Elutive Displacement of Precipitate Formed during Electromigration of Ions Joris Deman Kliniek uoor Radiotherapie en Kerngeneeskunde, Akademisch Ziekenhuis, Gent, Belgium The theory of chemical reaction during electromigration is extended to include the case where an ionic species is set free from the reaction product by the action of another ionic species that has greater affinity for the partner ion. Since precipitation reactions represent the simplest case with regard to theoretical treatment, computations and experiments are confined to this reaction type. When the precipitate is dissolved at one side of the precipitate region by the elution ions, the released ions, having the same sign of charge as the elution ions, move over the precipitate by electromigration, and react again with free partner ions at the other side of the precipitate region. Continuous elution will occur when the ratio between the concentrations of elution ion and partner ion does not exceed a certain value. This is an equilibrium state in which the precipitate region has a constant length and moves with constant velocity.
IN A PREVIOUS PAPER ( I ) , the assumptions underlying the theory, have been set forth. I will recall them briefly. The electromigration is thought to take place in a cylindric column of uniform cross sectional area, and a medium is used (e.g., gel), that impedes convective displacements that are not caused by direct electric current. Chemical reactions are occurring either between ionic species or between ions and neutral compounds. In order to avoid mathematical complexity, we limit our treatment to precipitation reactions and neutralization reactions. A nonreactive electrolyte is present in the column. The concentration of this electrolyte is uniform and exceeds largely the concentrations of the reactant ions. Consequently, the field strength may be assumed constant and uniform, the reactant ions contributing but a negligible part to it. We consider a part of the column that is far from the electrode compartments, so that the influence of products originating in electrode reactions can be neglected. Diffusion is left out of consideration. Ideal behavior with respect to constant activity coefficients and constant mobilities, is assumed. Variation of the effective mobilities with sorption, is neglected. We use a special nomenclature for the ionic species, according to their function in the elution process. “Competitor ions’’ (e.g. A”+) are those ions, which after having been precipitated, are brought in solution again. The ions to which they are bound in the precipitate, are oppositely charged, and are called “partner ions” (Rr-), “Elution ions” (Ee+), have the same sign of charge as the competitor ions and have greater affinity for binding with the partner ions.
initial concentrations: Le. concentrations with which the column was prepared electric field strength (dimensions: Voltjcm) the ionic velocity (cmlsec) the solubility product of the precipitation reaction: rAaf aRr- = A,R,, expressed in concentrations in gram equiv per cm8
+
KER=
P:
CEI’CR~
In the gel medium, the precipitate is very finely dispersed. Therefore, the term “concentration of precipitate” appears allowable. This concentration is denoted by the symbol P, and is expressed in gram equiv of either the cationic species or anionic species, present in the precipitate per cma gel
There are velocity symbols in which small lettered subscripts are used (up, ue, cd). They denote velocities of different fronts and will be explained further in the text. We use an imaginary x-axis parallel to the column. The positive side of it points in the direction in which the positive ions move. Consequently, the velocities can be either positive or negative depending on the direction of movement. FRONT FORMATION
It was demonstrated in our previous paper ( I ) that front formation occurs spontaneously when regions with reactive and oppositely charged ions, meet each other during electromigration. In Figure l a , the initial concentration distribution is shown. In Figure l b , the ionic regions have made contact with each other. Precipitation actually occurs only at section P. (The width of the reaction region is very small.) If the solubility product of the precipitate is small with respect to the concentrations of the reactant ions, then it can beshown that the velocity of displacement of the precipitation front is:
The precipitate remains immobile at the place where it was actually formed (Figure 2 ) . Owing to the movement of the front, a precipitate region is formed in which the concentration of precipitate is:
Symbols:
mA, mR, mE.’
effective electrophoretic mobilities of competitor ions, partner ions, and elution ions, respectively (dimensions : cmZ/ volt sec) concentrations expressed in gram equiv per cma
-
C A , C R , CE.‘
(1) J. Deman and W. Rigole, J. Phys. Chem., 74, 1122 (1970).
