This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This review is variously a presentation, reflection, synthesis and report with reference to more recent developments of an article – in a journal which has ceased publication – entitled “Some Electrode Theorems with Experimental Corroboration, Inclusive of the Ag/AgCl System”

The review and report here describes work which had a particular philosophical approach. There were several levels or hierarchies that went into this study, following a pattern that others have pioneered. The study was predicated on attempting to delineate in a broad manner phenomena concerning electrodes that could be quantified to arbitrary degree. Such types of research begins at the qualitative level [

Some of the key theories constructed to explain electrochemical phenomena seem to conform (understandably enough) to a linear superposition of independent physical concepts as a first approximation, without taking into consideration the possible special interference effects which might arise by juxtaposition of these concepts. An example of this approach is to be found in the general Stern modification of the basic Gouy-Chapman (GC) double layer theory. Here we observe a combination of the GC theory with the Helmholtz capacitance model, strictly without any interference. When interference effects are included, such as is suggested here (based on experimental and theoretical considerations) for the case of electrodes exchanging charge between the substrate electrode and the electrolyte across the Helmholtz inner layer, then the usual application of the Stern model to the electrode interface may not yield results compatible with experiment. The inadequacy here has nothing to do with the creation of “bad” theories or incomplete theories as opposed to better ones, nor the utilization of wrong concepts, but rather on not accounting for possible interference effects which would not be anticipated if the various key concepts were separately combined. Most of these cross-effects are of minor importance (such as those concerned with the electronic chemical potential), but others, such as those concerned with the double layer, can be quite dramatic, such as illustrated in

An example of minor interference is found in treatments of the solid state electronic chemical potential μ. In most electrochemical applications, it is inferred that since μ is an intensive quantity, it must be invariant with respect to system size, and since the electronic (or charge contribution) −Fψ_{m} accounts for the charge effects, it may be factored out of the pure, uncharged chemical potential contribution μ, which is in turn dependent on the free-electron (uncharged) number density ρ in the substrate lattice. On the other hand, there exists a charge transfer the instant the neutral electrode is inserted into the reactive electrolyte involved in the cell reaction, implying a change in the free-electron density ρ to ρ’. For fixed amount q of charge transferred, it is clear that ρ → ρ’ as the volume V of the substrate becomes large (V → ∞) since

The fundamental premise in electrochemistry is that μ(e), the chemical potential of the electrons in the metallic phase substrate electrode is invariant during measurements [^{ø}, the standard potential of the electrode relative to some other electrode. E^{ø} values are determined [^{ø} values when cells are concatenated [^{+} for the system considered here), then the fundamental premise is contradicted if a nonzero and varying cell e.m.f. is registered for (1a) for different concentrations m. Furthermore, it would be beneficial if any of these cell potentials could be quantified by recourse to theory to map out the approximate regime of electrode size, solution strengths and the assumed potential of zero charge required in the theory where extraneous electrode effects become noticeable, so that in actual measurements, these predicted effects might be reduced or estimated by suitable choice of physical conditions.

In what follows, we use the conventional notion of electrochemical potentials with reference to an arbitrary spatial point for all components of interest [_{d} between the metal inner (bulk) Volta potential ψ_{m} and bulk solution ψ_{s} electrical potential (Δψ_{d} = ψ_{m} − ψ_{s}) to be derived from equating the electrochemical potentials following the Gibbs equilibrium criterion:
^{ø} identified as:

Henceforth, F is the Faraday constant, the μ′s are chemical potentials, T the Kelvin temperature, R the gas constant and the a’s represent activities. E^{ø} is set constant for activity coefficient determination at a fixed temperature. The electrochemical potential of the electron is defined as _{e} = _{m}_{m}, and these are standard expressions [_{d} is not the Volta potential but the equilibrium potential difference. We note that the Volta coupling potential between the electrode and external circuit are opposed at either ends of the circuit during actual e.m.f. measurements and their effects cancel; and hence for single-electrodes even, we may absorb the effects of the external coupling in say the μ(Ag) term. In particular, the experimental e.m.f. measurements of the electrochemical cell (1a) used to compute properties such as the activity coefficients are deemed to arise solely from the electrodes with no interference from the external circuitry, and the following analysis accords with this viewpoint in that the external circuitry effects are not considered. _{l}(e, Met.) + μ_{F}(e) where μ_{l}(e, Met.) is the energy (per mole) required to transport an electron to the metallic substrate lattice at the lowest energy where the electrons are counted and arranged according to Fermi statistics, and μ_{F} is the “Fermi energy” region chemical potential which can be defined precisely for certain situations, such as when the free electron gas model is utilized, where μ(e, Met.) is the chemical potential of the electron gas in a uniform, positively charged ‘jellium’ background (where the lattice contribution is ignored). This partition is standard for many experimental purposes, not least the rationalization of the T^{3} dependence of the electronic specific heat at very low temperatures. Here, the precision of the μ(e, Met.) difference determination used to measure electrode effects are dependent on the invariance of μ_{l}(e, Met.), which implies non-distortion of the lattice; this can in principal be achieved by polarizing the electrode to the equilibrium value so that there is no change in lattice characteristics due to chemical (or metallic) deposition. The bulk potential ψ_{m} is linearly additive, i.e.

In order to test the regime of validity of E^{ø}, the following lemmas (all of which have further applications to be elaborated) will be used. The results which the low resolution experiments obtained (despite the fact that these experiments could not determine unambiguously the predicted e.m.f. change of order of 0.1 mV with KCl concentration change of 0.1 m, compared to the actual cell e.m.f. values of ~ 1.5 mV over the entire concentration range) are in quantitative agreement to the theory developed from the lemmas. Each of the following lemmas will be applied to specific quantitative experiments. For what follows, the Greek letter symbols used follows the convention of the standard literature and this is sometimes unfortunate because of multiple usage of alphabets such as

In what follows, the different electrode types must be distinguished. The usual nomenclature [_{d} is varied, the current i must remain zero for this electrode (the i _{d} curve is horizontal). Thus the equilibrium situation of the inert electrode is extrapolated to the dynamical regime of a varying Δψ_{d} defined by (2) where no electric c urrent flows. By ‘inert’ is meant that there is no net exchange of ions or atoms (except electrons) between solution and electrode. The other extrapolation of behavior refers to ii) when the ‘equilibrium value’ Δψ_{d} (when i = 0) of the electrode system does not vary as a current is passed through the electrode-solution interface (the i vs Δψ_{d} curve is vertical) ; iii) refers to the intermediate situation, where the departure of the electrode potential from the Nernstian equilibrium value due to the passage of Faradaic current is a measure η of its polarization, written η = E− Δψ_{d}′, where E is the measured electrode-solution potential for any current i and Δψ_{d}′ the equilibrium e.m.f. relative to a reference electrode. We here also define the iv) self-polarizing electrode, which refers to an electrode which is its own source of the electrode-solution bulk e.m.f. Δψ_{d}, without external impression. The many electrodes used in equilibrium electrochemistry is of this kind, when the electrode is reversible to one of the ions in solution, or provides a source for electron exchange, such as AgCl + e^{−}(metal) → Ag + Cl^{−}. The above definitions apply to what follows where only equilibrium steady state systems are considered. We also conform to standard usage of the definition of the inner and outer Helmholtz plane [_{1} from the metal substrate plane (MP) and x_{2} is the nearest distance of approach for the solvated ions corresponding to the locus of centers for the OHP (where 0 < x_{1} < x_{2}) and where the electric potentials at these points are ϕ_{1} and ϕ_{2} respectively. We note that for the GC and GCS models at _{s} = 0, ∂ψ_{s}/∂x = 0 to fit the boundary conditions. We define also ΔΨ_{d} = ϕ_{2} − ψ_{s}, where ϕ_{2} → ψ_{m} as x_{2} → 0 in the absence of specific adsorption (when x_{1} = 0), and in this case ΔΨ_{d} → Δψ_{d}. In this work both limits of x_{2} will be studied. The above setup is depicted in

If there exists a function Q = Q(ΔΨ_{d}, _{d} = ΔΨ_{d}(Δψ_{d}) is general and unspecified, it follows that if Δψ_{d} is the thermodynamically determined potential difference given by the Gibbs criterion (such as (2b)) then the potential ΔΨ_{d} gives the exact charge density on the electrode as Q(ΔΨ_{d}) for fixed _{d}). Under suitable conditions, one may (as discussed below) extend through superimposition the results of the inert electrodes of the GC and GCS theories pertaining to charge and potential distribution to that for self-polarizing electrodes provided the same mechanical structure is common to both. In particular if ΔΨ_{d} = Δψ_{d} (when x_{2} →0), the above formulation remains unaltered.

