Anal. Chem. lQ82, 5 4 , 2518-2524
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Calculation of Liquid Junction Potentials for Equilibrium Studies G. T.
Hefter’
Chemlsfry Department, University of Malaya, Koala Lumpur 22- 1 1, Malaysla
A detalled conslderatlon of the assumptlons underlylng the Henderson equatlon Indicates that, contrary to wldely held vlews, the ciilculatlon of llquld junctlon potentlals vla thls equatlon Is rellable under certaln well-deflned condltlons. I n particular thls Is shown to be true for the constant lonlc strength llquld junctions commonly used In equlllbrlum studles. The valldlty of the potentials calculated from the Henderson equatlon Is demonstrated dlrectly by comparlson of the calculated potentials wlth “experlmental”values and lndlrectly from equlllbrlum constant determlnatlons. The latter lndlcate that accurate thermodynamic lnformatlon can be obtained even In the presence of quite large llquld junctlon potentlals. Some of the llmltatlons and possible applications of the Henderson equatlon approach are brlefly dlscussed.
Since most potentiometric measurements for analytical and thermodynamic purposes are made on cells with liquid junctions ( I , 2), the renaissance of potentiometry due to the development of ion-selective electrodes (3-7) has heightened interest in liquid junction potentials (LJPs). As yet, liquid junction phenomena are not well understood and probably the most widespread technique for dealing with LJPs has been to simply assume them to be negligible. Unfortunately, this assumption is frequently untrue with the result that data so obtained may contain unsuspected but significant systematic errors. Such errors may well be a major cause of many of the frustrating differences in (for example) equilibrium constant or pH determinations, which are often encountered in the literature (7-9). Equations for the calculation of U p s have been in existence for nearly a century (10, 11). Despite some early successes in their use (12) it has become widely accepted that such equations are at best crude approximations and at worst grossly misleading (7, 13). Whilst it is true that LJP equations do break down badly under some circumstances (e.g., ref 13), nevertheless, reliable calculations of LJPs should be feasible for liquid junctions which conform closely to the models assumed by such equations. Clearly, before any confidence can be placed in calculated LJPs, the applicability of the particular model to the junction (or junction type) of interest must be very firmly established. Thus, before any experimental data can be considered it is necessary to review the basic premises underpinning the common LJP equations. This also seems a worthwhile task in view of the many misapprehensions in the literature regarding the calculation of LJPs. Once the basic assumptions are clearly identified, the applicability of the various models to liquid junctions of relevance to equilibrium and analytical studies can then be assessed both theoretically and experimentally. The latter is done in two ways in this paper: directly by comparison with “experimental” values and indirectly through equilibrium studies. ‘Present address: School of Mathematical and Physical Sciences, Murdoch University, Murdoch, W.A., 6150, Australia. 0003-2700/82/0354-2518$01.25/0
THEORY 1. Mathematical Description of Liquid Junction Potentials. When two dissimilar electrolyte solutions are brought into contact, an intermediate region is established between them in which the composition varies from that of one solution to that of the other. Whilst the exact nature of this region will depend on transfer processes such as diffusion, convection, etc., it can be considered conceptually as a series of infinitely thin layers (12). Using standard thermodynamic relationships and integrating over the entire boundary region for all species it can be shown that
Ej =
-RT F
-j
ti
A
C-Zi d In ai
where Ej is the liquid junction potential for the junction between solutions A and B, and ti, Zi, and ai are, respectively, the transference number, charge, and adivity of the ith species (12). Two important points should be noted. Equation 1is valid regardless of the physical nature of the liquid junction but, as it involves single ion activities, it cannot be evaluated purely within the framework of thermodynamics (9, 12). The nonthermodynamic assumption usually made to evaluate eq 1 concerns the nature of the liquid junction, i.e., the concentration profiles in the boundary region. The simplest approach mathematically is to assume linear concentration gradients which results in the well-known Henderson equation (11).More realistic physically, although computationally more complex, is the assumption of diffusion controlled gradients (10,14) as originally suggested by Planck (10). Both of these approaches will now be considered. a. The Henderson Equation. If linear concentration gradients are assumed across the liquid junction and activity coefficients and ionic mobilities (ui or Xi) are assumed constant, i.e., independent of solution composition, eq 1 becomes (11, 12, 15)
vi)
which is known as the Henderson equation. All symbols are as defined in ref 15, p 18; and it should be noted that both ui and Zi are taken as + for cations and - for anions. Although the Henderson equation was originally derived assuming fi = 1 (11, 12), it can be readily shown that an identical result is obtained for the more general case of fi = constant. b. The Planck Equation. If a liquid junction is formed by separating two solutions with an inert porous plug (a common procedure in practical potentiometry (12)), and the boundary concentration profile is determined solely by diffusion, then, again assuming f i and Xi are composition independent and constant, the following equation due to Planck is obtained (12):
gU2 - Ul - In ( C z / C l )- In t gCz - C1 (3) V2 - tV1 In ( C z / C J + In E CZ- ~ C I where Ej = (RT/F)In and all other symbols are as defined in ref 12.