When all competitor ions have reacted with partner ions, the bulk of the precipitate region thus established, has a uniform concentration given by Equation 2. At the borders of the region, there will be a deviation from this common value, since precipitate formed there arises from competitor ions that reacted in a concentration not equal to CA,O (see Figure la). Now we suppose that behind the region with competitor ions A&+,elution ions Ee+ are present. Since the latter ions
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1970
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---
r-----
Figure 1. a. Ionic distribution before precipitation b. Ionic distribution after formation of precipitation front F
moved from section Q to section F, leaving behind a precipitate region (hatched)
Rr-
4-
I
Figure 3. P: precipitation front E: elution front
+ eA,R,
=
aE,R,
+ erAaf
(3)
The reaction between Eef and RP-can be a precipitation reaction: e . g . , 2Fe3+ ~ C O ( O H= ) ~2Fe(OH)3 3c02+ or a neutralization reaction: e.g., 2Hf Co(0H)z = 2Hz0 Co2+ The elutive capacity of the Ee+-ions,implies that:
+
+ +
+
CEI'CR~
=
K E R< KAR
(4)
The elution reaction will give rise to a front, since in the theory ( I ) no objection exists to the fact that one of the reaction partners (A,R,) has a mobility zero, provided the other partner (Ee+) moves (Figure 3). ELUTION
We suppose that we are dealing with a column in which the ionic concentrations of elution ions and partner ions are such that a front of a direct reaction between Eef and R'- would move in the direction of the cathode. It will be shown in the next section that this condition is not only necessary, but also sufficient for displacement of the precipitate A,R, in the 1700
a
I
E
P
+
have greater affinity for the partner ions Rr-, they will bring in solution again, competitor ions from the precipitate. It is important to note that whereas the region of competitor ions is assumed to be of limited length, the regions of partner ions and elution ions are supposed to be very long, constituting in this way a quasi inexhaustible supply of ions. The elution reaction can be written as follows: arEeS
I
-
direction of the cathode whatever the initial concentration CA,O of the competitor ions might be. Suppose the elution front has moved during a small time interval d t , over a distance dx (Figure 4). If no precipitate had been present, the distance covered would have been : m~ E dt. This implies that mEECE ,oqdf
(a)
gram equivalent of elution ion has crossed the section SO with cross sectional area q (in cmz). The concentration P of A,R, is, generally speaking, a function of x. At each moment of time, equivalent amounts of elution ions and Precipitate react. Since P and cE both are expressed in gram equivalents per cm3, we may write that in a time dt Pqdx
(b)
gram equivalents of elution ions have disappeared from the solution. The amount of elution ions present in solution at the right of section So, is obtained by subtracting (b) from (a): mEECE,oqdt - Pqdx
(c)
This amount is also given by the expression : CE , o d x
( 4
We neglect the amount of elution ions that has passed section SI. Their concentration there is very small because it stands in equilibrium with precipitate A,R, in concentration P (Equation 3). Equalling (c) and (d) yields:
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1970
Figure 4. Displacement of the elution front. See text mEECE,oqdt - Pqdx
=
CE,oqdX
The velocity of the elution front is: dx dt
MECE,o
D e = - = -
+P
CE,O
E
(5)
Comparison with Equation 1 shows that P with velocity zero, has replaced CR,O. In order to derive a formula that gives the concentration of the eluted competitor ions, we suppose that the initial competitor ions with concentration C A , O , all have crossed the rear section So of the precipitate region, before elution takes place. Then the concentration CA will be determined only by the velocity of the elution front, the concentration of precipitate, and its own mobility MA. The amount of competitor ions eluted in a time interval dt, is : Pc,qdt
(a)
The distance covered in the same time by the elution front, is: v,dt by the eluted competitor ions: mAEdt. When the elution front is at section SI (Figure 4)) then the competitor ions that were released first, are at section T. The volume filled with competitor ions is:
1 mAE - ce 1
qdt
(b)
We have to take the absolute value of mAE - ue because the velocity of the competitor ions can be either greater or smaller than the velocity of the elution front. When (a) is divided by (b), we find: CA
=
-I
Po, ~ A E-Oue
'
mECE
= ~
I
(mA
~
Likewise for the concentration of precipitate formed at the precipitation front :
P-1
(MA
1
+
+ MR)CICR,O
?nACA
-
mRCR,O
(8)
It will be seen in the following section that transitory states can occur in which all precipitate A,R, has been eluted. The elution ions make direct contact with the partner ions Rr-. All competitor ions are then in solution and move behind the front of a direct reaction between elution ions and partner ions. The velocity of this front is found by substituting in Equation 1,ME and CE,O for mA and CA,O.
