In the Classroom
Electric Potential Distribution in an Electrochemical Cell Pierre Millet Laboratoire des Composés Non Stoechiométriques, Université de Paris Sud, Bâtiment 415, 91405 Orsay Cedex, France
Many students learning electrochemistry do not understand how current flows from one electrode to the other in an electrochemical cell. This is generally because interfacial phenomena and ionic conductivity in the electrolytic solution tend to be treated separately by chemistry teachers. An easy way to improve students’ understanding of “how it works” is to compute the two-dimension electric potential distribution in the cell and to make a gravitational analogy. Current flows from one electrode to the other just as a ball would do in a gravitational field.
i
∂ϕ ∂x
– x + dx
∂ϕ ∂x
=0
When dx → 0, we can write:
∂ 2ϕ ∂ 2ϕ ∂ 2ϕ + + =0 ∂x 2 ∂y 2 ∂z 2
∂ 2ϕ ∂x 2
∂ 2ϕ ∂y2
Let us consider a one-dimension problem. Under steady state conditions, the current passing at a distance x from the electrode surface is equal to the current passing at x + dx :
(5)
where each term of the sum can be different from zero. Given adequate boundary conditions (electrode potentials), eq 5 can be solved by using the finite difference method. To simplify the treatment, eq 5 is solved in two dimensions (the term ∂2ϕ/∂z2 is not considered). In the finite difference method (1), the region of interest (the interpolar volume “R”) is replaced by a horizontal grid (Fig. 1b). The grid consists of equidistant straight lines whose intersections are called mesh points. Then we use a difference equation approximating the partial differential equation (5), by which we relate the unknown values of ϕ at the mesh points to each other and to the given boundary values.
Primary Current Distribution
(1)
(4)
By considering the three-dimensional problem, one obtains the Laplace equation:1
cathode: H + + 1e2 → 1⁄2 H2 If the aim of the study is the anodic reaction, then the anode will be called the working electrode and the cathode the counter electrode.
The flux–force relation that applies to any electrochemical system is given by eq 1, where i is the current density, ϕ is the electric potential, and λ is the electric conductivity of the solution:
(3)
x
∂ 2ϕ =0 ∂x 2
An electrochemical cell can be used in two different ways: first, to make chemical transformations (an electrolyzer in which electric power is supplied to the cell); and second, to generate electric power (a generator supplying power to the user). We will consider here an electrolysis cell. At the laboratory scale, most electrochemical research is done in a glass cell as pictured in Figure 1a. Three different (most of the time metallic) electrodes are used: (i) the working electrode (WE), which is the electrode under study; (ii) the counter electrode (CE), which is used to collect electrons, and (iii) the reference electrode (REF), which is used to measure the potential difference between the working electrode and the electrolytic solution. The working and counter electrodes are disposed in parallel in the electrolytic solution and are connected to an external power supply device. Let us consider the case of water electrolysis in aqueous sulfuric acid solution using platinum electrodes. The electrode reactions are: anode: H2O → 2 H+ + 1⁄2 O2 + 2e2
956
(2) x
Combination of eqs 1 and 2 gives eq 3:
Description of the Cell
i = – λ grad ϕ
=i x + dx
= y
= x
ϕ x + 1,y + ϕ x – 1,y – 2 ϕ x,y ∆x 2 ϕ x, y+ 1 + ϕ x,y– 1 – 2 ϕ x,y ∆y2
(6)
(7)
This yields a system of linear algebraic equations. By solving it, we obtain approximations to the unknown values of ϕ at the mesh points. Figure 2 shows the solution of eq 5 when a potential difference of 2 V is applied between WE and CE. Secondary Current Distribution In the case of a primary current distribution as treated above, the rate at which electrons are transferred from (or to) the electrode is not considered. A more realistic picture of the potential distribution in the cell is
Journal of Chemical Education • Vol. 73 No. 10 October 1996
In the Classroom
obtained when the rate of electron transfer is taken into consideration. In electrochemical applications, high current densities are sought in order to improve the cell efficiency. This departure from equilibrium has a cost: the overvoltage. The overvoltage “η” at any given metal–solution interface where electrons are transferred is defined as the potential difference between the equilibrium potential and the rating potential. The product (η × i) is homogeneous to the J cm–2 scale and corresponds to a loss in the cell efficiency: heat is generated and exchanged with the surroundings. The relation between overvoltage and current density is given by the Butler–Volmer relation (2–4):
βnF i = i 0 exp α n F η – exp – η RT RT
(8)
where i0 is the exchange current density, α and β are the symmetry factors (related to the activation energy of the charge transfer process), n is the number of electrons exchanged in the process, F is the Faraday constant (96,487 C mol–1 ), R is the gas constant (8.32 Pa m3 mol –1 K–1 ), and T is the temperature (K). In most cases, especially when the cell is rated at elevated current densities, the reverse term of the reaction can be neglected. This results in the simplification of eq 8, in which one of the two exponentials can be neglected. Coupling of eqs 5 and 8 at the electrode–electrolyte interface results in a nonlinear relation that can be solved numerically using a Newton–Raphson method (5, 6). Results obtained in the case of water electrolysis are shown in Figure 3. The cathodic process is fast (a low overvoltage has to be applied to obtain a substantial current density), whereas the anodic one is slow. As a consequence, it can be seen in Figure 3 that the distribution of electric potential in the vicinity of the cathode is similar to that obtained in Figure 2 in the case of a primary current distribution. On the contrary, a very strong potential drop is observed between the anode and the electrolyte because the anodic process is slow.
