A thesis submitted for the degree of

"KINETICS OF ION-EXCHANGE AND WATER DIFFUSION IN CHAEAZITE"

A thesis submitted for the degree of

Doctor of Philosophy of the University of London

by

Pamela Jean Steele, B.Sc., A.R.C.S

Physical Chemistry Laboratoiles, Imperial College,

Errata

Page 39 Line 7: for reference:(76) read (77). Line 17: for reference(77) read (76). Pace 45 Equation 3.3.2(11) should read

DAB = [(z1c1L111) z c L )g1+(zic11,121) - z_e L )?!2 2 21 a, 22` .Ltz +(z1o

1L13 P-z2c2L23 ac 1 (i+p),

Page 53 Line3:for equation 3.51(6) read 3.5.1(6). Equation 3.5.1(8) should read

,2

g

[k(1

c,1)

1

ac

A7_,

Page 85 Table 4.3:- for 1-°Na read 133Ba.

Page 88 For equations 4.4.2(4) read V(c1-c2)/0.1M 4.4.2(5) read ....10mV(c1 -c2)/M 4.4.2(6) read ...=10-3mV(ci -c2)/M Page 140 Fig.6.17: heading - should read

K+--) +

--t1(+ 0 +

Page 147 Insert "Small Particles of" before "Sodium Chabazite". Page 157 Line 10: for entergy read energy.

Page 163 Line 9: for 2Na i Da2+ read Da2+ --)2ra+ Page 196 Line 6: for reference (98) read (97). Page 197 Line 15: for evaluted read evaluated.

NOTE In Figs.2.1, 6.9 to 6.16 and 6.23 to 6.24, the vertical axis should be labelled "TAU" not "TOR".

2 -

Abstract

Ion-erchan7e kinetics in the zeolite chabazite have been studied for the exchan7e systems sodium-potassium, sodium-lithium. sodium-calcium and sodium-barium by three different methods. lnte7ral interdiffusion coefficients were measured at a series of temperatures in each system, and were found to obey the "Arrhenius" equation. The values of the interdiffusion coefficients at 28°C ranged from 2.4 x 1C-15 m2s-1 for 21:a--> Ca21- e:7chan7e to

_17

66C x 10 'M s for Na h exchange'and activation energies were in the ran!,;a 35.6 to PP-O kJ rol. These values have been discussed in terms of the crystal structure and degree of hydration of the cations.

Water diffusion has been studied by a tracer techniqae in natural chabazite both with pure water and with a variety of aqueous salt solutions as the liquid phase. The self- diffusion coefficient of water in natural chabazite at 2,'°0 was found to be 1.3 r 10-11 m-2 s-1 , and the apparent

activation ener7y for the process was 34-9 kJ rol-1. The Presence of exchan7in7 ions in the solution phase was found to have no effect on the rate of :rater diffusion.

Experimental ion-erchan7e kinetic data were compared with predictions from the I:elfferich and Plesset theory using a computer rrorrram. _{In the sodium-potassium and}

sodium-calcium systems, the predicted rates were the reverse of the errerirertal ones, whereas in the sodium-lithiur

3

correct. The inclusion of experimental activity

correction terms in the theory was found not to improve the agreement with observed rates to any Treat extent.

Acknowled;Tements

I would like to extend my sincere thanks to my supervisor, Dr. L.V.C. Rees, for his unfailing help

and encouragement throughout the duration of this work. Thanks are also due to the technical staff of

the Department of Chemistry, particularly Mr. D. Alger of the Electronics Workshop.

I am grateful to my fellow students for their immensely helpful discussions and advice. Especial thanks go to Mr. H.A. Ramakers of the Technische Hochschule, Aachen, West Germany, for his invaluable support and help with computational problems.

I would like to thank Mrs. M. Grossman for her

patient typing of the manuscript, and Miss M. Pinches for her assistance in the production of this thesis.

Finally, I am indebted to the Science Research Council for the award of a research student-ship 1971 - 1974.

CONTENTS Page Abstract 2 Acknowledgements 4 Contents 5 6 21 38 59 96 113 Chapter 1: Chapter 2: Chapter 3: Chapter 4: Chapter 5: Chapter 6: Chapter 7: Chapter 8: Introduction Diffusion Ion-Exchange Experimental Section Computing Section Experimental Results and Discussion Computational Results and DiScussion Conclusions 167 199 References 200

6

Chapter 1. Introduction

1.1

General Introduction

1.2

Classification and Strr'ctures of Zeolitec

1.2.1

Structure of Chabazite

1.3 Properties of Zeolites

1.3.1 Cation Excharme and Diffusion

Chapter. 1: Introduction

1.1 General Introduction

By corron definition, ion-exchangers are insoluble materials which carry exchangeable cations or anions. References to phenomena which can now be explained as due to ion-exchange can be found in the Bible, Aristotle and Francis Bacon, for example. These are generally concerned with the purification of water by pass inn; it through layers

of rock, soil or sand. During the nineteenth century, studi-°5 of ion-exchange were made by Thompson(1) and Way(2), two

English agricultural chemists workinf7 on the chemistry of soils. Lemberg(3) reversibly converted leucite into analeite by treatment with sodium chloride solution. Lemberg, and later Wiegner(4) explained ion-exchange by

soils as due to the presence of both organic and inorganic species in the soil, including clays and zeolites . The earliest application of ion-exchange for industrial use was by Gans(5) in 1906. He used both natural and synthetic aluminium silicates (permutites) for softening water.

With the rapid development or organic ion-exchange resins from 1925 onwards, interest in zeolites and other

inorganic exchangers decreased. Organic exchangers were found tc have many advantages which made them particularly suitable for industrial uses: they exchange rapidly and can easily be converted into the hydronium form by treatment with acids, whereas aluminosilicates are generally unstable in acid solutions. The synthesis of ion-exchange resins

- 8 -

made it possible to vary the properties of the exchanger in a systematic manner, leading to an increase in the theoretical knowledge of ion-exchange.

However, zeolites have the advantage that, unlike organic resins they do not swell or shrink when in contact with water(6) are more stable at high temperatures(7), and are more

resistant to ionising radiation. The development in recent years of radiochemical methods of investigation has led to a renewal of interest in inorganic exchangers, and in

zeolites in particular.

Until recent years, most publications on ion-exchange

have concentrated on equilibrium rather than kinetic studies(8)

.

As a result, theories of ion-exchange kinetics are not as

well-developed as theories of exchange equilibrium, and agreement between theoretical predictions and experimental observations is often poor.

The broad objectives of this work were to measure interdiffusion coefficients in four exchange systems in chabazite and hence determine apparent activation energies for these processes, and in addition, to investigate the self-diffusion of water in chabazite. A third aim of this study w2^ to compare experimental measurements of ion-exchange kinetics with theoretical predictions by a computational method.

1.2 Classification and Structures of Zeolites

Silicates can be divided into several groups on a structural basis(9): the zeolites belong to the group known as the tectosilicates(1 , which also includes the felspars and feispathoids.

-9-

Tectosilicates are based on a three-dimensional framework in which each SiO4 tetrahedron- is linked to four other

tetrahedra. This framework has the formula (S100 -11 and is electrically neutral. The partial replacement of Si . by A134- results in a group of minerals known as aluminosilicates.

