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This chapter discusses theoretical considerations. A phase that separates two other phases to prevent mass movement between them but allows passage with various degrees of restriction of one or several species of the external phases may be defined as a membrane which when used as an electrode in an electrochemical cell constitutes a membrane electrode. The behavior of the membrane electrode will be determined by the properties of the membrane which can be a solid or a liquid containing ionized or ionizable group. Ion-selective membrane electrodes may be roughly classified according to the physical state of the substances—the electroactive materials—that form the electrode membrane in ion-selective electrodes with solid membranes. The membrane may be homogeneous as in a monocrystal, a sparingly soluble crystal-line substance or a glass that is considered to be a solid because of the immobility of the anionic groups. Alternatively, the membrane may be heterogeneous, by the incorporation of the electroactive substance within an inert matrix. The electrode membrane is represented by an organic liquid immiscible with water. The organic liquid contains the electro-active substance which may be either electrically charged or neutral ligand groups. The electroactive substance is capable of exchanging ions in solution for which the electrode is selective.

(i) Ion-selective electrodes with solid membranes. The membrane may be homogeneous as in a monocrystal, a sparingly soluble crystalline substance or a glass which is considered to be a solid because of the immobility of the anionic groups. Alternatively, the membrane may be heterogeneous, by the incorporation of the electroactive substance within an inert matrix.

(ii) Ion-selective electrodes with liquid membranes. Here the electrode membrane is represented by an organic liquid immiscible with water. The organic liquid contains the electroactive substance which may be either electrically charged or neutral ligand groups. The electroactive substance is capable of exchanging ions in solution for which the electrode is selective.

Fig. 1.1 Schematic representation of membrane electrode cell assembly: 1, membrane; 2, potentiometer; 3, internal reference electrode; 4. external reference electrode; 5, sample solution; 6, internal filling solution.

1.1 SOLID MEMBRANES

Relations based on investigations by Nicolsky (2) have been derived from experimental data for the e.m.f. of cells with liquid or solid ion-exchange membranes (3–5). These relations are as follows for glass and pure solid ion-exchange membranes (6):

(1.1)

where R = gas constant, T = absolute temperature, F = Faraday constant, aA(orB) = ion activitites in the sample solution (monovalent ions), a‘A(orB) = ion activities in the internal filling solution (monovalent ions), n = constant depending on the ions A and B and the membrane; kA,Bpot = selectivity coefficient (preference of sensor for ion B in relation to ion A).

In ion-selective electrodes having a given inner reference electrode system (a’A(orB) = constant) we have:

(1.2)

For a mixture of N monovalent ions with n=1, one could obtain from eqn. (1.1):

(1.3)

in general, and

(1.4)

for a given inner reference electrode system.

The selectivity coefficient, , which characterizes the preference of the sensor for the ion B as compared with the ion A, is given by:

(1.5)

where kA,B is the equilibrium constant of the exchange:

(1.6)

and uA and UB are the mobilities of the ions in the membrane. If the sensor responds to a divalent ion (aA) and a monovalent ion (aB), eqn. (1.3) becomes:

(1.7)

For a completely reversible cell assembly, can, in principle, be determined approximately by measurements carried out in a solution of the ion A and in a solution of the ion B.

For the theoretical interpretation of the behaviour of precipitate-based ion-selective electrodes, the electrodes based on silver halide may be used as a model (7). By using either a heterogeneous or a homogeneous ion-selective electrode at zero current in a solution containing the ion to which the electrode is reversible, the equilibrium between the solution and the solid phase is attained when the difference of the electrochemical potentials of the solvated ion and the ion bonded to the solid phase is equal to zero. If the electrochemical potential of the appropriate i-th ion in the solution is:

(1.8)

while in the membrane

(1.9)

where η is the electrochemical potential, μ the chemical potential, ψ the Galvani potential, zi the valency of ion i, F the Faraday constant and s and m denote the solution and membrane phase, respectively. At equilibrium, ηs = ηm:

(1.10)

(1.11)

(1.12)

where E is the electrode potential, Eo the standard electrode potential, μo the standard chemical potential and (ai)s and (ai)m are the activities of the i-th ion in the solution and in the membrane phase, respectively.

In this deduction of the electrode potential (7) the ion diffusion across the membrane is not considered as it normally has no effect on the membrane potential at zero current.

If the concentration of the appropriate ion is relatively low in the solution examined, then the electrode potential approaches a limiting value which can be expressed for univalent ions in the following way:

(1.13)

where Sji is solubility product of the precipitate used as electrode.

In a solution containing not only the ion to which the electrode is reversible but another ion, which also forms a precipitate with one of the components (K) of the membrane matrix, the following precipitate exchange reaction is established:

(1.14)

(1.15)

On the basis of this exchange equilibrium the following equation can be derived for the potential of the membrane electrode if the intra-membrane diffusion phenomena are neglected (7).

that is:

(1.16)

where ai, ci and ak, ck are the activities and concentrations of i-th and k-th ions, respectively, n is the number of ions taking part in the exchange reaction, f± is the mean activity coefficient and Kik becomes the selectivity coefficient of the electrode.

The selectivity coefficient can be deduced from the...