It is useful to consider a metal as an array of ions, Mm+, the valence electrons of each atom having been transferred to the crystal as a whole.
These “free” electrons account for the electrical conductivity of the metal and other electronic properties. The metal in aqueous solution also exists as an ion, and thus, the relative tendency for the ion to exist in the metal or in the solution depends, along with other factors such as the concentration, on the relative electrochemical free-energy of the ion in these two phases. The electrochemical free energy is used in this appli-cation rather than the Gibbs free energy because charged phases are in-volved. The electrochemical free energy per ion, gel, is composed of a chemical contribution, g, and a charge contribution, qφ, such that:
gel= g + qφ (Eq 2.48)
where q is the charge on the ion, andφ is the electrical potential at the ion in the phase (solid or liquid). The electrical potential at the ion is de-fined by the work required to move unit positive charge from an infinite reference state to the position of the ion. The difference in electrical po-tential between two points is therefore directly related to the work re-quired to move unit positive charge between the points; this difference of potential is the more important concept for the present discussion.
Just as the condition for chemical equilibrium is∆g = 0, the condi-tion for electrochemical equilibrium is∆gel= 0. This condition is now
applied to the transfer of ions across the metal/electrolyte interface. For convenience, the symbols gM0 andφM0are used to indicate the GFE and electrical potential of the ion in the metal; the symbols gM+ andφM+ ap-ply to the ion in solution. The change in electrochemical free energy on going from an ion in the solid to an ion in solution is given by:
∆gel=
(
gM+ −gM0) (
+q φM+ −φM0)
(Eq 2.49) At equilibrium,∆gel= 0, and therefore:(
gM+ −gM0)
= −q(
φ′M+ −φ′M0)
(Eq 2.50)where the primedφs indicate equilibrium values.
The charge transferred per ion is q = me+, where m is the valence and e+the unit positive charge. Therefore, per ion:
(
gM+ −gM0)
= −me+(
φ′M+ −φM′ 0)
(Eq 2.51)Multiplying by No, and with G = Nog and F = Noe+, the change in GFE per mole is:
(
GM+ −GM0)
= −mF(
φM′ + −φ′M0)
(Eq 2.52)These equations imply that metal ions tend to transfer from the solid across the interface to the solution due to a decrease in the GFE (i.e., GM+ < GM0). They tend to transfer in the opposite direction as a conse-quence of the difference in potential between the two phases (i.e., (φ′M+ >φ′M0). These concepts are summarized in Fig. 2.2. This result leads to the brief generalization: At equilibrium, the GFE driving force to transfer ions from the metal to the solution is exactly balanced by the electrical potential difference attracting the ions back to the metal.
It is not possible to calculate or experimentally measure absolute val-ues for GM+,GM0,φ′M+,orφ . However, relative potential differ-′M0
ences can be measured by connecting two electrode systems as indi-cated in the electrochemical cell of Figure 2.1, and also as indiindi-cated by the abbreviated cell representation of Fig. 2.3. In Fig 2.3, the right-hand electrode (RHE) is shown as the hydrogen reaction, 2H++ 2e = H2, oc-curring on platinum as an inert conductor. When the activity (effective concentration) of the hydrogen ions is unity (molality, m
H+ ≈ 1), the pressure of the hydrogen gas is one atmosphere, and the temperature is 25°C, this electrode is called the standard hydrogen electrode (SHE).
Its interface potential difference may be indicated as (φH2 φH+
′ − ′ )s, with the s subscript indicating standard conditions. This combination of electrodes is an electrochemical cell, the potential difference between the electrodes being defined as:
( ) ( )
E′M,Mm + = φ′M0 −φ′Mm + − φH′ 2 −φH′ + s (Eq 2.53) E′M,Mm + is called the single electrode or half-cell potential of the M,Mm+electrode on the standard hydrogen scale. It should be recalled that in this text, E denotes the potential in the general case, E′ the poten-tial at equilibrium, and Eo the potential at equilibrium under standard
Fig. 2.2 The metal/solution interface. Based on Ref 3
Fig. 2.3 Abbreviated cell representation showing absolute potentials
conditions, all relative to the standard hydrogen electrode (SHE). It is to be noted, based on Eq 2.53, that the half-cell potential of the hydrogen reaction under standard conditions is zero (i.e., EH Ho
2, += 0).
The sign or polarity of the electrode (M,Mm+) is determined basically by the difference in the work required to move unit positive charge from infinity to the metal, M, less the work required for transport to the SHE.
The electrode requiring the greater amount of work in moving the unit positive charge from infinity will be at a higher potential and is said to be positive relative to the second electrode, which is called the negative electrode. If the electrodes are connected externally through a conduc-tor, conventional positive current, I, will flow from the positive to the negative electrode, although the actual carriers are electrons flowing in the opposite direction. Practically, the polarity of the electrode whose potential is being measured relative to the SHE is given by the polarity of the terminal of a high-impedance voltmeter or electrometer that must be attached to the electrode to obtain a positive meter reading. Thus, if M spontaneously oxidizes to Mm+when coupled to the SHE, the M elec-trode will be negative relative to the SHE, and E′M,Mm + will be negative for the half-cell reaction, M = Mm++ me.
It is important to realize that the standard half-cell potential, Eo, or the half-cell potential at other than standard conditions, E′, is sign invariant with respect to how the equilibrium reaction is written or considered, for example, EFe,Feo 2+ = –440 mV (SHE) for both Fe = Fe2++ 2e and Fe2++ 2e = Fe. This point can be appreciated by examining the mea-surement of the difference in electrical potential of the cell in Fig. 2.1.*
Although these measurements are usually made with an electrometer (>1014ohms internal resistance), it is helpful to examine measurements with a potentiometer. The potentiometer is a variable potential device that is attached to the cell and adjusted until the current flow is zero. At this condition, the potentiometer is applying a potential to the cell that just equals the cell potential, for example, 440 mV for Fe = Fe2++ 2e with the negative terminal of the potentiometer connected to the Fe electrode, that is, EFe,Feo 2+ = –440 mV (SHE). If the potentiometer is ad-justed to slightly increase the potential of the Fe electrode relative to the SHE, for example, –430 mV (SHE), equilibrium no longer exists, the cell reaction occurs as it would spontaneously (but at a reduced rate), and net oxidation occurs (i.e., Fe→ Fe2++ 2e). Thus, for the M elec-trode in general, very slight increasing or decreasing of the potential of M relative to the SHE by the potentiometer upsets the equilibrium and causes net oxidation, M→ Mm++ me, or net reduction, Mm++ me → M, but with only a very small change relative to EM,Mo m +.
*The assumption is still made here as previously that the spontaneous hydrogen reaction on iron is negligible compared to that on platinum.