Uranyl-citrate Speciation Diagram in Aqueous Solutions

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    Uranyl-citrate Speciation Diagram in AqueousSolutions.

    Theoretical and Capillary Electrophoresis Studies at 25CSondes Boughammoura, Jalel Mhalla *

    Laboratoire dElectrochimie des solutions, Facult des sciences, Universit de Monastir 5000 Monastir, Tunisia

    * [email protected]

    Abstract- In this study, we develop new analytical and numericalspeciation calculations corresponding to all kinds of combinations of complexation constants and hydrolysis constantsgiven by more or less reliable literature data, and associated tofifteen probable uranyl-citrate species: M pL n, (with: M (UO 2)

    2+ ,L (HCit) 3- and p = 1, 2; n = 1, 2, 3), M pL nH m, (p = 1, 2; n = 1, 2;m = 1, 2, 4), M p(OH) q , (p = 1- 4; q = 1-7), M pLnH m(OH) q, (p = 1, 2;n = 1, 2; m = 1, 2; q = 1, 2) as well as to the improbable:[(UO 2)H 2(HCit) 2]2- specie (reported by only one reference), inorder to analyse deeply their impact on the speciation diagram of U(VI) in citrate aqueous solutions at 25C, in a wide range of pHand for citrate to uranium ratios R c = U(VI)/Citrate equal to 0.02,1 and 2. The main result of this work is that, although theemployed combinations present some significant dispersions,they leads to the same result: the hydrolyzed monomer[(UO 2)H(HCit)(OH)]

    -, the hydrolyzed dimer[(UO 2)2H 2(HCit) 2(OH) 2]

    2- and the monomer (UO 2)(HCit)]- are

    largely predominant in the pH range of our interest (2 pH 5.5),by comparison to the percentages of the other speciespresumably existing (except the [(UO 2)H 2(HCit) 2]

    2- specie). Thisinvestigation is therefore a supplementary proof of the reliabilityof the processes of protonation and hydrolysis of both themonomer and the dimer: [(UO 2)(HCit)]

    -, [(UO 2)2(HCit) 2]2-

    detected via the experimental electropherograms obtained by thecapillary electrophoresis method which was used because of itscapacity to separate the different hydrolysis products of theanionic U(VI)-citrate complexes. Consequently, it is thus possibleto underline that the hydrolysis of UO 2

    2+ which generally occursat relatively elevated pH is facilitated at lower pH by the citratecomplexation followed by the protonation.

    Keywords-Uranium Citrate Speciation Electrophoresis Analytical Numerical

    I INTRODUCTION The ability of uranium to form strong stable complexes

    with organic compounds has been extensively utilized for

    cleaning up uranium contamination in soils and in the nuclear industry. Citric acid is a common constituent which formsstrong water soluble complexes with U(VI) and U(IV) over awide range of pH. The U(VI) citrate complexes have already

    been investigated in several experimental and modellingstudies [1,2,5,6,10]. Accordingly, it was concluded that citratehas a strong affinity for uranium. However, it was also shownthat this affinity varies with the experimental method, theionic strength, the pH and the stoichiometry of the (M U(VI)

    UO 22+ )/(L ligand HCit 3-) ratio. This dependence could

    partially explain the fact that the reported data by the literatureconcerning the different stability constants pn for M pLncomplexes scatter markedly. In fact, the M pLn complexescould exist under different protonation and hydrolysis states:M pLnHm and M pLnHm(OH) q. However, most of the analyticaltechniques cannot differentiate between these different states.

    From recent analysis of all the previous works [8-10], alimited number of U (VI) citrate complexes have beenselected ([(UO 2)(HCit)]

    -, [(UO 2)2(HCit) 2]2-

    , [(UO 2)H(HCit)])and their corresponding stability constants pnm re-determined.More recently [16], we have underlined the advantage of thecapillary electrophoresis technique over most of the speciationmethods previously used because it should give the possibilityto specify all the anionic M pLnHm(OH) q entities, by separatingthem on the basis of the differences in their electrophoreticmobility even at low concentration closer to the level found innature. Accordingly, we have assigned some three peaksfound in the absorbance spectra of our capillaryelectrophoresis experiments for pH 5, to the following threespecies: the U(VI)-citrate monomer, the hydrolysis product of this monomer complex: [(UO 2)H(HCit)(OH)]

    - and the dimer of the hydrolysis product: [(UO 2)2H2(HCit) 2(OH) 2]

    2-. Thelatter two species were postulated in reference [2] andtherefore play a crucial role in our speciation calculationsdespite the controversy concerning them from a chemical

    point of view. In particular, we have signalled for pH 3 the

    occurrence of an inverted tendency accompanied by a strongmodification of the calculated speciation diagrams when wetake into account these hydrolysed complexes via their corresponding constants: 1111 and 2222 . Indeed, the use of these additional constants leads to a completely differentU(VI) distribution so that 60-70% of uranium is present in theform of hydrolyzed monomer rather than [(UO 2)2(HCit) 2]

    2- and 20-30% in the form of hydrolyzed dimmer rather than[(UO 2)(HCit)]

    -. Notice that the speciation diagrams given in

    our previous work paper (in the absence and in the presenceof the hydrolyzed monomer and dimmer) are calculated on the

    basis of two constants: 11 and 22 selected among some published sets of values currently more or less established.However, in spite of the detection of the three peaks abovementioned, this inverted tendency is not quantitativelycompletely in agreement with the electropherograms resultsand we have explained this difference in assuming that the

    protonation and hydrolysis effects are non-reproduciblekinetic processes.

    Instead, we will expose in the present work, newspeciation calculations corresponding to all kinds of combinations of complexation constants given by more or lessreliable literature data, and associated to fifteen probablespecies: M pLn, M pLnHm, M p(OH) q, M pLnHm(OH) q, as well asto the improbable : [(UO 2)H2(HCit) 2]

    2- specie (reported byonly one reference), in order to analyse deeply their impact onthe speciation diagram of U(VI) in citrate aqueous solutions.In the first step of this study we will therefore investigatetheoretically (analytically and numerically) the different

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    speciation diagrams obtained at 25C, in a wide range of pHand for different citrate to uranium ratios, following the use

    partly or completely of the uranyl-citrate complexes anduranyl- hydroxides mentioned above, and in assuming that thedifferent reactions of complexation, protonation andhydrolysis have reached their complete chemical equilibrium.In the second step, we will compare the similarities and thedifferences between them in the perspective of verifying thefact that: for the pH range of our interest (2 pH 5.5), despitetheir dispersion, the use of the several sets of values (more or less reliable), of the different complexation constants given bythe literature, confirms the large predominance of thehydrolyzed monomer: [(UO 2)H(HCit)(OH)]

    - and of thehydrolyzed dimer: [(UO 2)2H2(HCit) 2(OH) 2]

    2- by comparisonto the percentages of the other species presumably existing(except the [(UO 2)H2(HCit) 2]

    2- specie). This investigationcould be therefore considered as a supplementary proof of our

    previous interpretation of the experimental electropherogramsobtained by the capillary electrophoresis method (in particular the reliability of the processes of protonation and hydrolysisof both the monomer and the dimer: [(UO 2)(HCit)]

    -,

    [(UO 2)2(HCit) 2]2-

    ), and also as a test of the validity of thedifferent literature data as well as of the assumption of thechemical equilibrium.

