Butler’s equation

A generalized nonideal mixed monolayer model based on the pseudo-phase separation approach is presented. This extends the model developed earlier for mixed micelles (J. Phys. Chem. 1983 87, 1984) to the treatment of nonideal surfactant mixtures at interfaces. The approach explicity takes surface pressures and molecular areas into account and results in a nonideal analog of Butler s equation applicable to micellar solutions. Measured values of the surface tension of nonideal mixed micellar solutions are also reported and compared with those predicted by the model. 

Rearranging this expression gives the following nonideal analog of Butler s equation (29)... 

The authors have evaluated the surface tension in Equation (6) of A-B binary alloys on the basis of Butler s equation [38] as follows ... 

A direct consequence of the equality of all coi is that the adsorption ratio of two surfactants remains constant when their concentrations are varied in the same proportion, i.e., at constant ci/c2. However, for surfactant molecules with different coj, Butler s equation (2.7) predicts that with increasing surface pressure the adsorption of the smaller molecules is preferentially increased. This has been shown experimentally [16] and is conveniently theoretically illustrated for an ideal surface behaviour [19] by the equation... 

As there is no analogue of Butler s equation for ionised surface layers, the procedure used to derive the equation of state has to be based on the Gibbs adsorption equation and a model adsorption isotherm. The isotherm equation can also be derived from the theoretical analysis of the expressions for electrochemical potentials of ions. For the solution of a single ionic surfactant RX, with the addition of inorganic electrolyte XY, starting from Eqs. (2.2) and (2.21) for the electrochemical potentials, one obtains the adsorption isotherm... 

The application of Butler s equation has been illustrated in form of the equation of state and adsorption isotherm for non-ideal monolayers (with respect to both enthalpy and entropy) comprised of an insoluble 1 and a soluble 2 component. In this case the equation of state is Eq. (2.31), but the adsorption isotherm (2.32) is true for the soluble component 2 only... 

For an ideal monolayer and nj= 1, this expression is reduced to the adsorption isotherm (2.157) derived using Pethica s equation. It can be concluded therefore, that the two approaches lead to similar results. At the same time, Butler s equation (2.7) always leads to a logarithmic form of the equation of state for mixed monolayers, which often disagrees with the experimental results. For these systems, Volmer s or van der Waals equations of state are more appropriate [58, 98]. Therefore the method based on Pethica s equation is advantageous, enabling one to apply semi-empirical model equations of state for mixed monolayers. 

To consider the expression for the standard free energy of adsorption in more detail, one can take into account the contribution of surface tension to the standard chemical potential of the surface layer. This can be done by using Butler s equation (2.7) instead of Eq. (2.2). In this expression both standard chemical potentials of the surface and the bulk depend on temperature and pressure only. From Eqs. (2.11) and (2.2) one obtains Eqs. (2.9), (2.11)-(2.13) instead of Eqs. (2.170)-(2.173). Therefore, the standard free energy of adsorption is given by... 

All systems shown in these figures can be perfectly described by the model defined by Eqs. (3.28)-(3.31) which supports this theoretical model based on Butler s equation for the chemical potentials of the surface layer, and the regular solution theory. In addition, this agreement is due to the certain choice of the dividing surface after Lucassen-Reynders, and to the fact that Eq. (3.31) was used to calculate the mean molar area of the surfactants mixture. It is important to note that in some cases (for mixtures of normal alcohols. Fig. 3.62, and mixtures of sodium dodecyl sulphate (Ci2S04Na) with 1-butanol and 1-nonanol, Figs. 3.63 and... 

By substituting pi, from the Butler s Equation 4.16 into Equation 4.26 and integrating, we can derive... 

The two-step charge transfer [cf. Eqs. (7) and (8)] with formation of a significant amount of monovalent aluminum ion is indicated by experimental evidence. As early as 1857, Wholer and Buff discovered that aluminum dissolves with a current efficiency larger than 100% if calculated on the basis of three electrons per atom.22 The anomalous overall valency (between 1 and 3) is likely to result from some monovalent ions going away from the M/O interface, before they are further oxidized electrochemically, and reacting chemically with water further away in the oxide or at the O/S interface.23,24 If such a mechanism was operative with activation-controlled kinetics,25 the current-potential relationship should be given by the Butler-Volmer equation... 

Butler27 and Volmer28 advanced Tafel s equation by relating overpotentials to activation barriers. The quantitative relationship between current and overpotential is called the Butler-Volmer equation (eqn (32)), and is valid for electrochemical reactions that are rate limited by charge transfer. 

