The rotating disk electrode (RDE) is shown in Figure 6.20. Here an electrode is imbedded in the end of a cylinder, which rotates submerged in the electrolyte. The electrode is typically a smooth surface surrounded by an insulating material such as Teflon. The RDE merits special attention for several reasons. First, the hydrodynamics for the RDE are well known. Second, the velocity toward the surface of the disk depends only on the distance from the surface and not on the radial position or the angle. Thus, the surface is uniformly accessible to mass transfer, and the rate of mass transfer is constant as long as the bulk concentration does not change with time. Finally, compared to other devices with similar characteristics, experiments with the RDE are relatively simple to prepare and conduct. These features allow the RDE and the rotating ring disk electrode (RRDE) to find frequent use in measuring kinetic and mass-transport properties.

We will not go into the detailed development of the hydrodynamics here as that development is available in numerous references. Flow streamlines are illustrated in Figure 6.20 for laminar flow (Re < 200,000). Analytical solution of the flow field yields the following for the velocity in the z-direction:

(6.42)

where a = 0.51023 and the negative sign indicates that the velocity is toward the surface. Here, ν is the kinematic viscosity. The key result from this is that the z-velocity depends only on distance from the surface and does not depend on r or θ. Since the z-component of the velocity brings reactant to the surface, we might expect that the concentration also depends only on z. From Equation 4.25, the convective diffusion equation for the case of excess supporting electrolyte is

(6.43)

This equation has a relatively straightforward solution. The dimensionless flux at the surface can be expressed in terms of the Sherwood number, a result that follows from the solution of Equation 6.43 under laminar flow conditions. The resulting flux at the surface is uniform under mass-transfer control:

(6.44)

As we saw in Chapter 4, Equation 6.44 can be used to derive an expression for the current density, which is uniform across the surface when mass transfer is controlling:

(6.45)

Application of Equation 6.45 at the limiting current shows that i_lim varies with . This behavior is illustrated in Figure 6.21, and a plot of i_lim versus  yields a line, as shown in Figure 6.22, from which the diffusivity, for example, can be determined. This type of plot is known as a Levich plot.

At conditions below the limiting current, the RDE can be used to measure kinetic parameters. Typically, the current is high enough so that mass transfer is important, but not so high that the limiting current is reached. An example of such an analysis is that of Koutecký–Levich in which the reaction at the surface is assumed to be first order with respect to the concentration of the reactant. We shall investigate the current as a function of rotation speed when the electrode is held at a constant overpotential. Under these conditions,

where k(η) is the rate constant at the specific value of η. The concentration at the interface can be expressed in terms of the limiting current:

Thus,

(6.46)

Writing Equation 6.45 for the limiting current case where the surface concentration is zero,

(6.47)

If there are no mass-transfer limitations (concentration is constant and the surface concentration is equal to the bulk concentration), we can define a kinetic current for the specific value of η.

Substituting this equation into Equation 6.46 and rearranging gives

(6.48)

Since the limiting current is proportional to the square root of rotation speed, a plot of  versus the reciprocal of the square root of rotation speed gives a straight line with slope of  and an intercept of . This relationship is shown in Figure 6.23 where the different lines represent different potentials. The intercept of each of the lines, corresponding to Ω → ∞, represents the current density in the absence of any mass-transfer limitations. These values can be fit to a kinetic expression such as the Butler–Volmer equation. From this same experiment the slope is determined, which can be used to identify either the diffusivity or n for the reaction.

One of the attractive aspects of the Koutecký–Levich analysis is that the kinetic data obtained from the intercept of each of the lines are all at the bulk concentration, which simplifies the fitting. However, there is nothing about the RDE that limits the coupled kinetic mass-transfer analysis to the first order system considered above. The advantage of the RDE is that it provides a relationship between the bulk and surface concentrations, and provides a mass-transfer configuration that is uniform. Data taken with a RDE over a range of potentials and concentrations can be used to fit a kinetic expression that includes nonlinear concentration-dependent terms. A note of caution is appropriate at this point. While the current is uniform under mass-transfer limited conditions, the primary current distribution is not uniform and the secondary distribution may not be uniform. The above kinetic analysis assumes a uniform current distribution, and a nonuniform distribution would introduce error. Consequently, the current distribution should be carefully considered at currents below the limiting current when taking kinetic data.

ILLUSTRATION 6.6

Use the data from Figure 6.23, provided in the table below, to calculate the diffusivity of oxygen. These data represent oxygen reduction in acid media. The solubility of oxygen is 1.21 mol·m⁻³, and the kinematic viscosity is 1.008 × 10⁻⁶ m²·s⁻¹.

Rotationrate[rpm]	Rotationrate[s−1]	i[A·m−2]0.701 V	i[A·m−2]0.682 V	i[A·m−2]0.671 V	i[A·m−2]0.638 V
2500	262	13.33	20.41	26.67	45.45
1600	167	12.66	19.23	24.69	38.46
900	94.2	11.90	17.39	22.22	31.75
400	41.9	10.53	14.71	18.18	23.81

SOLUTION:

We first use the data to calculate the slope of each of the lines. Since each set of data is linear, we simply use the first and last points to estimate the slope of each line as follows:

Average slope (each curve) =

We then average these to get the average slope of the four curves = 0.216 .

For the Koutecký–Levich plot, the slope is equal to , where n = 4 for oxygen reduction, and  is equal to the solubility of oxygen. Substituting in the known values yields