INTRODUCTION
The objective of this study was to investigate the electrotransport of molybdenum in the multiionic system NaOH/Na_{2}CO_{3}/Na_{2}MoO_{4}. In the first part we have observed the electrotransport of molybdenum and carbonate ions through a commercial electrodialysis anion exchange membrane. We used different CO^{=}_{3}/MoO^{=}_{4} ratios. In the second part the concentration and the potential profiles in aqueous interfacial layers were determined by a numerical resolution of the NernstPlanck electrodiffusion equation coupled with the Poisson equation.
At the time of the NernstPlanck and Poisson equations resolution several important problems are encountered: Transport in the regions where is nozero charge density, flux of coion equal to zero, transport of solvent and ionion interaction and their impact on various transport phenomena when we studied the transport in multiionic systems. For the participation of the natural convection, it seems that this phenomenon is less important with the anion exchange membrane. These membranes are known for rising water splitting process (Zabolotski et al., 1985); Transport problem treatment in multiionic system seems to be demanding regarding numerical techniques. Consequently several numerical methods for NernstPlanck solution and Poisson equation system have been reported previously. As mentioned by certain authors (Choi and Moon, 2002) there is no numerical procedure applied to the description of ionic transport through both membrane and unstirred layers that takes into consideration the mentioned problems. The resolutions of boundary problems are done with considerable mathematical difficulties, especially if the transport involved more then two sorts of ions (Lebedev, 1999). Other problems arise from the mathematical complexity of the coupled differential equation ruling transport. Many researchers have solved the governing equations using simplifying assumptions such as electroneutrality (ρ = 0), constant field condition (Lim, 2007). His goal is to develop a Finite Element Method (FEM) to solve the NPP equations (NernstPlanckPoisson).
In this present study we propose an interpretative model. This model is based
on the one hand on electrodiffusion equations and on the other hand on the fact
that concentration amount ratio of both species in the membrane equals the one
of the species it selves within the solution. This theory will be employed to
elucidate the independence or the interdependence of CO^{=}_{3}
ion and molybdenum. However this hypothesis have to be justified given that
this equality is actually usual for unstirred layers extremity. That’ why,
the concentration and the potential profiles in aqueous interfacial layers (multiionic
system NaOH/Na_{2}CO_{3}/Na_{2}MoO_{4}) were
determined by a numerical resolution of the NernstPlanck electrodiffusion equation
coupled with the Poisson equation for. We consider two cases, the constant and
variable electrical fields. The nonconformity of the electric field in the
unstirred layers incites to take into consideration the space charges in this
layer, that can’t be neglected. The density of space charges is proportional
to Concerning
the flux of coion we have used the radiotracers to underline an eventual leak.
Unidirectional ions flux values are calculated from sample radioactivity variation,
measured by spectrometry γ (Packard Auto Gamma 500C). The acquired results
display that the coion Na^{+} leak through the anions exchange membrane
is a negligible phenomenon.
The numerical resolution used requires the following steps: discretization, linearization by the Newton method then the resolution by a direct method.
This theme of research which concerns the purification of the carbonate leach solutions of the uranium ores by electrodialysis process has been investigated since 1994 in the laboratory of analysis (Nuclear Research Center of Draria.
The first results have been presented during the international congress on membranes and membrane process (Lounis, 2002).
MATERIALS AND METHODS
Laboratory electrodialysis cells, as well as the experiment procedure and titration
method of OH^{–}, CO^{=}_{3} and HCO^{=}_{3}
have been previously described (Lounis, 1997). The effective area of each membrane
and each electrode is 10 cm^{2}. An ASTI pump provides the solution
dilution flow; its initial volume is 100 mL. The current density is 20 mA cm^{–2}.
The cell, schematically represented in Fig. 1, is composed
of five compartments, that are separated by ion exchange membranes. In the central
compartment P, a solution containing OH^{–}, CO^{=}_{3}
and MoO^{=}_{4} anions was circulating.
The membranes used are Selemion AMV (anion exchange membrane) and CMV (cation
exchange membrane) produced by Asahi Glass.