However, there is no proof that the converse follows, i.e. that from one coupled equation, the two separate equations may be recovered as is sometimes supposed[_{d}) is the simplified and well-known GCS Theory which gives:
_{2} to ∞) where n is the electrolyte concentration, ε the permittivity, e the electronic charge, z the charge number, and k Boltzmann's constant [_{d} = ϕ_{2} ≠Δψ_{d}, and x_{2} finite and non-zero):
_{2} is closest distance approach to the surface for charges and are of magnitude for hydrated ions of ionic radii ~ 3 – 7 Å (for the ions of relevance here). Both these expressions are of course special case applications of the general expression (in the limit ∂ψ_{s}/∂x = 0) for the electric field intensity dϕ/dx, which when integrated over the appropriate limits yields the charge via Gauss’ theorem, and is inferred by:
_{i} with bulk concentration
^{−}) as given in (1b), which we shall utilize, the charge on the metal substrate σ_{m} is given by solving the transcendental equation:

In particular, the hydrated ionic radius for the chloride ion is 3.5 Å [_{2} to demonstrate in Sec. III that for ‘dynamical’ interfaces, i.e. which is not ideally polarizable in that there is ionic exchange with the metal substrate, the model provided by Stern for ideal polarizability is simply not applicable, and direct experimental measurement also bears this out. The above lemma, together with Lemma 4 below was first derived and applied [_{2} as defined above was very much less than 0.001 Å (~ 10^{–4} – 10^{–7} Å), which does not correspond to realistic ionic radii dimensions quoted in the studies. It is also obvious that the various GC and GCS models of electrodes were derived strictly for a system in thermodynamical equilibrium. It will be postulated that for the equilibrium case studies here, the Stern model is not applicable since the fixed x_{2} plane does not exist because any ion at the supposed closest distance approach (OHP at x_{2}) may be absorbed via an electron transfer reaction that does not allow the construction of an average-distance x_{2} plane; the Stern model superimposes dynamical interactions of ions beyond x_{2} but fixes their position at averaged distance x_{2}; for specific adsorption, x_{1} is also fixed but is not considered here (implying that the basic theories here may possibly be extended to incorporate these effects). In what follows the electrode used is not ideally polarizable (inert) but reversible to the specified ion (e.g. Cl^{−}) because there exists an electronic pathway which exchanges the electrons on the substrate electrode with the chloride ions and AgCl, so that the invariant inner and outer Helmholtz planes of the Stern model (with no effective electronic charge distribution between them because of dynamical ion interactions) may be approximated as not existing, or its effects are relatively small.

For metals whose free electrons may be approximated by a highly degenerate electron gas, (and only a restricted set of metals may be so approximated) from (2c) we infer the introduction of electrons will only alter ψ_{m} and μ_{F}(e) if the other lattice parameters are constant (so that μ_{l}(e) is a constant).

We assume this to be the case since the volume change is theoretically never accounted for and is small compared to the bulk volume (V) of the electrode and from the statistical mechanics of the degenerate gas [_{o} + δν) is given for the highly degenerate electron gas without pairwise interactions by [_{o} = (h^{2}/8m) (3N/πV)^{2 / 3} and δN is the mean number of electrons introduced from the neutral state. When ψ_{m} = 0, ν = ν_{o} and N is the electron number in the neutral state; m is the electronic mass and h the Planck constant. From the various expansion theorems [_{F} of the electron gas relative to the lattice potential energy μ_{l} is given up to fourth order by:

This expression, amongst others, will be used to compute μ_{F} which varies as charge is transferred to the electrode. The other definitions also used for computing μ_{F} will be given in the next section.

Clearly, not all metals conform to the above analysis, because of the Brilloun Zones formed by a periodic potential [_{l} = 0) accords extremely well with experiment, where the ratio γ_{observed}/γ_{theory} = 1.0 (this agreeable ratio is unmatched by any other element in the periodic table) [

Other mechanical parameters cannot be directly correlated (such as the bulk modulus) since as noted μ_{l}(e), the lattice contribution to the free energy becomes an important factor when compressing a solid, and with strong binding by lower-level electrons, the effects from μ_{l}(e) will predominate over μ_{e}(e). However, we are only interested in the thermodynamical properties when there is no distortion of the lattice. The thermal properties such as the observed and calculated Hall coefficients and Lorentz numbers are in close agreement[

For the free electron gas jellium model, if the correlation and exchange energy is given (per particle) as ε_{xc}, then the Fermi-Dirac density distribution [^{–1} (and which can be derived from the grand canonical ensemble) gives to first order the chemical potential as:
_{F}(e) given by (5); n(ε) is the mean occupation number with energy ε, μ the chemical potential, k the Boltzmann factor and T the Kelvin temperature.

_{xc} is a total system property, then for each quantum number n within the Fermi sphere[

Consequently, the degeneracy g(ε) becomes:
_{o} is the Fermi energy. By the substitution ε − ε_{xc} = ε′, the total number of particles N in the system becomes [

Similarly, the Fermi-Dirac distribution transforms as:
_{xc} = μ′, 0 < ε′ < ε_{o}.

Since [

These expressions are now in standard form [

The above expansion techniques lead to the standard series by elimination of μ′ as:

Hence:

On the other hand [_{xc}]/∂n for the correlation and exchange part of the total energy (n being the density of the gas). This form can be traced to the use of the classical Gibbs type equation for intensive quantities by differentiation of the extensive free energy. The Fermi function and (15), on the other hand, is derived from the quantized grand canonical ensemble, subjected to the usual constraints of fixed particle number and energy, which utilizes the fundamental criterion of invariant chemical potential between any two members of the ensemble. It will be found that the experimental values of the cell potentials mentioned lie midway between those derived theoretically from the Seitz expression and the free electron model potential μ_{F}(e). A hybrid form is used in computations [_{S} is written as:
_{o} obtained from the expansion techniques mentioned above (_{o}/5 for the free electron gas for the non-interacting terms, the chemical potential for this system should be, according to the Seitz theorem ∂E/∂N ~ (3/5).ε_{o}. The discrepancy sometimes found between results from numerical computations and experiments (e.g. the work function for the noble metals where a 30% discrepancy has been reported [

For our computations, in addition to the form given by (5), which is in excellent agreement with the thermodynamical properties for Ag mentioned above, we will use two other expressions for the chemical potential for purposes of comparison, the often utilized [_{S} (e) and the form derived from Lemma(3), μ_{T}(e) which we write as:

The Seitz potential is [_{xc} is used above, where (in a.u.) [_{s} (in a.u.) is defined such that:
_{o} is the Bohr radius (in S.I. units as with

It is possible to determine the approximate value of the cell standard potential relative to the solution by electrocapillary maximum (e.c.m.) measurements (illustrated by the Ag/AgCl electrode).

The equilibrium single electrode Nernst equation relative to the solution ions is:
_{2} = 0 or otherwise), and there is implied a value for ℑ_{AgCl|Cl} which must be estimated (henceforth ℑ^{o} = ℑ_{AgCl|Cl}). At the electrocapillary maximum (e.c.m.), according to the normal interpretation of the Lippman equation [_{AgCl|Cl} may be determined because Δφ_{d} and the bulk activities of the ionic species are known in (21). This method was first applied to determine activity coefficient changes in a system whose two electrodes where varied in size [

The term capacitance has undergone development in meaning ever since the solution of Laplace’s equation ∇^{2}φ = 0 for a source free region with electrical potential φ. For a system of charges Q_{i} (i = 1, 2,…. N) in a region of a fixed relative permittivity ε_{o}, Laplace’s equation [_{ii} are called coefficients of capacitance, where these coefficients are all constant for a fixed geometrical configuration of charges; these expression are based on the early experimental results with conductors which showed remarkable linearity with regard to charge and potential variation within the available regime of variable values. In electrochemistry, the capacitance C_{el} of the electrode of charge Q with defined potential φ (relative to the bulk solution) is defined C_{el} = ∂Q/∂φ. Within the GCS theory the permittivity factor ε_{o}’ (where ε_{o}’ = ε.ε_{o} in 3(a–d)) is constant and C_{el} is not, implying that for any fixed number of ionic charges, the geometrical arrangement of these charges vary, and so the solution charge density ρ [^{2}φ = –ρ/ε_{o}’, and φ(x) in each case varies according to the boundary conditions of the externally “impressed” e.m.f. at φ(0). In normal electrical (engineering) applications, variation in the capacitance is due to the variation in the dielectric coefficient for a fixed geometry, whereas here the mechanical structure (geometry) is the cause of the change; the electrochemical definition of the differential capacitance may be interpreted as a shorthand for describing systems that show different geometrical structures at different potentials and concentrations; in order to retrieve the charge on the electrodes, the appropriate integrations must be performed [

The important question is whether in a self-polarizing electrode (where it is assumed that it is mechanically similar to the GCS models, except that in the GCS models the potential is externally impressed and may be varied) it is meaningful to define or deduce properties such as its ‘differential capacitance’ at its equilibrium potential, since it must be conceded that these potentials are invariant at any particular concentration of electrolyte. In what follows, it is proved that there is a meaningful definition of differential capacitance that is compatible with experiment provided the self-polarizing electrode is mechanically equivalent to a specified ideally polarizable (GCS usually but not necessarily) type. By ‘mechanical equivalence’ is meant that the ionic interactions are the same, and that the metal substrate of the electrode is also the same in all respects for the two electrode systems. With these assumptions, we model the ions in solution as minute “conductors” of arbitrarily small spherical radius δr_{1} and δr_{2} (δr_{1}, δr_{2} → 0) for cation and anion respectively having a fixed quantum (±e) of charge, which implies the normal surface field strength:
_{o}_{i,n}_{i,n}_{j} (j = 1, 2,... N) where N is fixed, large but finite, equal to the number of ions in the system; clearly these S_{j} are the internal surface boundaries of the system. The boundary conditions for the potentials are specified as φ_{l} = 0 for x = l, φ_{o} at x = 0 and as normal gradients ∂φ_{j}/∂n at S_{j} (either φ or ∂φ_{j}/∂n may be specified for each boundary at i).

