11.3: Coulometric Methods

Selecting a Constant Potential

To see how an appropriate potential for the working electrode is selected, let’s develop a constant-potential coulometric method for Cu²⁺ based on its reduction to copper metal at a Pt working electrode.

\[\ce{Cu^2+}(aq) + 2e^- ⇋ \ce{Cu}(s)\tag{11.28}\]

Figure 11.28 shows a ladder diagram for an aqueous solution of Cu²⁺. From the ladder diagram we know that reaction 11.28 is favored when the working electrode’s potential is more negative than +0.342 V versus the standard hydrogen electrode. To ensure a 100% current efficiency, however, the potential must be sufficiently more positive than +0.000 V so that the reduction of H₃O⁺ to H₂ does not contribute significantly to the total current flowing through the electrochemical cell.

Figure 11.28 Ladder diagram for an aqueous solution of Cu²⁺ showing steps for the reductions of O₂ to H₂O, of Cu²⁺ to Cu, and of H₃O⁺ to H₂. For each step, the oxidized species is in blue and the reduced species is in red.

We can use the Nernst equation for reaction 11.28 to estimate the minimum potential for quantitatively reducing Cu²⁺.

\[E = E^\circ_\mathrm{Cu^{2+}/Cu} - \dfrac{0.05916}{2}\log\dfrac{1}{[\mathrm{Cu^{2+}}]}\tag{11.29}\]

If we define a quantitative electrolysis as one in which we reduce 99.99% of Cu²⁺ to Cu, then the concentration of Cu²⁺ at t_e is

\[[\ce{Cu^2+}]_{t_\ce{e}} = 0.0001×[\ce{Cu^2+}]_0\tag{11.30}\]

where [Cu²⁺]₀ is the initial concentration of Cu²⁺ in the sample. Substituting equation 11.30 into equation 11.29 allows us to calculate the desired potential.

If the initial concentration of Cu²⁺ is 1.00 × 10^–4 M, for example, then the working electrode’s potential must be more negative than +0.105 V to quantitatively reduce Cu²⁺ to Cu. Note that at this potential H₃O⁺ is not reduced to H₂, maintaining 100% current efficiency.

Minimizing Electrolysis Time

In controlled-potential coulometry, as shown in Figure 11.27, the current decreases over time. As a result, the rate of electrolysis—recall from Section 11A that current is a measure of rate—becomes slower and an exhaustive electrolysis of the analyte may require a long time. Because time is an important consideration when choosing and designing analytical methods, we need to consider the factors affecting the analysis time.

We can approximate the change in current as a function of time in Figure 11.27 by an exponential decay; thus, the current at time t is

\[i = i_0e^{-kt}\tag{11.31}\]

where i₀ is the current at t = 0 and k is a rate constant that is directly proportional to the area of the working electrode and the rate of stirring, and that is inversely proportional to the volume of solution. (Problem11.16 asks you to explain why a larger surface area, a faster stirring rate, and a smaller volume leads to a shorter analysis time.) For an exhaustive electrolysis in which we oxidize or reduce 99.99% of the analyte, the current at the end of the analysis, t_e, is

\[i ≤ 0.0001 × i_0\tag{11.32}\]

Substituting equation 11.32 into equation 11.31 and solving for t_e gives the minimum time for an exhaustive electrolysis as

From this equation we see that a larger value for k reduces the analysis time. For this reason we usually carry out a controlled-potential coulometric analysis in a small volume electrochemical cell, using an electrode with a large surface area, and with a high stirring rate. A quantitative electrolysis typically requires approximately 30–60 min, although shorter or longer times are possible.

Instrumentation

A three-electrode potentiostat is used to set the potential in controlled-potential coulometry. The working electrodes is usually one of two types: a cylindrical Pt electrode manufactured from platinum-gauze (Figure 11.29), or a Hg pool electrode. The large overpotential for the reduction of H₃O⁺ at Hg makes it the electrode of choice for an analyte requiring a negative potential. For example, a potential more negative than –1 V versus the SHE is feasible at a Hg electrode—but not at a Pt electrode—even in a very acidic solution. Because mercury is easily oxidized, it is less useful if we need to maintain a potential that is positive with respect to the SHE. Platinum is the working electrode of choice when we need to apply a positive potential.