The extent of substitution of Si4 by A13+ is limited by Loewenstein's rule(11) which states that A104 tetrahedra

may only be linked to SiO4 tetrahedra and never to each other. As a result, the maximum alumina content must occur when the SiO4 and A104 tetrahedra alternate. The lengths of the Si-0 and Al-C bonds are similar, so that replacement causes little lattice distortion. However, the framework has an overall

negative charge which requires the presence of cations to preserve electroneutrality; these cations do not form rart of the framework and are exchangeable. They reside at the walls of the zeolite channels, preferably at positions of high negative charge density. The channels are of molecular dimensions and may be interconnected in one, two or three directions. As well as cations, the channels may contain water molecules or occluded molecules such as paraffins. The channel dimensions may vary periodically along their lengths and between one exchanger and another. Barrer(12)

introduced a graded series cf aluminosilicates on the basis of their degree of openness:

Faujasite > Y aA > > Vordenite Chabazite Gmelinite

> Levynite > Phillipsite Analcite, Sodalite Harmotome Cancrinite

- 10 -

Early attempts at classification of zeolites were made in terms of morphological properties. Bragg(13) considered three framework types: fibrous structures, with weakly

cross-linked aluminosilicate chains; lamellar structures, with aluminosilicate sheets weakly bonded to each other; and three-dimensional framework structures with uniform

bonding. However, several zeolites could not be classified into any of the above grouns.

Yeier(14)proposed a classification consisting of seven groups named after typical members, as shown in Table 1.1 Each group has a different arrangement of linked A.104 and SiOL tetrahedra. Yeier also suggested that zeolite structures could be characterised by eight building units, called "

secondary building units" (SBU) which are made up of primary building units of linked A104 and SiO4 tetrahedra.

Table 1.1: Classification of Zeolites(14)

Group Lame Typical Tembers Analcite analcite, wairakite

ratrolite natrolite, thomsonite, edingtonite

Chabazite chabazite, sodalite, cancrinite,levynite, gmcliAite Phillipsite phillipsite, harnotome, :ismondlite

Heulandite heulandite, clinontilolite,brewsterite r:ordenite nordenite, ferrierite, epistilbite Faujasite faujasite, X, Y, Linde Sieve A

Reviews of zeolite structure and classification are given by Neier(14) and Barrer(15) . Progress in accurately determining structures has been slow due to the difficulty in obtaining specimens in the form of single crystals

suitable for X-ray work. As a result, less informative X-rav powder patterns have been used. However, a number of

structural studies have now been carried out, such as those by Smith(16) on chabazite.

1.2.1 Structure of chabazite

Chabazite was first identified in 1772, and occurs mainly in Ireland and Nova Scotia. It has a density of 2.G5-2•10g cm-3 and a refractive index of about 1.47. The typical unit cell contents of the naturally occurring zeolite are

Cat (A102)4 (SiO2).8 13H20. Small amounts of sodium and potassium may also be present.

The structur3 of chabazite was first investigated by

Wyart(17), who proposed a system of four-and six-membered rings, giving a maximum cavity aperture of approximately.30nm in

diameter. It was later found by Kington and Laing,(1P) that argon (diameter .424nm) and methane (diameter '.394n m) were

easily adsorbed on to chabazite. This led to a redetermination of the structure by Dent and Smith( -1-9) and Nowacki et

al(

' 0)

.

They, independently, proposed a structure in which the most important sub-unit is formed

by

linking together two six-

membered rin

g

s to give a deformed hexagonal prism. The sides of the prism are formed of four-membered rinrrs. Four-membered rings are used to join the prisms to form a framework

containing, large ellipsoidal cavities: there is one cavity (or cage) per unit cell.. Each cage has six octarronal, two

- 12 -

hexagonal and twelve quadrilateral faces, and is connected to the six neighbouring cages by eight-membered oxygen rings. The unit cell is shown in Fig. 1.1. in skeletal form, that is,

each vertex represents a tetrahedral unit.

The unit cell is approximately cubic, but is in fact rhombohedral (space group R3m), with cell dimensions given by Meier as cx = 94° 29' and a = -924nm. The stacking sequence in chabazite is represented by AABFJCC, a sequence of double six-membered rinc,s repeating every fourth layer. Barrer and Kerr(21) give the major and minor dimensions of the care as approximately .1-10nm and-66nm, and the minimum dimension of the octagonal "windows" as-37nm (given by Dent end Smith(19) as-39nm). The hexagonal prisms have a roughly spheroidal free volume of diameter ,v-4Cnm, and the diameter of the six-membered rings is approximately-27nm.

Smith and co-workers( -P2 '"2324) have attempted to determine cation positions in both the hydrated and dehydrated forms and in the zeolite containing adsorbed chlorine using X-ray

diffraction and electron density measurements. It was

found that in the dehydrated form, the calcium ions occurred in three sites designated Cal, Ca2 and Ca3 respectively. Ca₁ was at the origin of the unit cell, Ca

2 on the body

diagonal of the unit cell near the centre of the six-membered rings, and Ca3, although not definitely located, was postulated as occurring on the inside walls of the cavities. Each unit cell contains one Ca

1 site, two Ca2 and twelve Ca₃ sites. Fractional atoms were assigned to the sites on the basis of a statistical distribution. Thus, peaks equivalent in

electron density to C.6 calcium atoms at Ca

- 13 -

Fii.1.1

Skeletal Diagram of the Cavities in Chabazite (Oxygen atoms not shown)

- 14 -

atoms at each Ca₂ site have been located, and 1/16 calcium atom assigned to each Ca

3 site, giving a total of two calcium atoms per unit cell. The preferred site is Cal since the cation is surrounded by a near octahedron of oxygen atoms at .238no. Each Ca2 atom has three oxygen atoms at .237nm and three at .2P4nm. The former are also bonded to the Ca

1 atom leading to a strong local charge imbalance. This results in an electrically active surface around each cavity which is responsible for the adsorption properties of chabazite.

Locations of cations and water molecules in hydrated chabazite were difficult to determine. Hydration is

accompanied by some distortion of the framework as calcium ions move to new positions. The six-membered rings are more planar than in the dehydrated form, and the eight-membered rings become more circular. The five nearest neighbours of each calcium ion are incompletely co-ordinated water molecules, and the unit cell contafuls two ions separated by

.33nm.

Four types of water molecule were postulated: (H20)1 - six molecules per unit cell, associated with the framework;

(H

20)2 - three molecules per cage, one at the centre of each octagonal ring, shared between adjr1cent cages;

(H

2

0)

3

- two molecules, on the triad axis near to the centre

of the six-membered rings (occupied by Ca2 ions in the dehydrated form);

(H

20)4 -two molecules, on the triad axis but near the centre of the cavity.

Due to the inherent rigidity of the framework the

- 15 -

. •

dehydration, the water molecules probably occupy the sites most closely bound to the framework so that the (H20)4 and (H

2C)2 molecules are removed first. The dielectric relaxation time for the movement of calcium ions (about 10-2s in hydrated chabazite at ordinary temneratures) has been found to decrease with dehydration and then increases considerably, indicating that the calcium ions are tightly held and relatively immobile in fully hydrated crystals. Nmr studies of hydrated zeolites (25) have been interpreted in terms of the contents of the cavity

behaving similarly to a concentrated electrolyte.

Attempts to locate the cations in forms other than

calcium chabazite have proved unsuccessful. however, Smith(26) in interpretinr; the sorption results of Barrer and. Baynhan(27) in various exchange forms of chabazite has pcstulated that (1) many, if not all, of the potassium ions in dehydrated

samnies may be placed in the eight-membered rings, and (2) the first two sodium ions can be placed in the six-membered rings or in the hexagonal prisms, the remaining ions in or near the octagonal rings. Highly polar molecules such as water will link with the cations forming their spheres of hydration.

Differential heats of ion-exchange, measured by '3arrer, (29)

Davies and. Rees _{indicate that two-thirds of the sodium} ions occupy one type of site and the remaining one-third occupy a second tyre. The size of the ca'c.ion determines to some extent the cation nosition. Site Ca

1 is approached through a six-membered ring of radius.13nm. _{Thus, large} ions such as caesium (radius

.1(:9nr),

rubidium (.14Pnm) and thallium (.1hLnIr) would probably not take up the Ca, nosition, and potassium (.133nn) and barium (.132nm) would do so only with difficulty.

