Enginnyring
Ion-Exchange Chromatography of Trace Components
Process development I
A DESIGN THEORY THEODORE VERMEULEN
AND
NEVlN K. HlESlERl
D E P A R T M E N T OF CHEMISTRY AND C H E M I C A L ENGINEERING, UNIVERSITY O F C A L I F O R N I A , BERKELEY 4, CALIF.
I
ON-EXCHANGE columns have been utilized increasingly to carry out separations of rare earths, amino acids, and other inorganic and organic materials that are not isolated easily by the older chemical procedures. The present paper will outline a quantitative calculation method for interpreting and predicting the performance of fixed-bed separation columns under the limiting condition of low relative concentration-i.e., under “trace” conditions. The proposed method is based upon a kinetic or diffusional approach and reduces under specified restrictions to a result given also by the equilibrium plate theory of Mayer and Tompkins (86). The relation of an ion exchange separation to a bulk saturation of ion exchange resin will be of interest to the general reader. As shown in the upper diagram of Figure 1, an ordinary bulk saturation operation as performed on a process liquor involves a large extent of displacement of an innocuous ion from the resin by an undesirable ion (adsorbate ion) from the feed liquor. When the column is saturated t o the extent that breakthrough of the undesirable ion occurs, the flow is interrupted and the column is then eluted or regenerated by passing through it an excess of solution of the innocuous ion. Saturation operations will be considered here only to the extent needed for deriving the theory of trace separations.
undersaturated with respect to the adsorbed component and continually takes it into solution. Passing beyond the peak of the zone to the downstream (leading) side, the same liquid is supersaturated relative to coexisting resin and hence gradually redeposits the solute component. OPERATING VARIABLES
The elution separation just described is a function of a large number of chemical and physical factors, which will be shown in this paper to be amenable to mathematical interpretations. DISTRIBUTION RATIO. This is the dimensionless ratio.
DA
which is given in Equation 4, of solute concentrations in the solid and liquid phases. The distribution ratio affects directly the rate of linear advance ( U A ) of a zone; the ratio ( R ~ ) of A zone movement to fluid movement (I;)is given h5-
In order to carry out a separation, it iF; obviously necerisary for the values DA, DB, etc., of the components undergoing elution to differ substantially from one another. Under trace conditions the zone behavior for each solute is unaffected by the presence of other solutes. RELATIVEZONE SHARPXESS.This may be described by the ratio of the volume of effluent in which a component is contained (the zone width) to the entire volume of effluent. This ratio is controlled by the following six variables. EQUILIBRIUM PARAMETER, TA. This dimensionless number depends upon the exchange equilibrium betveen a trace ion, A , and the carrier ion, G, and depends also upon the concentration level of A relative to G. It measures the extent of symmetry of the chromatographic zone formed by component A . The more closely T A approaches unity, the more symmetric the zone usually is, and the sharper (relatively) it becomes as it passes through the column. A value of T A = 1 represents the condition of linear equilibrium, in which the distribution ratio, DA,has avalue that is independent of local concentrations. In ion exchange T A can be greater than unity in an unfavorable equilibrium or less than unity in a favorable equilibrium. TOTAL IONCONCENTRATION, eo, IN ELUTANT FEED. This is ementially the concentration of the gross component, G . A higher value of co will reduce the total elutant volume, V , and hence will generally decrease the time required to effect a separation. COXCENTRATION O F A COMPLEXING AGENT I N ELUTAXT FEED. Complexing agents increase the rate of zone movement, increase the value of T (usually toward unity), and may widen the differences in D values. LINEARFLOW RATE, U , PARTICLE DIAMETER,d,, AXD BED h. The relative zone sharpness will usually be improved LEKGTH,
SAlURAllON
RESIN COLUMN
--I
IlUlIOH-ltPLRAlION
Figure 1.