Figure 11.29 Example of a cylindrical Pt-gauze electrode for controlled-potential coulometry. The electrode shown here has a diameter of 13 mm and a height of 48 mm, and is fashioned using Pt wire with a diameter of approximately 0.15 mm. The electrode’s surface has 360 openings/cm² and a total surface area of approximately 40 cm².

The auxiliary electrode, which is often a Pt wire, is separated by a salt bridge from the analytical solution. This is necessary to prevent the electrolysis products generated at the auxiliary electrode from reacting with the analyte and interfering in the analysis. A saturated calomel or Ag/AgCl electrode serves as the reference electrode.

The other essential instrumental need for controlled-potential coulometry is a means for determining the total charge. One method is to monitor the current as a function of time and determine the area under the curve, as shown in Figure 11.27. Modern instruments use electronic integration to monitor charge as a function of time. The total charge at the end of the electrolysis is read directly from a digital readout.

Electrogravimetry

If the product of controlled-potential coulometry forms a deposit on the working electrode, then we can use the change in the electrode’s mass as the analytical signal. For example, in we apply a potential that reduces Cu²⁺ to Cu at a Pt working electrode, the difference in the electrode’s mass before and after electrolysis is a direct measurement of the amount of copper in the sample. We call an analytical technique that uses mass as a signal a gravimetric technique; thus, we call this electrogravimetry.

Maintaining Current Efficiency

To illustrate why a change in the working electrode’s potential may result in a current efficiency <100%, let’s consider the coulometric analysis for Fe²⁺ based on its oxidation to Fe³⁺ at a Pt working electrode in 1 M H₂SO₄.

\[\ce{Fe^2+}(aq) ⇋ \ce{Fe^3+}(aq) + e^-\]

Figure 11.31 shows the ladder diagram for this system. At the beginning, the potential of the working electrode remains nearly constant at a level near its initial value. As the concentration of Fe²⁺ decreases, the working electrode’s potential shifts toward more positive values until the oxidation of H₂O begins.

\[\ce{2H2O}(l)⇋\ce{O2}(g) + \ce{4H+}(aq) + 4e^-\]

Because a portion of the total current comes from the oxidation of H₂O, the current efficiency for the analysis is less than 100% and we cannot use equation 11.25 to determine the amount of Fe²⁺ in the sample.

Figure 11.31 Ladder diagram for the constant-current coulometric analysis of Fe²⁺. The red arrow and text shows how the potential drifts to more positive values, decreasing the current efficiency.

Although we cannot prevent the potential from drifting until another species undergoes oxidation, we can maintain a 100% current efficiency if the product of that oxidation reacts both rapidly and quantitatively with the remaining Fe²⁺. To accomplish this we can add an excess of Ce³⁺ to the analytical solution. As shown in Figure 11.32, when the potential of the working electrode shifts to a more positive potential Ce³⁺ eventually begins to oxidize.

\[\ce{Ce^3+}(aq) ⇋ \ce{Ce^4+}(aq) + e^- \tag{11.33}\]

The Ce⁴⁺ that forms at the working electrode rapidly mixes with the solution where it reacts with any available Fe²⁺.

\[\ce{Ce^4+}(aq) + \ce{Fe^2+}(aq) ⇋ \ce{Ce^3+}(aq) + \ce{Fe^3+}(aq)\tag{11.34}\]

Combining reaction 11.33 and reaction 11.34 shows that the net reaction is the oxidation of Fe²⁺ to Fe³⁺

\[\ce{Fe^2+}(aq) ⇋ \ce{Fe^3+}(aq) + e^-\]

maintaining a current efficiency of 100%. A species, such as Ce³⁺, which is used to maintain 100% current efficiency, is called a mediator.

Figure 11.32: Ladder diagram for the constant-current coulometric analysis of Fe²⁺ in the presence of a Ce³⁺ mediator. As the potential drifts to more positive values, we eventually reach a potential where Ce³⁺ undergoes oxidation. Because Ce⁴⁺, the product of the oxidation of Ce³⁺, reacts with Fe²⁺, we maintain current efficiency.

Endpoint Determination

Adding a mediator solves the problem of maintaining 100% current efficiency, but it does not solve the problem of determining when the analyte's electrolysis is complete. Using our same example, when the oxidation of Fe²⁺ is complete current continues to flow from the oxidation of Ce³⁺, and, eventually, the oxidation of H₂O. What we need is a signal that tells us when there is no more Fe²⁺ in solution.

