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

Electrochemical cells

23

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. <10mV then absolute stability of the reference junction is vital and such a junction should be given serious consideration. An example would be the estimation of the variability of [Na+] in blood plasma, where even if it varied between extremes of 120 and 160 mmol.l−1 would generate <8mV. The free diffusion junction is more widespread as it is easier to construct. When excessive leakage of the bridge electrolyte into the experimental system is undesirable it can be constrained by immobilising it in an agar matrix or interposing a ceramic plug to restrict movement of the electrolyte. The latter method will be described here (section 4.6) and has been used in many commercial applications as it also forms a mechanically robust configuration. However, it is important that ceramic plug junctions are maintained to ensure a relatively constant liquid junction.

3. PRINCIPLES OF ION-SELECTIVE ELECTRODES 3.1. TYPES OF ION-SELECTIVE MATERIALS A useful ion-selective material includes any substance that alters physical or chemical properties in a quantitative, reversible fashion when it interacts with an ion. In addition it should usually display selective properties to a particular ionic species. Many ionselective materials are diffusible and thus are particularly useful in confined spaces such as the intracellular compartments of cells. Examples of such ion-selective materials include compounds which change their absorbance or fluorescence properties when an ion is bound. Fluorochromes in particular have found a widespread current use in measuring intracellular ion concentrations as they respond rapidly to ion concentration changes and are relatively easy to introduce into cells. They suffer however from the disadvantage that their distribution within the cell is uncertain and their accurate calibration can often be difficult. Other examples include molecules which emit light when they bind particular ions. The protein aequorin is an example of a family of proteins which emit a photon when they bind Ca2+. These photoproteins rapidly respond to changes of the ion concentration but are also difficult to calibrate and, due to their large size, cannot be introduced easily into the intracellular compartment. Such ion-sensing techniques are not the subject of this book and adequate reviews of their practical use are available (e.g. Mason 1993). Other ion-selective materials can be confined more readily to a solid or liquid phase (a membrane). They generate a potential in the membrane when they reversibly bind to an ion in the external phase—the test solution—and transport it across the membrane. Such constructions constitute an ion-selective electrode (ISE) and are known as potentiometric systems. In general there are three basic types of selective electrode: • Glass membranes. By varying the composition of glass it can be made sensitive to H+, Na+ and K+. H+ -sensitive glass is widely used today and glass pH electrodes are described in section 6.6. • Solid inorganic salt membranes. The membrane is an ionic solid with very low solubility to prevent it dissolving in the test solution. The ion to which it is sensitive absorbs onto the surface and conducts through defects in the crystal lattice generating a potential. An example is the AgCl-coated Ag wire (section 2.2) which serves as a Cl− -sensitive system. The relationship between ion activity and potential is complicated as the electrode can respond to both the cation and anion. A principal use is in the detection of metal ions or some anions which are difficult to detect by other electrodes. For example Ag+ with a Ag2S solid, Pb2+ with a PbS-Ag2S solid and Br−

Ion-selective electrodes for biological systems

26

with a AgBr surface. They will be considered briefly in appendix 4, but the theoretical basis of their action is similar to the third type of membrane: • Membranes based on ion-exchange. With these membranes the ion species to be sensed crosses the membrane from one side to the other with the help of a compound dispersed within the membrane. The compound can be either an ionic exchanger or a neutral carrier. This system has found widespread use in biological systems and forms the basis of the following chapters in this book: their practical construction; their calibration and examples of their utility.

3.2. ION EXCHANGE SYSTEMS The basis of these membranes is the incorporation of an ion-exchanger, capable of binding to the ion of interest, into a suitable solvent and often an additional inert polymeric matrix of polyvinyl chloride (PVC) or silicone. Ion transport is by exchange of the ion between contiguous ion-exchange molecules. The membrane can be made into macro-electrodes to measure ion concentrations in relatively large volumes of solutions (Chapter 4), or incorporated into micro-electrodes suitable for insertion into the intracellular space of cells or confined to small extracellular spaces (Chapter 5). Several other practical developments, but based on the same principle have included ion-selective coated-wire electrodes, ion-selective field-effect transistors (ISFET’s), enzyme-selective electrodes and chromoionophores. However, despite their theoretical advantages they have yet to find as widespread a use in biological systems as have the ion-exchange macro- and micro-electrodes. The principles of these other systems will be considered briefly in appendix 4. The first ion-exchange materials suitable for ion-selective electrodes were generally ionised materials which formed low resistance membranes, and often suffered from a relatively poor selectivity and degradation of the electrode. A relatively successful example is the ionic Ca2+ -sensitive material isooctylphenyl-phosphate (see Table 4.1, section 4.4). The introduction of neutral ion-exchangers ligands by Wilhelm Simon in the 1970’s and 1980’s revolutionised the field. When incorporated into a PVC membrane with a suitable plasticiser and solvent they generated reliable membranes with the properties of a liquid phase (polymer liquid membranes). When doped with a small concentration of an ionised material the membrane had a low enough resistance to make it practically useful.

3.3. COMMON IONS MEASURED IN BIOLOGICAL SYSTEMS— CATIONS AND ANIONS The ions that can be measured with ion-exchangers is largely limited by the needs of the experimenter. The common biologically important ions such as Na+, K+, H+, Ca2+, Mg2+ and Cl− are amenable to direct determination, as well as several useful ions used in various experimental systems, such as Li+, and

.

, Rb+, Cs+, Ba2+, Sr2+, Cu2+, Ag+, Pb2+,

Principles of ion-selective electrodes

27

3.4. THE RELATIONSHIP BETWEEN POTENTIAL DIFFERENCE AND ION ACTIVITY—BASIC EQUATIONS The potential developed across the ion-selective membrane is of real interest and the variation of this potential as a function of the ionic activity in the test solution. This potential can be simplified into three components: two are boundary potentials developed at the interfaces of the test and filling solution with the ion-selective membrane,

and

, the third is a diffusion potential across the ion-selective membrane, ED—these are shown in figure 3.1. The potentials at the two boundaries are given by; (3.1)

where

and

are the activities in the test and filling solutions respectively of the ion, i.

and are the concentrations of the ions on the boundary of the ion-selective membrane facing the test and filling solutions and ki is a constant describing the distribution of the ion, i, between solution and ion-selective membrane. The diffusion potential, ED, can be written in a form similar to that given above for the potential at a liquid junction, namely the Henderson equation. If it is assumed that the boundary potential at the ion-selective membrane/filling solution and the diffusion potential are constant then the overall potential is governed by the boundary potential at the ionselective membrane/test solution interface, Further if the concentration, at this interface is also assumed constant then the overall potential, E, is given by the equation.

Figure 3.1. A model of an ion-selective membrane exchanging the ion i+; symbols are in the text.

Ion-selective electrodes for biological systems

28

or (3.2.) The value of the gas constant, R is 8.314 51J.K−1 mol−1 and F—the Faraday equivalent— is 96,485 C.mol−1. The ratio 2.303.RT/F has a value of 59.1mV at 25°C and 61.5mV at 37°C. The term E° is a constant which is the sum of all the invariant terms in the system. Equations 3.2 show that an ion-selective membrane which is permeable to one ionic species alone shows a linear relationship between potential and the logarithm of the activity, ai, of the ion in the test solution. Plots of equation 3.2 are shown in figure 3.2 for an ion-sensitive electrode, selective solely to K+, where the concentration axis is shown as a linear axis in part A or a logarithmic axis in part B. Standard values of physical constants are listed in appendix 3. A more complete description of the voltage generated across the ion-selective membrane will take into account the contribution from other, less strongly but significantly interacting ions. The magnitude of the contribution from these interferent ions will depend upon how well they interact with the membrane and this is quantified in . terms of a selectivity coefficient, Let i be the primary ion, to which the ion-selective membrane is primarily permeable and j, k, l, etc. be other interferent ions which will also interact with the ion-selective membrane. Equation 3.2 is then modified to yield the Nikolsky-Eisenman equation given below—often just called the Nikolsky equation;

or (3.3.)

The Nikolsky equation is the key equation in describing the way that an ion-selective electrode responds to the primary ion. It will be referred to again when electrode calibration is considered in detail and methods described to estimate values of (chapter 6). The equation assumes that the interferent ions act independently of each other and also independently from the primary ion in generating the total potential, E. A derivation of the equation for the interested reader is given in appendix 1. Some of the arithmetic calculations require an ability to manipulate exponential and logarithmic functions—the necessary tricks are laid out in appendix 2.

Principles of ion-selective electrodes

29

Figure 3.2. The relationship between electrode potential and ion concentration to which an ion-selective electrode is perfectly selective, in this case K+. In part A, the [K+] is plotted on a linear scale; part B the [K+] is plotted on a logarithmic scale.

Ion-selective electrodes for biological systems

30

Figure 3.3. Plots of the Nikolsky equation for a Na+ -selective electrode. The linear plot is one for an ideal electrode; non-linear plots are those with interference from a second monovalent cation with various values of

above the different plots.

The larger the value of the greater is the contribution to the total membrane potential from interferent ions and thereby the less sensitive the membrane will be to the primary ion. In practice as the activity of the primary ion, ai, decreases its contribution to the membrane potential, E, diminishes—a plot of E versus ai approaches a horizontal line. Thus, it is important to know the range of activities over which the electrode responds predominantly to the primary ion. Because the value of

is often low it is

convenient to tabulate them as their logarithm or their value . Plots of the Nikolsky equation (eqn 3.3) are shown in figure 3.3. Calculations were made for a Na+ -selective electrode with the [Na+] varying from 0.1 µmol.l−1 to100 mmol.l−1. Interference was supplied from another monovalent ion, j, at a concentration of 1 mmol.l−1 and for various values of +

. The straight-line on this plot shows the

response for a perfect Na -selective electrode. As the value of becomes more insensitive to Na+.

increases the plot

4. MANUFACTURE OF DIP-CAST IONSELECTIVE ELECTRODES 4.1. INTRODUCTION There are several considerations when designing and manufacturing an ion-selective electrode. These include: i) the fluid space in which it is placed: such as the intracellular space, protein-containing solutions, e.g. plasma, or large volumes of electrolyte solutions. In biological solutions the solvent will generally be water. ii) the range of concentrations over which it is expected that the primary ion will vary, as well as the concentration of potentially interferent ions. iii) the speed of response of the electrode and its long-term stability; this factor can be minimised by optimising the experimental protocol. iv) the physical properties of the electrode, in particular the electrical resistance. This will determine the speed of response of the electrode and also the recording instrumentation to be used. In addition the ruggedness of the electrode may be important, if it is placed in situations where physical damage might occur. The physical characteristics of ion-selective electrodes will vary according to circumstance. Two types of ion-selective electrode will be described which will satisfy most of the requirements described above. One is a macro-electrode based on a dip-cast construction, the other is an ion-selective micro-electrode based on the intracellular micro-pipette used for recording intracellular potentials. The dip-cast electrode has dimensions of the order of one millimetre, has a relatively low resistance and is physically robust. It has thus been used in a variety of extracellular spaces some of which will be given as examples later. Ion-selective micro-electrodes are very small, with a tip diameter of the order of one micrometre or less and so have been used to make measurements in the intracellular space and regions inaccessible to larger ion-selective electrodes. The small size however increases their resistance thus limiting their speed of response, stability and selectivity.

4.2. FABRICATION OF DIP-CAST ION-SELECTIVE MACROELECTRODES Several stages are associated with the manufacture and use of ion-selective dip-cast electrodes. 1. Manufacture of an electrode template.

Ion-selective electrodes for biological systems

32

2. Manufacture of a stock ion-selective plastic. 3. Dip-casting a layer of ion-selective plastic onto the template. 4. Constructing suitable reference bridges and metal electrodes. 5. Connecting the ion-selective electrode and reference bridge to the recording apparatus. Each of these processes can be dealt with separately. Details of the materials used, and their suppliers, will be listed at the end of the book.

4.3. MANUFACTURE OF ELECTRODE TEMPLATES Several features of the template are required. 1. A robust support for a thin ion-selective membrane. 2. Mechanical flexibility so that the electrode can be placed optimally in the experimental system. 3. Polyvinyl chloride (PVC) tubing to ensure optimal bonding between the template and ion-selective membrane. Some authors have used polyurethane tubing which provides a more physically robust template but may provide less good binding between the ionselective membrane and the tubing itself. Such features can be conveniently incorporated into the design that will be shown below. This consists of a length of PVC tubing into which a short piece of ceramic rod has been wedged at one end. The ceramic rod will provide mechanical support for a thin layer of ion-selective plastic cast over it surface. Ceramic rod is suitable as the porosity of the material allows filling solutions inside the electrode to penetrate the support and make close contact with the ion-selective plastic. Thus it is possible to obtain a thin layer of

Figure 4.1. A schematic representation of a plastic dip-cast ion-selective electrode.

Manufacture of dip-cast ionselective electrodes

33

ion-selective plastic on a robust support with filling and test solutions in close proximity. Figure 4.1 is a schematic diagram of the finished electrode. The ceramic plug is made from frequentite rods, which can be machined easily to a size which will fit exactly into the end of the PVC tubing. It is important that the plug fits tightly into the tubing otherwise it is difficult to cast an adequate plastic membrane over the template. In addition the plug should be smoothed with fine grade emery paper to a rounded end to ensure equal deposition of the plastic.

4.4. MANUFACTURE OF ION-SELECTIVE PLASTICS Care and patience are required to achieve a usable ion-selective plastic. If the procedure is hurried then an inadequate plastic may be obtained, which is difficult to rescue. The essence of the procedure is to ensure that the ion-selective ligand is homogeneously distributed among a polyvinylchloride (PVC) matrix. This is conveniently achieved by adding a suitable plasticiser to the mixture. In addition, if so-called neutral ligands are used then the resistance of the plastic will be very high which will make the electrode electrically noisy and difficult to use. To overcome this the mixture is doped with a small quantity of a lipophilic salt which does not significantly degrade the ionic selectivity of the electrode, and generally improves it. The recipes employed for the preparation of ionselective plastics using both neutral ligands are detailed below, the actual weights of different materials are described in table 4.1.

Table 4.1. The composition of ion-selective plastics Ca2+ -selective plastic (neutral ligand). Pot 1.

Pot 2.

high molecular weight PVC

95mg

tetrahydrofuran

6ml

ETH 1001

10mg

potassium tetrakis(4-chlorophenyl)borate

2.5mg

bis(2-ethylhexyl) sebacate

190mg

2+

Ca -selective plastic (ionic ligand). Pot 1.

high molecular weight PVC

100mg

tetrahydrofuran Pot 2.

6ml 2+

isooctylphenylphosphate (Ca ligand)

20mg

dioctylphenylphosphonate (DOPP, plasticiser)

200mg

+

H -selective plastic (neutral ligand). Pot 1.

Pot 2.

high molecular weight PVC

165mg

tetrahydrofuran

6ml

tri-n-dodecylamine (TDDA)

5mg

Ion-selective electrodes for biological systems

34

potassium tetrakis(4-chlorophenyl)borate

2.5mg

bis(2-ethylhexyl) sebacate

330mg

+

Na -selective plastic (neutral ligand). Pot 1.

Pot 2.

high molecular weight PVC

270mg

tetrahydrofuran

6ml

ETH 227

10mg

bis(2-ethylhexyl) sebacate

550mg

potassium tetrakis(4-chlorophenyl)borate

6.5mg

+

K -selective plastic (neutral ligand). Pot 1.

Pot 2.

high molecular weight PVC

300mg

tetrahydrofuran

6ml

valinomycin

3.0mg

potassium tetraphenylborate

0.10mg

bis(2-ethylhexyl) sebacate

600mg

nitrobenzene

200mg

2+

Mg -selective plastic (neutral ligand). Pot 1.

Pot 2.

Pot 1.

Pot 2.

high molecular weight PVC

165mg

tetrahydrofuran

6ml

ETH 5506 or ETH 7025

5.5mg

potassium tetrakis(4-chlorophenyl)borate

3mg

bis(2-ethylhexyl) sebacate

330mg

high molecular weight PVC

100mg

tetrahydrofuran

6ml

nonactin

10mg

bis(2-ethylhexy) sebacate

200mg

i. Clean two small glass bottles (about 10ml volume) with tetrahydrofuran (THF) and allow them to dry. Prepare a solution of PVC in one bottle. Weigh out the desired amount of high molecular weight PVC (see table 4.1) and add this slowly to 6mls of tetrahydrofuran, stirring continuously to dissolve the PVC. Perform these procedures in a well-ventilated room or fume cupboard as THF should not be inhaled. ii. Into the second bottle add the neutral ion-selective material and the lipophilic salt Ktetrakis (4-chloro-phenyl)borate. Next add drop-wise the plasticiser such as bis(2ethylhexyl) sebacate (dioctylsebacate—DOS) and stir slowly for a few minutes to

Manufacture of dip-cast ionselective electrodes

35

ensure adequate mixing of the components. The K-tetrakis (4-chloro-phenyl)borate has K+ -selective properties and so should not exceed the molar quantity of the neutral ligand otherwise the interference by K+ will become excessive. In the case of the -sensitive material, nonactin, it is omitted as the discrimination between and K+ is already poor. The disadvantage here is to produce relatively high resistance electrodes. iii. Add the PVC solution in the first bottle to the second bottle containing ion-selective ligand, DOS and doping agent—stir slowly for a few minutes. iv. It is now important to allow the THF to evaporate away slowly leaving a clear plastic with the active components evenly distributed throughout the PVC matrix. If the THF evaporates too quickly then precipitates may develop which are difficult to re-dissolve in THF. Slow evaporation is most conveniently carried out by placing a thick wad of filter paper over the top of the bottle and held in position with a heavy weight. Evaporation will be complete after a few days leaving a clear ion-selective plastic. It is advisable to keep the prepared plastic away from direct light by wrapping the bottle in aluminium foil. Prolonged exposure to light makes the plastic turn brown and seems to impair the performance of the electrodes. v. Correct handling of the commercial neutral ligands is important in making good ionselective plastics, in particular when dispensing small quantities into the glass bottles. The authors have had poor experience using disposable pipette tips for this purpose. The most successful way is to draw out polythene tubing over a small flame to a narrow diameter (≈0.5mm) and suck up appropriate quantities with a syringe. Table 4.1 also lists the materials required for a Ca2+ -selective plastics prepared not from a neutral ligand but from a Ca2+ -binding ionic ligand distributed in a PVC matrix. In this case it is not necessary to add the tetraphenylborate derivative as a doping ion, but in other respects the preparation is similar except that a different plasticiser is used.

4.5. PREPARATION OF DIP-CAST ELECTRODES It will be necessary to have already prepared electrode templates of ceramic rod and PVC tubing as well as the appropriate ion-selective plastic. Dissolve the ion-selective plastic in a small amount (≈1ml) of THF and stir gently to allow the solvent to evaporate slowly. The solution is ready for use when it is a viscous fluid with the consistency of golden syrup. Dip the end of an electrode template into the mixture for a few seconds, allowing the small bubble of air that escapes from the ceramic plug to become detached from the template. The whole of the ceramic plug should be coated and a small portion of the PVC tube. If a slow stream of bubbles comes from the ceramic plug then the latter is not fitting tightly into the PVC tube and the template should be discarded as it may become mechanically fragile and allow the membrane to break. Take the template out of the mixture and hold it just above the fluid, in the THF vapour, for about 10 seconds to ensure that the solvent does not evaporate too quickly from the coat. Allow the coated template to dry in the air for about five minutes and then repeat the process to obtain a second coating. Repeat the process a third time and allow the electrode to dry completely for several hours.

Ion-selective electrodes for biological systems

36

The accumulation of three coats on the template will produce a rapidly responding electrode (<100ms) and with a life span of several months. Fewer coats may yield more rapidly responding electrodes but of a shorter life span. The electrodes, when dry, can be stored virtually indefinitely in a clean, closed container. Avoid allowing the coated end of the electrode to touch the side of the container. Before use, a suitable filling solution is placed in the PVC tubing—see figure 4.1. The magnitude of the potential difference developed across the ion-selective membrane will depend on the concentration gradient across the membrane and so upon the composition of the filling solution. Therefore, it is of value to have the primary ion in this solution at a concentration similar to that expected in the test solution to minimise large potential differences across the electrode. The composition of the filling solutions that could be used in various ion-selective electrodes is given in table 4.2. In addition, it of the utmost importance that the concentration of the primary ion in the filling solution does not alter as this would introduce drift into the system. This requirement is generally maintained as the composition of the filling solution changes little. The exception is potentially with H+ electrodes, when alteration of the pH of the filling solution will alter the electrode response. In this context it should be remembered that CO2 will readily diffuse across the PVC tube into the filling solution so that a strong buffer should be used. In table 4.2 the citric acid/ Na citrate mixture fulfils this role. A similar problem may also occur with

electrodes in alkaline solutions (at pH values

greater than about 8.5) when the equilibrium between

and NH3 will favour

generation of ammonia gas which may diffuse across the membrane and alter the of the filling solution.

Table 4.2. Filling solutions used for various dipcast ion-selective electrodes Primary ion

Filling solution

Calcium

1 mmol.l−1 CaCl2; 150 mmol.l−1 KCl

Hydrogen

58 mmol.l−1 Na citrate; 29 mmol.l−1 citric acid; 63 mmol.l−1 NaCl

Sodium

1 mmol.l−1 NaCl; 150 mmol.l−1 KCl

Potassium

4 mmol.l−1 KCl; 146 mmol.l−1 NaCl

Magnesium

1 mmol.l−1 MgCl2; 150 mmol.l−1 KCl

Ammonium

1 mmol.l−1 NH4C1; 150 mmol.l−1 NaCl

In addition to the primary and secondary ions listed above all filling solutions used in both ion-selective and reference electrodes should be saturated with AgCl. This can be achieved by putting a small lump of AgCl in the stock solutions. AgCl is conveniently prepared by adding HCl to AgNO3 then washing and drying the precipitate of AgCl. The reason for this procedure is that AgCl is slightly soluble, especially in moderately strong ionic solutions. This would eventually lead to the loss of AgCl from the Ag/AgCl electrodes, resulting in drift and an increase of noise. The saturation of the filling solutions with AgCl will prevent this process.

Manufacture of dip-cast ionselective electrodes

37

When filling an ion-selective electrode with a filling solution care must be taken to avoid the presence of air bubbles in the PVC tube which can break the contact between the ion-selective layer and the Ag/AgCl electrode. This is especially prevalent near the ceramic plug as addition of the filling solution will displace air trapped in the ceramic plug and cause its accumulation inside the PVC tubing. After filling the electrode with filling solution, check the liquid column after a minute and remove accumulated bubbles by gently flicking the electrode or adding more filling solution into the stem.

4.6. MANUFACTURE OF REFERENCE BRIDGES A convenient bridge can be made from an uncoated electrode template filled with a suitable reference solution. The ceramic plug allows excellent electrical contact between the solution and the test fluid without permitting an excessive leakage of the reference solution. If the volume of the test solution in which the electrodes are placed is relatively large then 3 mol.l−1 KCl can be used if the slight leakage of KCl is not large enough to alter the bulk [K] of the test solution. However, if the volume is more limited then isotonic KCl would be preferable so that the test solution would be contaminated less by leakage from the reference bridge. However it must be appreciated that in the latter case the liquid junction potential may be larger and perhaps more variable, a fact to considered below. The liquid junction potential, Ej which exists between the reference bridge solution and the test solution if they are not the same composition has been considered in section 2.3 and the most suitable equitransference solutions discussed. If the magnitude of this potential remains constant its presence would not be problem, as it is the difference between the reference and ion-selective electrode that is being measured. However alterations may occur if the test solution changes greatly, if the ceramic plug becomes dirty or the solution inside the bridge gradually changes composition and the use of concentrated equitransference electrolytes in the bridge must traded off with regard to possible excessive leakage of the bridge solution into the test solution. To calculate the magnitude of leakage from a reference bridge an experiment was conducted in which an electrode template, as described above, was used as a reference bridge and filled with 3 mol.l−1 KCl. This reference bridge and a K+− selective electrode were placed in 2ml of a solution containing 4 mmol.l−1 KCl and 146 mmol.l−1 NaCl. The [K+] increased to 5 mmol.l−1 in 30 minutes, representing a leakage of 4 µmoles of K+ per hour. If this leakage is considered too great or too rapid then less strong solutions should be used but with the possibility of a larger and more variable liquid junction potential.

4.7. MANUFACTURE OF Ag/AgCl ELECTRODES Connection between the ion-selective electrode filling solution or reference bridge and the recording apparatus is best made by a Ag/AgCl electrode. The electrode is relatively cheap to construct, exhibits low resistance and does not polarise (drift) when made properly. Poor response of an electrode system, either in noise or drift, often stems from badly made and maintained Ag/AgCl electrodes and care should be taken in this respect.