Schematic Diagram of Ion Exchange Column
The lower diagram of Figure 1 depicts an elution separation of the chromatogaphie type. I n this operation only a small amount of solution containing the components to be separated is admitted to the column. These components are then washed through (eluted) by an electrolyte solution (the elutant) that initially is free of them. The components travel through the column as bands or zones a t slightly different velocities. If the column is of sufficient length the zones will draw apart completely from one another and may be recovered in the effluent as separate solutions of each individual component in carrier electrolyte. Movement of a zone through the resin occurs by the following mechanism: Liquid on the upstream (trailing) side of a zone is 1
q;Pb/CAfE
Present address, Stanford Research Institute, Stanford, Calif.
636
March 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
by decreasing particle diameter at constant length and flow rate, by increasing the length, h, a t constant particle diameter and flow rate, or, to a lesser extent, by increasing the flow rate at constant particle diameter and constant residence time, h / U . However, the pressure drop, which is a function of these variables, must be kept within practical limits.
Figure 2. Variation in Symmetry of Elution Zone Concentration History Curves with Variations in r
AVERAGERATEOF INTAKE OF FEEDSOLUTION, Efaed.The zone sharpness required, together with the optimum combination of U , d,, and h, determines the time required for a cycle of feed and elution, and hence fixes the quantity of feed that can be separated per unit volume of resin. The latter value, taken with Rfeed,will determine the total volume of resin, v , required. A higher temperature leads generally to a sharper separation. However, m temperature enters implicitly in the mathematical treatment to be given, it will not be discussed as a separate variable. I n the design of an economical separation, further factors which improve the utilization of resin are listed as follows. These will be illustrated in the last part of this paper. 1. Recycle of an intermediate fraction of impure effluent 2. Multiple cycling in which a second feed charge is introduced into the column while the first charge is still being eluted 3. Multistage operation in which the separation is conducted in more than one column I n the design of a large scale separation it is obviously necessary to adjust the foregoing factors in order to achieve the most economic operzbting conditions. Even in cases where the theory developed for trace components does not apply exactly, it will still provide a semiquantitative interpretation and a pattern for future extension. SHAPES OF TRACE AND NONTRACE ELUTION CURVES
Saturation operations have been investigated a t length by rate theory approaches, but the case of chromatographic elution-i.e., of elution from an incompletely saturated column-has received little attention. An earlier paper by the present authors indicated a mathematical means of relating the concefitration history plot-effluent concentration us. time or effluent volume-of the chromatographic zone to the shapes derived on a kinetic basis for complete saturation and complete elution curves (16). As the equilibrium parameter, r, approaches unity, the concentration history becomes equal to the arithmetic difference of two separate saturation curves, drawn respectively from the times a t which saturation and elution were begun. In other words, the concentration history of the elution zone is a composite of suitably superposed saturation and elution curves. Since the r value for any ionic component changes toward unity as the ratio of its
637
concentration to that of total electrolyte in the solution decreases, the chromatographic behavior of a component can be described most simply when the component is present in trace concentration. The defisition of a trace component will thus be given on the basis of its value of r approaching unity within acceptably close limits. SYMMETRY OR ASYMMETRY. If r = 1, the concentration history for a chromatographic zone will take on a mirror image synimetry with reference t o its peak concentration, as the zone travels through a column of considerable length. Many asymmetric curves can therefore be attributed to deviations of r from unityLe., to nontrace conditions. (Other cases of asymmetry may be due to particle diffusion effects.) It will be seen that the r values for a saturation and a corresponding elution are reciprocals of each other. Also, for a column of given length, t,he smaller the value of r for saturation or for elution, the steeper is the saturation or elution curve. When r deviates from unity, the relative shapes of the leading and trailing edges of a zone will stili be indicated qualitatively by subtracting one saturation curve from another. If r < 1 for saturation, the leading edge of the zonethe first to emerge in the effluent-will be steeper than the trailing edge; if r > 1 for saturation, the leading edge will be more sloping than the trailing edge. Spedding and coworkers (39)have published a series of rare earth chromatograms in which the symmetry varies from the r < 1 type to the r > 1 type, as expected, when the effective concentration of complexing agent is increased by increasing the pH. Representative zone shapes for r < I, r = 1, and r > 1 are given in Figure 2. Because r depends upon concentration, the symmetry or asymmetry of an elution curve for any component depends upon the maximum concentration levels of that component in the solution and on the resin, relative to the concentration levels of the carrier component. The outstanding practical importance of using trace conditions for Chromatographic separations arises from the fact that at equivalent column lengths and equivalent flow rates, the symmetric or near-symmetric zones are usually much sharperLe., narrower-than zones that have a decided asymmetry.
