Ion-Selective Electrodes for Biological Systems

ION-SELECTIVE ELECTRODES FOR BIOLOGICAL SYSTEMS

ION-SELECTIVE ELECTRODES FOR BIOLOGICAL SYSTEMS Christopher H.Fry and

Stephen E.M.Langley Institute of Urology and Nephrology University College London UK

harwood academic publishers Australia • Canada • France • Germany • India • Japan • Luxembourg Malaysia • The Netherlands • Russia • Singapore • Switzerland

This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to http://www.ebookstore.tandf.co.uk/.” Copyright © 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Harwood Academic Publishers imprint, part of The Gordon and Breach Publishing Group. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. British Library Cataloguing in Publication Data Ion-selective electrodes for biological systems 1. Electrodes, Ion selective I. Fry, C.H. II. Langley, S.E.M. 541.3′724 ISBN 0-203-30474-8 Master e-book ISBN

ISBN 9058231070 (Print Edition)

TABLE OF CONTENTS PREFACE

1. Essential physical chemistry of solutions

V 1

2. Electrochemical cells

16

3. Principles of ion-selective electrodes

25

4. Manufacture of dip-cast ion-selective electrodes

31

5. Ion-selective micro-electrodes

41

6. Calibration of ion-selective electrodes

48

7. The presentation of data obtained with ion-selective electrodes

69

8. Instrumentation

81

9. Applications of ion-selective electrodes

94

APPENDICES

102

Appendix 1 The Nikolsky equation and selectivity coefficients

102

Appendix 2 Necessary mathematical tricks

106

Appendix 3 Recommended nomenclature for use with ion-selective electrodes, 114 the Systeme Internationale (S.I.) and standard values for selected constants Appendix 4 Other ion-selective electrode configurations 124 Appendix 5 Partial pressures and the determination of O2 and CO2 in solution

131

Appendix 6 Addresses of manufacturers and suppliers in the UK for raw materials and components described in the text Appendix 7 References

138

Index

140 145

PREFACE Accurate measurement of the concentration of different ions in biological fluids is of central importance to most areas of research. As a consequence a number of techniques have arisen, each of which has advantages in different conditions. Ion-selective electrodes are devices which generate a potential difference in response to a change in the concentration of one or several ions. The situations in which they can be used depend only on the ability of the experimenter to place such an electrode in the fluid compartment of interest, and to record and interpret the resulting signal. Therefore they have been used in compartments varying from the cytoplasm of a cell, to extracellular fluids in vivo, to bulk solutions on the laboratory bench, such as urine and plasma samples. The determination of ion concentrations with ion-selective electrodes has a long history; including the recognition that glass/electrolyte interfaces have ion-selective properties, the realisation that biological membranes themselves have ion-selective properties and the subsequent development of artificial ion-selective membranes by the synthesis of ion-selective carrier molecules. The physico-chemical basis of the subject owes its origin to the pioneers of the ionic theory of electrolytes in the 19th and early 20th century whose work is summarised in the early part of this book. The development of ion-selective electrodes themselves was dependent on the synthesis of selective carrier molecules which would function in an artificial membrane. Many have been thus involved, but the great variety of compounds currently available owes an invaluable contribution to Wilhelm Simon and his colleagues working at the technical polytechnic (ETH) in Zürich. The motivation for writing such a book is that the measurement of ion concentrations is of core importance to most research workers, even if it is only to ensure that the solutions which they use for their experiments contain what they ought to. However, many such researchers do not come across a formal treatment of how to measure ion concentrations in their early careers and as a consequence misconceptions of how to use electrochemical devices and incorrect interpretation of data can occur. We have tried wherever possible to guide the reader as to the correct approach in how to use ionselective electrodes properly, to avoid errors in the incorrect use of equipment and ancillary electrodes and how to interpret the data once it has been gathered. We are of course open to your suggestions and advice if we have failed in any of these objectives. Any such record of personal practical experience relies on the wisdom of others. In particular we would like to express our warmest appreciation to David Band and John McGuigan. They have taught us, tolerated our questions, given us invaluable advice, loaned (!) us chemicals when we have run out and brought us up in an environment of saturated KCl and silver wires.

1. ESSENTIAL PHYSICAL CHEMISTRY OF SOLUTIONS 1.1. SOLUTIONS—CONCENTRATION AND THE MOLE Many biological systems are concerned with the action of a chemical substance upon a particular process. Furthermore, these chemicals (solutes) are often dispersed in a suitable solvent such as water to form a solution. To quantify the actions of such solutes the amount in solution needs to be specified in a way which allows comparison between different solutions, and the most useful concept in this context is the mole. One mole of any substance contains the same number of particles (atoms, molecules, etc.). To define this number a ‘gold standard’ is required which is at present the common isotope of carbon, 12C. Therefore one mole of a substance contains the same number of particles as there are atoms in exactly 0.012 kilogrammes (12g) of 12C; this number, called Avogadro’s constant, NA, is about 6.022×1023. The molecular weight, Mr, of a substance is the mass of the molecule, m, relative to the mass of 12C, which is taken to be exactly 12; i.e.: (1.1) The atomic weight, Am, can be defined in an analogous way. An associated, more modern term is the molar mass, Mm, which is the mass per unit amount of substance (i.e. the mass per mole of specified particles) and in practice Mr is the numerical value of Mm in g.mol−1. Example The molecular weights, Mr, of NaCl, KCl and CaCl2 are 58.44 g.mol−1, 74.55g.mol−1 and 110.98g.mol−1 respectively—the atomic weights, Am, of Na, K, Ca and Cl are 22.99, 39.10/40.08, and 35.45 respectively. Note: atomic weights are not whole numbers due to different proportions of various isotopes. Some atoms such as Na contain a preponderance of one isotope, i.e. 23Na, whereas other atoms such as Cl contain a more general mixture of isotopes, i.e. ≈75.5%35Cl; ≈24.4%37Cl. Many chemicals, especially in a crystalline form, have water molecules associated with them and when bought from suppliers the label will specify the number of water molecules. For example, crystalline CaCl2 is often supplied as the hexahydrate, CaCl2.6H2O, so that 1 mole will be contained in 110.98 +6.(18.016)g, i.e. 219.08g.

