UDC 541.183.24

FEATURES OF THE MECHANISM OF ISOTOPE SEPARATION IN THE LIQUID ION EXCHANGER—SOLUTION SYSTEM

I. A. TIKHOMIROV, V. V. LARIONOV, A. P. VERGUN, L. A. AVDEEV

(Presented by the scientific seminar of the Faculty of Physics and Technology)

The mechanism of isotope separation on liquid ion exchangers is considered. On the basis of a calculation of the vibrational frequencies of the isotopic forms of liquid-ion-exchanger molecules, a number of changes in the ratios of the sums over isotope states are revealed, as well as changes in the enrichment coefficient according to the types of configurational transitions of symmetry \(T_d\), \(C_{3v}\), \(C_{2h}\), \(C_s\).

Illustrations 1, references 3.

It is known [1] that isotope redistribution between two compounds is produced by the energetic nonequivalence of the states of the isotopes in these compounds. The degree of such nonequivalence can be expressed as the ratio of the sums over the states of the isotopic formulae of the molecules entering into the isotope-exchange reaction (the \(\beta\)-factor). Calculation of \(\beta\)-factors, which makes it possible to reveal the mechanism of isotope separation, encounters a number of difficulties in each specific case. Let us consider the solution of this problem for the liquid-ion-exchanger—solution system.

Let us write the expression for the \(\beta\)-factor in the following form:

\[ \ln \beta = \frac{1}{n} \sum_i d_s \ln \frac{u_i^{(2)} \operatorname{sh} u_i^{(1)}/2} {u_i^{(1)} \operatorname{sh} u_i^{(2)}/2}, \tag{1} \]

where \(u_i\) is the reduced frequency \(\dfrac{\omega_i c h}{kT}\); \(n\) is the number of equivalent atoms of the given isotope participating in the exchange; \(d_s\) is the degeneracy multiplicity of the \(i\)-th frequency; \(\omega\) is the harmonic vibration frequency; \(c\) is the velocity of propagation of electromagnetic waves; \(h\) is Planck’s constant; \(k\) is Boltzmann’s constant.

The problem has a simple solution if the fundamental frequencies \(\omega_i\) or the observed frequencies \(\nu_i^{(1)}\) and \(\nu_i^{(2)}\) are known. Replacing the frequencies \(\omega_i\) by \(\nu_i\) is carried out without allowing for vibrational anharmonicity. The interaction of rotational and vibrational motions is likewise not taken into account. In general this does not lead to errors greater than 2–3%.

Finding \(\nu_i^{1,2}\) reduces to solving the secular equation describing the vibrations of the molecule.

As we have shown [2], a complex molecule of a liquid ion exchanger, consisting of 56 atoms, can be represented as a 9-atom molecule if the 2-ethylhexyl radicals are replaced by certain generalized atoms \(R\).

For a 9-atom molecule the secular equation still remains cumbersome; however, its formulation and solution are already possible.

The secular equation in the general case may be written as follows:

\[ \left\lVert \begin{array}{cccc} D_{11}-\omega & D_{12} & \cdots & D_{1n}\\ D_{21} & D_{22}-\omega & \cdots & D_{2n}\\ \cdot\ \cdot\ \cdot\ \cdot & \cdot\ \cdot\ \cdot\ \cdot & \cdot\ \cdot\ \cdot\ \cdot & \cdot\ \cdot\ \cdot\ \cdot\\ D_{n1} & D_{n2} & & D_{nn}-\omega \end{array} \right\rVert = 0 . \tag{2} \]

Let us single out in the equation the terms that give frequencies with an isotope shift. For a monomeric molecule \(AR_2PO_4\) of a liquid ion exchanger with isotopes \(A\) and \(A^*\), there will be two terms characterizing the frequencies of the valence vibrations \(A—O\) (here \(A\) is the isotope atom).

Ion exchange in the liquid ion exchanger–solution system leads to the formation of di-, tri-, and tetrameric molecules \(A_2(R_2PO_4)\), \(AR_2PO_4(HR_2PO_4)_2\), \(AR_2PO_4(HR_2PO_4)\).

Correspondingly, for each particular case the number of terms of the secular equation that determine the number of frequencies with an isotope shift increases.