Ion-selective electrodes for biological systems

38

Probably the best method to make such electrodes is to deposit slowly a layer of AgCl onto a segment of clean silver wire by electrolysis. The procedure is as follows; 1. Prepare several lengths of silver wire. Thin (≈0.002cm diameter) Trimel coated silver wire is very useful because it is possible to expose only a short length of bare silver, leaving the remainder covered by insulating material. It is necessary to scrape off a length (1–2cm) of the coating with a blunt scalpel blade to reveal the bare metal. Some people use commercial paint strippers to strip the Trimel away but in our experience is less effective. Such thin wire is used as it minimises the cost, produces a mechanically flexible electrode and the presence of an insulating coating above the prepared length helps keep the remainder of the electrode clean. Prepare also a second thicker piece of silver rod for the eleectrolysis process. 2. Place the prepared silver wires and rod into a solution of 100 mmol.l−1 KCl and connect them to a 9V battery in the manner shown in the upper panel of figure 4.2. A vigorous stream of bubbles will emanate from the silver wires and a whitish deposit of AgCl will form on the thick silver rod connected to the positive pole of the battery. This process serves to clean the silver wires that will be eventually coated. Note if the silver wires are soldered to other wires connecting with the battery take care to prevent the solder joint from coming into contact with the KCl solution. 3. Discard the KCl solution, clean the silver rod by scraping off the lumpy deposit of AgCl with tissue paper and then fine emery paper and place the silver wires and rod in fresh solution. Reverse the polarity of the connexions to the battery and place a large resistor (10–100 kΩ) in series to limit current through the circuit as shown in the lower panel of figure 4.2. The optimal current is often recommended to be about 10mA.cm−2, but this may not be true for the small surface areas to be coated here and a trial and error method for the best value of the resistor may have to be taken. Over a period of several minutes or tens of minutes a blackish deposit of AgCl will be deposited on the silver wires—the best coating has a slightly reddish tinge to it. Remove the Ag/AgCl electrodes from the KCl solution and store them connected together in a dilute solution of saline. Test the adequacy of the Ag/AgCl electrodes by placing a pair in a solution of 100 mmol.l−1 KCl and connect them to the two inputs of a voltmeter (an oscilloscope or other device). The potential difference between the two electrodes should be small and the variation of this voltage (drift) minimal. As a rough guide the potential difference should not be greater than 1mV and the drift should be less than 100 µV per hour. Despite the best efforts to manufacture adequate AgCl coated Ag electrodes, when placed in strong electrolyte solutions the potential difference between a pair of such electrodes can drift by several mV over a period of hours. One reason for this is that although AgCl has a very low solubility in water, the solubility is significantly greater in strong electrolytes. Therefore after some hours the AgCl coating can dissolve completely leaving bare Ag which will exhibit considerable drift. Saturation of the electrolyte solution with AgCl or AgNO3 will prevent such solubilisation and thus maintain a stable reference junction. It is recommended that all electrolyte solutions in which Ag/AgCl electrodes are placed be saturated with AgCl. AgCl pellets can be easily made by pouring

Manufacture of dip-cast ionselective electrodes

39

excess concentrated HCl over a AgNO3 solution and then washing thoroughly the precipitate before drying it in an oven.

Figure 4.2. Procedures involved in the manufacture of Ag/AgCl electrodes. In the upper diagram the procedure for cleaning Ag wire is shown. In the

Ion-selective electrodes for biological systems

40

lower diagram is the procedure for coating Ag wire with AgCl. 4.8. CONNEXIONS TO RECORDING DEVICES When all of the components of the system have been made (ion-selective electrode, reference bridge and Ag/AgCl electrodes) they may be connected as shown in figure 3.1 to form a complete system. The relatively low resistance of the dip-cast ion-selective electrodes—about 1MΩ—means that special precautions do not have to be taken when recording. Detailed circuits for use as recording systems will be given later—chapter 8— along with the suggested components. However, measurements can readily be made on the bench without recourse to Faraday cages which would exclude extraneous electric fields. It has been possible to record potential changes as small as 10 µV with such electrodes, stable over a period of several minutes. The response time of such electrodes is also rapid and other features of the experimental system, such as diffusion and the rate at which solutions can be exchanged, are likely in many cases to dominate the overall speed of response. In the authors’ experience response times <10ms have been recorded with these electrodes. The life-time of dip-cast electrodes depends upon their use. After manufacture and when stored dry the electrodes have shiny tips, but after exposure to electrolyte solutions for several hours the surface appears matt, presumably due to water absorbed at the surface. Removal from the solution (but not removing the filling solution) overnight restores the shiny appearance. If the electrodes are used for several hours a day and stored dry overnight the response time of the electrode and its sensitivity to the primary ion remains unchanged for 2–3 weeks. When used the next day after dry storage the p.d. developed by the ion-selective/ reference electrode pair may take tens of minutes to stabilise. After 2–3 weeks daily use the response time of the electrode tends to increase and the sensitivity may start to decline—this is usually accompanied by a change in the electrode appearance, i.e. it loses its shiny appearance after dry storage. This process may be the result of a gradual leaching away of the electrode components from the membrane, the electrode should then be discarded and a new one prepared with filling solution. However, if electrode longevity is less important than saving time on stabilisation of the ion-selective/reference p.d. after night storage it is better to store electrodes not dry but in a solution similar to the electrode filling solution (table 4.2).

4.9. DEVELOPMENTS IN ION-SELECTIVE ELECTRODE DESIGN Although the dip-cast electrode, and its variations, are the most widespread configuration for ion-selective macro-electrodes used in biological systems a number of alternative designs have been developed each with specific objectives and it is to be expected that they will eventually find greater utility in biological systems. Several different configurations are shown in appendix 4.

5. ION-SELECTIVE MICROELECTRODES 5.1. FABRICATION OF GLASS ION-SELECTIVE MICROELECTRODES The fabrication of ion-selective electrodes suitable for use in the intracellular space or in small extracellular spaces requires techniques different from those employed when making large dip-cast electrodes. The problem centres around introducing a small amount of ion-selective ligand into the tip of a micro-pipette and filling the remainder with a suitable ionic, filling solution. To achieve this it is necessary to render the inside of the pipette hydrophobic to allow the oily ion-selective ligand solution to fill the tip and prevent an electrical shunt via an aqueous layer on the inner wall of the pipette. Virtually every laboratory has different techniques for making such electrodes. The following procedure has been used in our laboratory for Ca2+, Mg2+, Na+, K+ and H+ -sensitive micro-electrodes. Micro-pipette blanks are pulled in a standard puller from borosilicate glass (e.g. GC150F, Clark Electromedical) with a fibre in the bore. The pipettes would have a resistance of 5–20MΩ if filled with 3mol.l−l KCl—i.e. suitable for intracellular potential recording. Generally about 20 blanks are pulled in a batch. The stages are as follows; 1. Heat the electrode blanks at 200°C for about 30min to drive off any water within the barrel and transfer them in a desiccator to a fume cupboard for silanisation. 2. About 2ml of a silane, such as dimethyldichlorosilane, is put into a 20ml bottle and heated to about 50°C so that the vapour condenses on the side of the bottle about half way up. The top of the bottle is covered with Parafilm and 4 to 5 micro-electrodes poked through holes in the Parafilm with the stems exposed to the vapour for 2 minutes—i.e. the fine tip is not exposed directly to the vapour but the silane reaches the tip by diffusion through the bore of the glass capillary. It is important that the silane does not get too hot, otherwise the vapour will be too dense and occlude the pipette tip on baking. 3. The pipettes are baked at 200° C for 90–120min and stored in a desiccator until ready for filling. Normally a batch of electrodes can be kept for 3–4 days if free of water vapour. 4. Filling the micro-pipette is achieved by introducing a small volume of the ion-selective ‘cocktail’ into the electrode shank, using a drawn-out piece of polythene tubing. The ligand may flow down and fill the tip in a matter of a few minutes leaving small air bubbles which may be teased out with a fine whisker. If filling is not evident immediately it may be possible to coax the ligand solution with a fine whisker, but

Ion-selective electrodes for biological systems

42

often leaving the micro-electrode to fill itself is the most practicable solution. Often electrodes are filled with ligand solution the evening before the day of use. 5. Various ion-selective ‘cocktails’ are commercially available (e.g. Fluka Chemicals). These cocktails are different from the mixtures used for dip-cast electrodes. They contain a large conducting anion and the solvent orthonitrophenyl octyl ether. The theoretical basis of the composition of such cocktails has been discussed previously (Ammann, 1986). 6. The rest of the micro-pipette must now be filled with an appropriate filling solution. If the electrodes are used in the extracellular space solutions similar to those used with the dip-cast electrodes are suitable (table 4.2). For intracellular electrodes different filling solutions can be used for Ca2+, Na+ and K+ electrodes because the intracellular concentration of these ions are different from those in the extracellular space. For Na+ and K+ -selective electrodes a suitable solution is 10 mmol.l−1 NaCl in 140 mmol.l−1 KCl. With H+ and Mg2+ -selective electrodes the same solutions as listed in table 4.2 have been used. With Ca2+ -sensitive electrodes a solution could be 150 mmol.l−1 KCl in which the [Ca2+] is maintained at 1 µmol.l−1 with a Ca2+ buffer, EGTA. Procedures for manufacturing a solution containing 1 µmol.l−1 free Ca2+ are considered later in the section on Calibration of Ion-Selective Electrodes, section 6.7. 7. Filling is achieved by injecting the solution through another thin piece of polythene tubing. It is best that the tubing is placed below the ligand column to avoid an air bubble between the ligand and reference columns. Filled pipettes are usable for 1–2 days after which their responsiveness to the primary ion decays. Connexion to a suitable recording device may be achieved by inserting a Ag/AgCl electrode into the filling solution.

5.2. CONNEXIONS TO RECORDING DEVICES Ion-selective micro-electrodes have a very high d.c. impedance, between 1010 and 1012Ω, so that special care must be taken over the recording system used and the arrangement of the experimental system. The particular electronic components used are critical and details will be given in the later section on Instrumentation (section 8.4). In addition great care must be taken in properly shielding the micro-electrode and immediate recording devices from extraneous electric fields and a Faraday cage will be essential. The prevention of ‘earth loops’ is adequately explained in several practical manuals of electrophysiology. When such micro-electrodes are used in the extracellular space the type of reference electrode can be similar to that employed when using dip-cast electrodes and the potential difference between the pair of electrodes measured. However if ion-selective micro-electrodes electrodes penetrate into the intracellular space from the extracellular compartment the total recorded potential difference, Eise, has two components: one is the membrane potential, Em, and the second results from the difference of ionic activity between the inside and outside of the cell. Only the latter component, Eise—Em, is of interest so that the membrane potential must be subtracted from the total signal. This is achieved by a separate measurement of the membrane

Ion-selective microelectrodes

43

potential with a conventional 3mol.l−1 KCl-filled micro-electrode and subtraction of this signal from the total p.d. recorded by the ion-selective micro-electrode. The membrane potential of a cell is negative on the inside with respect to the extracellular solution. If the intracellular cation concentration is less than the extracellular value a further negative potential is recorded—i.e. −Eise>−Em. Alternatively, if the intracellular concentration of the cation exceeds the extracellular concentration then Eise will be less negative than Em—i.e. −Eise<−Em. However, penetration of a cell with an ionselective microelectrode will generally result in a negative potential being recorded. A block diagram of the experimental arrangement is shown in Figure 5.1. Two microelectrodes are shown—a conventional one filled with a strong electrolyte such as 3 mol.l−1 KCl and an ion-selective electrode. In addition there will be reference electrodes, such as an electrolyte-filled bridges, in the extracellular space. The p.d.’s recorded by both micro-electrodes can be displayed on separate channels of a pen recorder. However, the p.d. recorded from the conventional micro-electrode, Em, is also subtracted from that recorded by the ion-selective microelectrode, Eise—the lower channel of the recorder in figure 5.1. Thus, the display in this lower channel, Eise−Em, will represent only the difference of ionic activity between the inside and outside of the cell.

Figure 5.1. Experimental lay-out for the use of intracellular ion-selective microelectrodes A practical problem arises with the use of intracellular K+ -selective microelectrodes. In most excitable cells the membrane potential is largely determined by the transmembrane K+ gradient, about 4 mmol.l−1 in the extracellular phase and 140 mmol.l−1

Ion-selective electrodes for biological systems

44

in the intracellular compartment. When the concentration of both phases are taken into account the Nernst equation (equation 3.2) is written as: (5.1.) where [K+]o and [K+]i are the extracellular and intracellular concentrations respectively. Substitution of the above values gives a value of Em=−95mV at 37°C, which is similar to that experimentally recorded. Thus the value of Em is roughly equal and opposite to the potential difference sensed by the K+ -selective microelectrode when it enters the intracellular compartment with the higher [K+] –i.e. −Eise≈0mV. Therefore it is difficult to know when the K+ -selective microelectrode has penetrated the cell. This problem can be tackled by using a double-barrelled electrode in which the second barrel records Em alone. Alternatively when it is thought that impalement has been achieved an increase of the extracellular [K+] from 4 to about 10 mmol.l−1 will depolarise the cell (Em will be less negative) and this will be recorded as a positive-going signal from the ion-selective microelectrode, as the intracellular [K+] will not vary significantly. The practical aspects are considered further below.

5.3. PRACTICAL ASPECTS IN THE USE OF ION-SELECTIVE MICRO-ELECTRODES The separate estimation of the membrane potential has made the use of ion-selective micro-electrodes difficult in many cell types. In some large cells, such as skeletal muscle fibres and certain neurones, it is possible to penetrate the same cell with several separate electrodes, but in many preparations this is impossible. Two broad approaches may be used, one is to use a double-barrelled electrode in which one barrel is filled with ionselective material and the other with electrolyte for measurement of membrane potential. Another approach has been used in multicellular preparations where the cells are electrically coupled by low resistance junctions. In this case different cells are penetrated with ion-selective and conventional micro-electrodes. In the latter method it is assumed that the membrane potentials of all cells are similar, due to electrotonic coupling, and it remains only to verify this. This is achieved with raised extracellular [K+] to depolarise the preparation; if the membrane potential recorded by the two electrodes is the same then the magnitude of depolarisation will also be the same. Thus there should be zero deflexion measured on the ion-selective electrode (Eise−Em) channel of the recorder. Figure 5.2 illustrates an experimental recording of the impalement of ventricular myocardium firstly with a Na+ -sensitive micro-electrode (upper trace) and then a 3 mol.l−1 KCl-filled micro-electrode (lower trace). When the ion-sensitive electrode penetrates the cell the overall p.d. has two components—Em and that due to the difference in the intracellular and extracellular [Na+]. Subtraction of Em, as recorded by the second micro-electrode, leaves a residual p.d. against which a calibration curve may be placed and the intracellular [Na+] estimated. After an interval, as represented by a break in the tracings, the superfusate [K] was raised from 4 to 30 mmol.l−1, depolarising the cell as seen on the conventional micro-electrode, Em, trace. The absence of a significant change in the p.d. on the Eise−Em trace indicates that the change of membrane potential as

Ion-selective microelectrodes

45

recorded by this micro-electrode was similar, thus indicating that the absolute value of Em as recorded by the two micro-electrodes was the same and justifying the subtraction procedure. The use of double-barrelled ion-selective micro-electrodes has not been widespread in biological measurement due to the difficulty of silanising only one barrel, to enable it to be filled with ion-selective material, but leaving the other intact so that it can be filled with an electrolyte. However, another approach has been stimulated by this problem, namely the possibility of manufacturing a different reference material, other than a strong electrolyte, that could be introduced into a silanised barrel. Thus it would be possible to silanise both barrels, by similar procedures to that described above, and fill one with an ion-specific material and the other which would respond equivalently to most ionic species and so behave as a reference barrel.

Figure 5.2. Measurement of intracellular [Na+] in ventricular myocardium with an ion-selective microelectrode. The 3 mol.l−1 KCl electrode output has been subtracted from the ion-selective electrode output. Superfusate [K] was raised from 4–30 mmol.l−1 as shown.

Ion-selective electrodes for biological systems

46

One such material is potassium tetrakis (4-chlorophenyl)bo-rate dissolved in 2-octanol and was first described in the literature by Thomas & Cohen (1981). This material responds equivalently to Na+ and K+ so that as long as the total concentration of these two ions remains the same there will be no change in electrode potential. In general this criterion is true for the intracellular and extracellular spaces of most cells, thus upon penetration of a cell such an electrode should record only the membrane potential, Em. Figure 5.3 shows an experiment using an isolated preparation of ventricular myocardium. Two separate micro-electrodes were used, one a conventional, 3 mol.l−1 KCl-filled electrode, and the other an electrode filled with a solution of potassium tetrakis (4-chlorophenyl) borate (K-TCB). The potentials recorded by the two micro-electrodes are designated Em and Etb, respectively. The upper trace is a low gain recording of Em and

Figure 5.3. Part A. The use of Ktetraphenylborate (tb) as a non-specific cation-sensitive (reference) electrode. The upper trace from a 3 mol.l−1 KCl microelectrode, the lower from a Ktetraphenylborate microelectrode. First the 3 mol.l−1 KCl microelectrode

Ion-selective microelectrodes

47

impaled a cell, followed by the Ktetraphenylborate microelectrode. Superfusate [K] was raised from 4 to 30 mmol.l−1 as shown. On the righthand side are electrode p.d.’s in the extracellular space. Superfusate [Na] and [K] were altered at constant Σ[Na]+[K], or solution pH altered by changing the CO2 percentage of the gas equilibrating with the superfusate. Part B. A diagram of the experimental arrangement. shows cell penetration by the conventional micro-electrode. The lower trace is a high gain recording of Etb−Em. The immediate left of this trace shows the situation when both micro-electrodes were in the extracellular space. When the conventional micro-electrode penetrated the cell this trace was deflected upwards. Penetration of a second cell by the non-specific K-TCB micro-electrode returned the high gain trace of Etb−Em back to the original position, suggesting that the two electrodes have recorded the same potential— i.e. the two electrodes record the same value of the membrane potential and thus the difference p.d. is zero. This was further supported by the observation that raising the extracellular [K] depolarised the preparation, as seen by the reduction of Em, but had no effect on the Etb−Em signal. The non-specific response of the K-TCB electrode was seen when it was taken out of the cell—right-hand side of figure 5.3. When the electrode was exposed to a solution containing either 140 mmol.l−1 NaCl+8 mmol.l−1 KCl or 148 mmol.l−1 KCl there was no change to the p.d., indicating that the electrode was equally responsive to NaCl and KCl. The final part of the trace shows the p.d. recorded by the K-TCB filled electrode was unaffected by altering the PCO2, and hence the pH, of the solution in which the electrode was placed.

6. CALIBRATION OF ION-SELECTIVE ELECTRODES 6.1. GENERAL NOTES ON CONVENTION-CONCENTRATION OR ACTIVITY STANDARDS The interpretation of signals obtained from ion-selective electrodes depends upon reliable calibration of the electrode system. It must always be remembered that it is a potential difference (p.d.) between the ion-selective and reference electrode that is measured and the interpretation of this p.d. in terms of ionic concentrations is achieved by a calibration curve. Thus note must be taken of the range over which the electrode will be reasonably responsive to the primary ion, the extent to which other ions will interfere with the signal and the stability of other potentials in the overall system (see chapter 2). Ion-selective electrodes measure the chemical activity of an ion in solution, which is related to concentration by the activity coefficient (section 1.4). However, most calibration solutions are made up as concentration standards and the calibration curve will use these concentrations to relate to a given electrode voltage. Thus, the p.d.’s recorded by the ion-selective/reference electrode pair will be a function of the concentration of a particular ion without any loss of generality provided that the relationship between activity and concentration does not alter. In general this can be achieved by working over a reasonably limited range of concentrations and ensuring that factors like the ionic strength are maintained constant and are similar to the test solution. Some authors quote activity values and do so by multiplying the concentration values from the calibration curve by a standard value for the activity coefficient. However, unless there are specific reasons for doing to, such as determining the precise kinetic conditions for ion binding to a ligand, no extra information can be gained by merely multiplying all values by a constant. Moreover, the activity coefficient will be a function of the counter ions present, temperature, ionic strength etc. so that unless the test solution was evaluated under the same conditions as for those at which the activity coefficient was determined the process will have an in-built error. For this reason concentrations standards will be used here. However, for the interested reader some discussion of activity standards will be given in section 6.3. An exception to the use of concentration standards occurs in the measurement of pH (see sections 6.2 and 6.7) where well-defined activity standards are used for electrode calibration.

Calibration of ion-selective electrodes

49

6.2. THE pH AND ANALOGOUS SCALES The concentration of certain ions, such as H+ and Ca2+, may change over several orders of magnitude in different situations and so a convention which avoids either very large or small numbers is in common usage as a means of expressing these concentration values. A linear concentration scale is changed to a logarithmic one so that geometric increments are converted to arithmetic changes. For example, let the concentration of H+ in solution be [H+], then the pH scale is defined as (see also section 6.7 below for a more detailed discussion): (6.1.) Thus, if the [H+] is 1.10−7 mol.l−1 (0.1 µmol.l−1) the pH is 7.0 and if the [H+] is 1.10−6 mol.l−1 (1 µmol.l−1) the pH is 6.0. The convention applies to other ions, so that pCa, pMg, etc. scales can all be devised, e.g. 3 µmol.l−1 [Ca2+] is equivalent to a pCa value of 5.52, 0.5 µmol.1−1 [Mg2+] is pMg 6.30, 10 mmol.l−1 [Na+] is pNa 2.0.

6.3. MAKING SOLUTIONS The use of a concentration calibration scale has been introduced above (section 6.1). When constructing calibrating solutions, the usual criteria of analytical chemistry should be used. The salt should be of as high purity as possible and not be left to attract water or other impurities. Many salts are hygroscopic when exposed to air, so that the quantity of salt in a given weight will diminish. Salts such as KCl and NaCl should be dried in an oven before use, but salts with associated water molecules, such as MgCl2.6H2O will change their formula weight if dried at high temperatures. Suppliers provide such information on the label of high purity compounds. For hydrated salts alternative approaches are possible: • suppliers provide accurate solutions of some salts (e.g. 1 mol.l−1 CaCl2) • a less hydrated or dehydrated salt can be used (e.g. MgSO4, rather than MgCl2.6H2O). However, in this case remember that the addition of will change the activity coefficient of Mg2+(and Ca2+) if added in large enough concentrations (see chapter 1). • accurate calibration solutions can be made de novo (e.g. by addition of HCl to a known weight of Mg ribbon to produce a solution of known molarity—this practice is the most accurate method). The accuracy of the solution can be checked by measuring the [Cl−] or a method such as freezing point depression. The best method for making a range of calibration solutions is to make one stock solution and prepare lower concentrations by serial dilution immediately before use. Thus if any error is made in the stock solution this will be perpetuated in a proportional manner with the weaker solutions. The diluent must stabilise the activity coefficient of the primary ion as well as perform other functions, such as maintain the pH of the solution. For example,

Ion-selective electrodes for biological systems

50

to dilute a CaCl2 stock solution down to a final concentration of about 0.5 mmol.l−1 a roughly isotonic solution containing 150 mmol.l−1 NaCl and 10 mmol.l−1 of a buffer such as HEPES (see section 6.8) can be used. For very small concentrations serial dilution will become difficult and the use of ion buffers will be required when possible. This will be discussed in detail in section 6.9. For those who wish to use activity standards a number of solutions have been formulated which should be used to allow conformity between different groups. The range of such solutions is limited as they are usually designed for specific purposes and for particular conditions, such as determinations in blood or plasma. Interested readers are referred specifically to the chemical literature (e.g. Analytical Chemistry, Analytica Chimica Acta).

