Title Page
 Table of Contents
 List of Figures
 A review of pertinent concepts...
 Synthesis of pertinent theory on...
 Solution of the convective-dispersion...
 Experimental objectives, materials...
 Results and discussion of studies...
 Summary and conclusions
 Biographical sketch

Title: Influence of dispersion, exclusion, and metathetical sorption on the transport of inorganic solutes in a calcium-saturated porous medium /
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00097476/00001
 Material Information
Title: Influence of dispersion, exclusion, and metathetical sorption on the transport of inorganic solutes in a calcium-saturated porous medium /
Physical Description: xi, 97 leaves : ill. ; 28 cm.
Language: English
Creator: Persaud, Naraine, 1944-
Publication Date: 1978
Copyright Date: 1978
Subject: Ion exchange   ( lcsh )
Soil absorption and adsorption   ( lcsh )
Soil Science thesis Ph. D
Dissertations, Academic -- Soil Science -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 91-96.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Naraine Persaud.
 Record Information
Bibliographic ID: UF00097476
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000012740
oclc - 04244058
notis - AAB5642


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Table of Contents
    Title Page
        Page i
        Page i-a
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
        Page ix
        Page x
        Page xi
        Page 1
        Page 2
    A review of pertinent concepts on solute transport during miscible displacement in discrete porous media and on inorganic ion exchange equilibria and kinetics
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
    Synthesis of pertinent theory on inorganic ion exchange and transport processes
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
    Solution of the convective-dispersion equation for a “pulse input” and its extension for a solute following a linear isotherm
        Page 32
        Page 33
        Page 34
        Page 35
    Experimental objectives, materials and methods
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
    Results and discussion of studies on exchange equilibria and transport of Na+, L1+, 45Ca2+ and Cl-
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
    Summary and conclusions
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
    Biographical sketch
        Page 97
        Page 98
        Page 99
        Page 100
Full Text







my ^arientS


The author wishes to express his sincere gratitude to Dr.

J. M. Davidson, chairperson of his supervisory committee, for the

unique and conscientious guidance and help provided throughout the

duration of this study.

The helpful suggestions made by members of the committee, viz.

Dr. J. G. A. Fiskell, Dr. L. Hammond and Dr. S. J. Locascio, during

their review of this manuscript are highly appreciated. The interest

shown by Dr. R. S. Mansell who was also a member of the co nittee until

he left on sabbatical leave is also highly esteemed.

A special word of thanks is reserved for Dr. P. S. C. Rao and

IMr. Ron Jessup for the useful advice given during many discussions on

diverse subjects pertinent to this study.

The cheerful companionship of Mr. Rick Janka did much to relieve

the tedium of the long hours spent in the laboratory.

The financial assistance provided by the Soil Science Department

is gratefully acknowledged.

Superlatives fail to express the author's deep feelings towards

his wife Savi whose constant encouragement and support were sin qua

non to the successful completion of this study.



ACKNOWLEDEMENTS . . . . . . . . iii

LIST OF FIGURES . . .. .. . . . . . . .. . vi

ABSTRACT . . .. . . . . . . . . . . ix

INTRODUCTION . . . .. .. . . . . . .. . 1


Miscible Displacement Processes in Discrete
Porous Media . . . . . .... . . ... 3
Dispersion of Solutes During Miscible Displacement
in Discrete Porous MIedia . . . . .... .. 4
The Convective-Dispersion Equation Including
Assumptions, Auxiliary Conditions, Analysis,
and Limitations .. ... .. . .. .. ... .. 5
Extension of the Continuum Mlass Balance Approach
for Single Interacting Solutes . . . . .. 11
Equilibria and Kinetics of Inorganic Ion Exchange
Adsorption . . . . . . . . . . 15

AND TRANSPORT PROCESSES. . . . . . . . . ... 23
Thermodynamic Conceptualization of Ion-Exchange
Processes . . . . . . . . . . 23
Disposition of Charged Species in Solution/
Exchanger Systems at Equilibrium . . . . .. 24
Physical Basis for Exchanger "Selectivity" ..... 28
Concepts on Inorganic Ion Transport During
Miscible Displacement. . . . ... . . 31

ISOTIIERII . .. .. .. . ...... . . . 332



AND TRANSPORT OF Na Ll ^5Ca2+ AND C]L. . . . .. 42
Exchange Adsorption Isotherns. ... ...... . 42
Miscible Displacement Experiments with Na .... 51
Miscible Displacement Experiments with Li.+ .... 70
Miscible Displacenent Experiments with 45Ca2. . 83

6 SUMPIARY AND CONCLUSIONS . . . . . . . . 87

LITERATURE CITED ......................... .. . 91

BIOGRAPHICAL SKETCH .................. ...... 97


Figure Pae

L Schematic of the flow system. . . . . . . 39

2 Exchange adsorption isotherms for Na+ in 0.05 II, 0.02
M, and 0.005 M Ca(NO3)2. ............... 43

3 Exchange adsorption isotherms for Li in 0.05 M, 0.02
M, and 0.005 M Ca(NO3)2 ................ 44

4 Dependence of the isotherm K values for Na and Li on
the concentration (C) of Ca in the equilibrium solu-
tion. . . . . . ... ... . . ...... 45
45 2+
5 Exchange adsorption isotherms for Ca2+ in 0.075 I1
and 0.05 11 Ca(O3)2 ................... 49

6 Elution curves for a "pulse input" of Na Cl and
HTO in 0.05 1 Ca(NO )2 at a pore-water velocity be-
tween 14 and 15 cm/r. . . . . . . . 52

7 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.05 M Ca(NO )2 at a pore-water velocity
between 7 and 8 cm/r . . . . . . . . 53
+ -
8 Elution curves for a "pulse input" of Na Cl and
HTO in 0.05 11 Ca(NO3)2 at a pore-water velocity
between 1 and 2 cm/hr . . . . . . . . 54

9 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.02 H Ca(NO ), at a pore-water velocity
between 14 and 15 cu/hr. . . . . . . . ... 55

10 Elution curves for a "pulse input" of Na Cl and
HTO in 0.02 I Ca(NO ) at a pore-water velocity
between 7 and 8 cm/r. ................. 56

11 Elution curves for a "pulse input" of Na+, Cl-, and
HTO in 0.02 M Ca(NO ) at a pore-water velocity
between 1 and 2 cm/r ................ 57

LIST OF FIGURES (continued)

gurc Page

I? Elution curves for a "pulse input" of Na C -, and
HTO in 0.005 M Ca(NO)2 at a pore-water velocity
between 14 and 15 ch ................. 58
between 14 and 15 m/hr. . . . . . . . . 58

13 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.005 11 Ca(N3 )2 at a pore-water velocity be-
tween 7 and 8 cm/hr................... 59

14 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.005 M Ca(NO )2 at a pore-water velocity
between 1 and 2 cm/hr. ............... 60

15 Linear dependence of the dispersion coefficient on
the pore-water velocity. . . . . . . . .. 62

16 Repeated elution curves for "pulse inputs" of Na
and HTO in 0.05 H1 and 0.005 11 Ca(NO3 )2 in a short
column at pore-water velocities between 7 and 8
cm/hr . . . . . . . . . . . . . 64

17 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.005 N Ca(NO3)2 in a column packed with
unsulphonated, macroporous, polystyrene beads .... 67

18 Elution curves for a "pulse input" of Na Cl and
HTO in 0.05 NI Ca(NO3)2 under steady-state unsaturated
water flow conditions. . . . . . . ... 68

19 Elution curves for a "pulse input" of Na Cl and
HTO in 0.02 M Ca(NO3)2 under steady-state unsaturated
water flow conditions. .. . ................. 69

20 Elution curves for a "pulse input" of Li and HTO
in 0.05 M Ca(NO3)2 at a pore-water velocity between
7 and 8 cm/hr. . . . . . . . . . 71

21 Elution curves for a "pulse input" of Li+ and HTO
in 0.02 11 Ca(NO )2 at a pore-water velocity between
7 and 8 cm/hr and without adjustment of the ionic
strength of the eluting solution. . . . . ... 72

22 Elution curves for a "pulse input" of Li and HTO
in 0.005 1t Ca(NO3)2 at a pore-water velocity between
7 and 8 cm/hr and without adjustment of the ionic
strength of the eluting solution. . . . .... 73

LIST OF FIGURES (continued)

Figure Page

23 Elution curves for a "pulse input" of Li and HTO
in 0.02 M Ca(NO )2 at a pore-water velocity between
7 and 8 cm/hr and with adjustment of the ionic
strength of the eluting solution. . . . . ... 75

24 Elution curves for a "pulse input" of Li and 11TO
in 0.005 M Ca(NO ), at a pore-water velocity between
7 and 8 cm/hr and with adjustment of the ionic
strength of the eluting solution. . . . . ... 76

25 Elution curves for "pulse inputs" of Li at 50 to 55
ppm in 0.02 M and 0.01 M Ca(NO3)2 at pore-water velo-
cities between 7 and 8 cm/hr and without adjustment
of the ionic strength of the eluting solution. .. 78

26 Obseryations on the elution pattern for displacement
of Li in 0.005 M Ca(N03)2 by deionized H10 followed
by 0.005 1 Ca(NO3)2 .. . . .. . 80

27 Breakthrough curve for a "step input" of Na in
0.005 M Ca(N03)2. ................... 84
45 2+
28 Elution curves for "pulse inputs" of 11TO and Ca
in 0.075 11 and 0.05 _i Ca(N03)2 ........... 86


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy



Naraine Persaud

June 1978

Chairperson: J. M. Davidson
Major Department: Soil Science

Classical thermodynamic concepts were used to derive general rela-

tionships to describe the equilibrium disposition of charged species in

aqueous contact with a cation exchanger. It was shown through these

relationships that anions would be excluded by the exchanger and solu-

tion phase cations of the same type as those initially saturating the

exchanger influenced the adsorption of counterions. The latter result

was verified with exchange adsorption isotherms for Na+ and Li+ in solu-

tions of 0.05 M, 0.02 M, and 0.005 M Ca(NO3)2 and a Ca-saturated ex-

changer. These results illustrated that Na4 and Li adsorption de-

creased with increasing concentrations of Ca2+ in the equilibrium solu-

tion. Similar results were observed for the adsorption of 45Ca2+ in

solutions of 0.075 M and 0.05 M Ca(NO )2. The adsorption isotherms were

linear over the range of concentrations studied for a given Ca(NO3)2

cu.ncontrationl. The thermodynamic relations in conjunction with the

DLbye-llucke] theory were used to predict the slopes for these isotherms.

The computed values agreed reasonably well with their experimental

counterparts. These results demonstrated inductively, that electro-

static ion-ion interactions as conceived and quantified by the Dehye-

Huckel theory, constituted the physical basis fox the observed equi-

librium behaviour.

The consequences of the foregoing results on the transport be-

haviour of Na+, Li+, 45Ca2+ and C"- were investigated using miscible

displacement experiments in laboratory columns packed with the exchanger

material. These studies were conducted using pulse inputs of tracer

solutions containing various concentrations of Ca(NO3)2. Tritiated

water (HTO) was introduced with all tracer pulses to evaluate hydro-

dynamic dispersion. An analytical solution to the convective-dispersion

mass transport equation for a reactive solute (linear adsorption iso-

thern) was used to describe the experimental data.

As expected, the elution curves for Cl- were displaced to the left

of those for HTO illustrating exclusion of Cl" by the exchanger. The

elution curves for Na, Li+ and 45Ca21 were displaced to the right of

those for HTO, the shift increasing with decreasing concentration of

Ca2+ in the tracer solution. Anomalous and unusual patterns were ob-

served in some of the elution curves. The elution curves for Na+ in

0.005 M Ca(NO )2 and for 45Ca2+ in 0.05 N Ca(NO3)2 showed deviations

that simulated the effect of kinetic mass transfer processes. Inflec-

tions were observed on the desorption side of the pulse elution data

for Li+ in 0.02 1! and 0.005 M Ca(NO )2 and for 45Ca2+ in 0.075 M

Ca(No ) 2. These elution curves were not described by the analytical
3 2

solution. Further experimentation demonstrated that these data could

be explained on the basis of differences between the parameters

characterizing exchange adsorption and desorption. A quantitative

treatment, based on the thermodynamic relations coupled with the

Debye-Huckel theory, showed that these differences were a direct con-

sequence of the metathetical nature of the sorption process.