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2

When making a solution a quantity of solute is dissolved in a given volume of solvent. In this case the solution has a molar concentration and is associated with the symbol, c. However, because the density, and hence volume, of a solvent varies with temperature so also will the molarity vary. In order to minimise this problem an alternative term is used in which a given quantity of solute is dissolved in a given weight of solvent, this is the molal concentration, denoted by the letter, m. Most theoretical considerations use molalities to express concentration values and this will be used in Chapter 1, but for the majority of dilute aqueous solutions the two terms are virtually equivalent. In the subsequent, more practical parts of this book, concentrations will be quoted in molar quantities at room temperature (≈18°C). The Systeme Internationale (S.I.) unit of molality (appendix 3), m, is moles.kg−1 solvent and of molarity, c, is moles.m−3 solvent. The units of molarity used here are derivative units namely moles per dm3 of solvent (mol.l−1, =mol.dm−3).Thus an aqueous solution of 1 molal NaCl is 58.44g per kg water and here a solution of 1 molar NaCl is 58.44g per dm−3 of water. Note: The word litre is a special non-S.I. unit name for the cubic decimetre, but the word should never be used to express high precision. The addition of an S.I. prefix, as in dl, cl, ml, etc. is a step away from the S.I., and the use of the term ‘c.c.’ is only slang. However, in harmony with general laboratory practise, but with this caveat in mind, the term ‘litre’ (1) and its derivative units will be used throughout this book.

1.2. OTHER UNITS OF CONCENTRATION The mole is the most generally useful and standardised way of expressing concentration. However, there are a number of other concentration units still in common usage and although they will not generally be used here it is useful to introduce them so that the reader can pass easily from one system to the other. 1.2.1. Gram equivalents and normal solutions This term is often, but not exclusively, used with acids and alkalis and is useful when considering chemical reactions. The gram equivalent is the weight in grams which contains one gram-mole (i.e. Avogadro’s number of particles) of the ion or atom involved in a chemical reaction. For example, with the neutralisation of a base by an acid, the gram-equivalent weight of acid contains one gram-molecular weight of hydrogen ions (i.e. 1.008g), it does not matter what is the source of the H+. For HCl, one mole of the acid will contain one gram-equivalent of H+, assuming it dissociates completely into H+ and Cl− (see section 1.4), whereas one mole of H2SO4 contains two gram-equivalents. A normal (N) solution by definition contains one-gram equivalent of the reactant. Therefore, a 1 molar (1M) solution of HCl and H2SO4 contains 36.46g and 98.08g respectively, whereas a 1 normal (1N) solution contains 36.36 and 49.04g respectively.

Essential physical chemistry of solutions

3

1.2.2. Electrical equivalents Many solutes (salts) dissociate in solution to their constituent charged particles, ions. Thus 1 mole of NaCl ionises to 1 mole of Na+ and 1 mole of Cl− and both ions have one unit of associated charge, i.e. they have a valency (z) of 1. One electrical equivalent (eq.) is 1 mole of the ion divided by its valency, thus for 1 mole of NaCl there is one equivalent of Na+ (22.99g) and one equivalent of Cl−(35.45g). In the case of Ca2+, 1 mole (40.08 gram-moles) will contain two electrical equivalents so 1 eq. of Ca2+ is 20.04g. 1.2.3. Percentage solutions These rather old-fashioned terms, still often used in clinical circles, give no information about the molar concentration. A solution is 1% weight-for-volume (w/v) when it contains 1g solute per 100ml solution. A 1% weight-for-weight (w/w) solution is 1g solute added to 99g solvent. For example, a nurse may be requested to make up a 0.9% saline solution, what does (s)he do? Add 0.9g NaCl to 100ml water (w/v); this is equivalent to 9.0/58.44 moles.l−1, i.e. about 154 mmol.l−1 NaCl. This solution is roughly isotonic with body fluids (section 1.3) and so will not cause excessive movements of water into or out of cells if administered to a patient. Closely allied is the term parts per million (ppm), a solution which contains 1 mg of solute per litre of solution. The concentration of solutes in biological systems is often very much less than molar so that a system is necessary to express small numbers in a convenient way. Table 1.1 shows the standard S.I. prefixes for numbers very much smaller (and for completeness greater) than unity.

Table 1.1. The S.I. prefixes and symbols fraction −1

10 10 10 10 10 10 10

−2 −3 −6 −9 −12 −15

10−18

S.I. prefix

symbol

multiple

S.I. prefix

symbol

deci

d

10

centi milli micro nano

c m µ n

pico

p

femto

f

atto

a

deca

da

2

hecto

h

3

kilo

k

6

mega

M

10

9

giga

G

10

12

tera

T

10

10 10

Example Find the molarity, normality and molality of an aqueous 3.0% (w/w) H2SO4 solution? The solution density is 1.020g.cm−3 (=1,020g.l−1) and the molecular weight of H2SO4 is 98.08.

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4

Answer: A 3.0% (w/w) solution is 3.0g H2SO4 in 97gm H2O. From the density of H2SO4, one litre of the solution contains: 30.60g.l−1H2S04. • Molarity is moles per litre. Molarity is: 312mmol.l−1 1 gram-mole of H2SO4 contains 2 gram-equivalents of H+. Thus the equivalent weight is 49.04g.

• Normality is equivalents per litre. Normality is: 0.664equiv.l−1=0.664N • Molality is moles per 1000g solvent. The solution contains 3g H2SO4 in 97g solvent Thus:

=0.315 mol.kg−1solvent.

