NATIONAL BUREAU OF STANDARDS REPORT

IK 024

Progress Report

on ADSORPTION THEORY INTERPRETATION OF

Ca

45

AND P

32

EXCHANGE WITH HYDROXYAPATITE

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

3 1 1.05- 11-31 1056 1 December 31, 1968 10 024

Progress Report

on ADSORPTION THEORY INTERPRETATION OF

Ca

45 AND P

32 EXCHANGE WITH HYDROXYAPATITE

* Research Chemist, Dental Research Section, National

Bureau of Standards, Washington, D. C. 20.234.

This investigation is part of the dental research program

conducted by the National Bureau of Standards in cooperation

with the Council on Dental Research of the American Dental

Association; the National Institute for Dental Research; the

Dental Research Division of the U. S. Army Medical Research

and Development Command; the Dental Sciences Division of the

School of Aerospace Medicine, USAF; and the Veterans

Administration.

by

William V. Loebenstein*

IMPORTANT NOTICE NATIONAL BUREAU OF <

for use within the Government

and review. For this reason, whole or in part, is not auth Bureau of Standards, Washini

the Report has been specitical

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015.

;ress accounting documents intended

is subiected to additional evaluation

ire listing of this Report, either in the Office of the Director, National

r, by the Government agency for which

I copies for its own use.

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

- 1 -

ADS ORPTI ON THEORY INTERPRETATION OF

Ca^5 and P

52 EXCHANGE WITH HYDROXYAPATITE

W. V. Loebenstein

ABSTRACT

A theory has been developed for characterizing adsorption from solution based on Langmuir kinetics. An unusual

application of the theory has been made to the simultaneous

interaction of Ca —calcium ions and of P —phosphate ions

in solution with synthetic hydroxyapatite crystals. The

saturated solution is comprised primarily of non-radioactive

calcium and phosphate ions in order to assure no net change

in the overall ionic concentration. In applying the adsorption theory to each radioactive species in turn, as soluteadsorbate, it is convenient to consider everything else in

solution as solvent. In analyzing the model, a distinction

is made between two processes:

(l) an Irreversible process where the tagged

ion is so strongly adsorbed that it is no

longer accessible. This process is assumed

to be confined to a certain weight fraction

of the substrate.

A reversible process where the tagged ion

occupies one of the surface crystal sites

( 2 )
- 2 -

yet is free to dissolve (desorb), and this

process is confined to the remaining weight

fraction of the substrate.

Modem high-speed digital computer techniques have been

employed to solve the problem in terms of physically-significant characteristic parameters. This has been accomplished

for a number of experiments over a range of pH values for

systems varying in slurry densities and tagged-ion concentrations. Based on the assumed model, no change in calcium to

phosphate ratio of the surface composition was evident,

regardless of variations in the solutions of pH, total ionic

concentration, and ionic ratio.
- 3-

1. INTRODUCTION

Studies of the solubility and crystallization of hydroxyapatite are of fundamental importance to an understanding of

the chemical dynamics of tooth structure. Batch adsorption

kinetics have been useful in earlier studies

1 ^

^ in characterizing the adsorptive properties of a surface with respect

to one or more components in solution. Through such studies

it has been possible, for example, to measure the population

of sites capable of adsorbing a solute without having to fill

them all in the process.

This measurement of the site population would constitute

one of the characteristic parameters of the system and should

be independent of variable quantities such as the mass of the

adsorbent, the amount of solution, the initial concentration

of solute, the time of measurement, or the amount of solute

initially adsorbed.

The possibility of distinguishing simultaneously the respective concentrations in solution of radioactive calcium

and phosphorous from their untagged counterparts affords an

unusual opportunity to observe the deposition of these radioactive ions from solution under conditions of equilibrium

between crystals of synthetic hydroxyapatite and saturated

solutions of their non-radioactive ions.

The so-called "adsorption" sites are in reality the

positions in the crystal lattice which are available to each
-4-

specific ion, since the crystal does not distinguish between

tagged and untagged isotopes of the same element.

The concentrations of radioactive ions in solution were

always several orders of magnitude smaller than their nonradioactive counterpart. This is a necessary condition for

a valid application of the proposed adsorption model to isotopic exchange. The model used in the present treatment

permitted an irreversibly adsorbed radioactive ion to migrate

to the interior of the substrate by diffusion, exchange, or

other mechanism, just as long as it retained its adsorbed

status. On the other hand, a reversibly adsorbed radioactive

ion was constrained to the actual interface layers always

accessible for desorption back into the solution.