    II METHOD OF CALCULATION

    A. DefinitionsWe will designate in the following definitions by [X i] the

    molar concentration of specie i and by Xi the activity

    coefficient of X i in the Henry reference in water at 25Cwhich is related to the ionic strength I according to thegeneralized Debye equation [ ] :

    log( Xi) = -0.5115(Z i)2I1/2/(1 + 1.5I 1/2) + jeij(C j) (1)

    I = 0.5 j(Z j)2C j (2)

    where Z j and C j are respectively the valence and the molar concentration of the j species, and e ij is a specific interactioncoefficient between species i and j.

    1) ComplexationWe define the complexation constant pn and the apparent

    complexation constant pn corresponding to the followingcomplexation equilibrium respectively by:

    pM + nL M pLn : pn = [M pLn ][L ]-n[M] -p (3a)

    pn = ( MpLn )( L)-n( M)

    -p pn (3b)

    2) ProtonationWe define the protonation constant pnm and the apparent

    protonation constant pnm corresponding to the following protonation equilibrium respectively by:

    M pLn + mH+ M pLnHm : pnm = [M pLnHm][M pLn ]

    -1[H+]-m

    (4a)

    ' pnm = ( MpLnHm )( MpLn )-1( H

    +)-m pnm (4b

    3) HydrolysisWe define the hydrolysis constant pnmq and the apparent

    hydrolysis constant pnmq corresponding to the followinghydrolysis equilibrium respectively by:

    M pLnHm + qH 2O M pLnHm(OH) q + qH+ : pnmq =

    [M pLnHm(OH) q] [H+]q[M pLnHm]

    -1 (5a)

    ' pnmq = ( MpLnHm(OH)q )( H+)q( MpLnHm )

    -1 pnmq (5b)

    a) Peculiar Case (n = 0; m = 0)We define the hydrolysis constant KH pq of the metallic

    ion M and the apparent hydrolysis constant KH pqcorresponding to the following hydrolysis equilibrium

    respectively by: pM + qOH - M p(OH) q : KH pq = [M p(OH) q][OH

    -]-q[M] -p (6a)

    KH pq = ( Mp(OH)q )( OH-)-q( M)

    -pKH pq (6b)

    4) AcidityWe define the acidity constant K a l and the apparent

    acidity constant K a l corresponding to the followingequilibrium respectively by:

    L + lH + HlL : K a

    l = [H lL][L]-1[H+]-l (7a)

    K a l = ( HlL)( L)-1( H

    +)-lK a l (7b )

    III EXPRESSIONS OF THE SPECIES CONCENTRATIONS IN TERMSOF [M], [L] AND [H+]

    A. General ExpressionsThe expressions of the different species concentrations in

    terms of [M] and [L] concentrations are given by:

    [M pLn] = pn [M] p[L ] n (8)

    [M pLnHm] = pnm pn [M] p[L ] n [H+]m (9)

    [M pLnHm(OH) q] = pnmq pnm pn [M] p[L ] n [H+]m[H +]-q (10)

    [M p(OH) q] = (KH pq) [OH-]q[M] p (11)

    [H lL] = (K a l)[L][H +]l (12)

    B. Equations of ConservationIf we define C M and C L as, respectively, the total

    concentrations of M and L, thus:

    C M = [M] + p p[M]

    p { n ( pn + m pnm pn [H+]m + m q

    pnmq pnm pn [H+]m[H+]-q )[L ] n}

    + p p[M] p{ q (KH pq) (K W)

    q[H+]-q} (13)

    CL = p [M] p{ n n( pn + m pnm pn [H

    +]m + m q pnmq pnm pn [H

    +]m[H+]-q )[L ] n } +{1+ l (K a

    l) H+]l }[L] (14)

    KW = [H+][OH-] is the apparent ionization constant of water.

    Now, if we note: x [M] and y [L], we will obtain thefollowing equations:

    C M = x + p px p nAnpy

    n + p px p E+ {1+ l (K

    al) H

    +]l }[L] (15)

    C L = p x p n nAnpy

    n + Dy (16)

    With : A np = ( pn + m pnm pn [H+]m + m q pnmq pnm pn

    [H+]m[H+]-q ) (17)

    E p = [ q (KH pq) (K W)q[H+]-q] (18)

    D = {1+ l (K a

    l) [H+]l} (19)

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    Note that the parameters A np, E p and D depend on thedifferent apparent equilibrium constants and are function onthe ionic strength I = 0.5 i(Z i)

    2C i and pH. We proceedtherefore to a recurrent self consistent method: As a firstapproximation we equal the apparent constants to their corresponding thermodynamic constants, we calculatetherefore for a given pH, the parameters A np, E p and Daccording Eqs. (17-19) in order to obtain a system of twoequations with two variables: x and y according Eqs.(15-16).The resolution of this system for a given total concentrationsC M and C L allows us to calculate the different concentrationsof all the species present in solution: [M pLn],

    [M pLnHm], [M pLnHm(OH) q],[M p(OH) q] and [H lL] according Eqs.(8-12)and therefore all the activity coefficients via the ionic strengthI. In the second step we recalculate the new apparentconstants via Eqs. (3b -7b) and we repeat the procedure untilconvergence.

    The resolution of the algebraic system at each step isachieved using a basic numerical program involving nsuccessive buckles and verifying the convergenceconditions:[(x n - xn-1 ) /x n] 10

    -5 and [(y n - yn-1 )/x n] 10-5. This

    method allows us to establish for a given C M and C L, thecorresponding speciation diagram giving the variation of thefraction in % of the different species with the pH.

    IV THEORETICAL URANYL -CITRATE SPECIATION DIAGRAM INAQUEOUS SOLUTIONS

    Now, we will apply this general method in the case of:

    M (UO 2)2+, L (HCit) 3-, in order to obtain the speciation

    diagram of Uranyl-Citrate complexes in aqueous solutions.For this, we will use different literature data for thecalculation of pnmq , pnm , pn , (K pHq) and (K

    al) constants.

    A. The Most Probable Uranyl-Citrate Complexes in AqueousSolutionsThe most probable uranyl species present in citrate

    aqueous solutions, according to the literature [2,3,7,10], arecollected in Table 1.

    B. Calculation of pnmq, pnm and pn Constants from Literature DataIn order to calculate pnmq and pnm constants, we

    proceeded as follows:

    According to the recent and less recent literature data[2,3,7,10], we can calculate the four protonation constants

    (111 ), ( 112 ) ( 122 ) , ( 224 ) and the two generalized constants(1111 ) and ( 2222 ), in terms of the intermediate constants (K I),(K 2I), (K 22I), (K 224I), (K II) and (K III) corresponding to theequilibria given in Table 2.