1.3. OSMOSIS—ISOSMOTIC AND ISOTONIC SOLUTIONS Although not directly relevant to the subject of this book, the concept of isosmotic and isotonic solutions are of fundamental importance in biological systems. Thus reference will be made to them throughout, which requires a brief explanation of these terms. Figure 1 shows an idealised osmotic system consisting of two chambers, one containing solvent only, say water, and the other a solution of a solute in water. The chambers are separated by a semi-permeable membrane which allows water to cross but is impermeable to solutes. Addition of a solute reduces the concentration of water and as a consequence water will flow down its concentration gradient, i.e. from the chamber with no solute (high solvent concentration) to the chamber with high solute concentration. The magnitude of the water flux depends only on the number of solute particles in the solution, n, and not on any physical property such as size, charge, etc., i.e. it is a colligative property of the solution. Application of a physical pressure to the solution will counterbalance the water movement so that the concept of an osmotic pressure, Π, (units, Pa) has developed. For most solutions encountered in biological systems, the magnitude of Π is given by the formula: (1.2.) where R is the gas constant (8.314 51J.K−1.mol−1), T the absolute temperature and V the volume of solution. For a simple solution such as glucose the ratio n/V is the

Essential physical chemistry of solutions

5

Figure 1.1. Schematic representation of osmosis. Water flow is from left to right through the semi-permeable membrane due to the presence of impermeable solutes in the righthand chamber. A physical pressure on the right-hand chamber will counteract the osmotic flux of water. concentration of the solute. Thus 1 mole of glucose will provide 1 osmole of particles to the solution. However, in following sections it is seen that a salt such as NaCl dissociates into two ions, Na+ and Cl− so that 1 mole of NaCl would provide 2 osmoles to the solution if the membrane was impermeable to the two ions. (The latter situation is a little more complex as the effective concentration of ions in solution is less than the number of electrical equivalents due to interaction between ions—see section 1.4). Two solutions which exert the same osmotic pressure with respect to each other are termed isosmotic. For two different solutions, the one exerting a smaller osmotic pressure is hypo-osmotic with respect to the more concentrated, hyperosmotic solution. The term tonicity is a more empirical, biological term which describes the behaviour of cells when placed in solutions of varying osmolarity. If a cell is placed in a solution which neither swells or shrinks, the fluid is isotonic. If the cell shrinks it is a hypertonic fluid as water is osmotically drawn from the cell, if the cells swells it is called a hypotonic solution. Normally, the intracellular fluid of cells is isotonic, as cells neither shrink or swell. In general, an isotonic fluid has an osmotic pressure which will be approximately the same as an isosmotic solution, but this may not always be the case. For example a cell may be induced to accumulate one of the components of an isosmotic

Ion-selective electrodes for biological systems

6

solution (e.g. glucose in the presence of insulin) in which case that component no longer behaves as a pure osmolyte. Examples Plasma normally has an osmolar concentration of 290 mosmol.l−1 Is a 0.9% (w/v) NaCl approximately isotonic? A 0.9% (w/v) solution is 0.9g NaCl dissolved in 100ml water. The molecular weight of NaCl is 58.44, i.e. a 1 mol.l−1 solution contains 58.44g NaCl per litre of water. A 0.9% solution is equivalent to 9g NaCl per litre or a concentration of 9/58.44 mol.l−1= 154 mmol.l−1. If NaCl dissociates completely into two ions this will yield 308 mosmol.l−1, near to plasma osmolarity. Is a 5% (w/v) glucose solution isotonic? The molecular weight of glucose (C6H12O6) is 180.16 so a 5% solution has a concentration of 278 mmol.l−1. Glucose does not dissociate significantly in solution so this concentration is approximately isotonic when first infused into the body. However, glucose eventually is taken up by cells and metabolised so the extracellular concentration falls yielding at later times a hypotonic solution.

1.4. IONIC SOLUTIONS—THE NEED FOR SO MUCH DETAIL When a salt is added to a suitable solvent, such as water, the solute components tend to dissociate into its constituent ions. Towards the end of the 19th century Arrhenius proposed that an equilibrium existed between undissociated solute molecules and ions which arose from electrolytic dissociation. For strong acids and bases, and salts such as NaCl and KCl this dissociation is almost complete. However, discrepancies between experimental data and the Arrhenius theory were found. It was concluded that strong electrolytes completely dissociate into their component ions and any deviation from an ideal behaviour, in which the degree of dissociation was apparently less than complete, could be ascribed to electrical interaction of ions in solution. Such deviations were anticipated to be more profound in concentrated solutions and for polyvalent ions. This review of the Arrhenius theory enabled Debye and Hückel in the 1920’s to devise a theory which is the basis for the modem understanding of electrolytes. A further problem with real solutions is a tendency for the dissociation to be incomplete, and for some degree of ion association to occur. This formation of ion-pairs would again tend to be greater at high concentrations. However, this is not a large problem when the solvent has a high dielectric constant, such as water (78.54 at 25°C; 74.02 at 38°C), and this situation, first formalised by Bjerrum, will not be considered further here. The rest of the chapter will go into some detail about the physical chemistry of ionic (electrolytic) solutions. But why do we need to go into such detail? The over-riding importance in this book is to remember that ion-selective electrodes measure the concentration of free ions in solution—the activity of a particular ion (see section 1.5.).

Essential physical chemistry of solutions

7

The relationship between ion activity and the total concentration in solution is what these sections will consider. Many other analytical techniques such as flame photometry or atomic absorption spectroscopy, measure the total amount of ion in solution—freely ionised and bound. However, it is probably only the freely ionised fraction which has biological activity in most cases. If the ionised fraction, varies at total concentration, this will influence the biological substrate and this change will also be measured by the ion-selective electrode. It is important therefore that we understand what alters the ionic activity in solution, to understand better the biologically-active agent and also to avoid mis-interpretation of the ion-selective electrode signals.

1.5. IONIC ACTIVITY The deviation of the behaviour of a solute particle from an ideal state—in this case a solute behaving simply as a completely dissociated system according to the Arrhenius theory—can be expressed in terms of the solute activity, denoted by the letter a. This term was originally introduced by Lewis in the 1920’s to describe the deviation in behaviour of a gas, liquid or solid state from an idealised pressure state, or fugacity. Whereas the concentration can be regarded as the number of ions present, the activity can be thought of as the availability of the species to take part in chemical reactions and to determine the properties of the particular phase. In the case of an electrolyte solution the activity of the solute, ai, will be related to the concentration, mi as: ai=γimi (1.3.) where γi is a proportionality constant, the activity coefficient. The activity, a, of an electrolyte is a property that can be measured by experiment, e.g. by measuring colligative properties such as osmotic pressure or freezing-point depression of the solution. It is not possible to measure the activity of a single ion in solution alone, because the activity of this will depend on the nature of the counter-ion to preserve overall electroneutrality in the solution, i.e. the activity of Na+ will depend on the nature of the . Therefore it is convenient to define a term for the activity of anion, such as Cl− or an electrolyte in terms of the ions into which it dissociates. A solute such as KCl will dissociate as KCl K++ Cl−. If the activity of the cation and anion is a+ and a− respectively then the mean activity, a±, of KCl is the geometric mean of a+ and a−; (1.4.) In general for an electrolyte that dissociates into v ions -v+ cations and v_ anions (v=v++v−). (1.5.)

