In this section, some examples of the diffusion of gases in solid media will be shown.

Diffusion of 222Rn in Soil

In porous media, the free volume can well be characterized by the diffusion of a gas that does not adsorb on the interfaces of the pores. For example, the diffusion of 222Rn gas is mentioned. Radon is a noble gas, and it has no interactions with the surrounding medium. Carrier-free ones can easily be produced (see Section 8.5.1) and their activity can be measured through the gamma radiation of its daughter nuclides. The diffusion of 222Rn has been studied in soil components, sand and clay, as porous media. The diffusion is described by the Fick’s laws; in this case, Fick’s second law, applied to linear diffusion, is used:

@C _ D

dt Sx2

where D is the diffusion coefficient (surface area/time), C is the concentration of the diffusing substance, and t is the time. When the concentration of the diffusing substance is measured by a radiotracer, the radioactive intensity (I) depends on the concentration, i. e.:

I _f (C)

Equation (9.32) does not have a general solution—only some solutions for special initial and boundary conditions. For linear diffusion at a constant temperature, a partial solution of Eq. (9.32) can be expressed by Eq. (9.34) when at t = 0, the total quantity of the diffusing substance is at a place x = 0 as a point source:

I = /° exp f—X ) (9.34)

4Dnt 4Dt

The diffusion coefficients can be determined in two ways. I values are measured as a function of time (t) at a given place (x = constant), or as a function of the dis­tance (x) at a given time (t = constant). This solution is frequently called “the para­bolic law of diffusion.” Depending on the method employed, the diffusion coefficients can be determined approximately from the slope of the ln I versus x2 or ln I versus 1/t, respectively. This gives a first-order relation; thus, it is a simple method for the determination of the diffusion coefficient. When, however, x = constant and t changes, a systematic error develops because the time is present in the intercept as well. Nowadays, computer programs are used for the estimation of diffusion coefficient from the original form of Eq. (9.34); in this way, this error can be ignored.

Simple laboratory equipment for the measurement of diffusion is shown in Figure 9.1, and the experimental results are plotted in Figure 9.2.

Figure 9.2 Linear diffusion of 222Rn gas in sand with different humidities. Humidity increases in the order: 1 > 2 > 3.

As seen in Figure 9.2, the volume of the free pores decreases when the humidity of sand increases, resulting in the decrease of migration rate of radon (Figure 9.3). Since the migration rate depends on the pore size, the determined diffusion coeffi­cient is a virtual diffusion coefficient.

Diffusion or migration measurements can be done in situ, under natural condi­tions. In this case, the solution of Eq. (9.32) applied to spatial diffusion has to be used. Under similar initial and boundary conditions as described in Eq. (9.34), the solution for spatial diffusion in an isotropic medium is:

Figure 9.4 Equipment for breaking a glass sphere containing 222Rn gas. (A) electric cable, (B) cap, (C) pushed spring,

(D) thin wire, (E) breaking iron disc, (F) gas outlet, and (G) glass sphere containing radon.

where r is the distance from the point source (considered as the origin of a sphere). In the studies of spatial diffusion, the diffusing gas (222Rn) is located as shown in Figure 9.4.

The spatial diffusion studies give similar results as linear diffusion studies, namely the virtual diffusion coefficient depends on the humidity—i. e., on the free volume of pores. Also, similar results are obtained in clay; however, clay contains less free volume pore space, and the virtual diffusion coefficients are about an order of magnitude smaller.