Elution of heavier rare earths with H.E.D.T.A. eluant at a high temperature

J. inorg,nucl.Chem..1969,Vol.31, pp. 2933 to 2949. PergamonPress. Printedin Great Britain

ELUTION H.E.D.T.A.

OF HEAVIER RARE EARTHS WITH ELUANT AT A HIGH TEMPERATURE

ZENZI HAG1WARA Faculty of Engineering, Tohoku University, Sendai, Japan (Received 23 December 1968) A b s t r a c t - I n order to study the detail behaviors on the separation of heavier rare earths by ion exchange with fairly high concentrations of H.E.D.T.A., the elutions have been performed at 60°C under steady state conditions. The theoretical plate height concerning the column efficiency was also investigated from the viewpoints of countercurrent extraction and rate theory, and the various experimental results relating to the mixed region of the adjacent rare earths were explained from the relation induced. Further, under different experimental conditions, the contamination of thorium present in a trace quantity in the heavier rare earth samples, was investigated at a temperature of 60°C. The small peak of thorium moved to the lutetium side with increase in the concentration of H,E.D.T.A. eluant. INTRODUCTION

UNDER the use of either ethylenediamine-N ,N ,N' ,N' -teteraacetic acid (E. D.T.A.) or N'-(2-hydroxyethyl)ethylenediamine-N,N,N'-triacetic acid (H.E.D.T.A.) as the eluant, the separation of the rare earths by ion exchange has been carried out on a large scale by Spedding and Powell[l]. These two chelating agents showed a great promise for the separation of rare earths in combination with cation exchangers. As compared to E.D.T.A., H.E.D.T.A. may not be so popular as the eluant. The latter, however, is effective for the separation of the heavier rare earth mixture containing Er, Tm, Yb and Lu as well as the lighter rare earth containing La, Ce, Pr and Nd, as understood from the differences in the stability constants[2] of rare earth-H.E.D.T.A, chelates. The elution of a mixture of yttrium and lanthanide with a dilute H.E.D.T.A. eluant was made at various temperatures in order to check the elution position of yttrium[3] in the exchange system. In the present research, higher concentrations of H.E.D.T.A. were used at 60°C for separating the heavier rare earth mixtures containing Tm, Yb and Lu as main species. The elution behavior of thorium in minute quantity was also investigated, using rare earth samples involving thorium less than 1 per cent as ThO~. Further some relations concerning theoretical plate height were induced from the viewpoint of rate theory, and the experimental data were explained on the basis of the relation introduced from rate theory as well as the countercurrent extraction. 1. J. E. Powell and F. H, Spedding, Trans. metall. Soc. A.I.M.E. 215, 457 ( 1959); F. H. Spedding and A. H. Danne, The Rare Earths. Wiley, New York ( 1961 ). 2. F. H. Spedding, J. E. Powell and E. J. Wheelwright,J. Am. chem. Soc. 78, 34 (1956). J. L. Mackey, M. A. Hiller and J. E. Powell,J. phys. Chem. 66.311 (1962). 3. J. R. Morton and D. B. James, J. inorg, nucl. Chem. 29, 2997 (1967). 2933

2934

Z. H A G I W A R A EXPERIMENTAL

Preparation of rare earth load solutions. The oxide samples of heavier rare earths were made in the Shinetsu Chem. Corp. After adding a slight excess of hydrochloric acid, these oxide samples were completely dissolved by heating, and the resulting solutions were then diluted to a proper volume with distilled water. The pH of the rare earth load solutions prepared was kept to approximately 2. Resin used. The Dowex 50W, X-8, 50-100 mesh was used as cation exchanger. Prior to the column elution, it was pre-conditioned with hydrochloric acid and citric acid for the removal of heavy metals in the resin, and finally washed with distilled water. The conditioned resin was inserted into the columns, followed by back-washing with distilled water in order to carry out uniform packing. In this case fine particles were also removed. Experimental apparatus. The jacket columns with a dimension of 5 cm in inner diameter and 120 cm in length, were connected in series. For these columns, hot water from a large thermostat was circulated. The bath was kept to 60°-0"3°C. The temperature gradient between the top and bottom of each column was less than I°C. The eluant, at first, passed through the de-aerator, auxiliary column with a temperature controlling device, and main exchange columns. Preparation ofeluants. The following eluants were prepared.