6.4. CALIBRATION IN UNBUFFERED SOLUTIONS OF THE PRIMARY ION Most applications of ion-selective electrodes in biological systems will use isotonic solutions, i.e. an osmotic strength of about 300milliosmoles.l−1. Apart from formulating accurate solutions of the primary ion, two other factors must be considered when interpreting the results of an electrode calibration. Firstly do other ions interfere with the ion-selective electrode itself and secondly is the chemical activity of the primary ion altered by the presence of other ions in solution? The first problem is due to the fact that other ions may be transported by the ligand within the electrode and so reduce the apparent sensitivity towards the primary ion. The second problem, most evident with divalent cations, occurs when certain anions reduce the mobility of the primary ion in , the latter tend to associate solution. An example is when Cl− are substituted with with Ca2+ in solution so reducing the activity of the Ca2+ ion. Interpretation of the p.d. recorded by the electrode system in terms of chemical concentration is only as good as the calibration process itself. Calibration solutions should reflect as closely as possible the solution under analysis: the concentration of interferent ions should be similar to that in the test solution and the effect that other ions may have on the activity of the primary ion quantified. Figure 6.1 shows the mean calibration curve obtained with a Na+ -sensitive electrode. The tracing is the recorded p.d. with respect to a 3 mol.l−1 KCl salt bridge. A zero potential is arbitrarily set at the p.d. recorded with the ion-selective electrode in a physiological saline (Tyrode’s) solution of the following composition (mmol.l−1): [Na+], , 24.0; [Mg2+], 1.0; [Ca2+], 1.8; Na pyruvate, 5.0. 142.0; [K+], 4.0; [Cl−], 127.6; + The [Na ] calibration solutions contained NaCl and KCl so that the sum of the NaCl and KCl concentrations was 147 mmol.l−1. The solution also contained 1 mmol.l−1 MgCl2, 10 mmol.l−1 HEPES to buffer the pH to 7.2 and a low [Ca2+] (0.4 µmol.l−1), maintained by a Ca2+ buffer, EGTA to mimic the intracellular space. The concentrations of NaCl were 147, 80, 40, 20, 8, 4, 2 and 1 mmol.l−1. Note that the p.d. recorded in Tyrode’s solution (Tyr) is not the same as that in the 147 mmol.l−1 [Na+] calibration solution, even though the [Na+] is the same in both solutions; this is largely due to the difference in the [Ca2+] between the two solutions. The plot shows the variation of p.d. as a function of the [Na+] in the calibrating solutions. The relationship between p.d. and log

Calibration of ion-selective electrodes

51

[Na+] is linear over the higher range of concentrations, but deviates to a greater extent as the concentration falls below about 10mmol.l−1. The arrow marks the limit of detection (see figure 6.2 below) and represents the [Na+] at which the calibration slope has been reduced to half the Nernstian slope (i.e. the linear region of this calibration curve) by interferent ions. Over the linear range the electrode therefore behaves as a perfectly selective Na+ electrode under these prevailing conditions, the behaviour described by equation 3.2. The reason for the deviation at low [Na+] is because of the increasing importance of interferent ions, such as K+. In order to compare different electrodes it is desirable to quantify the interference from other ions and this is achieved by estimating the potentiometric selectivity coefficient, , introduced as part of the Nikolsky equation (eqn 3.3). The measured selectivity coefficients do not appear to be constant under all conditions and should be estimated for the conditions used if they are to have any predictive value.

6.5. ESTIMATION OF SELECTIVITY COEFFICIENTS Two methods are in common use, the separate solution method (SSM) and the fixed interference method (FIM). 6.5.1. Separate solution method In this method the p.d. generated by a pure solution of the primary ion, Ei, is compared to the p.d. generated by a pure solution of the interferent ion of the same activity, Ej. From the Nikolsky equation; (6.2.) The method is not the one of choice as the concentration of interferent ion and primary ion may be very different from that encountered experimentally. 6.5.2. Fixed interference method This is a graphical method where the response of the electrode to variable concentrations of the primary ion, in the presence of a constant concentration of the interferent ion, is evaluated. This method has the advantage that concentrations of the interferent and primary ions can be used which are similar to those encountered experimentally. A plot of the recorded p.d. as a function of the ratio of the primary and interferent ion concentrations (both are monovalent here), mi/mj is obtained as shown in figure 6.2. In the two examples given here the ion-selective electrode is either 10 or 100 times more sensitive to the primary ions so that . Thus the two ions contribute equally to the electrode potential when the primary ion concentration is 10% or 1%

Ion-selective electrodes for biological systems

52

respectively that of the interferent ion—i.e. the abscissa value, mi/m=0.1 (10−1) or 0.01 (10−2).

Figure 6.1. The tracing shows the output of a Na+ -sensitive microelectrode in Tyrode’s solution and in calibrating solutions of various [Na] (see text for compositions). The plot is that of the calibration data obtained in the above experiment. The [Na+] is plotted on a logarithmic scale.

Calibration of ion-selective electrodes

53

The zero mV point (dotted line) is set arbitrarily at that measured in Tyrode’s solution. The line is a best-fit of the Nikolsky equation (eq 3.3).

Figure 6.2. The fixed interference method for the determination . The curves were calculated from the Nikolsky equation (equation 3.3) for a solution with the ratio of primary ion (i) to interferent ion (j) concentrations, m, varying from 10−4 to 10. Two values of have been chosen, 0.1 and 0.01 to yield the two curves (dotted and solid lines respectively). The limits of detection are obtained as the concentrations at which the two linear portions of the curve intersect.

Ion-selective electrodes for biological systems

54

Table 6.1. Potentiometric selectivity coefficients, , for various ion-selective micro-electrodes determined by the authors. The values of are given as their log10 values, i.e. is equivalent to log10 . The particular ionselective material is given in parenthesis in the first column. NaTPB=Na tetraphenylborate. n.d. not determined. Interferent ion +

K+

Ca2+

Mg2+

−10.4

−9.8

<−11.1

n.d.

−0.8

−

−2.2

0.4

−2.4

−4.4

−3.5

–

−4.4

−5.1

n.d.

−4.7

−5.2

–

−5.1

−2.8

−1.1

−1.4

−1.1

Primary ion

H

Na

H+ (TDDA)

–

+

Na (ETH 227+NaTPB) +

K (valinomycin) 2+

Ca (ETH 1001) 2+

Mg (ETH 1117)

+

—

At high concentrations of the primary ion the plot approaches a straight line with slope RT/ziF, whilst at very low concentrations the plot approaches a horizontal line. At some intermediate concentration the slope of the plot will be one half that at high concentrations when the primary ion and interferent ion will make equal contributions to the overall p.d. This concentration can be evaluated by extrapolating the two linear portions of the response curve and is known as the limit of detection. When Ej=Ei equation 6.2 reduces to; (6.3.)

Because the values of

are often very small it is sometimes convenient to tabulate

their values as the logarithm. Table 6.1 lists the values of log micro-electrodes.

some ion-selective

6.6. FACTORS INFLUENCING IONIC ACTIVITY The problem of the ionic activity of cations being dependent on the particular counteranion requires attention and a standard convention. This phenomenon is especially acute when multivalent ions are considered. For example, the measured free [Ca2+] of 1 mmol.l−1 CaCl2 in 150 mmol.l−1 NaCl is greater than in a solution containing 126 mmol.l−1 NaCl+24 mmol.l−1 NaHCO3 due to Ca2+ associating with

. The

Calibration of ion-selective electrodes

55

magnitude of the problem is plotted in figure 6.3, which shows the measured reduction of is partially substituted for Cl−. In this case substitution of 24 the free [Ca2+] as the −1 mmol.l of NaCl with NaHCO3 reduces the Ca2+ activity by 19%. For example, in the case of Ca2+ our laboratory uses a convention which relates all readings to those obtained

Figure 6.3. The influence of anion substitution on the ionised [Ca2+] in a physiological saline solution. The top line—line (1)—is that of identity—i.e. that measured in a solution containing only 150 mmol.l−1NaCl. The other lines are drawn through solutions in which NaCl is substituted by: (2) 24 mmol.l−1NaHCO3; (3) 118 mmol.l−1Na acetylglycine; (4) 118 mmol.l−1 Na isethionate; (5) 118 mmol.l−1Na2(SO4)2. Measurements at 25°C. in a standard solution—1 mmol.l−1 CaCl2 in 150 mmol.l−1 NaCl. Thus in physiological Tyrode’s solution, containing 24 mmol.l−1 NaHCO3+118 mmol.l−1 NaCl, the free [Ca2+] is reduced from 1.0 mmol.l−1 to 0.81 mmol.l−1 compared to that measured in 150 mmol.l−1 NaCl. Partial equimolar substitution of NaCl with other salts such as Na sulphate, Na isethionate or Na acetylglycine has a similar effect on the free [Ca2+]. Figure

Ion-selective electrodes for biological systems

56

6.3 plots the results from several determinations in these solutions. It is therefore essential to measure the free concentration of the ion of interest in various solutions if the composition is varied. In addition to the variation of the free [Ca2+] by altering the counter-anion, divalent ions in particular bind to other ligands, such as macromolecules like albumin. Figure 6.4 illustrates the magnitude of the effect as bovine serum albumin is added to a 150 mmol.l−1 NaCl solution to which 2.4 mmol.l−1 of CaCl2 had been added. Over the range of albumin concentrations used an almost linear decline of the [Ca2+] was measured—the concentration in normal human plasma is about 4% w/v (4gm.dl−1).

6.7. THE DEFINITION AND MEASUREMENT OF pH The measurement of H+ activity in electrochemistry deserves a special place as H+ play a central role in determining so many biological processes and accurate determination stimulated the need to define a standardisation of the electro-chemical processes. At the outset it should be stated that free protons do not generally exist in biological solutions, but rather the species H3O+ However, the H+ and H3O+ have become so intertwined in

Figure 6.4. Effect of bovine serum albumin on the [Ca2+] in a solution of 150 mmol.l−1 NaCl and 2.4 mmol.l−1 CaCl2. The straight line was obtained by least squares regression. Measurements at 25°C.

Calibration of ion-selective electrodes

57

the biological literature that the former term is almost exclusively referred to, except by the most rigorous. The definition of pH introduced by S.P.L. Sørensen, i.e. pH=−log aH3O, requires knowledge of the activity coefficient of the hydronium ion, H3O+, and a conventional pH scale is based on measurements of the p.d. of the cell: Pt(H2)/Test solution/saturated KCl/reference electrode where the reference electrode is a calomel electrode—i.e. 0.1 mol.l−1 KCl (Hg2Cl)/Hg. The left hand system is a Pt electrode maintained in a dry H2 pressure of 1 atmosphere. The test solutions are those which have a definite and reproducible H30+ activity, referred to as buffer solutions. Standard buffer solutions are now recommended by bodies such as the International Union of Pure and Applied Chemists (IUPAC) and the British Standards Institution and are commercially available. It is required that they have well-defined temperature characteristics (see below) and minimal liquid junction potentials with the aqueous components of the electrode systems. These so-called primary standards, of which potassium hydrogen phthalate is the most intensively investigated are used to characterise the properties of other, secondary standards (see section 6.8). The most usual way to measure pH at present is to use a glass electrode which consists of a thin-walled glass bulb filled with a buffered solution of known and constant pH. The bulb is immersed in a test solution of unknown pH. Two auxiliary electrodes placed in these two solutions, e.g. Ag/AgCl electrodes or calomel electrodes, act as connexions to the recording system so that the cell can be represented as:

The potentials, a and c, should be virtually equal and opposite in sign, or at least stable, so that the variation of the potential at b should determine the change of p.d. recorded by the whole cell when the pH of the test solution varies. The pH of the test solution is then calculated from the p.d. measured in this solution, EX, with respect to the p.d. measured in a buffer of known pH, EB and given by equation 6.4. (6.4.) A schematic representation of a glass pH electrode is shown in figure 6.5. The potential drop across the glass wall is assumed to be due to an exchange of Na+ with H3O+ at the two interfaces, and therefore depends on the electrochemical potentials of H3O+ in the glass and in the two solutions. This

Ion-selective electrodes for biological systems

58

Figure 6.5. A schematic drawing of a combination glass pH electrode. will explain several characteristics of these glass electrodes: i) the glass must be allowed to soak in aqueous solution before use so that it swells as water is taken up by the glass forming a gel-layer; ii) the electrodes cannot be used in strong alkaline solutions (as this dissolves the gel layer) or in strong acid solutions (where acid might be adsorbed into the gel; iii) the potential is unaffected by oxidising or reducing substances. There is some confusion as to the proper meaning of the term pH. As stated at the beginning of the section the accepted definition is in terms of H3O+ activity. The term pHa=−log aH3O+ (6.5a.) reflects this definition, and activity standards are described above to calibrate ionselective electrodes. However, an analagous concentration term, pHc, can be generated pHc=−log[H3O] (6.5b.)

Calibration of ion-selective electrodes

59

The relationship between the two requires knowledge of the activity coefficient of H3O+, : (6.6.) Unless otherwise stated a ‘pH value’ will imply the pHa definition in keeping with most conventions. Under some circumstances it is necessary however to differentiate rigorously between pHa and pHc. For example stoichiometric dissociation constants of H+ binding to ligands will be important. use the term [H+], and hence pHc in which case knowledge of is given Although this is not of direct concern here a brief method to estimate according to relationships described by Harned & Owen (1958). The variation of with solution ionic strength, I, at 20°C in a KCl medium is given by the empirical relationship: (6.7.) For example an increase of ionic strength from 0.15 to 0.20 will decrease by about 2.0% (0.764 to 0.749) and the difference in pHa and pHc will vary by 0.01 units. In addition, Harned & Owen also provide tables and formulae to estimate the change of with temperature at constant ionic strength. For example increasing temperature by about 0.7% per 10°C change. between 20 and 40°C decreases

6.8. BUFFER SYSTEMS A H+ buffer system is a solution of definite pH in which the H3O+ activity is approximately constant despite variations of small amounts of acid or base, dilution of the solution, atmospheric impurities, etc. A buffer system consists of a weak acid or base and its salt so that subsequent addition of a strong acid or base will only influence the acid (base)/salt ratio of the weak electrolyte, rather than the pH. The equilibrium constant, K, for a weak acid, say, in a sufficiently dilute solution H3O++A−↔HA+H2O′ =1) is: (where (6.8.) In general the activities of A− and HA are replaced by concentrations, so that the pH of the buffer depends only on the ratio of salt/acid and the value of the equilibrium constant, K (note pK=−logK). The buffer capacity, β, defines the effectiveness of a buffer in maintaining the pH. It is defined as the ratio of the quantity of acid (cA moles) or base (cB moles) required to change the pH of a buffer solution by one unit, equation 6.9. (6.9.)

Ion-selective electrodes for biological systems

60

Algebraic manipulation of equations 6.8 and 6.9 yields the result of equation 6.10. (6.10.) where co is the total amount of buffer salt (i.e [HA]+[A−]). It shows that: • the buffering capacity increases with the total concentration of buffer in the solution • the buffering capacity is maximal when the pH of the solution equals the pK value. Figure 6.6A represents equation 6.10 in a graphical form and illustrates these two points by plotting the magnitude of β as a function of pH around the pK value. The plot shows the buffering capacity of a solution containing 1 mmol.l−1 or 5 mmol.l−1 of a H+ buffer with a pK=7.0. Part B of figure 6.6 looks at the data in a different way, in which the change of pH is plotted as a function of the initial pH in the two buffer solutions when a given quantity of a strong acid (100 µmol.l−1) is added. This plot again emphasises that the change of pH is least around the pK value and is smaller in the solution with a greater buffer concentration. Table 6.2 lists some buffers recommended by various conventions and their values at 37°C; listed also is the value of a secondary buffer, Tris (2-amino-2-hydroxymethyl-1, 3propanediol, no. 8) at 37°C. Table 6.3 lists the temperature dependence of the pH values for the reference standard, 0.5 molal KH phthalate. This solution is the primary standard against which others are compared and thus its characterisation is of greatest importance (section 6.7). Figure 6.7 plots the temperature dependency of the buffers 1–7 in table 8 as the change of value from that at 37°C. Some of these buffers, such as KH phthalate, are relatively temperature independent and thus fulfil one criterion as a primary buffer, whilst others have a larger variation with temperature.

Table 6.2. The pH value of some buffers at 37°C, note that these are given as molal concentrations. 1. Saturated (@25°C) KH tartrate

3.548

−1

2. KH2 citrate (0.1mol.kg )

3.756 −1

3. KH phthalate (0.05mol.kg ) −1

4.022 −1

4. Na2HPO4 (0.025mol.kg )+KH2PO4 (0.025mol.kg ) −1

6.839 −1

5. Na2HPO4 (0.03043mol.kg )+KH2PO4 (0.008695mol.kg )

7.392

6. Na tetraborate (0.1mol.kg−1)

9.093

−1

−1

7. NaHCO3 (0.025mol.kg )+Na2CO3 (0.025mol.kg )

9.912 −1

8. Tris(hydroxymethyl)aminomethane (0.01667mol.kg ) +Tris(hydroxymethyl)aminomethane hydrochloride (0.0500mol.kg−1)

7.382

Calibration of ion-selective electrodes

61

Figure 6.6. Part A shows that variation of buffering capacity as a function of pH for a simple buffer with a pK value of 7.0: plots are shown at two buffer concentrations. Part B plots the change of pH, as a function of initial pH, for the same buffer solutions when a fixed quantity of acid is added.

Ion-selective electrodes for biological systems

62

Table 6.3. pH values of 0.5 molal KH phthalate at temperatures between 0°C and 95°C. Temperature, °C

pH

Temperature, °C

pH

Temperature, °C

pH

0 4.000

25

4.018

65

4.097

5 3.998

37

4.022

70

4.116

10 3.997

40

4.027

75

4.137

15 3.998

45

4.038

80

4.159

20 4.001

50

4.050

85

4.183

25 4.005

55

4.064

90

4.21

30 4.011

60

4.080

95

4.24

Commercial pH electrode meters are equipped with several controls for calibration: a temperature compensator, an offset and slope control. The temperature compensator alters the gain of the buffer amplifier in the meter (see chapter 9) to correspond to the change of the Nernst factor as temperature alters (i.e. 2.303RT/F—equation 6.4). Therefore it is necessary to calibrate the pH electrode with buffers at the same temperature as pH measurements are taken in the experimental solutions.

Figure 6.7. The temperature dependency of different H+ buffers. The difference of pH from that

Calibration of ion-selective electrodes

63

measured at 25°C is plotted. The inset shows the values of buffer pH at 25°C. The pH electrode is placed in a buffer near to the range of experimental pH solutions and the offset control used to adjust the meter reading to the test buffer pH value. If the experimental pH values are likely to vary over a wide range—i.e. several pH units—a second calibration buffer solution is then used and the slope control used to adjust the meter reading to the appropriate value. It is then advisable to return to the first buffer solution and check the meter reading again, adjusting with the offset control if necessary. This process may have to be repeated several times if the initial meter readings are substantially different from the buffer values.

6.9. CALIBRATION IN BUFFERED SOLUTIONS OF OTHER PRIMARY IONS Certain situations require electrodes to be calibrated and used in solutions in which the concentration of the primary ion is less than that which can be achieved by mere dilution of standards of the appropriate salt. This arises from the need to make solutions containing very small concentrations of the primary ion (
Ion-selective electrodes for biological systems

64

The first approach is to calculate the absolute binding constants of every possible combination of ions and ligands in solution, at different temperatures, ionic strength, etc., and to calculate the final free [Ca2+] by means of an iterative computer program. Various programs are available but are limited to the species for which the program has been written. These programs will therefore predict the free concentration of the ion of interest if the free concentrations of all other species in the solution are known. Examples of this approach include the estimation of Ca2+ binding to a single buffer (EGTA, Harrison & Bers, 1989) and Mg2+ binding to multiple ligands (Lüthi et al., 1999). The advantage of these techniques is that they have great predictive powers. However a significant practical problem is that the exact concentration of the reactants are not always known. For example, EGTA as supplied by chemical companies frequently has a small amount of water associated with it so that the concentration of pure EGTA will be underestimated if it assumed to be completely free of water. An error of about 5% will have significant effects on the calculation of the free [Ca2+] if this particular problem is ignored. The alternative is to calculate an apparent value of appropriate to the solution used and substitute the value into equation 6.12. The value obtained is merely an empirical description of the solution but has the merit that it is readily estimated experimentally and is appropriate to the solution that will be used in a given situation. This method will therefore be discussed and is considered to be successful as it is easy to perform and is reproducible. The method describes an empirical determination of the apparent equilibrium constant, , for the reaction between Ca2+ and EGTA as well as the total EGTA concentration in an aqueous solution containing several other ions that may bind to the buffer, so that the [Ca2+], or pCa, can be determined. The most important requirement is for an accurate CaCl2 standard solution for titrating the EGTA-containing solution. The presence of other ligands such as ATP will yield a more complex Ca2+ binding solution and may not give such a simple analysis. The example here uses a physiological solution containing; KCl 120 mmol.l−1; NaCl 10 mmol.l−1; MgCl2 1 mmol.1−1; HEPES 20 mmol.l−1 buffered to pH 7.2 with KOH and nominally 4 mmol.l−1EGTA. The experiment was performed at room temperature (22°C). The true EGTA concentration may be less than that calculated from the weighed quantities due to water and other impurities in the stock—the procedure will approximate the correct concentration. If the two values differ greatly the whole calculation can be iteratively repeated until a reliable estimate is achieved—this is not performed here. A Ca2+ and a H+ -selective electrode, along with a reference electrode, are placed into a well-stirred sample of the above solution. Samples from the CaCl2 stock solution are added to the EGTA solution and the pH titrated back to 7.2 after the addition of each sample. Additions are repeated until excess CaCl2 has been added. In this example CaCl2 is added until a total concentration of approximately 6 mmol.l−1 is achieved (nominally 2 mmol.l−1 free Ca2+). The precise volume is known after each step and the p.d. recorded by the Ca2+ -selective electrode measured. The results of such an actual experiment are shown in table 6.4. It is important to monitor the pH throughout this procedure. Addition of CaCl2 to an EGTA-containing solution releases H+ (equation 6.10) which diminishes the amount of Ca2+ which binds to EGTA. Thus the apparent binding constant will be a function of pH and must be carried out at a stable value. In addition, if working to a recipe in which a known amount of CaCl2 is added to EGTA to obtain a nominal pCa

Calibration of ion-selective electrodes

65

value, the solution will generate H+ and so increase the free [Ca2+] (reduce the pCa) compared to the desired value. The correct value will be achieved when the pH is titrated back to the original value (approximately 2 moles of H+ per mole of Ca2+ at pH values near to 7.0). The first two columns list the volumes of CaCl2, from a 1 mol.l−1 or 0.1 mol.l−1 stock solution, added to an initial 2ml of the solution. The third and fourth column list the volumes of KOH added to maintain constant the pH after CaCl2 addition. For the initial CaCl2 additions twice the volume of KOH must be added (using equal CaCl2 and KOH stock concentrations) but as the titration proceeds the amount of KOH required diminishes and eventually vanishes as the EGTA is fully in the Ca-bound form. The fifth, sixth and seventh columns show respectively the final volume after each set of additions, the quantity of Ca in the reaction chamber and the final Ca concentration. The eighth column records the potential difference, ECa, recorded by the Ca2+ -sensitive electrode, with the value at the final Ca concentration arbitrarily set at zero. It is assumed initially that the electrode is perfectly Nernstian (29.5mV per ten-fold change of [Ca2+] or 29.5mV per unit change of pCa value). Column 9 is the measured voltage, ECa, divided by 29.5. It is also assumed that the initial [EGTA] was 4 mmol.l−1 (8 µmol added) and a total of 12 µmol Ca2+ has been added. An excess of 4 µmol Ca2+ will be in a final volume of 2.085ml solution to give a [Ca2+] of 1.918 mmol.l−1 (final row of table 8), i.e the pCa is (−log10(0.001918)= 2.717. If this value is added to those in column 9 then the estimated pCa of the previous solutions are obtained as shown in column 10.