Some cursory observations were made regarding the transport be-

havior of Na IITO and Cl- under steady-state unsaturated water flow

conditions. The exclusion of Cl was again evident. Discrepancies

between the experimental and analytical curves for Na indicated that

portions of the water-unsaturated exchanger phase became either inacces-

sible to Na+ or was accessible only by diffusion.


The classic work of Thonpson and Way, reported during 1850-1852

on exchange reaction involving soil materials Created with inorganic

salt solutions, marked the beginning of systematic investigations of

ion exchange. Their experiments were conducted decades before Arrenhius

proposed the theory of electrolytic dissociation and many years before

the law of mass action was enunciated. There was, therefore, no exist-

ing theoretical framework within which the two English agricultural

chemists could interpret their observations. The principle that sub-

stances did not react except in a dissolved state was generally accepted

by chemists of their era. Whether the reactions reported were purely

chemical or physical became a debated question. This dichotomy in

views has still not been fully resolved.

For a period after its discovery, ion exchange remained of

academic interest until about 1907 when it was used to soften water.

Instability under non-neutral pH conditions and the low capacity of the

available natural and artificial siliceous ion exchangers provided the

impetus for the discovery and synthesis of organic ion exchangers. These

synthetic materials are now utilized in most industrial and laboratory

applications involving ion exchange. Extensive research concurrent

with their increasingly widespread use has resulted in the evolution

of more comprehensive and refined ion exchange theories.

The miscible displacement technique has in the past few decades

become an increasingly valuable experimental tool for studying

n'i:3ico-chcmical processes associated with solute transport in porous

nedia. Theoretical concepts on physico-chemical interactions of partic-

ular solutes have been successfully incorporated into classical mass

transport models. Comparisons between experimental and theoretical

results on solute transport have served to verity concepts and to

evaluate the influence of physical factors. The diversity of possible

physico-chemical interactions of inorganic ions with various soil

components makes natural soil materials a complex medium in which to

study ion exchange. This difficulty can be circumvented somewhat by

the use of synthetic exchangers.

The primary objective of this study was to evaluate some theore-

tical concepts on ion exchange equilibria and kinetics using the mis-

cible displacement technique and a porous medium prepared from a syn-

thetic organic ion exchanger.



Miscible Displacement Processes in Discrete
Porous Media

Miscible displacement is the term used to describe a process

whereby one fluid is displaced by another, both fluids being miscible

in each other. This process occurs in a soil when a solution is dis-

placed downwards by incoming rain or irrigation water or when sea water

displaces fresh water during drawdown in coastal aquifers. In the

petroleum industry, fluids miscible with crude oil are used to increase

the efficiency with which the crude oil is displaced from oil bearing

strata during secondary operations. Separation and recovery processes

in industrial packed towers or in chromatographic columns are other

examples of miscible displacement processes in porous media.

The transport behavior of materials dissolved in the displacing

or displaced fluids is influenced by the physical properties of the

medium (particle size, shape, and manner of packing), theological pro-

perties of the fluids, and both the equilibria and rates of reactions

between the solute and solid matrix of the porous medium. Miscible

displacement experiments where there is no interaction between the

medium and the solute can be used to evaluate the effect of the first

two factors. The combined effect of the first two factors has been

grouped under the term "dispersion" and a considerable body of literature

has grown out of these studies. Excellent reviews on dispersion have

been presented by Fried and Combarnous (25) and Bear (6).

Dispersion of Solutes During Miscible
Displacement in Discrete Porous Media

Dispersion is the result of a physical mixing between displacing

and displaced fluids. Its effect can be studied by measuring the rela-

tive concentration gradients between the displacing and displaced

fluids. Gradients are usually determined experimentally by withdrawing

samples at suitable time intervals at a fixed location in the flow region

under consideration. This technique and the relative concentration

versus time plots, termed breakthrough curves, have been described by

Nielsen and Biggar (53).

Dispersion in many practical situations involving solute transport,

especially in chromatographic and industrial separations, has provided

much of the initial motivation to quantify and describe the process.

Several theoretical approaches have been taken in this connection.

Attempts have been made to predict macroscopic dispersion effects based

on various geometrical models of the microstructure of the channels in

the porous matrix (23,33,64). These, however, require complex mathe-

matical analyses, are rather academic in nature, and have not proven to

be useful. Limitations of this approach have been discussed by Scheidegger

(66). Analysis of solute transport based on a physical description of

the flow region into theoretical plates has proved quite useful in eval-

uating dispersion in chromatographic separations (28,68).

An approach based on differential mass-balance equations for the

solute in the fluid flow field has been successful and has gained wide

acceptance. Taylor (67) used this approach to study solute dispersion

during laminar flow in straight capillary tubes. Analysis of the

differential mass-balance equation with appropriate initial and boundary

conditions yielded expressions which satisfactorily explained his experi-

me-ntal results. These results showed that dispersion was dependent on

the fluid velocity and the distance travelled by the displacing front.

During laminar flow of Newtonian fluids in straight tubes,a typical

parabolic velocity distribution is produced. This would not be the case

for a discrete porous medium with geometrically complex, tortuous channels

between the particles. For a porous medium it is assumed that the micro-

scopic velocities between particles would fluctuate continuously in a

random manner about some mean value. This viewpoint led to a concept

that was developed almost simultaneously by Scheidegger (65) and

Danckwerts (16). Although they used somewhat different approaches in

their arguments, they agreed that dispersion could be regarded as a

quasi-diffusion process. With this concept, the classical differential

ecnation for mass transport by simultaneous convection and diffusion

could be applied if an appropriate change was made in the physical

meaning attached to the diffusion coefficient. The validity of this

approach was demonstrated immediately by other investigators (4,19,52).

The Convective-Dispersion Equation Including
Assumptions, Auxiliary Conditions, Analysis,
and Limitations

Hass balance considerations for transport by simultaneous diffu-

sion and convection in a homogenous, isotropic porous medium yield

the classical convective-diffusion equation. In one-dimension this

equation (39) is

c = { vC} (1)
at ax ax -

where. C(x,t) is concentration of solute in the fluid of the flow re-

gion (M/L3), t is time (T), x is distance (L), D is the diffusion

coefficient (L2/T), and v is vector velocity of the fluid at any point

in the flow region (L/T).

Because of the pore size distribution, complex fluid velocity

fields arise within the channels of the porous medium. It is assumed,

however, that these velocity vectors are statistically distributed

about some fixed mean velocity vector v*. The probability distribu-

tion of the deviations about this average is then considered by the

introduction of a quasi-diffusion coefficient, D*, which characterizes

the unidirectional transport of the solute. This quasi-diffusion co-

efficient has been termed the "dispersion coefficient" by Scheidegger

(65). If it is also assumed that the dispersion coefficient is inde-

pendent of concentration, then equation (1) reduces to

a= D* a2C- *a (2)
at wx7 -ax

As shown by Scheidegger (65) and in a somewhat more heuristic manner

by Rifai et al. (59), this equation can also be obtained by arguments

based entirely on probability calculus. This equation has been appro-

priately called the "convective-dispersion" equation.

It is at once apparent from its description that v is the ex-

perimentally measurable average pore-water velocity of the fluid in

the porous medium. This is given by the relationship

v* =-
X- At (3)

where, Q is volume of fluid introduced (L3), A is the cross sectional

area of the flow region (L2), 0 is the fractional fluid filled poro-

sity of tne medium (L /L ), and t is time (T).

It is also clear that D is related to D* in the following manner:

lim D* = D.
vD, *- 0 (4)

For a fixed X = L, the solution of the convective-dispersion

equation would be C = C(t, D*, L, v*) where D*, L, and v* are para-

meters. It is advantageous to formulate the differential equation

and auxiliary conditions in dimensionless form. The problem then

becomes independent of units and the parameters are reduced to dimen-

sionless groupings. In addition, interactions among the parameters

become more obvious. The dimensionless variables are C* = C

X = and T = which, when substituted into equation (2) give
ac* D* a2C* ac*
T v*L X .' (5)
The reciprocal of the dimensionless grouping L has been termed

the Peclet number, usually denoted by P, and appears as a single

parameter in solutions of the transformed equation. The dimension-

less time T corresponds physically to the number of fluid-filled

void volumes introduced into the flow region.

The convective-dispersion equation in either form [equation

(2) or (5)] has been analyzed for various sets of boundary and ini-

tial conditions appropriate to different physical situations. These

have been surmarized by Nir and Gershon (54). The analyses are based

on the theory of equations in partial derivatives, a very difficult

and still incomplete branch of higher mathematics. Analytical solu-

tions can be obtained with comparative ease for simple sets of boundary

and initial conditions; however, for complex conditions or for finite

domains of definition, one is forced to resort to numerical integration


Analytic solutions to equation (2) have been obtained for two

simple sets of conditions. These can be written as

I. C(x > 0, t = 0) = 0 C(x = 0, t > 0) = Co,

C(x =- t > 0) = 0 (6)

II. C(x > 0, t = 0) = 0 C(x = 0, 0 < t < tl) = Co

C(x = 0, t > tl) = 0 C(x = t > 0) = 0 (7)

Physically, the first set of conditions corresponds to a solute

of concentration Co continuously displacing the solvent in a semi-

infinite medium. In the second case, the same displacement is allowed

to proceed for a time t and for all times thereafter the displacing

solution is replaced by pure solvent. These two situations have been

termed miscible displacement with a "step input" and "pulse input,"


The solution to equation (2) for a "step input" (59) is

given by
C(xt) I r x-vt v x+vt
C- = (erfc[ l + exp() erfc[-2- (8)

where the superscripts have been omitted from D* and v".

Except near the inlet (x = 0) or for very small tines, unless

D is large, the second term in equation (8) is small and can be

neglected. Applying this condition and the identity orfc(x) =

1- erC(x), equation (8) reduces to

C(xt) i-e -vt (9)
c =- Ii erf[ I] (9)

The cumulative distribution function P(::) for a standardized

Gaussian random variable of mean x and standard deviation o is given


P(x) = ( + erf[ 2 (10)

From the identity erf(-x) = -erf(x), it is apparent that equa-

tion (9) represents, for a given value of x = L, a Gaussian cumula-

tive distribution function of mean L and standard deviation '2Dt.

The solution for the "pulse input" is obtained by shifting the

solution for the "step input" by the time width of the pulse and sub-

tracting this from the original solution. Applying this rather in-

tuitive procedure to equation (9) gives
c(x t) 1 x-v(t-t x-vt
Co erf[ -U(t-tl)erf[X 2v ]} (11)

where U(t-t ) is the Heaviside unit function.

The previous equations have been used to analyze experimental

results from finite laboratory columns. As shown by Nir and Gershon

(54) only a small error is involved by assuming a finite rather than

a semi-infinite column.

The boundary conditions appropriate for a defined finite domain

are obtained by imposing the conservation of mass on the fluxes at the

boundaries. This results in the following boundary conditions for a

step input and a space domain of definition 0 < x < L

x= 0; t > 0; vCo = vCIx = D (12a)

x = L; t > 0; vCe = vCx= L D x = 12b)

where Ce is the exit concentration at a given time.

however, as argued by Danckwerts (16), if at x = L, xL 0

then CIx=L < C x
9 a=1 > 0. Hence x=L must be equal to zero otherwise a maximum
yy x=L hx
or a minimum exists in the interior of the column. The appropriate

condition at x = L must therefore be x= = 0.
ax x-L

An approximate solution to the convective-dispersion equation

in dimensionless form for the above boundary conditions has been pre-

sented by Brenner (14). In dimensionless form these conditions become
1 aC=3
S= 0; T > 0; C* = 1 (13a)

X = 1; T > 0; 1 = O (13b)

ehnler and Wilhelm (75) and Lindstrom et al. (46) have showm that,

for large values of P, the condition at x 0 reduces to C* = 1 for

t > 0.

