ANALYST, MAY 1987, VOL. 112 573 Temperature Compensation in Potentiometry: lsopotentials of pH Glass Electrodes and Reference Electrodes Part 1. Theory Derek Midgley CEGB, Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead, Surrey KT22 7SE, UK The reasons for the non-linearity of the temperature response of glass pH electrodes are explored theoretically and the relationship between pHisor the isopotential pH (at which the e.m.f. is invariant with temperature), and the different components of the potentiometric cell is demonstrated. In particular, the temperature coefficient of pH of the glass electrode's filling solution is shown to be paramount in designing electrodes with characteristics convenient for instrumental temperature compensation. The pH of most filling solutions varies parabolically with temperature and the linear correction applied by the temperature compensation circuits of pH meters will have a limited useful range.Filling solutions with linear temperature characteristics are proposed, but the resultant electrodes would be incompatible with the many pH meters having a fixed isopotential setting of 7. Liquid junction potentials are shown to make a small and essentially linear contribution to the temperature dependence of the cell. Keywords: pH; glass electrode; reference electrode; temperature compensation; isopotential Although the most accurate measurements of pH require sample and standard solutions to be at the same temperature, some applications have to cope with varying temperatures. Temperature compensation is, therefore, a normal feature of pH meters and is also found on some process analysers for use with ion-selective electrodes.Modern instruments may incor- porate digital thermometers reading to 0.1 "C and those incorporating microprocessors may have settings for the slope factor and isopotential pH (enabling corrections to be made for the temperature variation of the standard potential) as precise as 0.01 mV pH-1 and 0.001 pH, respectively. Instrumentally, therefore, it should be possible to compensate for temperature more accurately than before, provided that the thermal characteristics of sensing and reference electrodes can be defined with adequate precision and the theory implicit in the design of the temperature compensation circuitry is correct. In this paper, the temperature dependence of pH cells is examined theoretically and in Part I1 the performance of commercially available glass pH and reference electrodes will be considered.All pH meters incorporate some form of temperature compensation, although its exact nature is not always made explicit by the controls on the meter or the operating manual. In the equation relating e.m.f. to pH, both "constants" (the standard potential, Ecell, and the slope factor, k ) vary with temperature. The temperature coefficient of the slope factor should be constant and predictable for all electrodes and compensation for this is a feature of almost all pH meters, under either manual or automatic control. The temperature coefficient of the standard potential is the source of various approximations, errors and misapprehen- sions.In principle, it can be calculated for known reference elements in known solutions inside the glass and reference electrodes; in practice, the analyst does not know the composition of the solution inside the glass electrode. An indication of this temperature coefficient is sometimes sup- plied by the manufacturer in the form of the isopotential pH, but unfortunately this is often quoted with insufficient rigour (not being defined with respect to a named reference electrode) and may even be confused with the zero-point pH (at which the e.m.f. = 0 mV). For temperature compensation with most pH meters it is desirable that these points should coincide in practice, but they are theoretically distinct and may differ considerably. E=ECeI1-kpH .. . . * * (1) Even with electrodes whose temperature characteristics are well defined, errors may arise from the limitations of pH meters. Many pH meters lack an isopotential pH adjustment and work on the basis (often unstated in the instruction manual) that the isopotential pH is 7; errors are unavoidable if