In Section 4.3, we presented the following expression for the cell potential during discharge in the absence of concentration gradients:

(4.53)

We also examined how to solve problems with concentration gradients at various levels of approximation. The cell voltage comes naturally from the solution of the coupled equations for the potential field and concentrations. However, there are situations where we would like to estimate the cell voltage in the presence of concentration gradients without solving the fully coupled equations. In doing so, we assume that the other voltage drops associated with the reactions and the ohmic drop in solution still apply. We also assume that concentration gradients are limited to a mass-transfer boundary near the electrode surfaces. The task, then, is to determine the cell voltage that pertains to a certain current. Most frequently, the Tafel approximation is used to facilitate calculation of the surface overpotential for each reaction at the desired current. The ohmic drop between the electrodes is calculated assuming a constant conductivity at the bulk concentration, which accurately represents most of the domain between the electrodes. Following the treatment presented by Newman and Thomas, we now define a concentration overpotential to calculate the impact of the concentration gradients near each of the electrodes on the cell voltage:

(4.54)

where y is the distance from the electrode of interest, ∞ represents the bulk solution, i is positive for anodic current and negative for cathodic current, and activity coefficients have been neglected. The first term on the right side accounts for the error associated with use of the bulk concentration to calculate the ohmic drop near the electrode surface. The second term is the equilibrium potential difference between the electrode of interest and a reference electrode of the same type located in the bulk electrolyte. It is thermodynamic in nature and is commonly referred to as a concentration cell. The species of interest are only those that participate in the electrochemical reactions. Note that the stoichiometric coefficients in this term correspond to the cathodic reaction when calculating the concentration overpotential associated with either the cathode or the anode, consistent with the convention described in Chapter 3. The last term is the diffusion potential, which accounts for the fact that differences in the transport properties of the ions in solution impact the potential field. This last term is summed over all the charged species in solution. Because we have used the normal from the surface for both electrodes, concentration gradients that are in the same physical direction have the opposite sign at the anode and cathode.

It is possible for the terms in Equation 4.54 to have different signs, and hence contribute in opposite directions to the concentration overpotential. The sign of the cathodic and anodic concentration overpotentials, themselves, have significance. The concentration overpotential contributes in a negative way to the overall cell potential when its value at the cathode is negative and its value at the anode is positive.

If we assume a stagnant diffusion layer of thickness δ with linear variation of the concentration, the concentration gradients in the last term can be approximated as

(4.55)

Equation 4.54 then becomes

(4.56)

When there is excess supporting electrolyte, the conductivity in the boundary layer does not change significantly, and the first term in Equation 4.56 can be neglected. Also, since the conductivity scales linearly with concentration, the last term scales as the concentration of the minor species divided by that of the supporting electrolyte and is likewise small. With these assumptions, the concentration overpotential in the presence of a supporting electrolyte becomes

(4.57)

This is the equation most commonly found in the literature and the one that we will use most frequently in this text. The concentration overpotential expressed in Equation 4.57 will always be negative at the cathode and positive at the anode since the concentration of reactants and products at the surface will always be depleted or enriched, respectively, relative to the bulk. Consequently, Equation 4.57 represents a potential loss for both the anode and cathode. Therefore, the appropriate expressions for the cell potential during galvanic operation (discharge) and electrolysis (charge) become

(4.58a)

(4.58b)

One of the challenges with all of the above expressions for the concentration overpotential is that they require knowledge of the concentration at the surface of the electrode, which is a quantity that is not readily accessible. In the presence of excess supporting electrolyte where the bulk concentration is known, the current is related to transport across the mass-transfer boundary layer as follows:

(4.59)

Equation 4.59 applies to all species that participate in the reaction. As before, we treat this equation as a scalar rather than a vector. In doing so, we note that reactants are transported to the surface and have a surface concentration that is less than the bulk, while the opposite is true for products of the reaction. The n in this equation represents the number of equivalents per mole of the species of interest. Once the surface concentrations are known, Equation 4.57 can be used to calculate the concentration overpotential.

Finally, if there is only one species in the electrolyte that participates in the reaction (e.g., a metal cation that is deposited on the electrode) and the limiting current is known, the surface concentration can be written as

(4.60)

where  is most often zero, but can be the saturation concentration in situations where the surface concentration is higher than the bulk.

ILLUSTRATION 4.8

Consider an electrorefining cell similar to that of Illustration 4.7. In this case, however, there is forced convection (instead of natural convection) and Sh is dependent on the hydraulic diameter. The following data are known:

1. Electrolyte composition: 0.25 M CuSO₄ and 1.5 M H₂SO₄
2. Sherwood number (based on hydraulic diameter): 1200
3. Electrode dimensions: cell gap = 0.05 m, width = 0.5 m, height = 0.96 m
4. Operating current density: 200 A·m⁻²
5. Cupric ion diffusivity: 5.33 × 10⁻¹⁰ m²·s⁻¹

Kinetic parameters for copper reaction (apply at both electrodes):

1. i_o = 0.001 A·cm⁻²
2. γ = 0.42
3. 
4. α_a = 1.5
5. α_c = 0.5

Determine the surface and concentration overpotentials at both the anode and the cathode. Also determine the cell potential at the specified current density.

SOLUTION:

1. We need to find the concentration of cupric ions at both electrode surfaces. To do this, we will use Equation 4.59 to relate the current to the surface concentration, assuming that the bulk concentration is known.The characteristic length for Sh is d_h = 4A_c/P = 4(0.05 m)(0.5 m)/(2(0.05 m + 0.5 m)) = 0.0909 mFor the cathode:Assuming that the same mass-transfer coefficient and properties apply at the anode, c_surf,a = 397.3 mol·m⁻³.
2. To calculate the concentration overpotential, we need the value of s_i. The reaction written in the cathodic direction isTherefore, . This same value is used at both the anode and cathode since, in both cases, we are calculating the difference between the potential at the surface and that of a reference electrode of the same type as the working electrode located in the bulk solution.
3. The concentration overpotentials can now be determined:
   1. Cathode:    
   2. Similarly, for the anode: . Neither of these values is very large.
4. We can also calculate the surface overpotentials:From the BV equation with use of a nonlinear solver, .Similarly, for the anode,    From the BV equation with use of a nonlinear solver, .
5. We can now calculate the potential of the cell.From Equation 4.58b,  Equation 4.58b is used because this is an electrolytic cell. We will neglect the ohmic drop because of the supporting electrolyte. In practice, there will be a small ohmic drop, depending on the relative concentrations of the species in solution.

In summary, in this section we developed an expression for the concentration overpotential that can be used in conjunction with the surface overpotentials and ohmic loss to estimate the potential of a cell at a given current density, as illustrated previously. It applies in the presence of flow since it assumes that the concentration gradients occur near the electrodes across the respective boundary layers.