In order to make the condition at x = 0 homogeneous, Brenner

defines C = where C is the initial concentration in the

fluid at t = 0. The initial concentration was taken to be zero

for his solution. Brenner tabulated numerical values of his solu-

tion for a wide range of values of P/4. It was found that, with

increasing values of P, his solution asvyptotically approached

the simplified solution given by equation (9). He also discusses the

limiting behavior of his solution for P = 0 and P = -. These corre-

spond physically to complete instantaneous mixing and to no dispersion,

respectively. His solution has been used frequently for analysis of

experimental data from miscible displacement experiments (61).

Numerical procedures for integration of the convective-dispersion

equation are all based on methods whereby continuous systems are

matlhelatically reduced to equivalent discrete systems. Suitable algo-


rithms can then be developed and solved iteratively using high speed

computers. Serious and difficult mathematical questions of adequacy,

accuracy, convergence and stability arise for numerical solutions and

adequate answers are often not available. Procedures based on finite

differences and finite elements are discussed by Ames (1) and Finder

and Gray (55).

From the assumptions used in its development, the convective-

dispersion equation represents an idealized conception of the miscible

displacement process. Its shortcomings were demonstrated early by

its inability to describe non-reactive solute behavior in unsaturated

porous media (53). Also, it fails to predict the effects of density

differences between displacing and displaced fluids (62). Its major

weakness probably lies in the assumption of a fluid continuum. The

presence of dead-end pores and regions where the fluid is hydro-

dynamically immobile would clearly lead to unpredicted results. Re-

finements made to handle such cases (15,69) have resulted in better

agreement between predicted and experimental results.

Extension of the Continuum Mass
Balance Approach for Single Interacting

The convective-dispersion equation can be extended to describe

the transport of a single interacting solute through a porous medium.

This requires additional terms to cover the time rate of change of

solute concentration in the fluid as a result of mass transfer be-

tween the fluid and solid matrix of the medium. For a general case,

these terms constitute the difference between the instantaneous adsorp-

tion and desorption rates. Equivalently and more conveniently, this

can bc expressed in terls of the instantaneous net time rate of

accumulation of the solute on the particles of the medium. The

concentration of the solute in the matrix expressed as mass of

solute/unit mass of matrix can be denoted by S(x,t). The instan-

taneous net time rate of change of concentration in the fluid due

to mass transfer is then -in which p is the dry bulk density
ov @t
of the matrix and 6 is the fractional water-filled porosity. Con-

sidering only one mass transfer process, the resulting differential

equation is

DC ..@2C v P_ p S
21 a22 2 -_as
= D -7 x at (14)

The functional form of S(x,t) has profound consequences on the analysis

of the resulting differential equation. In general, the mass transfer

processes may involve purely diffusion kinetics, irreversible and

reversible chemical kinetics or both. Hence, it would be expected

that in the general case

t = f[C, S, 2, (1' 2"' n diffusion,!1' 2" )chemical (15)

where T is temperature, and A ..Xn are parameters characterizing

diffusion and chemical kinetics (57). The case of mass transfer

involving only irreversible chemical kinetics has been analyzed by

Amundson (2).

Processes involving only reversible chemical kinetics can be

represented in the general case as

a = 1 (S C, i X2f2(S, C, T( (16)

whlre At and A2 are constants characterizing the kinetics of surp-

Lion and desoiption, respectively.

An important simplification of chemical kinetics is based on the

assumption that, at a constant temperature, the processes occur fast

enough to insure that equilibrium is instantaneous. The functional

form of S(x,t) is then given by an equation which describes the equi-

librium isotherm. This equation can be experimentally determined

or deduced from kinetic equations, if these are known, by setting
-= 0. In general, these isotherms can be represented as

S = f(C, 1.l- ,n) (17)

where A 1... are parameters characterizing the equilibrium distri-

bution of the solute between the fluid and solid matrix of the medium.
as aS Cin the d e
With this assumption, introducing = in the differen-
at at at
tial equation gives

IC D* 12C v* IC
It RT R x (18)

where, R 1 + - S
0v -7

From this equation, some useful deductions can be made regarding

the transport behavior of the solute for certain generalized func-

tional forms for the isotherm.

In the case where the isotherm is a single-valued concave down-

ward function -S > 2 and thus c- < for C1 < C2
Icc c Ri ReL 2

Similarly the reverse is true for C1 < C2 when an adsorption isotherm is

represented by a concave upward function. For a linear isotherm,
i s independent of C. Typically, for a pulse input of a non-reactive

solute, dispersion causes the elution curves to have a gradual increase

in concentration from zero to a maximum on the front and falling again

to zero on the back side of the elution data. For a concave downward

isotherm, the concentration effect on the velocity would result in

a nullification of dispersion on the front and an enhancement on the

back side of the elution curve. This effect would not appear in the

case of a linear isotherm and the reverse would be true for a concave

upward isotherm. For a linear isotherm, the front and rear of the

elution curve would match exactly, but will not match -for non-linear

isotherms. If the assumption of instantaneous equilibrium was valid

for a particular situation, the shape of the elution curve can pro-

vide a valuable indication of the type of adsorption isotherm in-

volved in the displacement process. These arguments have been con-

firmed by analysis of the differential equation for various functional

forms of the isotherm, including cases where it was not single-valued

(45, 46,70).

As would be expected, the assumption of instantaneous equili-

brium is not valid for all situations. Exact solutions of the

differential equation incorporating two important models of rever-

sible chemical kinetics have been presented by Amundson(3). These

models are

klC k2S (19)

for linear adsorption kinetics, and

S- klC(Smax S) k2S (20)

for "Langmuir" kinetics, where Smax is the saturation capacity.

Numerical procedures have increased the range of possible theoretical

models that can be used.

Except for a few cases (24, 69), less effort has been devoted

to studying diffusion kinetics. In soils, however, where particles

exist as aggregates and dead end pores are present, diffusion kine-

tics may be important (69).

Investigation of the simultaneous transport of a number of

solutes which influence the mass transfer processes of each other

have not been attempted. In addition, comprehensive reviews have

not been made of the various equilibrium and kinetic models commonly

used by investigators. The equations derived from theory reflect

the effects of only those interactions for which the models have

been developed. Different theories may well attribute experimental

results to quite different causes. This constitutes the deficiency

in maay studies since it is often extremely difficult if not impossible

to independently measure the theoretical parameters introduced in

sorption kinetic models.

Equilibria and Kinetics of Inorganic
Ion Exchange Adsorption

Inorganic ion exchange was discovered more than a century ago.

For soils, ion-exchange properties were traced to their alumino-

silicate fractions (73,74). For many silicates, only ions from the

exposed surface layers are in a position to exchange. Typically

grinding increases the exchange capacity of these minerals (41).

For others, especially the zeolites, the lattice structure is open,

permeated by waterfilled channels, and possess internally accessible

exchange sites. Industrial applications of these and other synthetic

aluminosilicates were limited by their low exchange capacity and

instability under acid or alkaline conditions. The recent intro-

duction of synthetic organic ion exchangers with superior properties

led to widespread industrial and laboratory exploitation of ion

exchange and a resultant surge in research on ion exchange equilibria

and kinetics. With these exchangers, it was possible for the first

time to vary their properties systematically. Much of the theore-

tical advances made were due to this fact.

Ion exchange equilibria and kinetics are fundamental to an

understanding of ion transport and have been studied intensively.

As yet, no single comprehensive theory exists to explain all of the

results involving ion exchange. Certain characteristics are normally

common to all ion-exchange reactions. The reaction is usually revers-

ible and always involves an equivalent exchange of ions. In addition,

the exchanger preferentially adsorbs one ion over another, a property

appropriately termed "selectivity."

Electrical double-layer theory (71) advanced by Helmholtz and

modified by Couy and Stern as an explanation of the electrokinetic

properties of colloids has been utilized to explain the phenomena

associated with ion exchange by soil clays (7,22,37). This theory

adequately accounts for the preferential sorption of ions of higher

valence over those of lower valence, but fails to explain observed

exchanger selectivity among ions of equal valence. Bolt (8) attempted,

with limited success, to extend the double layer theory to account for

such behavior.

The overall ion exchange process can formally be represented as

a reverse ible chemical reaction. The exchange of ions 1 and J of

valence + zi and + zj respectively can be written as

zj i lil + zi -T. I zj mizil + 'i Jlzjl

where the bar signifies the ion associated with the exchanger. As

demonstrated by Kerr (42), the ion in combination with the exchanger

could not be considered as a precipitate with unit activity, He

assumed that the combination behaved as a solid solution, a view that

was subsequently supported (5,11,72), and has gained wide acceptance

by investigators studying ion exchange.

A useful quantity, the selectivity coefficient, characterizing

the relative preference of the exchanger for the ions I and J was

obtained by applying the law of mass action without activity correc-

tions. Thus

iC.zi C.j zi
K = l (21)
j zil Cilz j

where C represents molar concentrations.

Except when zi = zj, the numerical value of K depends on the

choice of concentration units. The total equivalent concentration

Co = z C + zj Cj of ions I and J in the solution and the corre-
1i J J
spending quantity Qo = z C.+ z C associated with the exchanger

must remain unchanged throughout the reaction. Defining Xi = ziCi/C

and Xi = ziCi/Qo as the "equivalent ionic fractions" of the ion I

in solution and exchanger respectively, and similar quantities for

the ion J, yields

K jl Xj i ( l zil zil
3 -Qo
xlzjl. xjlzil (22)

The selectivity coefficient as defined is thus dependent on the total

equivalent concentration of the exchanging ions in solution and the

capacity of the exchanger. In addition, as shown by Bonner and Bonner

and Payne (9,10), it is also a function of the extent of exchange and

assumes its highest value when the exchanger is completely saturated

with ion J. The selectivity coefficient is thus not a particularly

useful quantity for predicting ion exchange equilibria.

A thermodynanically rigorous application of the mass action law

results in an expression for the thermodynamic equilibrium constant.

This expression is

K* ailZJI ajziL

ailzj aj zi (23)

where "a" represents ion activity. Determination of K* requires

quantitative values for the activity coefficients of the exchanging

ions in solution and in combination with the exchanger. This con-

stitutes the difficulty in using the classical thermodynamic approach.

The Debye-Huckel theory and its extensions to predict activities of

electrolytes in solutions are applicable for concentration ranges

below those normally associated with the solid solution concept.

This difficulty is not insurmountable,however, because methods based

on classical thermodynamics are available to determine the activity

coefficients of ions in combination with the exchanger and the

equilibrium constant (17,26). This treatment is of more theoretical

than practical interest because it requires numerous measurements

before the equilibrium constant can be determined (21,30).

Ion exchange as a Donnan system was introduced by Mattson and

Larsson (47) and probably represents the most powerful concept in

explaining ion exchange. It was mainly through the work of Baunan

(5), Boyd (11) and Glueckauf (27) that this important concept has

gained general acceptance. The quantitative principle involved in

the Donnan approach is essentially a generalization of the double

layer concept, which is of universal occurrence whenever electrical

charge is confined within a definite region of space. The Donnan

approach imposes the thermodynamic condition that all other novable

charges must adjust themselves accordingly to produce a minimum in

the free energy of the system at equilibrium. The Donnan treatment

thus represents a partial fusion of the electrostatic and thermo-

dynamic aspects of the ion-exchange phenomenon.

Another approach likens the ion-exchange process to the phys-

ical adsorption of gases. Both the Langnuir and Freundlich equa-

tions have been used to describe data on ion exchange equilibria

(11,37). There is a general consensus that ion exchange processes

involve strong, long range electrostatic forces which were not con-

sidered in the conceptual development of the Langnuir and Freundlich

equations. Only vague physical meanings can be attached to the para-

meters obtained from the application of these equations, and there-

fore, they have not provided nuch insight into the adsorption pro-

cesses involved.

A novel and interesting simplified statistical approach has

been used by Jenny (38) and Davis (18) to derive general theoretical

equations for ion-exchange equilibrium. Although the results appear

interesting, these concepts have not been used widely.

