6.1 Potentiometry, Ion-Selective Electrodes, and Voltammetric Techniques

6.1 Potentiometry, Ion-Selective Electrodes, and Voltammetric Techniques

Quick Answer: Electroanalytical methods measure electrical signals (potential, current, charge, or conductance) to determine analyte concentration and chemical speciation. Potentiometry measures cell potential at zero net current ($i = 0$) using indicator electrodes (such as glass $pH$ and $LaF_3$ fluoride electrodes) referenced against standard half-cells (SCE, $Ag/AgCl$). Dynamic techniques apply controlled potentials: Cyclic Voltammetry (CV) diagnoses redox reversibility ($\Delta E_p = 59.2/n\text{ mV}$ at $25^\circ\text{C}$), while Anodic Stripping Voltammetry (ASV) preconcentrates trace heavy metals for sub-ppb quantitation. Coulometry applies Faraday's laws ($Q = nFN$) for absolute standardless quantification.

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Taxonomy of Electroanalytical Methods

Electroanalytical techniques are categorized based on whether an external current flows through the electrochemical cell and whether electrical control is maintained over potential or current:

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Potentiometric Cell Architecture and Reference Electrodes

A potentiometric cell comprises an indicator electrode (whose half-cell potential responds directly to analyte activity $a_A$), a reference electrode (with a known, invariant potential independent of solution composition), and a salt bridge that establishes electrolytic contact while minimizing the liquid junction potential ($E_j$):

$E_{\text{cell}} = E_{\text{indicator}} - E_{\text{reference}} + E_j$

Because the junction potential across typical saturated $KCl$ bridges is small and reproducible (typically $1-3\text{ mV}$), calibration against primary chemical standards absorbs $E_j$ and $E_{\text{reference}}$ into a constant cell parameter.

  High-Impedance Voltmeter (Input Impedance > 10^12 Ohms)
         [   E_cell = E_ind - E_ref + E_j   ]
               /                       \
              /                         \
      [Reference Electrode]      [Indicator Electrode]
      (SCE or Ag/AgCl)          (Glass, Metal, or ISE)
              |                         |
     [Sat. KCl Salt Bridge]             |
              \                         /
           [ Analyte Solution Containing Target Ion A ]

Standard Reference Electrodes

1. Standard Hydrogen Electrode (SHE):

   • Architecture: Platinized platinum foil immersed in an aqueous solution of unit hydrogen ion activity ($a_{H^+} = 1.00\text{ M}$) bubbling with pure hydrogen gas at standard state ($P_{H_2} = 1.00\text{ bar} = 100\text{ kPa}$).
   • Cell notation: $Pt(s) \mid H_2(g, 1\text{ bar}) \mid H^+(aq, a=1.00\text{ M}) \parallel$
   • Absolute assignment: $E^\circ = 0.000\text{ V}$ at all temperatures. Serving as the thermodynamic benchmark, it is rarely used in routine industrial or regulatory testing due to $H_2$ explosion hazards, difficulty maintaining unit activity, and platinum catalyst poisoning by trace sulfides, arsenic, or organic vapors.

2. Saturated Calomel Electrode (SCE):

   • Architecture: Liquid mercury in contact with a paste of mercury(I) chloride (calomel, $Hg_2Cl_2$) and a saturated solution of potassium chloride ($KCl, \sim 4.6\text{ M}$ at $25^\circ\text{C}$).
   • Cell notation: $Hg(l) \mid Hg_2Cl_2(s) \mid KCl(aq, \text{sat.}) \parallel$
   • Half-cell reduction: $Hg_2Cl_2(s) + 2e^- \rightleftharpoons 2Hg(l) + 2Cl^-(aq)$
   • Standard reduction potential: $E = +0.2444\text{ V}$ vs. SHE at $25^\circ\text{C}$.
   • Temperature coefficient: $E_t = 0.2444 - 6.6 \times 10^{-4}(T - 25)$. Note that calomel undergoes irreversible disproportionation at temperatures exceeding $65^\circ\text{C}$.