(a) NH4, H andCh Ch refers to the anionic species of H.E.D.T.A. A proper amount of H.E.D.T.A. was dissolved in water. For this solution, dilute ammonium hydroxide was added until a desired pH had been indicated by the glass electrode. After taking a definite volume of the eluant into a beaker, 5 ml of NH4CI-NH4OH buffer solution with a pH of l0 were added, and the total concentration of Ch was then determined by titrating with the standard solution of 0.04 M ZnCI2, using E.B.T. (Eriochrom Black T) as the indicator.

(b) Ca, NH4, H andCh This type of eluant was readily prepared by heating the mixed solution with CaCO3, NH4OH and H.E.D.T.A. After the decomposition of Ch by HNO3-HC104 (1 : 1) under boiling, calcium in the eluant was determined by chelometry with the standard solution of 0.02 M Na~H2Y. Y means the anionic species of E.D.T.A. In the titration, the pH was adjusted to 13 with 2 M NaOH, and Dotite N N (Commercial Name) was used as the indicator. On the other hand, the total Ch was determined as follows: after taking a definite amount of eluant into a beaker, the excess of Ch over the molar ratio of Ca/Ch = 1 was first determined by the described method in (a). As calcium in presence of chelating agent is able to determine by the above method, the total Ch can be finally confirmed. The total ammonium in the two types of the eluants was determined by usual Kjeldahl method. Experimental method. Four jacket column with I.D. 5 cm in series, were used as the rare earth adsorption bed. The pre-heated rare earth load solution was adsorbed by passing through the top of the leading column. During the adsorption, the temperature was held a t 60°C. After passing an excess of the load solution, the columns were washed with hot water until free from chloride. In each run, the heavier rare earth sample corresponding to about 1 kg as oxide was loaded. The rare earth adsorption columns were then connected with the retaining jacket columns with a hydrogen form of resin, and under a definite flow rate the loaded rare earth mixtures were displaced 2.5-3 band lengths with the de-aerated H.E.D.T.A. eluants. In the experiments, the elution temperature was kept to 60°C and the composition of the eluant was changed as described in later. The eluate of the rare earth from the column was collected into small fractions and precipitated with oxalic acid under heating on the water bath. The oxalate precipitates were ignited to the oxides, and analyzed by X-ray fluorescent spectrometry.

THEORETICAL

Analysis of the elution of a rare earth band from rate theory In general, the rates of particle- and film-diffusion controlled ion exchange process may be expressed approximately by linear-driving-force relations.

Elution of heavier rare earths

2935

For particle diffusion control [4], the relation is

k-~/z

(1)

0"071To 2

and for film diffusion control [4]

t - - y j,(°xq ___

C _ C: .

(2)

The material balance for an arbitrary species for a layer mentioned in Appendix, is given by

Introducing the term relating to the equivalent fraction of species i in both phases (J?i/)? = CJC), linear flow rate (v = 1/q(OV/Ot)z) and rate of the movement of a band (vb= v/(k/C+~)) (See Appendix), we have the following relation.

Ot /z

(4)

--vb\oZ/v"

Combination of Equations (1) and (4) gives

{0S2,\

14/)2 (2* -- 25).

(5)

Further the separation factor of the adjacent rare earths, Lnl and Ln2 in presence of chelating agent, is expressed by Ln2 ~. XLni • 3CI~nl aLni

(6)

XLn~• X~n,

where barred quantities represent the resin phase, unbarred the aqueous phase of corresponding species, and asterisks indicate equilibrium. The concentrations of Lnl and Ln2 in both phases are expressed by the equivalent fraction. Thus the following relations are held: XLn, -t- XLn2 1, ~7"m + JT*n,= 1. Introducing the relations into (6) =

-~L*n2 =

o/Ln2 Lna • XLn2

.

1 + t~Ln,t~Ln2__ 1)XLn2

(7)

Introducing.,7*~ = J?*.~/J~ into (7), J?*.~ is given by 4. E. Gluckauf, Ion Exchange and Its Applications, p. 34. Society of Chemical Industry, London (1955). F, Helfferich,Angew. Chem. Intern. Edn. 1,440 (1968).

2936

Z. H A G I W A R A

OILIIIXXLn -

~?~., = 1 +

/t O / L n. ,l_

.