- 16 -

In general, it may be said that the difficulties in

locating cations and water molecules is due to their mobility within the cavities, particularly for monovalent cations in open structures, and their association with each other.

1.3 Prorerties of Zoolites

The properties and uses of zeolites have been widely explored in recent years, with the result that zeolites are now used in many types of industrial processes. The molecular sieve properties are utilised in the separation and recovery of normal paraffin hydrocarbons, and sorption properties are employed in absorbing water vapour, carbon dioxide and other gases. The catalytic effect of zeolites is used in many

hydrocarbon reactions. These properties have been extensively discussed elsewhere(29)

The cation exchange behayicur of zeolites was first observed over a century ago. The ease with which zeolites and other minerals exchange cations led to an interest in

ion-exchange minerals as water softeners. Synthetic non- crystalline aluminosilicate minerals were mainly used, although more recently these have been replaced by organic exchange resins. Crystalline zeolites have not been used commercially as water softeners. One application for commonly occurring zeolite minerals is

in the

selective

removal of radioactive ions from radioactive waste materials(30)

1.3.1

Cation Exchange and Diffusion

The ion-exchange behaviour of zeolites depends upon a number of factors: the nature of the cation species, its concentration in solution, the anion species associated with the cation in solution, the nature of the solvent, temperature

- 17 -

and the structural characteristics of the particular

zeolite. Cation selectivities in zeolites do not follow the typical rules that are evidenced by other inorganic and organic exchangers. Cation exchange in zeolites may be accompanied by changes in stability, adsorption properties and catalytic activity.

A number of zeolites have been studied in detail including the natural zeolites chabazite, analcite and phillinsite, and the synthetic zeolites A, X and Y.

Zeolites X and Y are widely used as catalysts and hence have been studied by a number of workers, including Barrer and

(31,32,33)

associates 1easuremen.ts of exchange isotherms enabled the deduction of a selectivity series for each

zeolite: a comparison of these demonstrates the effect of the difference in cation density between X and Y. For exchange by monovalent ions and at low loadings of the incoming cation, the selectivity series for both

Na-X and Na - Y is Cs+ > Rbt > K+ > Ya+ > Li+. At 5C% exchange the selectivity series for zeolite X is

Na+ > h > Rb+ > Cs+ > Li+, and for zeolite Y is

Cs+ > Rb+ > K+ ) + > Li+. Zeolite X is more aluminous and therefore has a larger number of cations per unit cell. Hence, due to space restrictions, exchange by large cations is more difficult in zeolite X tha.n in zeolite Y. Exchange by alkaline earths has also been investigated by Barrer

co-workers, and the catalytically important rare-earth

exchanged forms of zeolites X and Y have also been investigated by Sherry(34)

- 18 -

• )

Barrer, Davies and Rees(2 •P studied the thermochemistry and thermodynamics of ion-exchange in chabazite. The

selectivity series was found to be

Tl+ > K+ > Ag+ > :Pit > NH + > Pb9+ ) Na+.3a2+ > Sr2+ > Ca2+> Li+. An interesting phenomenon yhich has emerged from

equilibrium studies on zeolites is the "ion sieve effect" which results in incomplete exchange in certain cases. For example, analcite can exchange all of its Nat ions for Lit,e and Ag+ but not for Cs+ (35). Various factors may produce this effect. The exchanging cation may be too large to enter the smaller channels and cavities within the zeolite

structure, or in some zeolites some of the exchangeable cations may be locked in during synthesis and cannot be replaced

e.g. potassium ions in zeolite L. A zeolite may have more than one type of exchange site.: Linde Sieve

4A

is a typical example. All of its thirteen Nat ions are replaceable by Ag , but only twelve are accessible for larger ions such as T1+(36)

In chabazite, the interconnected cavities in the crystals are rather small. Electroneutrality requires the presence of, on average,

3.3 to

4, univalent cations per cavity. The cavities may have sufficient room for, say, three lame and one small cation but not four large ions. _{Thus, exchange} stops when every cavity contains three large ions even though the intracrystalline space is not used up(36) A similar effect is shown by sodalite which can exchange only 6C,T of its Nations for h+ ions due to insufficient snace.

- 19 -

Interest in the kinetics of tracer diffusion and ion-exchange in zeolites has increased during recent years. Among the earlier studies on tracer diffusion was the investigation by 9arrer and Rees(37,38) of alkali-metal

diffusion in analcite. Yore recent work has been concerned mainly with the synthetic zeolites X, Y and A, and the natural zeolite chabazite. Levi and co-workers have developed two techniques for studying the fast tracer diffusion in these

40)

zeolites t390 4G), and Dyer and r.,,ettins have studied tracer-and exchange-diffusion kinetics in zeolites X, Y, A and ZIC-4 in non-aqueous and mixed solvents(41,42)

Rees and his associates have made a number of studies of self-and exchange-diffusion in chabazite. Brooke and Rees(43,44) compared the experimentally determined kinetics of calcium-strontium exchange with those predicted by the Helfferich and Plesset theory (see Chapter3), and extended the theory to include all forms of non-ideality of the exchanger. Brooke and Rees further suggested that useful information could be obtained from experimental studies of rates of forward and backward exchanges using starting materials which contain various ratios of the two ions at zero time. Investigations of this type have been carried out by Duffy and Rees(45) and by Kinshofer and co-workers(46)

1.3.2. Zeolitic Water

All zeolites have a certain amount cf water associated with their structures, although the positioning of the

molecules is not fully understood. Water molecules in large zeolite cavities behave similarly to the isolated

- 20 -

liquid, suggesting that molecules in the centre of the large cavities do not occupy definite lattice sites. In zeolites with smaller cavities, the molecules cluster around the cations. In chabazite each calcium ion has five associated water

molecules, and in analcite each sodium ion is in contact with two water molecules. During dehydration it appears that the water molecules line the inside of the zeolite cages(25)

It has been proposed (47) that the non-framework water

and cations behave as a concentrated electrolyte. For example, the intracrystallire phase in zeolite

X

corresponds to an

18 molal solution of NaCH, with a density of 1.45 r cm-3. Nmr measurements show that this water acts similarly to a viscous liquid. The water molecules retain their mobility down to a temperature of about 2OCK; below this, there is a steady decrease in nobility.

Diffusion and self-diffusion of water in zeolites has been studied by a number of workers using several different techniques. Tiselius(49) studied water diffusion in

analcite and heulandite by an optical method. Analcite has also been investigated by tracer techniques using

deuterium oxide(49), tritiated water and 190(5C) Barrer and Fender(51) studied diffusion of deuterium oxide in heulandite, gmelinite and chabazite.

Barrer and Lan ley(52) investigated the effect of cations on the water retentivity of chabazite using differential

thermal analysis and therrogravimetric analysis. For monovalent cations, the order of retentivity was found to be Li > La > > qb > Cs, suggesting an association between the cations and water molecules. It was also found that chabazites containing divalent ions show greater retentivity

- 21 -

..Chanter 2: Diffusion

2.1 Frarres of rieference and Diffusion Coefficients 2.1.1 Fickian Diffusion

2.1.2 Intrinsic Diffusion Coefficients 2.1.3 Interdiffusion Coefficient

2.1.4 Self-diffusion Coefficient 2.1.5 Summary •

2.2 Theoretical E:coressions for the Self-diffusion Coefficient D*

2.2.1 Brownian Yotion

2.2.2 Transition State Theory and Temperature Dependence of D*

2.3 Solutions of the Diffusion Enuation with Constant D 2.3.1 Diffusion from a Solution of "Limited Volume" 2.3.2 Diffusion from a Solution of "Infinite Volume" 2.3.3 A Semi-infinite System

2.4 Solutions of the Diffusion 7nuation with D a 7unctioll of Concentration

Fujita's Treatment 2.4.2 Boltzmann's Treatment

- 22 -

Chanter 2: Diffusion

2.1

Frames of Reference and Diffusion Coefficients

2.1.1 Fickian Diffusion

Diffusion is a process which leads to an equalisation of concentrations within a single phase by means of random molecular movements. In simple ionic crystals this may occur by the

"defect mechanism" through mobile interstitial atoms or by "holes" resulting from Frenkel or Schottky defects. In zeolites,cations or molecules diffuse down the channels by the process known as 'zeolitic" or "interstitial" diffusion.

