Study of the Kinetics of Isotope Exchange in the Ion Exchanger–Solution System
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STUDY OF THE KINETICS OF ISOTOPE EXCHANGE
IN THE LIQUID ION EXCHANGER–SOLUTION SYSTEM
I. L. TIKHOMIROV, A. P. VERGUN, V. V. LARIONOV, V. T. DORONIN
(Presented by the joint scientific seminar of Departments 21 and 23
of the Faculty of Physics and Engineering)
The isotope-exchange reaction in the liquid ion exchanger–solution system can be written in the following form:
\[ A_R + B_S = A_S + B_R, \]
where \(A\) and \(B\) are the exchanging ions, \(R\) is the ion-exchanger phase, and \(S\) is the solution phase. The concentration equilibrium constant of the indicated reaction is
\[ K = [A_S][B_R] : [A_R]\cdot [B_S] \]
\([A_R]\), \([B_R]\), \([A_S]\), \([B_S]\) are the concentrations of ions \(A\) and \(B\), respectively, in the ion exchanger and in the solution.
Three types of isotope-exchange kinetics can be distinguished in the liquid ion exchanger–solution system:
a) the determining process is the establishment of equilibrium at the phase boundary, i.e., the rate of diffusion of the exchanging ions in the phases is high;
b) the rate of ion diffusion in the phases is low, while equilibrium at the phase boundary is established rapidly;
c) the slowest process is the chemical reaction itself.
Let us consider each process separately.
a. The rate of the exchange process between the phases will be determined by the change in the content of the isolated isotopic ion in the liquid ion exchanger or solution. The isotopic concentration changes as a result of transfer of the isolated isotope from the solution into the ion exchanger or conversely.
Suppose that we have ion-exchanger and solution phases brought into contact. \(N\) is the total concentration of ions; \(C_1\) and \(C_2\) are the concentrations of the isolated isotopic ion in the ion-exchanger and solution phases, respectively. The amount of isolated isotopic ions transferred from the solution into the liquid ion exchanger will be proportional to the amount of the light isotope in the solution, \(N_2 C_2\), and to the amount of the heavy isotope in the ion exchanger, \(N_1(1-C_1)\). Consequently,
\[ \Phi_{2\to 1}=k_1\cdot N_2 C_2 N_1(1-C_1), \tag{1} \]
where \(k_1\) is the proportionality constant.
For the reverse transfer process, analogously,
\[ \Phi_{1\to 2}=k_2 N_1 C_1 N_2(1-C_2). \tag{2} \]
The total flux of the released isotopic ion will be determined by the difference between these two fluxes, i.e.,
\[ \Phi=\Phi_{2\to1}-\Phi_{1\to2}. \tag{3} \]
Substitute the expressions for \(\Phi_{2\to1}\) and \(\Phi_{1\to2}\) into equation (3):
\[ \Phi=k_2N_1N_2\left[\frac{k_1}{k_2}(1-C_1)C_2-C_1(1-C_2)\right]. \tag{4} \]
The quantity \(k_1/k_2\) is equal to the equilibrium constant of the exchange interaction, or, in the stationary case, is equal to the separation coefficient \(\alpha\).
Denoting \(k_2N_1N_2=K\), from equation (4), taking into account what was said above, we obtain
\[ \Phi=K\left[\alpha C_2(1-C_1)-C_1(1-C_2)\right]. \tag{5} \]
b. The kinetics of isotope exchange is determined mainly by diffusion of isotopic ions in the ion-exchanger and solution phases. The flux \((\Phi)\) of the released isotope can be estimated by Fick’s law [1]:
\[ \Phi=-D_1N_1\cdot\frac{dC_1}{dx}=D_2N_2\cdot\frac{dC_2}{dx}, \tag{6} \]
where \(D_1\) and \(D_2\) are the diffusion coefficients in the ion-exchanger and solution phases.
Let there be an ion-exchanger layer of thickness \(d_1\) and a solution layer \(d_2\). The concentrations at the phase boundary will be denoted, respectively, by \(C_1^0\) and \(C_2^0\) in the ion-exchanger and solution phases.
Then the concentration change is written as:
\[ \left. \begin{aligned} \frac{dC_1}{dx}&=\frac{C_1-C_1^0}{d_1}\\[4pt] \frac{dC_2}{dx}&=\frac{C_2-C_2^0}{d_2} \end{aligned} \right\}. \tag{7} \]
Taking into account the expressions given earlier, equation (7) is rewritten in the form:
\[ \left. \begin{aligned} \Phi\,\frac{d_1}{D_1N_1}&=C_1^0-C_1\\[4pt] \Phi\,\frac{d_2}{D_2N_2}&=C_2-C_2^0 \end{aligned} \right\}. \tag{8} \]
Adding these equations,
\[ \Phi\left(\frac{d_1}{D_1N_1}+\frac{d_2}{D_2N_2}\right)=(C_1^0-C_2^0)-(C_2-C_1). \]
We write the expression for the separation coefficient
\[ \alpha=\frac{C_1^0(1-C_2^0)}{(1-C_1^0)\cdot C_2^0} \quad \text{for } C_1^0\cdot C_2^0=0. \tag{9} \]
\[ C_1^0-C_2^0\simeq(\alpha-1)C_2(1-C_1)\simeq\Delta C, \tag{10} \]
where \(\Delta C\) is the maximum difference in isotopic concentrations between the phases.
Ultimately, after substituting expressions (10) into (9) and carrying out the transformation, we obtain
\[ \Phi=\frac{1}{\dfrac{d_1}{D_1N_1}+\dfrac{d_2}{D_2N_2}} \left[\alpha C_2(1-C_1)-C_1(1-C_2)\right]. \tag{11} \]
Comparing expressions (5) and (11), we conclude that they differ only in the values of the constants. In the general case, when both types of kinetics are present, the expression can be written in the form
\[ \Phi = \gamma_0[\varepsilon C_2(1-C_1) - (C_1-C_2), \tag{12} \]
where \(\gamma_0\) is the exchange-flux density of the isotope ion being released.
c. The rate-determining stage is the chemical reaction. In this case, as shown in [1], the transfer of the isotope ion being released can be defined by the expression
\[ \Phi = K'\{\varepsilon C_2(1-C_1) + (C_1-C_2)\}. \tag{13} \]
As can be seen, it differs somewhat from expression (11).
Depending on the experimental conditions, one of the processes considered in isotope exchange in the liquid ion exchanger–solution system will be the slowest in comparison with the other processes. The kinetics of isotope exchange under the conditions considered will then be characterized by this type of process.
REFERENCES
- F. Helfferich. Ion Exchangers. Moscow, IL, 1962.
- A. M. Rozen. Theory of Isotope Separation in Columns. Atomizdat, 1960.