q9 = 4.9 X 10~5 mol/m3

Notice the preferential concentration of DDT in sediment, which is hydrophobic due to its carbon content.

The amounts in each phase are given by the fZV values, i.e., the concentrations multiplied by the respective volumes. Then the number of moles of DDT

in air = 8.8 X 10^13 X 1 X 1010 = 0.0088 mol in water = 8.6 X KT10 X 7 X 106 = 0.0060 mol in sediment = 4.9 X 10~5 X 2 X 104 = 0.98 mol

Thus we see that, with air, water, and sediment accessible to it, 98% of the DDT will be found in sediment, and about 1% in air and in water. Notice that the concentration of DDT in water is greater than in air, but the total amount of it in air exceeds that in water because the air volume is so much larger. This sort of interchange in ordering between amount and concentration in different phases is common for pollutant chemicals.

The Parameters for the Model World in Fugacity Calculations Are Estimates

The volumes for the various phases used in the above calculation are based upon a model "world" (Figure 1) whose components are able to be in equilibrium with each other. Since only concentrations are obtained in the calculations, it is important only that the relative volumes, not their absolute values, be used. The model world has an area of 1 kilometer by 1 kilometer, whose characteristics are assumed to be average for the real Earth. The atmosphere is taken to be 10 kilometers high, which is a reasonable approximation to the troposphere. The air volume then is (1000 m X 1000 m) X (10,000 m) = 1010 m3. The 1-ktn square is assumed to be 70% covered by water and 30% by soil. The average water depth is taken to be 10 meters, which is relatively shallow since we are interested only in the part that achieves equilibrium with the air. Thus the water volume is 0.7 X (1000 m X 1000 m) X 10 m = 7 X 106 in3. The sediment in equilibrium with this water is assumed to be only 3 centimeters deep, giving it a volume of 0.7 X 1000 m X 1000 m X 0.03 m = 2.1 X 104 m3. In addition to air, water, and sediment, the model usually also includes soil, whose effec-3 3 3

tive volume is 9 X 10 m , plus 35 m of solids suspended in the water, and about 3.5 m3 of biota such as fish. The Z values for biota are usually of the same order of magnitude as those for sediment, so the concentration of a given chemical in biota is close to that in sediment.

### PROBLEM I

The Z values for hexachlorobenzene are 4 X 10"4 in air, 9.5 X lO-' in water, and 2.3 in sediment (and biota). Using the model world volumes above, calculate the equilibrium concentrations when 1 mole of hexachlorobenzene is distributed among air, water, and sediment.

### PROBLEM 2

In fugacity calculations, the Z values for diel-drin are 4 X 10-4 in air, 2.0 in water, and 2 X 10~5 in sediment (and biota). Using the model world volumes above, calculate the equilibrium concentrations when 1 mole of dieldrin is distributed among air, water, and sediment.

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