Electrochemistry

In order to be reasonably general we start off with a mixture of four salts with water. Let the four salts be the 1-1 electrolytes, KCl, NaI, NaCl, KI, for example. Any of these salts may be replaced by another 1-1 electrolyte. The description would not be much simplified if we used general symbols with suitable indices for the solution components. The author feels that reading is easier if the derivations given later are connected with real chemical species. The corresponding equations for polyvalent electrolytes will only occasionally be added. Their derivation is straightforward in all cases.

Assume that at t = 0 the mixing of the components H₂O, KCl, NaCl, and NaI is completed such that the system is uniform. This means that we have where x is the position coordinate in the system. For simplicity we assume that the cross-section A of the vessel containing the solution is constant. So it suffices to take the spatial coordinate as the variable x. We have also and likewise all the composition variables in the constituent coordinate system and the total solute-constitutents coordinate system have vanishing derivatives with respect to the position and the time. However the concentrations of molecular species present in the solution may change with time. In our system we have certainly the chemical reaction

where K ₁ and K ₂ are the rate constants for the reactions indicated. Molecular species are defined in the sense of eqs.(9)–(13). Of course, in the usual practical situation the reaction will be unmeasurably fast, however for the following it will be of great importance to clarify the physical situation in principle. Then in an uniform concentration distribution (∂C _i /∂x = 0, i = NaI, KCl, NaCl, KI) deviating from equilibrium with respect to the molecular concentrations we have

First we consider a system which has no electrodes but contains an electrolyte solution in which an electric current can exist. This in effect means that the electrodes are a very great distance away from the cell which we are observing. Alternatively we could have a closed loop of electrolyte solution, parts of which we can move through a static magnetic field. As we shall see below, the characterization of the system as a closed loop is a more fundamental picture than a cell construction where the electrodes are at very large distances from the section being observed. Assume now that we have switched on an electric current J _q by some device. The electric current can be identified by a magnetic field around the cell. When the electric current is switched on let it be constant in time. Since electric work is performed on the system, in order to keep the temperature constant, heat has to be continuously delivered to the surroundings. Of course, the electric current is constant along the total system. For simplicity we assume that the cross-section A of the tube containing the electrolyte solution is constant. J _q is the electric current density and we have Let the electrolyte solution be the same as previously in that it contains the four constitutents, Na, K, Cl, and I, and as before, the composition is not constant everywhere along the coordinate x which gives the location in the cell. The positive x direction is the same as the positive direction of the vector J _q . For the description of the composition we apply the total solute-constituents coordinate system.

The equations of the preceding section — no explicit considerations of the electrodes — have their most important application in conjunction with the moving boundary method which serves to measure transport numbers¹⁾. In a typical experiment we have two different cations and only one anion. In order to remain within the general framework of the main treatment we consider the special constituents Na, Li, and Cl. The replacement of K by Li is necessary for technical reasons as will be seen later. Now the set of eqs.(112)–(114) only contains two equations: the third one, eq. (114) is no longer necessary. Thus we have and where the concentration dependence of the diffusion coefficients has been neglected. The concentration distributions are sketched in Fig.7. The purpose of this arrangement is the measurement of the transport number of Na. Of course this is the transport number in the binary homogeneous electrolyte NaCl + water which we may denote by t_Na. To begin with, let us consider the first term on the right-hand side of eq.(126) which is − divj _Na .

In the previous chapter we described the changes of the concentration profiles in a liquid boundary in the presence of an electric current, in particular, the movement of the boundary was the main object of interest. When the electrodes are of solid material — or in other cases are not electrolytes and not miscible with the electrolyte — then it is more suitable to give the main emphasis to the description of the mass fluxes onto and away from the electrode material and to devote only a fairly brief discussion to the concentration variation in the electrolyte solution. It should be mentioned here that in this book the word electrode is reserved for one single metallic phase — sometimes in combination with a non-metallic solid phase — it is not understood to refer to the combined system metal-electrolyte, for instance to the “half-cell” Cu(metal)/CuSO₄ solution. Still, as has been demonstrated in Chapter 3, the boundary between two different liquid electrolyte phases in certain respects acts like an electrode although not an electrode in the sense of the definition just given. Therefore, in the last section of this chapter we shall also give an outline of the effect of the mass fluxes and the excess mass fluxes at a liquid junction between two bulk electrolytes.