1C'/c;
FLUID-CAPACITY,
t*, FROM
SGRT O F ELUTION
Figure 3. Elution Curves for Various Total Amounts of Trace Component
PEAKED us. FLATTENED ZONES. Both symmetric and asymmetric curves may be either peaked, rounded, or flattened in shape. The extent of flattening depends upon the total volume of solution or of resin within which the component was initially fed to the column, among other factors. Figure 3 skows the zone shapes predicted for different times of charging with feed solution, proportional to T, at a single flow rate which fixes s, with a feed concentration level, cg, small enough to produce synimetric curves ( r = 1). For the case shown, charging times proportional to T 5 15 will give curves of identical relative shape whose ordinates increase directly with T. A charging time proportional to T = 30 would still produce a peaked curve, but one more rounded than the previous group. A charging time propor-
638
INDUSTRIAL A N D ENGINEERING CHEMISTRY
tional to T = 80 would lead to a markedly rounded curve, in which the peak would maintain an effectively constant valuespecifically, eo-during a period of time proportional to At' 15. CHANGE IN ZONE SHAPE WITH INCREASING COLUMN LENGTH. A physical mechanism has been given for the movement of elution zones through a column. As discussed previously (10, 89, 41), a zone obeying a linear equilibrium-Le., one for which r = 1would retain a constant rectangular shape in the hypothetical case that complete equilibrium could be maintained between solution and resin a t each point in the column a t all times during its elution. Actually, equilibrium cannot be maintained, and hence the zone undergoes a continual spreading; an initial rectangular shape changes to a flattened-top bell shape, and this in turn develops into a fully peaked bell shape. A relative sharpening of each zone occurs because the spreading is proportional only to the square root of the column length, or still less if the zone is not fully peaked, whereas the total volume of elutant required for complete elution of the zone increases linearly with the column length. It can be shown that fully peaked zone shapes provide better separation between components than do flattened-top shapes. For maximum resin utilization, then, it is necessary to use the largest charging volume that will still give fully peaked zones in the effluent; in this situation, the zones will remain flattened or rounded as they travel through the column, and will not attain a fully peaked shape until just before they emerge. The effects of column length a t constant flow rate and of feed charge volume can be summarized as follows: The component which has the greatest tendency tomard flattening, generally the first component t o emerge from the column-the one with lowest D-will determine the optimum feed charge volume for a given column length. As d l be shown later, this optimum charge volume will increase with the square root of column length. Ordinarily an increase in column length will be made only t o improve the separation, because it actually represents a reduced utilization of repin, as based on volume of feed charged per unit time per unit volume of resin in cyclic operations. If the feed is introduced to the column as a narrov band of high (nontrace) Concentration, as is customary in many laboratory separations, a similar type of spreading will occur. The zone will first assume a flattened-top asymmetric shape, then a peaked asymmetric shape, and ultimately, because of the reduction of the peak concentration level, a peaked symmetric shape consistent with trace conditions It is observed that the optimum quantity of material that can be separated a t maximum efficiency by a given column, when fed in this manner, is nearly identical with the optimum quantity of material fed under trace conditions as an initially broader band of smaller amplitude; further, the column behavior for this optimum charge, or any smaller charge fed under pontrace conditions, becomes nearly identical with that predicted by the comparatively simple calculation methods derived below for fully trace conditions. Although these relations have a t present only an empirical validity, they are of great practical importance in making it possible to interpret and predict the behavior of separations which begin under nontrace conditions and end under trace conditions.