(8)

1)XLn 2

In the edge of the adsorption band of adjacent rare earths under a steady state, the following relations are held: )~Ln2 ~--- X l m 2 ,

SLnz

=

(SeeAppendix).

XXLn2

From the above relations and Equations (5) and (8)

a4O[XLo,X(~°, 1)(1--~.~)]

/ O.SLrl2"~

--

Ln2 __

-

-

T0 [

-

-

l+(aLn,

-

-

(9)

-

1)XLn,

J

Integrating (9), we have

1

14/3

a Ln~- 1 {In XL~ -- otLn~In ( 1 -- XLn~)} = -- V~T0zZ + C~.

(lo)

Iml

Substituting the term on the plate height from particle diffusion, Hp = vbToz/14D into (10) LHp ', {--lnXLn2q-

Z =

~Ln2 t'tLnl

O~Ln~ - - 1

In(1 --XLn~) +

(1 1)

, t a L1nL n 2 _ 1 ) C 1 } .

In case of f~ z dXL,~= 0, the integration constant C1 in Equation (11) should be equal to unity (See Appendix). Thus the following relation is finally induced for particle diffusion control. Z (XLn~) = H p

{

_

oL.

_

Lnl

•

t •

aLL~-- 1 l n X L n ~ + ~ m--t J - - , L . ~

X

x.

}

Ln~) + 1

(12)

where Z(XLn,) is the distance of the plane with concentration XLn~from the center of gravity of the boundary. As an example, Equation (12) is graphically represented in Fig. 1, in which the relations between Xvb and Z are shown for given values of

~ ~ \ \ \ V ~ J -

0.3

a~° =1.3

....? .

,,e

019

0

-IO

t

L

,

i

i

-5

t

,

i

t

J

0

5

Fig. 1. Theoretical curves obtained from Equation (12).

I0

Elution of heavier rare earths

2937

vb and Hp: The dimensions Of Xvb and Z am the equivalent fraction and centimeter, respectively. For film diffusion control, we have the following relation from Equations (2) and (4) O~Lu

[Of(i\

3D ( C , - C~*)

-

(13)

CLnz = CXLn,and C'n2 = CXLn2/OlLn , , Ln2 __ [OdLn, Ln,~__ 1)XLn~(See Appendix)

Introducing

Vbt~-~-=),

(14)

CXLn2 1 0 d L nn] -- ,[ o/Ln2Ln,-- 1)XLn "

--

Making the same treatment as particle diffusion, one obtains

1

t Ln, lnXLn_ln(l__xLn,)}.=--( \ 2 3DC ]Z+ ~ )

Ln2 __ 1 "tO/Ln'

O~Lnl

(15)

C2"

In case of fo~z . dXLn2 = 0 , the integral constant C2 should be equal to -1. Substituting H s = 28yoVbX/3DC into (l 5), the following relation is finally induced. Z(XLnz) = H

(

O/Ln2

InXLn2h

Ln2Ln--"--L__ 1 OgLnl

1

1

OdL n ~ ' Ira1

In(1 --XLn2) -- 1}

(16)

The above relation has appeared in Fig. 2, in which XTmis plotted against Z, using O gTm y b = 1"6 and Hfvalues given in the figure i .

.

.

.

I

.

.

.

.

I

.

.

.

.

I

l

,

0'7

!

|

,

,

,

,

I

'

•

'

"

IO

02

x

0'5

16

H vs Z

I-I

0 -15

I0

-5

0

5

z

Fig. 2. Theoretical curves obtained from Equation (16). SYMBOLS

T0 /5 D 8 .Y~

radius of resin particle diffusion coefficient in the exchanger diffusion coefficient in the aqueous phase film lhickness amotmt of species i in the resin per unit volume of bed

~0

~5

2938

,~* X X~ x~ $~ C~ C* C t Z q V fl v vb Hp Hr Ln~ OLLn 1 XLn £Ln

Z. H A G I W A R A

amount of species i in the resin per unit volume of bed at equilibrium with Ci total amount of species in the resin per unit volume of bed amount of species i in sorbent and solution per unit volume of bed equivalent fraction of species i in the solution equivalent fraction of species i in the resin phase concentration of species i in the interstitial solution concentration of species i in the interstitial solution in equilibrium with ,~ total concentration of species in the interstitial solution time distance from a reference point of the column column cross section solution volume passed through the layer void fraction of bed linear flow rate rate of the movement of adsorption band plate height due to particle diffusion plate height due to film diffusion separation factor of the adjacent rare earths, Lnl and Ln2 equivalent fraction of corresponding rare earth in the solution equivalent fraction of corresponding rare earth in the resin phase.