Then, subject to these boundary conditions, the Uniqueness Theorem [_{o}_{l}, the solution is unique to an additive constant relative to the postulated thermal averaged stationary distribution of charge at the specified temperature. Hence we reach the (remarkable) conclusion that for these electrodes, under appropriate conditions (such as fixed N, similarity of mechanical and electrical properties of the ions, and constant _{o}_{o}_{l}_{o}′ at any fixed temperature.

We infer that:
_{x} at distance x from the electrode surface under the above assumptions; hence the differential capacitance (when x = 0) of the two systems are the same if _{o} and _{l} are specified. Thus for self-polarizing electrodes, we may by inference from solutions of Laplace’s equation still define the differential capacitance for the self-polarizing electrode as for (inert) ideally polarizable ones under the assumption of physical equivalence of these two electrodes and all physical properties may thus be derived from the appropriate mathematical operation. However, there is a difference; under the assumption of ideal polarizability, one can experimentally measure the differential capacitance directly for instance, whereas for the self-polarizing electrode, one must specify the mechanical properties by specifying the Q charge density function that the electrode is assumed to possess on physical grounds.

With the above understanding, we can state a simplified capacitance theorem for self-polarizing electrodes that for electrodes reversible to the ion(s) in the solution, the differential capacitance is independent of the equilibrium potential but is dependent on the physical characteristic variables

_{o}–φ_{l}. Then dQ/dΔφ = Q′ (Δφ, _{d}, the differential capacitance. But Δφ = Δφ (

For instance, let the model of the electrode be according to (3a); then differentiating (3a) with respect to Δφ yields the differential capacitance C_{d} written:

Consider the equilibrium electrode equation:
^{o} − (RT/F)ln a[Cl^{−}].

Let ℑ^{o} = (RT/F)ln α′ where α′ = exp(ℑ^{o} F/RT).

Since Δφ = – (RT/F)ln (a[Cl^{−}]/α′), we get
_{d} is measured from experiment, then α′ may be determined by curve-fitting, so that the important parameter ℑ^{o} may be determined directly, if α′ does not vary appreciably with C*. Some experimental support for the above follows below. We note more elaborate expressions may be derived for extensions of the Gouy-Chapman model (e.g. that attributed to Stern) where Q the charge on the electrode is functionally written Q = f (Δφ,

The concentration of an ion in dielectric 1 may be determined in terms of its concentration in another dielectric 2 which is in thermodynamic equilibrium to phase 1 by the Gibbs equilibrium criterion.

The two different (approximate) methods developed here are based on the assumption (Method A) that the concentration of the ion at the boundary of the two dielectric phases is that of the bulk for the phase with the higher bulk concentration, and the other method (Method B) is derived using the Born equation; m represent the bulk concentrations in molality units, ρ the densities, μ^{o} the standard potentials, γ the activities, and c the concentration in moles per unit volume (liter if numbers are quoted), and subscripts refer to the phase concerned. The assumption in Method A is inspired by the fact that the concentration of the active species is much lower in the electrode dielectric than it is in the aqueous layer, and that rapid exchange of the active ion by diffusive processes in chemical equilibrium would make the concentration near the surface close to the one for the bulk aqueous layer. The equations below apply at ‘low concentrations’ (~ 0.05 m), of the same magnitude used in the experiments reported here.

The boundary conditions chosen are:
_{1}ρ_{1} = m_{2}ρ_{2} and therefore:
_{1}/ρ_{2}), and β is constant for variables ±x but differs for different bulk concentrations of the ion in phase 1. _{2} = ρ_{1}m_{1}/ρ_{2} ensues, implying (the absurd) conclusion that the c concentration of ions is invariant across the interface. The only approximation made here is the fixed value of the c ionic concentration at the interface; more sophisticated models might modify this concentration, as well as the value of γ_{2} at the interfacial boundary. If the permittivity of the phases follow ε_{1} > ε_{2} and c_{1} > c_{2}, then the other choice of the boundary conditions is unphysical, since it would require

This method is highly dependent on (a) highly accurate values of the effective radii of the ion in the dielectric and vacuum phase (to 3 decimal places), both of which need not be the same, and even until now little is known of these ionic radii under different ionic environments and concentrations and (b) the assumption that the activity coefficients at the standard states in both dielectrics 1 and 2 are approximately equal, or are small enough to be neglected, and that they are known for other ionic concentrations in the solid state dielectrics; in the relatively recent field of solid state ionics, the activity coefficients are customarily neglected by being set to unity [

The work done to convert a charged ion in vacuum (I_{0}) to a state of known molality concentration I_{4} may be done in a series of stages depicted as
_{1} is given by the Born expression:
_{i,vac}_{i,d}_{2} is q. From this state, the work done to compress the ions to any other arbitrary state I_{4} is:

Thus the concentration of the ions in equilibrium between dielectric phases 1 and 2 is governed by the equation:

Assuming no change in the compression energy q, the following equation obtains:

It does not immediately follow that if (3a) is true, that the converse always obtains, where the coupled equation may lead directly to ΔΨ_{d} i.e. Q(ΔΨ_{d}) ⇒ ΔΨ_{d}, as is sometimes supposed [_{2})=2Asinh(|z|f ϕ_{2}/2), then if | ϕ_{2} | << 1 and f is constant, the standard approximation [_{2} never approaches infinity) for low ϕ_{2} fulfilling the condition (zeϕ_{2}/4kT≤ 0.5) is ϕ = ϕ_{2} exp(−κx) with κ = (2nz^{2}e^{2}/ε.ε_{o}.kT)^{1/2}. If |ϕ_{2}| were allowed to be very large, so that Q(ϕ_{2}) ~ 2A′(n)/2 exp(|z|f ϕ_{2}/2), then this value of |ϕ_{2}| may not be small enough in some instances to correspond to the experimental setup where ϕ_{2} ~ (2/zf)lnQ + const - (1/zf)(ln n). The series expansion Q(ϕ_{2}) = 2A{|z|f ϕ_{2} /2 +....} does not immediately yield ϕ_{2} = constant ± (1/|z|f) lnC^{s} for non-constant [^{s} is varied. In the cases where the e.m.f.s do not correspond and/or Q is not definitely constant, it is suggested here that the reason for the Nernst-like relation found experimentally is that the electrode is also serving as a type of self-polarizing reversible equilibrium electrode with respect to the relevant ions at bulk concentration C^{s} whilst conforming to (3) as well; and it is suggested here that these types of self-polarizing electrode systems were essentially what Frumkin et al. were studying [_{2} = 0, ΔΨ_{d} = Δψ_{d}. For such cases, the Esin and Markov effects [_{C}^{s} = C^{s}γ_{±}, with bulk concentration C^{s} and (mean) activity coefficient γ_{±}, leading to the identity ln a_{C}^{s} = C^{s} + lnγ_{±}, then at the point of zero charge, Q = 0 (for the electrode) ⇒ Δψ_{d} = 0 ⇒ c′ + E^{pzc} - (1/|z|f)ln C^{s} = 0 from the Nernst equation (neglecting activity coefficients at low concentrations) given in Lemma 4 which is logarithmic if c′ is interpreted as ‘constant’ for the electrode at that particular concentration of electrolyte with the stated physical conditions and which arises for various reasons, including a residual potential due to charges on the interface and that due to the reference electrode, and of course, the presence of a mean activity coefficient which may be absorbed by the Nernst equation by the c′ term at higher concentrations. And in general if ΔΨ_{d} = ΔΨ_{d} (Δψ_{d}), a first-order Taylor expansion at the point of zero charge (ΔΨ_{d} = 0) implies a + bΔΨ_{d} = 0 (a,b are constants) or c” + 1/b(E’^{pzc} - (1/|z|f)lnC^{S}) = 0 and logarithmic behavior would still be observed under the assumption that these systems are essentially functioning as reversible (self-polarizing) electrodes, as discussed in Section II, Lemma (1); E^{pzc},E’^{pzc} and c” are all constants.

It appears that both the usual sinh expansion and/or Lemma 1 may be utilized to partially explain the many experimental phenomena connected with potentials of zero charge and the electrode interface. However, there is a way to distinguish the two rationalizations by carrying out measurements in a regime for which the sinh expansion is not valid.

If logarithmic behavior is still observed in these non-valid regimes, then it appears probable that the standard approximate relationship mentioned above may be open to question. The available data [_{2} (potential of the plane of closest approach of the solution electrolyte to the mercury electrode surface relative to the bulk sodium fluoride concentration c^{s}) against the logarithm of the bulk concentration of NaF for various potentials of the mercury electrode relative to the Normal calomel reference electrode (N.C.E.).

In the conventional analysis, the ϕ_{2} value as predicted by the conventional theories obtain when [_{2} | ≥ 0.1 V, but it is evident that linearity of the plots in _{2} ≈ Δψ_{d} when x_{2} = 0.

