SEPARATION OF ISOTOPIC MIXTURES BY THE METHOD OF TWO-DIMENSIONAL ELECTROCHROMATOGRAPHY

I. A. Tikhomirov, G. S. Tikhonov, V. T. Doronin

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

The method of two-dimensional separation of ionic mixtures under the conditions of an electric field was first developed and implemented by H. Svensson and I. Brattsten [1] on paper and showed good results in the separation of multicomponent mixtures.

We have attempted to apply and theoretically describe this process for the case of continuous electrochromatographic separation of isotopes. In this case the transfer of substance occurs in two mutually perpendicular directions (Fig. 1)—horizontally owing to the influence of a constant electric field and vertically owing to the hydrodynamic transport of the mixture being separated. The construction of the column is chosen so that the size of the gap between its side plates

Fig. 1. Schematic of a continuous two-dimensional electrochromatographic process for separating ionic mixtures. 1 — direction of the electric field; 2 — direction of the flow of the eluting solution; 3 — direction of motion of the mixture being separated; 4 — direction of motion of negatively charged particles; 5 — direction of motion of uncharged particles; 6 — direction of motion of positively charged particles.

of the order of a factor smaller than the height and length of the column. Thus a continuous two-dimensional process of separation of the initial mixture is ensured. In these columns we studied the separation of isotopic ions by methods of ion-exchange and partition electrochromatography.

We shall first consider the theoretical description of the two-dimensional electrochromatographic separation process when the column is filled with an ion-exchange resin (cation exchanger).

Two-dimensional ion-exchange electrochromatography

Questions of the theory of isotope separation on ion exchangers have been considered in works [2, 3], where estimates are given for the single-stage separation coefficients, the height of the equivalent theoretical stage (HETS), and the time required to establish a stationary state. In works [4—6] the theory of isotope separation by the method of ionic mobilities is considered in detail. The method investigated by us differs in that here both of the indicated processes are combined simultaneously—ion exchange and electromigration of ions. The theory of this method for the case of one-dimensional electrochromatography was considered earlier in work [7]. Therefore we shall dwell on the description of the two-dimensional separation process, when transfer of the substance in the column occurs in two mutually perpendicular directions. For this purpose we write the material-balance equation for the substance of a binary isotopic mixture passing through an ion-exchange bed. In the case of two-dimensional electrochromatography we shall have two fluxes of the substance being separated, one of which \((I_x)\) is associated with the transfer of isotopic ions under the action of a constant electric field and is directed in the horizontal plane, while the second flux \((I_z)\) is caused by the motion of the eluting solution passing through the bed in the vertical direction, perpendicular to the first flux. We write these equations as

\[ I_x = I_E - I_D - I_{D'} - I_{\Pi}, \tag{1} \]

where

\(I_E\) is the flux of the light isotope caused by the action of the constant electric field; \(I_D\) is the flux of the light isotope produced by diffusion of ions in the external solution; \(I_{D'}\) is the diffusion flux of the light isotope in the ion exchanger; \(I_{\Pi}\) is the amount of light isotope absorbed by the ion exchanger per unit time. Expanding the quantities entering into equation (1), we obtain:

\[ I_x = [v_{\mathrm{av}}+\varepsilon_1 v_{\mathrm{av}}(1-c)]cN - DN\gamma \frac{dc}{dx} - D'\frac{dm_1}{dx} - Vm_1 . \tag{2} \]

For the second flux we shall have

\[ I_z = UNc - U\frac{N}{m_0}m_1 = UN\left(c-\frac{m_1}{m_0}\right). \tag{3} \]

Here \(v_{\mathrm{av}}\) is the average velocity of motion of the isotopic ions under the action of the constant electric field, cm/sec,

\[ v_{\mathrm{av}}=\frac{(v_1+v_2)E}{2}. \tag{4} \]

\(v_1\) and \(v_2\) are the mobilities of the isotopic ions, cm\(^2\)/sec·V; \(E\) is the strength of the constant electric field, V/cm; \(\varepsilon_1\) is the enrichment coefficient due to the difference in the mobilities of the ions; \(c\) is the fraction of the light isotope in solution; \(N\) is the volume concentration of the electrolyte, g-equiv/cm\(^3\); \(D\) is the diffusion coefficient of the ions in solution, cm\(^2\)/sec; \(D'\) is the diffusion coefficient of the ions in the ion exchanger, cm\(^2\)/sec; \(\gamma\) is the coefficient of filling of the column; \(V\) is the velocity of motion of the stationary front of the sorption zone-

exchanged ions through the ion exchanger, cm/sec; \(m_1\) is the amount of the light isotope sorbed by the ion exchanger, g-equiv/cm\(^3\); \(U\) is the velocity of motion of the eluting solution through the ion exchanger, cm/sec; \(m_0\) is the total amount of substance absorbed by the ion exchanger, g-equiv/cm\(^3\). Moreover,

\[ m_0=m_1+m_2, \tag{5} \]

where \(m_2\) is the amount of the heavy isotope sorbed by the ion exchanger, g-equiv/cm\(^3\).