    TABLE 1 THE DIFFERENT URANYL SPECIES FORMED IN CI TRATE AQUEOUS SOLUTIONS

    Complexes: pn Protonated complexes: pnm Uranylhydroxides: KH pq

    General hydrolyzed complexes: pnmq

    [(UO 2)(HCit)] - [(UO 2)H(HCit)] [(UO 2)(OH)] + [(UO 2)H(HCit)(OH)] -

    [(UO 2)(HCit) 2]4- [(UO 2)H 2(HCit)] + [(UO 2)(OH) 2] [(UO 2)2H2(HCit) 2(OH) 2]2-

    [(UO 2)2(HCit) 2]2- [(UO 2)H2(HCit) 2]2- [(UO 2)(OH) 3]-

    [(UO 2)2(HCit) 3]5- [(UO 2)2H2(HCit) 2] [(UO 2)2(OH)] 3+

    [(UO 2)2H4(HCit) 2]2+ [(UO 2)2(OH) 2]2+

    [(UO 2)3(OH) 4]2+

    [(UO 2)3(OH) 5]+

    [(UO 2)3(OH) 7]-

    [(UO 2)4(OH) 7]+

    TABLE 2 HYDROLYSIS AND COMPLEXATION CONSTANTS OF URANIUM AT 25

    Equilibrium log(K) : Ref [2] log(K) : Ref [10]

    (UO 2)2+ + H(HCit) 2- [(UO 2)H(HCit)] log(K I) = 6.0 (I = 0.136) log(K I) = 4.23 (I = 0.1)

    (UO 2)2+ + H 2(HCit) - [(UO 2)H2(HCit)] + log(K 2I) = 2.79 (I = 0.1)

    (UO 2)2+ + 2[H(HCit)] 2- [(UO 2)H2(HCit) 2]2- log(K 22 I) = 11.2 (I = ?)

    2(UO 2)2+ + 2[H 2(HCit)] - [(UO 2)2H4(HCit) 2]2+ log(K 224 I) = 8.9 (I = 1)

    (UO 2)2+ + [H(HCit)] 2- + H 2O

    [(UO 2)H(HCit)(OH)] - + H +

    log(K II) = 2.84 (I = 0.136)

    2(UO 2)2+ + 2[H(HCit)] 2- + 2H 2O [(UO 2)2H2(HCit) 2(OH) 2]2- +2H +

    log(K III) = 7.68 (I = 0.136)and 9.04 (I = 0.)

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    The different expressions of the pnm and pnmq constantsare summarized in Table 3.

    TABLE 3 EXPRESSIONS OF PNMQ AND PNM CONSTAN TS OF URANYL -CITRATE COMPLEXES

    111 1111 112

    (K a 1)(K I) (11 )-1 (K II)(K I)-1 (K a 2)(K 2I)(11 )-1

    122 2222222 224

    (K a 1) (K 22 I) (12)- (K III) (K a 1) (22)- (K a 2) (K 224 I) (22)-

    C. Equations of Conservation in the Case of the UraniumComplexation with Citric Acid According to the data given in Table 1, the most probable

    values for p, n, m and q are: p = 1, 2; n = 1,2 ; m = 1,4; and q= 1,2 ; the corresponding parameters A np and E p are thus:

    A11 A = ( 11 + 111 11 [H+] + 112 11 [H

    +]2 + 1111 111 11 )

    (20)

    A21 = ( 12 + 122 12 [H+]2 ) (21)

    A12 = 21 (is neglected) (22)A22 C = ( 22 + 222 22 [H

    +]2 + 22422 [H+]4 + 2222 222 22)

    (23)

    E1 E = [3

    q (KH 1q) (K W)q[H+]-q] (24)

    E2 EE = [2

    q (KH 2q) (K W)q[H+]-q] (25)

    Now, in our case: x [UO 22+] and y [HCit 3-], therefore

    Eqs. (13-17) lead to:

    C L = x [Ay + 2A 21y2] + 2x 2Cy2 + Dy (26)

    C M = x [1 + Ay + A 21y2

    + E] + 2x2[EE + Cy

    2] (27)

    C = (C L - C M) = Dy - x[1 - A 21y2 + E] - 2x 2EE (28)

    The variables x and y which are the solutions of Eqs. (26 27), depend on the pH of the medium via the expressions of the parameters A, A 21, C, D and E, EE. On the other hand, for a given total concentrations C M and C L and for a given pH,the knowledge of x and y allows us to calculate the fractionin % of the different species in solution and, therefore, toobtain the corresponding diagram of speciation:

    %[M pLn] = 100.(C M )-1 pn [x]

    p[y ]n (29)

    %[M pLnHm] = 100.(C M )-1 pnm pn [x]

    p[y ] n (30)

    [H+]m %[M pLnHm(OH) q] = 100.(C M )-1 pnmq pnm pn

    [x] p[y ] n [H+]m[H+]-q (31)

    %[M p(OH) q] = 100.(C M )-1(KH pq) [OH

    -]q[x] p (32)

    V 5. RESOLUTION OF THE EQUATIONS OF CONSERVATION INTHE CASE OF THE URANIUM COMPLEXATION WITH CITRIC

    ACID

    A. PrincipleFor a given total concentrations C M and C L of M and L,

    the resolution of the system of the conservation equationsgiven above depends on five hydrolysis constants KH 1q ,three acidity constants K a l and on ten complexation constants (see Table 4 and Table 5). According to the literature, these

    last constants are regrouped in seven sets which aresummed up in Table 4. Indeed, this table lists different constants given in the literature [2,3,5,10], or calculatedaccording to t heir corresponding expressions given in Table3. In fact, the examination of this table shows that in realitywe have only two different independent sets: Set III [2,3,5]and Set V [2,10]. Indeed, the five other sets are some

    particular cases of these two sets obtained by neglecting some[M pLnHm(OH) q] entities (i.e. by taking their corresponding pnmq = 0).

    Note also that, in order to calculate the 122 constant givenin Set V , we used the K 22I constant given in reference [10]corresponding to the equilibrium: (UO 2)

    2+ + 2[H(HCit)] 2- [(UO 2)H2(HCit) 2]

    2- (See Tables 2 and 3). This K 22I constantwas in fact rejected by the authors of the cited review [10] for the following reasons: The original reference doesn't mentionexplicitly the experimental conditions used for thedetermination of this K 22I constant. Moreover, the complex[(UO 2)H2(HCit) 2]

    2- has not been reported by any other researcher.

    On the other hand, Table 5 gives the first five hydrolysisconstants KH 1q [10] and the three acidity constants K

    al

    defined in paragraph 2.1 [7, 10]. All these constants arecommon to the different sets.