Fig. 1: 
Electrodialysis cell schematic 
Sampling is made in the central
and anodic compartment at regular intervals of time. Molybdenum is titrated
by spectrophotometric method using thiocyanate (Gordon and Parker, 1983).
RESULTS AND DISCUSSION
On Fig. 2 are represented the number variations of meq of
OH^{–} and CO^{=}_{3} ions transferred from the
diluate of which initial compositions are NaOH 0,125 eq L^{–}^{1}
and Na_{2}CO_{3} 0,13 eq L^{–}^{1} to the
anodic concentrate. From the obtained straight lines, we can deduce that during
the two hours the flux represent 88% of the total current that is carried by
these two ions. The flux of electrical charges carried by OH^{–}
ions is almost two times higher than the one carried by CO^{=}_{3}
ions.
It results that in the central compartment, after two hours of ED, the concentration of OH^{–} ions is 10 times lower than that of CO^{=}_{3} while at initial time the concentration of these two ions was practically identical.
The Fig. 3 demonstrates that the transfer of hydroxyl ions
(OH^{–}) is not modified by the presence of molybdenum ions. However,
the electrotransport of carbonate ions is altered. During 90 min of ED we have
obtained similar result with the former case, the concentration of OH^{–}
ions is 10 times smaller than that of CO^{=}_{3}. The percentage
of ions recovery are 72, 33 and 13% for OH^{–}, MoO^{=}_{4}
and CO^{=}_{3}, respectively.
These results allow us to claim the following conclusion: The molybdenum is
present in alkaline solution as the molybdate ion (MoO^{=}_{4})
form, in case of the existence of molybdyl ion MoO_{2}(CO_{3})^{4}_{3}
we should observe a higher transfer of CO^{=}_{3} ions. This
result is similar to that of Pascal (1959). The action of an excess hard base
on the molybdenum gives in the solution a molybdate ion when the ratio of MOH/MoO_{3}
>2, in our case this ratio is as high as 6.

Fig. 2: 
Variation vs. time of the OH^{–}, CO^{=}_{3}
ions leaving the central compartment 

Fig. 3: 
Variation vs. time of the OH^{–}, CO^{=}_{3}
and MoO^{=}_{4} ions leaving the central compartment 
In order to confirm this result we have studied the electrotransport of molybdenum
and carbonate ions with different [CO^{=}_{3}]/[MoO^{=}_{4}]
ratio. The ratios are 1.16, 3.5, 7.33, 10, 14.5 and 18.29. The Fig.
49 assert these results.
The following assumptions are made to analyze the obtained data:
• 
Electro neutrality holds in all part of the membranesolution
system. 
• 
The two layers on both sides of the solutionmembrane interface are in
quasi equilibrium data. 
• 
The diffusion term is neglected. 
The NernstPlanck electrodiffusion equation is

Fig. 4: 
Variation vs. time of the CO^{=}_{3} and MoO^{=}_{4}
ions leaving the central compartment [CO^{=}_{3}]/[MoO^{=}_{4}]
= 1.16 

Fig. 5: 
Variation vs. time of the CO^{=}_{3} and MoO^{=}_{4}
ions leaving the central compartment [CO^{=}_{3}]/[MoO^{=}_{4}]
= 3.5 

Fig. 6: 
Variation vs. time of the CO^{=}_{3} and
MoO^{=}_{4} ions leaving the central compartment [CO^{=}_{3}]/[MoO^{=}_{4}]
= 7.19 

Fig. 7: 
Variation vs. time of the CO^{=}_{3} and MoO^{=}_{4}
ions leaving the central compartment [CO^{=}_{3}]/[MoO^{=}_{4}]
= 10 

Fig. 8: 
Variation vs. time of the CO^{=}_{3}, OH^{
}and MoO^{=}_{4} ions leaving the central compartment
[CO^{=}_{3}]/[MoO^{=}_{4}] = 14.5 

Fig. 9: 
Variation vs. time of the CO^{=}_{3}, OH^{–}
and MoO^{=}_{4} g.ions leaving the central compartment.
[CO^{=}_{3}]/[MoO^{=}_{4}] = 18.3 
J_{i} 
= 
Flux of I ion across the exchange membrane (g eq cm^{–}^{2}
s^{–}^{1}) 
D_{i} 
= 
Diffusion coefficient of ion (cm^{2} s^{–}^{1}) 
C_{i} 
= 
Concentration of i ion (g eq L^{–}^{1}) 
F 
= 
The Faraday constant (c) 
R 
= 
The gas constant (J mole K^{–}^{1}) 
T 
= 
The absolute temperature (K) 
dφ/dx 
= 
The gradient of electrical potentiel 
Under this assumptions the NernstPlanck electrodiffusion equation becomes:
The Nernst Einstein relation relates the mobility of the species i to its diffusion
coefficient:
Equation 1 and 2 give
From the Eq. (3) the expressions of are:
The ratio of these fluxes gives
In which the concentration ratio inside the membrane can be replaced by the
concentration ratio in the bulk.
Thus
On the Fig. 10 the value of were plotted as a function of . The values show that the transport theoretical model used is satisfied at
different concentration ratios.

