Electroanalytical Chemistry. Gary A. Mabbott

the capacitance of the dielectric medium separating the plates. (Here is the reason that the model is an over‐simplification. The double layer actually has a capacitance that varies with both the electrolyte concentration and with the double layer potential. However, this model does give results that set some upper limits. That is useful.) Empirically, the capacitance of the interface between a metal electrode in an aqueous salt solution is typically in the range of 10–40 μF/cm² [9]. For convenience, a value at the midpoint of that range will be used, i.e. 25 μF/cm² or 25 × 10⁻⁶ C/(V cm²), to estimate the charge in this example. (Notice that the units on the value for the capacitance indicate that total charge depends on the electrode area.)

      (1.22)

      Because there are 9.6485 x 10⁴ C/mole of charge:

      (1.23)

      The calculation indicates that 2.5 × 10⁻¹⁰ mol of anions will be required to charge the double layer to a potential of 1.0 V. Thus, on a mole basis, the number of ions required to charge the solution side of the interface is tiny. For comparison, consider the number of moles of anions present next to a square electrode 1 cm on a side. Consider the chloride ions in a volume of 1 cm³ solution of 0.1 M NaCl.

      (1.24)

      (1.25)

      Charging the electrode to 1.0 V would require less than 0.0003% of the chloride ions from the surrounding milliliter of solution to be recruited into the double layer. Clearly, that amount represents a negligible loss to the Cl⁻ concentration in the neighboring solution.

For every potential difference that appears across the double layer, there is a corresponding arrangement of charge. If the number of charges changes, the double layer potential changes. Likewise, if one is applying a voltage to the interface, then one must move electrons and ions to establish any new arrangement of charge. Because the movement of charge constitutes a current, then a current will exist until the new arrangement of charge is established. This phenomenon is called the double layer charging current. It can be a problem in some voltammetry experiments, because the signal current may be much smaller than the double layer charging current. In applied potential techniques one is usually interested in measuring the current that is related to the amount of analyte that is being oxidized or reduced at the electrode interface. The signal current associated with the oxidation or reduction of a chemical species is called a Faradaic current because the charge exchanged between the electrode and the electroactive species in solution is proportional to the number of moles of analyte that is oxidized or reduced according to Faraday's law (Q = ∫ i dt = nFN). The double layer charging current is non‐Faradaic; it represents a background component that one must remove from the signal in order to perform quantitative analyses. Methods for circumventing the double layer charging current are described in Chapter 5 on controlled potential techniques.

1.5 Conductance

Electrical conductance is a measure of the ability to carry current. Resistance is defined as the reciprocal of conductance. It is easily measurable. Because the measurement of the resistance of a solution depends on the area of the electrodes and the distance separating them, the standard method uses two square platinum plates, 1 cm on each edge separated by 1 cm of solution (see Figure 1.11).

Figure 1.11 Conductance cell.

      Of course, the interface between the solution and each plate develops an electrical double layer. As a consequence, the electrochemical cell behaves as a circuit with two capacitors in addition to the solution resistance. The resistance is measured using a special meter that applies an oscillating voltage to the electrodes and measures the current response. The resistance component has to be extracted from the response. The resulting resistance is called the specific resistance of the solution, ρ, and has the units of Ω cm. The electrical resistance for any other arrangement of electrodes is proportional to the length, ℓ, of solution between the electrodes of area, A.

      (1.26)

      where ρ is the proportionality constant. Because conductance is inversely related to resistance one can define the conductance, G in Siemens, as follows:

      (1.27)

where κ is the electrical conductivity of the solution in units of Ω⁻¹ cm⁻¹ or S cm⁻¹. Although the standard method defines the shape and separation for electrodes, commercial instruments often have a different geometry and correct for differences by applying a calibration factor.

      The solution resistance and conductance also varies with temperature [13].

(1.28)

      where T = the solution temperature in °C and r is a temperature coefficient in Siemens/degree for the solution. The temperature coefficient needs to be evaluated for different electrolyte solutions, but a representative value is r = 0.0191 for a 0.01 M KCl solution [13].

      The conductance of a solution also depends on the type of ions that make up the electrolyte. The important point here is that ions move at different speeds. Ions move by diffusion, the process that is conceptualized as a random walk of individual particles, but under the influence of an electric field, they also migrate in the direction of the oppositely charged electrode. The velocity of an ion caused by an electric field is sometimes called the drift velocity or the migration velocity. It is proportional to the strength of the electric field, ε, driving the current.

      (1.29)

      where the electric field, ε, has the units of V/cm. It is the voltage difference between the electrodes divided by the distance between them. v is the drift velocity of the ion in cm/s and the proportionality constant, u, is the ion mobility. The units for the ion mobility are cm²/(s V). The reason that the mobilities vary among ions is the fact that collisions with solvent molecules and other particles cause drag. Drag is related to the size of the ion. In this context, the size of the ion includes the sheath of solvent molecules that the ion drags with it, its solvent sphere. The bigger the solvated ion, the greater the viscous drag force opposing the ion's movement. All ions are slowed down by the viscosity of the solution. Because the viscosity decreases with temperature, the conductance increases with temperature as indicated in Eq. (1.28). The ion mobility is also proportional to the charge on the ion. Also, the bigger the charge, the greater the tug that the electric field exerts on the ion. Each ion carries a fraction of the total current in proportion to its mobility and its contribution to the total number of charges in solution. An important consequence of the variation in ion mobilities is the fact that the current is shared unevenly among the ions.

      There are two practical applications of conductance or conductivity measurements that are of interest to analytical chemists. The first application is a semiquantitative estimate of ion concentration. One can calibrate conductivity