Taking into account that there is no flux of substance along the \(Y\) axis, we obtain the general material-balance equation for the light isotope in the following form:

\[ \operatorname{div}(I_x+I_z)+\frac{dm_1}{dt}+\gamma N\cdot \frac{dc}{dt}=0. \tag{6} \]

Let us transform this expression somewhat. The quantities \(m_1\) and \(c\) can be related to one another by using the ion-exchange isotherm equation:

\[ \frac{c}{1-c}=K\cdot \frac{m_1}{m_2}=(1+\varepsilon_2)\frac{m_1}{m_2}, \tag{7} \]

where \(K\) is the ion-exchange constant; \(\varepsilon_2\) is the enrichment coefficient due to the ion-exchange process. Further, taking into account the works [2, 8, 9], and allowing for the lag time \(\tau\), for the fraction of the light isotope in the solution one may write the relation

\[ c=\gamma(1+\varepsilon_2)-\frac{v_{\mathrm{cp}}N}{m_0}\tau \frac{d\gamma}{dx}, \tag{8} \]

where

\[ \gamma=\frac{m_1}{m_2}. \tag{9} \]

Substituting expression (8) into equation (6), after certain transformations one may write:

\[ F\cdot \frac{d^2\gamma}{dx^2}-A\cdot \frac{d\gamma}{dx}=Q\cdot \frac{d\gamma}{dt}, \tag{10} \]

where

\[ F=\frac{N^2v_{\mathrm{cp}}\tau}{m_0}(1+\varepsilon_1)v_{\mathrm{cp}}+U+D N\gamma(1+\varepsilon_2)+D'm_0, \tag{11} \]

\[ A=[(\varepsilon_1+\varepsilon_2)v_{\mathrm{cp}}+\varepsilon_2 U]\cdot N, \tag{12} \]

\[ Q=m_0+\gamma N(1+\varepsilon_2). \tag{13} \]

Thus, we obtain a second-order partial differential equation. The solution of this equation in general form may be written as

\[ \gamma(x,t)=\gamma_0\exp(2\xi x)-\gamma_0\exp(\xi L_{\mathrm{eff}}) \frac{\sin \omega x}{\sin \omega L_{\mathrm{eff}}}\cdot \exp(\xi x)\cdot \exp\left[-\left(\frac{E\varkappa^2}{L_{\mathrm{eff}}^2}+\frac{A^2}{4F}\right)\frac{t}{Q}\right]. \tag{14} \]

Here

\(L_{\mathrm{eff}}\) is the effective length of the column, cm

\[ L_{\mathrm{eff}}=(L^2+H^2)^{1/2}, \tag{15} \]

where \(L\) is the length of the column, cm; \(H\) is its height, cm.

\[ \xi=\frac{A}{2F}; \qquad \omega=\frac{\varkappa}{L_{\mathrm{eff}}}, \tag{16} \]

\(\varkappa\) is the root of the transcendental equation

\[ \tan \varkappa=\frac{\varkappa}{\xi L_{\mathrm{eff}}}. \tag{17} \]

Let us now estimate the time required for the given process to reach the stationary state. It follows from expression (16) that, after a certain interval of time,

\[ t=\frac{Q}{\frac{F\varkappa^2}{L_{\mathrm{eff}}^2}+\frac{A^2}{4F}}, \tag{18} \]

the second term of the expression decreases by a factor of \(e\), i.e., the process approaches the stationary state. In this case

\[ t=\frac{4L_{\mathrm{eff}}^2\cdot Q\cdot F}{4\varkappa_1^2F^2+A^2L_{\mathrm{eff}}^2}. \tag{19} \]

With greater accuracy, the time for establishment of the stationary state should be taken as

\[ T=(2\div 3)t. \tag{20} \]

