The chemical potential of charged species

in terms of the chemical potential of the electrons, /Ue- if dn moles of electrons were transferred, we have 

The dn moles of electrons carry a negative charge dQ = —F dn, where F is the charge per mole of electrons, F = 96 484.56 C/mol. Combining these two equations yields, after division by dn, 

- be the chemical potential of the electrons in M when 0 is zero then, Pt- — fie- + F f). Subtracting this equation from the preceding one, we obtain 

Equation (17.6) is the relation between the escaping tendency of the electrons, fie-, in a phase and the electrical potential of the phase, 0. The escaping tendency is a linear function of 0. Note that Eq. (17.6) shows that if 0 is negative, Pe - is larger than when 0 is positive. 

By a similar argument, it may be shown that for any charged species in a phase 

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The right terms in the two last equations, namely (dpB/dcB z ) and (31n ttBz /31ncBz ), are both called the thermodynamic or -> Wagner factors [iii, iv]. The first of them can be determined experimentally from the ion concentration dependence on the chemical potential of neutral species (a-T-S diagram). The direct determination of the second factor is impossible as the chemical potentials of charged species cannot be explicitly separated from those of other components of a system this parameter can be assessed indirectly, analyzing activities of all components. 

The chemical potentials of charged species are not well-defined, and we need to represent them instead by chemical potentials of neutral species. For this purpose, we may assume equilibria between neutral and charged species and electrons, in the electrochemical redox reaction ... 

The electrochemical potential of single ionic species cannot be determined. In systems with charged components, all energy effects and all thermodynamic properties are associated not with ions of a single type but with combinations of different ions. Hence, the electrochemical potential of an individual ionic species is an experimentally undefined parameter, in contrast to the chemical potential of uncharged species. From the experimental data, only the combined values for electroneutral ensembles of ions can be found. Equally inaccessible to measurements is the electrochemical potential, of free electrons in metals, whereas the chemical potential, p, of the electrons coincides with the Fermi energy and can be calculated very approximately. 

Figure 3.1a shows a membrane that is permeable to water and K+ and Cl - ions but impermeable to colloidal electrolytes (polyelectrolytes such as charged proteins). Let a denote the interior of the cell and (3 the extracellular region. In the absence of the poly electrolyte, water, K + and Cl" partition themselves into the two sides such that the chemical potentials of each species are the same inside as well as outside, as thermodynamics would demand. Moreover, the requirement of electroneutrality in both ot and (3 demands that the concentrations of each species K + and CP be the same on either side of the partition. 

The electrochemical potential of the charged species i in a given phase contains terms originating from both the electrical and the chemical work. Thus, for the electrochemical potential of charged species i in the electrode we can write... 

Experimentally, it is often found that the anodic and cathodic charge transfer coefficients are about 1/2. This is typically the case for outer-sphere electron transfer. Values between zero and one are found for several more complex reactions. We now consider whether this behavior is reasonable in the framework of the phenomenological model presented here. In an outer-sphere process, the oxidized and reduced species are outside the electrochemical double layer. The chemical potential of these species is then not influenced by the electrode potential, and the following is valid ... 

Besides the dissolving power of a solvent toward substrate and supporting electrolyte, the ability to solvate intermediate cations and anions is also important. The chemical properties of charged species are dependent on whether there are formed tight ion pairs, solvent-separated ion pairs, or symmetrically solvated ions. Redox potentials of ions are... 

The electrochemical potential of charged species i, jl is defined as the work done when this species is moved from charge-free infinity to the interior of a homogeneous phase a which carries no net charge [6]. The opposite process is known as the work function and is familiar for the case of removal of an electron from a metal. In fact, the work function for single ions in electrolyte solutions can also be measured experimentally, as described in detail in chapter 8. This means that the electrochemical potential is an experimentally determinable quantity. However, separation of the electrochemical potential into chemical and electrostatic contributions is arbitrary, even though it is conceptually very useful. 

In order to relate the potantial gradient to the chemical potential of neutral species we introduce electrochemical equilibria between neutral and charged species of oxygen ... 