2. MATERIALS AND PROCEDURE*

The hydroxyapatite was prepared by a slow addition of

stoichiometric aliquots of Ca( 0H )2 in suspension with H-^PO^

in a C02 -free environment, the resultant precipitate being

stabilized by prolonged boiling. Its composition was confirmed by infrared absorption and by chemical analysis. Its

BET area was 26.4 m g” determined by N2 adsorption at 77°K.

Details of the preparation of the saturated solutions

in equilibrium with the hydroxyapatite (HA) suspension together

* Dr. Yoram Avnimelech (present address: Israel Institute of

Technology, Haifa, Israel) carried out the experiments described herein. He has kindly made available his experimental

data for the purpose of the present investigation.
- 5

-

with the stock solution of the tagged ions have been pror 4

1

vided by Avnimelech. J The isotopic exchange experiments

were carried out in a manner identical with batch adsorption. The suspension consisting of several hundred milliliters was kept agitated in a constant temperature bath held

at 25 ± 0.02°C during the course of the experiment. Zero

time was taken as the instant of innoculation with the radioactive tracer ions. Five milliliter samples of the suspension were withdrawn for activity analyses of the solution at

frequent time intervals beginning at one minute, or less,

and continuing for durations ranging up to nearly four hours.

In demonstrating the existence of reversibility,

Avnimelech introduced an unusual innovation which is seldom

rsi encountered 1 J in adsorption work because of the complexities it entails in the fitting of theory with experiment.

This consisted of making sudden additions, from time to time,

of tracer-free saturated solution to the reaction vessel.

This brought about an abrupt dilution of the radioactive ions

without disturbing the concentrations (or pH) of the nonradioactive components of the solution. As many as three

such dilutions were made in most of the experiments. The

several experiments differed from one another in weight of

solid, volume of solution, concentration of radioactive

tracer ions, concentration of non-radioactive saturated

solutions (which is related to pH at equilibrium), times and
- 6 -

frequency of sampling, and time and amount of dilution.

3. BACKGROUND THEORY

A three-parameter equation based on Langmuir kinetics

\2 31

was derived and tested in earlier work 1 and found to

apply rather well to reversible adsorption from solution.

That equation was expressed as follows

:

(M-N)--? _ (M- N >- qIN e

‘ 2(^)Nklt

(1)

(M+N)--q (M+N)-qIN

where

V

k.

M - — [j— + 0 (1 +

2W kl

0

Wq

( Vo"

•)]

and

N = (M

2

- q Q

c Q V/W ) 1 /

2

( 2 )

( 3 )
- 7

-

and where

q » the amount of solute adsorbed per unit mass of the

adsorbent at any time t

;

q Q « the maximum value q would have if all of the adsorption sites were filled;

c = the instantaneous solute concentration whose value

would be e if no ions were adsorbed;

W » the mass of the adsorbent;

V = the volume of the solution;

k^ = the specific adsorption rate constant;

kg » the specific desorption rate constant;

qIN * the amoun^ solute adsorbed per unit mass of the

adsorbent at onset of the sorption process.

Equation (l) can be rewritten in a more useful form in

which q is explicitly expressed;

(M-N)-qIN ^Nk-^ W/V

A

q

(M-N)-qIN

e ^2Nk

1

t W/V

(M+N)-qIN

+ 1

• N w

1
- 8-

A distinction is made between q and q in that the former

quantity is the theoretical or computed value,, while the

latter is used to designate the experimental amount adsorbed

per unit mass at any instant and is related to the concentration in solution according to the conservation equation

which is always valid; namely:

Equation (4) was derived from the differential equation

which expressed the rate of adsorption dq/dt as the difference between the forward and reverse processes:

The concentration factor c was eliminated between Eqs . (5)

and (6) prior to integration.

If 1<2 in Eq. (6) were set equal to zero, an irreversible

adsorption would result and the integrated form Eq. (4) subject to Eqs. (2) and (5) would simplify considerably.