Table 6.4. An experiment to determine for an EGTA-containing physiological solution. CaCl2

K KOH

1M 0.1M 1M 0.1M Vol µ1 µ1

µ1 µ1

ml

Ca

Ca

ECa −1

µmol µmol.ml

ECa/

pCa

mV 29.5

Ca free

Ca bound Ca −1

µmol.ml

µmol.ml−1 bd/free

0

0

0

0 2.000

0.0

0.000

136.0 4.610 7.328

1

0

2

0 2.003

1.0

0.499

135.5 4.593 7.311

0.000049

0.499

10184

1

0

2

0 2.006

2.0

0.997

132.0 4.475 7.193

0.000064

0.997

15578

1

0

2

0 2.009

3.0

1.493

128.0 4.339 7.057

0.000088

1.493

16966

1

0

2

0 2.012

4.0

1.988

122.5 4.153 6.871

0.000135

1.988

14726

1

0

2

0 2.015

5.0

2.481

115.0 3.898 6.616

0.000242

2.481

10252

0

5

1

0 2.021

5.5

2.721

110.5 3.746 6.464

0.000344

2.721

7910

0

5

1

0 2.027

6.0

2.960

104.5 3.542 6.260

0.000550

2.959

5380

0

5

1

0 2.033

6.5

3.197

98.5 3.339 6.057

0.000877

3.196

3644

0

5

1

0 2.039

7.0

3.433

90.0 3.051 5.769

0.00170

3.431

2018

0

2

0

4 2.045

7.2

3.521

84.5 2.864 5.582

0.00262

3.518

1343

0

2

0

3 2.050

7.4

3.610

71.25 2.415 5.133

0.00736

3.603

490

Ion-selective electrodes for biological systems

66

0

2

0

3 2.055

7.6

3.698

66.0 2.237 4.955

0.0111

3.687

332

0

1

0

1 2.057

7.7

3.743

59.0 2.000 4.718

0.0191

3.724

195

0

1

0

1 2.059

7.8

3.788

50.5 1.712 4.430

0.0372

3.751

101

0

1

0

1 2.061

7.9

3.833

42.5 1.441 4.159

0.0693

3.764

54

0

1

0

1 2.063

8.0

3.878

37.0 1.254 3.972

0.1067

3.771

35

0

2

0

0 2.065

8.2

3.971

30.0 1.017 3.735

0.1841

3.787

21

0

2

0

0 2.067

8.4

4.064

25.0 0.847 3.565

0.2723

3.792

14

0

4

0

0 2.071

8.8

4.249

18.5 0.627 3.345

0.4519

3.797

8

0

4

0

0 2.075

9.2

4.434

14.0 0.475 3.193

0.6412

3.793

6

0

4

0

0 2.079

9.6

4.618

11.0 0.373 3.091

0.8110

3.807

5

0

4

0

0 2.083

10.0

4.801

8.5 0.288 3.006

0.9863

3.815

4

2

0

0

0 2.085

12.0

5.755

0.0 0.000 2.717

1.914

3.841

2

1

2

3

4

6

7

11

12

13

5

8

9

10

Column 11 is the calculated free [Ca2+] from column 10, here expressed in units of µmol.ml−1. The total Ca added (µmol.ml−1) is known from column 7 thus the amount bound to EGTA is obtained from subtracting column 11 from column 7 to yield the values in column 12. Finally column 13 is the calculated ratio of Ca bound/Ca free. The next step is to plot the ratio Ca bound/free as a function of the Ca bound as shown in figure 6.8. The plot shows a linear region and a curved portion at top left; the latter points represent those obtained at the lowest free [Ca2+], i.e. the highest pCa values, and reflect the non-Nernstian behaviour of the electrode. The above calculations were made assuming the electrode to be Nernstian in behaviour and so it is reasonable to ignore these values. The slope of the linear part of the plot is an estimate of K’Ca and the intercept with the x-axis a value of the EGTA concentration. The box shows the derivation of these estimates. At constant pH: EGTA+Ca↔Ca.EGTA Thus: K’Ca=[CaEGTA]/ [Ca].[EGTA] The bound form (B) of Ca is the [CaEGTA]. The free form (F) of Ca is the [Ca]. The total (T) EGTA concentration is /CaEGTA/+[EGTA]—i.e.[EGTA]=(T −B). Thus: The final form of the equation is of a straight line (y=mx+c). Thus a plot of (B/F) on the ordinate as a function of B on the abscissa yields a line of slope—K’ca and intercept with the x-axis (B-axis) of value T—the total EGTA concentration. A straight line has been fitted through the linear part of the plot yielding a slope of −7.814×106M−1 (pK’Ca value is 6.89) and an x-axis intercept of 3.76 µmol.ml−1 Ca. The latter is an estimate of the true total concentration of EGTA, as this will be all in the CaEGTA (Ca-bound) form at the highest Ca concentrations used in the experiment. The

Calibration of ion-selective electrodes

67

amount of EGTA added was estimated to be was 3.84 µmol.ml−1 when all additions had been made, and was not greatly different from the calculated amount from figure 6.8. If this estimated total EGTA concentration [EGTA]T is greatly different from the assumed value it would be necessary to go through the cycle of calculation again using the new estimate. However, the similarity of values does not warrant this here. The values of K’Ca and the total [EGTA]T can now be inserted into equation 6.13 to obtain calibration solutions with different pCa values: (6.13) Remember that the [CaEGTA] is virtually identical to the added [Ca] and the total [EGTA]T=[CaEGTA]+[EGTA], from which the free [EGTA] is calculated. When these solutions are manufactured they can be used to calibrate a Ca2+-selective electrode, a calibration curve of which is illustrated in figure 6.9.

Figure 6.8. A plot of the ratio of bound Ca/free Ca2+ as a function of the bound Ca in a solution containing nominally 4 mmol.l−1 EGTA as a Ca2+ buffer.

Ion-selective electrodes for biological systems

68

Figure 6.9. The final Ca2+ -selective electrode calibration curve obtained from the above procedure to generate solutions of known pCa values. The straight-line was obtained using all the data points except the final three values at high pCa and the two arrowed points -see text for details. The straight line in figure 6.9 was obtained by least squares regression of all data points except the final three values where the electrode was deemed non-Nernstian from figure 6.8 and the two arrowed points. The slope was about 28.5mV per 10-fold change of [Ca2+] (i.e. 28.5mV per pCa unit). At pCa <5.0, When the Ca added was greater than the total EGTA (at pCa< 5.0 approximately), the pCa was calculated from the sum {Ca added-total [EGTA]}. The two arrowed points represent the region of the curve which is most inaccurate, as judged by the relative noise of the data points. This is because it is where the EGTA is just saturated with Ca, thus a slight error in the amount of added Ca will lead to large errors in the pCa.

7. THE PRESENTATION OF DATA OBTAINED WITH ION-SELECTIVE ELECTRODES 7.1. INTRODUCTION The calibration curve is used to translate the p.d. recorded by an ion-selective/ reference electrode pair into a concentration value of the primary ion. Previous sections have discussed the theoretical influence of interfering ions on the calibration curve and described methods of estimating the severity of the problem (sections 3.1 and 6.4). Appendix 3 shows the conventions used to report data obtained from ion-selective electrodes as well as the correct ways to present numerical data and units. It is recommended that these conventions be adhered to so that reports between different authors will be consistent. The following sections show some practical data which determine the accuracy to which an estimate of concentration can be made in a test solution with an ion-selective electrode. The degree of accuracy will obviously depend on the particular ion-selective electrode used and the data illustrated here are derived both from experiments using dipcast Ca2+ -selective electrodes and ion-selective micro-electrodes. Three factors are considered: • the effect of drift as a function of time and as a function of exposure to test solutions. • accuracy of the determination of the concentration of the primary ion in the presence of interferent ions. • the correct way to analyse concentration values obtained with ion-selective electrodes.

7.2. STABILITY OF ELECTRODES Figure 7.1 shows an experiment in which a Ca2+ -selective electrode/saturated KCl reference electrode pair was placed in a nominal 1 mmol.l−1 CaCl2+ 140 mmol.l−1 NaCl solution buffered to pH 7.0 with 10 mmol.l−1 HEPES. The left axis shows the recorded p.d., ECa, the right hand axis a ‘[Ca2+]’. The ECa values were recorded to an accuracy of 0.1mV, which represents about 2% of the total drift in this experiment. The 1 mmol.l−1 value on the [Ca2+] scale is placed opposite the final time point when the electrode p.d. had stabilised to a constant level. The remaining ‘[Ca2+]’ values are calculated from the earlier ECa values, using a calibration curve previously obtained by the electrode in the test solution. The important point to note is that the drift in the ECa values initially overestimates the true [Ca2+] by as much as 45%. The cause of the drift in such

Ion-selective electrodes for biological systems

70

experiments cannot be determined from such observations, but the most likely sources are drift of the potential at the Ag/AgCl interface with the filling solution and at the ionselective membrane with the test solution. The latter phenomenon is especially evident when a new electrode is placed in a test solution.

Figure 7.1. The drift recorded by a Ca2+ -selective electrode when placed in a test solution containing 1.0 mmol.l−1 CaCl2 and 150 mmol.l−1 NaCl. An electrode may also be affected in an unknown way by exposure to a test solution. One way to estimate the magnitude of such an effect is to place the electrode in a calibrating solution, containing a high concentration of the primary ion, before and after exposure to the test solution. The high concentration of primary ion minimises the effects of interferent ions. Any systematic difference in the readings before and after exposure to the test solution will indicate an influence of that solution on the electrode. Table 7.1 shows the difference in potential (∆mV, mean ±1 S.D.) recorded by several different types of ion-selective electrodes before and after exposure to a urine sample. Only the selective electrode showed a ∆mV which was statistically different from zero, suggesting that this electrode alone was systematically influenced by this test solution.

The presentation of data obtained with ion-selective electrodes

71

Table 7.1. The reproducibility of response after ionselective electrode exposure to a test solution (urine). The data are shown as the difference in potential (∆mV) in a standard calibrating solution after exposure to urine. *p<0.05, difference of ∆mV from zero, Mann-Whitney U-test. (S.J.Wood and C.H.Fry unpublished data). Ion-selective electrode

Ca2+

Mg2+

K+

NH4+

glass pH

Calibrating

CaCl210

MgCl210

KCl 50

NH4Cl 20

pH 7.0 Corning

solutions

NaCl 140

NaCl 140

NaCl 100

NaCl 130

standard

(mmol.l )

HEPES 10

HEPES 10

∆mV

0.78±1.26

0.76±1.24

0.36±2.92

2.01±1.57*

−0.06±0.1 3

n=22

n=23

n=22

n=22

n=23

−1

HEPES 10

7.3. ACCURACY OF A READING IN THE PRESENCE OF INTERFERENT IONS The deviation of an ion selective electrode p.d. from a Nernstian slope becomes continuously greater as the concentration of the primary ion falls in relation to the concentration of interferent ions. If the concentration of interferent ion changes this will generate a variable effect on the ion-selective electrode p.d. which will become more and more important as the concentration of primary ion reduces. This has been dealt with on a theoretical basis in section 3.1, with reference to the Nikolsky equation. The problem is illustrated in part A of figure 7.2 where calibration curves for an imperfectly-selective electrode are shown, in the presence of varying concentrations of interferent ion—highest for curve b and least for curve c. A measured p.d. (here −80mV) is transformed to different primary ion concentrations depending on the concentration of the interferent ion, [J]. Let the electrode be calibrated in a solution with a value of [J] to give curve a. If the test solution contains the same [J] then curve a can be used and the concentration of the primary ion read using line 1. If [J] is greater, curve b would be appropriate and the concentration should be read via line 2, but if curve a is used a falsely high value of the primary ion concentration would be obtained. Conversely if [J] is lower, curve c and line 3 would be appropriate but if curve a is again used the value of the primary ion concentration would be underestimated. As a practical example, measurements of the [Ca2+] were made in urine in which the concentration of the most important interferent ions, Na+ and K+, varied several-fold. It has been assumed that the interferent effects of Na+ and K+ are equivalent (which is approximately true), so that the interferent concentration is expressed as Σ([Na+]+[K+]). The Ca2+ -selective electrode was calibrated in a solution containing 150 mmol.l−1 NaCl. This was approximately equal to the urine Σ([Na+]+[K+]) of 151±125 mmol.l−1 (mean ±2

Ion-selective electrodes for biological systems

72

Figure 7.2. Part A: calibration curves of an ion-selective electrode in the presence of varying concentrations of an interferent ion, J; curve b has the highest [J], curve a intermediate [J] and curve c lowest [J], see text for

The presentation of data obtained with ion-selective electrodes

73

further details. Part b: the magnitude of error if estimating the [Ca2+] in a urine solution containing variable [Na] and [K], see text for details. S.D.’s). Thus the electrode would generate a smaller p.d. if placed in a urine with a Σ([Na+]+[K+]) less than 150 mmol.l−1, but with the same [Ca2+] as the calibration solution. This would therefore underestimate the [Ca2+] as read from the calibration curve. The reverse argument holds if the Σ([Na+]+[K+]) is greater than 150 mmol.l−1. Figure 7.2B illustrates the magnitude of the problem using such data; the electrode in this case have a preference for Ca2+ over Na+ or K+ of 500:1. The plot shows the percentage error in estimating the [Ca2+] as a function of the actual [Ca2+] plotted on the abscissa, and shows as expected that the error increases as the concentration of the primary ion decreases. The lines show the magnitude of over- or under-estimation of the [Ca2+] if the concentration of interferent ions in a particular sample is unknown but assumed to be within 2 S.D.’s (i.e. 95%) of the mean value. For example, if the actual [Ca2+] is 1 mmol.l−1, then the measured value can be under-estimated by up to 4% or over-estimated by up to 10%. Alternatively for a measured value of 1.0 mmol.l−1 the actual value may be between 0.90 and 1.04 mmol.l−1.

7.4. PRESENTATION OF DATA—LINEAR OR LOGARITHMIC DISTRIBUTIONS? If a series of determinations are made with ion-selective electrodes, the measured p.d.’s are converted to concentration values via a calibration curve. This manoeuvre however is not a linear transformation as there is a logarithmic relationship between p.d. and concentration—i.e. the Nernst or Nikolsky equations. The usual method of presenting such data is to calculate the mean value of the concentration terms with a standard deviation. This procedure however assumes that the set of concentration values is normally distributed, even though it has been derived from the experimental variable, the p.d., by a non-linear transformation. Other ways to treat the data would be to calculate the mean ±S.D. of the measured p.d.’s and then estimate the concentration, via the calibration curve from the mean p.d., with a range derived from the S.D. of the p.d. data set. Alternatively, the mean ±S.D. of the pIon values could be calculated and these converted to a mean concentration with a range derived from the S.D. of the pIon data set. Figure 7.3 illustrates the three methods. Is this a significant problem? A rather extreme example illustrates that it can be. A perfect K+ -selective electrode is initially calibrated in solutions of varying [K+] and the following p.d. (EK) values obtained, as shown in table 7.2. The K+-selective electrode is now placed in six test solutions to determine the mean value of the [K+]. The second experimental column lists the EK’s and the third column the corresponding [K+], as read from the calibration curve. The final column lists the pK values calculated in the same way as a pH—i.e. pK=−log10[K+] (note: do not confuse this with the pK term used in chemical reactions, (e.g. sections 6.8 and 6.9).

Ion-selective electrodes for biological systems

74

Figure 7.3. The three possible methods to obtain a mean value of an ion concentration from a set of ionselective electrode measurements. The mean value of the [K+] can be calculated in a number of ways. Method The mean EK± S.D. is calculated and this value used to estimate the [K+] from the 1. calibration curve. This is done in the first experimental column to give a mean [K+]=9.7 mmol.1−1. The values in parenthesis are the [K+] from the mean EK±1 S.D. Method The mean of the individual [K+] values is calculated (second experimental column) which 2. gives a value of 11.0±5.6 mmol.l−1. This is the method most people choose. Method The mean ±S.D. pK values calculated and these transformed to corresponding [K+] (third 3. experimental column). This gives a value of 9.8 mmol.l−1 with a range from 5.8 to 16.6 mmol.l−1 from ±1 S.D. of the mean pK value. These values are similar to those in the first

The presentation of data obtained with ion-selective electrodes

75

experimental column (they would be identical except for rounding-off errors). See also appendix A2.3 (p. 122) for the calculation of S.D. values from logarithmic variables.

Table 7.2. An experiment to determine the [K+] in a sample population of test solutions with an ionselective electrode. Calibration +

[K ] mmol.l

Experiment

−1

EK, mV

Test soln

[K+]

EK, mV

pK

1

10.0

1

50

4.5

2.35

2

28.5

2

60

5.6

2.19

5

53.0

3

70

9.5

2.02

10

71.5

4

75

11.4

1.94

20

90.0

5

80

13.8

1.86

50

114.5

6

90

20.0

1.70

100

133.0 Mean ±S.D.

70.8±14.3

11.0±5.6

2.01±0.23

−1

Measured

9.7mmol.l

11.0mmol.l

[K+]

(5.7–16.6)

(5.5–16.5)

−1

9.8mmol.l−1 (5.8–16.6)

Clearly there is a discrepancy, the mean [K+] of 9.7 mmol.l−1 calculated from the first and third methods is different from a [K+] of 11.0 mmol.l−1, obtained by the second method. The following analysis attempts to show that the value of 9.7 mmol.l−1 is a correct estimation—in brief this is because the measured EK values are a normally distributed data set—so that a mean and a S.D. can be safely calculated, but the [K+] values are not normally distributed due to the logarithmic transformation inherent in the calibration curve. The pK values, on this account are also normally distributed as they are linearly related to the EK values.

7.5. TESTING THE DATA SETS The following examines whether the experimentally determined p.d.’s, or the derived concentration values are normally distributed. Such tests are best performed on large data sets and the examples are accumulated data from many different experiments using ionselective electrodes. Figure 7.4 shows data obtained with intracellular Na+ -selective micro-electrodes when used to measure the concentration of the ion in the sarcoplasm of cardiac muscle at 37°C. Part A shows a histogram of the measurements from

Ion-selective electrodes for biological systems

76

Figure 7.4. Part A; histogram of values of the intracellular [Na+] measured with ion-selective micro-electrodes in mammalian cardiac muscle at 37°C. Part B: cumulative frequency distributions of a normal distribution

The presentation of data obtained with ion-selective electrodes

77

(open circles and continuous line) with mean and S.D. derived from the data in part A and the data set (closed circles). The value of the maximum difference, |d|, is shown. 181 determinations and it is evident that it is skewed with a tail of high values. This is verified by using a Kolmogorov-Smirnov test for normality which is shown in part B of the figure and is based on the absolute difference between expected and observed cumulative frequency distributions. The continuous line is a cumulative frequency plot for a normal distribution (open circles), calculated using the mean and S.D. values of the above data set by reference to an appropriate table. The actual data are shown as closed circles and the two data sets would superimpose if this data set was normally distributed. The degree of difference is a measure by which the observed data deviate from a normal distribution. The statistic in the test which expresses this deviation, |d|, is the value of the largest absolute difference between the two curves and has a value of 0.144 in figure 7.4B. This value of |d| can be used to accept or reject the hypothesis that the data are normally distributed by comparing it to critical values for different probability levels, p, and number of data points, n. If |d| is greater for a particular p value and n number, then the null hypothesis that the data set is normally distributed can be rejected at that level. In this case for |d|=0.144 and n =181 indicates that the observed distribution is significantly different from a normal distribution (p<0.005). In figure 7.5 is plotted similar data for the p.d. recorded by the intracellular microelectrodes, ENa, before the values were transformed to concentration values. In this case the histogram appears to be more normally distributed (part A), which is confirmed by the Kolmogorov-Smirnov normality test (part B). The test statistic has a smaller value of |d|=0.060 in this case, which for n=181 suggests that this set of data is indistinguishable from a normal distribution (p>0.10). Similar calculations have been done other data sets where intracellular Na+, H+, Ca2+, Mg2+ have been measured in cardiac or skeletal muscle at different temperatures. The results are summarised below in table 7.2. The |d| statistic for each set is shown along with the probability that the set is normally distributed (p>0.05) or is not distributed thus (p<0.05). The following conclusions can be drawn about each method illustrated in figure 7.3.

Ion-selective electrodes for biological systems

78

Figure 7.5. Part A; histogram of ENa values measured with ion-selective micro-electrodes in mammalian cardiac muscle at 37°C. Part B: cumulative frequency distributions of a normal distribution (open circles and

The presentation of data obtained with ion-selective electrodes

79

continuous line) with mean and S.D. derived from the data in part A and the data set (closed circles). The value of the maximum difference, |d|, is shown. Table 7.3. Summary of the Kolmogorov-Smirnov statistic |d|, for the data sets of ion-selective electrode measurements of different intracellular ions. The probability, p that the distribution is normally distributed as well as the number of measurements n, in each set is also tabulated. |d| ENa

0.081 +

p >0.10

n 181

Concentration 10.7(7.8–14.7) mmol.l

−1

0.130

<0.005

11.3(14.9–7.9) mmol.l

pNa

0.081

>0.10

10.7(7.8–14.7) mmol.l−1

ENa

0.060

>0.10

143

14.1(8.4–23.7) mmol.l−1

0.128

<0.025

15.9(8.1–23.7) mmol.l

pNa

0.061

>0.10

14.1(8.4–23.7) mmol.l−1

EH

0.045

>0.10

210

52.6(36.1–76.6) mmol.l−1

0.147

<0.001

56.9(31.6−82.7) mmol.l

pH

0.067

>0.10

52.6(36.2–76.6) mmol.l−1

ECa

0.126

>0.10

34

227(134–385) mmol.l−1

0.259

<0.05

285(116–453) mmol.l

pCa

0.147

>0.10

251(156–406) mmol.l−1

EMg

0.076

>0.10

[Mg ] pMg

0.149 0.129

>0.10 >0.10

22–25°C

Heart

22–25°C

Heart

37°C

Heart

18–20°C

Skeletal

−1

[Ca ]

2+

Heart

−1

[H ]

2+

37°C

−1

[Na ]

+

Tissue

−1

[Na ]

+

Temperature

38

0.93(0.35–2.45) mmol.l−1 −1

1.35(0.29–2.43) mmol.l

Muscle

−1

0.95(0.36–2.49) mmol.l

It is evident that concentration values derived from the EIon (method 1, section 7.4) or pIon (method 3) sets are less than from the mean of the [Ion] (method 2) data sets. Table 7.3 shows that large errors can be engendered; for example, the intracellular [Mg2+] is 45% larger by Method 2, compared to Method 1, whilst for the other ions the difference is between 6 and 15%. Method 3 is the one favoured by the authors, it allows a running average of a normally distributed set of data to be accumulated.

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Method The p.d.’s recorded by the ion-selective electrodes, EIon, are normally distributed sets, so 1. that the mean ±S.D. can be calculated. The mean concentration values in table 7.2, corresponding to each EIon row, are derived from the mean EIon value and using an averaged calibration curve. The values in parenthesis are the concentrations calculated from ±1S.D. of the mean EIon value. Method The sets of concentration values are not normally distributed, except for the Mg2+ data 2. where it cannot be rejected that it is normally distributed. The concentration values in table 7.2 are the means ±1 S.D., assuming the data sets are normally distributed. Method The data sets of pIon values are normally distributed. The concentration values are 3. obtained from the mean pIon value and the range in parenthesis from ±1S.D. of the mean pIon value.

8. INSTRUMENTATION A number of commercial systems are suitable for recording the p.d.’s generated by ionselective/reference electrode pairs. At their most basic they are based on the high input impedance voltmeters used in conjunction with commercial glass pH electrodes. Alternatively it is possible to manufacture measuring systems from individual electronic components. Financial considerations have a bearing on which course to take. However, if you feel confident enough to manufacture your own recording system, it gives you flexibility in tailoring the apparatus to your own experiment. Whichever course is taken it is important to understand the characteristics required of such measuring systems and this will be considered below. Most commercial systems are accompanied by detailed technical specifications and these should be checked that they comply with the requiremnts given here.

8.1. GENERAL REQUIREMENTS OF OPERATIONAL AMPLIFIERS The electrode system comprising an ion-selective electrode and reference electrode form a signal source that is recorded via a buffer amplifier. In general the voltages generated are large enough to be handled by recorders but the current is often too small. Therefore current amplification is the first requirement of the signal and is achieved by means of a

Figure 8.1. The circuit diagram of a unity gain buffer amplifier and the pin layout of an 8-pin commercially available operational amplifier integrated circuit.

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buffer amplifier positioned as close as possible to the electrode system. If this stage does not produce any voltage amplification (unity gain) this stage is often referred to as a voltage follower or impedance converter The circuit of such a unity gain buffer amplifier is illustrated in the left-hand diagram of figure 8.1. The triangle is a symbol for a linear integrated circuit operational amplifier (op amp). The ‘+’ sign represents the noninverting input to the op amp and the ‘−’ sign the inverting input. Connected in this way the output signal is of the same polarity as the input. Operational amplifiers are commercially available in general as 8-pin components and in the right-hand diagram of figure 8.1 the minimal practical connexions can be seen. In the arrangement shown in figure 8.1 the op amp behaves as a so-called voltage follower, the voltage gain is unity, i.e. Vin=Vout. The purpose of this arrangement is to isolate the source of the signal (the ion-selective electrode) from the recording systems and for this reason it is often called a buffer amplifier. There is a wealth of op amps available, at a variety of prices, and it is necessary to choose the appropriate one. In general, two specifications are worth looking at: input impedance and input bias current, if these are adequately met then others are unlikely to be limiting in this context.