    The resolution of the above system involving Eqs. (20 - 32)is generally achieved by the numerical methods explained in

    paragraph 3.2. Details are given in Appendix I. However, thissystem could be resolved analytically if we neglect theformation of the following species: [(UO 2)(HCit) 2]

    4-;[(UO 2)H2(HCit) 2]

    2-; [(UO 2)2(OH)]3+ and [(UO 2)2(OH) 2]

    2+ andtherefore their corresponding constants: (12), ( 122 ), KH 21 andKH 22 . In this approximation, both analytical and numericalcalculations lead to identical speciation diagrams. The mainadvantage of the analytical method consists in the fact that wecan resolve the system of Eqs. (20-28) by a Microsoft OfficeExcel program in order to calculate x and y for different pHand therefore to obtain automatically the graphs of thecorresponding speciation diagram: i.e. the % [species] infunction of the pH. Details of the analytical resolution aregiven in Appendix II. It is important to mention that bothanalytical and numerical calculations must verify thefollowing mass conservation conditions for (UO 2) and (HCit):

    p n m q p(%[M pLnHm(OH) q]) = 100 (31)

    p n m q n(%[M pLnHm(OH) q]) = 100 (32)

    However, analytical and numerical calculations show that,for some ranges of pH, and even if we use a double numerical

    precision, these conditions are not verified perfectly. Thereason for that is the very high sensitivity of the calculationsto the parameters used in these pH ranges. This explains whythe speciation diagrams presented below are given for different domains of pH between 0 and 14. Indeed, for eachfigure, we have chosen the best range of pH for which the Eq.(26) and Eq. (27) are verified better than 1%.

    This high sensitivity of the calculations to some parameters incited the authors to use their own analyticalExcel program and their numerical Basic program, in order toallow a better control of the different stages of calculations.Indeed, this kind of control is not always obvious with theutilization of some usual specific software.

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    Note finally that ionic strength corrections can be ignoredin a first approximation. Indeed more precise numricalcalculations on the basis of the self consistent procedure led tothe following conclusion: considering the uncertainties on thedifferent constants used in Set III , Set IV and Set V , theimprovement of the results by the recurrent method is notalways significant. Besides, in our calculations andexperiments, the ionic strength I never exceeded 0.02M.

    TABLE 4 DIFFERENT SETS OF COMPLEXATION CONSTANTS OF URANIUM INAUEOUS SOLUTION AT 25 C

    Set

    Ia

    Set

    I b

    Set

    I

    Set

    II

    Set

    III

    Set

    IV

    Set

    V

    11 1.0

    10 7

    1.0

    10 7

    1.0

    10 7

    6.3

    10 8

    1.0

    10 7

    6.3

    10 8

    6.3 10

    111 0 7.94

    10 4

    7.94

    10 4

    21.37 7.94

    10 4

    21.37 21.37

    1111 0 3.16

    10 -4

    3.16

    10 -4

    4.07

    10 -2

    3.16

    10 -4

    4.07

    10 -2

    4.07

    10 -2

    112 0 0 1.26

    10 6

    1.99

    10 4

    1.26

    10 6

    1.99

    10 4

    1.99

    10 4

    12 0 0 0 0 1.0

    10 11

    1.0

    10 11

    1.0

    10 11

    122 0 0 0 0 0 0 1.0

    10 12 ?

    22 1.0

    10 18

    1.0

    10 18

    1.0

    10 18

    7.94

    10 18

    1.0

    10 18

    7.94

    10 18

    7.94

    10 18

    222 1 1 1 1 1 1 1

    2222 0 6.3

    10 2

    6.3

    10 2

    79.43 6.3

    10 2

    79.43 79.43

    224 0 0 3.1610 11

    3.8910 10

    3.1610 11

    3.8910 10

    3.8910 10

    TABLE 5 HYDROLYSIS CONSTANTS KH 1QAND ACIDITY CONSTANTS KA L AT25C

    KH 11 KH 12 KH 13 KH 21 KH 22

    3.98 10 8 1.99 10 16 5.01 10 21 1.99 10 11 2.51 10 22

    K a l K a 2 K a 3

    7.94 10 1.99 10 1.99 10

    B. Results in Particular CasesFor a given concentration ratio, R c = C M / C L, the

    analytical method allows us to study easily the sensitivity of the diagram to the different constants. We used threeconcentration ratios: R c = 0.025, R c = 1 and R c = 2. Results aresummarized in Figs.1 to 11. Each figure corresponds to agiven set of constants and to a given ratio R c.

    1) Results Relative to Set I Constants in AqueousSolution at 25C

    We have first used both analytical and numerical methodsto calculate the uranyl speciation diagram according to theSet I a data giv en in Table 4 (see Fig. 1). This calculationcorresponds to the simplest case for which we neglected both

    protonation and hydrolysis of M pLn entities. Indeed, Set Iadiffers from Set I by the fact that , in this case, all pnm and pnmq are ignored. We have also neglected the formation of

    species[(UO 2)2(OH)]3+,[(UO 2)2(OH) 2]

    2+, [(UO 2) (H Cit) 2]4-

    and [(UO 2)H2(HCit) 2]2- (i.e. KH 21 = KH 22 = 12 = 122 = 0).

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    2 3 4 5 6 7 8 9 10

    pH

    %

    % UO22+

    % B11

    % B22

    %UO2(OH)+

    %UO2(OH)2

    Fig. 1 Speciation diagram of UO 22+ (4.10 -4M)) in aqueous HCit 3- (2.10 -2M) at

    25C with "Set Ia" constants

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

    pH

    %

    %UO2

    %B11

    %B22

    %UO2(OH)

    %UO2(OH)2

    %UO2(OH)3

    %B111

    %B1111

    %B222

    %B2222

    Fig. 2 Speciation diagram of UO 22+ (5.10 -4M) in aqueous HCit 3- (2.10 -2M) at

    25C, with "Set Ib" constants and ( 112 = 224 = 0)

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    0 2 4 6 8 10 12 14

    pH

    %

    %UO22+

    %B11

    %B22

    %UO2(OH)

    %UO2(OH)2

    %UO2(OH)3

    %B111

    %B1111

    %B222

    %B2222

    %B112

    %B224

    Fig. 3 Speciation diagram of UO 22+ (5.10 -4M)) in aqueous HCit 3- (2.10 -2M) at

    25C, with "Set I" constants and ( 112 > 0, 224 > 0)

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

    pH

    %

    %UO2

    %B11

    %B22

    %UO2(OH)

    %UO2(OH)2

    %UO2(OH)3

    %B111

    %B1111

    %B222

    %B2222

    %B112

    %B224

    Fig. 4 Speciation diagram of UO 22+ (5.10 -4M) in aqueous HCit 3- (2.10 -2M) at

    25C, with "Set II" constants and ( 112 > 0, 224 > 0)

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    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

    pH

    %

    %UO2

    %B11

    %B22

    %UO2(OH)