the electrodes have different characteristics. Commercially available pH meters invariably apply a correction that assumes that the standard potential varies linearly with temperature, but the non-linearity of the temperature variation of the standard potential has long been known.1 Isopotential pH values are usually determined experimen- tally and applied as if they were linear temperature coeffi- cients. Theoretical treatments have expressed isopotentials only in empirical coefficients.1-3 This paper relates the isopotential pH to the internal reference electrode and filling solution of the glass pH electrode and to the external reference electrode and its liquid junction with the test solution.The sources of non-linearity in temperature compen- sation are considered. Factors analogous to isopotential pH can also be derived for ion-selective electrodes394 and in these instances better linearity may be obtained. Theory The conventional potentiometric cell with a glass electrode for measuring pH may be represented as below. 1 MX I solution containing I 1 test solution I 1 X-ions,ax I I pH M'X I M' I glass I innersolution I /membrane1 dx,pH' 1 1 In practice, X- is the chloride ion and M and M' are either silver or mercury or, more rarely, thallium.If the membrane responds to hydrogen ions with perfect selectivity, the e.m.f. of the above cell may be represented as in equation (2). E = (E" - kloga'x + kpH')glass - kpH + E, -(E" - klog QX)ref . . (2) where E"' and E" are the standard potentials of the two reference elements M'j M'X and MIMX, Ej is the liquid junction potential and k is the Nernst slope factor, equal to RTlnlOIFwhere R is the gas constant, T K is the temperature and F is Faraday's constant. The terms in brackets are constant at a given temperature and Ej is assumed to be so for574 ANALYST, MAY 1987, VOL. 112 the sake of calibrating the pH meter (the reference electrode is chosen so that this is a reasonable assumption).Equation (2) can be re-expressed as equation (1), where EOcell contains all the constant terms. pH meter temperature compensation circuits2,s assume that EOcell and k vary linearly with tempera- ture. For k it may be seen that this is correct because Elk RlnlO aT- F ---- -constant . . . . . In the case of ITcell, however, this assumption is an approxi- mation. Subtracting equation (1) from equation (2) and differentiat- ing, we obtain ak + kapH' -1oga'x - - (aZZ' kaloga'x aT aT aT log ax - Rearranging equation (4) gives Examination of the terms in equation (5) shows the conditions necessary for obtaining an EOcel, that varies linearly with temperature. (i) The terms grouped as (-log + pH' + log ax) ak/aT will vary linearly because of equation (3).(ii) The standard potential terms dEOldT and dEO'ldT are not necessarily linear, but the curvature is negligible over a range of +15 "C for calomel and silver - silver chloride electrodes. With matching reference elements in the glass and reference electrodes, however, these terms cancel provided that the cell is isothermal. (iii) On a molal scale, the terms kaloga'x/aT and kalogaxl aT are approximately zero, provided that the solutions are not saturated. With saturated solutions, the introduction of a temperature-dependent solubility makes the terms non-zero and, generally, non-linear, e.g., with saturated potassium chloride solutions aaCl/a T is approximately linear, making (iv) The liquid junction term aEj/a T should be small for any junction likely to be used in a pH cell and it will be shown below that any variation should be almost linear over a range of 30 "C for concentrated, but not saturated, potassium chloride bridge solutions.If the junction contains a saturated reference solution, additional non-linear terms may arise. (v) The term kapH'/aT depends on the nature of the solution inside the glass electrode. With a solution of strong acid (pH <4), apH'/aT = 0 and this approximation is still fairly good for many weak acid buffers over a moderate range of temperature, e.g., acetate buffers. However, because the circuitry of some pH meters restricts the choice of pHiso, glass electrodes generally contain buffers closer to neutrality and for such solutions apH'/aT