Analysis of the eh:tion of rare earth from equilibrium theory After the rare earth band with adjacent rare earths, Lnl and Lnz, had been displaced for a distance corresponding to several band lengths, it may be considered to be equilibrium condition. In such a case, the length of a overlap region remains constant, and is independent of the elution band length. According to the treatment of Spedding and Powell[5], the following relation is given to calculate the H.E.T.P. (height equivalent to a theoretical plate), log~

Ln2

Ln2-- log OtLn~ _ = n log aLto - - H ~ . L

(17)

where L is the distance between the two points on the resin, n the number of theoretical plates, and aLn 1Ln~and R are the separation factor of the adjacent rare earths and their concentration ratio. In the present runs, the experimental data are plotted, using Equation (17). RESULTS AND DISCUSSION

Elution curves and separation efliciencies The experimental conditions of the heavier rare earth mixtures with a trace amount of thorium, are tabulated in Table 1, and further the typical elution curves appear in Figs. 3-6, in which the total concentration of the rare earths is showing a constant value within the errors in analysis, after steady state had been attained. The pH of each rare earth fraction is also constant. In the elution of the rare earth at room temperature, ca. 0.018M H.E.D.T.A.[6] buffered with 5. F. H. Spedding and J. E. Powell, J . A m . chem. Soc. 77, 6125 (1955). 6. D. B. James, J. E. Powell and F. H. Spedding, J. inorg, nucl. Chem. 19, 133 (1961).

Elution of heavier rare earths

2939

-~[-

0

aD , ~

o0

e-

,o

0

0

II

e-

i.

II

tl

e- ,a~

zzzE Z

0

0

,z _= O ? E

~L 0

tr~

e-

O

ZZZZ

6 Z

H .fi

I

I

I

I

I

I

I

I

[-- [--- ~-- [--

O

?

2940

Z.

HAGIWARA 9

HT-HR-II 60oc

40

7 3C

~-

"*

5 E

2(0

1-

Lu

.//

8 I0

0

rht .

~ 1350

fs 1400

1450

1500

i

i

i

i

| / 1800.

Eluate volume ,!

Fig. 3. Elution curves of a heavier rare earth mixture containinga smallamountof Th, using0-016 M H.E.D.T.A. bufferedwithammoniumhydroxide. ammonium hydroxide was used as the eluant. However, higher concentrations of H.E.D.T.A. mentioned in Table 1, are still available by raising the elution temperature up to 60°C. Even in such conditions, the ion exchange systems used have been arrived at a steady state. Further the addition of calcium in the eluant is made in order to reduce the ammonium ions through the exchange system. Thus the irregular diffusions due to ammonium in the rare earth band would be suppressed to a slight degree. The recovered rare earths showed the following composition: HT-HR-11 Lu~O~ > 98.5%, ThO2 -< 1.5%(9.1%); Yb203 > 99.9%, ThO~ =< 0.02%(26.6%); Tm203, Yb2Oa(3"l%)

Lu203, Yb203, ThO* (5.7%); Yb2Oa = 96-99%, THO*(55.5%);

HT-HR-12 Lu2Oa > 95%, ThO2 < 4"9%(12"2%); Lu203, Yb203, ThO* Yb203 > 99"9%, ThO2 < 0"02%(75"6%) ; Yb203, Tm~O~(0.7%)

(11"5%);

HT-HR-I3 Lu203, YbzOa, ThO* (9.7%); Yb203 > 99.9%, ThO2 < 0.03%(38.6%); Yb203, Tm203(15"8%) ; Tm~O3 > 99.8%, Yb203 =< 0.2%, ThO2 < 0.01%(31.9%); Tm203, Yb20~, Er203 (4.0%) HT-HR-14 Lu203 > 96%, ThOz < 4%(8.5%); Lu203, Yb203, ThO* (31-5%); Yb2Oa > 99.9%, ThO2 < 0.07%(52.0%) ; Yb203, Tm203 (8%). Asterisk and parenthesis are showing the presence of small amounts of thorium and the percent of the recovered rare earth and thorium, respectively. Behavior of trace amount of Th during the eultion According to the elution sequence [6] obtained by 0-018 M H.E.D.T.A. eluant with a pH of 7.4 and 25°C, the elution position of Th is located between Tm and