In the practical case of adsorption from solution, most

systems exhibit to some extent both reversible and irreversible attributes. Since this dual behavior was also observed

in the present investigation, it follows that a two-process

representation would be the very simplest combination that

could account for all the results observed.

q = (o

o

- c) V/W (5)

dq/dt = k

1 (q Q

- q) o - kgq ( 6 )
- 9

-

At this point it is necessary to distinguish between

different possible models or mechanisms. The choice of the

particular model for treatment in this paper was not based

on any preference of a fundamental nature. Other mechanisms

(one of which is discussed in the text) are equally plausible

from a theoretical point of view. They can also and, indeed,

should be tested against experiment before any choice is made

as to which mechanism is preferred. The principal purpose

of this paper is to illustrate how an assumed model can be

tested. The two most probable mechanisms for the adsorption

kinetics of the dissolved species are:

(a) Sequential Adsorption

(Solution) **

( Reversible v adsorption''

/Irreversible s 'adsorption '

and

(b) Competitive Adsorption

(Solution)

/Irreversible^

'Adsorption '

/Reversible v 'Adsorption-'

In the present paper, the latter mechanism was arbitrarily chosen for detailed study.

A crude model applicable to the present investigation is

illustrated in Pig. 1.
- 10 -

The first few surface layers of the solid are depicted

in some regular array at the bottom of the figure. Only the

calcium ions (small circles) and the phosphate ions are represented. The shaded circles identify the tagged ions. It

is evident that a tagged ion which is depicted as inaccessible

would correspond to an irreversibly adsorbed state. However,

an ion which occupies a site on the outermost layer in contact

with the s&ution might still be irreversibly adsorbed. Both

adsorption sites, reversible and irreversible, may be thought

of as competing for each ion in solution.

This situation is indicated mathematically by Eqs . (7)

and (8)

:

dq

2

/dt = k

11 (q 01 -q1

)c (7)

= ^12 ^02~^2

)

c “ ^22^2 ( 3 )

where the irreversibility of Eq. ( 7 ) is assured by the absence

of a desorption term. Equations (7) and (3) are logical extensions of Eq. (6) in which the additional subscripts 1 and

2 identify the process as either Irreversible or reversible.

The parameters q^ and q^j for example, now represent the

concentrations of irreversible and reversible adsorption

sites, respectively. The interdependence of the two processes is determined by the common factor, c, the concentration

of the particular tagged ion under investigation.
- 11 -

Before attempting to integrate this system of equations,

it is necessary to express c In terms of q^ and q2

, the

amounts irreversibly and reversibly adsorbed per unit mass,

respectively. The overall conservation Equation (5) is always valid and may be rewritten as

c = c o

- qW/V (9)

It was convenient to consider that the entire mas3 of

adsorbent W was partitioned into two parts and W

2

, such

that all of the irreversible adsorption q^ was confined to

while the reversible adsorption q2 could take place only on

W

2

. A partition parameter 6 was introduced to designate that

fraction of the adsorbent available for reversible adsorption

Thus

,

w

x = (i - e)w (10)

w2

= % w (11)

Now, since the total amount adsorbed is Wq, overall, it is

clear that

Wq = W

1 q 1 + W

2q2 (12)

from which it follows that

q = (l-®)q1 +®q2 U3)

This value of q may be substituted in Eq. (9) to get

c = c Q- [(1 - 6) q x + 0 q2

] W/V (lb)

for the conservation equation applicable to Eqs . (7) and (8).
- 12 -

If appropriate values could be found for the six

parameters qQ1 , q Q2 , k

i;L , k

l2 , k

22 , and 9, it would be

possible to integrate Equations ( 7 ) and ( 8 ) numerically 6,7 ^

/X /\ and thus obtain values for q 1

and q2

, the amounts computed

to be adsorbed per unit mass of each of the two respective

partitioned portions of the adsorbent. The experimental

quantities q^ and q2

, however, cannot be measured individually. The overall amount adsorbed per unit mass of total adsorbent q can be compared with its computed counterpart 'q'

using Eqs . (5) and (13). This affords a means of determining

goodness of fit according to the principle of "least squares".

Accordingly, a first estimate was made of each of the

six parameters, qQ1 , Qq2> k

li* k

l2* k

22 8X1 d Based on

these values. Equations ( 7 ) and (8) were integrated numerically and estimates of 'q^ and ^ were obtained. Using

these, together with Eq. (13), a first estimate of 'q was

determined corresponding to each time at which a sampling

was made

.

