Isotachophoresis : some fundamental aspects

Isotachophoresis : some fundamental aspects

Citation for published version (APA):

Beckers, J. L. (1973). Isotachophoresis : some fundamental aspects. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR80190

DOI:

10.6100/IR80190

Document status and date: Published: 01/01/1973

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SOME FUNDAM

E

N

TA

L ASPECTS

SOME FUNDAMENTAL ASPECTS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH-NISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. IR. G. VOSSERS, VOOR EEN COMMISSIE AANGEWBZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VER-DEDIGEN OP DINSDAG 19 JUNI 1973 TE 16.00 UUR

DOOR

JOZEFLEONARDUSBECKERS

geboren te Maastricht

1973

Prof.Dr.Ir. A.I.M. Keulemans, promotor Dr.Ir. F.M. Everaerts, co-referent.

CONTENTS.

INTRODUCTION.

THEORETICAL PART

1 PRINCIPLES OF THE ELECTROPHORETIC METHODS.

1.1 The principle of isotachophoresis.

1.2 The principle of zone electrophoresis.

1.3 The principle of moving boundary electrophoresis.

1.4 The principle of isoelectricfocusing.

1.5 Discussion.

2 GENERAL EQUATIONS IN ELECTROPHORETIC PROCESSES.

2.1 2.2

Introduction.

The general equations. 2.2.1 The equilibrium equations.

2.2.2 The electroneutrality equations.

2.2.3 The mass-balances for all ionic species. 2.2.4 The modified OHM's law.

3 A MATHEMATICAL MODEL FOR ISOTACHOPHORESIS.

3.1 Introduction.

3.2 Basic equations.

3.2.1 The equilibrium equations. 3.2.2 The isotachophoretic condition. 3.2.3 The mass-balance of the buffer. 3.2.4 The electroneutrality equations.

3.2.5 The modified OHM~s law.

3.3 3.4 3.5 Procedure of computation. Procedure of iteration. Discussion.

4 MOVING BOUNDARY ELECTROPHORESIS.

4.1 Introduction. 9 11 12 14 15 16 16 18 18 21 22 23 26 28 29 29 30 31 32 32 34 35 43 44

4.2 A model of moving boundary electrophoresis. 4.2.1 The electroneutrality equations.

4.2.2 The modified OHM's law.

4.2.3 The mass-balances for all cationic species.

4.3 Procedure of computation.

4.4 Exper imen ta 1.

5 VALIDITY OF THE ISOTACHOPHORETIC MODEL.

5.1 Introduction.

5.2 The concept of mobility.

5.2.1 Relaxation and electrophoretic effects. 5.2.2 Partial dissociation.

5.2.3 Solvation.

5.2.4 The relationship between entropy and ionic mobility. 5.2.5 The relationship between volume and ionic mobility.

45 46 46 46 47 50 55 55 57 58 59 60 62 5.2.6 Discussion. 63

5.3 The influence of the diffusion on the zone boundaries. 64

5.4 The influence of axial and radial temperature differences.64

5.5 The influence of the activity coefficients on the

concen-tration. 66

5.6 Some calculations.

6 SOME PHENOMENA IN ISOTACHOPHORETIC EXPERIMENTS.

6.1 6.2 6.2.1 6.2.2 6.3 6. 4. Introduction.

Some effects in the use of non-buffered systems. The HI-MI boundary.

The MI-MII boundary.

Enforced isotachophoresis. Water as a terminator. 67 74 74 74 75 80 83

EXPERIMENTAL PART. 89 7 8 8.1 8.2 8.3 9 9.1 9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9,3 9.4 9.4.1 9.4.2 9.4.3 9.5 INTRODUCTION. 90

DETERMINATION OF PK VALUES IN METHANOLIC SOLUTIONS.

The determination of the pH in methanolic solutions. 93

.The determination of the pK values in methanolic solutions.

. 99 97

Exper~ments.

THE QUALITATIVE SEPARATION OF CATIONS BY ISOTACHOPHORESIS.

Introduction. 100

Aqueous systems. 103

The system WHCL. 103

The system WHI0

3• 104

The system WKAC. 104

The system WKCAC. 107

The system WKDIT. 107

Combinations of systems. 108

Methanolic systems. 110

The system MHCL. 110

The system MKAC. 114

The system MTMAAC. 114

Discussion. 116

10 THE QUALITATIVE SEPARATiON OF ANIONS BY ISOTACHOPHORESIS.

10.1 Introduction. 119

10.2 Aqueous systems.

10~2.1 Separations according to mobilities.

10.2.1.1 The system Hist/HCl.

10.2.1.2 The system Imid/HCl.

10.2.2 10.3 10.3.1 10.3.2 10.3.3 10.4•

Separations according to pK values. Methanolic systems.

The separation of fatty acids.

The separation of dicarboxylic acids. The separation of inorganic ionic species. Discussion. 119 119 119 123 124 131 131 133 137 137.

11 THE SEPARATION OF NUCLEOTIDES BY ISOTACHOPHORESIS. 11. 1 11.2 11.3 11.4 11.5 Introduction.

The structure of the nucleotides. Experiments.

An enzymatic reaction. Discussion.

12 QUANTITATIVE ASPECTS IN THE SEPARATION BY

ISOTACHO-PHORESIS. 138 138 138 145 146 12 .. 1 Introduction. 148 12.2 Theoretical. 149 12.3 Reproducibility. 151

12.4 The determination of the calibration constant. 152

12.5 Quantitative aspects in the separation of mixtures. 153

12.6 Detection limits. 158

12.7 Discussion. 163

13 FURTHER DEVELOPMENTS. 164

REFERENCES. 168

LIST OF SYMBOLS AND ABBREVIATIONS. 173

APPENDIX A: The computerprogram X3. 177

APPENDIX B: Isotachophoretic equipment with sample valve. 182

APPENDIX C: Isotachophoretic equipment with injection block. 183

SUMMARY 185

SAMENVATTING 186

DANKWOORD 187

INTRODUCTION

In the middle of thenineteenthcentury WIEDEMANN1-2

and BUFF3 reported on the phenomenon that charged

par-ticles migrate as a result of an applied electric field. The charged particles have a characteristic velocity and their mobility is defined as: "The velocity in an elec-tric field E of unit-strength".

In general different ionic species have different characteristic mobilities and therefore different veloci-ties in an electric field. This can be used for their se-paration. Techniques based on this principle are known as electrophoretic techniques.

Four main types can be distinghuished in electrophoresis,

viz.: -Isotachophoresis

-Isoelectricfocusing

-Moving boundary electrophoresis -zone electrophoresis.

All these different types of electrophoresis can be carried out in different ways, e.g. on paper, on thin layers, in gels, in blocks and in capillary tubes. All these methods have advantages and disadvantages. The in-fluence of e.g. the production of heat, electroendosmosis and moreover the use of aggressive and volatile solvents can be troublesome. Limitations in the use of high volta-ges and electric currents are the result.