This general equation covers charge transfer at electrified interfaces under conditions both of zero excess field, low excess fields, and high excess fields, and of the corresponding overpotentials. Thus the Butler-Volmer equation spans a large range of potentials. At equilibrium, it settles down into the Nernst equation. Near equilibrium it reduces to a linear / vs. T) (Ohm slaw for interfaces), whereas, if T) > (RT/fiF) (i.e., one is 50 mV or more from equilibrium at room temperature), it becomes an exponential /vs. T) relation, the logarithmic version ofwhich is called Tafel s equation. 

Our chapter has two broad themes. In the first, we will consider some aspects of quantum states relevant to electrochemical systems. In the second, the theme will be the penetration of the barrier and the relation of the current density (the electrochemical reaction rate) to the electric potential across the interface. This concerns a quantum mechanical interpretation of Talel s experimental work of 1905, which led (1924-1930) to the Butler-Volmer equation. 

What conditions would be necessary for (9.38) to give Tafel s law (9.36) and replicate the Butler-Volmer equation (Section 7.2.3) Suppose (as with isotopic reactions) AG° = 0, then,... 

Fick s first and second laws (Equations 6.15 and 6.18), together with Equation 6.17, the Nernst equation (Equation 6.7) and the Butler-Volmer equation (Equation 6.12), constitute the basis for the mathematical description of a simple electron transfer process, such as that in Equation 6.6, under conditions where the mass transport is limited to linear semi-infinite diffusion, i.e. diffusion to and from a planar working electrode. The term semi-infinite indicates that the electrode is considered to be a non-permeable boundary and that the distance between the electrode surface and the wall of the cell is larger than the thickness, 5, of the diffusion layer defined as Equation 6.19 [1, 33] ... 

Charge transport is modeled by Ohm s law (Equation (3.10)) and the charge conservation equation (Equation (3.68)), while the current density distribution at the electrode/electrolyte interface is modeled through the Butler-Volmer equation (Equation (3.102)). It should be noted that, contrarily to Section 3.7, Equation (3.102) is here derived from Equation (3.37) rather than Equation (3.39), because the former allows for a better agreement between experimental and simulated results. Equations (3.40)-(3.42) are used to model, the exchange current density, the activation overpotential, and the ideal potential drop at the electrode/electrolyte interface, respectively. Heat transfer is modeled through Equation (3.6), and the appropriate heat terms for each domain. 

In the numerical model calibration phase, the unknown parameters are those contained in Fick s law and in the Butler-Volmer equation, i.e. the diffusion coefficients representing the porous micro-structural characteristics (e and r), and the electrochemical kinetics parameter (A and Ea). It should be noted that the calibration pro-... 

Here the subscripts s and m denote solid (electrodes and current collectors) and membrane (electrolyte) respectively. Note that these two equations can be treated as only one equation with variable a and source terms. The R s are the volumetric transfer currents due to electrochemical reaction which are non-zero only in the catalyst layers and can be calculated from the Butler-Volmer equation for anode and cathode sides as ... 

The basis of Butler s reasoning can be seen in the following equation, which refers to the rate of formation of charge carriers (electron, holes) at a distance x from the electrode/solution interface ... 

An outline of Butler s theory for the terms of the low surface state (transport-controlled) case is given in Section 10.3.5. Uosaki s 1977 theory of kinetics in the high surface state case was developed in greater detail by Khan (1984). Here, the beginning equation for the steady state (dnx/dt = 0) in the space charge region has a flux independent of distance, so that from the Nemst-Planck equation (4.226) and with dJIdx = 0, one obtains ... 

For n-A isotherms of insoluble monolayes of amphiphilic molecules, assuming the association or dissociation of these molecules in the surface layer, a generalised Volmer equation was derived (based on Butler s and Gibbs equations33 35), which has the form... 

If the adsorption step itself is rate-limiting, one must have available rate expressions for the adsorption and the desorption steps. The flux in (2.108) is then split into two opposing components. Using the notation of Delahay and Mohilner [201,403], there is a forward flux vj, adding to the adsorbate s surface concentration and backward flux tadsorbed substance. These obey rate equations rather analogous to those for electron transfer, the Butler-Volmer equation, in the sense that there are rate constants that are potential dependent. For the forward and backward rates, we have... 

Butler s quantitative development of the theory leads to an equation for the lowering of tension Ay by the organic substance... 

That is, for an irreversible electron-transfer process, the rate-limiting step over a wide range of potentials is the electron-transfer step rather than diffusion. The constant is related to the electrode potential and the standard rate constant, ko, by the Butler-Volmer equation described above. Use of the Butler-Volmer equation and Fick s laws of diffusion enables the voltammetric response of an irreversible process to be understood. 

The potential dependence of both terms Aads and ifdes is described by the well-known Butler-Volmer equation [3.308, 3.309]. Both terms involve the exchange current density, s/j g + ... 

---