is both non-zero and non-linear. This will be discussed after the consideration of isopotentials.It can be seen that careful consideration needs to be given to points (ii)-(v) if a linear variation of EOcell with temperature is to be achieved. Points (ii)-(iv) come readily within the control of the analyst, but point (v) does not. Isopotential Correction The temperature coefficient of the standard potential is usually expressed in pH methodology as the isopotential pH, i.e., the pH at which the e.m.f. of the electrode pair is invariant with temperature. aiogacl/a T curved. From equation (l), the e.m.f. at the isopotential point, Eiso, is given by By the definition of pHiso, aEi,,4aT = 0. Hence, Eiso = EOcell - @Hiso . . . . . . (6) aEOcell/aT=pHisOak/aT . . . . (7) Substituting from equation (3), 3EOCell/aT = pHi,J?lnlO/F .. . . (8) Equation (8) and the definition of PHis0 imply that EOcell varies linearly with temperature, hence EOc-11 may be expressed as where E, is the value of EOcell at reference temperature, T,. Hence Comparison of equations (6) and (9) shows that RTsln10 F Eiso = E, - - PHiso Substituting in equation (1) gives an expression that relates the experimental variables (E, pH) at (ideally) all temperatures by means of two temperature-independent constants (Eis0, pHis,) and the theoretically predictable slope factor (k) with its simple linear dependence on absolute temperature. E = Eiso - k(pH - pHis,) . . . . (10) Equation (10) forms the basis of temperature compensation in pH meters. Once Eiso, k and pHiso have been set at one temperature, through the "buffer," "slope factor" and "iso- potential'' controls, respectively, the meter thereafter requires adjustment only of the temperature setting for it to be used to measure pH at other temperatures.The analysis of the temperature-dependent components of the cell potential [equation ( 5 ) ] indicated that, in general, aE"cell/aT is not constant, and it follows that, in terms of the idealised pHiso defined above, equation (8) can only be an approximation. pHiso may be obtained graphically by plotting e.m.f. against pH for a number of solutions at a series of temperatures. If aE"cell/aT were a constant, all the isotherms would intersect at pHiso. In practice, non-linearity of the temperature dependence means that the intersecting iso- therms form a zone covering a range of pHiso.The minimum data for calculating pHiso are e.m.f.s in two standard buffer solutions at two temperatures, enabling intersecting isotherms to be drawn or solution by simultaneous equations of the form of equation (10). Hence, E - E' + kpH - k'pH' k - k' . . . . PHiso = where k and k' are obtained from AEIApH at temperatures T and T', pH and pH' are known and E and E' are measured. The precision of pHiso calculated in this way can be estimated from the standard deviation of the e.m.f. by the usual rules for combination of random errors. Neglecting random errors in the pH values of standard buffer solutions we obtain from equation (11) + . . . . . . " (k - k')2 where a(x) is the standard deviation in quantity x . Evaluating equation (12) for an ideal case with PHis0 = 7.0,ANALYST, MAY 1987, VOL.112 575 determined from measurements with potassium hydrogen phthalate buffer (pH 4) at 15 and 25 "C and with the standard deviation of all e.m.f. measurements assumed to be 0.1 mV, we obtain a (E - E') = 0.14 mV, a(k) = 0.05 mV decade-1 (for determination over a 3 pH span) and E - E' = -5.4 mV. Substituting in equation (12), -- a2(pHiso) - 5.2 x 10-4 + 2.47 x 10-3 49 Hence a(pHiso) = 0.38 and is dominated by the error ink - k' [the second term on the right-hand side of equation (12)]. Alternatively, equation (10) can be transformed into E + kpH = Eiso + kpHiso . . . . (13) If e.m.f.s are recorded in a buffer solution over a range of temperatures, the left-hand side of equation (13) can be plotted against k.Ideally the graph should be linear, with a slope equal to pHiSO, but it will generally be curved in practice. This method has the advantage that the curvature gives a qualitative indication of the range of usefulness of PHiso. pHi, as an Electrode Characteristic pHiso is a characteristic of the electrode pair