Author: admin
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Transference Numbers, Mobilities, and Migration
In Equation 4.3, both the ion mobility and a diffusion coefficient appear. Both the concentration and the potential impact the electrochemical potential (μi). While the details are beyond the scope of this text, it is the gradient of the electrochemical potential that is the true driving force for transport. Therefore, we might expect the diffusivity and mobility…
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Conservation of Material
In order to solve most transport problems, expressions for the flux, such as the Nernst–Planck equation, are incorporated into material balances or conservation equations. Here we derive a balance for a single species over a control volume of size ΔzΔxΔy as shown in Figure 4.3. The balance takes the form (4.9) The rate of accumulation in the…
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Nernst–Planck Equation
The most widely used expression for the flux in electrochemical systems is the Nernst–-Planck equation, (4.3) The flux of species, i, is the combination of three terms: migration, diffusion, and convection. The Nernst–Planck equation is similar to Equation 4.2 but adds a contribution that arises from the gradient in electrical potential called migration. For charged species, the force…
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Fick’s Law
The transport of material by diffusion is due to the random thermal movement of molecules and is described by Fick’s law: (4.1) where Ji is the molar flux [mol m−2 s−1] of species i. The flux represents the rate at which material passes through a plane of unit area. It is a vector quantity with both direction and magnitude. In…
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Current Efficiency
Several efficiencies are used to characterize electrochemical processes and systems. The faradaic efficiency was introduced in Chapter 1. Here, we present the current efficiency, which is slightly different: (3.39) Undesired reactions occur in both electrolytic and galvanic cells. These unwanted side reactions reduce the current efficiency. We can explore this concept with the charging of a lead–acid…
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Use of Kinetic Expressions in Full Cells
The purpose of this section is to help develop some initial experience and intuition with full electrochemical cells. By a full cell, we simply mean an electrochemical cell with the anode and cathode separated by some distance by an electrolyte. What’s more, we will consider the potential of the cell, namely, the potential of the…
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Direct Fitting of the Butler–Volmer Equation
In situations where current−voltage data are available only over a limited range that is not adequately addressed by one of the limiting cases above, a direct fit of the data to the full BV equation may be appropriate. This fitting can be done in a straightforward manner with use of a nonlinear solver or optimization…
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Simplified forms of the Butler–Volmer Equation
Tafel Approximation The BV equation has two exponential terms, one that represents the anodic current (i > 0, ηs > 0) and the other that represents the cathodic current (i < 0, ηs < 0). What happens to the relative magnitude of the two terms as ηs becomes more positive? What about as ηs becomes more negative? When ηs is large and positive, the anodic term of the BV equation dominates…
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Reaction Fundamentals
We will find the Butler–Volmer formulation of kinetics extraordinarily useful for the study of electrochemical systems. Nonetheless, it is largely a phenomenological or empirical equation with three parameters, io and two transfer coefficients, αa and αc. We saw vast variations in exchange-current densities based on the reactions and the electrode surface. What causes one reaction to be facile and others…
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Use of the Butler–Volmer Kinetic Expression
The purpose of this section is to help you understand and learn how to use Equation 3.17, the Butler–Volmer equation. The BV equation is a relationship between the current density (i) and the charge transfer or surface overpotential (ηs). It contains three parameters: the exchange-current density (io), the anodic charge transfer coefficient (αa), and the cathodic…