8.2. INPUT IMPEDANCE AND FREQUENCY RESPONSE The terms impedance and resistance are often used interchangeably. Resistance, R, is the ratio of voltage, V, to current, I, through a device when the current is a steady value— direct current, or d.c.. If however I varies as a function of time (say as a sine-wave, alternating current or a.c.) the V/I ratio is then called an impedance, Z. The value of Z is often frequency-dependent, usually diminishing as the frequency increases and so it is important to specify at which frequency Z is measured. Thus impedance is a more general term and resistance can be considered to be a particular value of impedance when the current frequency is zero. In general, the term impedance will be used in this chapter. The input impedance of an op amp is an indication of how well it prevents current from flowing in the input circuits and must be at least two orders of magnitude greater than the source impedance (ion-selective electrode impedance) so that the signal is not attenuated by the op amp. In general dip-cast electrode impedances are not greater than 10 MΩ (107Ω) and usually much less, so that an op amp input impedance of greater than 109Ω is needed. Field effect transistor (FET) input op amps have impedances of about 1012Ω and so will be adequate. The 741 op amp has an input impedance of comparable size to the impedance of the ion-selective electrodes and so will be inadequate for the purpose. Note also that conventional electrolyte-filled micro-electrodes used for intracellular recording have impedances of between 107 and 108Ω so that FET op amps may also be used for buffer stages with these micro-electrodes. The junction-FET (JFET) type 351 and the CMOS-FET type 7611 have both been used in circuits employing dip-cast ion-selective electrodes. However, ion-selective microelectrodes have much greater impedances, 1011−1012Ω and require op amps with a substantially greater input impedance (see sections 8.4 and 8.8). A quick check to ensure that the input impedance of the op amp has not diminished is to initially check the gain of the amplifier (sections 8.1 and 8.5) with no resistor in series

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with the input. Then place larger and larger value resistors in series with the input (source impedances) and record the magnitude of the output signal, Vout. When Vout starts to decline the input resistance of the op amp is similar to the source impedance. Check that the value of the resistor when Vout starts to decline is not smaller than that of the ionselective electrode you would normally have connected to this input. There are more accurate ways to measure the actual value of the input impedance, but this quick practical check is often all that is needed. Another characteristic that needs to be considered is the frequency response that is required of the system. Each circuit can be tested by applying a sine-wave to the input via a resistor that is a similar value to that of the ion-selective electrode and the output sine wave recorded. As the frequency of the signal is raised the output voltage remains constant until a certain frequency whereafter the magnitude gradually falls. The useful frequency range of the circuit is that over which the input signal is not attenuated. A plot of output signal amplitude as a function of frequency can be seen later in figure 8.7 (section 8.6) and is known as a Bode plot. In general, frequency responses of several hundred Hertz (cycles/second) can be obtained, which is adequate for most uses. Remember that the useful frequency range diminishes as the source impedance increases. Thus although the frequency range can look impressive with small source impedances these will be seriously reduced as source impedance rises. It is extremely important with any new device to check the frequency range with a source impedance that approximates that of the ion-selective electrode.

8.3. INPUT BIAS CURRENT With an ideal op amp the resistance between the two inputs should be infinite when they are unconnected (open circuit) and the voltage between them zero. In the real case the internal circuitry generates a small current at each input. If ib+ and ib− are these small currents at the non-inverting and inverting inputs respectively, the input bias current, ib, is the average of ib+ and ib−. If an impedance, Z, (i.e. an electrode) is placed on say the noninverting input a voltage ib+. Z will be generated. In general if either of these two variables did not vary there would be no problem because the output is referenced to another (arbitrary) voltage at the reference electrode. If however Z changed then this would alter this voltage and would manifest itself as drift if the change was slow. In general therefore it is important that ib is low enough not to generate a significant error. Examples will illustrate the case. FET op amps have an input bias current of ≈1pA (10−12A), or a bit greater. If the electrode resistance is 1MΩ a voltage of 1µV will develop. If the electrode resistance is 1GΩ (109Ω) this voltage would be 1mV. An increase in the electrode resistance to 5GΩ, because of gradual deterioration, would increase the input offset voltage to 5 mV. This drift of 4mV must not therefore be of a significant magnitude compared to the true change of potential at the ion-selective electrode. The input bias current can be measured using the circuit shown in figure 8.2. Resistors R1 and R2 can be shorted using the switches S1 and S2. The capacitors are used for practical reasons to reduce transient effects.

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When S1 and S2 are closed the circuit is a voltage follower. Any output voltage, Vout, will be due to internal imbalances in the two inputs and is termed the input offset voltage, Vio. When S1 is open and S2 is closed Vout is now Vio+R1.ib+ When S1 is closed and S2 is open Vout is now Vio+R2.ib− The values of ib+ and ib− can be calculated from the above measurements. Values of R1 and R2 have to be chosen appropriate to the likely values of ib. If ib is 1pA values of R1 and R2 should be about 109Ω and larger values for smaller ib.

8.4. SPECIAL CONSIDERATIONS FOR ION-SELECTIVE MICROELECTRODES When using ion-selective micro-electrodes the very much greater impedance of these electrodes demands different amplifiers for the buffer stage. Such ion-selective microelectrodes typically have resistances of about 1011Ω which is very much greater than the resistance of conventional electrolyte filled microelectrodes (section 8.2). Thus, the input impedance of a typical FET op amp would be too low and the input bias current too high for this purpose. The conventional solution is to use an operational amplifier with a very high input impedance, 1014Ω and greater, and an ultra low bias current, about 10−14A or 10 femptoamps (10fA). Such op amps—often called electrometer amplifiers-are more expensive and require careful handling but can be used adequately with ion-selective microelectrodes. Analog Devices op. amps. such as AD515L and AD 524L are suitable

Figure 8.2. The arrangement for measuring the input bias currents at the non-inverting and inverting inputs of an op amp.

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and both types can be used in a voltage follower arrangement to record from ion-selective micro-electrodes. However, it is advisable to measure the input characteristics of these op amps from time to time. Often the input impedance can deteriorate gradually, either by damage to the device of by accumulation of dust and moisture on the input pins of the op amp. This deterioration may not be large enough to prevent the recording system from ceasing to function completely but may merely attenuate the signal and especially reduce its frequency response. The consequence of this deterioration in input characteristics is that the ion-selective micro-electrode generates a sub-Nernstian and sluggish response to the primary ion. Frequency response with an appropriate source impedance, input resistance and input bias current should all be checked to see if they are in the reasonable range. Note that when testing electrometer amplifiers a very high source impedance will have to placed on the input stage 1010–1011Ω. It is possible to buy therse resistors from specialist suppliers but care should be taken when they are handled. A small amount of grease from fingers can reduce dramatically the resistance by forming a surface shunt pathway. Clean the resistor surface with alcohol and wear gloves otherwise you may be deluding yourself.

8.5. OPERATIONAL AMPLIFIERS WITH GAIN It is also possible to use an op amp as a non-inverting amplifier to increase the voltage signal. The arrangement is shown in figure 8.3. In this case the input resistance is again that of the op. amp but the output voltage, Vo, is greater than the input voltage, Vi, by a factor;

Figure 8.3. A non-inverting buffer amplifier with a gain of 10.

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Vo={l+(Rf/R1)}.Vi (8.2.) In the example of the circuit illustrated in figure 8.3 the gain is 10. Figure 8.3 illustrates the arrangement for recording from an individual electrode. Figure 8.4 illustrates a circuit diagram to record the p.d. developed by an dip-cast ionselective electrode with respect to a reference electrode in the same solution. The ionselective and reference electrode are each connected to a buffer amplifier and the outputs passed to a third differential amplifier. The final output represents the difference between the ion-selective and reference electrode voltages. If the system is arranged so that the buffer amplifiers have gain, it is important that each amplifier has exactly the same gain otherwise an error will occur at the inputs to the differential amplifier. The simplest way to ensure this is to impose an identical signal onto the inputs of both buffer amplifiers and check that the final output is zero.

8.6. DIFFERENTIAL AMPLIFIERS If signals from several ion-selective electrodes are being simultaneously recorded it is, in general, adequate to have only one reference electrode and one corresponding reference preamplifier. The output from the reference amplifier can be subtracted from all ionselective electrode amplifier outputs by the differential stages. It is however good practise to check that there is no cross-talk between different ion-selective electrode stages, by checking that a potential developed at one ion-selective electrode is not recorded by other channels. If this the case then separate reference electrodes may be appropriate. An example of a possible arrangement is shown in figure 8.5. The circuitry is similar to that shown in figure 8.3 with the output of the buffer stage from the reference electrode used for the inverting inputs to the secondary differential stages.

8.7. LOW-PASS FILTERS If a fast frequency response is not desired from the system then it may be profitable to attenuate high frequency components of the output to remove extraneous noise from the signal. This may be achieved with a low pass filter on the output, which selectively reduces the contribution of high frequency components but leaves low frequency components unaltered. However, care must be taken to ensure that the filtering does not alter the experimental measurements, especially if ion-selective electrodes are used to measure rates of ionic change. In general the filter should only attenuate components of the signal greater than ten times the fastest frequency required experimentally. For example, if it is thought that the electrode voltage will change maximally at 1Hz (1 cycle per sec) then the filter should be effective at frequencies only above 10Hz. Figure 8.6 illustrates the general arrangement for a so-called second-order low-pass filter.

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Figure 8.4. A unity gain differential amplifier using inputs from two buffer amplifiers. To determine the frequency range over which the filter is effective a simple design procedure can be followed. The actual component values may seem strange, but are socalled preferred values and are commercially available as single components. 1. Choose the frequency, f, above which components are to be attenuated—say 15 Hz to be on the safe side.

2. Put R1=R2=R and let Rf=2.R. 3. Calculate C1 from C1=0.707/(ωcR), where ωc=2 πf 4. Choose C2=2.C1.

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Figure 8.5. Arrangement for three separate ion-selective electrodes and a common reference electrode.

Figure 8.6. A second-order low-pass filter.

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Figure 8.7. The frequency response of second-order low pass filter with a nominal corner frequency of 15Hz. A 500mV peak-to-peak sine wave at frequencies between 0.1 and 200Hz was used.

Figure 8.8. Isolation amplifier AD 204 and clock driver AD 246 as a

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headstage suitable for ion-selective micro-electrodes. The minimum connections are shown. The clock is driven from a +15V power source. The gain of the amplifier is (1+(R1/R2)), Z is the source (ion-selective electrode) impedance. Remember the rules for adding resistors and capacitors. A 66 kΩ resistor is formed by connecting two 33 kΩ resistors in series. A 0.44 µF capacitor is formed by connecting two 0.22 µF capacitors in parallel. The frequency response of such a filter is plotted in figure 8.7—note the logarithmic axes. This Bode plot illustrates the essential features of the filter, at low frequencies the signal magnitude is unaffected but above certain frequencies is attenuated. The corner frequency is 15Hz in this example as desired from the above design equations and can be determined as the frequency at which the two extrapolated linear parts of the Bode plot intersect as seen in figure 8.7.

8.8. OTHER HEADSTAGE DESIGNS, INCLUDING OPTOISOLATED SYSTEMS Alternative designs to conventional electrometer amplifiers have been used with very high impedance ion-selective micro-electrodes. Two designs used with some success by the authors are described here. The first is a commercial isolation amplifier from Analog Devices (AD 204), i.e there is electrical isolation between input and output achieved in this case by transformer coupling between these stages. Power for the device is supplied by a clock driver (AD 246). The manufacturers specification quotes an input impedance of 1012Ω in practice it has been found to have a much higher value, especially when a FET op amp is placed in series with the input stage as a voltage follower. Figure 8.8 illustrates the general arrangement. The second design is the use of optical isolation between the head-stage and the recording apparatus. The input voltage is converted into a train of pulses, the pulse frequency depending on the value of the input voltage—the encoder module. This pulse train is then transmitted via an opto-isolator to a frequency-to-voltage converter to reconstitute the original signal—the decoder module. The decoded signal can then be amplified and filtered as required by the experimenter before final recordings are made. The advantage of the circuit is that the majority of the recording circuitry is electrically isolated from the input stage. This will reduce electrical noise and variations in reference electrode potential from extraneous sources as the encoder is not referenced to a fixed voltage. Indeed it is possible to use very high impedance micro-electrodes with a minimum of shielding. The circuit is illustrated below in figure 8.9.

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8.9. RESISTOR COLOUR CODES AND RULES FOR COMBINATION OF RESISTOR AND CAPACITORS Resistors of particular values have been recommended throughout the text and the value of commercial resistors is coded by a series of coloured bands. Figure 8.10 shows how to de-code these bands to values. The bands are asymmetrically placed on the resistor; the tolerance band lies near the centre, the other three bands are read from the end towards the centre. Examples: Refer to figure 8.10 reading from left to right. Brown, black, red, gold

10×100=1 kΩ ±5%

Orange, orange, orange, silver

33×1000=33 kΩ ±10%

Yellow, purple, yellow, red

47×10,000=470 kΩ ±2%

Blue, grey, green, gold

68×100,000=6.8 MΩ ±5%

Resistors of every possible value are not generally available, but are sold as a number of preferred values, if other values are required combinations of preferred values should be used according to the rules for combinations of resistors. Resistors in series:

Two resistors R1 and R2 are in series, the total resistance, R=R1+R2.

Resistors in parallel:

Two resistors R1 and R2 are in parallel, 1/R=1/R1+1/R2 R=(R1×R2)/ (R1+R2).

The values of capacitors are generally written on the component itself. Take care to read the subdivision of the unit (Farad) however as the small size of the components often means that the lettering is small. Note that different rules exist for combinations of capacitors Capacitors in parallel:

Two capacitors C1 and C2 are in parallel, total capacitance C=C1+C2.

Capacitors in series:

Two capacitors C1 and C2 are in series, 1/C=1/C1+1/C2 C=(C1×C2)/(C1 +C2).

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Figure 8.9. An optically-isolated headstage suitable for use with highimpedance ion-selective microelectrodes. The top diagram shows a block diagram of the circuit with the point of optical isolation illustrated. The middle circuit diagram shows the voltage-to-frequency converter, the encoder module, and the bottom diagram the frequency-tovoltage, the decoder module. CM: current mirror; D1/TR1: opto-isolated diode. The op amps -type 7611 and

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type 351—have FET input stages. All components are commercially available.

Figure 8.10. Resistor colour codes. The resistor shows four colour bands. The interpretation of the bands is shown in the tables below.

9. APPLICATIONS OF ION-SELECTIVE ELECTRODES Ion-selective electrodes can be used in situations limited only by the ingenuity of the user. A number of examples will be illustrated here which show some of their applications in biological systems.

9.1. MEASUREMENT OF ION CONCENTRATIONS IN ELECTROLYTE SOLUTIONS This perhaps is an obvious application by analogy with the use of a conventional glass pH electrode. Dip-cast electrodes are however, ideally suited for this because of their robustness, small size, cheapness and good electrical characteristics. For example, a valinomycin K+ electrode/NaCl bridge pair can be used to measure the [K+] in a series of standard and unknown solutions to illustrate the Nernst equation to students in a practical class, thus illustrating all of the above characteristics. The p.d. is measured using FET op amp input stages and the voltage displayed on a liquid-crystal display—the whole circuit run with a 9V battery. Other applications include the measurement of the concentration of ions such as Ca2+ in physiological solutions when it may be altered by added constituents (section 6.5, figures 6.3 and 6.4), or the measurement of Ca2+ and H+ when using Ca2+ buffers to construct standard Ca2+ -containing solutions (section 6.9).

9.2. MEASUREMENT OF IONIC EXCHANGE IN CELL SUSPENSIONS The relatively rapid frequency response that can be achieved with dip-cast ion-selective electrodes means that they can be used to measure the rate of change of ion concentration with time courses as rapidly as 0.1 seconds. One example is the measurement of Ca2+ and H+ fluxes in suspensions of cardiac myocytes (e.g. Fry et al., 1984, 1989). The experimental arrangement is seen in figure 9.1. Dip-cast ion-selective electrodes and a ceramic-tipped 3M KCl reference bridge protrude into the cavity of a water-jacketed chamber. In this example the floor of the chamber has been modified to form an oxygen electrode. The chamber is filled with a suspension of cells and the concentrations of Ca2+, H+ as well as the PO2 of the suspension medium monitored with the electrodes. If the cells accumulate an ion from the suspension medium, the electrodes will monitor a decline of that concentration and if the cells extrude an ion, the concentration will rise. The success of the technique relies upon the fact that very small proportional changes of ion concentrations can be monitored with

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these electrodes, due to their stability and low resistance. Changes of about 0.01% have been recorded with Ca2+ and H+ -selective electrodes, although in general changes of about 0.1% can be monitored.

Figure 9.1. A chamber suitable for the measurement of ionic fluxes in cell suspensions using dip-cast ionselective electrodes. The bottom of the chamber is a modified oxygen electrode. Ion-selective and reference electrodes pass through a Perspex plug to protrude into the chamber. An additional small hole in the plug allows addition of reactants to the chamber.

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Figure 9.2. Measurement of Ca2+ and H+ exchange in a suspension of cardiac myocytes. The example in figure 9.2 illustrates Ca2+ and H+ exchange, as well as oxygen consumption, by a suspension of cardiac myocytes in which the surface membrane has been rendered permeable to small molecules by the addition of the cholesterolprecipitating agent, digitonin. The suspension medium includes a high [K+], low [Na+] and a low [Ca2+] controlled by EGTA buffer, a composition similar to that of the sarcoplasm. Upon addition of digitonin (dig.) the intracellular organelles—which are relatively unaffected by digitonin—are exposed to the ions and small molecules in the suspension medium. In part A of the figure the mitochondrial fraction of the cell is active and upon addition of digitonin there is an accumulation of Ca2+ by the cell fraction, acidosis and a consumption of oxygen. Previous additions had included rotenone (rot.) and K succinate (succ.). In the lower tracing the sarcoplasmic reticulum was active

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(oligomycin and Na azide were added to ensure complete mitochondrial inhibition) and now Ca2+ accumulation upon addition of digitonin is accompanied by an alkalosis and no extra oxygen consumption. The tracing shows that changes in the concentration of several ions can be simultaneously monitored over a time scale of seconds and minutes in this example. The continuous nature of the output means that kinetic data, such as the initial rate of Ca2+ accumulation can be easily measured, which would be difficult by more conventional techniques, such as radioactive tracer exchange. Appendix 5 shows the principles underlying electrodes that measure dissolved gases, such as CO2 and O2, in fluids. The CO2 electrode is a potentiometric device like the ionselective electrodes described above—i.e the change in analyte is recorded as a voltage change. The O2 electrode is by contrast an amperometric device, which records a current change as a function of the analyte quantity.

9.3. MEASUREMENT OF PLASMA ION CONCENTRATIONS, IN VIVO The physical characteristics of dip-cast ion-selective electrodes means that they can be inserted into the blood stream of anaesthetised and awake animals or humans for continuous monitoring. One particular advantage is the fact that they can be used in solutions such as plasma where there are large quantities of proteins and other large (charged) molecules without evidence of drift. In the example shown in figure 9.3 plasma [K+] has been monitored in an artery of two human subjects. The rise of plasma [K+] is that observed during exercise on a bicycle ergometer and it is of interest to note differences before and after beta-adrenergic blockade of the subject. The use of such electrodes demonstrated, for the first time, that large and rapid changes in the [K+] occur during such manoeuvres. It would have been difficult to measure such changes by estimation of the plasma [K+] from serial blood samples using flame photometry.

9.4. MEASUREMENTS WITH ION-SELECTIVE MICROELECTRODES The experimental arrangement by which intracellular micro-electrodes have been used to measure the cytoplasmic concentration of ions has been discussed above. The limit to their regular use in this situation is the technical difficulty of such experiments because two micro-electrodes (an ion-selective and conventional, electrolyte-filled electrode) must be inserted into the preparation—the conventional one to measure the membrane potential of the cell. Thus relatively

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Figure 9.3. Measurement of arterial [K+] in a subject during exercise at 100W on a bicycle ergometer. Subjects stopped exercising at the arrows. Upper traces from a normal subject. The lower pair of traces were from a subject under control conditions and then after β-adrenergic blockade (upper trace of the pair (from Linton et al, 1984)). large cells have been usually used, thereby excluding many cell types in which investigation of intracellular ionic regulation is of importance. The recent development of fluorochromes, which change their fluorescence properties in the presence of different ions, and may be readily introduced into the cytoplasm have facilitated the measurement of intracellular ionic concentrations which will further limit the use of intracellular ionselective micro-electrodes. There are several areas in which ion-selective microelectrodes do however form an important measuring system. The first is their use to confirm the calibration of the signals obtained with ionsensitive fluorochromes. The calibration of fluorochromes is subject to several uncertainties, which include the precise intracellular compartments in which they are distributed—i.e. organelles as well as the cytoplasmic space—and any change in their ion-binding properties inside the cell compared to model extracellular solutions. Thus, initially it will be of interest to compare directly the values of ionic concentrations in the cytoplasm of cells as measured by ion-selective micro-electrodes and different fluorochromes. A second, more important use of ion-selective micro-electrodes is the measurement of ionic concentrations in restricted extracellular spaces which are not amenable to large probes and which would not retain fluorochromes but allow them to diffuse away from the site of interest. Two examples will illustrate their use.

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Figure 9.4 shows measurement of extracellular pH (part A) and the extracellular [K+] (part B) with ion-selective micro-electrodes in a bundle of ferret detrusor (bladder)

Figure 9.4. Simultaneous recordings of isometric tension and extracellular ion concentrations in an isolated smooth muscle preparation from the bladder. Part A: tension (upper trace) and pH (lower trace), part B: tension (upper trace) and [K+] (lower trace). Na azide was added at the vertical dotted lines to induce cellular hypoxia. The control extracellular pH and [K+] are shown by the horizontal dotted lines (from Thomas & Fry, 1996).

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smooth muscle. Each part of the figure contains two tracings, the upper one is a recording of isometric tension, and the lower trace the output of the ion-selective micro-electrode. The electrodes were broken so that the tip diameter was about 5 µm, thus ensuring that they would not penetrate a cell, and advanced into the core of the muscle strip. Tension was elicited by stimulating the muscle with a tetanic train of pulses delivered every 90 seconds. The small deflexions on the ion-selective electrode tracings are stimulus artefacts recorded by the electrodes. The superfusate surrounding the preparation was changed to one containing 5 mmol.l−1 Na azide, chemically mimicking hypoxia, but in which both the [K+] and the pH was kept constant. It may be observed that after a short delay the extracellular pH in the core of the tissue bundle was acidified by about 0.25 pH units and the [K+] increased by about 3 mmol.l−1, despite the fact that the bulk superfusate pH and [K+] remained constant. Several important consequences arise from such observations, the membrane H+ and K+ gradients are less than that which would be calculated if bulk superfusate values were used, extracellular pH and the [K+] are not a constant but vary throughout the tissue preparation and accurate and more realistic estimates of transmembrane H+ and K+ fluxes can be made by measuring the concentration of the ions immediately close to the cell. Such measurements have not been confined to smooth muscle preparations but have been made in many cell types, including skeletal and cardiac muscle.

Figure 9.5. Simultaneous measurements in an anaesthetised cat of arterial (plasma) [K+], with a dipcast ion-selective electrode, and the [K+] in the extracellular space of the carotid body with an ion-selective microelectrode. The upward deflexion on the plasma [K+] trace marks the time when a bolus injection of KCl into the bloodstream was made. Arterial blood pressure was monitored

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101

throughout (D.M.Band, C.B.Wolff and C.H.Fry, unpublished observations). The second illustration is the use of ion-selective micro-electrodes in the extracellular space of small organs which would be completely inaccessible to other techniques. In this case the measurement of the [K+] in the carotid body is shown in response to an arterial injection of KCl. In this experiment it was proposed that a rise of the plasma [K+] would raise the extracellular [K+] in the carotid body and thus stimulate the organ to increase afferent discharge along the sinus nerve. The carotid body is a small organ (about 1 mg weight) so that dynamic measurement of the extracellular [K+] would be difficult by any method other than the introduction of a small K+ -ion selective micro-electrode. The type of records that can be obtained in the carotid body of an anaesthetised cat are shown in figure 9.5. In this example are shown tracings from a dip-cast K+ -selective electrode in the femoral artery and from a K+ -sensitive microelectrode in the carotid body extracellular space. The dip-cast electrode is the more rapidly responding electrode and it can be observed that a transient rise of the plasma [K+] occurs after a small injection of 100 mmol.l−1 KCl into the bloodstream. The secondary, small rise of the plasma [K+] is due to recirculation of the injected KCl and the whole trace could be used to measure cardiac output. The more slowly responding transient is that recorded from the carotid body interstitium by the ion-selective micro-electrode and shows that the rise in the plasma concentration is reflected by a similar rise in this space. The other trace in this recording shows the arterial blood pressure.