Ion-selective electrodes for biological systems

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1.6. IONIC STRENGTH OF ELECTROLYTE SOLUTIONS The deviation from an ideal Arrhenius state results from the interaction between the component ions in solution. One factor which will influence this interaction is the charge on a particular ion—the force of which depends on the charge of a particular ion and one useful function in this context is that of ionic strength, I; (1.6.) where mi (or ci) and zi are the concentration and valency of each ion in the electrolyte solution. The approximation holds for dilute solutions, where ρo is the density of the solvent. In the context of aqueous solutions ρo is near unity and is usually ignored. For example, the ionic strength of 0.1mol.kg−1 (100mmol.kg−1) KCl is;

the ionic strength of 0.1mol.kg−1 (100mmol.kg−1) CaCl2 is;

1.7. THE DEBYE-HÜCKEL EQUATION An analytical approach to factors influencing activity coefficients is given by the DebyeHückel theory for ionic solutions. Electrolyte solutions are non-ideal because of the powerful long-range forces between ions. For two neutral molecules interactive forces fall off as a function of separation, r, approximately by 1/r6. However, electrostatic interactions decline only as a function of 1/r. Thus electrostatic forces are considered to be the predominant forces upon ions in solution. The Debye-Hückel approach assumes a number of conditions, and although many of these assumptions can be questioned the theory is at least adequate to describe the behaviour of ions in biological fluids. These assumptions include: • there is complete dissociation of electrolytes; • deviations from ideal behaviour of ions in solution are due to inter-ionic, coulombic (charged) attractions between ions; • the relative permittivity, εr, of the solution is assumed to be equivalent to that of the solvent alone; • the energies of inter-ionic attraction are small compared to the thermal kinetic energy; • ions are considered to behave as solid charged objects. The basic precept of the theory is that because of attraction between positive and negative ions, there are on average in the vicinity of a particular ion more ions of opposite sign and this will reduce the ‘effective concentration of ions’. Consequently if the solution is diluted the separation of ions involves doing an additional work to overcome these inter-

Essential physical chemistry of solutions

9

ionic attractions, and this represents the deviation from an ideal solution. Debye-Hückel theory generates a relationship between the mean activity coefficient and ionic activity, I, by: (1.7.) The term |z+z−| is a valency factor and emphasises the fact that it is only a mean activity coefficient, γ±, which can be determined rather than single ion activity coefficients. This valency factor is the modulus of the product of the valency of individual ions comprising the electrolyte and values of some electrolytes are listed in table 1.2. The other terms in the equation are constants, or combinations of constants which apply to particular values of temperature and pressure, i.e. where εr is the solvent relative permittivity and T is theabsolute temperature, °K (0°C=273.15°K).

Table 1.2. Valency factors for different electrolytes. Salt

Ionic charges

KCl

+

−

+

−

+

−

+

−

z =1, z =1

CaCl2

z =2, z =1 z =2, z =2

CaSO4 La2(SO4)3

z =3, z =2

Valency factor 1 2 4 6

a=the mean ionic diameter—the distance of closest approach of ions, the value is quoted in nm (nanometres). For water:

A has values of 0.5092 and B has values of 3.2872 and

at 25° and 38°C respectively at 25° and 38°C respectively

Notes in many older texts the value of a is the mean ionic diameter in Ångström units, Å(=10−10m, 10−8cm) and would be denoted as å in equation 5; in this case

Ion-selective electrodes for biological systems

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The term relative permittivity, εr, is a dimensionless constant which defines the ratio εr=ε/ε0, where ε and ε0 are the permitti vities of the solvent and of a vacuum. In some older texts the relative permittivity is called the dielectric constant and may be denoted by the symbol D. The value of a is difficult to calculate and for many instances when equation 5 is used it is assumed to have a value of 0.4nm (4Å). This value of a is greater than the sum of the ionic radii of two monovalent ions such as K+ (0.13nm) and Cl− (0.18nm) and presumes that some function of the hydrated ionic radii is the correct distance. Thus, although the value of a will vary for different electrolytes, and at various concentrations, the convention of a constant value is used unless alternative data are available. The value of εr is temperature dependent, ranging for water from 88.15 at 0°C to 55.90 at 100°C. Values at intermediate temperatures can be calculated from the empirical relationship: εr=α.exp(−βT), where T is the absolute temperature, α=304.9 and β=4.548×10−3. Example Use equation 1.7 to calculate the activity coefficient, γ±, of 200mmol.l−1 NaCl solution at 25°C Answer: The activity coefficient is given by: . At 25°C the values of A=0.5092 and B=3.286 from the above formulae, using a value for εr=78.54. The ionic strength, I,=0,20, the valency factor |z+z−|=1 and α=0.4 (remember the inserted number is the value in nm). Thus:

The value for γ± quoted in Robinson & Stokes (1955) is 0.735. The discrepancy may we lie in an imperfect guess at the value a: if a value of 0.5 nm is used a value of γ±=0.739 is calculated. However the example illustrates the difficulty in using calculated values of γ± and when possible experimentally derived values should be used. Two particular situations need to be considered: • dilute solutions • concentrated solutions. For dilute solutions—i.e. of low ionic strength—the denominator of equation 1.7 approaches unity so that it can be approximated to the so-called ‘Debye-Hückel limiting law’ (1.8.)

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11

At very high electrolyte concentrations equation 1.7 is unable to describe accurately the value of the mean activity coefficient. Hückel showed that the electrolyte produces a change of relative permittivity of the solvent in the vicinity of the ion. He assumed that the relative permittivity was a linear function of concentration so that an additional term, CI, was added to generate equation 1.9. C is an empirical parameter which is derived from experimental measurements of activity coefficients for various electrolyte solutions. (1.9.)