1)

In the course of time numerous workers have investi-gated the phenomenon of isotachophoresis and its

appli-5-11

cations. The separation of isotopes , the measurement

12-18 .

of transference numbers , the separat~on of ionic

. 19-28 29-34

spec~es , the use of counter-flow ,

. 36-38 . . 39-40

pH grad~ents · , and the use of spacers have been

dealt with, although optimum results often could not be obtained by defective equipment.

Better results are obtained by EVERAERTS. EVERAERTS41(1968)

and MARTIN and EVERAERTS35 (1967) described an analytical

method, based on the principle of isotachophoresis in capil-lary tubes. Ionic species migrate under the influence of an electric field in a closed system, filled with an elec-trolyte. Cooling is easy and even volatile and aggressive solvents can be used. A thermocouple serves as a detector.

Although several papers describing the isotachophoretic separation of ionic species have been published, a more de-tailed research on the possibility to separate ionic spe-cies by isotachophoresis has not been made.

The aim of this work is to give a contribution in the applicability of isotachophoresis for the qualitative and quantitative analyses of ionic species.

In the first part a mathematical model for isotacho-phoresis and moving boundary electroisotacho-phoresis is given and experimental values are compared with calculated values in order to check the models. In the second part data are gi-ven of separations of anions and cations with water and methanol as solvents.

"Anything will prove interesting as soon as you take an interest in it."

CHAPTER 1

PRINCIPLES OF THE ELECTROPHORETIC METHODS

1.1 THE PRINCIPLE OF ISOTACHOPHORESIS.

For the explanation of the principle of isotachophore-sis we will consider the separation of anionic species in capillary tubes. For the separation of anions, the capil-lary

ding than

tube and anode compartment, are filled with the lea-electrolyte. The leading anion has a mobility higher

b . f c i l) . f

any mo ill. ty o the sample an ons • The catJ.ons o

the leading electrolyte have a buffering capacity. The ca-thode compartment is filled with an electrolyte, called terminator. The anions of the latter must have a mobility lower than any of the sample anions. The sample is intro-duced by means of a sample tap, between the leading elec-trolyte and the terminating elecelec-trolyte (Appendix B).

After the introduction of the sample an electric current is passed. After some time a steady state is obtained with all ionic species of the sample separated in serried zones in order of their mobilities. The first zone contains the sample anionic species with the highest mobility, the last zone that with the lowest mobility. All these zones migrate with a velocity equal to the velocity of the leading zone. It follows that each zone has a characteristic electric

field, according to the relation

v

=

m. E, where the

velo-city V must 'be equalised with the velovelo-city of the leading zone. The boundaries between two zones are sharp, because of

the self-correcting effect of the isotachophoretic system41.

1)

Speaking about mobilities in experiments, always the effec-tive mobilities are meant, as these determine the actual velocities in an electric field; see also Section 5.2.

The producti.on of heat in a zone is determined by the product of E and I. Working at a constant current density, zones with ionic species of high mobilities will have a smaller production of heat, than zones containing ionic species of lower mobilities' this results in lower tempera-tures. As the zones are generally ordered according to de-creasing mobilities, the temperature of the succeeding zones will increase. The temperatures are detected with a thermo-couple. The step heights in the electropherograms are a

measure of the temperature and hence allow the identification of the ionic-species. All zones have a specific concentration

as already indicated by KOHLRAUSCH51 Therefore the length of

the zones is a measure for the amount of the ionic species present in the sample.

Figure 1.1 shows the voltages., the electric field strengths and the temperatures of the different zones. The stepheight H

•

-I

Al

I

A2

I

_A:3

I

A4 e rnA ) _{rnA >} mA3 > rnA 1 2 4 v

f

E

I

I

~ ~ ;,...__.;

r---:

T _ _ _ . temperature

t

of the zone differential signal

FIG. 1.1 The voltages, electric field strengths and tempe-ratures of the different zones in isotachophoresis.

is used for the identification and the length L is a measure for the quantities.

Because this method is characterised by equal velocities of all zones, in the steady state, the method is called "Iso-tacho-electrophoresis." In practice the name "Isotachophore-sis" is used. This method is comparable with displacement-chromatography.

1.2 THE PRINCIPLE OF ZONE ELECTROPHORESIS.

In zone electrophoresis the whole system is filled with one-electrolyte (back-ground electrolyte). The sample is in-troduced into this back-ground electrolyte. The separation of anionic species is considered. The ionic species of the back-ground electrolyte have certain mobilities and when an electric current is passed these ionic species will migrate with their specific velocities. Also the sample ions migrate under. the influence of the electric field applied, each io-nic species with its own characteristic velocity dependent on the conditions chosen.

A flow of ions of the electrolyte, supervened by a flow of sample ionic species is obtained. As the back-ground elec-trolyte can provide in the current transport, no serried zones of the sample ions can be expected and there is not a self-correcting effect of the boundary. Due to the diffusion the peaks are wide and unsharp (tailing) and adsorption phe-nomena can cause "trailing".

Figure 1.2 shows the voltages electric field strengths and temperatures of the different zones. The back-ground electrolyte supervened by a slow sample ionic species shows a higher electric field strength over the zone than in the case of a quicker sample ionic species. If the influence of the back-ground electrolyte on the conductivity of the zone, is large in comparison with that of the sample ions, a near-ly constant electric field strength and pH can be expected

v

I

Eri'---'----'---....:.__L.,___j

___l ~ ; : - . 1 ; : ! i---< '----' ----,

T'L~~;...____._i

. ~ :

~

i.. : - - - . .

u. : :

_:---f

FIG. 1.2 The voltages, electric field strengths and

tempera-tures of the different zones in zone electrophoresis.

and ~11 sample ions will have their own constant velocities

during the experiment. Identification is possible by diffe-rences in the "retention times" of the ionic species. This technique is comparable with elution chromatography.

1.3 THE PRINCIPLE OF MOVING BOUNDARY ELECTROPHORESIS.

In this method the sample fills the electrode compart-ment behind the leading electrolyte. A partial separation is obtained dependent on the time of the analysis. An elec-tropherogram may have the following shape (Figure 1.3):

T

I

Substance A₁, more mobile than the other substances of

the sample is separated from A₂ and A₃• Substance A

2 mixed

with A~ _{forms the second sample zone after the pure A1 zone.}

The 3t zone contains the mixture A₁, A₂ and A₃• This method

is comparable with the frontal analysis method in chromato-graphy.

In moving boundary electrophoresis, the zones generally contain more ionic species of the sample. The composition of the sample plays an important role in the determination of

the concentrations, pH's and conductivities of t~e zones. This

in contrast with isotachophoresis where all these quantities are independent of the quantitative composition of the sample.