The value of the tenth-normal Ag|AgCl electrode potential is approximately 0.5125 V at this maximum [^{o} ~ 0.512 V for aq. NaCl. Relative to the calomel electrode, the value is 0.557 V for 0.1 to 1.0 N solutions [_{2}Cl_{2}) electrodes (of Standard Potentials 0.2223 V and 0.2680 V respectively with difference 0.0457 V). Hence, for aq. KCl, we may infer with confidence the approximate ℑ^{o} value from the e.c.m. value relative to the calomel electrode by subtraction of ~ 0.045 V (the difference of these standard potentials) since no comparable data are readily available for the e.c.m. potential for KCl relative to the Ag|AgCl electrode. From the data [^{o} = 0.4489 V for the lower limit for ℑ^{o} since at higher concentrations a charge transfer from the external cell modifies ℑ^{o}. Higher values of ℑ^{o} would exaggerate the effects of the computed e.m.f. for cell (1a), calculated according to the prescription given in Section V below. The more accurate value for ℑ^{o} must be derived taking into account the spread of values over all concentration ranges of the electrolyte, from the constant value regime of ~ 0.557 V for 0.1 to 1.0 N solutions to the logarithmic regime at very low concentrations as determined for instance by the classic results of Grahame and others (^{str} for ℑ^{o}, where ℑ^{str} denotes the ‘stressed’ value, so that the measured value of the e.m.f. at the e.c.m. of Hg, E_{m} may be written:
^{−}] is the activity of Cl^{−} ions, and we also define:
_{m} → Δϕ, ℑ^{str} → ℑ^{o}. The general Taylor expansion for ℑ^{str} with these conditions are:
_{i} = 0 for i ≠ 1, and a_{1} = 1.

At the constant E_{m} regime, we note that a compensating (−RT/F)ln(a) factor to the strained value ℑ^{str} leads to a constant, so that a solution to this regime is:
_{m} = ℑ^{o}. On the other hand, (as explained below) at the logarithmic phase, E_{m} → Δφ, ℑ^{str} → ℑ^{o}, so that:

If the change of regime is reasonably abrupt, then at a[Cl^{−}] = a_{crit}, the critical activity, both (40e) and (40f) are jointly satisfied, and thus:
_{crit}) and:

From the data [^{o} = 0.4489 V, which is incidentally somewhat close to the ‘rational’ potential of Grahame [^{o}, the computed activity coefficient results given in the Tables of the _{.1N} = 0.394031 V which will be the value chosen to solve the coupled electrode differential equations.

Some writers have viewed the Hg electrode as a reversible cell [_{a} and Hg_{b}, the activities corresponding to the electrodes of the half-cells might cancel even if they are at different potentials and this construct demands amongst other things that such species as (KCl)_{a} [

(i) At sufficiently high concentrations of bulk solution electrolyte, the concentration of ions adsorbed within distance δl (where δl extends to a specified distance where negligible amounts of ions are present and which is expected from the Faraday demonstration of charges residing at the surface of the conductor) is approximately linear to first order in the bulk solution strength where the usual Gibbsian equilibrium conditions prevail written:
_{m} values up to and including the e.c.m. the following electrode reaction:

Applying the Gibbs criterion yields the electrode solution potential Δφ_{Hg-S} as:

Hence the total cell potential difference for the Hg and Ag|AgCl electrodes becomes:

However, at the e.c.m, the elementary Lippman theory in conjunction with the Gouy-Chapman description demands Δφ_{Hg-S} = 0, so that ℑ^{str}_{Hg} (E_{m}) = RT/Fln a[Cl^{−}] and thus (44) becomes:
^{+}, Cl^{−}, and e^{−} charges exists within the electrode to the extent that the net charge is zero.

(ii) At very low concentrations, the reversible anion or cation mechanism breaks down and the electrode may be viewed as “inert” but as having the same potential as the solution within the limits of the Gouy-Chapman theory of electrodes since residual charges and the capacitance becomes arbitrarily small as the concentration drops. Hence:

(1) Another gross verification of the present capacitance theorem may be observed from the experimental data provided by Grahame [^{−2}. We note in ^{−2}. Moreover, they all coincide at the point of zero charge. For a single-species ionic equilibria:
^{z+} approximately the same value for the C_{d} capacitance from _{z}) were polarized negative, it would encourage the cationic (45c) exchange reaction since species M would adsorb to the Hg interface according to such mechanisms as Hg + xNa^{+}(aq) → Hg.x(Na^{+})|_{electrode} and that described by (42). The determination of standard potentials for highly reactive metallic species in aqueous solution has been accomplished by Hg amalgam electrodes, which the above system might approximate at relatively high negative potentials. These conclusions accord with the data, such as depicted in ^{−}, NO_{3}^{−}, SO_{4}^{2−} and CO_{3}^{2−} with the approximate capacitance of 17 μFcm^{−2} with the interfacial voltage (relative to the N.C.E) of ~ 0.4 to −1.2 V. This capacitance does not obtain for the positively charged surface (~ 0 to + 0.6 V) and it is conjectured that the reason for this is because (45c) ceases to be the dominant electrode reaction at such positive potentials, because of the expulsion of the cations from the Hg interface.

If the mercury is envisaged as an electrode reversible to ions as outlined in (1) above and explained below, then, we might expect such a close fitting linearity (assuming similar values of activities) since C_{d} ~ constant for a fixed concentration of aqueous electrolyte, leading to the same gradients for the different curves. The parallel nature of the lines are due to changes in the physical conditions due to the different counter ions. For NaI, the curve is crudely of the form sinh(κφ) (constant κ, with potential φ), implying a net ideally polarizable inert electrode with non-reversibility of ions (no charge transfer).

(2) Concerning the constancy of the e.c.m. voltage, Grahame remarks [_{m} = 0 (i.e. at a value not corresponding to the e.c.m.) for the case of net Cl^{−} reversibility that:
^{o}_{Hg}(M) is a standard potential at molality M. Now, if the impressed voltage E_{m} conserves charge within its own circuitry, then the charge Q_{ref} extracted from the reference electrode (e.g.Ag|AgCl) equals that which is injected into the Hg electrode Q_{Hg}, i.e. Q_{ref} = Q_{Hg}. The change in potential Δφ′_{ref} for the reference electrode is Cd.Δφ′_{ref} = Q_{ref} since electrons are withdrawn. If we postulate that the impressed voltage on the reference electrode makes it more positive with respect to the solution, then from the reaction equilibria:
^{−} from the solution. Likewise, we observe that there is a net negative charge on the Hg electrode (relative to the reference electrode) and hence from (43) and (44), there will be an expulsion of Cl^{−} ions to the solution, leading to a neutral electrode (Δφ_{Hg-S} = 0) at the e.c.m..Hence, relative to the solution, the effect is the raising of the potential of the electrode. Another way of viewing this is to suppose the net absorption of Na^{+} ions (so that the solution remains uncharged) by the influence of the negative potential, which is written as:

In either case, the increase of potential Δφ′_{Hg-S} relative to the solution due to the ionic interchange is such that Cd. Δφ′_{Hg-S} = Q_{ref}, so that Δφ′_{Hg-S} = Δφ′_{ref}.

Hence at all moderate concentrations:

The similarity of value of potential in b′) for different electrolytes may be explained by postulating that the k values in (41 – 42) are approximately the same for each of the different types of anions (k_{a}) and cations (k_{c}), where k_{a} is not necessarily equal to k_{c} for strong electrolytes, and they might be treated as point charges independent of size within the surface of the mercury electrode. The above is clearly only an outline and other alternative net mechanisms may be adduced.

The setup of the system used for a low resolution experiment to determine e.m.f. contributions due to size effects is given in

Silver foil (Aldrich Chemicals, 99.98% purity) of square dimension 2 × 2 cm and thickness 0.025 mm and 1 mm were used for electrodes s2 and s1 respectively found in (1a). It was found that massive or heavy electrode holders or wiring [meaning ~2 gm (grams) excess weight for brass connectors] could alter the measured e.m.f. by ~ 0.5 mV if one of the electrodes were connected to such an arrangement. To avoid possible capacitive effects due to unequal metal mass contact, and also sheer mass of the connectors, the electrodes were directly spot soldered at one of the corners, (

The Ag foil electrodes were all electrolyzed under identical conditions, and the temperature was maintained at 25 ± 0.02 °C. This electrolysis to create the AgCl layer was carried out at 7.8830 mA for 1,200 seconds in HCl solution (1.5 M) using two platinum coated titanium gauzes as counterelectrodes for all electrodes s2 (under identical conditions) and the current corresponds to a total thickness of 2.5 μm (microns) of Ag which was converted to AgCl. The thickness mentioned refer to the Ag substrate and not to the AgCl deposit. The density of Ag was taken to be 10.501 gm cm^{–3} at 25 °C. The slight reduction in the average thickness of the Ag substrate was accounted for in the computations. The electrodes were dry weighed before and after electrolysis by a Mettler MT5 microbalance to determine the efficiency of conversion at ~ 92 ± 5%. The electrolysis utilized a commercially available potentiostat [EG & G Princeton Applied Research 363 Potentiostat (U.S.A)], and all e.m.f. measurements were made using a digital voltmeter [Keithley 197A Autoranging μV DMM (U.S.A.)]. Over 60 readings of cell e.m.f. E_{c} = ϕ_{10} – ϕ_{20} (where ϕ_{j0} represents the potential of the Ag substrate j) were taken each lasting at least 1/2 hour; further details of the measurements follow below. One result (without exception) was that E_{c} was negative to the order exceeding 1 mV for all the 60 measurements used in the e.m.f. determination under controlled conditions. The KCl solution was prepared from Analar grade KCl and doubly distilled water by weighing (since molality units were used). For the ordered set of solution concentrations m (molality units) {0.01, 0.03, 0.05, 0.07, 0.1}, the corresponding ordered set of averaged - E_{c} values (units of mV [millivolt], with a maximum experimental fluctuation per reading of ± 0.2 mV) are {1.46, 1.51, 1.57, 1.56, 1.77}. Altogether, 24 separate s2 electrodes were fabricated and electrolyzed, and studied and used for the qualitative and quantitative studies.