1.8. VARIABILITY OF ACTIVITY COEFFICIENTS Figure 1.2 shows the variation at 25°C of the mean activity coefficient, γ±, of some monovalent and divalent cations in combination with various anions. In parts A and B the values are plotted as a function of the concentration of the pure salt; in part C some of these data are plotted as a function of ionic strength. Mean activity coefficients are reduced as the concentration or ionic strength increases up to a value of about 1 mol.kg−1, but thereafter they increase sharply. The ionic strength of physiological fluids is about 0.15 mol.kg−1, the range where activity coefficients decline as concentration or ionic strength increases. In general, mean activity coefficients of divalent ions are smaller than those of monovalent ions, and results from the greater forces existing between ions which have a higher degree of ionisation. Note also that the concept of a single ion activity coefficient is not meaningful. The figures show that different K+ salts, such as KCl, KOH and K2SO4, and Na+ salts such as NaCl, NaOH and Na2SO4 have different γ± values, so that the particular counter-ion influences profoundly the mean activity coefficient. At physiological ionic strengths the γ± of KCl and NaCl are similar, i.e. about 0.75 and 0.74 respectively at 25°C, much greater than those of CaCl2 and MgCl2, 0.49 and 0.54 respectively. The increase of γ± values which necessitated the introduction of the final term CI in equation 1.8 is evident at ionic strengths which are greater than encountered in most biological fluids, except perhaps hyperosmotic urines or seawater environments. Is the original Debye-Hückel equation (equation 1.7) or even the limiting law (equation 1.8) adequate at biological values of ionic strength (about 0.15M)? Figure 1.3 shows a fit of equations 1.7–1.9 to the KCl data from figure 1.2. In this instance the x-axis is √I, which is the variable in equations 1.7–1.9 (and at least for 1:1 electrolytes like KCl, I is equivalent to concentration, c). The data is closest to that given by equation 1.9, the expanded Debye-Hückel equation: however the difference between equations 1.7 and 1.9 is small enough to be ignored, although the limiting law (equation 1.8) is clearly inadequate. The vertical line shows the magnitude of √I equivalent to isotonic fluids.

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Figure 1.2. Part A: the variation of the mean ion activity coefficient, γ±, for several salts of monovalent cations at 25°C as a function of solute molal concentration (moles of solute per kilogram of solvent). Part B; variation of γ± for several salts of divalent cations and anions at 25°C. Part C; variation of the mean ion activity coefficient for several salts of

Essential physical chemistry of solutions

13

monovalent and divalent ions at 25°C, as a function of the ionic strength of the solution.

Figure 1.3. The variation of the mean ion activity coefficient for KCl, γ±, as a function of ionic strength as calculated

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14

by the Debye-Hückel equation (equation 1.7); the limiting law appropriate in dilute solutions (equation 1.8); and the extended form of the equation (equation 1.9). Experimental data is shown as closed squares. The inset shows an expanded form of the main plot at ranges of ionic strength nearer to those found in most biological fluids (√I ≈ 0.4 for isotonic mammalian fluids). 1.9. TEMPERATURE DEPENDENCE OF ACTIVITY COEFFICIENTS Most activity coefficients have been measured at 25 °C which is different from the physiological temperature at which recordings are usually made. Thus if ion-selective electrodes were calibrated at room temperature would this difference alter the activity coefficient sufficiently to introduce an error. Fortunately the variation is very small and can be ignored. For 0.1 molal NaCl solution the γ± value varies from 0.781 to 0.776 between 0°C and 35°C, decreasing to 0.774 and 0.770 at 40° and 50°C. However, it is important to note that many other components of electrochemical cells are more temperature-dependent (chapter 3) and it is always good practise to calibrate at the same temperature at which experimental measurements are made.

1.10. MIXED ELECTROLYTE SOLUTIONS Thus far we have been studying the simple case of single electrolyte solutions, but this is rarely the case in biological systems. The physical chemistry of mixed solutions has been less extensively studied, but an example will exemplify the problem for HCl. At 25°C: γ± for 1.0molal HCl=0.809 γ± for 0.01molal HCl in 1.0molal NaCl=0.754 In other words despite the fact that the ionic strength of the two solutions is virtually the same, the γ± of HCl is reduced by the presence of NaCl. Similarly dilution of a NaCl solution by HCl at constant ionic strength would increase the γ± of NaCl. Thus the overall composition of the solution will also effect individual activity coefficients. This could present problems: for example the composition of calibrating solutions can differ substantially from test solutions with respect to ions other than the one measured by a particular ion-selective electrode. Apart from the fact that these other ions might interfere with the ion-selective electrode (see chapter 6), they might also change the activity coefficient of the ion of interest.

Essential physical chemistry of solutions

15

A semi-empirical approach has been adopted whereby for two electrolytes, labelled a and b, in a mixture the mean activity coefficients, γ±(a) and γ±(b), are given by: log γ±(a)=log γ±(0,a)+αa,bIa; log γ±(b)=log γ±(0,a)+αb,aIb 1.10) γ±(0,a) and γ±(0,b) are the activity coefficients of pure solutions of a and b respectively, Ia and Ib are the ionic strength of the solutions and the coefficients αa,b and αa,b are experimentally derived parameters. Fortunately at ionic strengths encountered in normal biological systems (I=0.15) the values of αa,b and αb,a will introduce only small errors and will be ignored. However, for more concentrated solutions or when there are fractionally large concentrations of complex polyvalent ions the errors will become significant and the reader is referred to advanced books on the physical chemistry of electrolyte solutions.

2. ELECTROCHEMICAL CELLS 2.1. INTRODUCTION The previous chapter has been concerned with the physical chemistry of electrolyte solutions. It is now necessary to consider the situation when electrodes (see section 2.2) are placed in these solutions and connected by conductors, usually of metal. A typical arrangement involving an ion-selective electrode system is shown in figure 2.1. An ion-selective electrode can be considered as a layer separating two electrolyte solutions, a test solution and a filling solution of known composition.