Fig. 10: 
Variation vs. 
A proportional variation is obtained as might
be expected from Eq. 10.
Concentration and potential profiles: Since necessary factors to resolve the NernstPlanck equation within the membrane are being unknown, we will only study the ions electrotransport in the unstirred layers (NernstPlanck film) close to the membrane.
The Nernst Planck equation is resolved keeping in mind, on one hand that the electrical field is constant and on the other hand using the Poisson equation, which explains the distribution of electrical charges.
For the NaOH/Na_{2}CO_{3}/Na_{2}MoO_{4} system,
the NernstPlanck electrodiffusion equation in general feature gives:
(It can be checked that
The relation between potential and distribution of electrical charges is ruled
by the Poisson equation.
Where, ε_{O} is the permittivity of vacuum, ε_{r}
the relative permittivity, Φ the electrical potential and ρ the local
charge density defined as:
N(x) is the concentration of fixed charges
in a medium with no fixed charge distribution
Reduced variables are defined to resolve the different equations.
Where, N_{O} is the normality of the bulk solution expressed in mol cm^{–}^{3}.
The membrane solution interfaces are given in Fig. 11.
With the reduced variables the above system of equation becomes:
Where the unknown factors are OH, CO, MO, NA and φ and the boundary conditions are:

Fig. 11: 
Concentraton profiles in unstirred layers (schematic) 
and
δ being constant the same for α, β and γ.
Case where the electrical field is constant: The concentration profiles
are obtained by resolving the system of equations differential in first order
with constant coefficients, which solutions are:
With θ =
Replacing the expressions OH; CO; MO and NA in the electroneutrality
Eq. 17 we get a third degree equation in which the unknown θ appears
both in the exponential term and in the quotient.
This equation presents a nonlinearity. It is solved by the dichotomy method
(Press et al., 1992) for a given value of ξ. Then, we obtain the
concentration profiles. The potential profiles are obtained from:
The following parameters were used for the calculations:
Case where the electrical field is not constant: The numerical resolution
used requires the following steps:
• 
Discretization with the finite difference method 
• 
Linearization by the Newton method 
• 
Equation system is solved using a direct method 
We note a slight discrepancy between the profiles obtained from the NernstPlanckPoisson equation solution and those obtained with the resolution of equation using the constant electrical field hypothesis.
The result accords with those proposed by Taky (1991) and Chapotot (1994) where the difference appears from a current density over or equal to 20 mA cm^{–}^{2}.
In the anodic interfacial layer the concentration and potential profiles are
given in the Fig. 1215 for the studied
respectively with the AMV Selemion membrane.
We observe that in the present conditions (current density 20 mA cm^{–}^{2} and complete concentration equal 0.265 mol L^{–}^{1}) the concentration profiles obtained from numerical resolution differ about 3 to 6% from to the values obtained by the analytical resolution method assuming the constant electrical field hypothesis for OH^{},CO^{=}_{3} ions, respectively. However, the potential profiles show that the constant electrical field hypothesis is strictly valid.
The concentration of OH^{–} in the diffusion layer decreases near the interface.
The concentration diminution of CO^{=}_{3} and MoO^{=}_{4}
ions near the membrane surface shows that those ions transfer across the membrane
in spite of their electrotransport mobility is less fast then that of the hydroxyl
ions.

Fig. 12: 
Concentration profiles for the case N°1 [CO^{=}_{3}]/[MoO^{=}_{4}]
= 14.5, ■ ∞ Constant electrical field hypothesis. +Δ+ NernstPlanckPoisson
resolution 

Fig. 13: 
Potentiel profiles for the case N°1 [CO^{=}_{3}]/[MoO^{=}_{4}]
= 14.5 

Fig. 14: 
Concentration profiles for the case N°2 [CO^{=}_{3}]/[MoO^{=}_{4}]
= 18.3, ■ ∞ Constant electrical field hypothesis. +Δ+ NernstPlanckPoisson
resolution 

Fig. 15: 
Potentiel profiles for the case N°2 [CO^{=}_{3}]/[MoO^{=}_{4}]
= 18.3 
The values of the transfer show that 97, 25 and 48% for the OH^{},CO^{=}_{3} and moO^{=}_{4} ions, respectively are reached.
We also note that when the ratio
ions in the solution increases the CO^{=}_{3} ions it contributes
more to current transport and the concentration of CO^{=}_{3}
decreases in the interface layer. This result is also confirmed with the percentage
recovery of CO^{=}_{3} ions which is 51 and 60.05%, respectively
for both cases
=
14.5 and 18.3. 
CONCLUSION
• 
The elimination of carbonate and molybdenum ions from aqueous
solution can be easily performed by electrodialysis. 
• 
The electrotransfer of molybdenum ion shows that molybdate ion is present
in alkaline solution. 
• 
The high values of the molybdenum flux obtained in this study show that
electrodialysis can be used as a separation technique for molybdenum extraction. 
• 
The results reported here indicate that the best value of the ratio CO^{=}_{3}
for molybdenum extraction through an anion exchange membrane is 1.16. 
• 
The concentration profiles obtained from numerical resolution differ in
3 and 6% from to the values obtained by the analytical resolution method
assuming the constant electrical field hypothesis for OH^{–},
CO^{=}_{3}, respectively. Meanwhile the potential profiles
show that the constant electrical field hypothesis is strictly valid. 