The experimental value of q from Eq. (5) was also evaluated at each measured point. Next, the "deviation", q-q', was
-13-

squared and summed over all points In an experiment. This

resultant "sum of squares" Is a measure of the extent of

agreement with "theory" and should be as small as possible

for the best fit. This Is a generalized statement of the

so-called "principle of Least Squares". A rather Involved

mathematical procedure (which is beyond the scope of this

paper) was used based on the Gauss-Newton^ ^ method of Nonr q l Linear Least Squares with Levenberg 1 J damping for obtaining

an up-dated second estimate of the six parameters. The entire

calculations were repeated and gave rise to a lower value for

the sum of squares. If the Initial choice of each parameter

was within certain broad bounds and the amount of change from

iteration to iteration was sufficiently restricted, a sum of

squares would ultimately be found which could not be reduced

by further change of any of the parameters. These limiting

values were then taken for the best fit of the theory based

on the assumed model.

It is significant that the method described for evaluation

of the parameters for the exchange of calcium and of phosphorous

arising from measurements made simultaneously from the same

solution were, nevertheless, treated independently as though

they were separate and distinct experiments.

4 . RESULTS

Pairs of isotopic exchange experiments are reported in
-14-

which the pH values were 5.64, 5.77, 5.82, 6.4l, 6.72, and

6.92 (see Table 4). These values had been selected to be

within the range In which hydroxyapatite Is the most Insoluble of the possible crystalline species in the calcium

phosphate series, such as dicalcium phosphate and octacalcium

phosphate, which could form from solution.

Experiment IE-15, which was carried out at the lowest

pH, is summarized in Tables la and lb for the isotopic ex45 52 change of Ca ^ and P , respectively. The first five columns

comprise the experimental measurements (variables). The last

five columns list the quantities computed by using the present

theory together with the best estimate that could be made of

the six parameters involved. Column (1C) contains the numerical values computed for the overall adsorption which may be

compared with the corresponding experimental values given in

column (l).

The computed values for and 'q^ shown in columns (8)

and (9) are especially interesting in light of the competitive aspect of the theoretical model. During the initial

stages of the experiment (before the first dilution) where

the overall uptake of tagged ions increased continuously, the

/\

amount computed to be reversibly adsorbed, qg, increased very

* Two additional pairs of Avnimelech's experiments were not

treated either because of insufficient points or inability

to obtain convergence.
-15-

rapidly at first from its initial value of zero at zero

time, then decreased . The reason for this reversal was the

Increasing inroads made by q^, the amount computed to be adsorbed irreversibly. As more of the tagged ions were irreversibly removed from solution, ions which had adsorbed

reversibly during the earlier stages of the experiment could

no longer be accommodated on the solid. Consequently, in

order to be consistent with the assumed model, they desorbed

to compensate. The overall adsorption, q, during this first

phase showed a continual increase with time. In subsequent

phases of the experiment after dilutions were made, the computed desorption as shown by the decrease of qg was more than

/N

enough to override any possible increase attributable to q^

/x

and the overall effect upon q was a net desorption. These

observations are graphically illustrated in Figures 2 and 5.

Experiment IE-8 (Tables 2a and 2b) differs from the other

experiments in that only one dilution was performed and that

was close to the end of the experiment. Accordingly, a somewhat different procedure was used in fitting the parameters.

A "best" fit was obtained based on the initial phase of the

exchange experiment which involved 25 points . The remaining

eight points were then compared with their extrapolated values

.

This experiment is graphically illustrated in Figures 4 and 5.

Fig. 4 shows the overall adsorption along with the agreement
-16-

bet ween experiment and theory while Fig. 5 compares the

computed values of irreversible adsorption with reversible

adsorption. A he agreement between average values of q with

A

q corresponded to a coefficient of variation of about two

to three percent in most of the experiments.

Values obtained for the characteristic parameters are

listed in Table 3 in the order of increasing pH values.

5. DISCUSSION

It is conceivable that many generalities could be made

concerning the physical significance of these parameters

either singly or in combination in attempting to correlate

the results with known experience. Some inconsistencies can

also be detected which will be discussed later. One relationship, R, which has been evaluated is shown in the last column

of Table 4 where R = (9 x Qo2^Ca/// ^ 6 x q02^p* The Produc't of

6 and qQ2 for the same adsorbate is proportional to the

maximum capacity for reversible adsorption per unit mass of

overall adsorbent. The ratio cf this product for calcium to

* that for phosphorous should be proportional to the relative

population of calcium to phosphate sites at the outermost

surface layers of the solid.