Although ion exchange can be represented formally as a chemical

reaction, the physical processes involved have little in common with

true chemical reactions. Evidence for this has appeared in equilibrium

studies (42,72) where the ion in combination with the exchanger was

treated as a solid solution. In addition, standard enthalpy changes

for ion-exchange reactions are often less than two kilocalories per

mole which is typical of the orientation energies involved in dipole-

dipole interactions. Such evidence indicates that ion exchange is

essentially a statistical redistribution of the exchanging ions be-

tween the exchanger and the solution.

Further support for this concept was provided by the pioneering

ion-exchange kinetic studies of Boyd, Adamson and Meyers (12). They

obtained close agreement between their experimental data on exchange

kiineics and predictions from theoretical equations based on the con-

cept of ion exchange as a diffusion process. They hypothesized that

either diffusion through a stagnant film around the particle or

diffusion into the particle were rate controlling. Integrated rate

equations for both cases were obtained by application of Ficks' laws

for constant diffusion coefficients. The hypothetical stagnant film

was assumed to have a finite thickness and was regarded as similar to

the 'Nernst' film encountered in reactions at electrode surfaces. They

found that either one or both mechanisms were rate-controlling and

depended upon the experimental conditions. Their findings were

immediately confirmed by other investigators (32,44,58). These

studies showed that, in general, particle diffusion kinetics were

favored for solution concentrations greater than 10-2 11,efficient

mixing (which reduces film thickness), large particle size and low diffu-

sion coefficients for ions in the exchanger. Opposite conditions were

conducive to film-diffusion kinetics. measurements of self-diffusion

coefficients of cations in synthetic organic exchangers were made by

Boyd and Soldano (13). In general, these were an order of magnitude

less than the corresponding values in solution and decreased with

increasing valence.

A significant improvement in the analysis of the two step

diffusion concept was introduced by Helfferich and his coworkers

(34,36,56). They observed that Fick's first law did not take into

account electrokinetic forces involved in the interdiffusion of two

charged species. They introduced the more appropriate Nernst-Planck

flux equation, which contained an additional electrical transference

term. This equation is

1* 3C z a1-0
J* - DC (24)

where J* is flux of any charged species, F is the Faraday constant

and 4 is electrical potential. Equation (24) was derived for diffu-

sion of charged particles in an electrical field assuming ideal

systems and is widely used in the analysis of electrochemical reactions.

The requirement that electroneutrality be maintained at all points in

the system implies a rigid coupling of the concentrations and fluxes

of the exchanging ions. Thus, for two ions I and 3, electroneutrality


Izil Ci + 2zj| Cj = constant (25a)
ziJ + z J = 0. (25b)

These conditions allow the derivation of a single flux equation

for either I or J. This equation is given (36) by

i DiD. (zi2C. + z2C) aCi
zi2CiDi + z2CjDj ax (26)

An immediate observation from this equation is that the interdiffu-

sion flux is dependent on the relative concentrations of the inter-

diffusing ions. With vanishing concentration of either I or J

the interdiffusion flux is controlled by the diffusion coefficient

of the ion in the minority. Further analysis of the kinetic be-

havior of ion-exchange reactions by these investigators for both

film and particle diffusion (36) led to the conclusion that the

rates of forward and reverse exchange reactions were not equal.

This conclusion was confirmed experimentally (34,35).

Incorporation of the simple diffusion-kinetics model based on

Fick's law into the differential mass-transport equations pro-

duces a mathematical problem of extreme complexity (40). As a

result simpler equations based on the linear diffusion concept

introduced by Glueckauf (29) have been utilized (63). As pointed

out by Helfferich in his comprehensive monograph on ion exchange

(35), for practical situations, the gain in accuracy does not warrant

the time and effort expended in solving the problem of greater com-




Thermodynamic Conceptualization of
Ion-Exchange Processes

A system is defined thermodynamically as a body or group of in-

teracting bodies intended for separate study. Any physically homo-

geneous body or set of identical homogeneous bodies is called a phase.

Phases are either pure or mixed, depending on whether they consist of a

single or several chemically individual species.

Systems are either homogeneous or heterogeneous depending on

whether they consist of a single or several phases. The existence of

physical boundaries (interfaces) and interphase regions are necessary

features of polyphase systems. A solution/exchanger system can

therefore be conceived thermodynamically as a heterogeneous system con-

sisting of two mixed phases.

A heterogeneous system may exist either in an equilibrium or non-

equilibrium state. In the former state all thermodynamic state

variables remain constant with time. If the system is non-equilibrium,

spontaneous phase interactions occur resulting in the establishment of

an equilibrium state, characterized by definite compositions of all the

phases. Interactions that do not involve the production of new phases

or new chemical compounds result in material or energy exchanges

across the interfaces. Such interactions are considered as physical

sorption processes and involve atomic and molecular interaction

energies distinct from those involved in chemical bonds. The term

adsorption refers to physical sorption in which the species trans-

ferred becomes either concentrated at the interface or distributed in

the bulk of the phase. If instead of being transferred, the species

is displaced by interactive forces back into the same phase it is

termed negative adsorption or loosely as exclusion.

A unique property of exchanger phases in the presence of fixed

electrical charge sites, which may be either restricted to the ex-

changer surface or distributed throughout its bulk volume. The

quantity of fixed charges, conveniently expressed as equivalents,

defines the absolute capacity of the exchanger. This property,

coupled with the restriction that electroneutrality be satisfied at

all points in either phase of the system, forms the basis for the

metathetical sorption phenomena in solution/exchanger systems. Were it

not for its fixed charges, the exchanger would lose its identity as a

distinct phase in the system at equilibrium. Ion exchange processes

occurring during equilibration of a solution/exchanger system can be

considered as physical adsorption, if it is hypothesised that no in-

teractions occur involving the formation of chemical bonds and produc-

tion of new chemical species or phases in the system.

Disposition of Charged Species in
Solution/Exchanger Systems at Equilibrium

For charged species in a heterogeneous system, a necessary con-

dition at equilibrium is equality of the electrochemical potential of

each species in the various phases. The electrochemical potential n

for a species in a phase is defined (31) by

n = u + zF (27)

in which p is the chemical potential, z the valence, F the Faraday

con,;tant and 0 the inner potential of the phase, The electrochemical

potential can be conceived as the sum of the reversible chemical and

electrical work required to transfer a charged particle from infinity

to any point in the interior of the phase. If the particle is un-

charged no electrical work is involved and n =.p.

Consider two ions I, J of valence zi, zj in the solution/

exchanger system at equilibrium. Then, ni = in and nj = ni, where the

bar signifies the exchanger phase. The chemical potential for a

species in a phase is given by

V = po + RT in a (28)

where p is the chemical potential in an arbitrary reference state, R

is the gas constant, T is absolute temperature and "a" is activity.

Substituting equation (28) into equation (27) and equating the

electrochemical potentials of I and J gives

S+ RT in ai + zi F + = + RT En ai + zi F (29a)

vi + RT in a. + z. F v = up + RT an a. + z. F i (29b)

The difference in the inner potential of the phases O i, at

equilibrium is invariable, and p = p., p = p'. Thus

RT -T a; RT (29c)
$ = 7 n i= n (29C)
z.F i F a


1 n Zi = 1 n (29d)
zi a.i zj a

flultiplying by z. zj equation (29d) becomes

z an ni = z. inA (29e)
3 ai 1 a

( )zj= (d)zi (29f)
a. aj

Similar reasoning for an ion X of valence -zx yields

(X)zi (a)zx and (Aj)zj ()zx (30)
x i x j

Insight into the usefulness of the above relations can be ob-

tained by considering some specific cases.

Consider an exchanger with fixed negative charges and absolute

capacity Qo satisfied by J ions in equilibrium with a solution of an

electrolyte J X Electroneutrality requires
Zx zj
zjC = Qo + zxCx and zjCj = zxCx (31a)

where C represents the molar concentration. From above equation

(29c) shows that
z.F a.
3 3

The potential difference 9 i is the equilibrium Donnan potential

across the interphase, and increases away from the solution/exchanger

interface. The interphase functions as a Donnan membrane in a ther-

modynanic sense because it is impermeable to the fixed exchanger

charges. This macroscopic potential is immediately established and is

the Force preventing the net transfer of J ions out of the exchanger

and of X ions into the exchanger despite concentration differences that

exist between the two phases. Increasing concentrations of J ions in

solution causes a lowering of the Donnan potential while increasing

the exchanger capacity would result in larger potentials. It is ob-

vious that lim (A--i) = 0. If a dilute solution Jx Xj is used for

equilibration then aj : C. and aj = Y jC, where y denotes the activity

confficient. Iroa equation (31a) it follows that

C = Qo + z Cx 9 (31b)
j zj zj
3 z

From above equation (30) is

x j

Substituting C. for a. and 4-' for a gives
3 zj j3

(J J = () l zx)x (31c)
x 'Y Q0o

Since y Q is much larger than Cjzj, the concentration of X in the

exchanger is lower than in solution. This effect is greater the more

dilute the solution of J xXj., and is enhanced with an increasing

valence of the ion X. It is commonly called "Donnan exclusion."

If a salt I XXzi is now introduced into the system described

above, exchange of I and J occurs. At equilibrium equation (29f) shows


(9i)z ( zi
a. a.

Rearranging gives

aj) 'I-" (a.-i (32)
3 i 3 -^
If I is introduced in a trace quantity, then a. is approximately

constant. With this condition, application of Le Chatelier's Principle

to equation (32) shows that if the concentration of J is increased in

the equilibrium solution then ai decreases and vice-versa. Thus, for

dilute solutions of I xXzi the presence of J ions in the solution

phase will lower the selectivity of the exchanger for I ions. This

effect will be greater the larger the difference between zj and zi.

From the general relation given by equation (29f)

(i)j= () zi

Dividing throughout by (dif)zi results in

(i)z (a i = 1. (33a)
ai a

Introducing activity coefficients gives
Cizj Y izj .
ci Ci i 1 (33b)
CiZj C.~i Yij I.

The term on the left side of equation (33b) is the definition

of the "selectivity coefficient" obtained by application of the mass

action law without activity corrections for the exchange of I and J.

Since the activity coefficients are functions of species concentration,

the selectivity coefficient depends on the experimental conditions.

Physical Basis for Exchanger "Selectivity"

The power of thermodynamics is its ability to produce general

relations among system variables without detailed knowledge of the

specific physical forces involved in the phase interactions. Those

are concealed in the thermodynamic activities of the system components.

The equilibrium composition of the phases is governed by the require-

ment that the free energy of the system be minimized. A charged

species in a non-equilibrium solution/exchanger system can lower its

free energy by interactions with the exchanger, solvent, and other ions

in the system. Quantitative theories do not exist to predict exactly

the effect of such interactions, generally termed solvation processes,

on the thermodynamic properties of the phases. The sum total of all

interactions reveals itself in the observed selectivity property of

the exchanger. In effect, selectivity is a measure of the relative in-

teractive effect of two exchanging ions on the thermodynamic properties

of the exchanger phase.

In an aqueous solution/exchanger system, electroneutrality re-

quires that the fixed charge on the exchanger be satisfied, at all

times, by an equivalent quantity of charges of opposite sign. This

fact determines the minimum equivalent concentration of the intersti-

tial solution of porous exchangers. If Qo represents the absolute

capacity in equivalents per gram, p the dry bulk density, and f the

fractional internal porosity of the exchanger particles, then pQ /f

equivalents/cm is the concentration of the interstitial solution. An

exchanger with Qo = 100 meq/100g, p = 1 g/cm3, and f = 0.5 gives a

concentration of 2 N for the interstitial solution. Similar concen-

trations would occur in the interphase regions of exchangers with sur-

face charge sites. In high capacity synthetic organic exchangers,

values as high as 10 N are encountered. Such concentration ranges are

beyond the scope of the Debye-HUckel theory for obtaining the thermo-

dynamic effect of ion-ion interactions.