Equations of diffusion connect the rate of flow of the diffusing substance with the concentration gradient responsible for the flow. These equations were put forward by Fick(53) and are analogous to the equations of heat conduction derived by Fourier(54) some years earlier.

The diffusion flow J of a substance in a mixture with

other substances is defined as the Lmount passing perpendicularly through a reference surface of unit area in unit time. If

x is the space co-ordinate normal to the reference surface, and c is the concentration of diffusing species, Fick's first law of diffusion in one dimension may be stated as

J = - D ac. 2.1.1 (1)

x

D is the diffusivity cr diffusion coefficient. If

J and c are expressed in the same units, D has dimensions of length- time-1.

- 23 -

In general, for diffusion in zeolites it is not possible to investigate diffusion under steady-state conditions, and it is necessary to measure the change of concentration with time, caused by diffusion. Fick's second law, in one dimension, is

ac ID3c1

at

ax t ay:

2.1.1 (2)

If it is assumed that D is independent of concentration and is constant, this can be rewritten as

ac =

D ( Lc 2.1.1 (3)

at

ax2-

The above equation can be extended to a three-dimensional system by including expressions for cc-ordinates y and z, giving

D plc

+ ac

2.1.1 (4)

at

. a,. 5:y:a

azy

In the following treatment, diffusion in one dimension only will be considered although the results may be directly applied to three-dimensional systems.

Before Fick's equations may be applied. , the reference surface through which diffusion is to be measured must be carefully defined. These frames of reference, and the resultinc relationships for diffusion coefficients, have

been described by several authors including Crank(55), Hartley and Crank(56) and Barrer(57) In the general case of a

two-component system, three principal diffusion coefficients may be considered:

(a) intrinsic diffusion coefficient; (b) interdiffusion coefficient;

-

2

4 -

2.1.2 Intrinsic Diffusion Coefficients

Consider the diffusion of two components A and 3 along the x-coordinate, with no mass flow. The origin of the x-coordinate must be in a plane normal to it which moves so that there is no mass flow across it. From nick's

equations, per unit area, two intrinsic diffusion coefficients can be defined:

JA = - DA

_ex

a°

A

2.1.2(1)

j

B - °3x ---Q B 2.1.2(2) 2.1.3 Interdiffusion Coefficient

Consider a closed binary system with two components A and B, and such that no volume change occurs on mixing the components at constant pressure. If V

A and VB are the specific volumes of A and B, and cA and cBare the

concentrations, then for unit volume of mixture V A cA + VB cB - - 1 Differentating,

2.1

.3(

1)

VA

_acx,

+ VB a 2- C 2.1.3(2)

In general there will be a flow due to pure diffusion across a fixed section normal to x, as JA * J In order to maintain a uniform pressure in the system, a compensatin7 mass flow must occur. The volume transfer of A per unit time across unit area of the section must be equal and opposite to that of

B.

The flows are

- 25 - J/ - Dv V ac A - A A --A ax Jt - - Dv V Do B B B B ax Hence, since JI = - Dv V ac Dv v

3

- 0 AA--A

-

ax

2.1.3(3)

2.1.3(4)

2.1.3(5)

If, as normally is the case, VA and V3 are non-zero, in order to simultaneously satisfy - equations 2.1.3(2,5)

Dv - DV

A - B 2.1.3(6)

Thus for a binary system with no change in total volume, a single interdiffusion coefficient may be defined:

DAB = DA = DD 2.1.3(7)

It can be shown that D A V = V AcA DB + VBcBDA and when V A = VB' D =XD +X AB .XB AB

_DA

AD 3

2.1.3(8)

2

.1.

3(

9

) where X

A and XB are the mole fractions of A and B. 2.1.4 Self-diffusion Coefficients

The self-diffusion coefficients D

A and D3 may be determined experimentally using radioactively labelled molecules which

have the same properties as unlabelled molecules. An exchange of labelled and unlabelled species occurs, he difference in mass of the isotopes being negligible for all but hydrogen so that no mass flow takes place.

Consider a mixture of uniform conposition but with a

gradient of labelled particles, containing nA labelled A molecules, n

-

26 -

Then,

• = C

a x`"

and a(n-A nA) c ax However, 6n11.* t

0

atx 2.1.4(1) 2.1.4(2) 2.1.4(3)

so that there is a pure diffusion flux of labelled particles per unit cross-section normal to x Friven by

* * J

A = D ac A --A _2.1.4(4)

where D

A is the self-diffusion coefficient of A.

Darken(58) has derived a relationship between the

self-and intrinsic diffusion coefficients, assuming that the gradient of chemical potential is the driving force for a component in a diffusion process. The relationship is

D

A = DA l a _{a ln c'}

n

a 2.1.4(5) A

where a

A is the activity of component A.

From equations 2.1.3 (9) and 2.1.4 (5) it can be seen that D .(X D + X D)aln a _{AB BA} AB aln c A 2.1.5 Summary

2.1.4(6)

For a constant volume binary system, five related diffusion coefficients may be defined: D

AB, DA' DB, DA and DB. Diffusion in zeolites can be considered as diffusion across a volume-fixed section. In r_reneral, component B of the binary system is the anionic framework of the zeolite and component A is either a sorbed molecule or cation.

Hence, DB = 0 and mixtures of A and•B have constant volume

- 27 -

2.2 Theoretical Expressions for the Self-Diffusion Coefficient D

2.2.1 Brownian Yotion

Diffusion is caused by random "Brownian" motions of Particles due to thermal agitation. _{These movements result} in an endless path as the particle changes position, this path being called a "Random Walk". Individual motions of a particle cannot be followed: the only properties which may be determined are the average net displacement 77), and the average of the square of the net displacement RNE), of an atom in time t along a given direction.

The following relationship between and D* has been derived by Einstein(59):

D = 0.5 "Ti _{2.2.1 (1)}

Kramers(60) _{Chandrasekhar(61) and LeClaire} (62) _have made studies of self-diffusion as a "random walk" process. It has been shown by Le Claire that the average ret flow of labelled particles per unit area cross a reference plane at x0 may be expressed as

J* =

I-

lac 2.2.1 (2) 2t ax

where C is the concentration of labelled particles and t is the time.

This corresponds to Fick's first Law with D as given by Einstein, but only applies if ac3 and higher derivatives

ax-1 are negligible.

- 28 -

2.2.2 Transition State Theory and Temperature Dependence of D Self-diffusion may be regarded as a transition state process(63) in which particles of the diffusing species "jump" between neighbOuring equilibrium positions over a potential barrier between the two. If d is the distance between neighbouring equilibrium positions,

D = otda r 2.2.2 (1)

r

is the number of jumps made in unit time, and ocis the reciprocal of the number of "nearest neighbour" positions surrounding each particle. In general,

r

is given by

r

=

p. . F* 2.2.2 (2)

h F

where p is the number of equivalent diffusion paths between adjacent positions, k and h are Boltzmann's and Planck's constants respectively, T is absolute temperature, and F and F are the partition functions for the particle in the initial and activated states respectively. It may be seen that

p = 1 2.2.2 (3)

Ek

If it is assumed that the particle may be represented by a linear oscillator of frequency v along the direction of diffusion, then

LA G /T

2.2.2 (4)

CiG is the free energy change in'moving one mole of the diffusing species from an equilibrium position to the top of the potential barrier between equilibrium positions, and. R is the gas constant.