    %UO2(OH)2

    %UO2(OH)3

    %B111

    %B1111

    %B222%B2222

    %B112

    %B224

    Fig. 5 Speciation diagram of UO 22+ (5.10 -3M) in aqueous HCit 3- (5.10 -4M) at25C, with "Set II" constants and ( 112 > 0, 224 > 0)

    In the second step, we replaced Set I a by Set I b inorder to take into account the protonation and the hydrolysisof the complexes [(UO 2)(HCit)]

    - and [(UO 2)2(HCit) 2]2-; we

    thus obtained the diagram reported in Fig. 2. Comparison between Fig. 1 and Fig. 2 shows that, in the pH range: 3 < pH

    < 9, the two diagrams are similar. In both figures we observetwo plateaus of which the levels are around 35% and 25%.The main difference consists in the fact that, in the case of Fig. 1 with Set I a, the y correspond respectively to[(UO 2)2(HCit) 2]

    2- (22) and [(UO 2)(HCit)]- (11) species,

    whereas in Fig. 2 with Set I b, they correspond respectivelyto the protonated and to the hydrolysed species:[(UO 2)2H2(HCit) 2(OH) 2]

    2- (2222 ) and [(UO 2)H(HCit)(OH)]-

    (1111 ). Besides, around pH 2, we observe in (Fig. 2) theapparition of a peak corresponding to a maximum of

    protonation of type [(UO 2)H(HCit)] ( 111 ).

    In the third step, we replaced Set I b by Set I in order totake into account the presence of the two species [MLH 2] and

    [M 2L2H4] ( 112 > 0; 224 > 0). The corresponding diagram for:C M = 5.0 10-4M and C L = 2.0 10

    -2 M (R c

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    In conclusion, we can say that, for R c=1, the species([(UO 2)(HCit)]

    -, [(UO 2)H(HCit)(OH)]- and

    [(UO 2)2H2(HCit) 2(OH) 2]2-) are observable by capillary

    electrophoresis, in the pH range: 3 pH 8 .

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

    pH

    %

    %UO2

    %B11

    %B22

    %UO2(OH)

    %UO2(OH)2

    %UO2(OH)3

    %B111

    %B1111

    %B222

    %B2222

    %B112

    %B224

    Fig. 6 Speciation diagram of UO 22+ (10 -3M) in aqueous HCit 3- (5.10 -4M) at

    25C, with "Set II" constants and ( 112 > 0, 224 > 0)

    0

    10

    20

    30

    40

    50

    60

    70

    8090

    100

    2 3 4 5 6 7 8 9 10 11pH

    %

    % UO22+

    % B11

    % B22

    %UO2(OH)+

    %UO2(OH)2

    %UO2(OH)3

    % B111

    % B1111

    % B222

    % B2222

    % B12

    % B112

    % B224

    %(UO2)2(OH)

    %(UO2)2(OH)2 Fig. 7 Speciation diagram of UO 22+ (4.10 -4M)) in aqueous HCit 3- (2.10 -2M) at

    25C with "Set III" constants; 112 = 0

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    2 3 4 5 6 7 8 9 10 11

    pH

    %

    % B22

    %UO2(OH)+

    %UO2(OH)2

    %UO2(OH)3

    % B111

    % B1111

    % B222

    % B2222

    % B12

    % B112

    % B224

    %(UO2)2(OH)

    %(UO2)2(OH)2

    % UO22+

    % B11

    Fig. 8 Speciation diagram of UO 22+ (4.10 -4M)) in aqueous HCit 3- (2.10 -2M) at25C with "Set IV" constants; 112 = 0

    On the other hand, Fig. 6 (for R c = 2) shows that theexcess of [(UO 2)]

    2+ ions changes appreciably the speciationdiagram in the sense that the respective heights of the plateaus:[ML], [MLH(OH)] and [M 2L2H2(OH) 2] are decreased, in the

    pH range: 3 < pH < 5, toward: 15%, 10% and 5%,respectively. Notice also that the variation with pH of the

    percentage of the species [(UO 2)]2+ is now completely

    different, by comparison to Figs. 4, 5. In particular, this percentage remains sensibly constant equal to 45 % in the pHrange: 3 < pH < 5 (for R c = 1). Nevertheless, we can say that,

    for R c=2, the species [(UO 2)(HCit)]-

    and [(UO 2) H(HCit)(OH)] - remain observable by capillary electrophoresis, in the pH range: 3 pH 5. However, and within the limits of the

    experimental precision, this fact is less true for [(UO 2)2H2(HCit) 2(OH) 2]

    2- species.

    C. Results in the General Case at 25C In the general case, and in addition to the previous entities,

    we have taken into account the formation of the followingspecies: [(UO 2)(HCit) 2]

    4-: (12), [(UO 2)2(OH)]3+: (KH 21),

    [(UO 2)2(OH) 2]2+: (KH 22). Their corresponding constants: ( 12),

    (KH 21) and (KH 22) are given in Table 4 ( Set III Set I +12 or Set IV Set II + 12) and in Table 5. Thequestionable case of the specie [(UO 2)H2(HCit) 2]

    2- : (122 ) will be analyzed at the end of this paragraph.

    The numerical result corresponding to Set III isrepresented in Fig. 7. Comparison with Fig. 3 (Set I) showsthat except for the two pH ranges: 3 < pH < 5 and 9 < pH < 11,the two diag rams are completely different. Indeed, with SetIII constants and for pH > 5, the percentages of both species:[(UO 2)H(HCit)(OH)]

    -: (1111 ) and [(UO 2)2H2(HCit) 2(OH) 2]2-:

    (2222 ) decrease with pH from 30% to respectively: 10% and5% in the pH range: 6 < pH < 9. This decrease is offset by theapparition in the same pH range of an important plateau of

    about 80 % height, corresponding to the specie[(UO 2)(HCit) 2]

    4-: (12).

    Now if we replace Set III by Set IV , so that 11(Set III) 11 (Set IV) and 111 (Set IV) 111 (Set III) (by analogy to

    the transformation of Set I into Set II ), we obtain thediagram given by Fig. 8. Comparison of Fig. 8 with the

    previously Fig. 7 indicates the apparition with a relativelyimportant percentage of the specie [(UO 2)(HCit)]

    - in additionto the species: [(UO 2)H(HCit)(OH)]

    - and [(UO 2)2 H2(HCit) 2(OH) 2]

    2- (by analogy to the comparison between Fig. 4 (SetII) and Fig. 3 (Set I)). Consequently, the height of theplateau corresponding to the specie [(UO 2)(HCit) 2]

    4-is nowof about 60 % rather than 80%.

    In conclusion, we can say that numerical calculation for R c

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    C M = 5.0 10-4 ; C L = 5.0 10

    -4 R c = 1.

    C M = 1.0 10-3 ; C L = 5.0 10

    -4 R c = 2.