Many of these electrodes were cleaned and re-electrolyzed to confirm the observations concerning the AgCl layer and the e.m.f. measurements. The measurements for different KCl concentrations were done slowly over a 10 month period (December 1997-September 1998), where two separate s1 electrodes were fabricated and used. The estimated cell e.m.f. observational errors is based on this relatively long time period of general experimental. It is deduced here that the inaccuracies are mainly attributable to the method of fabricating the electrodes and the inhomogeneity of the electrode materials, as discussed in the text. The cell e.m.f. results presented for any one KCl concentration is the average of measurements carried out at least twice at different times from different stock solutions. The input resistance of the Keithley instrument was > 1 GΩ for the range of e.m.f. measured, and so internal i-r drops and other associated effects causing extraneous e.m.f.s may be safely discounted. Further, the electrode e.m.f. was reset to zero volts by contact of the upper unelectrolyzed part of the electrode before immersing s2 into the solution, which would eliminate polarization e.m.f.s due to the circuitry other than the electrode system. That i-r drops are negligible for these measurements may also be inferred form the fact that no discernable change in e.m.f. (of 1 μV order) could be detected when distance between electrodes s1 and s2 were varied (to ~ 4 cm). Also, the observed e.m.f. polarity was the same for all of the measurements of the s1 – s2 couple, implying an intrinsic effect due to electrode size effects only. To confirm that the electrode e.m.f. differences were due to the Ag substrate size differences and not due to electrochemical differences caused by different coating thicknesses of the AgCl dielectric, five of the s2 electrodes were electrolyzed at half the current intensity for the same time (corresponding to total thickness of 1.25 μm of Ag converted to AgCl). It was experimentally found that the effects due to coating thickness was effectively negligible where the measured e.m.f. of cell (1a) is concerned, which accords with the common and fundamental electrochemical assumption (such as used in (2a)) where the solid dielectric or salt activity (in this case AgCl) is assigned value unity, with constant chemical potential for all electrodes regardless of amount. The surface area of s1 exposed to the solution was maintained constant, corresponding to a penetration depth (into the KCl solution measured from the tip of the electrode to the Ag-AgCl interface) of 1.920 cm measured by Vernier caliper, with an exposed solution area of approximately 6.90 cm^{2}, but s2 varied by ~ 6 ± 0.5 cm^{2} because the electrolysis involved vigorous stirring, so that the Ag-AgCl line boundary could not be exactly fixed; so all readings were normalized to 8 cm^{2} surface area exposure for s2 by linear extrapolation with s1 fixed with the stated extent of surface area. That this is a reasonable extrapolation may be inferred from _{S} = ε_{o} + d[nε_{xc}(n)]/dn (_{T}(e) = μ_{F}(e)+ ε_{xc}. (_{F}(e) (

This calculation was verified qualitatively by experiment, but because of the porosity of the AgCl dielectric, coupled with the effects of a solution film still present in areas not immersed in the solution, direct verification is difficult, but an e.m.f. change of ~ 0.5mV could be detected as a result of change in the degree of penetration of the s2 electrode into the KCl solution. The surface area are in units of square centimeter of electrode exposed to the reactive solution, and the potential that develops as a result of the charge transfer from the AgCl dielectric in contact with the solution.

Interestingly enough, a direct confirmation of this result is to be found in the award-winning work of K.L. Cheng [^{+} | Glass; where X represents the ion to which the reference electrode is reversible; the reference electrode is here defined to be the ‘inner electrode’ for this system. It must be pointed out that whilst the experimental results of Cheng concerning linear change of e.m.f. with exposed surface area of the electrode agree in terms of predicted effect for the specialized case of our Ag/AgCl electrodes, there are some theoretical differences. Cheng in particular seems to believe that the “Nernst equation has been misused” in nonfaradaic potentiometry, where “nonfaradaic processes deal with neither redox reactions nor current production”. Where only the glass membrane itself is concerned, Cheng’s analysis seems plausible by treating it as a linear capacitor (i.e. linear relation of charge and voltage), where the extent of adsorption of ions on the glass dielectric surface determines the net charge and voltage. Clearly, the ‘Nernst-like’ behavior, (which is theoretically derived from the Gibbs equilibrium criterion) of the electrodes in all probability originates form the coupling between the induced charge on the membrane, and the ion X which is reversible to the inner electrode. Cheng seems not to focus his attention nor his experiments on the inner reference electrode mechanism and the ions reversible to it (which he seems to treat as a black box potential sensor with no interfering role), but instead concentrates on the membrane potentials which are assumed to derive from the “activities” of the inner and outer layer of the membrane surfaces {his

For the bulky s1 electrode, calculations show that the cell e.m.f. varies by only ~ 0.02 mV for a 4 cm^{2} surface area change for a totally immersed s2 electrode, as expected. Hence it is sufficient to normalize just electrode s2. The normalization scale factor magnifies the raw experimental results by only an approximately further 0.3 mV. To conserve costs and to remove systematic errors, only one reference electrode s1 (0.1 cm thickness) was used for all the measurements involving s2. However, to check for consistency, another s1 electrode was made and immersed to the same extent into the solution, and after ½ hr (hour) equilibration, a maximum e.m.f. difference of 0.127 mV was observed between the two electrodes at 0.1 m KCl, in contrast to the value ~ 1.2 mV detected without fail when s2 is substituted. All the electrodes were stored in a solution comprising 0.05 M KCl and 0.05 M HCl. All e.m.f. cell measurements were conducted at the solution temperature of 25 ± 0.02 °C using a precision temperature controlled cooling unit with an inbuilt water circulation pump (Polyscience^{®} Temperature Control Unit (U.S.A.)), where the cell was placed in a lagged glass cooling jacket. The depth of penetration of the electrodes was measured by means of a vernier caliper (±0.01 mm) by observing the change of vertical distance of the upper flat surface of the Aluminum holder to a point marked on the glass rod of the electrode when the pointed electrode corner tip just touches the solution to the position when the electrode is at rest with the solution level coinciding with the boundary line between the AgCl dielectric and that of the pure Ag surface (_{2} degassing, since N_{2} degassing will increase the pH of the solution (normally at ~ 4.42 for the range 0.01 to 0.1 M KCl) to well over 7 because of the expulsion of CO_{2} gas which forms an acidic solution in water; from the Pourbaix diagram [_{2}O formation (line (6) of reference (

The cell was constructed from a 250 mL glass beaker (of diameter 7.0 cm) cut to approximate height 7.8 cm, where the KCl test solution reached a height of ~ 3.5 cm. The length of the hollow soft glass (0.56 cm diameter) used in the electrodes was 14.5 cm. The electrodes s1 and s2 were placed at a fixed distance of 3.5 cm, and where aligned so that their major surfaces were parallel because it was observed through trial and error that for at least this configuration, there was no detectable e.m.f. change as the distance of the electrodes varied (due to the solution ir drop caused by the external measuring circuitry). Two large aluminum holders of approximate height 3.4 cm and diameter 3.8 cm with a hole in the center to accommodate the electrodes, which were held by the pressure of Allen screws tapped at the sides of the blocks, both of which were planed at one vertical side (see

The experimental outcome for the above series of experiments, derived over a 10-month period after repeated trials with many samples is reported in the figures of Section 5 together with the theoretical predictions. Some comments concerning the accuracy of these measurements are in order, together with the reasons why these measurements were made over a range of KCl solution concentrations. Firstly, this experiment was aimed at detecting second-order effects which are assumed to be non-existent in normal electrochemical practice, where the measured cell e.m.f.s are typically 50 – 100 times greater in magnitude to that presented here, with error margins of the same magnitude as the averaged results given here, because in normal practice, size is not a controlled variable, and the variation in size for the different electrodes is approximately of the same magnitude as that between electrode s1 and s2, leading to uncertainty in the region of some fraction of mV, to approximately 1.0 mV; here we have controlled the geometrical structure of the electrodes, and the polarity of the e.m.f. was the same and the magnitude of the e.m.f. was approximately the same for all 60 readings that were made, implying a size effect. If no size effect were present, then the polarity of the measured e.m.f. would vary randomly, and this was not observed at all. The measurements over different KCl solution concentrations were done to verify the curious ‘non-Nernstian’ effect predicted by the theory, where the minute change of the e.m.f. for the cell due to change in the KCl solution concentration (from 0.01 m to 0.10 m) was in the order of 0.13 – 0.06 mV (depending on the type of chemical potential used). A typically Nernstian result would be in the order of 2.303 RT/F Volts (~59.2 mV at 25 °C), larger by a factor of ~700. This theoretical prediction is borne out in the experiments, where there was no dramatic change in the e.m.f. over the stated range of KCl concentrations. It was never the intention of the experiment to attempt to map out the actual change of the e.m.f. with concentration; the errors in observation would preclude that, although there is a hint, based on average values, of a very slight general increase which may not be considered statistically significant at all. The main intention was to detect a size dependent, fairly stable and constant e.m.f. presence in a variety of conditions, such as varying electrolyte concentration, so as to rule out other factors which might cause these changes. Another advantage of measuring over a concentration range is to give an indication of the most appropriate electronic chemical potential of the substrate electrodes, since the predicted e.m.f. differences based on the different chemical potential models are greater than the errors of measurement; i.e. the band (or spread) of experimental e.m.f. values over the concentration range can give an indication of the most appropriate theoretical model of the chemical potential. The results obtained here seem to especially support the free-electron gas chemical potential, which is corroborated by the low-temperature experimental specific heat capacity results for Ag.