1.4 THE PRINCIPLE OF ISOELECTRICFOCUSING.

In this method a column contains a buffer solution, that creates a pH gradient in the tube. When a sample, consisting of a mixture of amphiprotic molecules (with a particular pi value) is introduced, the particles will move until they reach a pH in the tube equal to their pi values.

At this point the effective mobilities are equal to zero. In the stationary state the particles will be sepa-rated, if they have different pi values, according to their pi values.

1.5 DISCUSSION.

Although in this chapter four main types of electro-phoresis have been distinghuished, often a sharp distinction between these types can not be made in practice. Disturbances during the experiments are often caused because not all con-ditions are fulfilled, required for a specific type of

elec-trophoresis. During isotachophoretic experiments all other types can exist.

The first stage in the separation by isotachophoresis is a moving boundary procedure in the sample compartment, i.e. all ionic species have a velocity determined by e.g. the actual pH, the ionic strength, the temperature, the visco-sity, the effective mobilities and the electric field strength. After some time, when a steady state is reached, the ionic species are separated and we can speak of isotachophoresis.

If the differences between the mobilities are too small and/or if the differences in concentrations are too large, mixed zones can be expected and we can not speak of isotacho-phoresis properly.

If the influence of a back-ground electrolyte (solvent effect at low and high pHs) is too great, zone electrophoretic phenomena can be expected. The use of spacers (ampholytes) during isotachophoretic experiments gives a combination of

isotachophoresis and isoelectricfocusing. Some phenomena will be discussed further on.

CHAPTER 2

GENERAL EQUATIONS IN ELECTROPHORETIC PROCESSES

2.1 INTRODUCTION.

Experiments based on the principle of electrophoresis 1-4 50-54

have been described for a long time ' . Already in

1897 KOHLRAUSCH51 gave a mathematical model for

electro-phoretic processes. Using the divergence theorem, the con-tinuity equations can be derived and using the principle of electroneutrality and assumptions such as constant re-lative mobilities, he formulated the socalled "Beharrliche funktion":

=

Constant.

This regulating function prescribes that at any point the sum of the concentrations divided by the mobilities must be constant.

In this chapter the general equations in electrophoretic processes will be discussed. They will be used for the mathe-matical models of isotachophoresis (Chapter 3) and moving boundary electrophoresis (Chapter 4).

2.2 THE GENERAL EQUATIONS.

For the derivation of the general equations in electro-phoretic processes we will consider the movement and forma-tion of zone-boundaries, when a electric field is applied over an existing zone-boundary between two electrolyte solutions. On one side of the boundary a mixture of several anionic and cationic species and on the other side a "single electrolyte" is present.

The anode is placed in the single electrolyte. Only the migration of the anionic species is considered, whereby the effective mobility of the anionic species of the single elec-trolyte is assumed to be higher than any of the anionic species of the mixture (Figure 2.1).

e

<

FIG. 2.1 A zone boundary between a mixture of several anionic

and cationic species and a "single electrolyte".

After some time all anionic species have the same

counterion (BL) because the cationic species B₁ to Br

are moving in the opposite direction.

Furthermore a number of boundaries will be formed. Two types of boundaries have to be distinghuished viz. the concentration and the separation boundaries.

For the concentration boundaries the number of anio-nic species is equal on both sides of the boundaries, whereas for separation boundaries one particular ionic species is present on one side of the boundary only. In general r+1 boundaries will be present if an electric current is passed across the original boundary as shown in Figure 2.1, considering the separation of anionic spe-cies, viz. one concentration boundary, r-1 separation boundaries and the boundary between the single electrolyte and the zone containing the anionic species with the highest effective mobility of the mixture (Figure 2.2).

The velocity of the concentration boundary is neglected

61-63 .

e

A -1 .• r A -1 .. r-1

-t

-concentration boundary A1 +A2

Al

AL--- - B L

'

_{r-1 separation}

t

t

t

I t boundary boundaries L.E.-A₁

<

FIG. 2.2 Zone boundaries formed when an electric current is

passed across a zone boundary as shown in Fig. 2.1.

to the velocity of the AL and A₁ ionic species. The

velo-cities of the separation boundaries are equal to the veloci.,.. ties of the ionic species with the lowest effective mobility in those zones. These anionic species are not present in the preceding zones.

Speaking about ionic species in the model we mean amphi-protic polyvalent particles, containing different chemical groups with different equilibrium constants. For such a particle, the following equilibria can be set up:

-ZA -(i-1) A r + H 0 ~ .. !::::====; .. r 2 ZA - (n-1) r + H 0 ~=::::p 2 _pK n ZA -1 A r r ZA - i A r r ZA -n A r r 2.1

ZA

The particle Ar r, with the highest positive charge zA ,

r

is taken as a reference in all computations.

The pK's are increasing from pK₁ to pKn. A similar

reac-tion can be given for the buffering counterions B. Nearly all general equations are similar for both the ionic species to be separated (anions) and the buffering counter-ions

(cations).

For the derivation of the equations the following assump-tions are made: the electric current is constant; the cross-section of the tube is constant; the influence of the diffu-sion, hydrostatic flow and electroendosmosis is negligible; the activity coefficients and the influence of the radial temperature differences can be neglected.

The generalequations describing electrophoretic processes

are: the equilibrium equations.

the electroneutrality equations

the mass-balances for all ionic species the modified OHM's law.

These equations will be considered in more detail.

2.2.1 The eguilibrium equations.

The chemical equilibrium equations determine all pH depending quantities such as the effective mobilities. Con-sidering the reaction 2.1 the general expression for the equilibrium constant will be (for the Uth zone):

So:

KAr,U,i • cA ,U,zA

-i+l

r r

CA ,U,zA

- i

=

r

r

cH,U

Substituting the expressions for c

-l.'+l

etc., up to

A 1 U 1 ZA

in eqn.

2.3 r r

c

=

A ,u,zA

- i

r r

The total concentration of an anionic species is:

c

A ,U,zA

r r

nA

.;;:--r

( 1

+

<-i=l i

j"[

KAr,u ,j )

(ca,u>i

2.3

2.4

2.5

Similar equations can be derived for the buffering counterions.

2.2.2

The electroneutrality equations.

In accordance with the principle of electroneutrality,

the arithmic sum of all products of the concentrations of·

all forms of all ionic species and the corresponding

valen-ces, present in each zone, must be zero.