Combining equations 2.2.2 (I) and

(4)

gives D = d 2 V ern(— LS, G*/2T ) _{2.2.2 (5)}

= d2 V exp 21a* exp _2.2.2

₍₆₎

RT

- 29 -

where A S and

A

are the entropy and enthalpy of activation respectively.

The dependence of D on temperature is usually represented by the Arrhenius equation:

D = D_o exp (- E/RT) 2.2.2 (7)

where AE is the activation energy for the process and Do, the pre-exponential factor, is constant.

If Ali* = AE - RT , 2.2.2 (8) (6h)

it may be shown ' , using equation 2.2.2

(6)

that Do marbe defined as D o = 2'72

d

2

kT exp S 1 2.2.2 (9) R

An expression for zeolitic diffusion has also been derived by Earrer(65)

D d2•%) /21f-1 1 N1.-n 1 exp (- 12.2.2 (10

—

6

-

rff

(f-1): 1 N'

RT where f is the number of degrees of internal freedom and N.

N. - n no. of vacant interstices . total no: of interstices

The pre-exponential term of the Arrhenius equation is then given by D = d 2 '1) (Al (f-1): f-1 1

6

1 RT ( ) 2.2.2 (11)

Barrer has estimated D_o to be ca. 1•7 x 1C-3 cm2 s-1 for zeolitic diffusion.

2.3 Solutions of the Diffusion Fauation with Constant D Consider a zeolite sample consisting of a number of spheres of radius r. If the diffusion is radial, the diffusion equation for a constant diffusion coefficient takes the form

- 30 -

ac = D(C)2c + 2 ac 2.3. (1)

at

a r2-

r

FT:

On substitUting u cr, this becomes

au = D eu 2.3. (2)

at

c7 r

This is the equation for diffusion in one dimension,

. but the solutions to most problems relating to radial diffusion in spheres can be deduced directly from the solutions to the corresponding linear problems.

In experimental systems which use radioactive tracers to follow the diffusion, there are two distinct types of experiment: diffusion from a solution of "limited volume", and diffusion from a solution of "infinite volume". The solutions of the diffusion equation are quite different for the two cases.

2.3.1 Diffusion from a solution of "Limited Volume"

In "limited solution volume" experiments, the amount of the diffusing species placed in solution is approximately equal to the amount in the solid. If the solution is tagged initially, the radioactively-labelled ions or molecules

exchange for unlabelled species in the solid so that the

solution count-rate, which is continuously monitored, gradually falls as diffusion proceeds. At equilibrium the count-rate

is roughly half the initial value.

Suppose the solution has a volume V and an initial concentration of active particles co, and the sphere

of radius a initially contains no tracer. At time t, the total amount of labelled species in the solid Yt, expressed as a

fraction cf the corresponding quantity at infinite time _{Y ee ,} is given by Crank(55) as

- 31 -

00 = 6 0( (1+ ) e A-Dg t/a 2.3.1 (1) 9-1-9o(+0 Y n oe _{TA r4 1} where the a

n 's are the non-zero roots of the equation -

tan •qn = cln 3 +c4q 121-

2.3.1 (2)

and Otis given by 1 2.3.1 (3)

1

+ « Vc o The quantity Dlt

/Y is known as the fractional attainment of equilibrium, F.A.E.

Equation 2.3.1 (2) is not soluble analytically but solution by a "successive approximation" method( 66)

gives rise to a series of values of qn- On substitution of these values into the full equation, 2.3.1 (1), an array of values of F.A.E. against /-is obtained, where -r is given by

*-1" = Dt a2

2.3.1 (4)

Since D and a are constants, lr is directly dependent on time. An alternative to equation 2.3.1 (1), suggested by

Carman and Haul(67) is

F.A.E.,-(1-1-0()

[1-

X, eerfct?5,1t eerfcpla,

175;

+ _{a2 X}t.+ _Y2. of a2 + higher terms 2.3.1 (5) 4cx where et = 1/41+ 3 ) 1 +13 , = Zr, - .1, 2.3.1 (6) and eerfc(z) = exp (z2) erfc(z) = exp (z2)(1-erf(z))

2.3.1 (7) The authors claim that the higher terms in equation(5) cancel almost completely so that the equation is applicable for long times, but this is doubted by Crank. A computer

program for the evaluation of the full equation was developed by Duffy(66). _{A comparison of F.A.E. values from equations} 2.3.1

(1)

_{and (5) gave differences only in the third decimal} place of F.A.E., and at F.A.E., values greater than 0'90.

- 32 -

Barrer

(68)

has derived an expression based on Henry's Law for the sorption of gas in a powder of small spheres, which can also be applied to tracer diffusion systems. The equation is

F.A.E. =

6

00

117-

2.3.1 (8)

a Qo - -co 11-

where Q0 is the amount of labelled particles initially in solution and Q,po is the amount in the solid phase at

equilibrium. Q0 and Q n. are related through

\ 2.3.1 (9) Qco = Q0 (amount of tracer in solid.

total amount in solid + solution For spherical particles, 6/a = 2A/0 2.3.1 (10) where A is the surface area per gram and /o is the density. This treatment is generally applicable only for small times.

2.3.2 Diffusion from a Solution of "Infinite Volume" In this type of experiment, the solid is initially labelled with radioactive ions or molecules which are then allowed to diffuse out into a sol:It'ion containing a large excess_ of the unlabelled diffusing species. If a 500-fold excess is used, it is obvious from equation 2.3.1 (9) that Q 00 / Q0 is very small, so that the concentration of tracer in the solution is kept almost constant. All the experiments carried out during the course of this work were of this type.

The solution of the radial diffusion equation under these conditions gives

? 00 F.A.E. = 1 -

6 < 1

e -Dn- 11 2 t/a2 2.3.2 (1) T 2- C n2 ,,2 = 1 -

6

1

e -n " 2.3.2 (2) TTY 2 n=1 -

- 33 -

A computer program to solve this equation has been written by Duffy_{( 66 ) which gives values of the F.A.E. for}

given values of1". A working maximum of n = 500 was used since no further change in F.A.E. values occurred at larger values of n. The solution of this equation is shown in Fig. 2. 1 , together with the solution obtained from the Carman and Haul equation (2.3.1 (5)) with a value of ocof 1.0. At values of c< of about 500, the curves obtained under the two sets of conditions are identical.

An expression analogous to equation 2.3.1 (8) has also been derived by Barrer(68)

F.A.E. = Qs - Lt .= 6 4

/

Dt 2,3.2 ₍₃₎ QS Q00 a QS Q- a.a ✓ Tr

where Qs, (4, Q 00 are the amounts of tracer in the solid

initially, at time t, and at equilibrium respectively. Hence,

Q,00 = _{Qs (} Amount _{of tracer in solid ) 2.3.2 (4)}

Amount in solid + solution

Again, this equation generally applies only for small times.

2.3.3 A Semi-Infinite System

Beattie and Davies(69) have derived an equation for self-diffusion using labelldd particles in a system having semi-infinite boundary conditions. Their equation is

= 2A Dt 2.3.3 (1)

co v Tr

where A is the external surface area of the particles,

V is the volume of solution, co is the initial concentration of tracer in the solid and c is the mean concentration at

time t in the solid. However, the equation is only applicable for small tines.