While a considerable spread in the values of R is

evident, no trend in the ratios was observed. The correlation

*The proportionality constant does not cancel out because a different proportionality exists between cgm and absolute

concentration for Ca^5 as compared with .
-17-

coefficients were not significantly different from zero

when attempts were made to correlate R with pH; with the

ratio of total [Ca] to total [PO^] in solution; or with

total ionic strength. These results may be verified from

the data in Table 4.

The specific surface (BET) of the synthetic HA used

2 here was 26 m /g. This is small compared with that of anorganic dentin which has been found to be as high

10 ^ as

2

l4o m /g. It would be interesting to know whether the

product of 9 and qQ2 for a given ion such as [Ca] or [PC/]

adsorbed from solution would be proportional to surface area

for different preparations of hydroxyapatite.

The maximum amount ^q

2 reversibly adsorbed per g of 6 fraction of the sample in any given experiment gave no indication of how much of its capacity, q Q2 , was used. For

example, almost the same capacities (22 to 23 cpm/mg) were

found for qQ2 corresponding to the calcium adsorption in

Expts. IE-13 and IE-8 (see Table 3). While Table la shows

that 17 1/2 cpm/mg of this capacity was attained in IE-13,

no more than about 51/2 cpm/mg was reached in IE-8, as is

seen from Table 2a. If, indeed, the 5-g samples of hydroxyapatite of IE-13 and IE-8 did possess the same capacities,

q Q2 , for calcium adsorption as the calculations show, these

results would lend credence to the theory that the determination of this capacity does not correlate with the solution
-18-

concentrations used in the experiments. The total concentration of calcium in the saturated solutions was 7.5 x 10"^

and 4.2 x 10“^ moles/4, respectively, (see Table 4); also,

the corresponding initial concentrations of the tagged calcium

ions (Tables la and 2a) were proportional to 56.2 and 25.5

cpm/u X, respectively.

A comparison of the relative sizes of qQ2 and qQ1 for

each experiment (Table 5) shows that the capacity for reversible adsorption, qQ2 , is usually about ten times larger

than for the irreversible process.

The forward rate constant for the reversible process,

k^ 2

, was considerably greater than for the irreversible process,

k^, in all cases (except for IE-10). This is consistent

with experience. Irreversible adsorption (as described

earlier) can involve a series of separate processes which

might be expected to slow it down. A preferred orientation

of adsorbate seems likely for irreversible adsorption which

would also be somewhat time consuming.

Tables 1 and 2 and Figures 5 and 5 show in each case that

the increase in q and the corresponding decline in q2 with

time are completed in a surprisingly short period of time -

usually within the first half hour or so of the experiment.

While the compensation effect would preclude this observation

in the overall change in q with time, it would seem that the
-19-

phenomenon would have to extend over a prolonged period of

time - perhaps even beyond the duration of the experiment.

It is, of course, an over-simplification to expect

that only two processes are sufficient to account for all

of the effects involved in adsorption or isotopic exchange.

It is reasonable, for example, to expect (neglecting the

influence of isotopic decay) that given sufficient time the

distribution of radioactive ions should be uniform throughout the entire mass of crystals and that the ratio of tagged

ions to untagged ions should be the same in the crystals as in the saturated solutions surrounding them. This long-range

aspect of equilibrium is not alluded to in the present results, nor is it approximated in the model. An experiment

whose duration is measured in hours within a single working

day cannot be expected to extrapolate to months or, possibly,

even years

.

Some inconsistencies have been noted in the observed

parameters. Their precision cannot justify the number of

"significant" figures used in Table 3. While the degree of

agreement between q and ^ corresponded to about 2 to 5 percent in most cases (as previously stated), no method has been

found to estimate the precision of the individual parameters.

Fortunately, the two parameters used in the test for stoichiometry (9 and Qq2 ) are among those found to be least sensitive
- 20-

to change from iteration to iteration. Nevertheless , it

iic; was to be anticipated that 9 for Ca J would be the same as

32 for P in the same isotopic exchange experiment regardless

of whether its value might have changed from experiment to

experiment. The reason for this is that 6 was defined as

that fraction of the weight of the adsorbent with which the

reversible process was concerned. While the parametric fit

32

was determined for Ca independently of P , we know from

crystallographic considerations and electroneutrality principles that constant ratios of sites must exist in the adsorbed layers as well as in the crystal substrate. Accordingly, an analysis of variance was performed on all the 9 values

in order to separate the "between Ca and P" contribution and

the "among six experiments" contribution from the total

variability. The "discrepance" variability with five degrees

of freedom was used as error. The null hypothesis that no

statistically significant difference was discernible between

9 for Ca and 9 for P was confirmed. Indeed, the "among

experiments" comparison also supported the null hypothesis.