While the first zone contains one ionic species of the

sample, each zone following always contains one ionic species

th

.

more. The U

zone will contain U ionic species of the sample

consequently. The ionic species are numbered in order of

de-creasing effective mobilities. For the Uth zone can be written:

u

~{

r=1 { ( ZA -i) • cA U - . } } + ' ,zA ~ r r r 0

Substituting eqns. 2.4 and 2.5 in eqn. 2.6, both for the sample ionic species and counterions:

i

1f

K . . _ 1 A ,U,J -·) J-_{~ .} r _i (cH,u> } + } + i 1 + i

~B

i=1 i 1 +

1f

KB,U ,j j=1 i=1

2.2.3 The mass-balances for all ionic species.

2.6

2.7

0

In the stationary state the amount of all ionic species passing .a separation boundary is equal to the amount rea-ching the separation boundary. For the Uth separation

boun-dary this means that U-1 balances for the anions and 1

ba-1)

lance for the bufferions can be obtained •

The zone-boundary U/U-1 has a velocity of EU.mA

u·

u'

The quantities written with a bar

,m,

indicate that they do

not apply to ions, but to the equilibrium mixtures of all forms of the constituent, consequently the effective mobili-ties of the ionic species are meant. As the boundary velocity is determined by the Uth ionic species, the subscript figure

r in rnA is replaced by a "u". r

For the effective mobility TISELius64 pointed out that

a substance consisting of several forms with different mo-bilities in equilibrium with each other will generally mi-grate as a uniform substance with an effective mobility:

n n

m

=

~

~

2.8

i=O i=O

provided that the time of existence of each ionic species is small in comparison with the duration of the experiment. In this effective mobility, factors such as the relaxation effect, the electrophoretic effect, the influence of the temperature are neglected (see also Section 5.2).

1)

The sample ionic species Au1 the ionic species with the

lowest effective mobility of the sample, determines the velocity of the Uth zone and is not present in the u-1th zone.

Substituting the eqns. 2.4 and 2.5 in the eqn. 2.8 delivers: ' t Eu t1 To EU-1 t cB,U cB,U-1 t _T T1 t cA c r'u 0 Ar,U-1

FIG. 2.3 Migration paths of the different ionic species over

a zone boundary. 1 } + m A ,U,zA r r n i Arlf

<

K . +

"<:

~j_=_1 ____ A_r~._,u __ ,_J i=1

The amount of the buffer ions, just passing the moving boundary is (Figure 2.3):

The amount just leaving the boundary is:

1!.2

B

2.9

2.10

Those amounts must be equal and the mass-balance for the buffer will be:

In a similar way for the mass-balances of the anionic spe-cies can be derived:

2.2.4 The modified OHM's law.

Working at an equal current density:

I/G = Constant = E₀ • Au

The electric conductivities for the zones are the somma-tion of all: ci . mi • 1zi1 , consequently:

( lzA -il .cA i.mA U ·

r ,u,zA - r' ,zA -~

r r r

+

Substitution of the eqns. 2.4 and 2.5 in eqn. 2.15 gives: 2.13

2.14

i

nAr

1f

K .

~ ~

. j = 1 Ar 1 U 1 J ~

lzA

-1.1 .

rnA . + lzA l·m

U i=l r i r 1 0 1 2_A

-1.

_{r Ar}1 0 1 2_A [

~

______________

(_c_H_~u

__ l

1

~.

____________ r _________________ r_ r=J

ci

U}

+ rl

~r

i=l 1 + i jill KBIUij i .mB1U1zB-i

~+lzBI

.mB1U1zB (cH 1U) t ---~~~l.---.cB

₁

U + 1 +

~

i=l

1T

KBIU I j j=l I/G 2.16

CHAPTER 3

A MATHEMATICAL MODEL FOR ISOTACHOPHORESIS

3.1 INTRODUCTION.

In Chapter 2 the general equations, describing the movement and formation of zone boundaries, are discussed for the case that a stabilised electric current is passed across a zone boundary, between a mixture of anionic and cationic species on one side and a single electrolyte on the other side. Generally r+1 zone boundaries were obtained for the separation of anionic species. No complete separa-tion of the anionic species can be obtained in this way.

In principle an isotachophoretic system is a similar one. The sample (mixture of anionic and cationic species) is introduced between a leadingelectrolyte and a terminator electrolyte (Figure 3.1).

The first stage is a separation procedure as will be described in Chapter 4. In the steady state all the ionic species of the sample are separated and each sample zone contains only one ionic species of the sample.

A.r AI. .r AL

e

----

---

(9

-

B.r

--

B1 .• r

---

BL

mA.r

<

mA

1.. r

<

mA L

FIG. 3.1 Original situation when a sample is introduced in an

Each zone has correlation formulae only with the zone in front of it. Calculations of pH, concentration and other parameters are possible. For the mathematical model of

iso-41-43 63 65

tachophoresis ' ' the general equations (Chapter 2)

will be combined with the isotachophoretic condition, which prescribes that all zone velocities must be equal.

3.2 BASIC EQUATIONS.

In analogy with the general equations and with the same assumptions we will give here the equilibrium equations, the mass-balance of the buffer, the electroneutrality equations

and the modified OHM's law, combined with the isotachophore~

tic condition, for the description of the isotachophoretic model.

Only the mass balance of the buffer will be used as the anionic species of the sample are only present in their own zone (the separation of anions is considered).

3.2.1 The equilibrium equations.

In a similar way as described in Chapter 2 we can derive:

KA__ .

--v'

~

1) _3.1

1)

3.2

The subscript figure refering to the Vth zone is used only for the hydrogen ions. For the other symbols this

indication is superfluous as the indication

Av

always

i n~

_1f

KIV .

t ( 1 +

~

j=1 ,J c A = c

.

₎

v

Av•ZAv _i=1 {cH,

v>

i 3.3

3.2.2 The isotachophoretic condition.

In the steady state all zones move with a velocity equal to that of the leading zone, therefore:

3.4

-The mAL and mAv are the effective mobilities of the leading

ion in the leading zone and the sample .ions

IV

in the Vth

zone respectively.

t

i=l n~ 1 +

<

&,1

i

JT

K . j=1

IV·J

(cH

v>

i

) +

mA_ -v'zAv 3.5

For all other ionic species a similar expression for the effective mobilities can be derived. The isotachopho-retic condition is the essential difference between iso-tachophoresis and other electrophoretic methods.

3.2.3 The mass-balance of the buffer.

The movements (AX) of the zone boundaries L V and V W per unit of time are equal (Figure 3.2):

AX 3.6 t _{tl to} t _t1 t 0 0 0

l

AX AX B2X I B1X vw 62

FIG. 3.2 Migration paths and movement of the zone boundaries

in an isotachophoretic system.

The distances over which the buffer ions move during one unit of time in order to reach the zone boundaries

are respectively:

B1X EL.mB

L 3.7

B2X

₌

_Ev.m~ 3.8

Therefore the amounts of the buffer that pass the zone boundaries L-V and v-w, are the amounts of the buffer

pre-sent in the volumes

6

₁ and

6

2 respectively, at t=O.