For our purposes, it is convenient to treat a controlled-current coulometric analysis as a reaction between the analyte, Fe²⁺, and the mediator, Ce³⁺, as shown by reaction 11.34. This reaction is identical to a redox titration; thus, we can use the end points for a redox titration—visual indicators and potentiometric or conductometric measurements—to signal the end of a controlled-current coulometric analysis. For example, ferroin provides a useful visual endpoint for the Ce³⁺ mediated coulometric analysis for Fe²⁺, changing color from red to blue when the electrolysis of Fe²⁺ is complete.

Instrumentation

Controlled-current coulometry normally is carried out using a two-electrode galvanostat, consisting of a working electrode and a counter electrode. The working electrode—often a simple Pt electrode—is also called the generator electrode since it is where the mediator reacts to generate the species that reacts with the analyte. If necessary, the counter electrode is isolated from the analytical solution by a salt bridge or porous frit to prevent its electrolysis products from reacting with the analyte. Alternatively, we can generate the oxidizing agent or the reducing agent externally, and allow it to flow into the analytical solution. Figure 11.33 shows one simple method for accomplishing this. A solution containing the mediator flows into a small-volume electrochemical cell, with the products exiting through separate tubes. Depending upon the analyte, the oxidizing agent or the reducing reagent is selectively delivered to the analytical solution. For example, we can generate Ce⁴⁺ using an aqueous solution of Ce³⁺, directing the Ce⁴⁺ that forms at the anode into our sample.

Figure 11.33 One example of a device for the external generation of oxidizing agents and reducing agents for controlled-current coulometry. A solution containing the mediator flows into a small-volume electrochemical cell. The resulting oxidation products, which form at the anode, flow to the right and may serve as an oxidizing agent. Reduction at the cathode generates a reducing agent.

There are two other crucial needs for controlled-current coulometry: an accurate clock for measuring the electrolysis time, t_e, and a switch for starting and stopping the electrolysis. An analog clock can record time to the nearest ±0.01 s, but the need to frequently stop and start the electrolysis as we approach the endpoint may result in an overall uncertainty of ±0.1 s. A digital clock allows for a more accurate measurement of time, with an overall uncertainty of ±1 ms being possible. The switch must control both the current and the clock, so that we can make an accurate determination of the electrolysis time.

Coulometric Titrations

A controlled-current coulometric method is sometimes called a coulometric titration because of its similarity to a conventional titration. For example, in the controlled-current coulometric analysis for Fe²⁺ using a Ce³⁺ mediator, the oxidation of Fe²⁺ by Ce⁴⁺ (reaction 11.34) is identical to the reaction in a redox titration (reaction 9.15).

There are other similarities between controlled-current coulometry and titrimetry. If we combine equation 11.25 and equation 11.26 and solve for the moles of analyte, N_A, we obtain the following equation.

\[N_\ce{A} = \dfrac{i}{nF}× t_\ce{e}\tag{11.35}\]

Compare equation 11.35 to the relationship between the moles of analyte, N_A, and the moles of titrant, N_T, in a titration

\[N_\ce{A} = N_\ce{T} = M_\ce{T} × V_\ce{T}\]

where M_T and V_T are the titrant’s molarity and the volume of titrant at the end point. In constant-current coulometry, the current source is equivalent to the titrant and the value of that current is analogous to the titrant’s molarity. Electrolysis time is analogous to the volume of titrant, and t_e is equivalent to the a titration’s end point. Finally, the switch for starting and stopping the electrolysis serves the same function as a buret’s stopcock.

Controlled-Potential Coulometry

The majority of controlled-potential coulometric analyses involve the determination of inorganic cations and anions, including trace metals and halides ions. Table 11.8 summarizes several of these methods.

Source: Rechnitz, G. A. Controlled-Potential Analysis, Macmillan: New York, 1963.

^a Electrolytic reactions are written in terms of the change in the analyte’s oxidation state. The actual species in solution depends on the analyte.