Ion-solvent effects can be evaluated by the Born theory. This

theory gives the solvation free-energy per mole of ions in solution


G = -NA(ze)2 (1 ) (34a)
2(r + 0.85)

where NA is Avogadro's number, z is the valence, r the crystal radius,

e the electron charge, e the dielectric constant, and 0.85 A an em-

pirical correction factor. The change in free energy in transferring

one mole of I ions from solution to an exchanger in the J form due to

ion-solvent interactions and electrical work would be

G = (2( 0) + zi F( ) (34b)

Since the exchanger solution is more concentrated e < e and the

solvent interaction term is positive. Thus the transfer does not

occur spontaneously unless the second term, which is negative, has a

greater absolute value.

The transfer of I into the exchanger must be accompanied by an

equivalent transfer of zi/zj moles of J out of the exchanger. For

this process

AG = 2r .85) ( ) + zF( ) (34c)
2(rt + GM5) t t j

In this case the leading term is negative and the electrical work term

is positive. The total free energy change for the metathetical reac-

tion is

AG = AG + AG. (34d)

When zi = zj, AG is negative if rj < ri. Thus, considering only ion-

solvent interactions, an exchanger with fixed charges neutralized

by J has preference among ions of equal valence which would increase

in order of increasing crystal radius. In part, ion-solvent interac-

tions serve to explain selectivity among ions of equal valence. When

zi i z., the electrical work term dominates the solvation term and

thus regardless of radius, the exchanger prefers the ion of higher


The concept of exchanger selectivity as competitive solvation

cannot be developed further because a complete understanding of these

processes is far from being realized. However, the above does

illustrate the complexity of the physical interactions involved in

solution/oxchanger systems,

Concepts on Inorganic Ion Transport
During Hiscible Displacement

Established concepts on ion-exchange equilibria and kinetics

allow some qualitative deductions regarding the transport behavior

of inorganic ions during miscible displacements.

If instantaneous exchange equilibrium is assumed, the transport

behavior of an ion depends on the shape and characteristics of the

adsorption isotherm. These would rest heavily upon the selectivity

properties of the exchanger. Over a small range of very dilute con-

centrations of the equilibrating solution phase, it can be expected

that the isotherms would be linear. In such ranges, for an exchanger

saturated with J ions, the presence of J ions in the equilibrating

solution would influence the sorption of another counterion in the

system. As a result, variations in the concentration of J ions in a

displacing solution of I ions would produce variations in the elution

tines for the ion I.

The two-step diffusion concept of exchange kinetics suggests that

for small diameter exchanger particles and trace concentrations of I

in the displacing solution, film-diffusion kinetics may control mass

transfer of I. Since the film is considered as a stagnant hydro-

dynamic boundary layer around the exchanger particle, its thickness

would be inversely influenced by flow velocity of the solution. Also,

as discussed in the foregoing review [equation (26)], mass transfer of

I by diffusion would be influenced by the presence of J ions. The

existence of such rate-controlled diffusion processes would reflect

in deviations from the predicted equilibrium shapes of elution curves

for I.



Although solutions to the convective-dispersion equation are

often quoted and used, detailed derivations of these solutions are not

usually presented. Details of an asymptotic solution to the convective-

dispersion equation for "pulse input" boundary conditions are given

below for a non-reactive and a reactive solute (linear adsorption


The mathematical formulation of the problem is given by the


ac C aC
-t = x for 0 < x < and t > 0 (35)

with initial and boundary conditions

C(x,o) = 0, for 0 < x < m (36a)

C(0,t) = Co[U(t) U(t tl)], for t > 0 (36b)

C(x,t) = 0, for t > 0 (36c)

where U(t) is the Hieaviside unit function and t1 is the time-width of

the pulse.

Let the Laplace transform of C(x,t) be denoted by u(x,s), The

above equation and auxiliary conditions under the Laplace transforma-

tion become

su(x,s) C(x,0) = D v (37)
Dd- dx

u(0,s) = (1 e-Stl). (38a)

u(x,s) = 0. (38b)

The characteristic equation of the ordinary differential equation is

Dm2 vm s = 0. (39a)

with roots v v+ 4Ds.

Denoting -v_ + 4Ds by R, the general solution is

u(x,s) = Cle( + R)x + Ce R)x (39b)

Applying the above initial and boundary conditions

u(0,s) = C1 + C2 = (1 eStl). (39c)

u(x,s) = 0=: C1 = 0. (39d)
x '


u(x,s)- = (l e-Stl)e( R)x (39e)

vx vx
C e2 e-R Ce etl e-Rx (39g)
s s

Consulting a table of Laplace transforms (60), transform pair #3.2-80 is

listed as

-ea(s + b2 1/2 -ab bt/2
a 2 erfc(2t 1/2

+ e- erfc(- + bt1/2

valid for real s >0


e-,= e sx = e + s)

e 'D (40)
S v
letting a = and b the first term inverts to
[if 2/UD
vx vx t1/2 vx 1/2
C_ e2D e- -2 erfc(---- t ) + eM erfc(- + vt )}(41a)
S2/t 2 2/i 2/T

= {erfc( ) + e erfc( X )} (41b)
S2/t 2/Dt

The second term is inverted by noting that it is equal to the first

term x e-st By the shifting property of the Laplace transforms, if

L{f(t)} = f(s), and g(t) = f(t tl). U(t tl) then, L{g(t)}

= e-st f(s) The second term inverts to

C {erfc[x v(t tl)] + e erfc[X + v t tl) U(t tl) (42)
2 2(t t) 2D(t tl)

The second term in both inverse transforms is small except near the

inlet where x = 0 and for small values of t unless D is large. As a

result it can be ignored without introducing a serious error. The

solution reduces to

C /2{erfc( t) erfc [ (t U(t t) (43a)
Co 2/ Dt 2/D(t t1)

Using the identity erfc (x) = 1 erf (x), this can be written as

C = 1/2{erf[ v(t- t)] U(t t) orf( Vt)} (43b)
C 2/D(t tl) 2/Dt

It is convenient to use a transformed variable 6 = which is

physically equivalent to the number of pore volumes, for experiments

with a fixed value of x = L. Setting x = L and dividing the arguments

of the error functions in equation (43b) top and bottom by L, the

solution given by equation (43b) transforms into

-= 1/2{erf[l-- ] U(6 ) erf( --I-)} (43c)
Co 2(e e) 2 DP

For a solute following a linear isotherm, S = KC, and = K The
It at
differential equation becomes

S= a C C pK DC
a3t 2 -v- 9 wt (44a)

Transposing -L and rearranging

ac D Ba2 v ac
at R ax2 R ax (44b)

where R = 1 + -p
For the same initial and boundary conditions it is obvious that the

solution to equation (44b) can be obtained by setting D = D/R and

v = v/R in the solution given by equation (43b), This yields after
introduction of 8 = and rearrangement

= 1/2{erf[(R 8 + 61) ( 0)
o 2/D( )R(45)

erf[ ) -
2/ D6R

Equation (45) reduces to equation (43c) when R = 1. The former

equation can therefore be used to generate theoretical elution curves

for both reactive and non-reactive solutes, with given values of the

parameters v, L, D, R and 01. Equation (45) was obtained assuming a

semi-infinite space domain but it can be used to analyse displacement

experiments in finite columns. It has been shown (14, 46, 54) that

for large values of v and L no serious error is involved in using equa-

tion (45) for this purpose.



The primary objective of this study was to examine the transport

behavior of selected inorganic cations during steady saturated or un-

saturated flow in a reactive porous medium. It was anticipated that

these results would provide insight into the nature of the metathetical

mass transfer processes.

From the onset, it was clear that soils were too heterogeneous to

study exchange processes in detail. Therefore, a mixture was prepared

with a synthetic exchange material and sand. Synthetic exchangers

were utilized previously in miscible displacement studies by Day and

Forsythe (20). In addition to providing the required homogeneity, the

overall exchange capacity could be controlled.

A rigid, analytical grade, iacroporous, granular (50 100 mesh),

organic exchange resin of a highly crosslinked sulphonated copolymer

of styrene with divinylbenzene was purchased (Bio-Rad Laboratories).

This material is both thermally and chemically stable and has a cation

exchange capacity of 4.9 meq/g. In addition, special treatment during

polymerisation results in a low resistance to intra-particle mass trans-

fer by diffusion. Preparation of synthetic organic exchangers in gen-

eral has been described by lelfferich (35). Their general physical prop-

erties have been reviewed by Heyers et al. (48) and the special physical

and chemical properties of the macroporous resins have been described

by Miller et al. (49,50). Because of its high exchange capacity, it


was necessary to dilute this resin with an inert material. Soil from

the I m 1.3 m horizon of a Lakeland sand (Typic Quartzipsamment)

was passed through a nest of sieves and the fractions retained on the

500, 250 and 105-micron sieves were combined in the ratio of 25 : 50 :

25, respectively. This material was then treated with hydrogen peroxide

to destroy any organic matter present. The resin was treated repeatedly

with I N calcium acetate solution until no further pH change was observed,

and then packed into a plastic column and eluted with 1 N Ca(NO3 )

solution. This procedure was considered sufficient to Ca-saturate

the resin. The resin was then dried and sieved and the fraction between

200 and 105 microns combined with the sand to yield a computed exchange

capacity of 30 40 meq/100 g. This porous exchange material was used

in all subsequent studies.

Cations selected for investigation were Li+ and Na+ because of

their ease of detection at low concentrations by flame spectrophoto-

metry. Calcium was chosen as the common ion because the Ca-saturated

exchanger was expected to exhibit low selectivity for Li and Na

resulting in low residence times. Because of this, longer columns

could be used. The exclusion of Cl by the exchanger was also studied

and a cursory investigation was made of the transport behavior of


The columns to contain the exchange medium were prepared from a

single length of rubber-cast plexiglass tubing with an internal dia-

meter of 5.0 cm. The columns were designed to provide unsaturated

flow conditions. To achieve unsaturation, 3-mm holes were drilled

in the walls of the column, and small pieces of wire gauze (less than

100-micron mesh) were placed on the inside to retain the material in

the column. Porous end-plates with an air-entry pressure of 30 40

cm of water were used to retain the material tightly in the columns.

The dead volume of the end plates did not exceed 7 cm The holes in

the column were plugged during saturated flow studies. For unsaturated

flow studies the holes were not plugged and the column was sealed into

a 10-cm diameter column. The outer column was pressurized to achieve

the desired unsaturated soil-water potential and the pressure was kept

constant with a bubble tower.

The solid matrix material was packed into the column under water

to insure complete water saturation. An adjustable peristaltic pump

was used to deliver solution at predetermined rates to the columns.

A fraction collector was used to sample the effluent at equal time

intervals. A schematic of the experimental apparatus is given in

Figure 1.

Molar stock solutions of Ca(NO0) NaC1, LiCI, LiNO and NaNO

were prepared. Tracer solutions were prepared by combining suitable

portions of the required stock solutions and making up to volume.

The concentration of Na+, Li+ and Cl- in the tracer solutions were

kept between 80 85 ppm. Concentrations of the common ion (Ca2+)

used were 0.05 M, 0.02 M and 0.005 M. Tritiated water (HTO) was added

is a non-reactive tracer to evaluate hydrodynamic dispersion. The

transport of 45Ca2+ was studied using tracer solutions prepared by

dissolving one gram of 45CS04 in 2 liters of 0.075 It and 0.05 M

Ca(NO( )
Exchange isotherms for Na+ and Li+ in 0.05 1. 0.02 N and 0.005 M

Ca(:KO3)2 were determined over a 0 200 ppm concentration range. Ten

C) P

01 .0:

0. C
C) )

U 0
C) C)
4- C

C) 0

.01 .,.

C) '-C C)
0 0

C) -'-
-'C-, C

00 '
.0 CC
C)" CC
a0 C
CC )
0 x
C)Is .