The 9mounts of the buffer entering and leaving a zone must be equal, therefore:

Combining the eqns. 3.9 and 3.4 gives:

t -

-cB • { 1+mB /rnA_)

L L --r,

3.10

3.2.4 The electroneutrality equations.

In accordance with paragraph 2.2.2 for the electroneutrality can be written: i Tr KA.__ •

j~:\

---v'

J i {cH,V)

3.2.5 The modified OHM's law.

Working at an equal current density:

+

= 0

The electric conductivities for the zones are the somma-tion of all: ci.,zil'mi1 consequently:

nA

(cOH L.mOH L+cH L"mH L+ <L ( lzA -i\.cA_ _ .• rnA -i> +

I I I I

&o

L --L I

z

A l. L I

z

A

L L

I/G. 3.13

Substitution of the eqns. 3.2 and 3.3 gives:

i 1T KB ]' j!d1 Ll ) +

IZB I

i L (cH,L) nB

~

L 1 + i=1

=

I/G 3.14

A similar expression can be set up for the sample zone. Assuming the left-hand side term of the eqn. 3.14, QL and QV for the leading and Vth zone respectively, the function RFQ defined as:

must be zero according to equation 3.12.

3.3 PROCEDURE OF COMPUTATION.

The procedure of the computation is the following.!) If all mobilities and pK values are known, and the to-tal concentration of the leading ionic species and the

pHL are chosen, all computation constants2) of both the

leading ions and the buffer ions in the leading zone can be calculated.

From an equation similar to 3.3 thecA z can be

L' A_

calculated out of the total concentration L and with

eqn. 3.2 all partial ionic concentrations of the ionic species AL. With eqn. 3.11 the total buffer concentration in the leading electrolyte zone can be obtained, and with an eqn. similar to 3.3 and 3.2 the partial concentrations of the buffer. Furthermore QL and the left-hand side term of the buffer correlation (eqn. 3.10) can be acquired. All quantities of the leading electrolyte are known now.

1) With the equations derived in section 3.2 a computer program has been developed. In Appendix A the program is shown. An example of the in- and output is given. The language used is ALGOL 60.

Calculations were made with the P9200 time sharing computer.

2) Computation constants are e.g. the effective mobili-ties and the continual products in the equilibrium equations.

Assuming a certain Pliv for the following zones, all computation constants for those zones can be calculated, in a similar way as indicated for the leading zone. With the eqn. 3.10 the total concentration of the buffer can be found and with the eqns. 3.2 and 3.3 all other par-tial concentrations. With eqn. 3.11 the total concentra-tion of the sample ionic species and with the eqns. 3.2 and 3.3 all partial concentrations can be obtained.

With equation 3.14 the QV can be obtained and the eqn. 3.15 will give the value of the function RFQ for the

assumed pH. This value must be zero for the correct Pllv· In fact more zero-points are possible. The way found the correct Pllv zero-point will be dealt with in the next

sec-tion.

3.4 PROCEDURE OF ITERATION.

As mentioned in Section 3.2.5., the function RFQ must be zero for the correct Pllv value. For several cases this function RFQ is computed as a function of the Pllv·

In Figure 3.3 this function is plotted for .the se-parations of univalent cations and anions. Also the buf-fering counterions were univalent. In Figure 3.4 the function is shown for polyvalent sample ionic species and bufferions. In Figure 3.5 the function is shown for a system, where in the leading electrolyte zone, the leading ion buffers in stead of the counter ion. Only in the sample zones, the counter ion acts as a buffer and in general this means that a larger pH shift between pHL and Pliv is present. This is used in disc.-electrophoresis according to ORNSTEIN and DAVIS 25 , 26 •

In the Figures 3.3, 3.4 and 3.5, the anionic and cationic

func-TABLE 3.1 pK values and ionic mobilities of the ionic species, used for the calculation of the relationship between

RFQ and p~.

Fig. Leading zone

Buffer ionic s:eecies Leadin9: ionic s:eecies

m.105 pKs n z cone. m.105 pKs n z pHL cm2(_vs mole(_l cm2t._vs 3.3.a 0,50 3 1 0 0.01 75,0 14 1 1 3 3.3.b 19,0 11 1 1 0.01 0,76.5 -2 1 0 11 3.3.c 0,50 4 1 0 0.01 75,0 14 1 1 4 3.3.d 30,0 10 1 1 0.01 0,76.5 -2 1 0 10 3.3.e 0,50 6 1 0 0.01 75,0 14 1 1 6 3.3.f 19,0 6 1 1 0.01 0,76.5 -2 1 0 6 3.3.g 0,50 10 1 0 0.01 75,0 14 1 1 10 3.3.h 30,0 4 1 1 0.01 0,76.5 -2 1 0 4 3.3.i 0,50 11 1 0 0.01 75,0 14 1 1 11 3.3.j 30,0 3 1 1 0.01 0,76.5 -2 1 0 3 3.3.k 0,50 12 1 0 0.01 75,0 14 1 1 12 3.3.1 30,0 2 1 1 0.01 0,76.5 -2 1 0 2 3. 4. a 50,0,50,70 2,4,8 3 1 0.01 75,0 14 1 1 5 3.4.b 0,40 4.75 1 0 o.o1 75,0 14 1 1 5 3.4.c 19,0 6 1 1 o.o1 0,76.5 -2 1 0 6 3.4.d 19,0 6 1 1 0.01 0,76.5 -2 1 0 6 3.5 19,0 8 1 1 0.01 0,40 4.75 1 0 4.75

Fig. Sam:ele ionic s:eecies Fig. (a)

m.105 pKs n z pKs 2· em (_Vs 3.4.a 50,0 .!! 1 1 3.3.a 3,5,6,7 50,0 10 1 1 3.3.b 9,10,11,12 50,0 14 1 1 3.3.c 3,4,5,6,8,10,12 70,30,0,30 4,6,8 3 2 3.3.d 1-6,9,12 3.4.b 70,30,0,30 4,6,8 3 2 3.3.e 3,5,7,9,13 70,50,0 5,7 2 2 3.3.f 1,6,10,11,12 50,0 14 1 1 3.3.g 4,8,10,13 3.4.c 0,50 4 1 0 3.3.h 1,4,5,10 0,50,70 4.5,5 2 0 3.3.i 2,4,8,10,11 50,0,30,60 3,5,7 3 0 3.3.j 4,5,8 3.4.d 50,0,30,60 2,4,8, 3 1 3.3.k 2,4,8,12 50,0,50 3,9 2 1 3.3.1 1,4,5,8 70,70,0,50,70 2,4,6,8 4 2 3.5 30,0,30 2,9 2 1

(a) because in this cases the assumed mobilities for the mono-valent cations and anions were resp. 50,0 and 0,50, only the pK values of the sample ionic species are given.

j·

2 1 0

~

tl

5 -1 3 2 1 0

~

t

-1 7 10 pH =3 L pHV FIG. 3.3.a pH =4 L 2

-pll.v

FIG. 3.3.c

e

pHL=ll 3 12 2 11 10 1 0

~

9 5 10

-

pHV FIG. 3.3.b -1

e

12 3 2

I

9 1 0

~

1-6 5

-

PRy

FIG. 3.3.d -1

3 2 1 0.