-
Vol. 44,
No. 3
tions, T A depends both upon the exchange equilibrium of ion A with carrier ion G and upon the concentration level of A relative to G. When ions -4 and G have the same valence, 01, and the exchange reaction is
A"+
+ G-resin ;=* A'resin + G"T
it can be shown ( 1 6 ) that
>There ( C A ) O = entering concentration of ion A during the feed period, ( c a ) ~= entering concentration of elutant ion G during the feed period (in the event that G is present in the feed), and co = ( c o ) ~ ( C A ) O . Csually co = (CC)O because ( C G ) O >>(cA)O. In the event ,4 is not charged as a trace component in a solution of G, co and ( C A ) Omust be evaluated for the elutant period, denoted by primee. Then CO' mill equal ( c b ) ~ , and (c;)o will represent a mathematical rather than actual maximum concentration of A in the effluent that d l correspond to a specified ?A. I t is desirable to consider also the more general exchange reaction in which ion A has valence and ion G has valence y
+
o(
?A"+
+ aG.resiny F== yA.resina +
aGY+
The equilibrium constant for this exchange is given by (3) and the partition coefficient, DA, satisfies the relation
where the concentration terms, expressed in gram- or pound , on unit weight of airequivalents, are: (9.resin) = q ~ based dried resin; (G.resin) = PG = - qA; (A"+) = CA, based on unit volume of solution. (cy+)= ca = CQ ca; * denotes concentration in equilibrium with the adjacent fluid; p b = resin bulk density, expressed as Yeight of air-dried resin per unit column volume (wet) as packed; and j~ = ratio of void space outside resin particles to total bulk volume of packed column. A t low concentrations of A relative to G (see Assumption l), the q z / c A ratio may be used to define an effective second-order equilibrium constant IC;' vhich applies only a t the special Q value
-
from which This relation enables the second-order rate theory for Tvhich r was first derived (16) to be extended to ions of unequal valence under trace conditions by use of the relation 1
r.4
A
1
+ [ K y - 11 (cA)O
(5B)
CO
DYNAMIC THEORY OF COLUMN PERFORMANCE
EQUILIBRIUN PARAMETER AS A CRITERION OF TRACE CONDITIONS. The equilibrium parameter, ?A, enters into ion exchange and adsorption calculations as one of three dimensionless parameters involved in solving the second-order kinetic equation, or diffusional equations which can be reduced to the second-order kinetic form (16). I n the exchange of gross components, ?A = ~ / K A .I n bulk adsorption, for comparison, rzd = 1/(1 -t Ktdco). ( I n these relations K A or K%dis the appropriate mass action equilibrium constant.) It has already been indicated that for chromatographic separa-
A trace cQmponent has already been defined as one for which r approaches unity within acceptably close units. As has been mentioned, elution zones for which r is near unity can become much sharper than those with a larger or smaller r. The reason lies in the fact that the volume of effluent solution AV within which the zone is completely contained is limited, if r > 1, by AV
or if r
> U ~ B D (-T 1)
ufzD
(:
- 1)
(6-4)
March 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
These ranges result from the nonvertical limiting shape of the saturation history for C A / ( ~ A ) Oat values of ?-A > 1, or of the elution history for saturation values of ?-A < 1. It is still assumed here that the elution zone history is given qualitatively by a superposition of saturation and elution curves for bulk exchange. Cbnsider now t h a t two components, for which the D values differ by lo%, are to be separated. A complete separation cannot be obtained unless the half-width volume of each zone, assuming they are of similar shape, is less than 5% of the total effluent volume. This requires also that the effluent volume AT' within which each zone is contained must not exceed about 10% of the total effluent that is required to complete their elution from the column-Le., 1(3% of v f E D . Equations 6A and 6B indicate that r must lie between 1.10 and 0.90 in order t o achieve this separation in even a very long column. These valuw of r may be taken as working limits, but must be further restricted if the separation factor-ratio of D's-is still nearer to 1.00, and may be widened somewhat if the separation factor is more favorable. By reference to Equation 5B, it is seen that the limits selected for ?"A correspond t o the restriction
ILK;'
-
11 ( C ~ ) o / C o l
< 0.10
(7A)