C) C).C
*s4 C
C) .
C) 0.
C)o rl
rI C)


solutions in increments of 20 ppm were prepared by weighing out the

required amounts of 2000 ppm stock solutions and making it up to volume

with the appropriate Ca(NO3)2 solution. Ten-gram portions of the dry

solid matrix material, previously washed with deionized water, were

mixed with 10-ml portions of each solution and shaken at frequent

intervals during a 12-hour period. This time was shown to be sufficient

for equilibration. A sample of the supernatant was then withdrawn and

analyzed. The amount adsorbed was calculated from the concentration

difference between the sample and original solution. A similar tech-

nique was used to determine exchange isotherms for Ca2+ in 0.075 M

and 0.05 M Ca(NO )2.

The 'ulse input' boundary condition was utilized in all miscible

displacement studies. The columns were leached with appropriate

tracer-free solutions of Ca(NO3)2 and then a pulse of the tracer solu-

tion containing the cation plus chloride and HTO was introduced. This

pulse was subsequently eluted with the tracer-free Ca(NO3 )2 solution.

Changeover of solutions was achieved in approximately 2 minutes during

which time the outlet was sealed, and the front end-plate and delivery

tubing were flushed and refilled with the new solution. The total

amount of solution introduced during a displacement was determined by

weight differences in the bottles containing the solutions. These

together with the measured time, were used in computing an average

pore-water velocity.

Analysis for Li and Na in the effluent samples were made using

a Beckman model B flame spectrophotometer. Sets of standard solutions

for these analyses were prepared by accurate dilution of a 2000 ppm


stock with the appropriate Ca(NO )2 solutions. Analyses for Cl- were

made using an "Orion" specific-ion electrode. Activity of HTO and

45Ca2+ was determined by liquid scintillation counting of l-ml samples,

in 10 ml of a commercial phosphor (Aquasol II).

The pore volume of the columns was determined by drying the material

in the column at the end of a series of displacements. The dispersion

coefficients were extracted from the breakthrough data for HTO using

a least squares curve-fitting procedure and the simplified asymptotic

solution [equation (43c)]. These coefficients were then used to gen-

erate analytical curves for the reactive solutes using the sorption

parameters from the adsorption isotherms.



Exchange Adsorption Isotherms

The adsorption isotherms for Na+ and Li+ in 0.05 i, 0.02 M and

0.005 M Ca(NO3)2 on the exchange mixture are given in Figures 2 and 3.

The adsorption isotherms were linear for both Na and Li over the 0 to

200 ppm concentration range. The slope of the adsorption isothern in-

creased as the Ca2+ concentration in the equilibrating solution de-


The data in Figures 2 and 3 were fitted by the least squares pro-

cedure to the equation for a linear isotherm, S = K C. The resulting K

values for Na+ in 0.05 M, 0.02 M and 0.05 M Ca(NO3)2 were 0.1324,

0.2196 and 0.3770, respectively. The corresponding values for Li were

0.0822, 0.1122 and 0.1534. These K values increase in a non-linear

fashion with decreasing Ca2+ concentration in the equilibrating solu-

tion. As shown in Figure 4, an assumed general exponential relation-

ship of the form K = Ko exp [P CCa] fitted the data reasonably well.

For Na the values of K and B were 0.3883 and -9.7 and for Li the

corresponding values were 0.1568 and -5.80. These values provide use-

ful quantitative insight into the damping effect of the Ca2+ on the

adsorption of Na+ and Li .

From the theoretical considerations discussed previously [equation

(32)] for equilibrium conditions the following relationship holds:



\ (D


o, O

00 00

o o
\ t \\ 0 c


\\00 000 0



0 CC

N 0 s

LO 0

2 M I.w
0 0 0

(I)e II

4 tJ) c~

:3 5


Do, C


C Q.

+ 0


U .H

0 0-
c3 -
^ Oi


c 0 G
I -H

0 0
+ 4

-~rl U


0 H









2 -2
a4 L (46)
aCa aCa

For the concentrations used in this study, both aNa and aCa are accessi-

ble from the extended Debye-Hickel theory. This theory allows calcula-

tion of the activity coefficient from the following relationship:

log y = (47)

1 2
where y is the activity coefficient, I = 2 EC.z. and is the ionic
i i-

strength of the solution, A = 0.507 at 20 C, B = 0.328 at 20 C and a is

the ion-size parameter. The value of a is 6 for Li+ and Ca2+ and 4 for

Na (43). If only small quantities of Na are adsorbed in the exchanger

phase one can assume that aCa is constant. If it is further assumed

that yN remains constant, then CN = pp may be substituted into

equation (46) to give

a= -Y S = K S (48a)
Na 1 Na
Ca Ca


YN = SNa (48b)

Ca Ca

S = -- a C(48)
Na Na (48c)

If K1 is known, the K value for each adsorption isotherm can be computed.

It is recognized that since I varies with the Na+ concentration in

the equilibrating solution, both yNa and aCa are not true constants;

however, their range of variation can be determined. For the experimental

isotherm with Na+ in 0.05 11 Ca(NO3)2 over the 0 200 ppm Na range,

a minimum calculated value of 0.15 M for I is obtained using the above

formula. From the measured data for this isotherm a maximum value of I

can be obtained. When the highest concentration of 200 pg/ml was

used, the measured equilibrium values were S = 22 pg/g and CNa = 178

pg/ml. Assuming that anions are excluded completely by the exchanger,

the concentration of ionic species in the solution phase was Na+ =

0.0077 1, Ca2+ = 0.0505 M, Cl- = 0.0057 M and NO& = 0.10 M giving

1 = 0.161 M. The computed values for yNa for I = 0.15 M and I = 0.161 M

are 0.740 and 0.735, respectively. Corresponding values for iCa are

0.114 and 0.132. Thus, the values of yNa and /aa do not vary appre-

ciably between the maximum and minimum I values. However, it is clear

that with decreasing concentration of Ca(NO3)2 in the equilibrating

solutions, the gap between the maximum and minimum values of yNa and

aCa does increase. As a realistic approximation, an average value of

I = 0.155 M can be used to compute values of yNa and aCa with the ex-

tended Debye-HUckel formula. These calculations yield yNa = 0.7381 and

aCa = 0.1331. Utilizing these values and the experimental K value of

0.1324 for this isotherm, a value of K1 = YNa /K Ca can be calculated.

If the above reasoning and assumptions are correct, this value of K1

may be used to predict the experimental K values for the adsorption

isotherms using 0.02 M and 0.005 1 Ca(NO3)2. Similar arguments can be

advanced for the equilibrium isotherms of Li +

With the above approach, K values for the isotherms of Na+ in

0.02 11 and 0.005 N Ca(NO3)2 were predicted using a calculated K value

of 41.9 and mean I values of 0.064 M and 0.020 M. These predicted K

values were 0.201 and 0.347 which compares favorably with the measured

values of 0.2196 and 0.3770, respectively. Similar computations for

Li+ using K1 = 70.18 and mean I values of 0.079 I and 0.025 1,

yielded predicted K values for the Li+ adsorption isotherms in 0.02 M

and 0.05 M Ca(NO3)2 of 0,121 and 0.199 which compared reasonably well

with the measured values of 0.112 and 0.153.

Implicit in the use of the Debye-Hiickel theory is the assumption

that the activities of the ions in the exchanger and solution phases

are predominantly the result of electrostatic ion-ion interactions.

The foregoing theoretical results confirm the validity of this assump-

tion and underscores the importance of such interactions in ion-

exchange equilibria.

The exchange adsorption isotherms for Ca2+ in 0.075 M and 0.05 M
45 2+
Ca(N03)2 are given in Figure 5. These were linear over the Ca2+ con-

centrations used and illustrate a similar decrease in the K value with

an increase in 40Ca2+concentration in the equilibrating solution.

Assuming that 45Ca2+ and 40Ca2+ are indistinguishable, then y45Ca

74Ca and y45Ca = Y40Ca This implies that there are no differences in

their physical interactions to produce selectivity. With the previous

theoretical considerations [equation (33b)], and using a dagger to dis-
45 2+ 40 2+
tinguish between properties of 4Ca2 and 4Ca2

CC 1 (49n)


Ct (49b)

At low concentrations of 4Ca2 both C and C can be regarded as con-


c* C(49e)
at = C c)


S = C t




V0 0


\ \0 o


Qo o

o o

0O ,

0 O\O 0 0


\ \ -0 "
\ \ "

in wh;ch K embraces C and all factors involved in adjustment of units.

Employing the same approach as that used for Na and Li+, it is possible

to compute K1 using the experimental K value of one adsorption isotherm.

This value can then be used to predict the K value of other isotherms.

This approach provides an evaluation of the validity of equation (49d).

The experimental K value of the isotherm for 45Ca2+ in 0.075 1 Ca(NO)2

is 0.764 which gives K1 = 0.057. The predicted K value of the isotherm

for 45Ca2+ in 0.05 M Ca(NO3)2 was 1.146 which compares favorably with

the measured value of 1.041.
45 2+
The exchange adsorption isotherms for 4Ca2 are of interest in
40 2+
explaining the basis for the effect of increasing Ca concentrations

on the K values. Consider specifically the exchanger in equilibrium

with 0.05 M Ca(NO3)2 solution. If a Donnan electrostatic potential

difference is set up across the interphase region, this essentially

equalizes the diffusion rates of Ca2+ ions into and out of the exchanger

allowing the continued existence of a higher concentration of Ca2+ in

the exchanger phase to maintain electroneutrality. Although macro-

scopically the composition of either phase remains fixed, microscopic

exchange of Ca2+ ions continues to occur at a fixed rate across the
45 2+
phase boundary. If some fixed amount of 4Ca2+ ions is introduced in
45 2+
the solution, it is reasonable to expect that at equilibrium the 4Ca2

ions entering the exchanger depend solely on their relative abundance

to 40Ca2+ in the solution. Increasing the solution concentration of

Ca2+ to 0.075 1 would reduce the relative concentration of Ca2+ to

40C2+ on the exchanger and thus the macroscopic adsorption of Ca2

is decreased. It is therefore reasonable to assume that the ratio of

Ca2+ in the exchanger and Ca2+ in the solution is inversely propor-

tional to the concentration of 40Ca2+ in the solution. Expressed

quantitatively, K.05/K.075 = 0.075/0.05 which is what was concluded

above [equation (49d)] using a different approach. The foregoing re-

sults serve to give credence to the concept of ion exchange as a

Donnan-type redistribution of ions between the solution and exchanger


It was not possible to determine the negative adsorption isotherms

for Cl- in batch studies because the increases in C1- concentration were

too small to detect above random variations associated with the specific-

ion electrode.

Miscible Displacement Experiments with Na'

Elution curves are presented in Figures 6 through 14 for a series

of input pulses containing Cl-, HTO and Na+ in 0.05 M, 0.02 H1 or

0.005 M Ca(NO3)2,using three pore-water velocities ranging from 1.5 to

15 cm/hr for each Ca(NO3)2 concentration. These displacements were

made using a 30.4-cm long column. The medium was packed to a bulk

density (p) of 1.786 g/cm3 and had a saturated, fractional volumetric

water content, 6 of 0.344. Areas under the breakthrough curves in

Figures 6 through 14 were determined by trapezoid rule integration and

in each case indicated complete recovery of the material injected.

Thus, complete reversibility of the mass transfer processes was achieved

in these column studies.

It was not possible to maintain identical values of the three

pore-water velocities for each concentration of Ca(NO3)2. However, in

no case did the measured pore-water velocity deviate by more than 2% of

the mean values (14.69, 7.29 and 1.52 cm/hr) taken over the three

Ca(N03)2 concentrations, Dispersion coefficients were determined from


.0 0Ii





0 0 0


000 I- u1

00I 0




c i

06 6

* >n >

i II .0 o
0. 0 0


&, 0' 1"

.4 O





Ii '-1
z -

' o

__ I I I I 1 0 o


n OI L


a0 ~ 000





ri II



o I





0. 0

U g






:4 OCC


C-'~ -I 0C

0 0
044 0-


r C)X

0 00
00 u

ii II -4
a 4 0

9qcr( y C cs 'O




O Cu



0I i





0 '

~ 0
C ^-


Cu 0


m- ^

C -3

u u

C ( .,

I co g

0 0 0 oco


I- C' -0C0

-F- 0

F- ci-

(5 Q -jLFL N

I I 2 2 1 1




3 II

II i






m II
3 yj





the HTO Clution curves and were not markedly affected by the Ca(NO3)2

concentration. The dispersion coefficients averaged over the three

Ca(NO )2 concentrations and corresponding to the mean pore-water

velocity values given above were 0.453 + 0.025, 0.263 0.040, and
0.082 0.007 cm-/hr, respectively. These values plotted against the

average velocity, as shown in Figure 15, gave a linear relationship.