APPENDIX 1 THE NIKOLSKY EQUATION AND SELECTIVITY COEFFICIENTS A1.1. THE NIKOLSKY EQUATION An electrochemical equivalent of an ion-selective membrane has been modelled as two boundary (interface) potentials and a diffusion potential in Chapter 3 (section 3.4). A more detailed consideration of the system will be considered to derive the Nikolsky equation and the various equations used to calculate the potentiometric selectivity coefficients. As the simplest case consider two monovalent cations, i+ and j+ which interact with the membrane, i+ will be called the primary ion and j+ a secondary (interferent ion). of any ion depends upon its The energy state, or electrochemical potential location; in particular its chemical and its electrical environment. In general for ion i, can be described as: (A1.1.) where ai is the activity of i, z is the valency and E the local potential; k is a constant. The term RT In ai is equivalent to the chemical potential of i. Consider the interface between one side of the membrane (m1) and the test solution (s). At equilibrium the electrochemical potentials of both ion i and ion j in the test solution and membrane face will be equal, the Donnan equilibrium, i.e. (A1.2.) Expanding the terms for µi: RT ln ai, s+zFEs+ks=RT ln ai, m1+zFEm1+km1 (A1.3.) which on re-arrangement yields: (A1.4a.) A similar result can be obtained for the interface between the filling solution (f) and the other membrane face (m2).

Appendix 1

103

(A1.4b.) The constant k is called the standard chemical potential of ion i, in a particular phase. As ks and kf both refer to an aqueous phase then ks=kf; similarly km1= km2. These equivalences will be used later to simplify the final equation. The potential across the membrane (Em2−Em1) is due to diffusion of the ions i and j across the membrane and can be obtained from the Henderson equation (equation 2.3). (A1.5.) The potential across the whole membrane (Em=Ef−Es) can be obtained from summing equations A1.4 and A1.5, i.e. (Ef−Em2)+(Em2−Em1)+(Em1−Es).

Algebraic manipulation of this equation leads to: (A1.6.) where U=(uj/ui), A1=(aj,m1, .ai,s/ai,m1) and A2=(aj,m2.ai,f/ai,m2) At the membrane interface, m1, with the test solution, s, the electrochemical potentials of the two ions are equal in the two phases (equation A1.2). Moreover as there is a free energy balance in the exchange of ions i and j between the two phases: i+(s)+j+(m1)=i+(m1)+j+(s) then (A1.7.) As the reaction involves no net charge transfer, it is insensitive to the potential at the membrane face and can be described by an equilibrium constant (A1.8.) A numerically equivalent constant can be derived for the other membrane face and substituted into equation A1.6. (A1.9.)

Appendix 1

104

The term is a constant of the system called the potentiometric selectivity coefficient. Moreover if the composition of the filling phase (f) is constant equation A1.9 reduces to the Nikolsky equation (equation A1.10) for a monovalent with a single monovalent interferent ion—E0 is a constant. (A1.10a.) If additional monovalent interferent ions, k, l, m,…, are present then equivalent potentiometric selectivity coefficients can be derived for each ion and their effect summed to produce a more general form of the equation (A1.10b.)

Note that as then equation A1.9 reduces to the Nernst equation (equation 3.2) If the interferent ion has a valency, zj, different from that of the primary ion, zi, the equilibrium constant will be defined as:

The more general form of the Nikolsky equation then becomes: (A1.11.)

A1.2. SELECTIVITY COEFFICIENTS The formulae used to estimate the potentiometric selectivity coefficient by either the Separate Solutions or Fixed Interference methods can be derived from the Nikolsky equation. A1.2.1. Separate solutions method In a pure solution of either ion i+ or ion j+ the potential at the electrode, Ei or Ej, is: (A1.12.) (A1.13.)

Appendix 1

105

It will convenient here to convert from natural logarithms (In) to logarithms to base 10 (log), i.e. In(x)≈2.303 log(x)—see appendix 2. Rearranging (A1.14.)

(A1.15.)

(A1.16.)

A1.2.2. Fixed interference method In this method the concentration of aj is kept constant and the concentration of ai is varied. The value of ai* used to calculate is obtained from the intersection of the two linear parts of the calibration curve (see figures 6.2 and A5.1). At this value of ai*, in the presence of aj, both the primary and interferent ions contribute equally to the electrode response, i.e.:

thus

APPENDIX 2 NECESSARY MATHEMATICAL TRICKS The purpose of this short appendix is to explain some of the mathematical operations carried out in the text. It is not intended to be complete and to act as a course in mathematics for biologists, but rather to explain the specific manoeuvres used above.

A2.1. POWER FUNCTIONS, EXPONENTIALS AND LOGARITHMS These functions have occurred throughout the text to manipulate data, derive equations and as operators in fundamental equations. This appendix shows what these functions do to numbers. A2.1.1. Power functions An operation when a number, x, is raised to a certain power. One function is where a variable number x is raised to a fixed power, e.g. x2. The mathematical representation of this is f(x)=x2. An alternative is when a fixed number is raised to the power of the variable, e.g. f(x)=10x. Note these are not equivalent functions. A2.1.2. Exponential functions The number represented by the series: where say 3! is 3×2×1=6 is called e. The xth power of e is written ex or exp x and is represented by a similar series:

Some basic rules can be written down for exponential functions (as other power functions): exp x×exp y=exp(x+y) thus: exp x×exp (−x)=exp(x+−x)=exp(0)=1 exp(−x)=1/(exp x) thus: exp x/exp y=exp x×exp(−y)=exp(x−y) (exp x)2=exp x×exp x=exp(2x)

Appendix 2

107

thus: (exp x)n=exp(nx). The exponential function has a further interesting property in that the exponential function is equal to its own derivative (and integral).

A2.1.3. The logarithmic function The logarithmic function follows from the preceding power and exponential functions. Let: y=10x. Then log10 y=x. y=ex. Then loge y=x. Thus there are a number of logarithmic functions which refer to the base number (10 or e) which is converted to a power function. Note that a logarithmic function reverses the action of a number used in a power function, i.e.

Unless it would be confusing, in the above text the following notation is used: log10x is written as log x and logex is written as In x. The following operations can be carried out with logarithms: log(x.y)=log x+log y

ln(x.y)=ln x+ln y

log(xn)=n log x

ln(xn)=n ln x

In addition : thus: and lnx=log10x. ln 10=2.3026.log10x The logarithmic function is useful when numbers over a wide range need to be expressed on a single graph. This has been encountered when plotting ion concentrations when the ‘p’ notation has been used—i.e. for pH and pCa. In this case the concentration is expressed as a negative logarithm if the value. Thus 1 µmol.l−1 [H+] is expressed on the ‘p’ scale as −log(10−6)=pH 6.0; 0.1 µmol.l−1[H+] is −log(10−7)=pH 7.0 and 10 µmol.l−1 [H+] is −log (10−5)= pH 5.0. Thus a geometric progression (10−7 to10−6 to 10−5) in converted to an arithmetic progression (7.0 to 6.0 to 5.0). It is important to note that intermediate points that are equally spaced on the number scale are not equally spaced on the logarithm scale, i.e. pH 6.5 (half way between pH 7.0 and pH 6.0) is not equivalent to 0.5 µmol.l−1 [H+]. The antilogarithm of −6.5=0.3×10−6, i.e. pH 6.5= 0.3 µmol.l−1 [H+].

Appendix 2

108

Example Construct an experiment where the concentration of an agent, X, increases from 1 nmol.l−1to 1 µmol.l−1 in equally spaced logarithmic increments, what concentrations are used? Answer: Call the substance X, then the range of |X| will be from pX=9.0 to pX=6.0. increasing by 0.5pX unit increments—i.e. 9.0 to 8.5 to 8.0… …6.0. Successivc concentrations will be 1 nmol.l−1, 3 nmol.l−1, 10 nmol.l−1. 30 µmol.l−1 . 100 nmol.l−1, 300 nmol.l−1 and 1 µmol.l−1. If finer gradations than 0.5pX unit increments are the table below shows the concentration equivalents of raising the pX from 9.0–8.0, from 8.0 to 7.0 and from 7.0 to 6.0 in 0.1pX unit increments. pX

Conc nmol.l−1

pX

Conc nmol.l−1

pX

Conc mmol.l−1

9.0

1.00

8.0

10.0

7.0

100

8.9

1.26

7.9

12.6

6.9

126

8.8

1.58

7.8

15.8

6.8

158

8.7

2.00

7.7

20.0

6.7

200

8.6

2.51

7.6

25.1

6.6

251

8.5

3.16

7.5

31.6

6.5

316

8.4

3.98

7.4

39.8

6.4

398

8.3

5.01

7.3

50.1

6.3

501

8.2

6.31

7.2

63.1

6.2

631

8.1

7.94

7.1

79.4

6.1

794

8.0

10.0

7.0

100

6.0

1000

A2.2. STRAIGHT-LINE GRAPHS Many mathematical manipulations are designed to generate a relationship between two variables and produce a straight-line graph. This is straightforward to analyse and is often a useful visual guide to a process. For example the relationship between the primary ion concentration and the ion-selective electrode potential (figure 3.2) is a curvilinear process which can be linearised by the transformation of the abscissa to a logarithmic scale The equation of a straight-line is: y=mx+c and the two parameters m and c are explained in figure A2.1A. Linearisation of a nonlinear curve will generate an equation with the above form, from which the parameters m and c can be interpreted in terms of the other parameters of the original equation. Some examples of this have been already encountered, for example in section 6.9 where the EGTA binding constant for Ca2+ was estimated.

Appendix 2

109

A2.2.1. Linearisation of exponential functions. The three functions: y=A.exp(kt); y=A.exp(−kt); y=A.(1−exp(−kt)) are shown in figure A2.1B, where t is the independent variable, and A and k are a constants. These are linearised as follows: y=A.exp(kt)

y=A.exp(−kt)

ln(y)=ln(A. exp(kt))

ln(y)=ln(A.exp(−kt))

ln(y)=InA+ln(exp(kt)) ln(y)=InA+kt

ln(y)=InA+ln(exp(−kt))

y=A.(1−exp(−kt)) 1−(y/A)=exp(−kt) ln(1−(y/A))=−kt

ln(y)=In A−kt

The final forms are all equation of a straight line, with ln(y) on the ordinate and t on the abscissa. The slope, m, of the line is equal to k in the three cases and in the first and second cases the constant, c, is equivalent to In A. Figure A2.1C shows these linearised forms with the various parameters. Note that in the first two cases both A and k are estimated, whereas in the third case A must already be known. If the x variable in the above equations is time (unit second, s), then the constant k is called a rate constant and has units of s−1. The reciprocal of k is now a time constant, τ, with units of s. If two temporal processes are compared the slower one therefore has a smaller rate constant or a larger time constant.

A2.3. ERRORS AND SIGNIFICANT FIGURES Any measurement in analytical chemistry is associated with an error. It is important that the magnitude of the error is known and that quoted values of the measured variable are expressed to the appropriate degree of accuracy. Errors can be either systematic or random. A systematic error is reproducible and results from an experimental deficit. For example, a calibration standard may be wrong so that its repeated use will always give an erroneous result. This error may be difficult to identify but may be detected if a different calibration standard is used or a different analytical method if available. Regular checking of calibration solutions, either by making up a second solution or comparing the solution with one from another source, is therefore always good experimental practise. Random errors are generated by uncontrolled factors in the experimental system and cannot be corrected. Precision and accuracy are two terms which are often confused in this context. Precision means how reproducible is the result and accuracy how close to the ‘actual’ value is the result. Figure A2.2 illustrates these concepts. Four archers shoot arrows at their owns targets and the ‘hits’ recorded. Archer 1 is precise but inaccurate; archer 2 is imprecise and inaccurate; archer 3 is precise and accurate; and archer 4 is accurate but imprecise.

Appendix 2

110

Figure A2.1. Part A: plot of the equation of a straight line, y=mx+c. c (=10) is the intercept with the y-axis. m(=2) is the gradient of the line. Part B: plots of the three exponential functions listed in the text: y=A.exp(kt)—upper trace; y=A.exp(−kt) middle trace; y=A.(1−exp(−kt))—lower trace. For all plots the parameters have values A=10, k= 0.5. Part C: linearised transforms of

Appendix 2

111

the equations in part B, showing how the parameters A and k are determined.

The uncertainty of a variable is another important measure. Let the absolute uncertainty of a weighing balance by 0.01 g; this means that if the recorded weight is 15.55g, the actual value could be anywhere between 15.54 and 15.56g. In this particular example the relative uncertainty would be 0.01/15.55=0.006 and the percentage uncertainty is 0.6%. Note that if the absolute uncertaintty remains constant the relative uncertainty will vary according to the actual value. Uncertainty is usually expressed as a standard deviation or confidence interval and usually implies that random errors are being referred to. When several readings are taken each with an error and arithmetic operations are performed on these readings, what is the final error. Consider three readings x±e1, y±e2, and z±e3. Let the three readings be added together then the result is (x+y+z)±e4. the same calculation would apply if addition or subtraction was carried out. Let the three readings be multiplied together then the result is (x.y.z)±e4. To calculate e4 now it first necessary to calculate the percentage uncertainties, then: the same calculation would apply if multiplication or division was carried out.

Appendix 2

112

Figure A2.2. Precision and accuracy. Four targets showing ‘hits’ be different archers. A is precise but inaccurate; B is inaccurate and imprecise; C is both precise and accurate; D is accurate but imprecise. Errors may also be calculated when more complex operations on numbers are carried out. For each case a value x has an uncertainty ex, it is required that the uncertainty in y, ey, is calculated Let y=exp (x), then ey=y.ex.

Let y=ln (x), then ey=ex/x.—relative error in x

Let y=10x, then ey=ln(10).y.ex.

Let y=log(x), then

Let y=xa, then %ey=a.%ex—it is assumed here that a has no associated uncertainty. Example The pCa of a solution has a mean (±S.D.) value of 5.10±0.05. Find the [Ca2+] and its uncertainty Answer: Let pCa=−log [Ca2+], then the [Ca2+]=10−pCa=10−5.10±0.05

Appendix 2

113

The mean [Ca2+]=10−5.10=7.94×10−6 mol.1−1 From the above the uncertainty is ln(10).y.ex=2.303×7.94×10−6×0.05= 9.14×10−7 mol.l−1, The [Ca2+] is 7.9±0.9 µmol.l−1. The example above raises a further question, which is the number of figures necessary to quote a value. The number of significant figures is the minimum required to quote a value without loss of accuracy. Figure A2.3 shows, for example, a ruler place opposite a rod whose length is to be measured—the smallest divisions of the ruler are in mm.

Figure A2.3. Measurement of a length of a rod using a ruler with cm and mm scales. The length of the rod is in this case 25.4 mm. The final figure is estimated from an interpolation between the smallest ruler divisions—in this case mm marks and the minimum uncertainty is ±1 in the last digit. The value for the rod length is quoted to three significant figures, with the first two numbers known with certainty and the last value an estimate. In scientific notation the value is 2.54× 10−2m. It would be wrong to write the number as 2.540×10−2m, as this implies the number is known to four significant figures, and it would be too imprecise to quote it as 2.5×10−2m. When numbers are added or subtracted the sum should be rounded off to a number of significant figures corresponding to that available to the number with the fewest significant figures. Add: 23.2543–35.6582+16.35=3.9461 rounded off to 3.95 Similarly with multiplication and division—note that powers of 10 have no effect on the number of figures. Multiply: 3.462×10−6 by 4.53×10−9=1.57×10−14.

APPENDIX 3 RECOMMENDED NOMENCLATURE FOR USE WITH ION-SELECTIVE ELECTRODES, THE SYSTÈME INTERNATIONALE (S.I.) AND STANDARD VALUES FOR SELECTED CONSTANTS A3.1. THE IUPAC CONVENTION The International Union of Pure and Applied Chemists (IUPAC) has listed recommended terms and definitions for use with ion-selective electrodes and experimental details when reporting results in the literature. A3.1.1. Ion-selective electrode and membrane An ion-selective electrode is an electrochemical sensor which generates a potential which is proportional to the logarithm of the activity of a particular ion in solution. The potential has as its main component the free energy change associated with mass transfer, by ionexchange, adsorption or by another mechanism—across a phase boundary. The term ‘selective’ rather than ‘specific’ is preferred as no electrode is wholly specific for one ion but rather more selective towards one ion than others. A membrane is a continuous layer which separates two electrolytic solutions. The membrane is responsible for generating the potential response and for the selectivity of the electrode. A3.1.2. Calibration curve A calibration curve is a plot of the p.d. of a ion-selective electrode/reference electrode pair (called an ion-selective electrode assembly) against the logarithm of the ionic activity or concentration of a particular species. The p.d. should be plotted on the ordinate (vertical axis) with the more positive potentials at the top of the axis and the logarithm of the ion activity (or concentration) plotted on the abscissa with increasing activity (or concentration) to the right (i.e. decreasing PIon values to the right). A3.1.3. Limit of detection The limit of detection is estimated from the calibration curve as the concentration defined by the intersection of the extrapolated two linear regions of the curve (see figure A3.1). It

Appendix 3

115

is the concentration where the slope of the Nernstian response (below) to the primary ion is reduced by 50%. Experimental conditions, such as composition of the test solution, history of the electrode and stirring rate should be recorded. A3.1.4. Nernstian response An ion-selective electrode assembly has a Nernstian response over a given range of ion concentration, ci (or activity, ai) if the calibration curve is linear with a slope of 2.303×103 RT/ziF mV per 10-fold change of ci (per unit change of pci). At 25°C this is 59.16/zi mV per unit change of pci. A3.1.5. Interfering substances and potentiometric selectivity coefficient An interfering substance (interferent) is any substance other then the ion being measured whose presence in the solution affects the p.d. of the system. There are two classes: electrode interferents and methodological interferents. Electrode interferents are substances: i) that give a similar response to the ion of interest and result in an apparent increase in the concentration of that ion, e.g. Na+ interfering with a Ca2+ -selective electrode; ii) such as organic solvents, which interact with the membrane and change its chemical composition; iii) electrolytes which generate significant liquid-junction potentials. Methodological interferents include substances which interact with the ion of interest and reduce its activity or apparent concentration but leave the electrode response unaffected. The potentiometric selectivity coefficient, , defines the ability of an ion-selective electrode to distinguish between two ions, i and j in solution. The coefficient is determined from the ion-selective electrode response using mixed solutions of the primary ion, i, and the interferent ion, j, by the fixed interference method (see below). The concentrations, or activities, of i and j at which the coefficient is determined should is determined from the Nikolsky equation. The smaller the be stated and the value of value the greater is the selectivity of the electrode to the primary ion. Because the values are usually less than 1, they are often expressed as the negative logarithm of , or as the value. The terms selectivity factor or selectivity constant are sometimes used, but are not preferred. A3.1.6. Fixed interference and separate solutions methods The fixed interference method is when the p.d. of an ion-selective electrode assembly is recorded in solutions containing a fixed activity (or concentration) of an interfering ion, aj, and with varying concentrations of the primary ion, ai. A plot of the calibration curve is used to determine the limit of detection, (see above) from which the potentiometric selectivity coefficient is determined, zi and zj are the valencies of i and j: (A3.1.)

Appendix 3

116

Figure A3.1. A calibration curve of an ion-selective electrode showing the limit of detection and the region of Nernstian response. The separate solution method is when the p.d. of ion-selective electrode assembly is recorded in two separate solutions, one containing i (but no j) and the other containing j (but no i) and where ai=aj. If the measured p.d.’s are Ei and Ej then the potentiometric selectivity coefficient is: (A3.2.)

The former method is preferred as the is determined under conditions more similar to where the ion-selective electrode is actually used. Note that the form of equation A3.2 is an alternative one to that quoted in chapter 6 and derived in appendix 1. A3.1.7. Drift and hysteresis Drift is a slow, non-random change of p.d. recorded by the ion-selective electrode assembly in a test solution of constant composition and temperature. Hysteresis is when a different p.d. is recorded on second exposure to the same solution, (containing a particular concentration of the primary ion) and between when the ion-selective electrode assembly was exposed to another solution containing a different concentration of the primary ion.

Appendix 3

117

A3.1.8. Practical response time The time between when the ion-selective electrode assembly is brought into contact with a new solution and when the measured p.d. is within 1mV of the final steady-state value. The composition of the new solution and the previous solution, the stirring rate, previous history of the electrode and temperature should be stated. A3.1.9. Isopotential point For an ion-selective electrode assembly there is often a particular activity or concentration of an ion for which the potential is independent of temperature. That activity (concentration) and the potential constitute the isopotential point. A3.1.10. Reference electrode An electrode which has a virtually unchanging potential under the experimental conditions when electrochemical measurements are being made (such as measurements made with an ion-selective electrode) and which permits the measurement or control of potential of the indicator (ion-selective) electrode. An internal reference electrode is contained within an ion-selective assembly, e.g. a Ag/AgCl electrode in contact with a filling solution containing Cl− and a fixed concentration of the ion to which the electrode is sensitive. A3.1.11. Combination electrode An electrochemical system incorporating an ion-selective electrode and a reference electrode in a single assembly.

A3.2. THE WRITTEN REPORT When writing a paper or thesis using either a new ion-selective electrode or a novel application where the concentration of an ion is measured, the following information should be included in the first instance. Details of Construction. Membrane composition and method of construction, filling solution and internal electrode. For commercial electrodes the manufacturers catalogue number. Calibration Range. Range of concentrations tested, electrode slope, detection limit, effect of pH. Stability. Rate of voltage drift, storage conditions and any compounds known to affect drift. Response Times. Details of their determination and any change with electrode age. Interferents. Interferents should be listed in the two classes described above. Potentiometric selectivity coefficients should be quoted and the method used—fixed interference (preferable) or separate solutions. Variation in the value of the coefficients with time should be quoted if known.

Appendix 3

118

Temperature coefficient. Any effects due to temperature variation should be noted, except the expected change of Nernst slope with temperature. Comparison of values. Comparison of the value obtained with the new ion-selective electrode should be made with values obtained by more established or alternative techniques. Recording apparatus. Electrode resistance should be quoted as well as the characteristics of the potentiometer (manufacturer’s details or characteristics described in chapter 8).

A3.3. PHYSICAL QUANTITIES AND MEASUREMENTS—THE INTERNATIONAL SYSTEM OF UNITS (SI) A physical quantity is defined by a complete specification of the operations used to measure the ratio (a pure number) of two instances of the physical quantity and a unit: numerical value=physical quantity/unit The International System of units (SI units) is based on a selected set of base units, one each for a set of dimensionally independent physical quantities. Derived units are obtained from the base units by multiplication or division of two or more base units without the introduction of any numerical factors. Two supplementary units for plane angle (radian) and solid angle (steradian) are defined for completeness but will not be considered here. The SI Prefixes are used to form decimal multiples or submultiples of any SI Unit and a table has been given in chapter 1. By convention seven particular physical quantities are regarded as dimensionally independent and each has a corresponding SI base unit. Physical quantity

Symbol

name of SI unit

symbol for SI unit

Length

l

metre

m

Mass

m

kilogramme

kg

Time

t

second

s

Electric current

I

ampere

A

Thermodynamic temperature

T

kelvin

K

Amount of substance

n

mole

mol

Luminous intensity

Iv

candela

cd

A3.3.1. Definition of SI base units Metre. The length equal to 1 650 763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the 86Kr atom. Kilogramme. The mass of an internationally recognised prototype. Second. The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the 133Cs atom.

Appendix 3

119

Ampere. That constant current which, if maintained in two straight parallel conductors of infinite length, of negligible cross-sectional area, and placed 1 metre apart in a vacuum, would produce between these conductors a force equal to 2.10−7 newton per metre of length Kelvin. The kelvin unit of thermodymanic temperature is the fraction 1/273.15 of the thermodynamic temperature of the triple point of water Mole. The amount of substance which contains as many elementary entities as there are atoms in 0.012 kilogramme of 12C. The elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of particles. Candela. The luminous intensity, in the perpendicular direction, of a surface of 1/600 000 m2 of a black body at the temperature of freezing platinum under a pressure of 101 325 N.m−2. A3.3.2. Standard values for selected constants and useful relations Molar Gas Constant, R

8.31451 J.K−1.mol−1

Faraday Constant, F

9.648 5×104 C.mol−1

RT/F, at 25°C

25.693mV

2.303 RT/F, at 25°C

59.159mV

Boltzmann Constant

1.380 66×10−23 J K−1

Absolute Temperature

−273.15°C

Avogadro’s Number, NA

6.022 14×1023 mol−1

Atomic Mass Unit, mu

1.660 56×10−27 kg.