The ability to control selectivity by carefully adjusting the working electrode’s potential makes controlled-potential coulometry particularly useful for the analysis of alloys. For example, we can determine the composition of an alloy containing Ag, Bi, Cd, and Sb by dissolving the sample and placing it in a matrix of 0.2 M H₂SO₄ along with a Pt working electrode and a Pt counter electrode. If we apply a constant potential of +0.40 V versus the SCE, Ag(I) deposits on the electrode as Ag and the other metal ions remain in solution. When electrolysis is complete, we use the total charge to determine the amount of silver in the alloy. By shifting the working electrode’s potential to –0.08 V versus the SCE, we deposit Bi on the working electrode. When the coulometric analysis for bismuth is complete, we determine antimony by shifting the working electrode’s potential to –0.33 V versus the SCE, depositing Sb. Finally, we determine cadmium following its electrodeposition on the working electrode at a potential of –0.80 V versus the SCE.

We also can use controlled-potential coulometry for the quantitative analysis of organic compounds, although the number of applications is significantly less than that for inorganic analytes. One example is the six-electron reduction of a nitro group, –NO₂, to a primary amine, –NH₂, at a mercury electrode. Solutions of picric acid—also known as 2,4,6-trinitrophenol, or TNP, a close relative of TNT—can be analyzed by reducing it to triaminophenol.

Another example is the successive reduction of trichloroacetate to dichloroacetate, and of dichloroacetate to monochloroacetate

We can analyze a mixture of trichloroacetate and dichloroacetate by selecting an initial potential at which only the more easily reduced trichloroacetate reacts. When its electrolysis is complete, we can reduce dichloroacetate by adjusting the potential to a more negative potential. The total charge for the first electrolysis gives the amount of trichloroacetate, and the difference in total charge between the first electrolysis and the second electrolysis gives the amount of dichloroacetate.

Controlled-Current Coulometry (Coulometric Titrations)

The use of a mediator makes a coulometric titration a more versatile analytical technique than controlled-potential coulometry. For example, the direct oxidation or reduction of a protein at a working electrode is difficult if the protein’s active redox site lies deep within its structure. A coulometric titration of the protein is possible, however, if we use the oxidation or reduction of a mediator to produce a solution species that reacts with the protein. Table 11.9 summarizes several controlled-current coulometric methods based on a redox reaction using a mediator.

^a The analyte is the underlined species in each reaction.

For an analyte that is not easily oxidized or reduced, we can complete a coulometric titration by coupling a mediator’s oxidation or reduction to an acid–base, precipitation, or complexation reaction involving the analyte. For example, if we use H₂O as a mediator, we can generate H₃O⁺ at the anode

\[\ce{6H2O}(l) ⇋ \ce{4H3O+}(aq) + \ce{O2}(g) + 4e^-\]

and generate OH^– at the cathode.

\[\ce{2H2O}(l) + 2e^- ⇋ \ce{2OH-}(aq) + \ce{H2}(g)\]

If we carry out the oxidation or reduction of H₂O using the generator cell in Figure 11.33, then we can selectively dispense H₃O⁺ or OH^– into a solution containing the analyte. The resulting reaction is identical to that in an acid–base titration. Coulometric acid–base titrations have been used for the analysis of strong and weak acids and bases, in both aqueous and non-aqueous matrices. Table 11.10 summarizes several examples of coulometric titrations involving acid–base, complexation, and precipitation reactions.

In comparison to a conventional titration, a coulometric titration has two important advantages. The first advantage is that electrochemically generating a titrant allows us to use an unstable reagent. Although we cannot easily prepare and store a solution of a highly reactive reagent, such as Ag²⁺ or Mn³⁺, we can generate them electrochemically and use them in a coulometric titration. Second, because it is relatively easy to measure a small quantity of charge, we can use a coulometric titration to determine an analyte whose concentration is too small for a conventional titration.

Quantitative Calculations

The absolute amount of analyte in a coulometric analysis is determined by applying Faraday’s law (equation 11.25), with the total charge given by equation 11.26 or equation 11.27. Example 11.10 shows the calculations for a typical coulometric analysis.

The best way to appreciate the theoretical and practical details discussed in this section is to carefully examine a typical analytical method. Although each method is unique, the following description of the determination of Cr₂O₇^2– provides an instructive example of a typical procedure. The description here is based on Bassett, J.; Denney, R. C.; Jeffery, G. H.; Mendham, J. Vogel’s Textbook of Quantitative Inorganic Analysis, Longman: London, 1978, p. 559–560.