Figure 2.1. A schematic diagram of an electrochemical cell containing an ionselective electrode and a reference electrode. A potential will be generated at the interface between the test solution and the ionselective electrode, the magnitude of which will depend upon the electrolyte composition of the test solution. This potential cannot however be measured in isolation but only with respect to another, stable potential as a potential difference (p.d.). The second, stable potential is generated at a so-called reference electrode also placed in the test electrolyte

Electrochemical cells

17

solution to complete the circuit. An example of a reference system is a metal, often Ag, coated with AgCl and placed in a solution of strong electrolyte such as 3 mol.l−1 KCl. The p.d. can be measured by a suitable voltmeter. Connexion of the ion-selective electrode to the voltmeter can also be achieved via a AgCl coated Ag electrode. At each interface in this system a potential, Vi, will develop and the sum of these potentials, p.d., will be recorded by the voltmeter. An electrochemical equivalent of the cell in figure 2.1 is shown below and the recorded p.d. is

Each slash (/) represents a junction, and hence a place where a potential develops. The potential at the ion-selective membrane/test solution interface, V4, is the one of interest and which is a function of the electrolyte composition of the test solution. V1 and V7 are potentials between the Ag wire and AgCl coating; V2 and V6 are potentials between the AgCl coating and the filling solutions of the reference and ion-selective electrodes; V3 is a liquid junction potential between the reference electrode electrolyte and the test solution. All these other potentials must remain constant otherwise changes may be interpreted as an alteration of V4. It is important to appreciate the nature of all interfaces in this system so as to ensure that, except for V4, they do remain constant. It cannot be emphasised too much that it is vital for the user of ion-selective electrodes to give proper attention to the stability of these remaining junctions. Failure of the ionselective electrode system to respond in a predictable manner is more often than not due to variable potentials at the reference electrode. For this reason reference electrodes have been investigated extensively by many authors in an attempt to understand the origin of the various potentials and no complete theoretical and practical solution has yet been devised to cover all circumstances. The next two sections provide an overview of the more central theoretical and practical aspects in the context of ion-selective measurements in approximately isotonic biological fluids—the references provide access to more detailed accounts.

2.2. REFERENCE ELECTRODES An electrode is a component of an electrochemical cell where charge is carried by movement of electrons. They can be made of a metal or semiconductor, solid or liquid. In general, metal electrodes are used in biological systems and when such a metal electrode is placed in an electrolyte solution a potential may develop at the interface. The value of this potential will depend upon the nature of the electrode and the solution in which it is immersed and an ideal would be a system in which the passage of a current through the electrode causes no potential change at the interface, a nonpolarisable electrode. A standard, reference electrode has been designated, which by definition has a zero potential at all temperatures. This is the standard hydrogen electrode (SHE) whereby H2 gas at one atmosphere is bubbled over a platinum black foil and thus enables H2 gas to be in equilibrium with H+ in solution. If another, secondary reference electrode was placed in the same solution, a potential difference between the two would develop. For example,

Ion-selective electrodes for biological systems

18

a Ag electrode is positive by 0.7996V at 25°C. However, the SHE is impractical in most biological situations and for that reason secondary reference electrodes are commonly used. Although these secondary electrodes do have a potential at their surface with the aqueous solution it must be stressed that the only requirement is that this potential is unchanging. What is measured in the potential difference (p.d.) between the ion-selective and reference electrodes, and this p.d. should be a function only of the potential between the ion-selective electrode interface with the test solution. The need to measure the actual value of a secondary electrode potential value relative to the SHE arises rarely in biological systems. It may be that the stability of a secondary electrode potential needs to checked in a particular environment. When this is required the use of the SHE is described in books of practical electrochemistry. The most commonly used secondary reference electrode in biological measurements is the AgCl coated Ag wire—the Ag/AgCl electrode—and is shown in figure 2.1. Ag/AgCl electrodes are easy to manufacture (see section 4.7) and the electrode potential has a relatively small temperature coefficient. If the Ag/ AgCl electrode is placed in saturated KCl it is written as Ag/AgCl (solid)/KCl (saturated, aqueous). Other non-polarisable electrodes can also been used, such as a Pt black electrode where PtCl4 is coated onto platinum metal by a similar electrolytic process as described for the manufacture of Ag/AgCl electrodes (section 4.7) Briefly, a 3% (w/v) chloroplatinic acid (H2PtCl6) solution, containing 0.005% Pb acetate (Kohlrausch’s solution) is used to deposit the PtCl4 onto the platinum metal. These electrodes have slightly better characteristics than Ag/AgCl electrodes but because the raw materials are more expensive they are used less often. Other reference electrodes which are often used are the calomel electrode—Hg/Hg2Cl2/KCl (saturated, aqueous), and when Cl− are unavailable in the ionic solution in which the electrode is placed it is possible to use a mercurous sulphate electrode: Hg/Hg2SO4/K2SO4 (saturated, aqueous). However, it is very important to understand the reactions at the surface of these secondary electrodes with the aqueous solution as this will determine the conditions under which they can be used in biological systems. The reaction at the interface for an Ag/AgCl electrode is: AgCl+e−↔Ag+Cl− Current is carried to the electrode by exchange with the common anion in solution and on the electrode, namely Cl−. Therefore the precise value of the potential at the electrode surface will depend upon the Cl− activity in solution. It is important therefore that the concentration of the common anion remains constant in the test solution if the electrode potential is not to alter. The magnitude of the potential change as a function of the [Cl−] is given by the Nernst equation (section 3.3), and is approximately 61.5mV for a ten-fold change of the [Cl−] at 37°C or 59.1 at 25°C. For this reason it is not recommended that a reference electrode such as a Ag/ AgCl electrode (or Pt black electrode) is in direct contact with the test solution, if the [Cl−] is likely to change. This can happen for example in vitro when the superfusate around a cell is altered and the [Cl−] changes. It can also happen in vivo when the reference electrode is placed in a body fluid compartment where the [Cl−] may change, e.g. with haemoconcentration or in contact with sweat of varying compositions.