    Comparison of Fig 10 (with: 12 > 0, 122 > 0) with Fig. 5(12 = 0, 122 = 0) shows that the formation of species[(UO 2)(HCit) 2]

    4- has no significant influence on the diagram,and that the influence of [(UO 2)H2(HCit) 2]

    2- species is alsonegligible by comparison with Fig. 9 (R c 0

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    1 2 3 4 5 6 7 8 9 10pH

    %

    % UO22+

    % B11

    % B22

    %UO2(OH)+

    %UO2(OH)2

    %UO2(OH)3

    % B111

    % B1111

    % B222

    % B2222

    % B12

    % B122

    % B112

    % B224

    %(UO2)2(OH)

    %(UO2)2(OH)2 Fig. 10 Speciation diagram of UO 22+ (5.10 -4M) in aqueous HCit 3- (5.10 -4M) at

    25C with "Set V" constants; 112 > 0

    0

    10

    2030

    40

    50

    60

    70

    80

    90

    100

    0,5 1,5 2,5 3,5 4,5 5,5 6,5 7,5 8,5 9,5 10,5pH

    %

    % UO22+

    % B11

    % B22

    %UO2(OH)+

    %UO2(OH)2

    %UO2(OH)3

    % B111

    % B1111

    % B222

    %B2222

    % B12

    % B122

    % B112

    % B224

    %(UO2)2(OH)

    %(UO2)2(OH)2

    Fig. 11 Speciation diagram of UO 22+ (10 -3M) in aqueous HCit 3- (5.10 -4M) at

    25C with "Set V" constants; 112 > 0

    Note also that the two gaps corresponding to both[(UO 2)]

    2+ and [(UO 2)(OH) 3]- species, for R c = 1, are also

    comparable to those of Fig. 5 (R c =1).

    In the same manner, comparison of Fig. 11 (R c = 2, 12 > 0,122 > 0) with Fig. 6 (relative to Set II: 12 = 0, 122 = 0 and

    R c = 2), shows that the formation of the two species[(UO 2)H2(HCit) 2]2- and [(UO 2)(HCit) 2]

    4- is always negligible.Indeed, the heights of their corresponding peaks are

    respectively less or of about 3% in the pH range: 2 < pH < 3. Notice finally that, according to Fig. 11, the relative excess of (UO 2)

    2+ ions (characterized by the ratio: R c = 2), increases themaxima of the bell-shaped peaks relative to the species[(UO 2)2(OH) 2]

    2+ and [(UO 2)(OH) 2], respectively up to 43%

    in the pH range: 3 < pH < 8, and to 40% in the pH range: 6 < pH < 10. However, these peaks cannot be observedexperimentally in the present case of our experience.

    In conclusion, the formation of [(UO 2)H2(HCit) 2]2-: (122)

    and [(UO 2)(HCit) 2]4-: (12) species seems to appear with

    important and constant percentages (respectively: 95% for:1.5 < pH < 6.5 and 60% for: 4.5 < pH < 10), only for lowratios: R c

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    -1500

    -500

    500

    1500

    2500

    3500

    4500

    5500

    1 1,5 2 2,5 3 3,5 4 4,5 5 5,5

    Migration time (min)

    A

    b s o r b a n c e

    ( a . u . )

    pH = 3.5

    pH = 4.0

    Fig. 13b Electropherograms of 1.10 -5M nitrate and 5.10 -4M U(VI) in 0,02M

    citrate buffer at pH=3.5 and pH=4.0

    1 2 3 4 5 6

    Migration time (min)

    A b s o r b

    a n c e

    ( u . a

    )pH = 5

    pH = 4.5

    pH = 4

    pH = 3.5

    pH = 3

    pH = 2.5

    2000 a. u.

    Fig. 14 Electropherograms of 5.10 -4M U(VI) in 0.02M citrate buffer at

    various pH values

    VI EXPERIMENTAL

    A. Apparatus

    The capillary electrophoresis apparatus is a modular system consisting of a Spectraphoresis 100 injector (hydrodynamic mode) coupled with a high voltage (0-30kV)

    power supply (Prime Vision VIII from Europhor) and ascanning UV-visible detector (Prime Vision IV fromEurophor). In the present study, a voltage of 30kV wasapplied and a fixed wavelength of 200 nm was chosen for theabsorbance measurements. The external temperature wasmaintained equal to 25C. The acquisition of theelectropherograms and the data treatment were performedusing the chromatography software BORWIN (JMBSDveloppements).

    Each measurement was repeated two or more times in

    order to assure the reproducibility condition. B. Chemicals

    All solutions were prepared with deionised water andchemicals of analytical reagent grade. For the sodium citrate

    buffer solutions, we have used different Fluka solutions for HPCE (210 -2M) having pH values between 2.5 and 5.5. Theuranyl stock solution (concentrated UO 2 (ClO 4)2) was

    prepared by dissolving 1g of uranyl nitrate hexahydrate salt,UO2(NO 3)2.6H 2O (FLUKA), in 25 ml of concentrated

    perchloric acid. The resulting mixture was then almostcompletely evaporated at 150C in a sand bath and this

    procedure was repeated several t imes in order to eliminate allthe nitrates. This acidic (HClO 4 + UO 2 (ClO 4)2) stock solutionwas then diluted in order to adjust the uranyl concentration toaround 10 -2M (the exact concentration was determined byfluorescence spectroscopy).

    For each pH value of the citrate buffer, a correspondingsample was prepared by mixing appropriate volumes of diluted uranyl stock solution and sodium citrate buffer solution. The final pH of this sample solution was adjusted byadding a small quantity of NaOH solution (0.1M). The totalfinal volume of the different prepared samples is equal to1000 l.

    It is important to mention that, before performing anyelectrophoresis experiment, the sample solutions were kept atrest for at least two days, in order to allow the differentreactions of complexation, protonation, and hydrolysis, toreach their complete chemical equilibrium.

    C. ProcedureThe silica capillary (SUPELCO, 75 m of internal

    diameter, 75 cm long), was first conditioned by successivewashes with 0.1 NaOH, then by deionised water, and finally

    by 0.02M sodium citrate buffer solution of a given pH.

    The cleanliness of the capillary is checked by a first runusing only the buffer solution under study (check of the

    baseline). The corresponding electropherogram should not present any peak but only a smooth baseline (or noise line).

    In order to control the eventual effect of theelectroosmotic flow (neglected at low pH values [11]), as wellas other experimental fluctuations, which can shift the peak

    positions of the anionic species under observation, nitrate ions(at very low concentrations) were added to some samples.This anion has already been used in previous works as areference anion for mobility measurements [12, 13, 16]. Fig.12 gives an example of such electropherograms obtained at

    pH = 3 (NO 3- peak appears around 2 min).

    Each run consisted in a hydrodynamic injection of 2nnanoliters during n sec, of a sample solution (uranyl solutionin buffer citrate with or without nitrate ions) into the capillary.The corresponding electropherogram was obtained byapplying 30 kV during an acquisition time of at least 10 min.Finally, the capillary was carefully rinsed with the buffer under study between two runs and kept filled with deionisedwater during the night.