CE,O
1- C R , O
THE EQUILIBRIUM STATE
Two possibilities are considered. A. The initial precipitate concentration is such that the velocity of the elution front is greater than the velocity of the competitor ions. (Ue)init
0
- mE)cE,O P
c.4 -f C R . 0
> MAE
The precipitate region will be dissolved completely, and a direct front will form between the elution ions and the partner ions. The velocity of this front, as given by Equation 9, depends upon the concentrations CE,O and CR,O. We suppose that these concentrations are such, that:
ue is replaced by its value from Equation 5 : CA
(distance) and time. For this reason, we have derived Equations 5 and 6 in a general manner, taking P as variable. Equation 1 for the velocity of the precipitation front must be changed accordingly. The uniform concentration CA,O must be replaced by the variable concentration CA:
ud
mA
Let us take an example. Suppose the initial concentration
P is such that the velocity of the elution front is nearly equal to the velocity mAE of the competitor ions. Then, according to Equation 6', a high concentration CA will be produced. This concentration moves in a narrow band together with the elution front. The precipitate region will be eluted and disappear until this new c*-concentration reaches the precipitation front, giving rise to a precipitate with another concentration. This in turn, alters the value of f i e . It becomes obvious that in the course of elution, the values of P formed at the precipitation front, and CA eluted at the elution front, must be taken as variables, changing with section
< mAE
In this case, the competitor ions will overtake and cross again the direct reaction front. Precipitate A,R, is formed again and the elution ions become separated from the partner ions. A new elution front is formed of which the velocity is dependent upon the new concentration of precipitate. To get some insight into the phenomenon, one can consider the fact that the velocity Dd is constant (cE,0, CE,O, ME, and m R are constants), whereas the velocities ue and v, are dependent, respectively, upon the concentrations P (Equation 5) and CA (Equation 7). When the competitor ions are captured again between the two fronts, then this implies that P i s this time so great that Ue
< Ud
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1970
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$6
Figure 5. Dependence of velocity of precipitation front on Concentration of competitor ions. E = 2.5 Voltlcm; c R , ~ = 5 x 1 0 - 6 gram eq./cma; mA = 0.5-10-a cm2/ Volt.sec.; mR = 1.0 x cm2/Volt.sec
0
The mechanism for the adjustment of the P-value is described in the following paragraph B. Since it has been assumed that ud < mAE, it follows that: ce < m.4E
Consequently, the Competitor ions remain captured in the precipitate region A,R,. Brought in solution by the elution ions, they pass over the precipitate and react at the precipitation front. Summarizing, we conclude that the initial condition v e > mAE will change t o the condition v e < mAE, if the competitor ions are electromigrating with greater velocity than the front of a direct reaction between elution ions and partner ions. B. We suppose that the velocity of the elution front is lower than the velocity of the competitor ions.