Extrapolation to zero pore-water velocity gave an intercept of 0.0458

cm2/hr (1.3 x 10-5 cm2/sec) which represents the diffusion coefficient

of HTO. Recent values reported by Mills (51) for the self-diffusion

coefficient of HTO are 1.724 0.003 x 10-5 cm2/sec at 15 C and 2.236
-5 9
0,004 x 10 cm2/sec at 25 C. The experiments reported in this study

were conducted at approximately 20 C. A linear interpolation yields a
-5 2
value of 1.88 x 10- cm /see which compares favorably with the experi-

mental value when tortuosity factors are considered.

The elution curves for Cl- appeared, in all cases, slightly to the

left of those for HTO confirming the expected exclusion of Cl by the

exchanger. Because no exclusion isotherm was measured for Cl-, the

excluded volumes determined by the magnitude of the left-hand shift

of the Cl- curve from C/Co = 0.5 at V = V for these nine elution

curves were averaged. The excluded volume was calculated to be 12.8

cm using the Cl- data. This information was used to calculate a

"retardation" coefficient of R = 0.938 for C1- from the relationship

R = 1 12.8/Vo. As shown in the figures 6 through 14 this value gave

analytical curves which described the elution curves for Cl-.

Analytical curves for Na utilizing R values (1 + ) calculated

with the K values from the appropriate isotherms, described the elution

curves for a+ in 0.05 M and 0.02 1 Ca(N03)2 reasonably well. These



+ L

cd oU o


> ;



E 5

results indicate that equilibrium was instantaneous and followed the

experimental isotherm. The somewhat poorer agreement between the

analytical and experimental curves for Na in 0.02 M Ca(N03)2 was

probably the result, in this case only, of not using the same batch

of 0.02 M Ca(N03)2 solution as that used to determine the adsorption

isotherm. Concentration differences between the two batches would ex-

plain the consistent left displacements observed in this case. It also

underscores the sensitivity of the system to changes in the concen-
tration of Ca2

The linear equilibrium model failed completely to describe the

elution curves for Na+ in 0.005 It Ca(NO3)2. This was unexpected and the

displacements in 0.05 M and 0.005 1 Ca(NO3)2 were repeated using a

shorter column with p = 1.764 g/cm3 and 0v = 0.344. These elution

curves are presented in Figure 16 and are shown to behave in an identi-

cal manner as that observed for the longer column. This rules out the

possibility that the observed results were experimental artifacts.

Failure of the equilibrium model to describe experimental data is

usually construed as an indication of non-equilibrium. As previously

discussed, it was expected that, at the concentrations of Na+ used,

film diffusion would constitute the main resistance to mass transfer.

According to this concept, a stagnant fluid film is thought to exist

around the exchanger particles across which interphase mass transfer

takes place by molecular diffusion. The thickness of this film would

vary inversely with fluid velocity. However, no marked shifts or

changes were observed in the shapes of the elution curves for Na or

HTO in 0.05 N, 0.02 1 or 0.005 i Ca(NO3)2 although the pore velocity

was varied by almost an order of magnitude.


r r-

9 L

z z 00- (D0





Whatever the process associated with the observed results in

Figure 16, it clearly depends in some fashion upon the concentration of

Ca2+ in the system, since the concentration of Na was held constant at

80 85 ppm in all cases. The theoretical and experimental studies of

Helfferich and his coworkers, reviewed previously, indicated that the

interdiffusion coefficient of Na in either phase would depend upon

the relative concentrations of Na and Ca2+ in that phase [equation

(26)]. Their theory showed that at low Na to Ca2+ ratios, the inter-

diffusion coefficient is close to the diffusion coefficient of Na+ and
approaches that of Ca as the ratio increases. However, it was in-

conceivable that the diffusion coefficients of Na and Ca2+ differed

by an amount large enough to account for the drastic changes in the

elution curves observed when the concentration of Ca2+ was decreased

from 0.02 M to 0.005 M. This argument coupled with the observed null

effect of variations in pore velocity leads to the conclusion that in-

terdiffusion in films, as conceived above, was not the process

responsible for the observed behavior.

The possibility existed that intra-particle diffusion nay be

the factor responsible for the results presented in Figures 12 to 14.

However, if this were true, it was difficult to explain why such

results appear only during the displacement of Na in 0.005 M Ca(N03)2

and not with the two higher concentrations. In order to investigate

the resistance to intra-particle mass transfer of the exchanger par-

ticles, unsulphonated, spherical, 20 50 mesh, macroporous copolymer

beads were used. Their mean diameter was at least 5 times larger than

that of the sulphonated exchanger particles used in the previous ex-

periments. The identical tracer solution of Na in 0.005 i Ca(NO )2

was displaced through a 20.5-ca long column packed with these beads. The

elution curves for Cl-, HTO and Na are shown in Figure 17. These

curves showed no indication of diffusional mass-transfer processes thus

negating the possibility that intraparticle diffusion kinetics were

responsible for the previously observed results. The results in Figure

17 also confirm the assumption that a low resistance to intraparticle

mass transfer existed for the macroporous, polystyrene exchanger


From the foregoing studies, it was now clear that diffusional

mass transfer kinetics was not the major factor responsible for the

failure of the equilibrium model to fit the elution data for Na in

0.005 M Ca(N03)2. Experiments for the transport of Li+ were next used

in an effort to gain a deeper insight and provide a reasonable explana-

tion to this anomaly.

Before proceeding onto these experiments, displacements of two

tracer pulses consisting of Na Cl- and HTO in 0.05 1 and 0.02 H

Ca(NO3)2 were done for steady-state unsaturated flow conditions in the

same column used in the foregoing experiments. The column was kept

under a constant pressure of 23 cm of water using the method described

previously. The fractional volumetric water content, v was reduced

to 0.2740. The column was positioned vertically in order to give the

highest possible flow rate without incurring large water-content

gradients in the column. The flow rate was adjusted so that a constant

pressure head of 5 6 cm of water was maintained at the inlet.

The elution curves for these displacements are presented in

Figures 18 and ]9. Areas under these curves by the trapezoid rule in-

tegration indicated that the material injected with the pulse was


0 -
0 3 0

0 +
- a
I z
* 0

I-- I-- I I- I-- I-- I I I -

Ocr ron







*H 40


ifl"' o

0 ill


0 *



u ^
0 rl 3







c II

II rl






0 -

C 0,
00 0

'So I I
o y SC,


0 (Y
=" o,0


.o> t

Q ) p O j T q C\

> uu

recovered completely in the effluent. The R values used for the

saturated flow experiments were suitably modified to take into account

the reduction in 0 The equilibrium model [equation (45)] was used to

generate the analytical curves shown in Figures 18 and 19.

Rather poor agreement was obtained between the analytical and

experimental curves for Na in 0.05 N Ca(N03)2 and was worse for Na

in 0,02 M Ca(N03)2. The general tendency was a displacement of the

experimental data to the left of the predicted analytical curve.

It is clear that better agreement is obtained if a lower R value were

used. It is possible that with unsaturation a portion of the porous

matrix becomes either inaccessible or was accessible only at an ex-

tremely slow rate. Subtle arguments are required to justify a shift to

the left of the predicted equilibrium curve in the second case. These

fortunately were discussed elsewhere (69, 70) and are not presented

here because it was the intention of this study to make only cursory

observations of Na+ transport under unsaturated flow conditions.

Miscible Displacement Experiments with Li+

As a consequence of the foregoing results, it was decided to ob-

serve the elution behavior of pulse inputs of Li+ and HTO in 0.05 M,

0.02 M, and 0.005 M Ca(NO3)2 at a single fluid pore velocity. The

experimental and analytical curves for these displacements are presented

in Figures 20, 21, and 22, respectively. Areas under these curves by

the trapezoid rule indicated complete recovery of materials injected

with the pulse. A 20.5-cm long column packed to a bulk density (p) of

1.764 g/cm3 and with a fractional volumetric water content (0 ) of 0.344,

was used in these and all subsequent displacements involving Li .




U C)


I I1u
00 0

o 0 0


r 0
0 u

u ua

I 0
0 0




^0 & ^ ~ ^ ^ ^ ~ =


-3 O 7TOF Co




cJ cO.
0 0









o .



U tJ



.~-C I

o .-
0 40-

S 0--

= 1^
in o

*. 0


44 C

*r4 Cf

0 0 rl
C 1i
*H -I
aj o

The behavior of Li in 0.05 t Ca(NO3)2 agreed with the theoretical

curve, but the linear equilibrium model failed to describe the elution

patterns of Li+ in 0.02 M and 0.005 I Ca(NO3)2. A mild discrepancy was

apparent in the front portions of these latter curves becoming more

pronounced as the Ca2+ concentration was decreased. In order to obtain

a concentration of 80 85 ppm Li in the tracer solution, it was

necessary to add 12.5 cm3 of 1 M LiCl/liter of tracer solution. This

resulted in density differences and may have caused the observed dis-

crepancies which were similar to those reported by Rose and Passioura


The results on the back side of the breakthrough data were in

disagreement with the predicted analytical curves. A clue to the

reason for this anomalous behavior was provided by the observation that

the deviations commenced at approximately one pore volume after change-

over to the tracer free eluting solution.

As discussed above, for analytical purposes, it was necessary to

keep the concentration of Li+ in the tracer solutions between 80 85

ppm. This concentration of Li+ resulted in a larger numerical contri-

bution to the ionic strength of these solutions than the same concen-

tration of Na+. It was, therefore, reasonable to assume that

differences in ionic strengths between the eluting and tracer solutions

were the reason for the observed behavior. Figures 23 and 24 present

elution curves for the displacement of the same tracer solutions of Li

in 0.02 1 and 0.005 I Ca(NO3)2, but with the ionic strength of the eluting

solution adjusted to match that of the Li+ plus Ca(NO3)2 tracer solu-

tion. The observed inflections in both cases were altered, but did not

disappear completely; however, the change was towards better agreement

- 0
* 0

0 *


0 1


g n"

u wb


In U



w 0~

0 -' >

C 1








- -H

S 0 0 0
0 0 0

o 0

0 0 0


O u
"-^_ 0 o CM *^

with the calculated curves. In order to pursue this idea further and

to confirm that the concentration of Li+ was a contributing factor,

displacements were conducted using 50 55 ppm Li in 0.02 It Ca(NO0)2

and 0.01 M Ca(N03)2. These elution curves are shown in Figure 25. The

analytical curves were generated using a value of D obtained from a pre-

vious displacement at almost the same pore-water velocity. The R value

for the displacement of Li+ at 50 55 ppm in 0.01 M Ca(N03)2 was com-

puted using the empirical exponential relationship for the dependence
of K on the concentration of Ca (Figure 4). The inflection in the

elution curve for the displacement of Li+ in 0.02 M Ca(N0)2 was less

pronounced than for the displacement of Li+ at 80 85 ppm. The in-

flection in the elution curve for the displacement in 0.01 M Ca(N03)2

was more pronounced than in 0.02 ti Ca(NO )2; showing similar enhance-

Iment with decreasing Ca2+ concentrations as those observed previously

(Figures 21 and 22).

The foregoing experiments served to isolate some factors associated

with the observed results, but did not provide any insight into the

mechanism. This was needed to explain why the effect appeared only

on the desorption portions of the elution curves and why the inflections

did not disappear with ionic strength corrections. In addition, cal-

culations utilizing the extended Debye-HiUckel formula show that the

activity of Ca2+ in the tracer solutions containing 80 85 ppm Li

in 0.05 M, 0.02 M and 0.005 11 Ca(N03)2 were respectively 2.4, 4.8 and

11.4% less than the corresponding activities in the pure solutions.