Elution of heavier rare earths

2941

o

tlJ

= to

e~

~z

to

~5 ~0 eI--

-e

.=

e-

d ~ >.-

e-

;12

r~

_= i

,

,

I

i

o I/6' ep!xo-3EI

,

,

,

,

o

2942

Z. HAGIWARA

I00

F

HT-HR-12 60"C

E 50

=o 8

Th

o

i

i

450

~ i

r c. -~

L~

Yb

i

| 500

•

550

Eu l otevou l me,.l

,

,

• 600

curves of a heavier rare earth mixture containing a small amount of Th, using 0 . 0 4 1 M H . E . D . T . A . buffered with ammonium hydroxide.

F i g . 5. E l u t i o n

Yb in the lanthanide series. As seen in the elution curves, the contamination of rare earth with a trace amount of Th is observed over a fairly wide range, and the degree is different by changing the experimental condition. The present experiments show that Th-position shifts to Lu side with increasing both the concentration of the eluant and temperature. A schematic representation of the rare earth contamination by minute Th is given in Fig. 7, in which the movement of a small peak of thorium is seen by changing the concentration of eluant. This tendency may be understood from the comparison of the stability constants of rare earth and Th with H.E.D.T.A. The latter shows a value of 10 la'~ greater than LuH.E.D.T.A. chelate. One of the major reasons for Th-contamination, is due to 15o

ioo .%

Yb o 5o

Th

\

.

.

i , 525

3,50

375

400

Eluote volume,I

F i g . 6.

Elution curves of a heavier rare earth mixture containing a small amount of Th, using the eluant of Ca-NH4-H-Ch type C h = 0-05 M ) .

Elution of heavier rare earths

I'

Lu

' I'

Y0 ..

,,

Tm

I

.

~

Th-peak

2943

,I I O016M (25°6)

Th-peok

I

(60° C )

I.

1

I ooo-oo M (6O o C )

:~

Elutionsequence

Fig. 7. Schematic explanation on the contamination of the rare earth by a trace amount of Th.

the fact that as the diffusion coefficient inside the resin decreases with increase in ionic charge, the value of Th a+ would be one or more orders of magnitudes smaller than tripositive Yb and Lu ions, and the tailing of Th into the heavier rare earth takes place. This phenomenon will be expected to improve to some extent by using higher temperature, lower crosslinked resin and slower flow rate.

Analysis of mixing region with adjacent rare earths The elution experiments were carried out under the described conditions in Table 2. The theoretical analysis of overlap portions was made, using the experimental data obtained after the elution of the rare earth mixtures to a distance Table 2. Experimental condition used for the elutions of heavier rare earth at 60°C

Run No. HT-HR-I HT-HR-2 HT-HR-3 HT-HR-4

Composition of eluant same same same same

as as as as

Heavier rare earth* adsorbed

Flow rate? (ml/min)

Elution band length

Tm-Yb-Lu Tm-Yb-Lu Er-Tm-Yb-Lu Tm-Yb-Lu

30 40 40 40

3.0 3.0 2.5 2-5

HT-HR-11 HT-HR-12 HT-HR-I 3 HT-HR-14

*Rare earth loaded = Ca. 1 kg as oxide. ?I.D. 5 cm-column.

corresponding to 2.5-3.0 band lengths. According to Moeller's study[7], the temperature dependency of the stability constants ofTm-, Yb-, and Lu-H.E.D.T.A. is small in the range of 15-40°C, and these chelates show a same tendency for the change in temperature. For the present calculations, the separation factors are found from the values [2] of Mackey and Spedding at 25°C. Taking the mean value, the separation factor of adjacent rare earths in presence of H.E.D.T.A. is as f o l l o w s : aLuvb = 1"3, These

values are assumed

a vbTm =

1"6.

to be constant through the overlap zone.