The general solution of equation (10) also makes it possible to estimate the height of the equivalent theoretical plate. For this purpose let us turn to expression (14). As \(t\to\infty\), i.e., upon establishment of the stationary state, the second term in this equation tends to zero. Therefore, if we take into account that in our case \(\varepsilon\ll 1\), we may write

\[ \frac{\nu}{\nu_0}=\exp\left(\frac{A}{F}x\right)=(1+\varepsilon)^{\frac{Ax}{\varepsilon F}}=\alpha^n, \tag{21} \]

where

\[ \varepsilon=\varepsilon_1+\varepsilon_2 \quad\text{and the single-stage separation coefficient}\quad \alpha=1+\varepsilon, \tag{22} \]

\(n\) is the number of plates. Then the height of the equivalent theoretical plate is

\[ \text{HETP}=\frac{L_{\mathrm{eff}}}{n}=\frac{\varepsilon F}{A}. \tag{23} \]

Relation (21) makes it possible to determine easily the change in concentration of the light isotope along the length of the column. Taking into account the work [7], one may write:

\[ c(x)=\nu_0(1+\varepsilon_2)\exp(Ax/F), \tag{24} \]

and at the sampling point the expression for the concentration of the light component has the form:

\[ c_k=\nu_0(1+\varepsilon_2)\exp(AL_{\mathrm{eff}}/F). \tag{25} \]

Analysis of the formulas obtained shows that noticeable enrichment effects may be expected at a feed-solution concentration of the order of \(0.5\,N\) and at ion velocities of the order of \(10^{-3}\ \text{cm/sec}\). Therefore we carried out a comparison of calculated and experimental data for the two-dimensional electrochromatographic process of separation of isotopic ions of light alkali elements in a column filled with KU-2 cation exchanger, under the following initial data:

\[ \begin{gathered} N=0.5\cdot10^{-3}\ \text{g-eq/cm}^3;\quad \nu_{\mathrm{av}}=1\cdot10^{-3}\ \text{cm/sec};\quad U=2\cdot10^{-3}\ \text{cm/sec};\\ m_0=3.24\cdot10^{-5}\ \text{g-eq/cm}^3;\quad D=10^{-5}\ \text{cm}^2/\text{sec};\quad D'=10^{-6}\ \text{cm/sec};\\ A=1.22\cdot10^{-8}=1.4;\quad \gamma=0.5;\quad \tau=10\ \text{sec};\quad L_{\mathrm{eff}}=23\ \text{cm};\\ F=5.62\cdot10^{-7};\quad \varepsilon_1=1.1\cdot10^{-2};\quad \varepsilon_2=4.5\cdot10^{-3}. \end{gathered} \]

It turned out that the time for establishment of the stationary state \((T)\) and the HETP, calculated from formulas (20) and (23), respectively, are \(T=86\) hours and \(\text{HETP}=7\ \text{mm}\). Experimentally, the time for establishment of the stationary state was found to be \(T=91\) hours, while the HETP at ob-

with the separation coefficient \(K_{\mathrm{cb}}=1.47\), proved to be equal to \(9\ \mathrm{mm}\).

Thus, the calculated values agree well with the experimental ones.

Two-dimensional partition electrochromatography

In the case of partition electrochromatography, the process of separation of isotopic ions is characterized by their redistribution between two different solvents and by electromigration of the ions under the action of a constant electric field. Quartz sand is usually used as the packing.

Let us first consider the distribution of the substance and of the isotopic ions contained in it between two solvents. When any salt is dissolved in an aqueous–organic mixture, part of the ions of the isotope mixture being separated will associate with water molecules, and part of the ions with molecules of the organic solvent. If \(N\) is the volume concentration of isotopic ions bound to water molecules \((\mathrm{g\text{-}eq}/\ell)\); \(N_0\) is the volume concentration of isotopic ions bound to molecules of the organic solvent \((\mathrm{g\text{-}eq}/\ell)\), then \(\frac{N_0}{N}=K\) is the distribution coefficient of the substance between the solvents. Let us assume that \(\frac{N'}{N}=c\) is the mole fraction of the light isotope bound to water and \(\frac{N'_0}{N_0}=\mu\) is the mole fraction of the light isotope bound to the organic solvent. Then \(\frac{c}{\mu}=\alpha'_2\) is the separation coefficient due to redistribution of the ions of the isotope mixture between the solvents. \(N'\) and \(N'_0\) are the volume concentrations of the light isotope in water and in the organic solvent.