in the poten- tiometric cell, but if the body of the reference electrode is separated from the junction by a long tube and held at a constant temperature, the cell pHiso can be equated with the glass electrode's pHiso. Such an assumption neglects the contribution to pHiso from the temperature coefficient of the liquid junction potential (discussed below). Provided that a remote junction reference electrode of the above sort is used, the same pHiso should be found for the glass electrode, regardless of the nature of the reference electrode or its temperature (provided that it is constant).With a non- isothermal cell of this kind, therefore, pHiso can be regarded as a characteristic of the glass electrode. If the glass electrode in the non-isothermal cell is replaced by a second reference electrode and the temperature of the test solution is varied, the change in e.m.f. can be attributed to the second reference electrode alone and its temperature coefficient calculated. This temperature coefficient can be expressed as an isopotential factor characteristic of the reference electrode; in a pH cell this factor would be formally equivalent to a pHiso value, although it is unrelated to any real solution pH. This terminology is adopted below: a graph of E against k gives pHiso for the reference electrode as the slope of the line represented by equation (14).E = Ei, + kpHiso - - . . . . (14) pHiso for an isothermal cell consisting of glass and reference electrodes may now be calculated as the difference of individual pHiso values obtained as above. pHiso(cell) = pHiso(glass) - pHiso(reference) Relationship between the Isopotential and Zero Points Glass electrodes should6 be marked with their zero point, denoted by E X , with respect to a stated reference electrode at 25 "C, i.e., Xis the pH at which the glass - reference electrode pair gives 0 mV. Almost all commercial electrodes are nominally E7. Manufacturers rarely quote pHis0 and it is often assumed that pHiso = 7 also. However, these quantities are not identical.Hence pHiso for a non-isothermal cell with a remote junction exceeds that for the corresponding isothermal cell by pH 2.2 or 0.4 for reference electrodes with calomel or Ag - AgCl elements in 3 moll-1 potassium chloride solution, respectively. With the remote junction electrode at 25 "C, the zero-point pH is the same for both configurations of cell. From equation (2), at E = 0 mV the zero-point pH is given by equation (15), where AEO = E"' - E". AE'+Ei a'x k ax -log-++H'. . . . (15) pH" = From equations (4) and (7), aAE" + aEj ~ k apH' k alog ax k alog a'x a T +-- a T a T aT a T PHiso = akia T . . . (16) - a x ~ The relationship between pH" and pHiso will be considered for a range of electrode types. Isothermal cells with non-saturated inner and reference solu- tions In this instance, where fx is the activity coefficient.Concentrations defined as molality are independent of temperature and activity coeffi- cients' are observed to vary by about 0.01% K-1, which is equivalent to 0.0025 mV K-1 and less than can be detected with a pH meter reading to 0.1 mV. Similarly, aloga'x/aT = 0. Further, any small non-zero components of the two terms tend to cancel. On a molar scale alogax/aT # 0, and this would involve additional terms in the density of the solutions and different aE"IaT terms from those which follow. The tem- perature coefficient of the e.m.f. is itself independent of the concentration scale and use of the molal scale gives simpler expressions. The NBS standard buffers are defined on a molal scale.Combining equations (15) and (16) and eliminating alogax/ a T terms, aAE' + aE, + kapH' If the two reference elements are of the same type, AE' = 0 and aAE"/aT = 0. Hence In practical pH cells, Ej is arranged to be small and its variation with temperature will make only a small contribution (see below). As an approximation, therefore, apH' a T pHiso pH" + T- . , . . (19a) or 1 apH' pHiso == pH" + - - . . (19b) lnlO alogT ' . Equation (19a) shows that even with identical inner and external reference electrodes and favourable assumptions about liquid junction potentials, pHiso will not coincide with pH" unless apH'/aT = 0. The latter condition