Vacuum Permittivity, εo

8.854 19×10–12 C2.N−1.m−2

Relative permittivity, water, εr

78.54 at 25°C; 74.02 at 38°C

(Dielectric constant) Elementary Charge, e

1.602 18×10–19C

Units of Pressure

1 Pascal (Pa)=1N.m−2; 1bar=105 Pa=100kPa

the inter-relationship of

1 atmosphere=101.325kPa=

several units of pressure

1.013 25×105 N.m−2 1mmHg=133.3224Pa 1Torr=133.322Pa (exactly) 1atm=760Torr (exactly)

The number, e

≈2.718 282

The number ln(10)

≈2.302 585

The number, π

≈3.141 593

Appendix 3

120

A3.3.3. Some useful SI derived units Note that vector quantities are printed in bold type; e.g.energy, E and electric field strength, E. Physical quantity, brief definition

Symbol

SI unit

area

A…S

m2

volume

V…v

m3

speed: u=ds/dt—s, distance

u

m s−1

acceleration: a=du/dt

a

m s−2

mass

m

kg

density (mass density): ρ=m/V

ρ

kg m−3

force: F=md2s/dt2

F

kg m s−2=N

weight

G…W

N

pressure

p…P

kg m−1 s−2=Pa

work: F s (force times path distance)

w…W

kg m2 s−2=J

energy

E

J

power: E/t

P

J s−1=W

viscosity:

η…µ

kg m−1 s−1

diffusion coefficient

D

m2s−1

electric charge (quantity of electricity)

Q

A s=C

elementary charge

e

C

electric current: I=dQ/dt

I

A

electric potential

kgm2s−3A−1=V

electric potential difference

V

capacitance: C=Q/U

C

C V−1=F

electric field strength

E

V m−1

electric displacement

D

C m−2

permittivity: D=ε E

ε

F m−1

permittivity of a vacuum

εr

F m−1

relative permittivity: εr=ε/ε0

ε0

1

resistance

R

V A−1=Ω

conductance: G=1/R

G

Ω−1=S

thermodynamic temperature

T…Θ

K

Celsius thermodynamic temperature interval.

Appendix 3

121

θC=T −273.15 K

θC

K

amount of substance B

nB

mol

number of molecules

N

1

Avogadro constant

L, NA

mol−1

molar mass: M=m/n

M

kg mol−1

molar volume: Vm=V/n

Vm

m3 mol−1

molality of solute B; amount of B/mass of solvent

mB

mol kg−1

concentration of B: cB=nB/V

cB

mol m−3

charge number of ion (+ve or −ve)

zi

1

I

mol kg−1

vi

m s−1

ionic strength: I=1/2 Σimizi

2

speed of ion, i

Physical quantity, brief definition

Symbol

SI unit

electric mobility of ion i: ui=vi/E

ui

m2 s−1V−1

electrolytic conductivity: k=j/E

k

S m−1

molar conductivity of electrolyte: Λ=k/c

Λ

S m2 mol−1

transport number of ion i:

ti

1

γC,A, γ±

1

ti=|zi|civi/Σi|zi|civi activity coefficient of an electrolyte solute Cv+Av− :γC,A=(γv+Cγv−A)1/(v+v−)

A3.4. PRINTING OF SYMBOLS FOR PHYSICAL QUANTITIES, UNITS AND NUMBERS • The symbol should be a single letter (with odd exceptions) of the Latin or Greek alphabet. • Symbols for physical quantities should be printed in italic type. e.g. amount of substance, n • Abbreviations (e.g., p.d.) should not be used as symbols in mathematical equations; printed in text they appear as upright (roman) type. • Subscripts or superscripts which are symbols for physical quantities are printed in italic type, others are in upright type, e.g. Cp, heat capacity at constant pressure but xB, mole fraction of substance B. Two or more subscripts/ superscripts are separated by commas. • Symbols for a unit are printed in roman type, are unaltered in the plural and are not followed by a full stop, except at the end of sentence. E.g.: cm but not cm or cm or cm or cms or cm.

Appendix 3

122

• Symbol for a unit derived from a proper name should begin with a capital roman letter. E.g. J, Hz. Otherwise units for SI Units are printed with a lower case letter • Symbols for prefixes to units are printed in roman type with no space between prefix and unit; avoid compound prefixes. E.g. ps but not p s or µµs for 10−12 s. A combination of unit and prefix is a single unit and is raised to a power without brackets, e.g. cm2. • A product of two units can be represented by: N m or N m or N . m or N×m, but not Nm. • A quotient is written as or m/s or m s−1. However, to avoid ambiguity never use more than one solidus. Example: J K−1 mol−1 or J/(K mol) but not J/K/mol. • Numbers are printed in upright type. The decimal sign should be a dot (.) or a comma (,) but never a centred dot (·). To facilitate reading of long numbers digits may be grouped in sets of three about the decimal sign, but no point or comma should be used except the decimal sign. Example: 2 573.421 736 or 2 573,421 736 but not 2,573.421,736 •When the decimal sign is placed before the first digit, a zero should always be placed in front. It is often convenient to print numbers with just one number before the decimal sign. Example: 0.2579×104 or 0,2579×104 or 2.579× 103 but not .2579×104 •Rules for multiplication and division of numbers are similar to the expression of products and quotients of physical quantities and units. Do not use a centred dot with a decimal sign which is a dot.

A3.5. EXPRESSION OF NUMERICAL VALUES FOR CONSTANTS, HEADINGS TO TABLES AND GRAPHS The value of a physical quantity is equal to the product of a numerical value and a unit (section A3.3). Neither the physical quantity, nor the symbol used to denote it, should ever imply a particular choice of unit. Example: the physical quantity called the ionic mobility of Na+ in water, u, has a value at conventional temperature (298.15 K): u=5.19×10−3cm2s−1V−1 or u/cm2 s−1V−1=5.19×10−3 or any other way of expressing the equality of u and 5.19.10−3 multiplied by cm2 s−1 V−1. Example: Correct: The conductivity k is defined by the relation k=l/A R where l is the length, A is the area and R is the resistance Incorrect: The conductivity k is defined by the relation k=l/A R where l is the length in cm, A is the area in cm2 and R is the resistance in ohms. The expression which is placed at the head of a column of numerical values of a physical quantity in a table or used to define these numerical values plotted on a graph should be a pure number, i.e. the quotient of the symbol for the physical quantity and the symbol for the unit used, eg p/MPa. Example: Plot the dependence of the vapour pressure of water pw on thermodynamic temperature T. Temperature in degrees Celsius is denoted, θC (=T-273.15). Plot the ordinate in units of 1/T and the abscissa as In p. The data will appear as follows.

Appendix 3

θC/°C

123

T/K

103K/T

pw/kPa

0

273.15

3.661

0.611

−0.4927

20

293.15

3.411

2.337

0.8489

40

313.15

3.193

7.375

1.998

60

333.15

3.002

19.92

2.992

80

353.15

2.832

47.36

3.858

100

373.15

2.680

101.3

4.618

ln(pw/kPa)

APPENDIX 4 OTHER ION-SELECTIVE ELECTRODE CONFIGURATIONS Apart from the glass pH electrode (chapter 6), the ion-exchange membrane, immobilised in a PVC matrix or in the tip of a microelectrode, has found the most widespread utility in biological systems and their manufacture and use have been described in detail here. However, a number of other designs have been developed which have uses in particular situations or are undergoing development and evaluation. It will be of value to describe briefly their principles of action.

A4.1. INORGANIC SALT ELECTRODES These electrodes formed the generation of sensors which followed the glass electrodes. The electrodes are responsive to a number of cations and anions, many of which are less amenable to measurement with ion-exchangers. The sensing surface consists of a solid salt of the primary ion pressed into a pellet, and sometimes incorporated into an inert base material, such as PVC. The salt should have a low solubility to avoid its dissolution when placed in the test solution. The electrodes function by ion conduction through the crystal of the pellet, taking advantage of lattice defects caused by local excess of cations or anions. Contact between the sensing pellet and the reference system consists of a Ag/ AgCl electrode immersed in a filling solution containing a salt of the primary ion. Thus the F− electrode is filled with a solution containing 1 mol.l−1 KF, saturated KCl and saturated with AgCl. For some Ag+ sensing electrodes direct contact is made by a Ag wire soldered onto the sensing pellet. These constructions are shown in figure A4.1. Table A4.1 lists some electrodes; the primary ion, the pellet sensor and the approximate limit of detection in unbuffered solutions of the primary ion. In general the anion sensitive electrodes work well in the pH range 3–12. The listed cation sensitive electrodes function better in the acidic range, pH 3.7, except the Ag2S electrode which has an extended range up to pH 9. The halide electrodes suffer from strong interference by CN− and OH− and to a lesser extent from cross-interference from other halide ions. In addition it is the authors’ experience that glucose causes considerable drift in the Cl− electrode, in much the same way as some carboxylate ions have been reported to cause similar effects on the F− electrode. In general the divalent cation selective electrodes suffer interference to some extent from other divalent cations, the AgS suffers serious interference only from Hg2+. The interested reader is referred to more comprehensive information from suppliers data sheets.

Appendix 4

125

Table A4.1. Inorganic salt electrodes Primary ion Sensor Useful range, mol.l−1 Primary ion Sensor

Useful range, mol.l−1

F−

LaF3

saturated—10−7

Ag+

Ag2S

saturated—10−8

Cl−

AgCl

1–10−5

Cd2+

CdS/Ag2S 10−1–10−7

Br−

AgBr

1−10−6

Cu2+

CuS/Ag2S 1−10−9

I−

AgI

1−10−8

Pb2+

PbS/Ag2S 10−1−10−7

SCN−

AgSCN 1−10−5

Figure A4.1. Constructions of inorganic salt ion-selective electrodes. The left hand diagram shows the more common system which uses a filling solution and Ag/AgCl electrode in the interior of the electrode. The right hand diagram shows an alternative construction in which a silver wire is soldered directly onto the sensing Agcontaining pellet.

Appendix 4

126

A4.2. COATED-WIRE ELECTRODES This is a simple design whereby an ion-selective plastic is cast directly over a metal wire, such as silver, platinum or even the copper core of a coaxial cable and is shown schematically in figure A4.2. The design is similar to the dip-cast electrode above but without the internal filling solution. The theoretical basis of their function is not completely understood but there must be some interface between the plastic layer and the wire to generate a stable internal reference system. As with conventional ion-selective electrodes the potential generated by the electrode should be proportional to log . These electrodes offer the advantages of robustness and ease of manufacture and often show a superior selectivity towards the primary ion compared to conventional ionselective electrodes. However, the reproducibility of response can be poor as it can take a considerable time for the membrane to come to equilibrium with the solution, and is very dependent on the thickness of membrane. In addition the electrodes can drift considerably, as membrane constituents leach from the membrane and as water is absorbed into the membrane. Despite these problems electrodes sensitive to several cations and anions, including Ca2+ and K+ have been described.

Figure A4.2. Diagram of a coated-wire electrode in which a layer of ionexchanger in a PVC matrix is cast directly over the tend of a Pt wire.

Appendix 4

127

A4.3. ENZYME ELECTRODES These are an example of a class of devices known as biosensors, that use biological compounds as sensors for particular moieties. An example is the determination of urea by the enzyme urease:

The enzyme is incorporated into a membrane which then is placed over an selective electrode. The

produced by the action of the enzyme on urea in the

-selective electrode and provided the urea external solution is detected by the concentration is not great enough to saturate the enzyme reaction a quantitative estimate detected by the ionof the urea concentration can be made from the activity of selective electrode. A similar principle is used for the detection of glucose but uses an amperometric reaction scheme with hydrogen peroxide (H2O2) as an electron mediator. Amperometric systems for the determination of dissolved O2 will be described in Appendix 5. The sensor utilises glucose oxidase in one membrane which reacts with a glucose molecule to form a reduced form of the enzyme. The oxidised form of the enzyme is reconstituted by the consumption of two electrons at an O2 permeable second membrane. A schematic diagram of the reaction scheme is shown in figure A4.3. The system suffers from the disadvantages that two separate membranes must be used and the whole reaction is slightly dependent on the prevailing PO2. Simplified versions have been described where electrons are removed directly from the reduced form of the enzyme.

Figure A4.3. Amperometric reaction scheme at the membrane of a glucosesensitive electrode

Appendix 4

128

Figure A4.4. An ISFET electrode in which an ion-selective membrane is cast over the gate of a field-effect transistor. The potential generated at the ion-selective membrane modulates the current between the sink and drain. A4.4. ION-SELECTIVE FIELD-EFFECT TRANSISTORS (ISFETS) The development of the ISFET has been motivated by attempts to miniaturise ionselective electrodes for biomedical applications and a diagram is shown in figure A4.4. A normal field-effect transistor is a three-terminal device in which current flow between the source and drain is modified by a potential at the ‘gate’ terminal. In this case the normal metallic gate is replaced by a thin film of ion-selective material, which is exposed directly to the test solution. The potential between the solution and the ion-selective material is proportional to the logarithm of the ion activity in the test solution, just as with a conventional ion-selective electrode. Most applications use the ISFET in conjunction with a separate reference electrode, although some applications have attempted to incorporate a reference junction into the body of the ISFET. The advantages of such a system are the small size that can be achieved and the removal of conduction cables between the sensor and the electronic recording systems. However, in practise their use has been limited to a more restricted range of ion concentrations than conventional designs and a fairly short lifetime due to degradation of the thin ion-selective membrane. Suitable constructions using valinomycin-based K+ electrodes, dioctylphenyl-phosphonate Ca2+ electrodes, tri-n-dodecylamine H+ electrodes and inorganic salt membranes to Cl− and CN− and I− have been described.

Appendix 4

129

A4.5. OPTODES AND CHROMOIONOPHORES A recent development in ion sensors is to couple the ion-sensing phase of an electrode to a second phase which consequently alters its optical properties. The change in optical properties can then be monitored as the experimental variable. This system, known as an optode, has several advantages, in particular the minimisation of electrical interference in the recording system, but at present suffer from a relatively small concentration over which they are effective and a slower time response.

Figure A4.5. Membrane configuration of optodes. The upper panel shows a cation selective system. The membrane contains a chromoionophore, C, which binds H+ and an ion-exchanger, L,

Appendix 4

130

which will bind the test ion, I+. The membrane also has added lipophilic anionic sites, B−. Overall electroneutrality will ensure that [C−H+]+[L−I+]=[B−]. The lower panel shows an anion sensitive system whereby the chromoionophore, C, and an anion exchanger, L, are incorporated into the membrane. Most current configurations (figure A4.5) use two ion-exchange systems within a membrane (PVC) phase, one sensitive to the ion of interest and the other an ionophore which changes its optical properties when an ion, usually H+, is bound—the latter is called a chromoionophore. The optical change is generally an alteration to the absorption spectrum of the chromoionophore, so that the absorption of a particular wavelength will be a function of the protonated form of the chromoionophore. The upper panel configuration shows a cation-sensitive system whereby two cationexchange ionophores are present in the membrane, each will extract their particular ion from the test solution. The membrane also contains a lipophilic anion which ensures that the total cation content of the membrane phase must remain constant so that the entire membrane remains electroneutral. The lipophilic anions are usually borate salts, such as K-tetrakis-(4-chlorophenyl) borate (which are used in the dip-cast membrane electrodes described earlier, see chapter 6) or Na tetrakis[3, 5-bis(trifluoromethyl)phenyl]borate. Thus if the concentration of the sample ion, i, increases in the test solution, more will be extracted into the membrane phase and consequently less H+ will be taken up. Therefore fewer H+ will bind to the chromoionophore and the optical signal will change. The lower panel configuration shows an anion-sensitive system, where electroneutrality is again the requirement of the membrane phase. An increase of the anion concentration in the test solution will increase the extraction into the membrane phase, consequently there will be an increased extraction of H+, which will bind to its chromoionophore. In all of these solutions it is clear that the pH of the test solution must remain constant to avoid extraction of H+ onto the chromoionophore independently of change in concentration of the test ion. An example of a chromoionophore is a Nile Blue derivative ETH 5294 which has an absorption maximum at 545nm for the unprotonated form and 614/600nm for the protonated form.

APPENDIX 5 PARTIAL PRESSURES AND THE DETERMINATION OF O2 AND CO2 IN SOLUTION Although the estimation of dissolved CO2 and O2 in solution is not directly concerned with the topic of this book, it is often carried out in conjunction with the measurement of ion concentrations in solution and the methods employ many of the techniques dealt with previously. An example is the blood gas analyser used in clinical settings, where not only is blood pH and PCO2 measured, but so too is the PO2. Derivative calculations are also made such as the bicarbonate, be explained.

, concentration. The terms PCO2 and PO2 must first

A5.1. THE PARTIAL PRESSURE OF A GAS The concept of a gas partial pressure can be used in two contexts in biological systems. In a gaseous mixture it refers to the proportion occupied by an individual gas. In the liquid phase it refers to a quantity of gas in solution. A5.1.1. Partial pressures in gas mixtures A mixture of gases occupying a given volume will exert a particular pressure and this total pressure is the sum of the individual pressures which would be exerted by each gas, if it was alone in that volume. Each individual gas will therefore exert a partial pressure, the magnitude of which will depend only on the relative proportion it occupies in the total mixture. The pressure (P) exerted by a gas is related to the quantity (n, moles), the volume (V), the absolute temperature (T, °K) and a constant (R, the Gas Constant=8.314.51 J K−1 mol−1) by the Universal Gas Law: P.V=n.R.T (A5.1.) The modern standard condition for reporting data is at 25°C (T=298.15°K) and a pressure (P) of 1 bar (=100kPa and equivalent to 0.987 atmospheres) and is called standard ambient temperature and pressure, SATP. Under this condition the volume occupied by 1 mole of gas is therefore approximately 24.79 dm−3. SATP has tended to replace STP (0°C and 1 atm). Note: the subtle differences between the units of pressure cause confusion; their inter-relationships are explained in appendix 3.

Appendix 5

132

If a gas mixture at total pressure P has several components present in the amounts nA, nB, nC, etc. moles, the partial pressure exerted by each gas, PA, B, C,…, is described by Dalton’s Law of partial pressures: PA=nA.R.T/V, etc. and P=PA+PB+PC+. . . (A5.2.) Example A sample of dry atmospheric gas contains by mass approximately 23.1% O2, 75.6% N2 and 1.3% Argon (Ar), what are the partial pressures of these gases if atmospheric pressure is 760 mmHg? The molecular weights of the gases are: O2 32.00 g.mol−1; N2 28.02 g.mol−1; Ar 39.95 g.mol−1; Answer: In 100 g air:

nO2=23.1/32.00=0.722 moles. nN2=75.5/28.02=2.698 moles. nAr=1.3/39.95=0.032 moles. Total: n=3.452 moles of the gases.

The mole fraction of O2 (xo2)=no2/n=0.209 and xN2=nN2/n=0.782; xAr= nAr/n=0.009. The partial pressures are:

PO2=(0.722/3.452)×760=159 mmHg; PN2=(2.698/3.452)×760=594 mmHg; PAr=(0.032/3.452)×760=7 mmHg.

Note: the S.I. unit of pressure is the Pascal (Pa). 1 mmHg=0.133 kPa. Thus the above partial pressures would be: PO2=21.2kPa; PN2=79.2 kPa; PAr= 0.9 kPa. Example: A sample of exhaled gas is analysed immediately at 37°C and found to contain in molar proportions 3.5% CO2 and 17.5% O2. What are the partial pressures of these gases? Answer: Exhaled gas is saturated with water vapour which at 37°C has a partial pressure of 47 mmHg. If atmospheric pressure is 760 mmHg the dry gas pressure is (760– 47)=713 mmHg. Therefore: PO2=0.175×713=125mmHg and PCO2=0.035×713=25mmHg. A5.1.2. Partial pressures in solution Consider the system, shown in figure A1.1 whereby a liquid is in contact with a gas mixture above it. Some of the gas molecules will dissolve in the liquid and the quantity dissolved will larger if the partial pressure in the gas phase is greater. The extent to which a gas, A, dissolves in a liquid is summarised in Henry’s Law: PA=xA.KA. (A5.3.)

Appendix 5

133

KA is a constant with units of pressure and xA is the mole fraction of A. At 25°C KO2=3.30×107 mmHg; KN2=6.51×107 mmHg; KCO2=1.25×106 mmHg. Example Calculate the mole fraction of CO2 in an aqueous solution in equilibrium with a gas at atmospheric pressure containing 5% CO2 by molar proportion. Calculate the concentration and volume of CO2 in this solution. Answer: From Henry’s Law: xCo2=PCO2/KCO2. where PCO2=0.05×760=38 mmHg Thus: xCO2=38/1.25×106=3.04×10−5. xCO2=nCO2/(nCO2+nH2O) ≈ nCO2/nH2O. In 1 kg H2O there are 1000/18.02=54.95 moles H2O (molecular weight of H2O=18.02) Thus there are 54.95×3.04×10−5=1.67×10−3 moles CO2 (=1.67 mmoles) in 1kg H2O. At 25°C 1 mole of gas occupies 24.47 litres. Thus 1kg H2O will contain 37.5ml CO2 In biological systems the relationship between the quantity of gas in the gaseous and liquid phase is often expressed slightly differently: CA=αA.pA (A5.5a.) where CA is now the concentration of gas in the liquid phase. The constant αA is called a solubility coefficient, but note that it is not the same as the constant KA in Henry’s Law. The value of αCO2 in plasma is 0.03 mmol.l−1.mmHg−1, which is equivalent at 25°C to about 0.7 m1.1−1 .mmHg−1 For O2 the value is less: 1 litre of plasma will dissolve about 0.03 ml O2 per mmHg—equivalent at 25°C to 0.03/ 24470 mol.l−1.mmHg−1 (=0.0013 mmol.l−1.mmHg−1). The solubility coeffi-

Figure A5.1. A liquid in equilibrium with a gas, where gas molecules

Appendix 5

134

dissolve in the liquid. The larger partial pressure of gas (right) results in a greater dissolved quantity in the liquid. cients, αA, have great utility in biological systems to relate the concentration of gas within plasma to the partial pressure, an example of which is shown below.

A5.2. MEASUREMENT OF DISSOLVED CO2 CO2 hydrates in aqueous solutions to form carbonic acid which in turn dissociates into H+ . The pK of the overall reaction is 6.36; in plasma it is 6.1 as CO2 is slightly and more soluble. Thus the more CO2 dissolves in solution, the more acid will be the solution. The precise relationship is given by the Henderson-Hasselbalch equation: (A5.5a.) where [CO2] is the total amount of CO2 dissolved in solution, PCO2 is the partial pressure of CO2 in mmHg and α is the solubility of CO2 in solution. For plasma, α=0.03 mmol.l−1.mmHg−1 (note here that the units of α mean that the mmol.l−1).

are expressed in

The equation can be re-arranged to yield a solution for the (A5.5b.) Example A sample of plasma has a pH of 7.4 and a PCO2 of 40 mmHg, using equations A5.5a and A5.5b, calculate the . The subject then breathes more rapidly and the PCO2 falls to 20 mmHg as more CO2 is exhaled per unit time. What now is the plasma pH (assume the remains the same—this is not quite true but is convenient for this calculation)? Answer: pH=7.4; pK=6.1 and αPCO2=0.03×40=1.2 mmol.l−1. • Use the Henderson-Hasselbalch equation (equation A5.5a) to practise manipulation of logarithmic functions: pH—pK(=1.3)=log( /1.2)

/1.2)→ antilog1.3(=20)=(

→ =24 mmol.l−1 • Use equation A1.5b for a more straightforward calculation: [HCO3]=101.3.1.2=24 mmol.l−1.

Appendix 5

135

• If the PCO2 falls to 20 mmHg,→ /αPCO2=40→pH=6.1+log (40)=7.70. This is a respiratory alkalosis. Derivation of the Henderson-Hasselbalch equation The overall reaction for the hydration of CO2 in aqueous solution is:

By the Law of Mass Action, the equilibrium constant, K, is:

and pH=−log[H+] pK=−logK The [H2O] is assumed to be relatively large and unchanging and set to unity. Thus:

The principle of the technique is to measure the change of pH in a solution of varying . In practise a combination glass pH electrode cannot be [CO2] and constant placed directly in the test solution as factors other than a changing [CO2] may alter the pH. This may be overcome by placing a gas permeable membrane over the pH-sensitive tip of the glass electrode and trapping a containing solution between the membrane and the pH electrode. The membrane is generally made from polytetrafluroethane (PTFE or Teflon). CO2 will diffuse across the membrane, whereas ionic and other constituents of the solution will not do so. The amount of CO2 leaking across will depend upon the concentration in the test solution. The diffused CO2 will alter solution which will be measured by the electrode. If the the pH of the in the membrane-electrode gap is known then the [CO2], or PCO2 can be calculated from the Henderson-Hasselbalch equation. Most solutions in the gap contain 5–10 mmol.l−1 NaHCO3 plus 20–100 mmol.l−1 NaCl. A schematic drawing of such an electrode is shown in figure A5.2.

