The Paneth surface determination postulates that the heterogeneous isotope exchange is much faster on the surface of the bulk solid phase than inside it. Thus, the specific surface area of a solid substance can be determined from the ratio of the radioactiv­ities of the solid and its saturated solution. For example, the specific surface area of lead sulfate can be determined using radioactive lead ions (e. g., 212Pb21). The isotope exchange takes place between the lead ions in the solution and on the surface of the solid lead sulfate:

PbSO4 1 *Pb21 з* PbSO4 1 Pb21 (9.115b)

Since there are no chemical reactions, the process is directed by the change of the mixing entropy. When the exchange takes place only on the surface, the ratio of the activities of the solution and solid is determined by the ratio of the atoms/ ions in the solution and on the surface of the solid, as follows:

Activity of solution cV

= 9.116)

Activity of solid X

where X is the number of the atoms/ions on the surface, c is the concentration of saturated solution (solubility), and V is the volume of the solution. After multiply­ing X by the cross section of the surface atoms/ions, the specific surface area is expressed in area units.

The radioactive atoms/ions can be buried into the bulk by isotherm transcrystal­lization. This effect is corrected by the kinetics of the isotope exchange; the radio­activity is measured as a function of time (Figure 9.15). The kinetic curve has two sections: a sharp and a slight increase; and the fast surface exchange is followed by a slow exchange (transcrystallization) with the bulk phase. Extrapolating the slight

Figure 9.15 The determination of the surface exchange from the radioactivity versus time function by a heterogeneous isotope exchange.

increase to the initial time (t = 0), it cut the vertical axis at the radioactivity belong­ing to the pure surface exchange.

For crystalline substances, only a portion of the surface sites, the so-called active surface, exchanges with the radioactive atoms/ions in the solution. The active sur­face depends on the temperature. According to Imre, the relation of the total surface (X) and the active surface (X*) is as follows:

X* = X exp (-R0 (9.117)

where E is the activation energy of the isotope exchange and T is the temperature. The total surface area can be determined from the active surface areas measured at different temperatures.