Scale of Operation

A coulometric method of analysis can be used to analyze a small absolute amount of an analyte. In controlled-current coulometry, for example, the moles of analyte consumed during an exhaustive electrolysis is given by equation 11.35. An electrolysis using a constant current of 100 μA for 100 s, for example, consumes only 1 × 10^–7 mol of analyte if n = 1. For an analyte with a molecular weight of 100 g/mol, 1 × 10^–7 mol of analyte corresponds to only 10 μg. The concentration of analyte in the electrochemical cell, however, must be sufficient to allow an accurate determination of the endpoint. When using a visual end point, the smallest concentration of analyte that can be determined by a coulometric titration is approximately 10^–4 M. As is the case for a conventional titration, a coulometric titration using a visual end point is limited to major and minor analytes. A coulometric titration to a preset potentiometric endpoint is feasible even if the analyte’s concentration is as small as 10^–7 M, extending the analysis to trace analytes.¹²

Accuracy

When using controlled-current coulometry accuracy is determined by the current efficiency, by the accuracy with which we can measure current and time, and by the accuracy of the end point. The maximum measurement errors for current and time are about ±0.01% and ±0.1%, respectively. The maximum end point error for a coulometric titration is at least as good as that for a conventional titration, and is often better when using small quantities of reagents. Together, these measurement errors suggest that an accuracy of 0.1%–0.3% is feasible. The limiting factor in many analyses, therefore, is current efficiency. A current efficiency of greater than 99.5% is fairly routine, and it often exceeds 99.9%.

In controlled-potential coulometry, accuracy is determined by current efficiency and by the determination of charge. If the sample is free of interferents that are easier to oxidize or reduce than the analyte, a current efficiency of greater than 99.9% is routine. When an interferent is present, it can often be eliminated by applying a potential where the exhaustive electrolysis of the interferents is possible without the simultaneous electrolysis of the analyte. Once the interferent has been removed the potential can be switched to a level where electrolysis of the analyte is feasible. The limiting factor in the accuracy of many controlled-potential coulometric methods of analysis is the determination of charge. With electronic integrators the total charge can be determined with an accuracy of better than 0.5%.

If we cannot obtain an acceptable current efficiency, an electrogravimetic analysis may be possible if the analyte—and only the analyte—forms a solid deposit on the working electrode. In this case the working electrode is weighed before beginning the electrolysis and reweighed when the electrolysis is complete. The difference in the electrode’s weight gives the analyte’s mass.

Precision

Precision is determined by the uncertainties in measuring current, time, and the endpoint in controlled-current coulometry or of charge in controlled-potential coulometry. Precisions of ±0.1–0.3% are routinely obtained in coulometric titrations, and precisions of ±0.5% are typical for controlled-potential coulometry.

Sensitivity

For a coulometric method of analysis the calibration sensitivity is equivalent to nF in equation 11.25. In general, a coulometric method is more sensitive if the analyte’s oxidation or reduction involves a larger value of n.

Selectivity

Selectivity in controlled-potential and controlled-current coulometry is improved by carefully adjusting solution conditions and by properly selecting the electrolysis potential. In controlled-potential coulometry, the potential is fixed by the potentiostat, and in controlled-current coulometry the potential is determined by the redox reaction with the mediator. In either case, the ability to control the electrolysis potential affords some measure of selectivity. By adjusting pH or adding a complexing agent, it may be possible to shift the potential at which an analyte or interferent undergoes oxidation or reduction. For example, the standard-state reduction potential for Zn²⁺ is –0.762 V versus the SHE. If we add a solution of NH₃, forming Zn(NH₃)₄², the standard state potential shifts to –1.04 V. This provides an additional means for controlling selectivity when an analyte and interferent undergo electrolysis at similar potentials.

Time, Cost, and Equipment

Controlled-potential coulometry is a relatively time consuming analysis, with a typical analysis requiring 30–60 min. Coulometric titrations, on the other hand, require only a few minutes, and are easily adapted for automated analysis. Commercial instrumentation for both controlled-potential and controlled-current coulometry is available, and is relatively inexpensive. Low cost potentiostats and constant-current sources are available for approximately $1000.