Lastly, can the accuracy of the experiments be further improved by suitable precautions? It may be improved, but the precautions taken would probably involve methods and facilities that go beyond the techniques routinely employed in college Electrochemical and Analytical laboratories and the experimental capabilities and facilities of our laboratories. For instance, it is known [^{−2} (milliAmperes per square centimeter) in 0.1N halide solutions the deposits observed under an electron microscope are non-porous, whereas at other current densities, other different growth mechanisms are observed, with the deposited halide films having different physical properties (such as the specific conductance) and hence the average chemical potential of these substances would be expected to vary for each of the different halide structures. Under normal aqueous electrolysis, the field gradients would vary with factors such as the position of the counter-electrodes and the turbulent motion of the vigorously stirred solution; with a magnetic stirrer, one can achieve an “averaged” halide film structure, but there will be fluctuations about the mean for the different electrodes that will cause slightly different e.m.f. readings. (It is also very difficult to control the height of the AgCl/Ag interface because of the turbulent electrolysis, and clearly this is one source of the varying e.m.f. because of the different surface areas exposed.) In principle, one way of overcoming the problem of irregular film physical properties is to utilize some form of high temperature vapor deposition of the AgCl layer on the same form of Ag surfaces. The commercially procured Ag foils were presumably milled, implying again an average surface structure which could contribute to the e.m.f. fluctuations; in principle this problem could be overcome by using silver sheets with fixed crystal orientation, which are not readily available, if at all. There is also the problem of spot soldering or welding. The amount used would vary unless equally weighted amounts were used for all the electrodes. Then again, depending on the Ag crystal orientation of the metallic grains, one could expect small differences in the Volta contact potentials, leading to very small possible e.m.f. fluctuations. Although Analar reagents were used, it has been demonstrated that in normal ‘first order’ electrochemistry, trace amounts [_{2} and N_{2}) and to the formation of oxides on the Ag surface during measurement in a non-acidic medium, and to control these factors would require specialized methods not normally employed in most electrochemical laboratories. Nevertheless, despite the above unavoidable circumstances, it was possible to detect an e.m.f. of ~ 1.2 mV (unscaled) with the same polarity without exception whatsoever for all the KCl cell concentrations and this value is well within the range predicted form theory. Attempts at more accurate values (which is not the immediate and main objective of this report) would involve long-term collaboration with specialist laboratories which have the means of imposing the necessary controls on the various variables which affect the e.m.f., including the means of producing the electrode substrate and halide films of unambiguous molecular composition and physical (crystal) orientation. Alternatively, other electrode systems with a much larger predicted cell e.m.f. could be explored, such as the pure metal in equilibrium with its cation, but here the major problem is the definite interference of the oxide layer at the metallic surface which may considerably reduce the second order cell e.m.f. effect.

For cell (1a), the e.m.f. E is given by (2a-b) as:
^{ø} is the standard potential of the cell concerned, and which is deemed to cancel in practical applications, leading to zero net potential; for what follows, γ_{–} refers to the single ionic activity coefficient which is calculated from the Pitzer equation[_{mx} and C_{mx} are rather complicated functions[^{ø} = (−μ(Ag) – μ^{o}(Cl^{−}) + μ(e) + μ(AgCl))/F and μ_{l} (e) is assumed invariant in μ(e) = μ_{F}(e) + μ_{l} (e), (7) becomes by virtue of subtraction at fixed (25 °C) temperature:
_{F} refer to the electronic chemical potential of the cell concerned. If A is the total surface area of the electrode then:
_{d} = ℑ_{AgCl|Cl} − (RT/F)ln m.γ_{–} (where ℑ_{AgCl|Cl} = ℑ^{o}), then for any fixed m, ΔΨ_{d} = ΔΨ_{d} (γ_{–}). From (3a,b,d), we may write:
_{gen} is a generic function that is utilized to determine the charge Q on the electrode; the forms utilized here are given in 3(a,b,d), so that for fixed m, Q = Q(γ_{–}). Further, we note that at 0.1 N, we wrote:
^{o} = (−μ(Ag) – μ^{o}(Cl^{−}) + μ(e) + μ(AgCl))/F where ℑ|_{neutral} refers to the neutral state when Q = 0, we write:
^{o} = δμ(e)/F is due to the change of ℑ^{o} due to charge transfer. At 0.1 N ℑ^{o} is known precisely (the value 0.394031V is used here for the theoretical calculations), so that Q|_{m} = 0.1 may be calculated, and δℑ^{o} may be determined from (4) and (5). Hence:
^{o} = δℑ^{o} (Q) and the functional form δℑ^{o} may be determined, then for any other value of m (other than 0.1 N where N refers to Normality units of concentration), δℑ^{o}(mγ) may be computed from (50c). Thus, Q = Q(γ_{–}) even if ℑ^{o} is allowed to vary and clearly, μ_{F,2} = μ_{F,2}(γ_{–}).

The above sub-iterations are used to solve the following equation derived from

It will be noted that for each secant iteration in (37), the charge Q must be known, and this information is obtained by solving one of

The entire cell may be characterized as having the following potentials for the entire surface perpendicular to the line direction x (from the surface of the metal substrate of the electrode concerned) as depicted in

In _{ij} represents the interfacial distances of phase boundary j of electrode i, and the ϕ′s are the corresponding potentials at these interfaces; ϕ_{s}, ϕ_{d1} and ϕ_{d2} are the bulk solution potentials of the respective regions. Each of the above electrodes i consists of the Ag substrate coated with the AgCl dielectric, which shares a boundary at x_{i3} with the aqueous KCl solution; x_{i1,2} are the inner and outer Helmholtz planes within the dielectric matrix. The model (abbreviated M1) that is favored here and described immediately below focuses on the net electrode reaction:
^{−} ions which are freed from the Ag-AgCl interface (x_{j0}) to distribute themselves according to the usual Debye-Huckel and Gouy-Chapman assumption of the Boltzmann energy distribution, where Φ, the potential within the dielectric obeys the limit Φ → 0 and dΦ/dx→ 0 as x → ∞. Furthermore, there is overall electroneutrality in (52) because the charge on the Ag metal interface equals the charge of the freed Cl^{−} ions; the actual mechanism of ionic migration of Cl^{−} ions is of no concern here, although it is of central importance in kinetic studies when various defect mechanisms (eg. Frenkel defects) [

It was discovered that for concentrations from 0.01 – 0.1 M KCl, and ℑ^{o} ~ 0.39 – 0.5 V, the symmetrical neutral electrolyte M^{+}Cl^{−} electrode charge ^{m}, i.e.:
_{d} ranging from 3.5 to 10^{–7} Å, with a maximum % deviation of ~ 0.8% at high σ^{m} (~ 20,000 μCcm^{–2}). Hence these two electrode types are practically indistinguishable for the same single ion concentration _{o}_{i} = ϕ_{s} = ϕ_{i2}. This accords with normal expectation since the average potential at a long distances from a system of dipoles is zero and also with the analysis of Grimley [_{s}(l) to the dielectric given by Λ + V_{c}(l) = V_{s}(l) (his ^{–7} μC (his ^{–2}).

In order to utilize M1, by the Uniqueness Theorem a boundary condition must be specified. Here we state that within the AgCl dielectric, when x → ∞, the concentration of the Cl^{−} ions must correspond to that at the AgCl-Aqueous interface, implying analytical continuity of the chloride ion concentration; hence we set [Cl^{−}] ≡ [Cl^{−}]_{aq} with the dielectric constant value of that for AgCl at this interfacial region, and we also set the potential gradient ∂φ/∂_{i}_{i}_{d} = 0 (corresponding to x_{i2} = x_{i1} = 0) is given in _{AgCl} and for the three chemical potentials mentioned here μ_{F} [_{T} [_{S} [

It will be observed that the experimental results depicted in _{T} and μ_{F} (but closer to μ_{F}). This result would accord with the low-temperature specific heat ratio γ (^{+} and Cl^{−} ions.

It may be reasonably argued that the AgCl dielectric does not play a prominent role (despite its occluding the Ag substrate surface) in determining the cell e.m.f., apart from providing a reservoir for Ag and Cl^{−} ions. This situation corresponds to solving for the cell e.m.f. with the H_{2}O dielectric constant (ε = 78.54) at the vicinity of Cl^{−} exchange at the electrode, the results of which are presented in

The types of electrode reactions that are applicable to the results presented in

It is clear that the computed values are far in excess of the experimental results and so it is safe to infer that AgCl does play a prominent role in distributing the Cl^{−} ions within the dielectric to produce the observed e.m.f. apart from its role as a dielectric ionic reservoir.