    D. Results and DiscussionAs expected, since citrate ions are absorbing species in the

    UV domain, the stability of the baseline is very sensitive tothe buffer pH value, as can be seen in Figs.13, 14. The

    background absorbance was observed to increase with pH andstrong fluctuations to appear for pH > 4.5. These changes inthe level and quality of the baseline can be assigned to anaugmentation of the charged citrate ions of the buffer when

    pH increases. Consequently, the current passing through thecapillary increases sharply with pH: (from 20 A at pH 2.5 3 to 135 A at pH 5). For these reasons, the best results wereobtained, for: 2.5 < pH

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    (see Fig.3 or Fig. 7), only cationic ( [(UO 2)H2(HCit)]+,

    [(UO 2)2H4(HCit) 2]2+) or neutral ([(UO 2)H(HCit)]) species,

    exist for pH < 3. The two anionic species: ([(UO 2)H(HCit)(OH)] -, [(UO 2)2H2(HCit) 2(OH) 2]

    2-), are thus expectedto appear quantitatively only for pH >3.5. Notice that,according to Set II constants ( 11, 111 , 1111 , 112 , 22,2222 , 224, KH 11 , KH 12 , KH 13 ) or Set IV {Set II + 12}constants (see Fig.4 or Fig. 8), the three anionic species:([(UO 2)(HCit)] -, [(UO 2)H(HCit)(OH)] - and [(UO 2)2 H2(HCit) 2(OH) 2]

    2-) appear also quantitatively, only for pH > 3. Thesetheoretical results are in good agreement with the fact that, for

    pH = 2.5 and pH = 3, the corresponding electropherograms donot point out any peak associated with anionic uranyl species.The only apparent peak shown in Fig. 13a, with a migrationtime t 2 min, is due to the added nitrate ions.

    2) Electropherograms for: 3 < pH < 5In contrast, for buffer pH values around 4, the

    corresponding electropherograms show three separate peaksat migration times of 3.2 min, t 4.7 min and t 5 min (seeFig. 13b, 14). Owing to our theoretical results, these peaks

    could be assigned to the three following species:[(UO 2)2H2(HCit) 2(OH) 2]2-, [(UO 2)H(HCit)(OH)]

    - and [(UO 2)(HCit)] -.

    Their electrophoretic mobility, u i, can be derived from theone of NO 3

    -: u NO3 , by applying the following expression:

    u i = u NO3- + (d L/U) (1/t Mi 1/t NO3

    -) (33)

    where u NO3 is the ionic mobility of NO 3- (7,4 10 -4 cm 2 V-1

    s-1); U is the applied voltage (30 000 V); L is the capillarylength (75cm) and d, the length from the capillary inlet to thedetector (35 cm). The parameters: t Mi and t NO3 are respectivelythe times taken by the ionic i specie and by the NO 3

    - specieto reach the detector [12].

    TABLE 6 ELECTROPHORETIC MOBILITIES , UI, CORRESPINDING TO THE PEAKSOBSERVED IN FIG . 13 B ( NO3- BEING TAKEN AS REFERENCE )

    pH = 3.5 pH = 4.0

    t(migration)(sec)

    10 u i (cm 2 V-1 s-1)

    t(migration)(sec)

    10 u i (cm 2 V-1 s-1)

    NO 3- 122.2 (7.4) 128 (7.4)

    species1

    180.9 5.08 195 5.06

    specie 2 256 3.66 279.5 3.73

    specie 3 294.8 3.54

    The calculation leads to the following electrophoreticmobility values: 5.1 10 -4, 3.7 10 -4 and 3.5 10 -4 cm2 V-1 s-1 (seeTable 6), corresponding to the three uranyl complex formsdetected in Fig. 13b. However, the exact correspondence

    between each of these values and the above cited anionicspecies is not a priori evident. Nevertheless, the ionicmobilities of monomer species are generally assumed to beapproximately proportional to their charge over size ratio [13].Indeed, according to the Nernst-Einstein relation: u iDi

    -1 =(Z ie)(kT)

    -1 and Stokes-Einstein relation: D i = kT(6 R i)-1,

    where: D i is the self diffusion coefficient, kT is the thermalenergy, is the viscosity and Z ie, R i are the charge and theradiu s of the considered i specie, the expression of themobility is therefore: u i = (Z ie)/ (6 R i). Consequently, wehave attempted to make the following qualitative predictionson the basis of the different structures attributed by recentliterature to the three complexes under consideration:

    [(UO 2)H(HCit)(OH)]- and [(UO 2)(HCit)]

    - they have thesame charge and similar radii; therefore, they should havenear mobility values. The two peaks observed at pH > 3.5 atmigration time t 4.8 min, and characterized by a weak separation and a comparable height(see Fig. 13b, 14), can thus

    be attributed to these two species. Indeed, if we assume thevali dity of the Set II constants, these two species are alsoexpected to have close complexation rates (or %), inaccordance with the theoretical predictions: 40% and 35%, for respectively [(UO 2)(HCit)]

    - and [(UO 2)H(HCit)(OH)]- (see

    Figs. 4, 8). We can note, however, that at pH = 3.5, and incontrast with pH =4, only one of the two close peaks appearsclearly on the corresponding electropherograms (the second

    peak could only be guessed on the electropherogram of Fig.14). Comparison between Fig. 13a and Fig. 13b shows thatthe percentage of the two above cited species increases from

    pH 2.5 to pH 4 with a gap at pH 3. This tendency isreflected in the diagrams given in Figs. 4, 8 but with a gap at

    pH 2.5.

    The [(UO 2)2H2 (HCit) 2(OH) 2]2- the complexion has a

    double negative charge (Z i = -2) and it exhibits three possibleconformations [14]: two of them are rather sphericalcharacterized by a large radius R i and therefore by a weak diffusion coefficient: D i; In contrast, the third most probableconformation, is a cylindrical chain presenting a weak viscousfriction and therefore, a large mobility u i [15]. However, sinceits corresponding complexation rate is about 10%, (accordingto our theoretical calculations: Figs. 4, 8), its relativeelectrophoretic peak should be weak. This fact is confirmed

    by the electropherograms performed at pH 3.5, pH4 and pH4.5 after a migration time of about: t 3.0-3.5 min.

    Note moreover that, for pH 4.5, this peak (at t 3 min)is always observable, whereas the other two peaks are, moreor less hidden by the noise of the baseline (see Fig.14). On theother hand, according to our calculation using Set IIconstants, the increasing of the U(VI) concentration (i.e. theratio Rc), will not give more information concerning thesethree species since their percentages decrease with increasingthe ratio Rc.