P etc., and is the chemical potential of these species in the different phases. (The summation is over all the chemical species i, except one which is selected as a reference component.) Since in Eq. (1) y is used, the equation is vahd for fluid phases. For phases including a solid dy has to be replaced by surface energy (see -> interfacial tension). For charged interfaces (adsorption of charged species)... 

The general way in which a Galvani potential is established is similar in all cases, but special features are observed at the metal-electrolyte interface. The transition of charged species (electrons or ions) across the interface is possible only in connection with an electrode reaction in which other species may also be involved. Therefore, equilibrium for the particles crossing the interface [Eq. (2.5)] can also be written as an equilibrium for the overall reaction involving all other reaction components. In this case the chemical potentials of aU reaction components appear in Eq. (2.6) (for further details, see Chapter 3). 

For uncharged species z =0 and = jUz. The first term in Eq. (3.1.2) is thus the chemical potential of a charged species in the phase considered. 

However, a more realistic model for the phase transition between baryonic and quark phase inside the star is the Glendenning construction [16], which determines the range of baryon density where both phases coexist. The essential point of this procedure is that both the hadron and the quark phase are allowed to be separately charged, still preserving the total charge neutrality. This implies that neutron star matter can be treated as a two-component system, and therefore can be parametrized by two chemical potentials like electron and baryon chemical potentials [if. and iin. The pressure is the same in the two phases to ensure mechanical stability, while the chemical potentials of the different species are related to each other satisfying chemical and beta stability. The Gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases reads... 

Nevertheless, the chemical potentials of SE s are frequently used instead of the chemical potentials of (independent) components of a crystalline system. Obviously, a crystal with its given crystal lattice structure is composed of SE s. They are characterized much more specifically than the crystal s chemical components, namely with regard to lattice site and electrical charge. The introduction of these two additional reference structures leads to additional balanced equations or constraints (beside the mass balances) and, therefore, SE s are not independent species in the sense of chemical thermodynamics, as are, for example, ( - 1) chemical components in an n-component system. 

If several ( ) charged species i equilibrate across the phase boundary, the set of Eqns. (4.116) has to be solved simultaneously for i = 1,2,..This does not lead to an over-determination of Atpb but ensures that the chemical potentials of the electroneutral combinations of the ions (= neutral components of the system) are constant across the interface. The electric structure (space charge) of interfaces will be discussed later. 

Due to the chemical potential difference for species in the electrolyte and the photoelectrode, and by virtue of the fact that the electrode can be run in forward and reverse bias configurations, a number of important processes at the interface can be discerned. In each case, we will be concerned with the energy required for the process under consideration to occur and its resulting effects on photoelectrode performance. We can think of these processes as being of four basic types chemisorption, the desired electron or hole charge transfer, surface decomposition and electrochemical ion injection. In the rest of the paper we will briefly summarize our present understanding of each. 

This is the general condition of the equilibrium partitioning process. As we show later, it applies to both electrically neutral and electrically charged species. The chemical potential of species X in a phase (gas, solid, or solution) is (c.f. (A. 16), (A. 19), and (A.25))... 

This partial molar change of free energy is called the chemical potential of species i. If this species is charged (i.e., an ion) it is called the electrochemical potential of species i because the change of the free energy of the system includes a component of electrical work. 

Throughout the discussions in Sections 8.15-8.18, we have emphasized methods for obtaining expressions for the chemical potential of a component when we choose to treat the thermodynamic systems in terms of the species that may be present in solution. A complete presentation of all possible types of systems containing charged or neutral molecular entities is not possible. However, no matter how complicated the system is, the pertinent equations can always be developed by the use of the methods developed here, together with the careful definition of reference states or standard states. We should also recall at this point that it is the quantity (nk — nf) that is determined directly or indirectly from experiment. 

A fundamental hypothesis of the Gouy-Chapman theory is that the interaction of an ion with all the other charges can be described by a mean potential Jt(jc), where x is the distance from the surface. The chemical potential of an ion of species T with charge [Pg.383]

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