Thus, for a value of K y w k c h is large compared to unity, the peak value of CA (in the effluent, a t least) will need to be very small relative to a. However, if KY should approach unity, a second restriction must be applied in order to ensure that interactions between two or more of the trace components not cause other deviations from the present theory. CONSTANCY OF CARRIERCOMPONENT. It is implicit in this discussion that the gross component G must remain unchanged throughout the charging and elution steps. If bulk exchange of one gross component for another having different rate and equilibrium properties were superimposed upon the elution of the chromatographic zones, prediction of their behavior would become a much more difficult problem. For the present, therefore, the theory is restricted to the case where ion exchange occurs in saturation between a resin containing only component G, and a solution of component G that contains also a trace quantity of A ; or in elution between a resin containing G and a trace amount of A , and a regenerating solution that contains only G. When the trace criteria given in Equations 7A and 7B are satisfied, the following assumptions may be utilized in the mathematical analysis : Assumption 1. The changes brought about in quantity of component G both on the resin and in solution, by exchange with component A , are small enough that the concentrations of G can be considered essentially constant. Assumption 2. Any exchange of trace component A with another trace component B will be negligible in effect compared to exchange with carrier component G. Because of the nearly conDIFFERENTIAL RATEEQUATIONS. stant exchange capacity of a resin, the ions in the solid phase may be considered to form a stoichiometric compound, and the exchange may thus be treated as a heterogeneous chemical reaction. A sequence of three ionic processes will then be involved in order for an ion, A , in the flowing fluid to replace a gross ion, G, originally on the solid, if any. These are: 1. External counter-diffusion of A and G through the fluid film surrounding the resin particle (also, in some cases, through the fluid phase in open pores within the particle). 2. The exchange reaction, in which A from the fluid displaces G from the solid. 3. Internal counter-diffusion of A and G through the solid phase, provided the latter is a homogeneous gel. The effective rate of exchange will be controlled by the step that exerts the largest resistance to transfer. The mathematical
639
problem is simplified by neglecting all but this rate-controlling resistance or by combining all three steps into one effective step with a single apparent rate. I n external diffusion controlling, the rate is governed by the ionic diffusivities, and .also, in turbulent flow, by the eddy diffusivity whichcontrols the effective thickness of the residual laminarflow layer. The rate equation for ion exchange a t a given particle surface may, in the case of component A , be expressed as
Here r is the time variable, ( k p ) is~ the mass transfer coefficient, and uF is the transfer area per unit bulk volume. Under Assumption 2, the value of C A in equilibrium with a given qA is a function of the concentrations of component G only. Thus, applying Equation 5, Equation 8 becomes
Similar equations can be written for other trace components present. Since such relations will be identical in formfor each trace component, the subsequent derivations will consider only component A . For internal diffusion controlling, a n approximate treatment can be obtained by averaging the extent of saturation over the entire particle and by taking the rate of approach to complete saturation as proportional to the extent of saturation a t the outer surface of the particle minus the average extent of saturation within the particle. Thus
where ( k s ) is~ the transfer coefficient and as is the effective particle area per unit volume. By use of Equation 5 for the relationship between q A and C A , Equation 10 becomes
The rate of surface reaction will normally be very rapid and thus its resistance will generally be negligible. For the general case (CY # y ) , under Assumption 2, its rate when controlling may be given as
where ( k k h n ) A is the specific rate factor. introducing Assumption 1, there results
By rearranging and
It is a particular property of the r = 1 case that the rate equation has nearly the same form, regardless of which rate step is controlling. Equations 9, 11, and 13 can all be expressed by
For flow through a fixed-bed column, this becomes
where 7 = V / R , with V the cumulative influent volume and R the volumetric flow rate of the solution, and where ( P A ) - is the limiting resin concentration in equilibrium with a solution concentration of ( C A ) ~according to Equation 5. I n the event that more than one of the controlling steps will occur a t comparable rates, the resistances can be added in the customary manner.