~·

1

-

_PlV

FIG. 3.3.e -1 1~ 4 l 13 2 1. 0

~

5 10

-

_PlV

)

;)/

FIG. 3. 3 .g -I

e

3 2 1 1 0

~

t

I

-1

e

3 2 1 0

~

t

-1 6 5 4 ₁₀ 5 12 10 1 pHL=6 FIG. pH =4 L

-FIG,

PlV

3.3.f

Pllv

3.3.h

3 0 11 2 1

r

\

01, r...:

!l:l

t

5 1

-Pflv

FIG. 3.3.1 -1 pHL=12 2 4 12 8 1

\

I 01

I

~

\

i

5 10

_Pflv

-1 FIG. 3.3.k

e

8 4 5 2 1 01

~

5 -1

\\.\

e

2 1 4

~

a

1 01 I \

~

t'

\ 5\ -1

\~

10 pH =2 L 10 pH =3 L

-FIG.

Pflv

3.3.j - pHV FIG. 3. 3. 1

2 1 0

~

-1 -2

e

3 2 ill'l

\~

I'"-ll'l ("f) 1 0 ...,_, cz: -1

~I\

pH =5 L

-

_PJV

FIG. 3.4.a pH == 6 L 10 -plfv FIG. 3.4.c 14 p~=S 2 1 0

~

t

10

-

_PJV

-1 -2

~n

I

I

FIG. _3.4.b

e

pHL=6 3 co

_..

2 co

..

1.0

..

<;!' <;!'

I

..

N N 1 0

~

t

---..p -1 FIG. 3.4.d

tions are indicated by a number, representing the pK values of the sample ionic species. All assumed pK values and ionic mobilities for the leading electrolyte and the sample ionic species are given in Table 3.1.

For all these electrolyte systems different functions were obtained. Some show no real zero-points, sometimes two

zero-points are present and some show discontinuities. All those properties depend on quantities such as pK values and mobilities. Although not all possible functions have been computed, we can conclude that all systems have one common property viz., in the case of a cationic separation the correct zero-point was always the transition between a

e

2 1 0

~

t

5

- Pf\r

-1 -2

FIG. 3.5 The relationship between the function RFQ and the

negative and a positive value of the function RFQ in the direction of higher pHs and for the anionic separations it was the transition between a positive and a negative

va-lue of RFQ. (For the false zero-points negativ~

concen-trations were obtained).

The way to find the correct zero-point is therefore:

In the computer program first a

Pliv

is searched for, with

a positive (resp. negative) value for RFQ and then for a

Pliv

with a negative (resp. positive) value for the

Plivr

for anionic (resp. cationic) separation. The correct

Pliv

at which the function QV is zero, within a certain deviation, is obtained by iterating between those two values. If no pair of positive-negative resp. negative-positive QV values can be obtained in a traject of 6 pH values from the pHL then "NO REAL ZERO-POINTS" will be printed.

The procedure of iteration is shown in Figure 3. 6.

PRINT RESULTS

FIG. 3.6 Flow chart of the iteration procedure of the

3.5 DISCUSSION.

Sometimes, the function RFQ shows no real zero-point,

i.e. the function is always positive (e.g. Fig. 3.3. a,

3.3. band 3.3. c). Mainly this effect can be observed

at low pHs for the cationic and at high pHs for anionic

separations. The exact pHs at which this phenomenon occurs

depends on the pK values and mobilities of all ionic species

and a general treatment to determine them can not be given.

The importance of this fact is that theoretically the

mathematical model is not valid at those pHs. Practically

it means that at those pHs the influences of the hydrogen

and hydroxyl ions are such that we do not have real

isotacho-phoresis. The isotachophoretic condition is lost, i.e.

isota-chophoresis is transferred into e.g. a moving boundary procedure.

In the next chapter a. model of moving boundary

electro-phoresis will be given. This model is necessary in order to

understand some other phenomena in isotachophoretic

experi-ments.

CHAPTER 4

MOVING BOUNDARY ELECTROPHORESIS

4.1 INTRODUCTION.

If the separation in isotachophoresis is completed,

only one ionic species of the sample is present in each

sample zone. The parameters of each zone are related with those of its preceding zone. Calculations of the pH, concentration and other parameters are possible. A mathematical model for the buffered systems already has been given in the previous chapter.

If the separation is not completed, i.e. mixed zones

are present, and/or if the influence of the back-ground ions is too great, the conditions for real isotachophoresis

are lost and the model described is not valid any more.

Especially this can occur in non-buffered systems. In this case the separation procedure can be better understood by using a model similar to the moving boundary technique.

Several authors55-60 gave already a mathematical model

for the moving boundary system, but i t is very difficult to work with an exact model. Some simplifications have to be made. Each zone does not consist of one ionic species of the sample, but the number of ionic species in the zones increases to the rear-side. Only the first sample zone, following the leading electrolyte zone, contains one ionic species from the sample. All zones have correlations with both the preceding and following zone, which explains the difficulties in computation.

A simpler model was used by BROUWER and POSTEMA61• They

which is moving boundary electrophoresis in principle. Con-centration effects, the influence of the pH and the tempe-rature were neglected. Although this is not a general model, i t can be used for non-buffered systems of monovalent, fully ionised ionic species.

In this chapter we will describe a model similar to that 61

of BROUWER and POSTEMA . The influence of the temperature

is taken into account. With the formulae a computer program is made and calculations are compared with the results of experiments. Moreover some phenomena in isotachophoresis of non-buffered systems can be explained.

4.2 A MODEL OF MOVING BOUNDARY ELECTROPHORESIS.

To carry out experiments with moving boundary electro-phoresis the capillary tube can be filled with an electrolyte of a strong acid, when a separation of cations is desired. The cation present has a mobility higher than the mobility of any other cation of the sample. The sample is situated at one end of the capillary tube, i.e. in the anode-compartment.

For the derivation of the formulae the following assump-tions are made: fully ionised monovalent caassump-tions and anions are considered; the contribution of the back-ground ions to the conductance of a zone is negligible; the influence of differences in pH, and concentrations are negligible; the electric current is stabilised; the diffusion, hydrodynamic flow and electroendosmosis are negligible; the solution ini-tially present in the capillary tube and anode compartment is of well known constant composition.

The formulae needed to be considered are; the

electro-neutrality equations; the modified OHM's la~·; the mass

4.2.1 The electroneutrality equations.

If the influence of the back-ground ions can be neglected and when all ionic species are fully ionised the concentra-tion of the counter ions will always be equal with the con-centrations of the cations present in a zone. This if mono-valent ionic species are considered of course.

