Transport of hazardous compounds or ions in ground water flowing through soils may be modeled using chromatography theory. The theory of linear dynamic chromatography is quite simple and much can be understood about the transport of hazardous waste with a little effort.

First, consider the soil made up of many particles of sand, clay, etc., and either an aquifer with water is flowing around these particles, or water flowing downward through the soil to the aquifer. In both cases, there is a net flow of water in a particular direction.

The hazardous compound will be in the water, and the water will be in contact with the soil. For simplicity, consider the soil to be saturated with water. We then only have to consider two phases; the water and the soil.

If the flow of water is sufficiently slow, the compound will be in equilibrium between the water (aqueous phase) and the surface of the soil (solid phase). The equilibrium equation is

K_D = C_S/C_W

where K_D is the partition constant and C_S and C_W are concentrations on the soil surface and in the aqueous phase, respectively.

The water is flowing but molecules are periodically sticking to the soil surface. The soil does not move. Subsequently, the molecule does not move when it is sticking to the soil surface, but it does move along with the water when it is in the aqueous phase. Thus, one would expect that the rate at which the molecule moves should be less than (or equal to) the flow rate of the water.

We can calculate the rate of travel by first realizing that the fraction of time that a molecule or ion, in dynamic equilibrium between the soil surface and the aqueous phase, is equal to the number of molecules in the aqueous phase, divided by the total number of molecules. In other words, the fraction of species in the aqueous phase is equivalent to the fraction of time spent in the aqueous phase.

The fraction of species in the aqueous phase can be calculated from the equilibrium equation given above, if we know the relative volumes or water and the surface area and "identity", e.g., kaolinite clay, sand, humus, etc., of the soil. Note that real soils are a collection of these surfaces while the equilibrium equation is for only one. (Though the case of mixtures of soil constituents can still be modeled using this simple approach as long as the equilibrium is linear in concentrations).

The number of molecules in the water phase is concentration times the volume of water in a volume element in the saturated soil.

# Molecules in Water = C_WV_W

The number on the soil surface is the surface concentration times the surface area, again, per unit volume of saturated soil

# Molecules on Soil = C_SA_S

The units for C_S are in moles-per-unit-area (moles/m²). The total number of molecules is sum of those in water plus those adsorbed on the soil

Total # Molecules = C_WV_W + C_SA_S

Thus, the fraction of time a molecule spends in the aqueous phase is

Fraction of Time in Aqueous Phase = C_WV_W/[C_WV_W + C_SA_S]

Using the distribution (equilibrium) equation, and the above,

Fraction of Time in Aqueous Phase = 1/[1+K_D(A_S/V_W)]

Some things that this result predicts are;

1. The rate of travel of the hazardous waste is always less than or equal to the rate that the water is moving.*
2. The rate of travel increases with increasing water flow rate.
3. The rate of travel decreases with increasing K_D.
4. The rate of travel decreases with increasing A_S.

The first prediction is only true of the water flows much faster than mass diffusion. Mass diffusion is pretty slow. However, if the water is stagnant, diffusion transport dominates.

The second prediction is true only up to the point where equilibrium between soil surface and water is maintained. At really fast water flow rates, the equilibrium condition may not have time to establish itself (a kinetic limitation). In this case, the rate of hazardous material transport may increase nonlinearly with increasing flow. For example, if you put a toy boat in slow-flowing stream, it bounces from shore to shore as it is transported down the stream. If you put the same boat in a "raging river", it may not touch the shore before moving several kilometers.

The third point predicts that the rate of travel will decrease if the hazardous material prefers to be adsorbed on the soil surface. That makes sense. The more the material likes the soil particle, the slower its overall rate of transport. However, this trend only holds as long as equilibrium is maintained. Moreover, the equilibrium has to be linear with respect to concentrations. If it is not, all bets are off.

The fourth point predicts that the greater the surface area (per unit volume of saturated soil) the slower will be the transport of the hazardous substance. Notice that this prediction is independent of distribution coefficient (K_D). In fact, we do know this to be true. But there are reservations regarding the absolute validity of the prediction.

What is true is that transport is much more rapid through relatively low surface area soils, such as sand, than it is through high surface area soils, such as clay.

What is not true is that there is a linear relationship. This is because not all of the surface area will be in direct contact with the flowing water. The soil may consist of particles with cracks, pours, etc., and the water may not flow through the cracks, etc. To make maters worse, the fraction of the soil surface "sampled" by the flowing water make change with the flow rate. In general, faster moving water will effectively "sample" less of the total surface area than slow moving water.

Nonetheless, given all the uncertainties, the dynamic theory does predict general trends in the transport of hazardous substances through the ground water or down to the ground water. The four predictions stated above will hold, but the quantitative aspects may be flawed.

Many corrections to this simple model are known to correct for nonlinear adsorption and these give more reasonable quantitative predictions. These can be found in the theory of chromatography as discussed in most modern Analytical Chemistry textbooks.

More details will be given in lecture.