INDUSTRIAL AND ENGINEERING CHEMISTRY
640
where Iois the modified Bcssel function-Le., Bessel function with imaginary argument-of the first kind and zero'th order. J is related to Thomas's function +, as follom
This provides the completely general definition of K a :
For an evaluation of column behavior, the appropriate rate equation must be solved jointly with a conservation equation. The latter expresses the fact that any loss of component A from the solution flowing through the section must equal the gain of component A in the solid phase and in the solution remaining in the section
By a transformation of variables =
Pb
[ b ( V dqA -
VfE)
1"
Vol. 44, No. 3
J
(SJ)
= 1
- e-g-z+(z,y)
(20)
T'aluei; of the J function for a wide range of values of s and t (or x and y) are plotted in Figure 4 on somewhat different coordinates than those used by Furnas and others (12, 17, 20). I n the more pertinent case of elution from an incompletely saturated column, as shown in the lox-er diagram of Figurr 1, the result has been partly anticipated in Stene's equilibrium-stage calculation ( 3 3 ) and in the discussion of a column completely saturated with a mixture of trace and carrier components by Boyd et al. ( 4 ) . The present authors (16) have obtaincd the following kinetic result for this t a s r
(l6.A)
The explicit solution of Equations 14 and 16 under the boundary conditions corresponding to either a saturation, or an elution following complete saturation, has been given by Thomas ( S 4 ) , and is reviewed in another paper by the present authors ( 1 6 ) .
I n this equation, the primes denote values in the elution period. For example
The flow rates R and U , and with them, the column capacity, s, should be based on the elution period but these variables are not primed because it is necessary to assume in the derivation that their values remain constant during both saturation and elution. By Equations 4 and 5
Further, T Ais the solution capacity for the saturation period and is given by
50
100
10,0130
t . F l u i d - C a p o c i t i Partrrnoter
Figure 4.
Master Plot for Concentration History i n Saturation
INTEGRAL RESULTS. CoLu~m CAPACITYASD SOLCTIOX CAPACITY P A R A n m T E R s . The mathematical solutions for C A / ( C A ) ~are best expressed in terms of two dimensionless parameters or moduli, the column capacity parameter SA
= RAV.~E/R = KAhSfE/R =
&h/U
(17)
where S = 7rdW2/4 = over-all cross-sectional area of the column, with d, = internal diameter of the (cylindrical) column, and the solution capacity parameter
where V ~ Eor h/U corrects the volume or time variable from an influent to a n effluent basis. In the case of a saturation operation in which r = 1, which corresponds to the upper diagram of Figure 1, the concentration leaving a column of volume v when a total volume, V,of snturating solution has entered the column is given by CA/(CA)O
=
J (sA,~A)
where
J (s,Y)=
1
- Jz
10( 2 f i € ) d €
wheie TTaat ( = n A / ( r A ) o ) is the volume of solution fed duling the charge period, Tsat , under actual or hypothetical trace charge conditions, and 124 is the total equivalents of A charged. Thus, Equation 21 describes a bell-shaped concentration history curve as the difference between a J curve drawn from the start of the saturation period and a second J curve drawn from the start of the elution period. C A R R I h R I O N C O N C E h T R A T I O K , C O l l P L E X I N G AGEWPS. Variations in the total ionic concentration of the solution, and thufi of the concentration of the cairier component G, influence elution by alteiing the solution capacity or column capacity parametrrs. Combination of Equations 15) 18A, and 22 gives
(19)
for external diffusion controlling, or
for internal diffusion controlling. The physical significance of Equation 23 is as follows: For external diffusion or surface reaction controlling, incrcaqing LO' at constant R / S will reduce the number of column volumes of solution required for a n elution, in the proper (or/? p o u e ~ inverse ) proportion, and will similarly reduce the time required without altering s or the relative sharpness of the elution zone. For internal diffusion controlling, Equation 23 shows that t a will be unchanged by increases in concentration; since in this case S A is proportional t o (&/cO')"/Y by Equations 15 and 17, increasing the carrier ion concentration xi11 lead to smaller S A values and thus to a diminished zone sharpness. The time required for elution will be reduced in proportion to the decrease i? s, because material balance ties t a t o S A , and T' is linked to f ~
.