It is well known that the Stern modification (GCS systems) with x_{d} ≠ 0 is widely utilized as a refinement to the Gouy-Chapman theory [_{i2} of atomic radii dimension in

The results of using the Stern model for the electrode processes in our equilibrium electrochemical system (1a) with ε = ε_{AgCl}, written:
_{e}(x_{d}) (where x_{d} = x_{i1}), the plane of closest approach in the electrode systems for fixed aqueous KCl ionic concentration (0.05 m) ; x_{d} must correspond to atomic (radius) dimensions (~ 3.5 Å for the hydrated chloride ion).

It is clear from _{d} must be much less than 10^{–3} Å, (between 10^{–3} – 10^{–7} Å) for there to be fair agreement with experiment; i.e. x_{d} is far below atomic (or even electron-nuclear atomic core) dimensions, implying that the GC double layer certainly appears more appropriate for our equilibrium study, which incidentally yields good agreement (to at least order of magnitude accuracy) with experimental values for model M1. We can conclude that in those schemes which replicated the present methodology, non-equilibrium effects may well have perturbed the steady-state ionic distribution to such an extent that the GCS equilibrium electrode model appeared to be the most appropriate one, since when we applied this model to a true equilibrium system, the results were not compatible with the experimental results where atomic dimensions are concerned. Although tentative physical considerations might rule out ionic penetrants for the case of the AgCl dielectric, theories have been successfully developed [

In order to illustrate the feasibility of the method (even if it is not very applicable here) we solve _{±,2} in the AgCl dielectric (phase 2) relative to parameters from the aqueous phase 1, given by:

With the above (56) activity coefficient, ^{–3} and ~ 2 × 10^{–4} molal KCl present in the AgCl phase, for aqueous KCl concentration in the range 0.01 – 0.1 m. The average branches from these two branches were taken, because it was noted that by slight adjustment of the ionic radii, a unique solution could be attained by any one fixed concentration of the aqueous KCl. The average solution was monotonically increasing and was linearized to yield the form c = 0.3342 m (c is the KCl concentration in moles L^{–1} in the dielectric layer, m is the aqueous molality). The corresponding values for the ionic radii (Å units) were derived by scaling from Shanon’s crystallographic data [^{−}, and 1.29357 (1.38) for K^{+}. By a process of trial and error, (36) was solved (no bifurcations were observed at low aqueous molalities < 0.05 m) yielding a monotonically increasing function with increasing aqueous concentration, and it was linearized to c = 0.064246 m (again c is the KCl concentration in moles L^{–1} in the dielectric layer, and m is the aqueous molality) ; the ionic radii used was 1.6966 (1.81) for Cl^{−}, and 1.3077 (1.38) for K^{+} for the aqueous concentration range 0.01 – 0.1m KCl. We then use these concentrations in the AgCl phase to predict the cell e.m.f. where the bulk ionic concentration corresponds to that in the AgCl dielectric layer, and where the electrode used is either of the kind described by

We may conclude from the above discussion that the so-called standard potentials E_{i}^{ø} for the various half-cells used to compute the overall cell e.m.f. E with standard E_{s}^{ø} written:
_{i} the stoichiometric coefficient for each half-cell reaction must be modified for a net z-electron reaction to:
^{ø} (p,^{ø} = 0 and ^{o} ” as reported by Bates and Macaskill in their extensive tabulation and derivation of standard potential values, especially for the Ag/AgCl electrode [

Having observed the rough magnitude of the cell e.m.f. which is incompatible with the predictions made on the basis of the GCS interfacial model, we may perhaps attempt to understand why this is so from the point of view of charge accumulation on the electrodes. For the electrode reaction AgCl + e^{−}(Met.) ↔ Ag + Cl^{−} (aq) (where we assume that the AgCl dielectric is only a reservoir for ions with no other effect) the charged species in solution includes the cation, Cl^{−} and electrons which are able to percolate through a pathway to the substrate Ag electrode, whereas the Gouy-Chapman (and Stern modification) assumes the ions being the only charge carriers (with finite size for the Stern modification of the Helmholtz double layer). This process pathway still obtains even if we should choose model M1 (but the charges on the electrode would be scaled down by a factor of approximately two, due to the lower value of the dielectric constant for AgCl). For purposes of calculation and comparison, if the (theoretical) substrate-solution potential were fixed at ℑ^{o} = 0.394031 V, then for model M1, the charge per unit area on the electrode may be computed for distances x_{2} = 3.5 Å (GCS model) and x_{2} = 10^{–9} Å (effective GC model). For the For the ordered set of KCl normalities {0.001; 0.005; 0.01; 0.05} with activity coefficients set to unity, the corresponding charge density (μCcm^{–2}) for the electrode at x_{2} = 3.5 Å and x_{2} = 10^{–9} Å are the respective ordered sets with values {27.32, 33.46, 36.19, 42.70} and {398.37, 890.77, 1,259.74, 2,816.87}. If it is maintained that at very low concentrations, the Gouy-Chapman model may be accepted as fairly accurate [^{–1}cm^{–2}, which differs from the Stern model prediction of ~ 100 μCcm^{–2}V, which is ~ 30 times less than the value predicted by the simple GC theory at its regime of validity. Hence we can expect from these observations that the Stern theory may not be a suitable description for these types of electrodes, even at low concentrations.

From a strictly theoretical angle, there appears to be a (subtle) and probably minor peculiarity which is assumed in using the GCS model as an ideal polarized electrode. By definition, no charge transfer is envisaged with change of potential, yet there is a constant setting^{4} of σ^{m} = –σ^{s} in order to determine the charge on the electrode (σ^{m}) where –σ^{s} = –εε_{o}(dϕ/dx)_{x = x2}, where ϕ is the solution potential at x_{2} (x_{2} = 0 for the GC model). The reason is attributed to total charge neutrality; if this were the case then one must allow for charge transfer. On the other hand, if there were absolutely no charge transfer then a counter-electrode would have to be placed in the solution for electroneutrality, which would negate the model boundary conditions. As discussed below, this charge neutrality between electrode and solution is accounted for directly in the ideal self-polarizing electrode. These secondary considerations, in addition to those derived from experiment, make us choose the simplified Gouy-Chapman model of the interface because the electrode reaction mentioned above provides an electronic pathway across the interface for charging the electrode; the electronic factor is not considered in the Stern modification, which could lead to a much smaller x_{2} than predicted for an electrode system not reacting with the fluid ions.

For instance, consider the component electrode reactions for the AgCl|Cl electrode:
_{2} averaged between the hydration radii of Ag^{+}(s) and Cl^{−}(s). On the other hand, b’ shows that this static OHP cannot exist at x_{2} because of the transfer of charge from the Ag^{+}(s) ion to the lattice, implying a smearing of charge between x_{2} and the metal substrate which is precluded in the standard treatment of the OHP layer, which serves as a capacitance layer with no charge distribution between them (when there is no specific adsorption, i.e. x_{1} = 0 in _{2} = 0; also, the σ^{m} = –σ equation would obtain here without a need for a counterbalancing electrode since it is assumed that the charges separate in thermodynamical equilibrium from a neutral (electrode plus solution) combination, i.e. σ^{m}+σ^{s} = 0 at all times up to equilibrium. Lastly, according to Sparnaay [_{o} is the permittivity of free space. Further modifications to this would follow along similar lines as for previous discussions on this topic such as excluded volume effects and the finite size of ions, but it is anticipated that the ε(x) term would partially account for some of these factors.

Most of the theorems presented here may be corroborated by well established experimental data. The preliminary experiment presented here indicate that electrode size effects which affect the cell e.m.f. does exist, and despite the 20% uncertainty in the measurement, due mainly to the unavoidable variation of physical properties of the substrate silver metal and the silver chloride constituents, are of a magnitude which does not contradict the standard theories, such as the postulated electronic chemical potential of Ag inferred from data such as its low temperature specific heat. Where change of e.m.f. with size electrode size is concerned, the problem then becomes one of determining the precise magnitude of these differences with the change of the thermodynamical variables (including size), so that a clear electrochemical standard may be prescribed. From a practical point of view, the standardization of electrodes following the recommendation of people like Bates and Macaskill [

The preceding theoretical results concerning the e.m.f. differences may conceivably be quantitatively improved further by using modified theories of the electrode-solution potential[^{+}(aq) + e^{−} ↔ Ag (solid)) of different - up to thinfilm - dimensions to attempt to observe possible size effects on the cell e.m.f. (if the oxide-layer and Ag structural problems can be overcome!). Then, an empirically based set of Tables may be constructed for all such electrodes if for the purposes at hand the changes of potential are considered significant since at present exact calculations on the (Fermi) electron distribution for many metallic elements only seem to be tentative and semi-quantitative and are also based on single-crystal assumptions, none of which are strictly valid for real electrodes. We also expect the effects to vary depending on the nature of the Fermi surface; some metals like Ag are predicted to have discernable effects on the standard potential with size of the electrode, whereas others may show less variation of size effects mentioned above due to the effects of the zone boundaries partitioning the electronic states.

Recently, it has been shown [

One of the consequences of the electrode theorems above, corroborated by the experimental results is that ^{–4} V, i.e. ℑ^{o} is approximately constant. If C_{d} is determined from capacitance studies, then the elusive single–ion activity coefficient may be determined exactly if either ℑ^{o} is known, or is assumed constant, as indicated here.

The other lemmas, such as those concerning the self-polarizing electrode are readily applied to a variety of problems such as those concerning capacitance; in particular, these electrodes also suggest, by way of operation another 3-dimensional treatment of the interfaces, where at present the 2-dimensional utilization of the Gibbs surface excess and surface tension equations are the main descriptions.