7. T. Moellor and R. Fen-us, J. inorg, nucl. Chem. 20, 261 (1961 ).

2944

Z. H A G I W A R A

The plots of log [Yb]/[Lu] and log [Tm]/[Yb] against L, are represented in Figures 8 and 9, in which linearities are clearly observed. From the slope and o~-value, the H.E.T.P. is calculated as listed in Table 3. The results explain well Equation (17). On the contrary, when considered ion exchange from rate theory, the over-all mass transfer is divided into the following three steps: film diffusion, particle diffusion and chemical exchange reaction. As the chemical exchange reaction is very rapid, the rate controlling step is either film diffusion or particle diffusion or the combination of these two steps. Of course the rate controlling step depends on the exchange system employed. When the rare earth band had been displaced for a distance proportional to several band lengths by the use of the eluants mentioned in Table 1, the length becomes constant and it is independent of the elution distance. The overlap region with adjacent rare earths is also considered 0 31

HT-HR-I

i0 z

t

HT-HR-2

"~ i01

HTH -.~ IRI4-

iO°

60"C

IO'

10 2

i

I

50

I00 L:cm

Fig. 8. Relation between log [Yb]/[Lu] vs. L.

i

150

Elution of heavier rare earths

2945

HT-HR-3

t0°

Id'

60%

~ , , ~ 1 150

20O

25O

300

L,cm

Fig. 9. Relation between log [Tm]/[Yb] vs. L.

Table 3. H.E.T.P, values found in the elutions at 60°C

Exp. No. HT-HR-1 HT-HR-2 H T - H R-3 H T - H R-4

Overlap (a~ region Lu-Yb Lu-Yb Yb-Tm Lu-Yb

Slope of straight line 0-190 0.125 0-108 0.114

H.E.T.P. (u' H.E.T.P. (He) (° (cm) (cm) 0.6 0.9 1.9 1.0

~")Overlap region with adjacent rare earths listed is used for calculation. (bJCalcd. by Equation (17). (¢)Found by curve fitting method. *I.D. 5 cm column,

0.4 0-6 1"45 0-8

Flow rate* (ml/min) 30 40 40 40

2946

Z. H A G I W A R A

to be unchanged. In such a state, the total concentrations of rare earths, ammonium and hydrogen in the aqueous and resin phases are maintained to a constant value. If we consider only adjacent rare earths as main species, the shape of a rare earth boundary of Lnl and Ln2 is dominated by the separation factor in the presence of a chelating agent, and by the column operating conditions. In adjacent rare earths, behaviors resemble each other. Thus the diffusion coefficients of adjacent rare earth ions are approximately equal in the resin phase or in the aqueous phase, and furthermore the separation factor for the rare earth pair is considered to be approximately equal over the whole region with mixed rare earths. If the present exchange systems for the separation of heavier rare earths will be predominantly controlled by film diffusion, the height of the theoretical plate due to film diffusion, H s will be able to find from the curve fitting method, using theoretical curves (Fig. 2) obtained from Equation (16). The experimental data plots appear in Figs. 10 and 11, in which various data from the same run come on the same theoretical line. Therefore the results indicate that the exchange systems used favor rate control by film diffusion. On the other hand, when particle diffusion will be the rate I.O

x

05

HT-HR-3 . . 60°C . . .

I ,

~

-

. I

-I0

0

+10

+20

Z Fig. 10. Concentration of thulium, XTmvs. Z. I0

- Hfo4 HTHR

05

60 ° C

oJ

I

I

I -5

I

I

I 0

i

I

I

I

I + 5

Fig. 11. Concentration of ytterbium, xvb vs. Z.

I

Elution of heavier rare earths

2947

limiting step, H , will be found f r o m the theoretical lines in Fig. 2. It is clear f r o m T a b l e 3 that in spite of the complicated ion exchange system, H . E . T . P . values calculated f r o m equilibrium theory are slightly greater as c o m p a r e d to H r v a l u e s found f r o m film diffusion. SUMMARY

U n d e r steady state conditions, the separation of the heavier rare earth mixtures has b e e n carried out at 60°C with higher concentration of H . E . D . T . A . up to 0.05 M. T h e contamination of the rare earths due to a trace a m o u n t of thorium was also investigated under various experimental conditions. T h e experimental data were analyzed f r o m the points of rate theory and countercurrent extraction, and the theoretical plate height was finally found. On the other hand, s o m e relations concerning the plate height contribution f r o m film and particle diffusions were theoretically induced. According to the derived relations, the e x p e r i m e n t a l data were treated. It was concluded that the ion exchange s y s t e m s e m p l o y e d were mainly controlled by film diffusion as the rate limiting step.