, '
March 1952
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
If, instead, the zone sharpness is prescribed and the concentration is increased, then a smaller flow rate must be used so as ,to maintain S A constant and equal to its value a t the previous c, , and no change in the time results. The lower limit to the cycle time is reached a t a carrier concentration a t which external diffusion becomes as fast as internal diffusion. Operations a t still higher concentration levels-i.e., in the internal diffusion region-would be justified, if at all, by the smaller volumes of the effluent fractions. As the zone will contain the total number of equivalent; charged TLA, the material balance requires the concentration, cA, during elution to be increased in inverse proportion to the number of column volumes. Thus
This effect of gross concentration makes possible the recovery of trace components by elution as relatively concentrated solutions even if they were originally adsorbed onto the column from very dilute solutions, as in the copper-hydrogen exchanges studied by Beaton and Furnas (9). The direct effect of variations in concentration is often overshadowed by the more pronounced effect of complex formation. Frequently the presence of a complexing agent will alter the form in which the trace component exists in solution without changing the form in which it is attached to the resin; the result is a reduction in the effective partition ratio, D,and thus in the number of column volumes passed to reach the elution peak. The over-all equation becomes
The constant for complex formation is given by
with, now, C A = ( A X P - p X ) ) > > ( A ( * ) ) .The equilibrium constant for the exchange reaction as written above then becomes, analogously to Equations 3 and 4, Const. =
KX'Y KAX,(X(-x))p
($>"' D A X-~ 'E Pb
(25A)
which is a constant for any specific concentration of complexing agent; K A is the constant for exchange of the uncomplexed ion, as given specifically in Equation 3. It may be assumed that the diffusivities of the complexed and a n d uncomplexed trace ion are of the same magnitude, and hence that the trace component in the liquid phase diffuses mainly in complexed form. Ki'Y in Equations 4 and 22 will be replaced by the constant of Equation 25A, and hence
for external diffusion controlling. Thus, an increase in concentration of complexing agent (by decreasing D;, similarly to an increase in concentration of carrier component) will reduce V' or 7 for the elution without altering the re!ative sharpness of the zone. For internal diffusion controlling, t, is again $dependent of D; while S A decreases in direct proportion t o D A or t o 1/ ( X - x ) p . At a constant flow rate, S A and T are reduced in proportion when ( X - X ) is increased, or, by reducing the flow rate, S A and T may be kept constant. If surface reaction controls and it involves only the uncomplexed form of the trace ion, SA will again be proportional to l/(X-X)p, and the zonesharpness behavior will correspond to the behavior in the internal diffusion case.