ve
< mAE
We leave it an open question whether this condition was fulfilled from the beginning, or has been reached by the mechanism described above. We now will demonstrate that the velocities 0, and ve automatically adjust themselves to the same value. For the help of the reader, it is shown in Figure 5 that v, varies with CA. For this figure, we have chosen arbitrary values for mA, m ~ C R, , O , and E, and have inserted them in Equation 7. Only the positive part of v, must be considered. Let us assume a situation in which the precipitation front moves faster than the elution front. up
t
v, in cm/hr
15
20
precipitate reg'on decreases until the two front velocities are equal. The condition ue = up characterizes the equilbrium state of the system. In this state, competitor ions in concentration c ~ , ~ ,are , precipitated at the precipitation front, giving rise to a precipitate with concentration Pea, This concentration is such that, when eluted at the elution front, competitor ions come free in the same concentration C A , ~ , as that present at the precipitation front. Hence the value P,, is found by equalling the concentration CA at the elution front, with that at the precipitation front (Equations 6 and 8). This yields: (ME
peg
=
+
~R)CE,OCR,O
mECE,O
- mRcR,O
(1 0)
This value of P is brought in Equation 6 :
These results are remarkable since they indicate that the concentrations cAand P, after equilibrium is installed, are dictated by the concentrations and effective mobilities of ions that are outside the precipitate region. The equilibrium velocity of the system is found either by substituting P,, from Equation 10 into Equation 5 , or C A , ~ , fromEquation 11 intoEquation 7. The result is:
> De
The condition implies that the length of the precipitate region will be increasing. This means that the concentrations CA and P of the competitor ions, both are decreasing, since it can be seen from Equation 6 that CA and P vary in the same sense. When P decreases, then the velocity of the elution front increases according to Equation 5. The decreasing cA-value, which after electromigration reaches the precipitation front, gives rise to a decrease of its velocity (Figure 5 or Equation 7). The increase of ue and decrease of up will continue as long as 0, > f i e . This implies that ultimately up will become equal to
The elution front moves with the same velocity behind the competitor ions, as would be the case when no competitor ions were present, and the elution ions were in direct contact with the partner ions. In paragraph A, it was pointed out that m d > 0,sis the condition for the competitor ions to remain captured between the elution front and the precipitation front. This now becomes evident since in the equilibrium state, the whole system moves with velocity ud. When the denominator and numerator in Equation 12 both are divided by C R , ~ ,we can write the unequality mAE >. ud as follows:
08
up = ve
The same reasoning can be followed, mutatis mutandis, for the case that u, < u,. In that situation, the length of the 1702
25
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1 9 7 0
CR.0
cR,O
1
KCI 0 . 2 t ~ HCI 1.5I O - ~ N
J
)3Iml: KCI 0.2N t CoCI,
WN
3 KCI 0 . 2 ~+ KOH 4 ioe3@
Figure 6. Composition of gel column before elution
The ratio
CE 0
I I I I I I K
is the only variable in this expression. It can
CR,O
be varied at will in experiments. The length dA,eqof the precipitate region in the equilibrium state, can be computed in function of the length ~ A , Oand concentration CA,O of the original region of competitor ions before precipitation. We may write : d.4,0c.4,0= ( p e q
f
CA,eq)dA,eg
or
c ~and, P,,~can~be replaced in this expression by their values given by Equations 10 and 11.
EXCHANGE REACTIONS DURING ELECTROMIGRATION During elution, the free competitor ions stand in chemical equilibrium with the bound ions. Hence, each competitor ion will spend some time in the immobile phase and will acquire an actual mobility equal to
The ratio
CA
~
CA
+
gives the time fraction the ions have been P A
free. In the formula for the velocity of the precipitation front, Equation 7, the electrophoretic mobility mA and not the actual mobility, has to be used, because at the precipitation front, it is only the free ions that react, and it does not matter to what extent they have undergone previous exchange reactions. Also the elution ions will exchange with the product E,R, before they reach the elution front. For the same reason as above, it is not the actual mobility rn; but mE that must be used in Formula 5. Elution essentially is an exchange reaction. At the elution front, the competitor species is displaced by the elutor species.
Figure 7. Development and movement of the Co(OH)z precipitate. Dotted area: red (CoZ+); hatched area: green [Co(OH)2]. I: after 20 min; 11: after 98 min; 111: after 180 mia; IV: after 240 min
The free ions and the bound ions can be considered as a whole, i.e., as an entity that is moving with velocity m*E. When this velocity is multiplied by the total concentration (c P),we obtain the amount of species that is transported per unit of square section and time. It was shown in another paper (2), that for sections where E and A are pregent together:
+
When it is supposed that a = e, Le., the competitor ions and elution ions carry equal charges, the expression simplifies, and the condition for the competitor species to be displaced at a higher velocity than the elution species, becomes : m.k&R'/'
> ~EKER"'
(1 6 )
The condition for Ee+ to be a true elution ion is given by the above expression, rather than by the unequality KAR> KER as assumed so far. It was shown (3) that when ions undergo exchange reactions during electromigration, their actual mobility exhibits a random distribution around an average value. We have pointed out (2) that an additional cause for fluctuation exists when exchange occurs with precipitates, because the bound ions exhibit different degrees of accessib in the precipitate granules. In the paper mentioned, we also demohstrated that the fluctuation effect does not impair the validity of condition 16, although it will be the cause of some blurring of the elution front. The precipitation front remains sharp, because there, the reaction is a direct precipitation between competitor ions and partner ions.
