The rate-determining step of the heterogenous isotope exchange is very frequently the transport of the substances from the bulk to the interface. The transport means the convection, the mixing, and the diffusion of the dissolved substance from the solution phase to the interface, i. e., the reaction zone. The transport in the solution phase has two steps: the movement of the dissolved substances in the bulk solution and through a so-called adhesion layer. The convection and mixing influences only the transport in the bulk phases; in the adhesion layer, only diffusion (called “film diffusion” in this case) is possible, governed by the concentration gradient through the adhesion layer. Mixing decreases the thickness of the adhesion layer.

Since the concentration of the radioactive nuclide in radiotracer experiments is frequently very low, the diffusion plays an important role. In the adhesion layer,
the diffusion can be described by Fick’s first law, assuming that the concentration gradient is constant through the adhesion layer:

dc dc

= D

dt dx

In this case, the net kinetics of the isotope exchange is of the first order.

9.3.3.1 The Empirical Equation of the Heterogeneous Isotope Exchange

The heterogeneous isotope exchange in solid/liquid systems cannot be described by the usual, theoretically correct kinetic equation. For the interfacial reaction of very insoluble salts, an empirical equation was derived by Imre in 1933:

where xt and xN are the relative amount of the radioactive isotope on the solid sur­face at time t and in equilibrium, respectively; kj, k2, k3 are rate constants; and Ab A2, and A3 are empirical coefficients. The members in Eq. (9.112) refer to simulta­neous first-order reactions. However, the equation is formal; thus, it gives no infor­mation on the mechanism of the net reaction. In the literature, there are many frequently speculative interpretations for the rate constants and the empirical coef­ficients. These interpretations include speculation about the rate-determining step. Similar equations are used for the kinetic description of the interfacial processes of the crystalline powder/solution or the metal/solution. The experimental data are interpreted by including additional members to the kinetic equation.

Many heterogeneous isotope exchange processes between metal and the ions of the same element were studied to determine the exchange rate in equilibrium. The Ag/Ag1 system was intensively studied. The kinetics of the exchange process was interpreted as follows:

and

___ VoVj 1 VdVj 1 VoVd

T xN

VoVjVd

where v0 is the rate of surface diffusion, Vj is the rate of the electron exchange, and vd means the rate of diffusion in the solution. Equations (9.113) and (9.114) take into consideration the steps and mechanism of the heterogeneous isotope exchange on the interface of the metal/electrolyte solution. The t versus concentration func­tion gives information on the rate-determining step and the mechanism of the isotope exchange.

The kinetics of the heterogeneous isotope exchange was described by empirical equations that take into account the heterogeneity of the surface. For example:

xt = btn (9.115a)

where b and n are empirical values. These values are interpreted by the energy distribution of the exchange sites of the surface.

The main conclusions of the heterogeneous isotope exchange studies on metal surfaces are as follows:

• In equilibrium between the metal and the solution phase, continuous isotope exchange takes place.

• The rate of the electron exchange (exchange current) can be measured only in special cases; namely, when the rate-determining step of the surface reaction is the electron exchange, such as in the Fe/Fe21 system.