641
The present theory applies to cases of complex ion formation in solution only when a negligible proportion of the complexing agent is consumed in complex formation-that is, when the solute is a trace component with respect to the complexing agent as well as to the carrier component. Otherwise 0;will not be constant through the entire concentration range of the elution zone and a more complicated analysis will be required. Tompkins ( 3 6 ) has pointed out that flat-topped zones can arise when KAX,is large and the eluting solution is deficient in complexing agent-Le., ( X ( - ' ) ) < p (c1)O; experimental curves corresponding to this situation have been reported by Spedding and coworkers (SI). GAUSSIAN-SHAPED ELUTION ZONES
NECESSARY CONDITIONS.Two basic assumptions have already been stated which make possible the quantitative evnluation and interpretation of elution curves in terms of the J values of Figure 4. As discussed earlier, however, the conditions favorable for chromatographic separations will generally lead to fully peaked, symmetric curves for the elution zones. The mathematical analysis of such curves is greatly simplified, and it will be seen shortly that their shape is predicted to be that of a Gaussian probability distribution, in the event that two further restrictions apply: Assumption 3. The volume of the resin bed, and hence the residence time of any element of the solution, should be relatively large. This leads t o a large value of s. Assumption 4. The period of partial saturation is relatively small, Le., T / s is small. The precise quantitative limitations involved on s and on T / s will become evident during the derivations. EXPONENTIAL FORM EQUATIONS. Xlinkenberg and others (20,Q, 39) have shown that for values of 2 and y > 5 the value of J (z,y) can be closely approximated by a simple error function. Thus, under Assumption 3, with s > 50 (for a maximum deviation in Jof *0.02)
where
r
r z
1
This converts Equation 21 to
e-J-2dS
(29)
If the upper limit is close to the hwer limit, Le., if TA is small compared to SA and t ~ the , theorem of the mean ( 4 8 ) allows the integral to be approximated by the product of an intermediate value of its integrand (here, e-") multiplied by the interval of the - f ~ ) . A suitable intermediate value is independent variable that a t the midpoint of the interval. Thus, under Assumption 4
(r~
(30) Additional simplification can be made by use of the binomial expansion (with €2 < 1 )
+
Provided that (I e) lies between (l.%)-l and 1.85, thefir&dws terms, Le., 1 0.5q will provide an approximation to eL/(e~)b,
+
whose maximum deviation will be no greater than 2% of the peak value [ For convenience the subscripts will be abandoned temporarily. Let A = t' s; and e = ( T A)/s, or E = A/s. The limitations on (1 E ) then introduce Assumptions 3 and 4. It follows that
( d G i dz) /4] . +
+
dT
Vol. 44, No, 3
INDUSTRIAL AND ENGINEERING CHEMISTRY
642
+ t'
6(+
=
+ 1)
t
+
"")2s
with
DERIVATION FOR ZONEWIDTH. TOTAL RECOVERY.The zone width is readily related t o the value of the column capacity parameter, S A . Equation 39B transforms to
(31)
and also I n terms of measured volumes, Equation 39C transforms similarly to This leads to (43)
(33) Let (Ii:),,' = the elutant volume entering the either time that the effluent concentration of A is maximum value; Le., a t e; = 0 . 6 ( ~ a ) m a x . Then (VL)max,]will be the half-width of the zone a t its and S A will be given by
Likewise
A more convenient Gaussian form is then obtained (35) As shown also in Figure 3, the location of the maximum on the scale is at the value of s diminished by 0.5 T. Equation 35 can be further simplified by neglecting 0.5 T in the exponent t o give the less accurate relation
column a t one-half its
[(Va)l/2
-
half-height,
If, instead, the half-width is measured a t the l / e height (36.8% of the peak)
't'
Equation 35 shows that the maximum of the concentration curve will be encountered a t (tA)[na~.
=
SA
- 0.5TA
(37)
Similarly, the less accurate Equations 36 and 40 lead to
It should finally be noted that Equations 38C, 39C, and 44 permit the concentration history curve t o be expressed in typipal Gaussian form without recourse to any extraneous variablr? whatever
with
with the interrelation between
%A
and (c;)~&