- 34 -

Fig.2.1

Comparison of Solutions of the Diffusion Eouation with Constant D

(Curve 1 : Limited Solution Volume Conditions, 0( = 1.0 Curve 2 : Infinite Solution Volume Conditions)

•0 4

•08 •12

•16

•20

TOR

- 35 -

2.4 Solutions of the Diffusion Fquation with D a Function: of Concentration

The diffusion coefficients considered in Section 2.3 were all independent of time and concentration. However, much experimental interest is centred cn systems in which the diffusion coefficient varies with concentration, ion- exchange being an example of this type of behaviour. In this section, the solution of the diffusion equation with D a function of concentration will be discussed briefly. A fuller discussion is given by Crank(55)

2.4.1 Fujita's Treatment

The equation for one-dimensional diffusion when the

diffusion coefficient D(c) is a function of concentration c is ac =

a (D(c) ac

2.4.1 (1)

at

ax

ax

ac

where

at

is the amount of diffusion at time t, and x is the distance from the reference plane.

The above equation has been solved by Fujita(70,71,72) who gives exact solutions for D(c) given by

13' =

Do

, D - --1-

Do

1-

)4,c

(1- 1.c) 2

D_ Dn (T-T2ac+bc"-) 2.4.1(2) where D = Do at c = co. These solutions are for

a semi-infinite solid with initial and boundary conditions

o=15,x)o,t=o 2.4.1 (3)

C = c o , X =o,t

2.4.1 (4) Solutions are given for c as a function of x and t.

2.4.2 f3oltzmann's Treatment

Boltzmann(73) showed that the concentration may be expressed as a function of a single variable xt so that

- 36 -

equation 2.4.1 (1) becomes an ordinary differential equation. The boundary conditions for a semi-infinite medium are

c = c

o, x = o, t > d 2.4.2 (1) c = o, x > o, t = o 2.4.2 (2)

_1

by the transformation y = xt ", equation 2.4.1 (1) reduces to x de = d /D do) 2.4.2 (3)

2 dy dy

l

dy J

On integration with respect to y this becomes

-2

_{f c}

1

ydc = (D do ) e:"-'el 2.4.2

(4)

o dy / c=o

On rearrangement and reintroduction of x and t this becomes cl

D = 1 dc Jr c x d 2.4.2 (5) el 2t dx o

where c

1 is any value of c between o and co.

Hence, D ray be obtained as a function of c at a given time t from the curve of concentration versus distance.

2.4.3 Crank and Park's Treatment

By means of an analogy between diffusion and the sorption and descrption rates of vapous in polymers, Crank and Park 74) have shown that an experimentally determined inte7ral

diffusion coefficient may be approximated by cl

D=

D dc 2.4.3 (1) el - c2 Jc 2 where c

1 and c2 are the concentration limits in the solid. This approximation was shown to be valid by comparison with the numerical solution of the diffusion equation for

several types of variable diffusion coefficient( (5). If D

s and Dd are the mean diffusion coefficients for sorption and desorption respectively in a vapour-polymer

- 37 -

system, then

D = 3-7 (Ds + Dd) 2.4.3 (2) D

s and Dd are measured from the initial slopes of a

"root-tine" kinetic plot. The analogous expression for an ion-exchange system is

D = (Df + Db) 2.4.3 (3) where D

f and Db are the mean coefficients for the forward and backward reactions of the reversible exchange system

AX + B ;,===t BX + A 2.4.2 (4)

- 38 -

Chanter 9: Ion-Exchange

3.1 Introduction

3.2 Early Ion-exchange Theories

3.3

Forms of the Interdiffusion Coefficient D AB 3.3.1 Helfferich and Plesset's Theory

3.3.2

The Irreversible Thermodynamic Treatment

3.4 The Numerical Solution of the Diffusion Equation 3.4.1 The Helfferich and Plesset Theory

3.4.2 The Irreversible Thermodynamic Treatment

3.5

The Activity Correction Term g

3.5.1 Kielland Activity Correction

3.5.2 Activity Coefficients Determined from Kielland Plots

3.5.3

Rees and Brooke's Treatment

3.5.4 D

- 39 -

Chapter 3: Ion-Exchancze

3.1 Introduction

Consider the general ion-exchange reaction An+Z Bri+(aq) > Bri+Z + An+(aq)

where A and B are cations of valence n and Z is the zeolite framework. The reaction can be regarded as two processes which occur sequentially(76), the slower step being rate- controlling. The two processes are, for the forward

exchange (1) diffusion of Bn+ ions through the solution to the surface and across the zeolite,-solution interface, and

diffusion of A n+ ions in the opposite direction; (2) diffusion of Bn+ ions through the zeolite to the exchange site, and

the simultaneous diffusion of An+ ions through the zeolite towards the solution.

If (1) is the rate-controlling process, the rate of exchange depends on the diffusion of ions An+ and Bn+ through the solution. Even in a well-stirred solution, Nernst(77) postulates the presence of a completely stagnant layer of liquid surrounding each particle of exchanger. If exchange

is controlled by diffusion through the Nernst layer, the problem is of the "film diffusion" type, whereas if the rate is controlled by diffusion through the zeolite the problem becomes one of "particle controlled diffusion".

It has been stated(78) that the nature of the rate-controlling step is a function or the film thickness cr the radius r of the particles (assumed to be spherical), the diffusion coefficients of the cation in the solid and

- 40 -

solution phase, Dz and Ds respectively, and the equilibrium ratio of Bn+ in solid to solution, k.

Then, for particle diffusion,

Itnz

l

.k. e

1

3.1(1)

3Ds r

and for film diffusion, TraDmEk >1 3.1(2) 3Ds r

Similar expressions have been derived by Helfferich(6) However, these equations are not the best method for determining which is the rate-controlling step.

3.2 ftarlyIonmfxchange Theories

Helfferich et al (79,80) have postulated ideal limiting laws for the kinetics of particle and film diffusion for

processes involving ions of different mobilities between

spherical ion-exchanger beads of uniform size in a well-stirred solution. However, the following discussion will be confined to particle diffusion in a well-stirred solution.

Until

_1957,

it was assumed tl:ct diffusion of one of the exchanging ions in one direction was dependent on the diffusion of the other ion in the opposite direction. Not only were the fluxes equal and opposite, but also the

concentration gradients:

c

A =

-LB

3.2 (1)

6x

Hence, for a given pair of ions, there would be a

single exchange diffusion coefficient D for both the forward and reverse exchanges. If

An+

were intrinsically more mobile than Bn- , a diffusion potential would be set up which would retard A and accelerate B so that the velocities of the ions were equal. Fick's first Law with constant D could then be

- 4

1 -

applied, and the solution of this with the appropriate

boundary conditions would, in the case of spherical exchanger particles and for short time t, give rise to

FAE = 2A rict 3.2

₍₂₎

Tr

Thus, the same exchange rate was predicted for both tie forward reaction (AZ ---)BZ), or the reverse reaction (BZ----AZ)

3.3

Forms of the Interdiffusion Coefficient D AB

3.3.1 Helfferich and Plesset's Theory(6,81,82)

The Heifferich and Plessot theory of ion-exchange was a major advance on previous theories since it included the gradient of electrical potential for the first tire. This is built up by the diffusion process itself, and has been referred to in the preceding section as the diffusion

potential. Together with the concentration gradient, the electrical potential gradient provides the major contribution to the-chemical potential gradient which is the driving force for diffusion of an ionic species. The model was set up with particular reference to ion-exchange resins, with the following assumptions.

(1) The resin was treated as a quasi-homogeneous phase, with constant cross-linking.

(2) Self-diffusion coefficients of the exchanging ions were assumed to be independent of the ionic corrosition of the

exchanger(79' 3'94),

and were taken to be those measured in the pure homoionic forms of the exchanger by tracer techniques

- 4

2 -

(3) The ideal Einstein relation between self-diffusion coefficients and electrochemical mobilities was assumed. (4) Any coupling of ionic fluxes other than by

electrical forces was neglected.

(5) The concentration of fixed ionic groups was assumed to be constant throughout the exchanger.