3. Silver/Silver Chloride Electrode ($Ag/AgCl$):

   • Architecture: A silver wire coated with a porous layer of silver chloride ($AgCl$) immersed in a $KCl$ electrolyte solution.
   • Cell notation: $Ag(s) \mid AgCl(s) \mid KCl(aq, \text{sat. or } 3.5\text{ M}) \parallel$
   • Half-cell reduction: $AgCl(s) + e^- \rightleftharpoons Ag(s) + Cl^-(aq)$
   • Potential values vs. SHE at $25^\circ\text{C}$:
     • Saturated $KCl$: $E = +0.197\text{ V}$
     • $3.5\text{ M } KCl$: $E = +0.205\text{ V}$
     • $1.0\text{ M } KCl$: $E \approx +0.235\text{ V}$ (the standard potential at unit chloride activity is $+0.222\text{ V}$)
   • Practical advantages: Usable above $60^\circ\text{C}$, where calomel electrodes become unreliable; non-toxic compared to mercury calomel electrodes, and easily miniaturized into combination $pH$ probes.

Converting Between Reference Scales

Reduction potentials are often reported relative to an SCE or $Ag/AgCl$ reference rather than SHE. The conversion requires adding the reference potential:

$E_{\text{vs. SHE}} = E_{\text{vs. SCE}} + 0.244\text{ V}$ $E_{\text{vs. SHE}} = E_{\text{vs. Ag/AgCl (sat.)}} + 0.197\text{ V}$

Worked Example: A cyclic voltammetric reduction peak is recorded at $-0.452\text{ V}$ vs. SCE at $25^\circ\text{C}$. Calculate this peak potential relative to (a) the Standard Hydrogen Electrode (SHE), and (b) a saturated $Ag/AgCl$ reference electrode.

• Part (a): Potential vs. SHE $E_{\text{vs. SHE}} = -0.452\text{ V} + 0.244\text{ V} = -0.208\text{ V}$
• Part (b): Potential vs. Saturated $Ag/AgCl$ $E_{\text{vs. Ag/AgCl}} = E_{\text{vs. SHE}} - 0.197\text{ V} = -0.208\text{ V} - 0.197\text{ V} = -0.405\text{ V}$

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Indicator Electrodes: Metallic and Redox Types

Indicator electrodes respond dynamically to the activity of an electroactive species in solution:

1. Electrodes of the First Kind: A pure metal in equilibrium with its own cation (e.g., $Ag/Ag^+$, $Cu/Cu^{2+}$, $Pb/Pb^{2+}$): $M^{n+} + ne^- \rightleftharpoons M(s) \implies E = E^\circ_{M^{n+}/M} - \frac{0.0592}{n}\log \frac{1}{a_{M^{n+}}} = E^\circ + \frac{0.0592}{n}\log a_{M^{n+}}$ Limitations: Poor selectivity (easily oxidized by dissolved $O_2$), slow electron transfer kinetics, and restricted to soft metals ($Ag, Cu, Pb, Cd, Zn, Bi, Tl$). Hard metals ($Fe, Co, Ni, Cr$) exhibit irreversible, non-reproducible potentials.

2. Electrodes of the Second Kind: A metal in contact with an insoluble salt of its cation and excess anion, responding to anion activity (e.g., $Ag/AgCl/Cl^-$, $Hg/Hg_2Cl_2/Cl^-$, $Ag/Ag_2S/S^{2-}$): $AgCl(s) + e^- \rightleftharpoons Ag(s) + Cl^-(aq) \implies E = E^\circ_{AgCl/Ag} - 0.0592\log a_{Cl^-}$

3. Electrodes of the Third Kind: A metal electrode whose potential responds to a different cation via two coupled homogeneous equilibria (e.g., a mercury electrode in a solution containing $Hg-EDTA^{2-}$, $Ca-EDTA^{2-}$, and free $Ca^{2+}$ ions responds to $a_{Ca^{2+}}$).