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0 1982 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982
c. Other Equations. Various alternative equations for calculating LJPs have been derived over the years but in general they may be regarded as more rigorous versions of the Henderson or Planck approaches (e.g., ref 14 and 16). Their mathematical complexity puts them outside the scope of the present paper and in any event the results often offer little improvement over simlple models (see below). 2. The Calculation of LJPs for Liquid Junctions of Practical Importance,,The electrochemical cells with liquid junction used for most thermodynamic or analytical purposes (2,7,15) may in general be represented as follows: ref electrode and ref solution
/ soluition salt bridge I/ sensing electrode and test solution
Ej 1
(4)
Ej 2
where the double vertical lines represent the liquid junctions and Ejlk2are the associated LJPs. Often, the reference electrode and salt bridge solutions are identical in which cast? the left-hand junction in cell 4 is eliminated, e.g., when using a saturated calomel electrode (SCE) for pH determinations (2,9).Even for cells where this is not so, the compositions of the reference electrode and salt bridge solutions are effectively invariant under normal experimental conditions and Ej, is a constant which can therefore be incorporated into the “standard” cell potential. Thus, for all practical purposes only the nature of the junction between the salt bridge and the test solution is of interest. In equilibrium aqd analytical measurements two broad types of liquid junctions are used. The essential difference between them is in the nature of the salt bridge. a. Salt Bridges of High Ionic Strength. Salt bridges of this type, exemplified by the SCE in pH measurements, e.g., in cells Hg, Hg,C12(s), KC1 (sat. soln) 11 test soln, H+-responsive electrode ( 5 ) essentially involve the “benign neglect” of any LJPs. That is, it is assumed that a high concentration of an inert electrolyte (KC1 in cell 5) in the salt bridge will render the LJP negligible or at least invariant. Despite the ubiquity of this approach in analytical ( I , 9) and equilibrium studies (16)and some direct experimental evidence for its reasonableness (17), there are very real limitations on its reliability as recently emphasized in an excellent review by Bates (7). Furthermore, the LJPs in such cells are without doubt the most difficult to describe theoretically since the salt bridge and test solution are usually of widely differing ionic strength and composition. Apart from the glaring inapplicability of the constant f i and X i assumptions of the Henderson and Planck models, the junction boundaries are often very poorly defined physically (15,1’7‘).This appears to be especially so with commercial reference electrodes (7). Suffice to say that LJPs calculated under such conditions, by simple or sophisticated approaches, are unlikely to have physical significance or practical utility (7). Thus, it must be accepted that attempts to refine this approach by the calculation of Ej are probably not worthwhile and no further consideration to this type of junction will be given in this paper. b. Salt Bridges Involuing Constant Ionic Medium Liquid Junctions. Many of the problems associated with high ionic strength salt bridges would appear, at least in principle, to be overcome by the use of salt bridges involving constant ionic medium liquid junctions. This approach, where the salt bridge is chosen to have essentially the same ionic strength and composition as the test solution, is far less common than the use of high I salt bridges. Its usage has until now been largely confined to thermodynamic (equilibrium constant) studies
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where it has arisen naturally out of the use of the constant ionic medium principle (16, 18). Cells utilizing this type of salt bridge may be represented as ref electrode
salt bridge
at I( AB)
at Z(AB)
/
sensing electrode and test solution at Z(AB)
(6)
where I(AI3)represenb a solution maintained at constant ionic strength (=I = 1/2CCiZt)by the addition, normally in large excess, of a supposedly inert “swamping” electrolyte AB. As shown above only Ej2(hereafter simply Ej) is of practical importance. The potential superiority of this approach lies not in reducing the magnitude of Ej (LJPs at constant ionic medium junctions would in general be greater than those at high I salt bridge/test solution junctions) but rather in being able to calculate Ej reliably. This is because such liquid junctions would appear to conform reasonably closely to the various models assumed in deriving simple LJP equations. However, before confidence can be placed in UPSso calculated, it must be shown to what extent these models are applicable to real liquid junctions. 3. The Applicability of the Henderson and Planck Models to Constant Ionic Medium Liquid Junctions. As implied earlier there are three basic assumptions made in deriving both the Henderson and Planck equations. These are as follows: the nature of the concentration profiies through the liquid junction (i.e., the nature of the liquid junction boundary) and the constancy of both activity coefficients and ionic mobilities through the liquid junction boundary. The applicability of each of these assumptions to constant ionic medium junctions will now be considered in detail. a. Nature of the Boundary. Despite the problems which can be associated with the physical definition and reproducibility of heteroionic junctions generally (7,12), under some circumstances U p s are essentially independent of the physical nature of the junction boundary. For example, seven independent studies (12)employing five distinct junction types for the junction 0.1 M KCl 11 0.1 M HC1 (which may be regarded as a “limiting case” constant ionic medium junction) obtained values of Ej = 27.65 f 0.49 mV; i.e., a standard deviation of