(2) J. Deman, ANAL.CHEM., 42, 321 (1970). (3) P. C. Scholten and K. J. Mysels, Trans. Favaduy SOC.,56, 994 (1960).
ANALYTICAL CHEMISTRY, VOL, 42, NO. 14,
DECEMBER 1970
a
1703
A
+
KCI 0.2N HCI 75 Id3 &
Figure 9. High precipitate concentration, 1: first appearance of precipitate; 11: after 170 min KCI 0.2N -I-KOH 2
I-
2tm
162fj
1 Cd"
II:
Cos'
4 . i . Figure 8. Preparation of a gel column giving a precipitation band with high concentration EXPERIMENTS
All experiments were performed with agar gel columns (1 %, w/v), (Special Agar-Noble, Difco Laboratories, Detroit, Mich., U.S.A.), prepared in glass tubes of 1-cm internal diameter. The mobility of the ions is decreased because sorption to the agar molecules occurs. Since there is uncertainty about the extent of sorption for the different ionic species (also variable with ionic concentration), we did not know the exact values of the effective mobilities. Consequently, the experiments required some preliminary trial and error, and it was not possible to verify quantitatively Equations 12, 13, and 14. However sufficient evidence is present to confirm the general conclusions of the theory. 1. A column is prepared of which the composition is shown in Figure 6. The cobalt ions (competitor ions) initially were present between section A and B. Below section B, there is a long region that contains hydroxyl ions (partner ions) which give a blue-green precipitate with eozT. Above section A, hydrogen ions (elution ions) are present which dissolve the precipitate; 35 mA of direct current is applied. The red color of the cobalt region is displaced downward and simultaneously a region with green precipitate is formed (Figure 7). The width of the precipitate region becomes larger with time, whereas the red Coz--region disappears. When all cobalt ions have placed themselves between the elution front and the precipitation front, the length of the precipitate region. becomes constant (2.8 cm), and the green band moves downward with a constant velocity of 2.1 cm/hr. 2. We choose another ratio between the concentrations of the elution ions and the partner ions: Above A: [HCI]: 2 X 10-aN UnderB: [KOH]: 3.5 X 10-3N At first we get a moving precipitation band. After 90 min, the precipitate has disappeared completely. Later on, the precipitate is formed again at a greater distance from section B. This time the formation of the precipitate region is definitive. The precipitate density is higher than that of the foregoing experiment. 1704
e
Figure 10. Separation between CO and Cd
W+l
Because the ratio -__ this time is higher than that of the [OH-] first experiment, the elution front moves faster than the cobalt ions. However, since the reaction front H+-OH- is moving with a lower velocity than the cobalt ions, these ions overtake the direct reaction front and place themselves again between the elution ions and the partner ions. 3. If the ratio
M+I is enhanced again: [OH-] e.g.
[HCl]: 3 X 10-3N [KOHI: 3 x 10-3~
then the cobalt ions are not precipitated again after dissolvation of the initial precipitate. This means that the velocity of the reaction front H+--OM- is greater than the velocity of the cobalt ions. 4. From Equation 10, it appears that the precipitate concentration will increase with increasing C E , O and C R , O , on condition that VZECE,O - ~ R C R , Oremains small, i.e., the velocity of the precipitate band must be kept small. Appropriatl conditions were present in the column given in Figure 8. A current of 35 mA is applied. After 40 min, the first precipitate appears. The precipitate concentration is high and the width of the band grows to 0.4 cm. The band moves downward with a velocity of about 0.7 cm,thr. (Figure 9). 5. The elution procedure can be used for the separation of ions. We prepare a column in which the under gel layer con3 X IOVN MQH. Above it, 1 ml of gel is tains MG10.2N placed containing 0.05N Cd2+ -C 0 , O W Co2+. The upper gel layer contains KC1 0.2N 1 X 10-3N HC1. A current is applied of 30 mA. At first the blue-green CO(OH)~ precipitate is formed, After displacement of the precipitation front, the white Cd(QH)? is precipitated behind. After the bands have separated and have assumed constant lengths, they are displaced with a constant velocity of about 5 cm/hr (Figure IO).
+
+
RECEIVED for review May 14, 1970. Accepted August 17, 1970.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1970