could only be expected from a solution of a strong mineral acid. From equation (15), with a 3 moll-1 KCl reference electrolyte, AE = 0 and Ej = 0, it follows that for pH"(= pHis,) = 7.0 as desired, the internal filling solution of the glass electrode would have to be 4 x 10-4 mol 1-1 hydrochloric acid (or a solution having the same activity of hydrochloric acid).576 ANALYST, MAY 1987, VOL. 112 With non-identical reference elements, equation (18) shows that pHiso = pH" only if (again setting Ej = 0 and aEj/aT = 0) aAE" kapH' +- AE" a T i3T - k ak/a T I .. . Equation (19b) shows that pHis, is a constant only if pH' changes linearly with the logarithm of the absolute tempera- ture. However, this is not a property of real solutions (see below), except in the trivial case apH'/alogT = 0. The present state of pH meter technology treats pHiso as a constant and hence meters have a limited range in which temperature compensation can be applied with a given accuracy.The next stage of sophistication (which would require a microproces- sor-based pH meter only slightly more complicated than present models) would be to apply a linear correction to pHis, by equation (19a): a prerequisite for this would be a glass electrode whose internal pH changed linearly with tempera- ture. There are probably no such electrodes available at present, although suitable solutions could be devised. With present pH meters, requiring pH" = pHiso = 7, ideal conditions inside the glass electrode can be calculated from equations (15) and (20). This has been done in columns (3) and (4) of Table 1 for calibration at 25 "C versus reference electrodes with 3 moll-1 KC1 reference electrolyte (neglecting liquid junction potentials). Microprocessor-based meters should be able to cope with a much wider range of electrodes, because the equalities pH" = pHis, = 7 are unnecessary, although some microprocessor meters imitate the limitations of older meters.Table 1 gives mathematical solutions, but it does not follow that real chemical compounds exist to fulfil them, particularly when apH'/aT is required to be constant over the desired range of temperature. Even when the reference elements are identical and apH'/aT = 0, the concentration of hydrochloric acid required may be considered too low for chemical stability inside the electrode. Isothermal cells with at least one saturated solution Because the solubility of potassium chloride varies by about 0.04 mol kg-1 K-1, we have aloga&T = 0.005 (equivalent to an increase of 0.14 in pHis,).Reference solution saturated with potassium chloride Derivations can be carried out exactly as before, except that the term apH'/aT in equations (17) and (18) is replaced by (apH'/a T + aloga& 7). The conditions required for PHis0 = pH" = 7.0 with a saturated potassium chloride reference solution can be obtained from Table 1 by (i) subtracting 0.17 from the pH' - loga'a column and (ii) subtracting 5 X 10-3 from the isothermal apH'/aT column. Inrier solution saturated with potassium chloride The derivation proceeds as before except that in equations (16) and (17) apH'/aT is replaced by (apH'/aT - aloga'cl/ 32'). The conditions required for pHiso = pH" = 7.0 are obtained from Table 1 by (i) adding 0.40 to the pH' - loga'cl column to obtain pH' at 25 "C and (ii) adding 5 X 10-3 to the isothermal apH'/a T column.Both inner and reference solutions saturated with potassium chloride In this instance alogacl/a T = aloga'cl/i3 T and equations (16) and (17) are still valid. Conditions for pHiso = pH" = 7.0 are obtained from Table 1 by (i) adding 0.23 to the pH' - loga'cl column to give pH' at 25 "C and (ii) retaining the isothermal apH'/a T values. Non-isothermal cells The reference electrode is kept at a constant temperature as that of the rest of the cell is varied. The temperature coefficient of the e.m.f. is obtained by differentiating equation (2), omitting the terms in the ( )ref parentheses. ak kapH' ak l o g a ' x z + + pH'-) a T dass Non-saturated inner and reference solutions As before, aloga'x/aT = 0.At pH = pHis, we have, by definition, aE/aT = 0 at constant pH, hence aE. ak apH' ak a T a T a T aT a T aE"' + - - (loga'x - pH') + k - - pHis, - = 0 