Appendix 5

136

Figure A5.2. Schematic diagram of the electrode system for the measurement of dissolved CO2. A5.3. MEASUREMENT OF DISSOLVED O2 The methods described above for measurement of ions in solution (and the [CO2]) have involved recording the potential difference between a sensing electrode and a reference electrode. These methods are therefore known as potentiometic systems. However, another class of measurement systems relies on recording the number of electrons generated by a reaction. The flow of electrons will constitute an electric current and such methods are known as amperometric systems. An example is the measurement of the amount of O2 dissolved in aqueous solution. Figure A5.3 shows a schematic drawing of a Clark-type O2 electrode and uses a similar system to that of the CO2 electrode, whereby a gas permeable membrane covers a sensing element. This again restricts the number of species that can come into contact with the sensor but limits its potential poisoning by solutes. The membrane is separated from the sensing electrode by a solution, usually saturated or 3 mol.l−1 KCl, to dissolve the diffused O2 and facilitate its transfer to the sensors. The sensing electrodes consist of a central metal (usually platimum) cathode surround by a metal ring anode. The anode and cathode are held at a fixed potential difference of

Appendix 5

137

about 0.6V to reduce the oxygen in solution: one molecule of oxygen will consume 4 electrons in a quantitative manner. In acidic solutions: O2+2H++2e−→H2O2 H2O2+2H++2e−→2H2O overall: O2+4H++4e−→2H2O In neutral and basic solutions: O2+2H2O+2e−→H2O2+2OH− H2O2+2e−→2OH− overall: O2+2H2O+4e−→ 4OH− (A5.6.)

Figure A5.3. Schematic diagram of an O2 electrode. The Clark electrode produces a very linear response down to a PO2 of about 1–2 mm Hg and therefore provides a reliable and accurate determination of the O2 content of the solution. An example of its use in measuring mitochondrial consumption of solution O2 is seen in figure 9.2. Most important is to keep the electrodes clean so that the reduction process is constant and this can be achieved by either coating the electrodes with AgCl, if they are made of Ag, or rubbing them with a fine emery paper at regular intervals (daily).

APPENDIX 6 ADDRESSES OF MANUFACTURERS AND SUPPLIERS IN THE UK FOR RAW MATERIALS AND COMPONENTS DESCRIBED IN THE TEXT Ion-selective materials and general chemicals. Fluka Chemika-BioChemika/Sigma Aldrich The Selectophore® Catalogue The Old Brickyard—New Road GILLINGHAM Dorset SP8 4JL Tel. 0800 373731 http://www.sigma.aldrich.com/ Ceramic rod (Frequentite) Morgan Matroc, Bewdseley Road, STOURPORT-ON-SEVERN, Worcestershire DY13 8QR Tel. 01299 827000 http://www.matroc.com/ PVC tubing (translucent) Portex Ltd, HYTHE, Kent CT21 6JL Tel. 01303 260551 Micro-electrode glass Clark Electromedical Instruments, P.O. Box 8, PANGBOURNE, READING, Berkshire RG8 7HU Tel. 01189 843888 http://www.clark.mcmail.com/ Metals and high quality ceramics Goodfellow Chemicals, Cambridge Science Park, Milton Road,

Appendix 6

139

CAMBRIDGE Cambridgeshire CB4 4DJ Tel. 01223 568068 http://www.goodfellow.com/ Oxygen electrode Rank Bros, BOTTISHAM, Cambridgeshire Tel. 0223 811369 http://www.rankbrothers.co.uk/ Electronic components RS Components Ltd, P.O.Box 99, CORBY, Northamptonshire NN17 9RS Tel. 01536 201201 rswww.com Electrometer operational amplifiers for ion-selective micro-electrodes Analog Devices, Station Avenue WALTON-ON-THAMES Surrey KT12 1PF, Tel. 01932 232222 http://www.analog.co/

APPENDIX 7 REFERENCES Physical chemistry of ions in solution There are several standard works which describe in fairly detailed terms the subject Harned, H.S. and Owen, B.B. The Physical Chemistry of Electrolyte Solutions. 3rd ed Reinhold Publishing Corp, New York, 1958. Murrell, J.N. and Jenkins, A.D. Properties of liquids and solutions, 2nd ed, John Wiley and Sons, Chichester, New York, 1994 Robinson, R.A. and Stokes R.H. Electrolyte Solutions, Butterworths, London, 1954

Ion-selective electrodes and electrophysiology There are a number of practical books to help the cell physiologist, these include: Ammann, D. Ion-selective electrodes. Principles, design and application. Springer-Verlag, Berlin, 1986. Brown, K.T. & Flaming, D.G. Advanced micropipette techniques for cell physiology. John Wiley and Sons, Chichester, New York, 1986. Purves, R.D. Microelectrode methods for intracellular recording and ionophoresis. Academic Press, London, New York, 1981. Thomas.R.C. Ion-sensitive intracellular microelectrodes. Academic Press, London, New York, 1978.

Electronics There are numerous books on op amps and their uses. One of the more useful ones is: Lenk, J.D. Manual for operational amplifier users. Reston Publishing Co Inc., Reston, VA, 1976.

Statistics From the vast number of choices useful ones include: Moroney, M.J. Facts from Figures, Penguin, London 1990. Snedecor, G.W. and Cochrann, W.G. Statistical Methods, Iowa State University Press, 1967

A7.1. SELECTED REFERENCES Conventions International Union of Pure and Applied Chemistry. Recommendations for nomenclature of ionselective electrodes. Pure and Applied Chemistry, 48, 129–132, 1976. International Federation of Clinical Chemistry (IFCC). Scientific Division. Committee on pH, Blood Gases and Electrolytes. Guidelines for transcutaneous PO2 and PCO2 measurement. Wimberly P.D., Burnett R.W., Covington A.K., Maas A.H., Mueller-Plathe O., SiggaardAnderson O., Weisberg H.F. & Zijlstra W.G. Clinical Chimica Acta, 190, S41–S50, 1990. International Federation of Clinical Chemistry (IFCC) Scientific Division. IFCC recommendation on sampling, transport and storage for the determination of concentration of ionized calcium in

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whole blood, plasma and serum. Boink A.B., Buckley B.M., Christiansen T.F., Covington A.K., Maas A.H., Muller-Plathe O., Sachs C. & Siggaard-Andersen O. Clinical Chimica Acta, 202, S13–21, 1991 International Federation of Clinical Chemistry (IFCC). Scientific Division. Committee on pH, Blood Gases and Electrolytes. Approved IFCC recommendations on whole blood sampling, transport and storage for simultaneous determination of pH, blood gases and electrolytes. Burnett R.W., Covington A.K., Fogh-Andersen N., Kulpmann W.R., Maas A.H., Muller-Plathe O., Siggaard-Andersen O., Van Kessel A., Wimberley P.D. & Zijlstra W.G. European Journal of Clinical Chemistry and Clinical Biochemistry 33, 247–253, 1995. International Federation of Clinical Chemistry (IFCC), Committee on pH, Blood Gases and Electrolytes: approved IFCC recommendation on definitions of quantities and conventions related to blood gases and pH. Burnett R.W., Covington A.K., Fogh-Andersen N., Kulpmann W.R., Maas A.H., Muller-Plathe O., Van-Kessel A.L., Wimberley P.D., Zijlstra W.G., Siggaard-Andersen O. et-al., 33, 399–404, 1995. International Federation of Clinical Chemistry (IFCC). Recommendation on mean molar activity coefficients and single ion activity coefficients of solutions for calibration of ion-selective electrodes for sodium, potassium and calcium determination. Burnett R.W., Covington A.K., Fogh-Andersen N., Kulpmann W.R., Maas A.H., Muller-Plathe O., Siggaard-Andersen O., Van Kessel A., Wimberley P.D. & Zijlstra W.G. European Journal of Clinical Chemistry and Clinical Biochemistry 35, 345–349, 1997. Lehmann H.P., Fuentes Arderiu X. & Bertello L.F. Glossary of terms in quantities and units in clinical chemistry (IUPAC-IFCC recommendations 1996). Pure and Applied Chemistry, 68, 957–1000, 1996. Coplen T.B. & Peiser H.S. IUPAC recommendations on nomenclature and symbols and technical reports from commissions Pure and Applied Chemistry, 70, 237, 1998.

Techniques Ammann D., Morf W.E., Anker P., Meier P. C, Pretsch E. & Simon W. Neutral carrier based ionselective electrodes. Ion Selective Electrode Reviews, 5, 3–92, 1983. Cardwell T.J., Cattrall R.W., Cross G.J., Mrzljak R.I. & Scollary G.R. Determination of calcium in waters, milk and wine by discontinuous-flow analysis. Analyst, 115, 1235–1237, 1990. Covington A.K. & Ferra M.I. Calculation of single-ion activities in solutions simulating blood plasma. Scandinavian Journal of Clinical Laboratory Investigation. 49, 667–375, 1989. Covington A.K. & Rebello M.J.F. Reference electrodes and liquid junction effects in ion-selective electrode potentiometry. Ion Selective Electrodes Review, 5, 93–128, 1983. Devlin C.L. & Smith P.J. A non-invasive vibrating calcium-selective electrode measures acetylcholine-induced calcium flux across the sarcolemma of a smooth muscle. Journal of Comparative Physiology, B, 166, 270–277, 1996. Durst R.A., Koch W.F. & Wu Y.C. pH theory and measurement. Ion Selective Electrodes Review, 9, 173–196, 1987. Harrison S.M. & Bers D.M. Correction of protons and Ca association constants of EGTA for temperature and ionic strength. American Journal of Physiology, 256, C1250–1256, 1989. Ito N., Saito A., Kayashima S., Kimura J., Kuriyama T., Nagata N., Arai T. & Kikuchi M. Transcutaneous blood glucose monitoring system based on an ISFET glucose sensor and studies on diabetic patients. Frontiers in Medicine and Biological Engineering, 6, 269–280, 1995. Mason W.T. Fluorescent and luminescent probes for biological activity. Academic Press, London 1993. Morf, W.E., Seiler K., Lehmann B., Behringer C., Hartman K. & Simon W. Carriers for chemical sensors: design features of optical sensors (optodes) based on selective chromoionophores. Pure and Applied Chemistry, 61, 1613–1618, 1989.

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Mulchandani A., Mulchandani P., Kaneva I. & Chen W. Biosensor for direct determination of organophosphate nerve agents using recombinant Escherichia coli with surface-expressed organophosphorus hydrolase. 1. Potentiometric microbial electrode. Analytical Chemistry, 70, 4140–4145, 1998. Partanen J.I. & Minkkinen P.O. Equations for calculation of the pH of buffer solutions containing sodium or potassium dihydrogen phosphate, sodium hydrogen phosphate, and sodium chloride at 25 degrees C. Journal of Solution Chemistry, 26, 709–727, 1997 Rouilly M., Rusterholz B., Spichiger U.E. & Simon W. Neutral ionophore-based selective electrode for assaying the activity of magnesium in undiluted blood serum. Clinical Chemistry, 36, 466– 469, 1990. Saha H., Harmoinen A., Karvonen A.L., Mustonen J. & Pasternack-A Serum ionized versus total magnesium in patients with intestinal or liver disease. Clinical Chemistry and Laboratory Medicine, 36, 715–718, 1998. Smit A., Pollard M., Cleaton-Jones P. & Preston A. A comparison of three electrodes for the measurement of pH in small volumes. Caries Research, 31, 55–59, 1997. Thompson J.M., Smith S.C., Cramb R. & Hutton P. Clinical evaluation of sodium ion selective field effect transistors for whole blood assay. Annals of Clinical Biochemistry, 31, 12−17, 1994. Warwick W.J. & Hansen L. Measurement of chloride in sweat with the chloride-selective electrode. Clinical Chemistry, 24, 2050–2053, 1978.

Applications to biological measurements Band, D.M., Kratochvil, J., Poole-Wilson, P.A. & Treasure, T. Relationship between activity and concentration measurements of plasma potassium. Analyst, 103, 246–251, 1978. Bourdillon P.D., Bettmann M.A., McCracken S., Poole-Wilson P.A. & Grossman W. Effects of a new nonionic and a conventional ionic contrast agent on coronary sinus ionized calcium and left ventricular hemodynamics in dogs. Journal of the American College of Cardiology, 6, 845–853, 1985. Brookes, C.I. & Fry, C.H. lonised magnesium and calcium in plasma from healthy volunteers and patients undergoing cardiopulmonary bypass. British Heart Journal, 69; 404–408, 1993. Buri A., Chen S., Fry C.H., Illner H., Kickenwicz E., McGuigan J.A.S., Noble D., Powell T. & Twist V.W. The regulation of the intracellular Mg2+ in guinea-pig heart studied with Mg2+selective microelectrodes and fluorochromes. Experimental Physiology, 78, 221–233, 1993. Fry C.H., Hall S.K., Blatter, L.A. & McGuigan J.A.S. Analysis and presentation of intracellular measurements obtained with ion-selective microelectrodes. Experimental Physiology, 75, 187– 198, 1990. Fry C.H., Harding D.P. & Mounsey J.P. The effects of cyanide on intracellular ionic exchange in ferret and rat ventricular myocardium. Proceedings of the Royal Society, series B, 230, 53–75, 1987. Fry C.H., Harding D.P. & Miller D.J. Non-mitochondrial calcium ion regulation in rat ventricular myocytes. Proceedings of the Royal Society, series B, 236, 53–77, 1989. Fry C.H., Powell T., Twist V.W. & Ward J.P. T. Net calcium exchange in adult rat ventricular myocytes: an assessment of mitochondrial calcium accumulating capacity. Proceedings of the Royal Society, series B, 223, 223–238, 1984. Fry C.H., Ward, J.P.T., Twist V.W. & Powell T. Determination of intracellular potassium ion concentration in isolated rat ventricular myocytes. Biochimica Biophysica Research Communications. 137, 573–578, 1986. Heining M.P., Band D.M. & Linton R.A.F. The effect of temperature on plasma ionized calcium measured in whole blood in vitro. Scandinavian Journal of Clinical Laboratory Investigation, 43, 709–714, 1983. Langley S.E.M. & Fry C.H. The influence of pH on urinary ionized [Ca2+]: differences between urinary tract stone formers and normal subjects. British Journal of Urology, 79, 8–14, 1997.

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Linton R.A.F., Lim M. & Band D.M. Continuous intravascular monitoring of plasma potassium using potassium-selective electrode catheters. Critical Care Medicine, 10, 337–340, 1982. Linton R.A.F., Lim M, Wolff C.B., Wilmshurst P. & Band D.M. Arterial plasma potassium measured continuously during exercise in man. Clinical Science, 67, 427–431, 1984. Linton R.A.F., Band D.M. & Haire K.M. A new method of measuring cardiac output in man using lithium dilution. British Journal of Anaesthetics, 71, 262–266, 1993. Lüthi D., Günzel D. & McGuigan J.A.S. Mg-ATP binding: its modification by spermine, the relevance to cytosolic Mg2+ buffering, changes in the intracellular ionised Mg2+ concentration and the estimation of Mg2+ by 31P-NMR. Experimental Physiology, 84, 231–252, 1999. Thomas P.J, Fry C.H. The effects of cellular hypoxia on contraction and extracellular ion accumulation in isolated human detrusor smooth muscle. Journal of Urology, 155, 726–731, 1996 Thomas R.C. & Cohen C.J. A liquid ion-exchanger alternative to KCl for filling intracellular reference microelectrodes. Pflügers Archives, 390, 96–98, 1981.

INDEX A Absolute temperature (value), 128, 129 Activity, ionic, 7, 55, 111–114 factors affecting, 61 Activity coefficient, 7, 130 calculation, 8–11 variation, 11–14 Activity standards, 55, 57 Ag/AgCl electrode, 19 manufacture, 42–44 Ag+ -selective electrode, 134 Ammonium-selective electrode -selective electrode) (see Amperometric system, 136 Anion-selective electrodes, 134 Arrhenius theory, 6 Atomic weight, definition, 1, 128 Avogadro’s constant (value), 1, 128, 129 B Biosensors, 136–137 Bode plot (low pass filter), 97–98 Boltzmann constant (value), 128 Buffer (see pH buffer, Ca2+ buffer) Buffer amplifier unity gain, 89–90 with gain, 93–94 Buffering capacity, 66–67 C Ca2+ -buffer, 69–75 Ca2+ -selective electrode, 38, 61, 62, 78, 80, 86, 103–106, 137 calibration, 69–75 Calibration curve, 59, 60, 74, 77, 80, 82, 114, 123, 125 Calomel electrode, 19, 63 Capacitance, 130 Capacitors—rules for addition, 101 Cardiac muscle, 86, 103–106 Carotid body, [K+], 106, 108–109 Cation-selective electrode, 52 Cd2+ -selective electrode, 134 Chromoionophores, 28, 138–139

Index Cl selective electrode, 20, 134, 137–135 Clark-type O2 electrode, 147 Coated wire electrodes, 28, 135 Combination electrode, 64, 126 Concentration, molal, 2, 129 molar, 2, 129 Concentration standards, 55, 56–57 Conductivity, solution, 22–24, 130 Cu2+-selective electrode, 134 Cumulative distributions, 84–85 D Dalton’s law of partial pressures, 142 Data sets of ISE data, 83–87 Debye-Hückel equation, 8–11 Dielectric constant, 7, 10 (see also permittivity) Differential amplifiers, 94–95 Dip-cast ISE’s, 35–41 applications, 103, 109–106 Donnan equilibrium, 111 Drift, of ISE’s, 77–79 E EGTA Ca2+ buffer, 69–75 estimation of, 73 concentration Electric charge, 129 Electric current, 129 Electric potential (difference), 129 Electrical equivalent, 3 Electrochemical cell, 17–26 Electrochemical potential of ion, 11 Electrolyte solutions (see ionic solutions) Electrometer amplifiers, 92–93 Elementary charge (value), 128, 129 Enzyme electrodes, 28, 136–137 Equitransference solution, 22, 25 Errors accuracy of estimate, 120–121 and interferent ions, 79–81 estimation of errors, 118–122 linear & logarithmic sets, 81–87 precision of estimate, 120–121 uncertainty of a measure, 120–122 Exponential functions, 115–116 linearisation, 118 F Faraday constant (value), 128

146

Index

147

FET operational amplifier, 91–93, 103 Fixed interference method, 60, 114, 125 Fluorochromes, 27, 107 Frequency response (op amp), 90–91, 93 G Gas constant (molar)—value, 128 Glass electrodes, 28 (see also pH electrode) Glucose detector, 136 Gram equivalent, 3 H H+ -selective electrode, 38, 61, 86 (see also pH electrode), 103–106, 107–108, 137 Henderson equation, 23, 29, 112 Henderson-Hasselbalch, 144–146 equation Henry’s law, 143 Hydronium ion, H3O+, 63, 64–65 I Inorganic salt electrodes, 28, 133–135 Input bias current (op amp), 91–92 ion selective microelectrodes, 92–93 Input impedance (op amp), 90–91, 93–94 ion selective microelectrodes, 92–93, 98–99 Instrumentation, 89–101, 127 Interferent ion, 30–33, 59–61, 79–81, 111–113, 124, 127 Internal reference electrode, 126 Ion exchange systems, 28 Ionic activity (see activity, ionic) Ionic diameter, 10 Ionic exchange in cells, 103–106 Ionic mobility, 22–25, 130 Ionic solutions physical chemistry, 1–15 measurements, 103 Ionic strength, 8, 10, 13, 129 Ion-selective electrode accuracy of a reading, 79–81 applications, 103–110 calibration, 55–76, 126 connection to recorder, 44–45, 49–51 definition, 123 drift and hysteresis, 126 isopotential point, 126 manufacture, 35–41, 47–48, 126 model, 29 practical response time, 126, 127 principles, 27–33

Index

148

stability, 77–79, 127 temperature coefficient, 127 (see also dip-cast ISE, microelectrodes—ion selective) Ion-selective materials, 27, 38, 149 Ion-selective membrane, 30, 123 Ion-selective plastics manufacture, 37–40 table, 38 ISE (see ion-selective electrode) ISFET’s, 28, 137 Isolation amplifier, 98–99 Isosmotic, isotonic solutions, 6 IUPAC, 63 IUPAC convention, 123–126 K K+ -selective electrode, 31, 38, 49, 61, 79, 106–110, 137 KH phthalate, 63, 67, 68 K tetrakis (4-chlorophenyl) borate (K-TCP), 38, 139 as reference electrode, 52–54 Kolmogorov-Smirnov test, 84–86 L Law of Mass Action, 70 Lewis, G.N., 7 Lewis-Sargent relationship, 23 Limit of detection, 60, 114, 124 Liquid junction potential, 17, 20, 21–26 Logarithmic data sets, 81–83 Logarithmic function, 116–117, 118 Low pass filter, 95–98 component values, 96 M Mg2+ -selective electrode, 38, 61, 79, 86 Microelectrodes, 49, 51 Microelectrodes, ion selective, 47–54 applications, 51–54, 106–110 connection to recorder, 49–51 instrumentation, 92–93, 98–99 manufacture, 47–48 Mixed solutions, 14 Molar mass and volume, 129 Mole, 1 Mole fraction of gas, 143 Molecular weight, definition, 1 N Na+ -selective electrode, 32, 38, 58, 61, 79, 83–87 Nemst equation (eq, 3.2), 30, 50

Index

149

demonstration of, 103 Nernstian response, 58, 71, 73, 79, 93, 124 -selective electrode, 39 Nikolsky-Eisenman equation, 31, 58, 59, 60 derivation, 111–113 Normal distribution, 84–85 Normal solution, definition, 3 O Operational amplifier (op amp), 89–99 8-pin numbers, 90 see also buffer amplifiers, differential amplifiers isolation amplifiers, opto-isolated amplifiers) Optodes, 138–139 Opto-isolated amplifier, 98–99 Osmotic pressure, 5 Oxygen electrode, 104 P Partial pressure of gas, 141–144 in gas mixtures, 141–142 in solution, 143–144 Pb2+ -selective electrode, 134 PCO2 (partial pressure CO2), 142 measurement, 144–146 Percentage solutions, 3 Permittivity (solution), 9, 10, 128, 129 pH (H+) buffer, 65–69 PH definition, 56, 62, 65 of test solution, 64 pH electrode (glass) calibration, 64, 79 pIon definition, 56, 117 data sets, 81–87 estimation of errors, 122 pK of pH buffer, 66, 144 temperature dependence, 68–69 values, 68 Plasma, [K+] measurement, 106, 108–109 Platinum black electrode, 19 PO2 (partial pressure O2), 142 measurement, 104, 146–148 Potassium-selective electrode (see K+ -selective electrode) Potentiometric selectivity , 30–33, 58–61, 113–114, 124–125 coefficient, definition, 113 Potentiometric system, 146

Index Power functions, 115 Primary buffers (pH), 67 Pressure (definitions), 128, 129 (see also partial pressure) R Rate constant-definition, 118 Reference electrode, 17, 19–26, 104, 126 K-TCB/octonol, 52 manufacture, 41–42 Resistors-colour code, 99–101 rules for addition, 99 S Salt bridges, 20 Selectivity coefficient, constant or factor (see potentiometric selectivity coefficient) Separate solution method, 59, 113–114, 125 S.I. (Systeme Internationale), 2 base and derived units, 127–128 prefixes, 4 S.I. derived units (table), 129–130 standard values (table), 128 Significant figures, 122 Silanisation, microelectrodes, 47, 51 Silver chloride electrode (see Ag/AgCl) Simon, W., 29 Skeletal muscle, 86 Smooth muscle, 108 Sodium-selective electrode (see Na+ -selective electrode) Solubility coefficient of gas, 143–144, 144–145 Sørensen, S.P.L., 63 Standard ambient temperature, 142 and pressure (SATP) Standard hydrogen electrode, 19, 63 Straight-line graphs, 118 T Teflon (PTFE) membrane, 146, 147 Time constant-definition, 118 Transport number, 21, 130 U Unbuffered solutions calibration of ISE’s, 57 V Valency factor, 9

150

Index

151

W Written reports, 126–127 rules for printing symbols, physical quantities, numbers 130–131 rules for table headers and graph axis labels, 131