(6) Any concentration change and flux of the co-ions was disregarded, since they were small in comparison to those of the exchanging ions if the concentration of the solution

was not too high and that of the fixed ionic groups not too lbw. (7) The ionic activity coefficients were assumed to be

unity throughout, that is, an ideal exchanger was assumed. (8) Swelling of the resin during exchange was ignored.

With the exception of the inclusion of electrical potential gradients, these assumptions were

the same

as those made in previous theories.

The fluxes of the two ionic species A and B are given from the Nernst-Planck equations(85,P6,87) as

JA - - DA (grad cA + zAcA F grad Q) RT

J_{B - -DB -} (grad c_B + BeB F grad Q) RT

3.3.1(1)

3.

3

.1(2)

where D Is the self-diffusion coefficient, c is the molar concentration, z is the electrocheliical valence, F is the Faraday, R the gas constant, T the absolute temperature and Q the electrochemical potential.

Electroneutrality requires that

ZA + Bz O.„ = constant,

_3.3.1(3)

and in the absence of an electric current, the fluxes of A and B must be equal and opposite.

D* c + D* B c AA B

3.3.1(6)

3.3.1(7)

°A CB = 1, and zA = zB , * * then D AB = D D A B

- 43 -

Hence, zAJ A + zBJB = 0

3.3.1(4)

Combining equations 3.3.1 (1,2,3 and

4)

gives * * f 2

JA = - DA DB (zA cA + z2B"cB)

[

The quantity'in square brackets is termed the interdiffusion coefficient D

AB' and depends not only on the self-diffusion coefficients of the exchanging ions, but also on their

relative concentrations.

If the concentrations are expressed as equivalent cation fractions i.e.

* 2 grad c * a _A 3.3.1(5) D

A zAcA + DB zBoB

This expression is identical to that derived by Beattie and Davies(69).

3.3.2 _The Irreversible Thermodynamic Treatment

In an aqueous solution/exchanger system there will be coupling effects between the exchanging ions A and B, and also one due to the flux of water. These effects are

ignored in both the Helfferich - Plesset and Beattie-Davies theories, bUt are considered in the following treatment.

The Nernst-Planck equations, written in terms of irreversible thermodynamics (88,9) are

[

J

1 = - °1 (L _11T,i ()du _{12 TR} + L adtz,..+ L _{1 3} 6 duj _{a x} )

f

(L

11 L12 -) z Fav

- 44 -

=

-

c2

(L22

axt.

4.

1_21

ilki +

1... 2a

81,us)

ax

ax

ax

z

+(L22 -1- L21

1) z2

F av 3.3.2(2)

z2 ax

where J1 and J2 are the fluxes of ions A and B of concentrations el and c2 respectively, the L's are the phenomenological

coefficients, v is the electrical potential at the plane x = x normal to the direction of flow, andp., , /42 3 are those parts of the chemical potential of A, B and water which (at constant temperature and pressure) are functions of

concentration only.

Writing Fick's first Law in one dimension. Jl _{= - DAB}

d

el

3.3.2(3)

J2 = - DAB

_--

a

cp

ax

3.

3.

2(

4)

where DAB is the differential interdiffusion coefficient. Hence, from equations 3.3.2 (1, 2,

3

and

4) ,

DAB = el L11814 ± L12 21' [

_{a c_}

i a el

+

.T.,1 a,u.. + ( z:ILli + zl.,12 )F_:7 - ) a e1 6 c1

3.3.20

)

auz.

L

c2 L 22a c_{2 aca}

a#, +

L 23 s

_a

L + z L )F 2 22 1 21

---

av

c2 ac,,

3.3.2(6) The electroneutrality condition,

zlci + z2c2 = constant may be rewritten as

z aci + z2 ilc, = o _:5x Z5x-

combining equations 3.3.2(6) and ( ) gives DAB - c z L22 _{+ L + L. a,u ;}

a ci

_{u cl} 2 2 22 --- _{23 a r:}

+ (z2L22+z1L21)

F a v a e

3.3.2(7)

3.3.2(R)

3.3.2(9)

— 45 —

Equations 3.3.2(5) and (9)

then lead to

F

av = - (

zcli +zcL)

₁₁

_{11 2 2 21}

3,C4 _(Z1C1

L12 + z2c2L22 ) aAl,

a el

-

a

ao c

c l

[

zl yz,Lii

+ z21.12 ) + z2c2 (z2L22 + z 1,21 )]

Substitution

D

AB = of equation

3.3.2 (10)

z1c11111P z2c2L21 )

into 3.3.2

(z1ciL12P -

3.3.2(10)

(9) p;ives

z2c2L22 ) ilL&I

a

c1

a

_c1

+ ( z1c1L13P - z2c2L23)

_c

I•F1(1 + P)

II -1

3.3.2(11)

where P = z2c2(z2L

22 -

+zL1

1 21'

z1c1( z1L11 + z2L12 )

3.3.2(12)

Consider a system in

which

the water flux is negligible so that aixs may he neglected, and in which all cross -coefficients

a el

are zero. Hence,

P = z

2c2L

22

a

z1c1L11

Applying the identities

3.3.2(13)

llA

= HTL11

3.3.2(14)

and

14:.=:.RTL22

3.3.2(15)

*

P =

_DB

_c2z

a

₂ * ₂

_3.3.2(16)

DA c1z1

Equation

3.3.2(11)

with

3.3.2(5)

then becomes DA * ,72,

a ]

na,

2 27

1 nal ‘ 3.3.2(17)

D

AB = -A DB(`-'1'1

a

inc." + z2°2

a

lnc-1

2

1

* 2 *

a

D Az1c1 + DBz2

c2

through the

identity

= RT d ln a

3.3.2(18)

where a is

the

activity.

+(z1c1L13 + z2c2L23) P2u

I

- 46 -

3.4 The Numerical Solution of the Diffusion Equation The diffusion equation applied to systems having spherical symmetry may be written as (7,55)

ac

1(

a

z3-TA

_raar

r DAB LA) or

3.4(

1) where c

A is the concentration of species A initially present in the exchanger, r is the distance from the centre of the sphere of radius r

o DAB is the interdiffusion coefficient, and t is the time.

Initial and boundary conditions are taken to be those of infinite solution volume conditions with A uniformly distributed within the sphere, that is

r

o- C 3.4(2)

o r < r0, t = o, cA(r) = constant 3.4(3) r = r0, t > o, cA(t) = 0 3.4(4) The above equation may be solved by application of the "finite-difference" method(55'90), which employs expansion

of functions using Taylor's theorem. The values of the function f at points a + 1, a, and a - 1 may be expressed as

f(a + 1) = f(a) + l.f'(a) + 12 f " (a) + .... 3.4(5) 2:

f(a - 1) = f(a) 1.f'(a) + l f " (a) - 3.4.(6) 2:

where 1 is a small increment. Approximate formulae for successive derivatives may be deduced by elimination between these two equations.