Therefore, (23) ak/a T pHis, = pH' - logalx + At E = 0, equation (16) is still valid for the non-isothermal cell, hence substituting in equation (23) gives Evaluating equations (16) and (24) for electrodes calibrated at 25 "C and with neglect of liquid junction potentials gives the results in columns 3 and 5 of Table 1. Inner solution saturated with potassium chloride For aloga'&T = 0.005 and wherever apH'aT appears in equations (22)-(24) it should be replaced by (apH'/aT - aloga'&T). The conditions for pHiso = pH" = 7.0 are obtained from Table 1 by (i) adding 0.40 to the pH' - loga'cl column to obtain pH' at 25 "C and (ii) adding 5 x 10-3 to the non-isothermal apH'/a T column. Table 1.Conditions required inside glass electrodes to give pHiso = pH" = 7.0. (Calibrated at 25 "C versus reference electrodes with 3 moll-' KC1 reference solutions) Reference element apH'Ja T apH'JaT required for required for Glass Reference isothermal non-isothermal electrode half-cell pH' - loga'cl cell cell AgCl . . . . . . AgCl 6.77 0 -2.7 x 10-3 AgCl . , . . . . HgzC12 7.54 3.5 x 10-3 -5.3 x 10-3 HgzC12 . . . . . . Hg2C12 HgZC12 . . . . . . AgCl 5.99 -3.5 x 10-3 -6.2 x 10-3 6.77 0 -8.8 x 10-3ANALYST, MAY 1987, VOL. 112 577 Reference solution saturated with potassium chloride As the temperature of the reference electrode is constant, pHiso is unaffected, but 0.17 is subtracted from column 3 to give pH' - logalcl (non-saturated inner solution) or 0.23 added to give pH' at 25 "C (saturated inner solution).Influence of the Liquid Junction Potential on the Isopotential Point The contribution of the liquid junction potential, evident in equations (18) and (24), has so far been neglected, mainly because it was expected to be small for practical liquid junctions, but also because the liquid junction potential of an unknown sample is not directly calculable. However, evalua- tion of the liquid junction contribution in some instances is instructive. Equations (18) and (24) for expressing the isopotential pH each contain the terms -Ej/k and (aEj/aT) (aT/ak).Calcula- tions by Picknett8 show that for a reference electrode with a concentrated potassium chloride electrolyte, E, is likely to vary within the range -3 to -7 mV for a wide range of sample electrolytes in the concentration range 10-2-10-6 mol 1-1. (Although the basis of these calculations is not strictly rigorous,9 experimental e.m.f.s in dilute solutions were predicted to within k0.5 mV.) The approximate maximum effect on pHiso, therefore, is 7/k = 0.12, but as only the variation of Ej with respect to the calibrating solution can be discerned, the effect is unlikely to exceed 0.06 units of pHiso. If PHis0 is determined from measurements in the same buffers as are used for calibration, the contribution to pHiso from this term should be even smaller.However, at the worst the Ej/k term is a fraction of the experimental variation in determining An estimate of aEj/aT may be obtained by differentiating the Henderson equation for the liquid junction potential, which is conveniently simple but involves assumptions that do not always apply to real junctions in pH measurements.10 PHiso. where hi, Ci and zi are the molar conductivity, concentration and charge (with sign) of the ith ion, respectively, and the superscripts R and L denote the right- and left-hand sides of the liquid junction. Letting P, Q, X and Y equal the sums in the numerators and denominators of the pre-logarithmic and logarithmic terms, we obtain Hence P a Q a T + -. - (25) aQ - P- I A aT Q2 - --- *-log-- k log-. a T Q Y \ Y P Y Q X Y2 + --.