4.2.2 The modified OHM's law.

+

The influence of the H and OH ions are neglected. It

follows that:

1)

I/G

=

Constant

4.2.3 The mass balances for all cationic species.

In the stationary state the amount of each ionic species passing a separation boundary is equal to the amount reaching the separation boundary. For each ionic species and all sepa-ration boundaries can be written:

4.2

Substituting:.

v

₀ 4.3

1)

The subscript letter U refers to the uth zone. The Uth zone contains U ionic species of the sample. The temperature

cor-rection for the mobilites TC₀ is taken uniform for all

4.4

Introducing:

Su-

₁

,u

4.3 PROCEDURE OF COMPUTATION.

Combining egns. 4.1 and 4.6, for a separation boundary will be obtained: U-1

~

r=1 4.5 4.6

The left-hand side term will be zero for r=U, because the

ionic species U is not present in the U-1th zone. This means its concentration is zero. Therefore the left-hand sum can be

extended to

u.

After simplification eqn. 4.7 will give:

~

=

0

4.8

or:

This is a modification of the "Dole-polynomals" (ref.66,55). Solutions are valid if:

<

1

If the composition of the leading electrolyte and the sample solution are known all parameters can be computed

with the eqns. 4.9 and 4.10 if the Su-₁,u were known.

4.10

The velocity of the concentration boundari~s can be neglected.

ted.

The parameters of the first zone can be calculated in two ways: both with the eqns. 4.9 and 4.10 and with the isotachophoretic conditions as described in Chapter 3.

In the computation we chose arbritarily a Su-₁,U of 1 and

computed all quantities. If the parameters of the first zone in this way obtained did not agree with those of the isotachophoretic calculation, we recomputed up to the last zone with the quantities obtained for the first zone with the isotachophoretic calculation (the S's are constant).

With the formulae a computerprogram is made. Experiments are carried out in order to check this model. To this end, all concentrations should be determined in each zone.

Be-cause this gives difficulties another possibility is to measure the speeds of the zones by means of a detector.

Each zone has its specific constant speed: Vu=Eu.mA .TCU. For practical reasons we use the relative speeds in ste}id of the absolute speed:

4.11

If the distance between the injection-point and the point of the detection is called P, the time needed for each ionic species to be detected will be:

P/VU = P/(mA .TCu.EU) or P u

4.12

The relation between speeds and times for the detection is:

The times of the detections can be measured, taking the time from the starting-point of the analyses up to the time of appearing of the step height of that specific ionic spe-cies in the electropherogram.

The speed of the leading electrolyte is equal to the speed of the first zone following the leading electrolyte

( isotachophoretical condition ) • Thus:

4.14

So we can use the ratio t

1/tu from the electropherograms

r

4.4 EXPERIMENTAL.

As indicated in the previous section the time of

de-tection can be used as a parameter, characteristic for mo-ving boundary systems. The relative time tL/tU is a mea-sure for the voltage drops over the zones and therefore for all other quantities such as the concentrations and the conductivities of the zones.

To check the model some experiments have been carried out and the experimental values of tL/tU are compared with the theoretical values of Vu/VL obtained with a computer program.

The values of tL/tu were taken from the

electrophero-f d 'electrophero-felectrophero-f . f + + .+ T + d T +

grams o 1 erent m1xtures o Na , K , L1 , ma an ea .

The leading electrolyte was 0.01 N HCl in water. The elec-tric current was stabilised at 70

1

uA.

The experimental data are given in Table 4.1. In Figure

TABLE 4.1 Theoretical and experimental values of the relative time of detection for some cations in a moving boundary electrophoretic system.

K Na Tma Li Tea a) concentrations 0.01 0.01 0.01 0.01 0.01 tL/tU theoretic 1. 000 0.904 0.863 0.793 0.717 measured 1. 00 0.90 0.85 0.79 0.70 b) 0.02 0.01 0.01 0.01 0.01 1.000 0.848 0.805 0.736 0.664 1. 00 0.84 0.79 0.73 0.65 c) 0.02 0.02 0.02 0.01 0.01 1. 000 0.889 0.845 0.753

o.

671 1. 00 0.88 0.83 0.75 0.66 d) 0.02 0.01 0.01 0.02 0.02 1. 000 0.872 0.837 0.793

o.

723 1. 00 0.87 0.83 0.78 0.71 e) 0.02 0.01 0.02 0.01 0.02 1.000 0.877 0.845 0.767 0.708 1. 00 0.87 0.84 0.76 0.70

4.1 those results are represented in a graph (the dotted lines represent the experimental values).

The experimental values agree very well with the calcu-lated values, and it may be concluded that the model is a suitable one. G. I

f.

"

_"

' IJ

l

e. K I• I _llo

....

_u (dl ' (bl Tao ' ' (c

FIG. 4.1 Graphical representation of the theoretical and

ex-perimental values for the times of detection for some cations in a moving boundary electrophoretic system (see Table 4.1).

' '

The relative time of detection for a mixture of 2 cations of a certain known concentration is constant and depends on the mobilities. By this i t is possible to determine the mo-bility of a cation from its relative time of detection.

In Figure 4.2 all relative detection times (calculated) are noted as a function of a mobility of an ionic species, if introduced as a mixture with K+ (O.OlN) for some concentrations of the sample ionic species. (The leading electrolyte is O.OlN HCl). If the relative time of detection is measured the

mobi-so

=

1

t

D.S 1,8

FIG. 4.2 Graphical representation of the calculated relative

times of detection as a function of the mobilities for different concentrations of the sample, mixed with 0.01 M KCl, after the leading electrolyte 0.01

N HCl.

lity can be found in this graph. In this way measurements were carried out with Na+, Tma+ and Tea+. The results are given in Table 4. 2. As can be seen this procedure is corre'ct for the measurement of mobilities. Figure 4.2 shows that for smaller concentrations of the ionic species, mixed with 0.01 M KCl in one sample, the relationship is a linear one.

This corresponds with the theory, as in that case the

elution phenomena prevail i.e. a uniform voltage gradient

the relative times of detection are a linear function of the mobilities.

TABLE 4.2 Theoretical and experimental mobilities of some

cations.

concentration _tL/tu m. 10 5 m. 105

in the sample theoretical measured

K-Na: 0.01 0.01 0.8375 50.5 51.25 0.01 0.005 0.7930 51.5 K-Tma:0.01 0.01 0.7900 45.0 45.7 0.01 0.005 0.7200 45.5 K-Tea:0.01 0.01 0.6770 30.0 32.2 0.01 0.005 0.6000 33.2

f

Trru:

FIG. 4.3 Separation of a mixture of cations in moving

boun-dary- electrophoresis. All initial concentrations were 0.01 M. The electric current was stabilised

at 70 ₁uA. The leading electrolyte was 0.01 M HCl

With moving boundary also separations of mixtures can be carried out. In Figure 4.3 the electropherogram is given of the separation of a mixture of Tma, NH

4, K, Na, Ca, Li,

+

Co, Mn, and Cu after the leading ion H •

The separation is quite good, but interpretation will be difficult if the sample is unknown due to the fact that the retention times are not constant and the step heights are dependent to both the mobilities and the concentrations in the sample.

