Metallic electrodes, free energies

We now consider briefly the matter of electrode potentials. The familiar Nemst equation was at one time treated in terms of the solution pressure of the metal in the electrode, but it is better to consider directly the net chemical change accompanying the flow of 1 faraday (7 ), and to equate the electrical work to the free energy change. Thus, for the cell... 

This difference is a measure of the free-energy driving force for the development reaction. If the development mechanism is treated as an electrode reaction such that the developing silver center functions as an electrode, then the electron-transfer step is first order in the concentration of D and first order in the surface area of the developing silver center (280) (Fig. 13). Phenomenologically, the rate of formation of metallic silver is given in equation 17,... 

The standard electrode potentials , or the standard chemical potentials /X , may be used to calculate the free energy decrease —AG and the equilibrium constant /T of a corrosion reaction (see Appendix 20.2). Any corrosion reaction in aqueous solution must involve oxidation of the metal and reduction of a species in solution (an electron acceptor) with consequent electron transfer between the two reactants. Thus the corrosion of zinc ( In +zzn = —0-76 V) in a reducing acid of pH = 4 (a = 10 ) may be represented by the reaction ... 

The two elements have similar electronegativity. (Note electronegativity is the power of an element to attract electrons to itself when present in a molecule or in an aggregate of unlike atoms it is a different property from the electrode potential, which depends on the free energy difference between an element in its standard state and a compound or ion in solution (see Section 20.1).) In addition a metal of a lower valency tends to dissolve a metal of a higher valency more readily than vice versa. 

In the introductory chapter we stated that the formation of chemical compounds with the metal ion in a variety of formal oxidation states is a characteristic of transition metals. We also saw in Chapter 8 how we may quantify the thermodynamic stability of a coordination compound in terms of the stability constant K. It is convenient to be able to assess the relative ease by which a metal is transformed from one oxidation state to another, and you will recall that the standard electrode potential, E , is a convenient measure of this. Remember that the standard free energy change for a reaction, AG , is related both to the equilibrium constant (Eq. 9.1)... 

Only if one takes into account the solvent dynamics, the situation changes. The electron transfer from the metal to the acceptor results in the transition from the initial free energy surface to the final surface and subsequent relaxation of the solvent polarization to the final equilibrium value Pqj,. This brings the energy level (now occupied) to its equilibrium position e red far below the Fermi level, where it remains occupied independent of the position of the acceptor with respect to the electrode surface. 

It represents the case of the reaction at the metal electrode in which ions of the same metal discharge at the electrode from the electrolyte. It can be said that copper ions in the electrolyte (copper sulfate solution) possess a free energy GCu(ej, and those in the copper metal electrode possess a free energy Ci(ll(-ril.. Then, if a copper ion is to leave its place in the copper sulfate electrolyte structure and occupy a position in the structure of the copper electrode, the free energy change accompanying this process will be ... 

The decrease in free energy (—AG) which provides the driving force in a cell may ensue either from a chemical reaction or from a physical change. In particular, one often studies cells in which the driving force is a change in concentration (almost always a dilution process). These cells are called concentration cells. The alteration in concentration can take place either in the electrolyte or in the electrodes. As examples of alterations in concentration in electrodes, mention may be made of amalgams or alloy electrodes with different concentrations of the solute metal and in gas electrodes with different pressures of the gas. 

Figure 8. Dependence of the symmetry factor a on the free energy of the transition for the reaction of hydrogen ion discharge on a metal electrode.

Most cations are strongly solvated, since their radii are small, and the free energy of solvation is approximately proportional to z2/r +, where ze0 is the cation charge in coulombs and r+ its ionic radius. The result of this is that even if the charge on the electrode is negative, there is usually little tendency for these cations to shed their water molecules and adsorb directly on the metal surface. Thus, the distance of closest approach of cations is determined by the radius of the inner solvent coordination sphere, and if the metal surface itself constitutes a plane, then the cation nuclei, at the distance of closest approach, will also constitute a plane termed the outer Helmholtz plane (OH P). 

In terms of understanding the mercury/electrolyte interface, it is clear from the above discussion that the measurement of the surface free energy (in terms of the surface tension), is central. If the clectrocapillarity technique could be applied to solid electrodes, then it is capable of supplying information extremely difficult to obtain by any other technique. Sato has indeed developed a technique to measure the surface tension of a metal electrode which he terms piezoelectric surface stress measurement and is based upon the previous work of Gokhshtein (1970). 

These points indicate that the continuum theory expression of the free energy of activation, which is based on the Born solvation equation, has no relevance to the process of activation of ions in solution. The activation of ions in solution should involve the interaction energy with the solvent molecules, which depends on the structure of the ions, the solvent, and their orientation, and not on the Born charging energy in solvents of high dielectric constant (e.g., water). Consequently, the continuum theory of activation, which depends on the Born equation,fails to correlate (see Fig. 1) with experimental results. Inverse correlations were also found between the experimental values of the rate constant for an ET reaction in solvents having different dielectric constants with those computed from the continuum theory expression. Continuum theory also fails to explain the well-known Tafel linearity of current density at a metal electrode. ... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

The difficulty of such treatments are that they do not provide expressions for the rate of die ET reactions at electrodes which can be compared with experiments. They involve complicated computer simulations to determine the free energy profile. Such simulations generally use adjustable parameters to make the results fit experiments. Conversely, these treatments include both short- and long-range ion-solvent interactions and the interaction of the ion and the solvent with the metal electrode at a molecular level. 

---