Concerning the electronic chemical potential, if accurate cell e.m.f. measurements are possible, then the (solid state) electronic correlation energy ε_{xc}(n) may be determined by using the expression such as μ_{T} [_{xc} (which is generally a function of the free electron density only, at fixed temperature, and pressure) to problems, since for Ag, its effective correlation energy may be taken to be zero where some thermodynamical properties are concerned, where, on the other hand, ε_{xc} yields a non-zero value.

The above work shows that with even very elementary methods, and relatively non-specialist skills, one can still pose and solve problems in a qualitative sense that suggests quantitative directions, and within a context that might appear attractive to those fragile political, social and scientific environments. But such solutions are not possible without in the first place the existence of a

I thank the management of MDPI for personally soliciting contributions to this Special Volume and to TWAS (Third World Academy of Science, Trieste, Italy) (1998) for financing this research and University of Malaya PJP grants for helping with preparation and publication of the work. Finally, I am grateful to Tan Hsiao Wei (postgraduate student, University of Malaya (2009)) for her help in typesetting.

The tables below represent the raw numerical values that were used in the quantitative Figures found in the main text. They are provided here merely to aid in quantitative analysis of the plots. The dimensions and scales of these numbers can be inferred from the corresponding Figures and plots in the main text and so column captions are superfluous.

Numbers used for plotting _{2} with ionic concentration.

−3.0 | 0.2018 | −3 | −0.2188 | −3.0 | −0.1469 | −3.0 | 0.1689 |

0.0 | 0.068 | 0.0 | −0.0719 | −0.0 | −0.0125 | 0.0 | 0.0347 |

Numbers used for plotting

−0.9536 | −24.0 | −1.0150 | −20.0 | −1.0731 | −20.0 | −1.1077 | −20.0 |

−0.5076 | −16.0 | −0.7385 | −16.0 | −0.8308 | −16.0 | −0.8539 | −16.0 |

−0.3115 | −12.0 | −0.5192 | −12.0 | −0.6000 | −12.0 | −0.6115 | −12.0 |

−0.1615 | −8.0 | −0.3000 | −8.0 | −0.3577 | −8.0 | −0.3692 | −8.0 |

0.0000 | 0.0 | 0.0000 | 0.0 | 0.0000 | 0.0 | 0.0000 | 0.0 |

0.1269 | 8.0 | 0.3000 | 8.0 | 0.2539 | 8.0 | 0.2539 | 8.0 |

0.2054 | 12.0 | 0.4385 | 12.0 | 0.3462 | 12.0 | 0.3000 | 12.0 |

0.2308 | 16.0 | 0.6000 | 16.0 | 0.4270 | 16.0 | 0.3577 | 16.0 |

Numbers used for plotting

0.05 | −0.2414 | −0.1438 | −0.1146 |

1.00 | 1.9329 | 1.1536 | 0.9194 |

2.00 | 4.2022 | 2.5123 | 2.0033 |

4.00 | 8.6818 | 5.2084 | 4.1575 |

6.00 | 13.0853 | 7.8765 | 6.2939 |

8.00 | 17.4150 | 10.5176 | 8.4127 |

9.00 | 19.5543 | 11.8281 | 9.4656 |

Numbers used for plotting _{AgCl}.

5.0000^{−3} |
1.6284 | 0.9826 | 0.7858 | xexp | yexp |

0.0100 | 1.6501 | 0.9959 | 0.7965 | 0.0100 | 1.4640 |

0.0200 | 1.6790 | 1.0135 | 0.8107 | 0.0300 | 1.5140 |

0.0300 | 1.6996 | 1.0263 | 0.8208 | 0.0500 | 1.5660 |

0.0400 | 1.7162 | 1.0364 | 0.8290 | 0.0700 | 1.5600 |

0.0500 | 1.7302 | 1.0450 | 0.8358 | 0.1000 | 1.7680 |

0.0600 | 1.7423 | 1.0524 | 0.8418 | ||

0.0700 | 1.7531 | 1.0589 | 0.8471 | ||

0.0800 | 1.7627 | 1.0649 | 0.8519 | ||

0.0900 | 1.7716 | 1.0703 | 0.8562 | ||

0.1000 | 1.7797 | 1.0752 | 0.8602 |

Numbers used for plotting _{H2O}.

5.0000e-3 | 4.1163 | 2.5309 | 2.0358 | xexpt | yexpt |

0.0100 | 4.1686 | 2.5640 | 2.0628 | 0.0100 | 1.4640 |

0.0200 | 4.2380 | 2.6084 | 2.0985 | 0.0300 | 1.5140 |

0.0400 | 4.2877 | 2.6396 | 2.1242 | 0.0500 | 1.5660 |

0.0500 | 4.3275 | 2.6649 | 2.1447 | 0.0700 | 1.5600 |

0.0600 | 4.3610 | 2.6861 | 2.1620 | 0.1000 | 1.7680 |

0.0700 | 4.3899 | 2.7046 | 2.1771 | ||

0.0800 | 4.4156 | 2.7210 | 2.1904 | ||

0.0900 | 4.4387 | 2.7357 | 2.2023 | ||

0.1000 | 4.4598 | 2.7491 | 2.2133 | ||

0.0300 | 4.4791 | 2.7614 | 2.2238 |

Numbers used for plotting _{AgCl}.

−20.0223 | 173.1564 | 104.5700 | 83.6511 |

−15.4172 | 173.1228 | 104.558 | 83.6343 |

−13.1146 | 172.8186 | 104.3690 | 83.4821 |

−11.5052 | 171.4856 | 103.5420 | 82.8155 |

−10.8120 | 169.8613 | 102.5360 | 82.0041 |

−9.2026 | 158.3099 | 95.4024 | 76.2588 |

−8.5095 | 146.6338 | 88.2300 | 70.4922 |

−6.9000 | 98.6750 | 59.0810 | 47.1330 |

−6.2068 | 74.1631 | 44.3246 | 35.3413 |

−4.5974 | 30.2894 | 18.0615 | 14.3911 |

−3.9043 | 19.1034 | 11.3867 | 9.0717 |

−3.2112 | 11.6658 | 6.9520 | 5.5382 |

−2.8057 | 8.6350 | 5.1459 | 4.0993 |

−2.6515 | 7.6868 | 4.5803 | 3.6487 |

−2.2949 | 5.8486 | 3.4848 | 2.7760 |

Numbers used for plotting

5.0000e-3 | 0.9565 | 0.5742 | 0.4584 |

0.0100 | 0.9696 | 0.5821 | 0.4648 |

0.0200 | 0.9869 | 0.5926 | 0.4732 |

0.0300 | 0.9993 | 0.6001 | 0.4792 |

0.0400 | 1.0093 | 0.6062 | 0.4848 |

0.0500 | 1.0178 | 0.6113 | 0.4881 |

0.0600 | 1.0252 | 0.6158 | 0.4917 |

0.0700 | 1.0317 | 0.6197 | 0.4949 |

0.0800 | 1.0376 | 0.6233 | 0.4977 |

0.0900 | 1.0430 | 0.6266 | 0.5003 |

0.1000 | 1.0479 | 0.6296 | 0.5028 |

Numbers used for plotting

5.0000e-3 | 0.4239 | 0.2534 | 0.2015 |

0.0100 | 0.4297 | 0.2569 | 0.2048 |

0.0200 | 0.4375 | 0.2615 | 0.2086 |

0.0300 | 0.4430 | 0.2649 | 0.2112 |

0.0400 | 0.4475 | 0.2676 | 0.2134 |

0.0500 | 0.4513 | 0.2699 | 0.2152 |

0.0600 | 0.4546 | 0.2718 | 0.2168 |

0.0700 | 0.4575 | 0.2736 | 0.2182 |

0.0800 | 0.4602 | 0.2752 | 0.2194 |

0.0900 | 0.4626 | 0.2766 | 0.2206 |

0.1000 | 0.4648 | 0.2780 | 0.2217 |

Miscellaneous: For theoretical calculation, the exposed area of s1 corresponded to 6.8926 cm^{2} for the penetration depth of 1.920 cm. All computations were in double precision (32-bit machine).

_{2}/Electrolyte Solution Interface: Calculation of zeta Potential Using Non-Equilibrium Theories

_{6}

^{4–/3–}redox system on platinum standard-size and ultramicroelectrodes

Schematic of the regions considered at the electrode interface.

Concentration profile of ion between two dielectrics.

Variation of ϕ_{2} with bulk ionic concentration.

Charge on Hg in contact with 1M electrolytes at 25 °C.

(a) General setup of cell. (b) Plan view of square polyethylene top for electrode; (c) Electrode used in the experiments.

Variation of cell e.m.f. with area of electrode s2 exposed to 0.05 m aq. KCl.

Sketch of interfacial region with associated potentials.

Computed cell e.m.f.s for single Cl^{−} ion electrode for various μ and experimental data.

Cell e.m.f. based on H_{2}O dielectric constant μ.

Cell e.m.f. computed for various μ and Helmholtz Inner layer distances X_{d}.

Cell e.m.f. for various μ, based on solution of dielectric equation for KCl concentration in AgCl dielectric.

Cell e.m.f. for various μ, based on Born dielectric equation for KCl concentration in AgCl dielectric.