In the case of two-dimensional partition electrochromatography we shall likewise have two fluxes of substance in the column

\[ I'_x=\left[v_{\mathrm{cp}}+\varepsilon_1 v_{\mathrm{cp}}(1-c)\right]cN-DN\gamma\frac{dc}{dx}-V'N_0\mu, \tag{26} \]

\[ I'_z=UNc-UN\mu=UN(c-\mu), \tag{27} \]

where \(V'\) is the velocity of motion of the chromatographic zone, \(\mathrm{cm/sec}\).

After certain transformations, analogous to those described in the preceding section, we obtain the differential equation describing this process:

\[ B\cdot\frac{d^2\mu}{dx^2}-M\cdot\frac{d\mu}{dx}=R\cdot\frac{d\mu}{dt}, \tag{28} \]

where

\[ B=v_{\mathrm{cp}}^2\frac{\tau_2'}{K}(1+\varepsilon_1)+U^2\cdot\frac{\tau_2'}{K}+D\gamma\alpha'_2, \tag{29} \]

\[ M=v_{\mathrm{cp}}(\alpha_1\alpha'_2-1)+U(\alpha'_2-1), \tag{30} \]

\[ R=\gamma\alpha'_2+\gamma K'=\gamma(\alpha'_2+K). \tag{31} \]

Here \(\tau\) is the retardation time, sec; \(\alpha_1\) is the separation coefficient due to the difference in ionic mobilities; \(\alpha'_2\) is the separation coefficient due to redistribution of ions between solvents.

The general solution of differential equation (28) is written in the form:

\[ \mu(x,t)=\mu_0\exp(2\eta x)-\mu_0\exp(\eta L_{\mathrm{eff}})\frac{\sin\beta x}{\sin\beta L_{\mathrm{eff}}}\exp(\eta x)\cdot \exp\left[-\left(\frac{B\Theta^2}{L_{\mathrm{eff}}^2}+\frac{M^2}{4B}\right)\frac{t}{R}\right], \tag{32} \]

where

\[ \eta=\frac{M}{2B};\qquad \beta=\frac{\Theta}{\eta L_{\mathrm{eff}}}, \]

\(\Theta\) is the root of the transcendental equation,

\[ \tan\Theta=\frac{\Theta}{\eta L_{\mathrm{eff}}}. \tag{33} \]

Using formula (32), one can estimate the time required for establishment of the stationary state, taking into account the considerations expressed in the preceding paragraph. For the given process we shall have

\[ T=\frac{10L_{\mathrm{eff}}^2\cdot B\cdot R}{4\Theta_1^2B^2+M^2L_{\mathrm{eff}}^2}. \tag{34} \]

The height equivalent to a theoretical plate is determined from the formula: HETP \(=\dfrac{\varepsilon B}{M}\), where \(\varepsilon=\varepsilon_1+\varepsilon_2;\ \varepsilon_2^{1}=a_2^{1}-1\).
\[ \tag{35} \]

The final concentration of the light isotope at the sampling point is

\[ c_K=c_0\exp\left(\frac{ML_{\mathrm{eff}}}{B}\right). \tag{36} \]

Analysis of the expressions obtained shows that the efficiency of separation by this method depends on the velocities of the flows \(v_{\mathrm{av}}\) and \(U\), on the single-stage separation coefficients \(\alpha_1\) and \(\alpha_2\), and also on the distribution coefficient \(K\). Moreover, the separation efficiency in the given process is maximal at values of the velocities \(v_{\mathrm{av}}\) and \(U\) of less than \(1\cdot10^{-3}\) cm/sec and decreases as \(K\) decreases.

In the work, a comparison was made of calculated and experimental data for the process of two-dimensional distribution electrochromatography in the separation of isotopic ions of light alkali metals in a mixture of \(n\)-butyl alcohol with water, with the following initial values: \(v_{\mathrm{av}}=0.7\cdot10^{-3}\) cm/sec; \(U=0.2\cdot10^{-3}\) cm/sec; \(L_{\mathrm{eff}}=23\) cm; \(\tau=50\) sec; \(D=10^{-5}\) cm\(^2\)/sec; \(\gamma=0.2\); \(\varepsilon_1=1.1\cdot10^{-2}\); \(\alpha_1=1.014\); \(\alpha_2=1.018\); \(K=0.028\). It turned out that the time for establishment of the stationary state, calculated by formula (34), is 40 h, whereas the time for establishment of the stationary state obtained from the experiment proved to be 43 h. The HETP calculated by formula (35) was 1.14 cm, and the HETP obtained from the experiment at the overall separation coefficient \(K_{\mathrm{ob}}=1.76\) was 1.16 cm. Thus, the calculated data agree well with the experimental data. The method considered can be used for separating ionic and isotopic mixtures.

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