- 47 -

- 3.4.1 The Helfferich and Plesset Theory

According to the Helfferich and Plesset theory the interdiffusion coefficient D AB is given by * * DAB = DA DB (zAcA + z;"cB) 3.4.1(1) * * a D AzAcA + DB zBeB

The comentrat ions may be written as fractions of exchangeable cation concentration, so that

CA + CB = 1 3.4.1(2) Using the transformations

a = D z A A - 1, b = zA - 1 D BzB zB

3.4.1(3,4)

and Zr = zAcA

3.4.1(5)

zAcA zBcB DA_B may be rewritten as D AB = DA (3- + +

'The dimensionless variables 1" = DAB

VT:

and p = r/re

3.4.1(6)

3.4.1(7)

3.4.1

(

8)

are introduced, enabling the diffusion equation (3.4(1)) to be transformed to read

a

2f - 1 _{[ (1 +} b ) + b-a )2+2 (1+b 2S ) =C

WY 1+ a Zr

a

pe- 1+Wfkap) _{P (DP}

3.4.1(9)

When zA = zB, b = 0 so that upon introduction of the variable u= in (1 + a .2S ) , 3.4.1(10)

equation (9) reduces to

au - ) =C 3.4.1(11)

- 48 -

This may be approximated by the finite-difference scheme

u( p ,"r ) =

u( p ,

"t ) i(p,T)• 3.4.1(12)

where

R

+( )[u(f+Aft-r ) R-(p) u(p,"T)-u(p-tIpt er) 3.4.1(13) and j (p, ) = exp ( -u (p, ) ) ZVT _3.4.1(14)

(g1P)1'

vp) = _/p

_3.4.1(15)

p) p

- 1

gy p)

/p3-z

3.4.1(16) Hence, u(p,1- + ) may be computed from a knowledge of

U(p+4f),17. ) U(p

T

) and u(p- p 'sr ) ,

using

equation 3.4.1 (12 ) . The initial and boundary

conditions

are now

u(p)1 = o) = In (1 + a) _3.4.1(17)

u(p=1,'Y ) =

o

_3.4.1(18)

and at the centre of the sphere

( au)

= 0

ap Ip=o

3.4.1(19)

Hence, u(f)=0;Y) = u(p=

0+ 4p, -r)

3.4.1(20) Solution of 3.4.1(11) produces a series of values of u at

equally spaced points in the range 0.C io C.; 1, for -1r

increasing from zero. Helfferich and Plesset used an initial interval of4p= 1/640, which was successively doubled as the solution proceeded to reach 1/10 in the final stages. The spacing of

T

was selected in accordance with the stability condition(91)

0.4 (Ap)'- eu

3.4.1(21)

3.4.2 The Irreversible Thermodynamic Treatment

The .irreversible thermodynamic treatment gave * ',.

amp

-3 a

lra A A

D (z c alrc + za c. alncA )

3 A A - 7-

T3 3.4.2(1) * 2_ i; 2. D AzAcA + DB zBcB

- 49 -

It has been-shown(89) that for all c_A'

a

lna A

a

lnc A

a

lnaB _{= g 3.4.3(2)}

a

lnc_B

Hence, using cA + CB = 1, DAB reduces to

* *

D

AB = DADB EcA( zA z ) + g

* * 2 * CA(DA ZA D3Z3) DBzB

3.4.2 (3)

According to the Helfferich and Plesset theory, DAB is a smoothly varying function of cA. However, it has been found(43) that the activity correction factor g may change its value

abruptly at certain values of cA, resulting in a discontinuity in D

AB at these specific points. The effects of such an interdiffusion coefficient may be reduced by the introduction of a parameter S which is defined as

S = I'

CR

D

AB DABdc/ 3.4.24)

The "S transform"method was suggested by Crank(92) although the transform was originally put forward by Eyres et al(93).

When the S - transform is applied to the diffusion equation 3.4.(1), together with the introduction of the dimensionless time and position variables

= (

dc ) t

AB 3.4.2(5)

rI andp= r/r

o, 3.4.2(6)

the equation siniplifes to

= 1 .

a leas1

3.4.2(7) air /32

ap

J

The finite-difference scheme for the solution of this equation is cA(/), 17+ _{) = cA (p+ (Ri.S.1. -R_S.. ) 3.4.2(8)}

(ZAP?

-

50 -

where

(

R. =

(p+ 4

P)'" S = S(p+zip,1- ) s (p , "sr )

R

-

=

(P 4P) a

S- =

S(13

t °r- ) -

S(p- )

P 2-

The expanded form of equation 3.4.2(7) is

3

_.4.2(9)

b

_c

_{A = 2 aS +}

_a's

_3.4.2(10)

(31' P

76,0 apa

The finite-difference scheme for the solution of this equation is

cA (p + CA ( p

) +

s (P4 _{AP,T )}

op

_p

+s( p+ np,'T )-2s(p,^r )+s(p-p,1- )

3.4.2(11)

The initial and boundary conditions are given by

cA(p,^r = 0)= const, S(p,1- = o) = const.

3.4.2(12)

cA( p= i• -r) = o, s(p=1,-1- ) =

3.4.2

(

13)

At the centre of the sphere

( a c

A =

= 0

3.4.2(14)

aP IP=

0

P= 0

or cA(p.=

0,1r) = cA (p= o

T)

3.4.2(15)

S(p

= o,"r ) = S(p = 0+ ap,'Y )

3.4.2

(16)

This solution yields a series of values of cA at equally spaced points in the range o p1, asTadvances from zero. The stability condition may be expressed as

0•4 (&p)1- (

_

A

3.4.2(17)

D

AB )min

Equation 3.4.2(7) shows that the solution of the diffusion equation depends only on the form of S as a function of cA, and the initial and boundary conditions. Thus,

the

S-transform method may be applied to a wide variety of

expressions for DAB, which will be considered in section

3.5.

A computer program has been written(94) to solve the diffusion

- 51 -

equation using both the finite-difference schemes give above.

(This will be discussed more fully in Chapter 5). The program

also calculates the fractional attainment of eauilibrium at

each stage of the solution as follows.

The quantity of species A remaining in the solid phase

after a time

T

can be shown to be given by

qA _{) = 3} fl

J cA(P' )

₀

12. d P

3.4.2(18)

The finite-difference solution gives a series of n discrete

values of c A

in the range o

p

1 at a fixed time -r , where n = 1/Ap + 1. These values may be converted to a

series of n discrete values of

c A ,0 2 = A(n) 3.4.2(19)

A numerical integration procedure can be carried out on this

series using a Newton-Cotes type of quadrature. Using the

fact that when n = 1,p= o, A(n) = o 3.4.2(20)

and when n =( 1 + 1) ,f) = 1,cA = o, A(n) = o, 3.4.2(21)

the follc,aing may be written:

(I/Ap)

qA('r ) = 3(0p) A(n) 3.4.2(22) rtza

(%'n,P) ('Q P' 1 )

qA ('Y) = L

p

t/-1, A(n) + 2 A(n)

Equation 3.4.2(22) represents the Trapezoidal Rule, and

equation 3.4.2(23), Simpson's Rule.

Wilen-r= o, cA(p,1") = constant 3.4.2(24)

Hence,

qA (1' = o) 3cA p2,0110 = cA = const. 3.4.2(25) Under "infinite solution volume" conditions, complete

equilibrium occurs when all of the species A, initially

present in the exchanger, has been replaced by B ions from

- 52 -

solution. Hence, the fractional attainment of equilibrium may be written as

F(T) = ciA(1r= o) - 0A(1r ) 3.4.2(26) q_A(1- = 0)

3.5

The Activity Correction Term rr

The activity correction term g is found from the solid phase activity coefficients f

A and fB of the ions in the solid. Unfortunately, these coefficients cannot be measured directly. However, in the case of zeolites, the rigid structures, absence, of swelling and relatively small changes in water content

between d.ifferent cationic forms of the zeolite often lead to simplifications in the law of mass action. In the most extreme case, the ratio fA/fB is constant, but in general it depends on the exchanger composition.

3.5.1 Yielland Activity Correction

Kielland(95) suggested the following expressions for the activity coefficients of the exchanging ions within the exchanger phase: in f A = kcB 3.5.1(1) in f B = kcA

_3.5.1(2)

k is known as the Kielland coefficient. Strictly, these equations apply only to uni-univalent exchanges.

All values of the Kieliand coefficient reported have been negative and in the range -. 1 to o.

The activity correction term g can be written in the form g = a lnal

a

lnc -A .

= aln (f A cA )

a incA

= a

_{lnfA + 1} lnc A

3.

5.

1(3)

3.

5.

1(4)

3.5.1(5)