Of course we would like to know the information of all

step heights and all retention times in the elec~ropherogram

but practically this is too difficult and in this way moving boundary electrophoresis hardly can be used.

CHAPTER 5

VALIDITY OF THE ISOTACHOPHORETIC MODEL

5.1 INTRODUCTION

In Chapter 3 a mathematical model of isotachophoresis has been given and based on this model a computer program has been developed for the computation of quantities such as the concentrations of sample and buffer ionic species, the electrical conductivities of the zones, the pH's of the zones and the effective mobilities of the ionic species in the zones during the steady state. For the calculations the composition of the leading electrolyte zone and data on

ionic mobilities and pK values of all ionic forms must be Rnown.

In this model the activity coefficients, the influence of the temperature (different in each zone), the relaxation and electrophoretic effects, the diffusion, the hydrostatic flow and the electroendosmosis were neglected.

In this chapter some of those factors will be discussed. For some of them corrections will be made in the calculations and the results of these calculations will be compared with the results of some experiments in order to check the validi-ty of the model.

5.2 THE CONCEPT OF MOBILITY.

The concept of mobility plays an important part in elec-trophoretic techniques. Differences in effective mobilities determine whether ionic species can be separated or not. The concentrations and voltage gradients of the different

zones in relations with the quantities of the leading zone are also fixed by the mobility values.

The absolute mobility (m

0) is defined as the average

velocity of an ion per unit of field strength. This absolute ionic mobility is a characteristic constant for each ionic species in a certain solvent and is proportional to the equivalent conductance at zero concentration:

A;t.

0 = 5.1

The effective mobility of an ionic species is related

with the absolute mobili.ty. TISELius64 pointed out that the

effective mobility was the summation of all products of the degree of dissociation and the ionic mobilities.

Other influences on the effective mobility are the

re-laxation and electrophoretic effects as described by ONSAGER67.

By the formulae of ONSAGER a correction is made for the ion-ion interaction-ions. The influence of the solvent (e.g. solvation-ion and influence of the dielectric constant) is also very impor-tant.

Summarising we can state that the effective mobility of an ionic species depends on factors such as the ionic radius, solvation, dielectric constant and viscosity of the solvent, shape and charge of the ion, pH, complex-formation, concen-tration, degree of dissociation and temperature. All those factors can influence each other and therefore i t is very difficult to give a mathematical expvession for the effective mobility. Speaking about effective mobilities we will use the expression:

=~

i

where ai =the degree of dissociation; Yi= a correction factor according to the influence of relaxation and

elec-trophoretic effects and m.

=

the absolute ionic mobility.

~

These effects will be described in more detail.

5.2.1 Relaxation and electrophoretic effects.

ONSAGER derived for the rela~ation and electrophoretic

effects the following expression:

5.3

where:

~~.·

0.98S·I0

6

.~·(1n+l·ln-I)A*

+

29(ln+l +In_ I)

(DT)+ I +

.J

q o (DT)t'lo

q = ln+l·ln-1

lri

+A;

ln+l + ln-1 In+ lA; + ln-l..:tri 5.5

For water as a solvent:

5.6

For methanol as a solvent:

5.7

To compare the effects in different solvents for diffe-rent charges of the cations, we calculated the effective mobility according to this expression for monovalent and divalent cations in water and methanol, for a hypothetical

absolute mobility of 50.10-5cm2/Vs at a concentration of

0.01 N. The results are shown in table 5.1. Those effects are even stronger for solvents with smaller dielectric con-stants and for cations with higher charges.

TABLE 5.1 Theoretical effective mobilities of mono- and di-valent cations in water and methanol (95,% b.w.).

1-1 2-1 50 50 Water 5 meff" 10 46 43 5.2.2 Partial dissociation. Methanol 50 50 37.5 25

Two main types of interactions can be distinghuished, protolysis and complex formation.

PPotoZysis.

A proton takes part in the dissociation reaction. The degree of dissociation depend's on the pH and the equilibrium constant, e.g.:

5.8

K (pK 4. 75) 5.9

CompZe~ foPmation.

Now a particle different from a proton takes part in the

The degree of complex formation depends mainly on the partial concentrations. Sometimes however, both types affect

3+

the mobility such as for Al :

---

-

Al ( OH) ( H 0) 2+

2 5

1)

If the value of the dielectric constant decreases, the interionic forces increase. This results especially for ca-tions with a higher charge in a stronger complex formation. Therefore the pK values of the dissociation depend on the dielectric constant.

5.2.3 Solvation.

To describe the exact effect of the solvation is difficult. In general, ions with large radii and a low charge have a

small degree of solvation, whereas highly charged ions with

small radii have a large degree of solvation~ In general, ions

with a large degree of solvation have a small mobility.

In water and methanol, the mobilities of the alkali metal ions decrease in the sequence Cs+:>Rb+:>K+:>Na+:>Li+, i.e. in the order of their decreasing radii. The differences between the mobilities however, seem to be favoured in methanol. In water Cs+, Rb+ and K+ ions are very difficult to separate, while in methanol the differences in mobility are such that these cations can be separated easily. A similar effect shows

also the series of

J-,

Br-, Cl- and F-. Also the mobilities

2+ 2+ 2+ 2+ •

of Ba , Sr , Ca and Mg ions diminish 1n the order of

their decreasing-radii.

1)

Organic cations often have high mobilities in methanol. Even the large cation Tba+ has a rather high mobility, about equal to the mobility of the very small cation Li+. This indicates that the Tba ion is hardly solvated, probably due to the screening effect of the groups surrounding the charge. In water, however, the Tba+ ion has a rather low mobility. The cation Tma+ has the highest mobility in methanol, except for H+.

When the absolute ionic mobility is ~nown and when for the

influence of the degree of dissociation and electrophoretic and relaxation effects can be corrected, the effective mobility can be computed. As the exact data for many ionic species are unknowl many authors have looked for correlations between the ionic

mo-bilities and parame~ers such as the radius of the molecule, the

ionic volume and the entropy of the ions. Some of those approaches will be discussed.

5.2.4 The relationship between entropy and ionic mobility.

E.K. ZOLOTAREv68 has tried to relate the entropy to the

ionic mobility in aqueous solutions. Combination of the by

KAPUSTINSKIIS 103 derived formulae:

s

A/rw + B 5.13 and + m

₌

n:-e

I

6. 1r .n 0• rw gives:

s

k 1.m + k2