-ln10 - The only temperature-dependent variables in P, Q, X and Y are the molar conductivities.Hence aEj ak Ej ( ncp; PY lnlO J \--/ Y2 Equation (26) can be evaluated by replacing ahi/a T by ALi/A T from tabulations by Rpbinson and Stokes7 which show that conductivities vary approximately linearly with temperature over a range of about 30 "C. With concentrated (but not saturated) potassium chloride electrolyte in the reference electrode, both Ei/k and the variation in log X / Y are negligible and with dilute test solutions Cf. >> CiR, so that the terms P and Q are influenced only by CKL and C&. Hence With a 3 mol 1-1 potassium chloride reference solution at 25 "C, a Ej -(mVK-1)=0.0410gXhiCiR- 0.10 aT Table 2 shows the terms in equation (25) for two different test solutions.The dominance of the middle term is obvious. From equations (18) or (24) the variation of junction potential with temperature contributes about -0.5 unit to pHiso for 10-4 mol 1-1 hydrochloric acid solutions or -0.2 for 0.1 mol 1-1 acetic acid - acetate buffer. In the empirical determination of PHis0 these contributions would only be apparent if pHiso were determined in solutions different from the calibration solu- tions. The results in Table 2 show that pHiso determined in a buffer solution is likely to be about 0.3 unit too high for application to dilute sample solutions. From equation (26) and the known conductivities we may also infer that aEj/aT makes an almost linear contribution to the over-all temperature coefficient provided that the tem- perature range does not exceed about 30 "C.The first term on the right-hand side of equation (26) is truly linear, but is dominated by the others in ahi/aT. The calculations show that liquid junction potentials make a small contribution to the temperature coefficient of the e.m.f. of a cell. However, empirically determined values of isopoten- tial pH should largely compensate for this effect and the likely error in pH is small compared with that from other sources. Temperature Variation of pH of Electrode Filling Solutions The appearance of the term 3pH'IaT in equations (18)-(24) shows that the internal filling solution has a significant effect on the thermal characteristics of an electrode. Restricting consideration to the conditions required to produce Table 1, it can be seen from column 3 of that table that pH' = 7 + logdcl k 1.The practical limits for a'cl are about lo-3-100 moll-1, hence the limits of pH' are 3-8. Inspection of the properties of standard buffer solutions11 shows that most carboxylate and phosphate buffers go through a minimum in pH somewhere in the range 0-50 "C, i.e., neither pH' nor 3pH'IaT is constant. Over a wide range of temperature (040 "C), a 0.1 moll-1 acetic acid + 0.1 moll-' sodium acetate buffer comes nearest to constancy, with aANALYST, MAY 1987, VOL. 112 578 Table 2. Components of temperature coefficient of liquid junction potential and contribution to isopotential pH x a P ? aT k Y a T Q Q aT Y Test solution mV K-1 mVK-1 mVK-1 mV K-1 to pHiso -k- p a -logXt/ -1 8Ej ak. “I -/clog- - (-) I aT Contribution 10-“moll-1 hydrochloric acid .. . . -0.015 -0.086 -0.002 -0.10 -0.5 0.1 moll-1 sodium acetate + 0.1 moll-’ acetic acid . . . . . . -0.006 -0.037 +0.001 -0.043 -0.2 * Reference solution 3 moll-1 potassium chloride. t Symbols refer to equation (25). maximum spread of 0.05 pH, but in the ambient and sub-ambient range (0-30 “C) 0.05 moll-1 potassium hydrogen phthalate has a spread of only 0.012 pH. Buffers in which apH‘IaT is more nearly constant are those with amino groups of suitable strength, e.g., N-tris(hydroxy- methyl)methyl-2-aminoethanesulphonic acid (TES) and N-2- hydroxyethylpiperazine-N’-2-ethanesulphonic acid (HEPES) buffers12 (pH = 7.5). 3-(N-Morpholino)propanesulphonic acid (MOPS) would seem to be a good candidate for a pH 7.1 buffer.13 However, these buffers generally would give too negative a value of apH’/aT in Table 1 for PHis0 = pH” = 7 to be achieved.Other buffers of this type are known” but they have not been studied at different temperatures. The more acidic buffers might be especially worthy of study in relation to glass electrode filling solutions. Amine buffers in aqueous glycerol have also been suggested for this purpose.14 Calculation of apHtaT The only instance considered is that of a solution of a single monobasic acid, HA, partially neutralised by sodium hydrox- ide. The system is defined by the mass and charge balance equations, the protonation constant, K = [HA]/[A][H], and the autoprotolysis constant, K,. For convenience, charges have been omitted and activity coefficients neglected.Total acid = TA = [HA] + [A] = [A](l + K[H]) Total base = TB = [Na] Charge balance: [HI + [Na] = [OH] + [A] i.e., Rearrangement followed by differentiation with respect to temperature gives { 3K[H]2 + 2[H](l+ KTB) + TB - TA - KKw] = aT aK aKW aT aT --([HI’ + [HI’ TB - [HI K,) + - (1 + K[H]) Expressing the temperature differentials in logarithmic form gives 3[H]2 + 2[H] (K-1+ TB) + (TB - TA) K-1- K , (27) Equation (27) shows that the variation of pH with tempera- ture is fairly complicated, as alogK,/aT is almost linear whereas 3logKla T may be parabolic. In certain circumstances, the above general equation can be greatly simplified. For moderately concentrated solutions of moderately weak acids (such that K2KW < 1 and KTA >> 1) [HI = K-1 at half-neutralisation and then apWa T = alogK/a T.Discussion The relationship between the zero point and the isopotential point, and the conditions necessary to make them coincide, have been demonstrated. The non-linearity of an electrode’s temperature coefficient is shown to depend mainly on the temperature coefficient of the pH of the solution inside the glass electrode, but the use of saturated solutions in the glass or reference electrodes is another cause. The contribution of the liquid junction potential (and its temperature coefficient) to the isopotential correction has been shown to be small for the usual concentrated potassium chloride bridging solutions. In an alternative approach (the Ross electrode), the non- linearity of the pH buffer’s temperature response is compen- sated by an oppositely-responding reference electrode based on a redox couple.15 With the factors listed above, the approximate nature of the linear temperature corrections applied by pH meters becomes apparent.For linear characteristics to be achieved, new filling solutions need to be devised for glass electrodes, but such solutions are unlikely to give pHiso = 7 and a meter with an adjustable isopotential control would be necessary. Over a limited temperature range, certain weak acid filling solutions have attractive properties for electrodes intended for use in the environmental range (0-30 “C) and could give pHiso = 7. Non-ideal Electrodes Midgley and Torrance16 observed with symmetrical pH cells, in which the glass electrode was filled with a more concen- trated potassium chloride solution than usual, that the temperature-dependent terms did not cancel as expected unless the test solution also contained concentrated potassium chloride.As the anomaly was observed only with very dissimilar solutions on each side of the membrane, it is inferred that differences in hydration, ion exchange or adsorption between the inner and outer surfaces of the membrane could be induced by the nature of the two bathing solutions and give rise to an additional source of temperature dependence which has not been considered in this paper or elsewhere. This work was carried out at the Central Electricity Research Laboratories and is published by permission of the Central Electricity Generating Board. 1. 2. 3. 4. 5. 6. References Jackson, J., Chem. Znd. (London), 1948, 7. Mattock, G., “pH Measurement and Titration,” Heywood, London, 1961. Covington, A. K., CRC Crit. Rev. Anal. Chem., 1974,3, 355. Negus, L. E., and Light, T. S., Znstrum. Technol., 1972, 19 (Dec.), 23. Westcott, C. C., “pH Measurements,” Academic Press, New York, 1978. British Standard BS 2586, “Glass and Reference Electrodes for the Measurement of pH,” British Standards Institution, London, 1979.ANALYST, MAY 1987, VOL. 112 579 7. Robinson, R. A., and Stokes, R. J., “Electrolyte Solutions,” Second Edition (Revised), Buttenvorths, London, 1965. 8. Picknett, R. G., Trans. Faraday SOC., 1968, 64, 1059. 9. Covington, A. K., “Specialist Periodical Report, Electro- chemistry,” Volume 1, Chemical Society, London, 1970, p. 72. 10. MacInnes, D. A., “The Principles of Electrochemistry,” Reinhold, New York, 1939, p. 231. 11. Perrin, D. D., and Dempsey, B., “Buffers for pH and Metal Ion Control,” Chapman and Hall, London, 1974. 12. Bates, R. G., Vega, C. A., and White, D. R., Anal. Chem., 1978,50, 1922. 13. Sankar, M., and Bates, R. G., Anal. Chem., 1978, 50, 1922. 14. Simon, W., and Wegmann, D., US Pat., 3445363, 1969. 15. Ross, J. W., US Pat. 4495050, 1985. 16. Midgley, D., and Torrance, K., Analyst, 1982, 108, 1297. Paper A61281 Received August 14th, 1986 Accepted December 1 Oth, I986