It was shown above [equation (48c)] that the K value is inversely re-

lated to the activity of Ca2+. The above results therefore imply a

depression in adsorption with a resulting increase instead of a decrease

in the Li+ concentration of the solution phase.




/-2 .42


0 4
+ .o



-I "

r, H
fl0 02.

20 42 V
+ 42

n30 02


02 42
-a o oJ


22.3 02

02 C

4-1 bD

mi o

& T3

+~ u
02 22

i -l il

4- U11
(U 0 01



M (

20 cn 4

02 --

These facts led to the following experiment which provided a

reasonable explanation for both the observed inflections and the

anomalous behavior of Na in 0.005 rl Ca(NO3)2. The solution in the

column was displaced with a tracer solution containing 80 85 ppm of

Li+ in 0.005 M Ca(NO3)2 until the concentration of Li+ in the effluent

was equal to its concentration in the influent solution. This im-

plies that equilibrium was achieved at all points within the system.

This solution was then displaced with deionized water and the con-

centrations of all components in the effluent were monitored. The

appearance of Ca2+ and Cl- was monitored qualitatively by precipitation

with Na-oxalate and AgNO3, respectively. Nitrate was identified by the

brown ring test. At exactly one pore volume, the concentrations of all

components including Li+ fell sharply to zero. It thus became clear

that the electroneutrality requirement peculiar to ion exchange adsorp-

tion processes resulted in all components behaving as non-reactive

solutes once equilibrium was achieved. This result implies that an
i+ equivalent toK
amount of Li equivalent to- VoCo remained adsorbed by the exchanger.

The adsorbed Li+ was then eluted with a 0.005 M Ca(NO3)2 solution.

The elution curve showed the appearance of Li shortly before one

pore volume, rising sharply to a steady maximum concentration and then

falling sharply to zero at approximately 2.56 pore volumes. This be-

havior is analogous to that observed on the back side of the pulse

elution data for Li+ in 0.005 1 Ca(NO3)2 (Figure 22). These results

are presented in Figure 26, and indicate that the exchange adsorption

for Li+ in 0.005 M Ca(NO3)2 was distinct from its exchange desorption.

If the same K value characterized both processes, the elution volume for

desorption should have been RL = 1.79 pore volumes instead of observed

-~-, T




0+ +

0 0

0 @ 0

0 0 0


1 F) Iz I I I L
*- uj6


9Q )

* CM

cb CO

L 21
c uj

0 C
[C) Z

0 *

o m

value of 2.56. This indicates that a higher K value was associated with

the desorption process.

The samples collected from the desorption experiment were tested
2+ 2+
for Ca with Na-oxalate solution. The tests showed that Ca2+ appeared

in the effluent with the Li+. It was also observed by comparing the

turbidity developed in the effluent samples with a standard (0.005 IH
Ca(N03)2) that the concentration of Ca rose sharply to a maximum

concentration which was less than 0.005 11. The Ca2+ concentration

remained at the lower level until the elution of Li+ was complete,

after which it rose to 0.005 M. This was expected because electo-

neutrality necessitates that an equivalent amount of Ca2+ replace the

Li+ desorbed. These results indicate that the desorption of Li+

occurs with a concentration of Ca2+ in the solution phase less than

0.005 M. This implies a lower Ca2+ activity and therefore a higher K

value [Equation (48c)].

The concentration of Li in the saturating solution was 0.0122 iH and

from the exchange desorption curve the maximum value reached was 0.0067

M. Applying the electroneutrality requirement, the concentration of

Ca2+ would have to decrease from 0.005 M to 0.0016 M. Thus, during

desorption the solution phase constitution is Li = 0.0067 1, Ca2

0.0016 M, NO3 = 0.01 M. This yields a value for the ionic strength of

0.0117 M. With this value the extended Debye-HUckel formula gives

values for YLi = 0.9012 and aaCa = 0.0325. The empirical formula K =

y i/70.1837 'i4C, used previously to calculate K values of the ad-

sorption isotherms for Li+, yields a value of K = 0.3952 which gives
P- = 2.03. This value is greater than the experimental value of 1.56
(Figure 26) but it is in the right direction. Observations from the

elution curve for Li+ in 0.02 M Ca(NO3)2 given in Figure 21 suggest

that if a similar experiment as that described above were conducted

with 0.02 M Ca(NO3)2, a maximum concentration of Li+ = 0.82 Co would

result for the desorption process. By the same reasoning, the con-

centration of the various species in the solution phase would be Li =
2+ -
0.01 M, Ca2 = 0.015 M, NO3 = 0.04 i yielding an ionic strength of

0.055 M. Similar calculations yield values of yLi = 0.8292, 4 Ca

0.0842, a predicted K value of 0.1403, and s K = 0.7205. The value of
pK v 0 g
compares favorably with the observed value of 0.60 (Figure 21).

The inflections observed in the elution curves for Li (Figures

21 through 25) become clear in light of the above results. When lower

concentrations of Ca(N03)2 were used in the tracer pulse, the quantity

of Li+ adsorbed is increased. The desorption process is characterized

by a K value that is dependent on the activity coefficient of Li and

the activity of Ca2+ in the desorbing solution. If a low concentration

of Ca(NO3)2 was used in this solution, the equivalence of exchange re-

sults in a reduced value for the activity of Ca2+ and hence a corres-

pondingly larger K value. When the concentration of the desorbing

Ca(NO3)2 solution is increased, the K value decreases. In addition, at

higher concentrations of Ca(NO3)2 the effect of the adsorption of
Ca would be less pronounced. The desorption K value then approaches

more closely the K value for adsorption. These facts explain the re-

duced degree of inflection when the concentration of the eluting

Ca(NO3)2 was increased to match the ionic strength of the tracer solu-

tion, or when the concentration of Li in the latter was lowered. In

addition, it explains why the inflections did not disappear after the

ionic strength adjustment and why they were not observed for the

displacement of L.+ in 0.05 1 Ca(NO3)2'

Thi foregoing results also clarify the apparent kinetic effect

observed for displacements of Na in 0.005 II Ca(NO3)2. Examination of

the condition under which the data in Figures 11 through 14 were obtained

reveals that because of large R values, the tracer front for Na in

all cases, had not reached the Covalue in the effluent at the time of

changeover to the eluting solution. It is obvious from the foregoing

that this would result in a change in the ionic composition of the

solution phase before the column was completely equilibrated with Na.

Thus, instead of reaching a maximum concentration of Co, the effluent

would reach a maximum lower than Co. This idea was tested by a step

input displacement experiment with Na in 0.005 M Ca(NO3)2. The result

given in Figure 27 shows that the above is indeed a reasonable ex-

planation since the effluent now shows none of the previous apparent

kinetic effect. A further point in favor of the correctness of this

hypothesis is that it explains why the apparent kinetic effect was

associated with only the front portion of the curves. If true

kinetic processes were operating and the adsorption isotherm was linear

the effect would have been symmetrical.

In light of the foregoing, the data for Na+ in 0.02 I Ca(NO3)2

may seem anomalous since they did not show inflections. It is likely

that inflections such as those for Li in 0.02 I Ca(NO3)2 may have

existed but were overlooked because they were too small.

Miscible Displacement Experiments with 45Ca2+

Two displacements of pulse inputs were conducted with 45Ca2+ in

0.05 M and 0.075 M Ca(NO3)2 to verify that the adsorption isotherns for

ACa2+ do reflect appropriately in their transport behavior. Because

of large R values, these displacements were conducted in a shorter




Cc C
UK)> '

it I ~c, I 1-. C

5.7-cm long column packed to a bulk density (p) of 1.769 g/cm3 and

saturated to a fractional volumetric water content (6 ) of 0.347.

The results of these displacements are given in Figure 28. Because

of the shorter column, experimental error was greater. In both cases,

if higher R values were used, closer agreement would have been obtained

between the experimental and analytical curves.

It is interesting to observe that for the displacement in 0.05 M

Ca(NO3)2 changeover to the eluting solution before the tracer front had

completely appeared in the effluent, resulted in a similar disagreement

between the analytical and experimental curve as was obtained for Na

in 0.005 M Ca(NO3)2. Further, the use of a much larger pulse for the

displacement in 0.075 M Ca(NO3)2 produced the expected inflection at

approximately one pore volume after changeover and a shift to the

right of the analytical curve.

These results confirm that the behavior of Na in 0.005 M

Ca(NO3)2 cannot be attributed to the effect of relative concentrations
+ 2+
of Na to Ca on the interdiffusion coefficient of either ion. Exa-

mination of the relevant equation of Helfferich and his coworkers

[equation (26)],reveals that in the case of isotapic exchange, the

interdiffusion coefficient is equal to the diffusion coefficient of

Ca2+ and independent of relative concentrations. Also, increasing

the relative concentration of Na to Ca2 should result in the inter-

diffusion coefficient approaching the diffusion coefficient of Ca2+

If this were indeed responsible for the behavior of Na in 0.005 M

Ca(NO3)2, then similar effects should have appeared in both the elution

curves for 45Ca2+ This however was not observed.

0 In


> C,

II 2

o C



















Theoretical approaches to describe cation exchange processes are

fairly recent although the topic spans a period of over a century.

Despite the acknowledged importance of this process in soil systems,

there are only a few references which deal with transport of ionic

species. In this study, the miscible displacement technique was

utilized to investigate the influence of exchange adsorption on the

transport of selected inorganic ions in a porous medium. The influence

of dispersion and exclusion was also studied. A Ca-saturated organic.

cation exchanger was used as the medium, which eliminated the possi-

bility of interactions other than ion exchange.

Solution and exchanger was conceived as a heterogeneous system

consisting of two mixed phases. A thermodynamic treatment based on

this concept predicted the exclusion of anions. It was shown that, in

general, solution-phase ions common to those initially saturating the

exchanger influence the adsorption of other charged species. This was

adequately verified by experimental exchange adsorption isotherms deter-

mined for Na+ and Li+ in 0.05 _, 0.02 I and 0.005 M Ca(NO3) and for

45Ca2+ In 0.075 M and 0.05 Ii Ca(N03)2. For the range of concentrations

studied, the adsorption isotherms for Li+ and Na+ were linear and

their slopes increased in a non-linear fashion with decreasing Ca2+

concentration in the equilibrium solutions. Similar trends were evi-

dent from the 45Ca2+ exchange isotherms. It was demonstrated inductively


thnt these results could be attributed to ion-ion interactions as

conceived and quantified by the Debye-lluckel theory.

Miscible displacement experiments involving Na+, Li+, 4Ca2

and C1- were conducted to examine the consequences of the foregoing

results, and to determine the presence of kinetic mass-transfer pro-

cesses. All experiments were performed using pulse inputs of the

tracer solution. Tritiated water (HITO) was present in all tracer

solutions in order to evaluate dispersion. An asymptotic solution

to the convective-dispersion transport equation for a reactive

solute (linear adsorption isotherm) was used to predict the experi-

mental breakthrough data.

A series of displacements involving pulses of Na+, HTO, and C1-

in 0.05 M, 0.02 M or 0.005 i Ca(NO3)2 were conducted under steady-state

water-saturated flow conditions at three pore-water velocities ranging

from 1.5 to 15 cm/hr. Dispersion coefficients obtained from the HTO

elution data were not influenced by Ca(NO3)2 concentration and were

linearly related to pore-water velocity. The elution curves for Cl-

were all displaced to the left of those for HTO confirming its exclu-

sion by the exchanger. Reasonable agreement was obtained between the

computed and experimental elution curves for Na+ in 0.05 1i and 0.02

M Ca(N03)2 at all pore-water velocities studied. The analytical

solution, however, failed to describe any of the experimental elution

curves for NaF in 0.005 1 Ca(NO3)2. These results could have been

used to support the concept of diffusional mass-transfer kinetics, but

it was shown experimentally that such processes were not involved.

A series of similar steady-state, water-saturated column experi-

ments were conducted to study the transport behavior of Li+ and JITO

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