The standard deviation for error determined from the discrepance is 0.059. The mean value of 9 for all 12 determinations is 0.372 which, together, correspond to a coefficient

of variation for 9 of 16$. Furthermore, the variability

among the six R-values of Table 4 which have been used in the
- 21 -

stoic hi ometric comparison is indicated by a standard deviation

of 0.273 and a mean value of 1.66 which again corresponds to

a 16# coefficient of variation. Despite the higher-than-hopedfor experimental error, no variation in stoichiometry at the

surface over the range of solution characteristics (ionic

concentration, ionic ratio or pH) is consistent with these

findings

.

6. CONCLUSIONS

Probably the most important outcome of the present investigation is the fact that a meais has been developed: (l)

for fitting to the experimental points an assumed model for

relating isotopic exchange with adsorption theory; and (2)

that the reasonableness of the assumed model can be tested by

examining and comparing the physical significance of the derived parameters

.

The results of the present investigation were consistent

with the assumption that the overall exchange (adsorption)

process might be represented by two independent processes,

one irreversible - the other reversible, both competing for

the same ions in solution.

i

hls type of study would then throw light upon the solubility-recrystallization equilibrium of hydroxyapatite, since

the chemical behavior of radioactive Ca and P are indistinguishable from their non-radioactive isotopes.
22 -

A constant stoichiometric relationship between sites

available to Ca and those available to P on the surface of

the crystallizing solid appeared to be borne out by the

theory, in that neither changes in pH nor changes in ionic

concentrations nor their ratios in solution had any

observable effect upon this relationship.
-23-

BIBLIOGRAPHY

[1] W. V. Loebenstein, Proc . of Fifth Tech. Session Bone

Char, 233 (1957)-

[2] W. V. Loebenstein, J. Res. NBS 66 A (Phys. and Chem.

)

No. 6 , 503 (1962).

[3] W. V. Loebenstein, J. Res. NBS 67A (Phys. and Chem.)

No. 6 , 615 (1963).

[4] Y. Avnimelech, Israel J. of Chemistry 6 , 375 (1963).

[5] H. Freundlich, Z. Physik. Chem. 57, 385 (1907).

[ 6 ] Runge-Kutta Method: S. S. Kuo, Numberical Methods and

Computers, Addison-Wesley Publishing Co., Inc., Reading,

Mass. 1965 (Chapt. 7 ).

[7] Picard's Method: R. E. Langer, Ordinary Differential

Equations, John Wiley and Sons, N. Y. (1954), p. 91.

[ 8 ] J. B. Scarborough, Numerical Mathematical Analysis, 2d

Ed., The Johns Hopkins Press, Baltimore, Md. (1950),

pp. 451-469.

[9] K. Levenberg, Quarterly of Applied Mathematics 2 , 164

(1944).

[10]

W. V. Loebenstein, Surface Studies of Natural and Synthetic Bone Mineral and Teeth, Adhesive Restorative

Dental Materials - II, National Institute of Dental Research, H.E.W. (1965), P. 213.
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CAPTIONS FOR FIGURES

Pig. 1

Fig. 2

Fig. 3

Fig. 4

Pig. 5

. Simplified Representation of the Solution-Solid

Interface showing Non-radioactlve Ions (open circles) and

Tagged Ions (shaded circles) of Calcium and

Phosphate

. Computed Overall Adsorption Plotted against Time

for p32_ Phosphate in Expt . IE-13 showing Three Dilution Desorption Steps. The Agreement between Values

Computed and Measured at each Sampling Time is shown

at the Bottom.

. Computed Reversible and Irreversible Adsorption

Processes vs. Time for the Same Experiment whose

Composite Adsorption is Depicted in Fig. 2.

. Computed Overall Adsorption Plotted against Time for

Ca^b. Calcium in Expt. IE-8 showing one Dilution Desorption Step. The Agreement between Values Computed

and Measured at each Sampling Time is Shown at the

Bottom.

. Computed Reversible and Irreversible Adsorption

Processes vs. Time for the Same Experiment whose

Composite Adsorption is Depicted in Fig. 4.
SOLUTION
Composite

Adsorption,

Minutes
Reversible

CL

O

o

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Irreversible

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Composite

Adsorption , o < CT O N

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