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 routine to minimize the error between the data and the desired BV expression. This process is illustrated for Microsoft Excel in the example that follows:

ILLUSTRATION 3.4

The following data were taken for a NiOOH electrode, which involves a single-electron reaction. Please fit the data to the appropriate kinetic expression.

Overpotential [V]	Current density [A·m−2]
−0.1	−4.20
−0.09	−3.36
−0.08	−2.40
−0.07	−2.30
−0.06	−1.80
−0.05	−1.25
−0.04	−1.00
−0.03	−0.80
−0.02	−0.50
−0.01	−0.22
0.01	0.24
0.02	0.45
0.03	0.80
0.04	1.00
0.05	1.45
0.06	1.80
0.07	2.10
0.08	2.80
0.09	3.50
0.1	4.10

SOLUTION:

We first plot the data to get a feel for what we have. Note that the maximum overpotential is 100 mV, so we would not expect to be able to fit the data with just Tafel expressions for the anodic and cathodic currents.

We first plot the data using linear scales for both the current and potential. We next plot the same data with the absolute value of the current on a log scale.

From the semi-log plot, we see that most of the data are not in the Tafel region, although we can get an estimate of what the Tafel slopes might be from the last few data points that correspond to the highest values of either the anodic or the cathodic current. Therefore, we will fit the data to the full BV equation. We will do this by using the Solver in Microsoft Excel to determine i_o, α_a, and α_c that best fit the data. Please note that the Solver may not come loaded by default, although it is included within the software as delivered.

When loaded, it is found under the “Data” ribbon in Excel. Before demonstrating, we will use the last few points of the cathodic curve to get an estimate of α_c and i_o. Note that the α values are bounded and relatively easy to guess with reasonable accuracy, but i_o can vary over orders of magnitude. Consequently, we will fit the last few points to a Tafel expression to get starting values for the desired parameters. A linear fit of ln |i| versus η_s for the last four points of the cathodic curve yields a slope of −0.0435 and an intercept of −0.0376. The corresponding values of α_c and i_o are 0.59 and 0.42, respectively, where i_o has the same units as the current (A·m⁻²) (see Equation 3.24). These will be our starting estimates for fitting the full BV equation. Since we have a single-electron reaction, we will assume α_a = 1 − α_c as the starting value. The setup on the spreadsheet is as follows:

Excel Solver Fit of BV Equation

Physical Constants (25 C)

F/RT

38.924

Fitting Parameters

αc	0.5900
io	0.4200
αa	0.4100

Overpotential	Measured Current	CalculatedCurrent	Error
−0.1	−4.20	−4.0894	0.111
−0.09	−3.36	−3.2181	0.142
−0.08	−2.40	−2.5200	−0.120
−0.07	−2.30	−1.9586	0.341
−0.06	−1.80	−1.5047	0.295
−0.05	−1.25	−1.1350	0.115
−0.04	−1.00	−0.8306	0.169
−0.03	−0.80	−0.5763	0.224
−0.02	−0.50	−0.3596	0.140
−0.01	−0.22	−0.1704	0.050
0.01	0.240	0.1589	0.081
0.02	0.450	0.3126	0.137
0.03	0.800	0.4670	0.333
0.04	1.000	0.6276	0.372
0.05	1.450	0.7996	0.650
0.06	1.800	0.9883	0.812
0.07	2.100	1.1994	0.901
0.08	2.800	1.4387	1.361
0.09	3.500	1.7130	1.787
0.1	4.100	2.0295	2.071
11.866 Sum of Error Squared

The “Calculated Current” is calculated with the desired BV equation using the parameters above. The “Error” is simply the difference between the calculated value and the measured value. The Sum of Error Squared was calculated with the SUMSQ function in Excel. Solver will minimize this value by changing the fitting parameters. The fitting is done by selecting the cell containing the sum of the squared errors (initial value 11.866), choosing Solver under Data in Excel, and then using the Solver to minimize the value of this computational cell by varying the three fitting parameters. The resulting values are shown in the following table:

αc	0.5035
io	0.5893
αa	0.5087

The Solver fit reduced the error from 11.866 to a final value of 0.186. Note that the fitting did not constrain α_a + α_c = 1. Therefore, the two parameters were fit independently and validated the fact that we have a single-electron reaction. The above process works well as long as the values of the current do not vary widely over orders of magnitude. In those cases, errors associated with the high current values are weighted more heavily and bias the fit. The bias can be offset through the use of a normalized error, but it is better to just use a Tafel fit, since you would undoubtedly have data in the Tafel region and a Tafel fit adequately handles the wide range of data.

3.7 The Influence of Mass Transfer on the Reaction Rate

As noted above, the exchange-current density and, hence, the current depends on concentration at the surface (see Equation 3.20). Species can be consumed and/or generated at the surface, leading to surface concentrations that can vary significantly from the bulk concentrations, as illustrated in Figure 3.8. In this section, we consider the situation where the reaction rate depends linearly on the reactant concentration at the surface. As the applied potential is increased, the reaction rate increases, the surface concentration decreases, and the mass transfer resistance to the surface becomes important. A simplified expression for mass transfer can be used in the presence of supporting electrolyte, as will be discussed in the next chapter, or for a reactant that is not charged. Under such conditions, the rate of mass transfer of the reactant to the surface can be approximated as

(3.31)

Normally the mass-transfer coefficient is represented by k_c. In this chapter, k_m is used for clarity. At steady state, the rate of mass transfer must equal the reaction rate at the surface, where the Tafel equation is assumed to apply:

(3.32)

For simplicity, we assume  These equations can be used to eliminate the unknown surface concentration. Solving for the current density yields

(3.33)

The expression in Equation 3.33 reflects the fact that the current depends on two resistances in series, one associated with mass transfer and the other with the reaction at the surface. If the kinetics is slow relative to the mass transfer, the second term in the denominator is large relative to the first, and i is limited by kinetics. In contrast, if the mass transfer is slow relative to the kinetics, the first term in the denominator dominates (k_m is small, so its reciprocal is large) and the current depends only on mass transfer. In between those extremes, both resistances influence the current density. The overall impact is shown in Figure 3.9, where the current rises exponentially (kinetically dominated) and then levels off due to mass transport limitations. At high overpotentials, the reaction is no longer limiting, and the current depends only on mass transfer, which is not affected by the overpotential. At high values of surface overpotential, the reaction at the surface is so fast relative to the rate of mass transfer that the reactant is consumed at the surface as quickly as it arrives, yielding a surface concentration of the reactant that is essentially zero. The mass transfer limited or limiting current can be calculated from the transport expression by assuming a zero concentration at the surface:

(3.34)

where n is the number of electrons transferred per mole of  reacted as determined from the electrochemical reaction. In practice, there are many industrial reactions that are carried out at the limiting current in supporting electrolyte.

The above problem is difficult to solve explicitly when the dependency of i_o on concentration is nonlinear (e.g., a fractional power). We also allow i_o to be known at a concentration other than the bulk concentration. In such cases, the following equation can be solved numerically for the surface concentration over a range of overpotentials:

(3.35)

Once the surface concentration is known, either the mass transport expression or the rate expression can be used to calculate the current.

Although the expression for mass transfer has intentionally been kept simple, this section provides an initial exposure to the limiting current, critical to the design and operation of electrochemical processes. A more detailed treatment of transport is found in the next chapter.