Electrochemical cells

19

Example Let the total [Cl−] of a physiological saline solution at 37°C containing 2 mmol.l−1 CaCl2 be 128mmol.l−1. During the experiment the CaCl2 is raised to 10 mmol.l−1 so that the total [Cl−] is now 144 mmol.l−1. From the Nernst equation (section 3.3) a Ag/AgCl electrode potential at the interface would change by about 3.1mV. This is large enough to be measured and would be mis-intcrpreted as a change of potential at the ionselective electrode interfaee if it was assumed that the reference electrodc potential remained constant. A solution to the problem is to have a so-called salt bridge between the metal electrode and the test solution. The composition of the salt bridge is constant so that the electrode potential will not change despite variations to the composition of the test solution. In figure 2.1 the salt bridge consists of the 3 mol.l−1 KCl column on the left-hand side. The salt bridge now makes the contact with the test solution at the interface between the two electrolyte solutions. However, removing one problem generates another as an additional potential will be generated at the interface between two dissimilar electrolyte solutions— a liquid junction potential, denoted by voltage V3 in section 2.1. Minimisation, or at least stabilisation, of this potential is the next task.

2.3. LIQUID JUNCTION POTENTIALS When two solutions come into contact with each other there is diffusion of their component ions from one compartment to another, the magnitude of which is dependent upon the particular components and their concentrations. Because some ions have a greater mobility than others there is a tendency for either cations or anions to move more rapidly, so generating a diffusion or liquid junction potential at the interface. It is important to consider the properties of a solution which determine the liquid junction potential because the composition of salt bridges will vary in different experimental situations. For example, a concentrated solution of KCl would be inadvisable in a small volume, as leakage of KCl might affect the [K+] in the test solution, whereas in a large volume this would be less important. In addition, if the composition of the test solution varied this would affect the value of the liquid junction potential and it will be necessary to calculate the magnitude of the variation. It is convenient to consider three types of liquid junctions; Type 1.

Two electrolyte solutions of the same type, say KCl, at different concentrations.

Type 2.

Two electrolyte solutions at the same concentration but of different salts, say HCl and KCl.

Type 3.

Two electrolyte solutions of different salts and at different concentrations.

Type 1 Junctions

Ion-selective electrodes for biological systems

20

A type 1 junction is more straightforward than the others. When current passes through an electrolytic solution different fractions are carried by the various ions in solution. The fraction of current carried by each ion is called the transport (or transference, t) number. For a solution of KCl let t+ be the (cation) transport number of K+ and t− the (anion) transport number of Cl−. Since the total current carried by the ions is the sum of the cation and anion currents, then: t++t−=1 (2.1a.) and for a solution of many ions, i. (2.1b.) Transport numbers can be evaluated by a variety of methods and the values of several cations in association with different anions are given in table 2.1. Note that the transport number for a given cation, such K+, is not a constant number but depends on the particular anion with which it is associated. For a 1:1 electrolyte, such as KCl, the liquid junction potential, Ej, of a type 1 boundary is given in equation 2.2, where a1 and a2 are the activities of the salt in the two compartments.

Table 2.1. Transport numbers of different cations in water at 25°C in combination with various anions. Concentration, mol.l−1 Electrolyte

0.01

0.05

0.10

0.20

HCl

0.825

0.829

0.831

0.834

KCl

0.490

0.490

0.490

0.489

NaCl

0.392

0.388

0.385

0.382

LiCl

0.329

0.321

0.317

0.311

NH4Cl

0.491

0.491

0.491

0.491

K2SO4

0.483

0.487

0.489

0.491

KNO3

0.508

0.509

0.510

0.512

0.50 0.489

0.287

(2.2.) When t+ and t− are approximately equal the value of Ej will be small. Such so-called equitransference solutions will therefore be useful to minimise the value of a liquid junction potential. Table 2.1 shows that such a condition is true for salts such as KCl, NH4Cl, KNO3 and K2SO4. A further advantage of these solutions is that the transport number is relatively independent of concentration over the range shown above,

Electrochemical cells

21

permitting their use in a variety of conditions where the osmolarity of the test solution may vary widely. Example Calculate the liquid junction potential between 0.01 mol.l−1 and 0.1 mol.l−1 KCl solutions at 25°C. Answer. From table 2.1 the t+ for KCl=0.490. therfore, t=0.510. In (0.1/ 0.01)=2.303, and at 25°C RT/F=25.7mV. The estimated value of Ej≈ 1.2mV. Example. Calculate the liquid junction potential between 0.01 mol.l−1 and 0.1 mol.l−1 HCl solutions at 25°C. Answer. From table 2.1 the t+ for HCl ≈0.83, therefore, t−≈0.17. The estimated value of Ej≈39.1mV. The transport number of an ion depends on two intrinsic properties of an ionic solution which need to be considered when using type 2 and 3 junctions; these are the equivalent conductivity of a solution, Λ, and the mobility of an ion in solution, u. The passage of current through an ionic solution depends upon the speed with which different ions can move and the index of movement is the ionic mobility of the ion, ui with units of cm2.V−1.s−1. The resistance to current flow in a solution is a closely allied term usually expressed as its reciprocal, conductivity κ (units Ω−1.cm−1, or S.cm−1; S=siemen). The conductivity depends on the number of ions in the solution so that the molar conductivity, Λm, is: Λm=κ./c (units, S.cm2.mol−1). Because some ions are multiply charged the equivalent conductivity, Λeq (or simply Λ) can now be introduced, where for an ion, i Λeq=Λm/zi. Type 2 and type 3 junctions These junctions are more complex and the liquid junction potential depends upon the physical junction that is formed, i.e. whether the solution is slowly flowing out of the salt bridge or is static (section 2.4). The usual expression of the liquid junction potential for a type 2/3 junction is given by the Henderson equation (equation 2.3). A number of assumptions are made in the derivation of the equation not least that the concentration terms used are equivalent to ionic activities throughout and that there is a linear concentration profile throughout the junction between the electrolytes. The over-reliance on the Henderson equation to estimate the value of Ej can be criticised and at least in more complex solutions its value is to estimate the magnitude of Ej rather than accurately calculate the value. (2.3.)

where ui, zi and ci are the mobilities, valencies and concentrations of each ionic species, i, in the two solutions. The term |zi| is the modulus of zi, that is the numerical value is always positive, regardless of whether the valency itself is positive or negative, i.e. |zNa|=|zCl|=1.