For a dimeric molecule the frequencies of the valence vibrations are written as

\[ \nu_A^2 = \varepsilon_O \cos\alpha\, K_{q_1Q_2} +(\varepsilon_A+\varepsilon_O)K_Q -\varepsilon_A\cos\beta\, K_{Q_1Q_2} -\varepsilon_O\frac{r_H}{r_{A-O}}\sin\alpha\, K_{Q_1a}, \tag{3} \]

where \(\varepsilon_A\) and \(\varepsilon_O\) are the reciprocal “spectroscopic” masses of the isotope atom \(A\) and of oxygen, which is the donor for this system; \(\alpha\) is the angle \(A—O—P\), formed by the oxygen–isotope atom and oxygen–phosphorus bonds; \(K_Q\) is the force constant of the oxygen–isotope atom bond; \(K_{q_1Q_2}\) is the force constant of the interaction of the two \(O—A\) and oxygen–phosphorus bonds; \(K_{Q_1Q_2}\) is the force constant of the interaction of the bonds \(A—O\) and \(A\ldots O\); \(\beta\) is the angle \(O—A\ldots O\), formed by the valence bond oxygen–isotope atom and the coordination bond oxygen–isotope atom; \(r_H\) is the length of the \(C—H\) bond in Å; \(r_{A-O}\) is the length of the \(A—O\) bond in Å.

As is seen from equation (1), the magnitude of the \(\beta\)-factor is affected by the absolute value \(\nu_i\), the difference of isotope frequencies \(\Delta\nu_i\), and the number of frequencies \(\nu_i\) undergoing an isotope shift. Thus, the magnitude \(\Delta\nu_i\) is produced by the difference in reciprocal masses of the isotope atoms \(A\) and \(A^*\) (the 2nd and 3rd terms of equation (3)). The absolute value of the vibrational frequencies \(\nu_i\) is affected by the magnitudes of the force constants \(K_{q_1Q_2}\), \(K_{Q_1Q_2}\), \(K_Q\), \(K_{Qa}\), the masses of the donor atoms, and the magnitudes of the equilibrium angles.

Further, a series of polymerized states can be represented by the symmetry groups of the molecules as \(C_s \to C'_s \to C_{3v} \to T_d\).

In the aqueous phase the aquacomplexes of isotopes \(A\) have the symmetry type \(T_d\) or \(O_h\).

These series of configurational transitions from system to system, which set the number \(\nu_i\), determine the isotope effect.

This is seen from the formula

\[ \alpha = \frac{\beta_{T_d}}{\beta_{C_s}} > \frac{\beta_{T_d}}{\beta_{T_s}} > \frac{\beta_{T_d}}{\beta_{C_{3v}}} > 1; \qquad \frac{\beta^{1}_{T_d}}{\beta_{T_d}} \approx 1, \tag{5} \]

where \(\alpha\) is the single-stage separation coefficient in the ion exchanger–solution system.

The conclusions given make it possible to conduct a targeted selection of systems for isotope separation. A semiempirical search can be replaced by a preliminary theoretical analysis.

With some approximation, the force constant can be expressed through the electrostatic interaction of two atoms in a molecule of a liquid ion exchanger; then the expression for the force constant takes the form

\[ K=[(2S+M)/2S]^2\{2A/\rho^2 \exp[-(r-b)/\rho]-12\mu/r^4-20e^2\alpha/r^6-144\alpha_i\mu^2/r^8\}, \tag{6} \]

where \(A\) is the energy of electrostatic interaction of atoms (\(\approx 1200\,eV\)); \(\mu=4.3D\) is the dipole moment; \(\alpha=7\cdot10^{-24}\,cm^3\) is the polarizability of the liquid ionite molecule; \(\alpha_i=0.03\cdot10^{-24}\,cm^3\) is the polarizability of the isotope aqua complex; \(b=0.9\,\text{\AA}^0\); \(\rho=0.32\,\text{\AA}^0\) (literature data [3]); \(M\) is the mass of the isotope atom; \(S\) is the mass of the oxygen atom.

Fig. 1. Change in the single-stage enrichment coefficient by types of configurational transitions: 1 — experimental data, 2 — theoretical calculation.

Expression (6) shows that a poorly oriented search for systems by means of widespread correlations with the dipole moment and dielectric constant of the medium can be replaced by a directed search through a mass calculation of the \(\beta\)-factors and vibration frequencies of the exchanging molecules.

The calculations carried out, which reveal the mechanism of isotope exchange in such systems, have been verified experimentally for the series described by formula (5). A comparison of the data is presented in Fig. 1.

REFERENCES

1. Ya. M. Varshavskii, S. E. Vaisburg. “Uspekhi khimii,” vol. 26 (1957), 1435.
2. I. A. Tikhomirov, V. V. Larionov, A. P. Vergun, V. T. Doronin. ZhFKh, vol. 45, 1805 (1971).
3. W. F. Edgell et al. J. Am. Chem. Soc., vol. 92, 2240 (1970).