Acknowledgements-The author is grateful to the Shinetsu Chemical Industry Corp. for supplying heavier rare earth samples. Thanks are due to Mr. lsao Terashima and Mr. Yoshihiro Koyama for the technical assistance through the experiments. APPENDIX 1. Derivation of Equation (4). As shown in Fig. 12, when a solution volume (V) has passed through the layer of Z and Z + A Z , the layer involves species i corresponding to q A Z X i ( V ) = {X~(V)+ /3C~(V)}qAZ. After the passage of further dV volume of solution, it contains q A Z X i (V + d V). Thus the change dX~is given by dX, = qA Z { X~( V + dV) --X,(V) } = q A Z [ {..Y,( V + dV) + flC~( V + dV) } -- {)(,(V) +fiG(V)}]

~(~' qAZtk-~"/zdV

+~ (~c,~ kOV/z

dr}

(l-I)

x i --_~

z =o

xi--I

Z

Z+AZ

×i=O xi = 0

xi,~

Fig. 12. Material balance in the ion exchange column.

2948

Z. H A G I W A R A

The solution enters the layer with C~(Z) and leaves with C~(Z+AZ). Therefore the change of ispecies, 8X~, is given, taking only the first term of Taylor's expansion. 8 X , - - - - d V { C , ( Z ) - - C , ( Z + A Z ) } -~ --dv(OC_'~ AZ

\OZ/v

"

(1-2)

F r o m dX~ = 8X~, one obtains

O.~l OC~ OCt q{(-~)z+fl(~)z}+(~)v=O.

(1-3)

When the band moves down the column proportional to AZ during the time (At), the material balance is expressed as follows: q A Z ( . ~ + tiC) = qvCAt. Thus the rate of the band movement is given

After the steady state, the band m o v e m e n t is kept to constant. Thus XdC~ = const. At boundaries, we have the following relations: x~ = 0 ~ = 0 (front boundary) xt = 1 ~ = (rear boundary). F r o m the above relationships Ct 2 i x, = X,, ~ - = ~ .

(1-5)

F r o m the relations (1-5) and v = 1/q(OV[Ot)z, (1-3) reduces to

1IX

\ IOXi\

IO.~i\

Using Equation 1-4, the following equation is induced

-~-/z = - k " ~ ) vv""

(1-6)

This is the same as Equation (4).

2. Derivation of Equations ( 1O) and (12). Using (1-5), -~Ln, is written, -YL., = .YXL.,. Therefore Equation (9) is expressed

o£Ln2Ln, --

1

1 -- XLn,

dxLn, = --

VbYoz

(2-1)

I ntegrating

1 [ f 1 A~ + f aLnni ] _ 14/) f L,,_---L--q" - - '~L,~ - dXLn,j = dZ. O~LnI -- 1 [ J XLn, J 1 -- XLn, VbYoz J

(2-2)

The above relation leads to Equation 10. 141) 1 1 [InXL~,-- aLL~ In(1 --XL~,) ] = -- v--~0zZ+ C~

oLLn~ -Lnl

F r o m Equation l I, we have fo 1 Z dXLn2

HPfol_ [ Ln*__ O/LflI 1

Ln2 lnXLn2+aLmln(1--XLn~)+(aLn2--1)Cl]dXLn2"

(2-3)

E l u t i o n o f h e a v i e r rare earths T h i s is e x p r e s s e d by

f/ ZdXLn2

Hp [ 4[1

al .... Lnl

|

OLTM Lnl)~-(

--

a[t.,,, . . . . I)C,].

W h e n the integration f~ Z . dXLn 2 is equal to z e r o , the integration c o n s t a n t Ca s h o u l d be equal to 1. n2 Lnz _ I)XL.~} 3. Derivation o f C*n~ = CXLnJ{o l LL.,-(aLn~

aLn2

x*,Xt ....

1 - x'L~h~ )CLn2

Lnl Xl~ll 2 X L n 1

XI~II2

1 --- X L n 2

XLn~

X[r,2-lntroducing.~Ln~ = xt.,~ and c~*n2=

Ln __ o~Ln __ 1)XLn~

OtLni

(

Ln~

cxi%into the a b o v e equation, w e have CXLn~ Ci%~ - - °t lL,.m. . .

Ln2 (O~Ln,

__

1 ) X ,~n~ '

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