Ion-selective electrodes for biological systems

22

For a type 2 junction between 1:1 electrolytes, such as KCl and NaCl, ci(1)= ci/(2) and κ(1)/κ(2)=Λ(1)/Λ(2). It can also be shown that Equation 2.3 now simplifies to the so-called Lewis-Sargent relationship (equation 2.4) (2.4) a positive sign refers to a junction with a cation which is common to the two solutions, a negative sign if the anion is common; Λ1 and Λ2 are equivalent conductivities of the two solutions. The values of the ionic mobilities and equivalent conductivities for several electrolytes are given in table 2.2.

Table 2.2. Ionic mobility, u, of various ions and the equivalent conductivity, Λ, of different salts at various molal concentrations (in parenthesis). The unit of mobility here is cm2.s−1V−1 and the unit of equivalent conductivity is 10−4m2.S. mol.l−1. Ion

u

+

H

Na

3.63×10 +

+

K

1/2Ca Cl

2+

− 2−

1/2 SO4 HCO3

−

−3

Salt

Λ (0)

Λ (0.001)

Λ (0.01)

Λ (0.1)

Λ (1) 332.80

HCl

425.95

421.15

411.80

391.13

5.19×10

–4

NaCl

126.39

123.68

118.45

106.69

7.62×10

–4

KCl

149.79

146.88

141.20

128.90

6.17×10

–4

1/2CaCl2

135.77

130.30

120.30

102.41

7.91×10

–4

1/2MgCl2

129.34

124.15

114.49

97.05

8.27×10

–4

1/2Na2SO4

129.80

124.09

112.38

89.94

4.61×10

–4

KHCO3

117.94

115.28

110.03

Example Use equation 2.3 to calculate Ej between 0.01 mol.l−1 KCl and 0.1 mol.l−1 KCl? Answer. zk=1, zCl=−1 and |zk|=|zCl|=1; uk=7.62×10−4 and uCl= 7.91×10−4cm2.V−1.s−1 The left-hand term in the equation is:

The right hand term is: 25.7× In (0.01/0.1)=–59.2mV Therefore Ej=−0.0187×–59.2=1.1mV. This is similar to that obtained using equation 2.2. Example. Use equation 2.4 to calculate Ej, between 0.1mol.l−1 KCl and 0.1mol.l−1 NaCl?

111.90

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Answer. In (ΛKCl/ΛNaCl)=In (141.2/118.45)=In 1.192=0.176 Ej=25.7×0.176≈4.5mV. Example. What is the value of Ej between 0.1 mol.l−1 KCl and a test solution containing 0.1 mol.l−1 NaCl, 0.01 mol.l−1 KCl and 0.001 mol.l−1 CaCl2? Answer. Let solution 1 be 0.1 mol.l−1 KCl and solution 2 be the more complex solution.

Table 2.3. The individual terms in equation 2.3: ci(2) ci(2)−ci(1) |zi|/ui/zi. |zi|ui/zi. |zi|ui. |zi|uici(1) |zi|uici(2) ci(1) zi|ui. −1 −1 −1 −4 mol.l mol.l mol.l ×10 ci(2)−ci(1) ×10−4 ci(2)−ci(1) ×10−4 ×10−4 0.01

0.09

7.619

−0.686

7.619

−0.686

0.7619

0.07619

Na+ 0

0.1

0.1

5.193

0.519

5.193

0.519

0

0.5193

Ca+ 0

0.001

0.001

6.166

0.0062

12.332 0.0062

0

0.0123

Cl

0.112

0.012

7.912

0.095

7.912

0.095

0.7912

0.885

−0.066

1.553

1.493

K+

Σ

0.10

0.10

−0.256

Collection of the terms gives The question remains which is the best bridge solution to use? When leakage from the bridge solution into the test solution is a significant problem then one similar to the test solution should be used and the above calculations will give the experimenter an idea of the absolute magnitude of the liquid junction potential that will be encountered. However, it is important to remember that the absolute magnitude of Ej is not as important as its stability and when leakage is less important, or can be restricted, then concentrated equitransference salt solutions are preferred. One important reason is that their resistance is lower, thus lowering the overall resistance of the system. The most commonly used bridge solutions are concentrated KCl solutions: saturated KCl (4.16 mol.l−1 at 25°C); 3.5 mol.l−1 KCl or 3.0 mol.l−1 KCl. Saturated solutions are the easiest to prepare but the tendency to form crystals inside the reference assembly can reduce the reproducibility of the junction. In some cases a very small drift of this small liquid junction potential may still cause significant errors. For example, a pH electrode drifts over a period of time by 0.6mV when placed in a buffered solution of unchanging pH. However, if the drift is interpreted as a change of pH this would represent alteration of about 0.01 pH units, using the Nernst equation (section 3.4)—a change which could well be significant in a biological experiment and certainly within the sensitivity of a pH electrode. To overcome this possible problem other equitransference solutions have been proposed, including: 1.8 mol.l−1 KCl and 1.8 mol.l−1 KNO3 or 3.0 mol.l−1 KCl and 1.0 mol.l−1 KNO3

Ion-selective electrodes for biological systems

24

When K+ or Cl− containing solutions are a problem concentrated lithium trichloracetate or lithium acetate have also been proposed.

2.4. PRACTICAL ASPECTS OF FORMING A LIQUID JUNCTION Apart from theoretical aspects about the superiority of various liquid junctions it is important to consider practical aspects such as whether excessive leakage of the bridge electrolyte would contaminate the experimental system and the physical robustness of the reference electrode. Some of these aspects have been introduced above. A number of different junctions have been described of which the most widespread include: • the free diffusion junction—the two electrolytes initially form a sharp boundary and then are allowed to freely intermix by diffusion • the flowing junction—an upward flux of a dense electrolyte meets the downward flux of a less dense electrolyte The flowing junction has been found to be a more reproducible and stable junction but it is relatively impractical in most situations and so has not been commonly used. However, if the total range of potentials generated by an ion-selective electrode in different test solutions is relatively small, i.e.