Figure 1. (a) Schematic diagram of an electrochemical cell. WE = working electrode; CE = counter electrode; REF = reference electrode; G = gas (to de-aerate the cell). (b) 2-D grid. The electric potential is computed at each mesh point.
Effect of Size and Position of Electrodes on the Potential Distribution The distribution of electric potential in the cell will be altered by any modification of the geometry of the system. We shall consider the size of the electrodes and their position in the cell.
Figure 2. 2-D electric potential distribution in an electrochemical cell. Solution to the Laplace equation.
Size of the Electrodes Figure 4 shows the distribution of electric potential in two different cases. In Figure 4a, the two electrodes have the same area. In Figure 4b, one of the electrodes is twice the area of the other. It can be seen from Figure 4b that the distribution of electric potential is more regular in the vicinity of the smallest electrode. This is why the counter electrode is generally larger than the working electrode. Position of the Electrodes The distribution of electric potential in an electrochemical cell is directly related to the position of the electrodes in the cell. Figure 5 shows the result when we consider two square electrodes perpendicular one to another. A strong distortion of the electric field is observed, in direct relation with the disposition of the electrodes.
Figure 3. 2-D electric potential distribution in an electrochemical cell. Secondary current distribution. ηcathode = 17.5 ln i + 7.4; ηanode = 17.4 ln i – 6.9.
Vol. 73 No. 10 October 1996 • Journal of Chemical Education
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In the Classroom
Summary The way current flows from one electrode to the other in an electrochemical cell is easier to understand by presenting the calculated 2-D electric potential distribution in the cell. The curves obtained can be compared to gravitational fields, and the electrons (or ions) are analogous to small balls jumping out of the anode and falling downhill to the cathode. It appears clearly from the figures that the current can go from the back of the anode to the back of the cathode, even though it flows most directly through the interpolar region. The current lines flow close to the boundary of the cell and the entire solution is affected. From these figures, it is easy to understand that the position of the reference electrode in the cell must be chosen carefully in order to measure the interfacial overvoltage precisely. This is especially the case for fast electrochemical reactions where the overvoltage is small in comparison with the electric conductivity of the solution. The size and the position of the electrodes in the solution is also of importance and determines the distribution of electric potential.
(a)
Notes 1. Pierre Simon, marquis de Laplace (1749–1827), French mathematician, was a professor in Paris. He developed the foundation of potential theory and made important contributions to celestial mechanics, astronomy in general, special functions, and probability theory. Napoleon Bonaparte was his student for a year.
Literature Cited 1. Kreyszig, E., Advanced Engineering Mathematics, 5th ed.; John Wiley & Sons: New York, 1983. Tafel, J. Z. Physik. Chem. 1905, 50, 641. Butler, J. Trans. Faraday Soc. 1924, 19, 729. Erdey-Gruz, T.; Volmer, M. Z. Physik. Chem. 1930, 150, 203. Ortega, J.; Rheinboldt, W. Iterative Solution of Nonlinear Equations in Several Variables; Academic: New York, 1970. 6. Press, W. H.; Flannery, B. P.; Teukolsky S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University: Cambridge, 1986.
(b)
Figure 4. 2-D electric potential distribution in an electrochemical cell. Effect of the electrode size. (a) Same size. (b) Factor 2 difference in size.
2. 3. 4. 5.
Figure 5. 2-D electric potential distribution in an electrochemical cell. Effect of the position of the electrodes: the two electrodes are perpendicular.
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Journal of Chemical Education • Vol. 73 No. 10 October 1996