4. Inert Redox Electrodes: Chemically inert conductors ($Pt, Au$, glassy carbon, graphite) that do not participate in the redox reaction but act as an electron source or sink for redox couples in solution (e.g., $Fe^{3+} / Fe^{2+}$ or $Ce^{4+} / Ce^{3+}$): $E = E^\circ_{Fe^{3+}/Fe^{2+}} - 0.0592 \log \frac{a_{Fe^{2+}}}{a_{Fe^{3+}}}$

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Ion-Selective Electrodes (ISE) and Membrane Potentials

Ion-Selective Electrodes respond selectively to specific ionic activities across a semipermeable, ion-conducting membrane without involving redox transformations. The potential developed across the membrane is a phase-boundary potential generated by ion-exchange equilibrium at the two membrane interfaces.

The Glass Membrane pH Electrode

The standard glass electrode employs a thin bulb ($0.03-0.1\text{ mm}$) of lithium silicate or Corning 015 glass ($72\%\text{ } SiO_2, 22\%\text{ } Na_2O, 6\%\text{ } CaO$).

• Hydrated Gel Layers: When soaked in water, alkali ions ($Na^+$ or $Li^+$) in the outer $10-100\text{ nm}$ layer of the three-dimensional silicate lattice exchange with $H^+$ from the solution: $\text{Glass}^- - Na^+(s) + H^+(aq) \rightleftharpoons \text{Glass}^- - H^+(s) + Na^+(aq)$
• Both the outer and inner glass surfaces form hydrated silicic acid gel layers separated by an anhydrous central glass core ($~0.05\text{ mm}$) that conducts current solely by interstitial hopping of $Na^+$ or $Li^+$ ions under high electrical resistance ($10^7-10^9\text{ }\Omega$).
• The total membrane boundary potential ($E_b$) depends on the ratio of hydrogen ion activities in the external test solution ($a_1$) and the internal reference solution ($a_2$, typically $0.1\text{ M } HCl$): $E_b = E_1 - E_2 = 0.0592\log \frac{a_1}{a_2} = 0.0592\log a_1 - 0.0592\log a_2$

Internal Buffer/Cl- | Glass Gel (Inner) | Anhydrous Glass Core | Glass Gel (Outer) | External Analyte
    (a2 = constant) |   (H+ Exchange)   |     (Na+ Hop)       |   (H+ Exchange)   |    (a1 = variable)

Systematic Operational Errors of the Glass Electrode

1. Alkaline Error (Sodium Error): At high $pH$ ($pH > 10-12$) in solutions with low $[H^+]$ and elevated $[Na^+]$ or $[K^+]$, the glass membrane loses exclusivity for $H^+$ and exchanges competitively with $Na^+$. Because the electrode responds as if extra $H^+$ were present, the instrument registers a falsely low $pH$ (e.g., true $pH = 12.00$, measured $pH = 11.65$). Lithia glasses ($Li_2O$ replacing $Na_2O$) significantly reduce alkaline error up to $pH\ 13.5$.
2. Acid Error: In concentrated acidic media ($pH < 0.5-1.0$), water activity ($a_{H_2O}$) drops below unity, altering the structure of the outer hydrated gel layer and causing the electrode to register a falsely high $pH$.
3. Asymmetry Potential: A residual potential ($1-5\text{ mV}$) observed across the membrane when identical solutions are placed on both sides of the glass bulb. It arises from unequal mechanical strain during blowing, structural defects, inner vs. outer surface etching, and contamination. Asymmetry potential is completely eliminated by routine two-point calibration (bracketing) using standard buffer solutions (e.g., $pH\ 4.01, 7.00$, or $10.01$).
4. Dehydration: Storing a glass electrode dry dehydrates the outer silicate gel, leading to erratic, drift-prone potentials and sluggish response times. Electrodes must be stored in saturated $KCl$ or commercial storage solution.

The Nikolsky-Eisenman Equation

When an interfering ion $B$ with charge $z_B$ competes with primary analyte ion $A$ with charge $z_A$, the membrane potential follows the Nikolsky-Eisenman equation:

$E = \text{constant} + \frac{0.05916}{z_A} \log \left( a_A + \sum_B K_{A,B}^{\text{pot}} (a_B)^{z_A / z_B} \right)$

where $K_{A,B}^{\text{pot}}$ is the potentiometric selectivity coefficient:

• $K_{A,B}^{\text{pot}} < 1$: The electrode preferentially detects target analyte $A$ over interferent $B$.
• $K_{A,B}^{\text{pot}} > 1$: The electrode responds more strongly to interferent $B$ than to target ion $A$.
• For a high-performance electrode, $K_{A,B}^{\text{pot}}$ should be as small as possible (ideally $< 10^{-4}$).