    It is interesting to underline at this stage, and more particularly for the speciation measurements under consideration, the interest of the capillary electrophoresismethod by comparison to some other analytical methods: thelatter ones are rather sensitive to the coordination state of U(VI), (for example, because of the variation of the U-Odistance with pH). It ensues that these techniques cannot ingeneral distinguish between protonated and non- protonatedcomplexes: M 1L1, M 1L1H1 , M 1L1H2, M 2L2, M 2L2H2, andM2L2H4, as well as between hydrolyzed and non-hydrolyzedcomplexes: (for example: M 1L1H1(OH) 1 or M 2L2H2(OH) 2). Inother terms, and for the present study, they could not informus sufficiently on the protonation state nor on the hydrolysisstate of M U(VI). On the other hand, the capillaryelectrophoresis method, which is very sensitive to both thecharge Z ie and/or the size of the species, allows the separationof the different anionic complexes expected to be formed onthe base of their relative mobility.

    VII CONCLUSION In order to complete more deeply our previous work [16]

    in selecting the more appropriate complexation constantsamong those published by the literature, and following thecontroversy concerning the processes of protonation and the

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    hydrolysis of the uranyl-citrate species, we undertook tocompare our new uranyl-citrate speciation diagramscalculated analytically or numerically with experimentalresults obtained by the capillary electrophoresis method, indilute citrate aqueous solutions (0.02M) at 25C and for: 2.5 9, thespecie [(UO 2)(OH) 3]

    - is predominant. Notice that if weassume the existence of the [(UO 2)H2(HCit) 2]

    2- specie(reported by only one reference), the speciation diagram isthen completely modified in the range: 1.5 < pH < 6, since its

    percentage in this domain is evaluated to 95%. However, for R c 1, both [(UO 2)(HCit) 2]4- and [(UO 2)H2(HCit) 2]2- species

    become negligible; consequently, the respective proportionsof the three [(UO 2)(HCit)]

    -, [(UO 2)H(HCit)(OH)]-, [(UO 2)2

    H2(HCit) 2(OH) 2]2- species become constant (3 plateaus) in the

    region: 3 < pH < 8. For: R c = 1, these percentages are about:35%, 30% and 10%. For: R c = 2, they decrease to about: 20%,17% and 5% because of the predominance of the free (UO 2)

    2+ ion for pH < 4, and also due to the apparition of thehydrolyzed ions: [(UO 2)2(OH) 2]

    2+ for 4.5 < pH < 7 and[(UO 2)(OH) 2] for 7 < pH < 9. It is interesting to notice thatless recent constant data (Set Ib), lead to the same results,except that the specie [(UO 2)(HCit)]

    - is substituted by theneutral specie [(UO 2)H(HCit)], which appears only for pH 0, thecorresponding system of conservation equations is given byEqs. (26-28) with: x [UO 2

    2+] and y [HCit 3-], therefore:

    y = 0.5[-D + 1/2](xA 21)-1 (I.1)

    with : = D 2 + 4(xA 21)[ C + x(1+E) + 2x2EE] (I.2)

    C = (C L - C M) = Dy - x [1 - A 21y2 + E] - 2x 2EE (I.3)

    C L = x [Ay + 2A 21y2] + 2x 2Cy2 + Dy (I.4)

    CM = x [1 + Ay + A 21y2 + E] + 2x 2[EE + Cy 2] (I.5)

    Recall that: x, y, A, A 21, C, D, E, EE are functions of the pH.

    For a given pH, the numerical method consists to startfrom an initial x 0 value. x 0 can be chosen equal to the finalsolution x of the above system (I.1-I.5) obtained for the

    previous pH = pH pH. After n iterations, we define:

    x (n) = x 0 nx. Thus: y(n) = = 0.5[ -D(n) + (n)1/2](x(n)A 21)

    -

    1 (I.6)

    The convergence is reached when both x(n), y(n) valuesverify Eqs.(I.4 and I.5); within a given precision: %Px =100[x(n) x(n-1)]/x(n). If it is not the case, we proceed in thesame manner with the next x(n+1), y(n+1) values. It isobvious that the precision %Px of these solutions is increasedwhen the step x is very small. However, this precision islimited by the precision of calculation of the computer. Ingeneral, our convergence precision is less than 1% which iscomparable to the estimated error on the concentrations.

    APPENDIX II ANALYTICAL RESOLUTION OF THE EQUATIONS OFCONSERVATION

    The system of the conservation equations given in paragraph 4.3 could be reduced to an algebraic equation inx of degree 4, if we ignore the following species:[(UO 2)2(OH)]

    3+, [(UO 2)2(OH) 2]2+ [(UO 2)(HCit) 2]

    4- (12 = 0)and [(UO 2)H2(HCit) 2]

    2- (122 = 0). Therefore, EE = A 21 = 0.With such an assumption, the system of the conservationequations becomes:

    C L = Axy + 2x2Cy2 + Dy (II.1)

    C M = x [1 + Ay + E] + 2x2Cy2 (II.2)

    C = (C L - C M) = Dy - x[1+ E] (II.3)

    Therefore, y = ( C + x[1+ E])D-1

    (II.4)

    And: x 4 + a 1x3 + a 2x

    2 + a 3x + a 4 = 0 (II.5)

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    The International Journal of Nuclear Energy Science and Engineering IJNESE

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    a1 a2 a3 a4

    2 C(1+E) -1 D -2[2C( C) 2+A(1 +

    E)D](a 0)-1

    [1 + E+A( C)D -

    1](a 0)-1

    (-

    CM)(a 0)-1

    With a 0 = 2C(D-2

    )(1+E)2

    The previous Eq. (II.5) can be factorized as following:

    x4 + a 1x3 + a 2x

    2 + a 3x + a 4

    = (x 2 + x S m+ P)( x2 + xSm + P) = 0 (II.6)

    After identification we obtain:

    (Sm + Sm) = a 1; (S m.Sm + P + P) = a 2; (P.Sm + PSm) = a 3;

    P.P = a 4 (II.7)

    If now we define the parameter z as:

    z = (P + P) (II.8)

    We can therefore express the parameters S m, Sm, P and Pin terms of z and the a 1, a2, a 4 coefficients:

    Sm = 0.5[z - (z2- 4a 4)

    1/2] ; Sm = 0.5[z + (z2- 4a 4)

    1/2] (II.9)

    P = 0.5[a 1+ {(a 1)2- 4a 2 + 4z}

    1/2] ;

    P = 0.5[a 1- {(a 1)2- 4a 2 + 4z}

    1/2] (II.10)

    Now, the condition on the a 3 coefficient (P .Sm + PSm)= a 3 shows that z is solution of the following equation (of third degree) in z:

    z3 + A 1z2 + A 2z + A 3 = 0 (II.11)

    With: A 1= - a 2 ; A2 = (a 1a3 - 4a 4) ;

    A3 = [4a 2a4 (a3)2 - (a 1)

    2a4] (II.12)

    The z real solutions depend on the sign of thedeterminant D t:

    D t = Q3 + R 2; Q = [3A 2 (A1)

    2]/9 ;

    R = [9A 1A2 -27A 3 -2(A 1)3]/54 (II.13)

    D t >0 D t = 0 D t