In the preceding chapters we have given a detailed analysis of the local rates of change of the partial mass densities characterizing the analytical field. In the next step it is necessary to present a similar analysis for the local rate of change of the internal energy.

We have now to compare our treatment of the mass flux, the electric current, and the excess mass and energy fluxes in inhomogeneous systems with the theory generally accepted and found in the literature^1,2,3). We wish to give as simple an analysis as possible and therefore we consider a binary electrolyte-water system, for instance NaCl + H₂O. Of course the NaCl concentration is not uniform throughout the system. In the conventional treatments the fluxes are the centre of interest, and thus we shall mainly discuss the interconnection between electric current density and mass flux on the one hand and the field properties determining the system on the other hand. Usually the mass fluxes are referred to the solvent fixed coordinate system and instead of the gradient of the solute particle density the gradient (or gradients) of the chemical potential are written in the equations. We assume that the electrodes in the conventional sense — if at all present — are remote and processes at the electrodes do not enter explicitly into the treatment. Comparison of the circumstances at the boundary metal/electrolyte will be postponed to the next chapter.

As described previously, the galvanic cell is a system constructed from a set of thermodynamic phases such that an electric current can exist and a chemical reaction can proceed in the system. It seems to be a reasonable starting point to describe a system in which two different electrolyte solutions are placed in contact with another. The remaining two boundaries are formed from solid materials and the loop is closed by some inert metal which does not enter in the reaction scheme. Fig. 25 gives a schematic representation of a special system which involves the constituents Na, K, Cl, and I together with water. In Fig.25 the

galvanic cell has indeed been drawn as a closed loop. This is to remind the reader that the galvanic cell is a device shaped in a particular way in which a chemical reaction proceeds. In the following pages we shall repeatedly give this schematic representation of a galvanic cell as a “reaction vessel”. This is in contrast to the abbreviated form of representation which is commonly used in conventional treatments and also in the heading of this section.

We now turn to the class of simpler galvanic cells which are built up of only three constituents. Thus we may have two different cationic constituents and only one anionic constituent. As a first example, we choose the NaCl + KCl system. The galvanic cell with an electrolyte containing two different anionic species and only one cationic constituent should be treated in an analogous way.

We consider a system as depicted in Fig.33.

Let us consider the following galvanic cell (Fig.34). In the presence of an electric current the chemical reaction is:

As is well-known the salt bridge is a device which has the purpose of eliminating or at least reducing the diffusion or liquid junction potential. The set-up is simple. The two half cells from which the galvanic cell is to be constructed, are not brought into direct contact, rather they are connected via an electrolyte phase which in most cases consists of a concentrated KCl solution. Experimental details need not be given here. The conventional explanation of the observed effect, namely a drastic reduction of the diffusion potential, is based on the fact that the transport numbers in the bridge electrolyte, t_Cl and t_k, are almost equal. As a consequence of the equal mobilities of the K⁺ and Cl⁻ ions an electrical potential difference presumably inherent in any ionic diffusion process does not arise. In the present chapter we shall give the theory of the salt bridge within the framework of our treatment. It will be seen that the physical reason for the strong reduction of the liquid junction potential differs from that given in the classical treatment.

We begin with the simplest example. We have two NaCl solutions and these solutions are separated by a membrane in which some compound of the constituent Na is formed but which does not contain the constituent Cl. Then, in the left-hand and right-hand compartments we have Cl₂ electrodes or also Ag/AgCl — or calomel-electrodes —. Thus the system has the construction scheme as shown in Fig.42.