Worked Example: A potassium ion-selective liquid membrane electrode exhibits a selectivity coefficient $K_{K^+, Na^+}^{\text{pot}} = 2.5 \times 10^{-4}$. Calculate the percentage relative error in determining potassium activity in a serum sample containing $a_{K^+} = 4.0 \times 10^{-3}\text{ M}$ and $a_{Na^+} = 0.140\text{ M}$.

• The effective apparent potassium activity is: $a_{K^+, \text{apparent}} = a_{K^+} + K_{K^+, Na^+}^{\text{pot}} \cdot a_{Na^+} = (4.0 \times 10^{-3}) + (2.5 \times 10^{-4} \times 0.140)$ $a_{K^+, \text{apparent}} = 4.0 \times 10^{-3} + 3.5 \times 10^{-5} = 4.035 \times 10^{-3}\text{ M}$
• Percentage error: $\%\text{ Error} = \frac{4.035 \times 10^{-3} - 4.000 \times 10^{-3}}{4.000 \times 10^{-3}} \times 100\% = \frac{3.5 \times 10^{-5}}{4.0 \times 10^{-3}} \times 100\% = +0.875\%$ Conclusion: The sodium interference causes an error of less than $1\%$, which is analytically acceptable for physiological serum electrolytes.

Liquid Membrane and Solid-State Crystal Membrane Electrodes

1. Liquid Membrane Electrodes (Polymer Matrix ISEs):

   • Potassium Ion-Selective Electrode: Employs the neutral macrocyclic depsipeptide valinomycin dissolved in a plasticized PVC membrane. Valinomycin features a hydrophobic exterior and a hydrophilic cavity lined with six carbonyl oxygens whose coordination dimensions match the unhydrated crystal ionic radius of $K^+$ ($1.38\text{ }\text{\AA}$) with extreme precision. Its selectivity over $Na^+$ ($K_{K^+, Na^+}^{\text{pot}} \approx 10^{-4}$) is unmatched, permitting direct clinical assay of potassium in whole blood.
   • Calcium Ion-Selective Electrode: Employs an immobilized liquid ion exchanger, such as calcium bis(di-n-octylphenylphosphate), dissolved in PVC. It responds to $Ca^{2+}$ down to $10^{-6}\text{ M}$ and discriminates against $Mg^{2+}$ and $Na^+$.

2. Solid-State Crystal Membrane Electrodes:

   • Fluoride Electrode ($LaF_3$ Crystal): Employs a single crystal disk of lanthanum trifluoride ($LaF_3$) doped with europium difluoride ($EuF_2$, $\sim 0.5\%$). Doping divalent $Eu^{2+}$ for trivalent $La^{3+}$ introduces structural anion lattice vacancies through which $F^-$ ions migrate across the crystal lattice by vacancy hopping.
   • Chemical Interferences: Hydroxide ion ($OH^-$) is the sole significant chemical interferent because its ionic radius ($1.33\text{ }\text{\AA}$) and single negative charge nearly mirror fluoride ($1.36\text{ }\text{\AA}$). At $pH > 8$, $OH^-$ precipitates $La(OH)_3$ at the crystal face, releasing excess $F^-$ and causing a massive positive error. Conversely, at $pH < 5$, $F^-$ protonates to form weak acids ($HF$, $pK_a = 3.17$, and $HF_2^-$), which do not register on the crystal membrane, yielding a falsely low fluoride readout.
   • Total Ionic Strength Adjustment Buffer (TISAB): Routine determination of fluoride in fluoridated municipal water and dental formulations requires a $1:1$ volume dilution with TISAB. TISAB contains:
     1. Acetate Buffer ($pH\ 5.0-5.5$): Maintains solution $pH$ strictly between $5$ and $7$ to avoid both $OH^-$ interference and $HF$ formation.
     2. High Concentration $NaCl$ ($1.0\text{ M}$): Fixes the total ionic strength ($\mu$) so that the activity coefficient $\gamma_{F^-}$ remains constant across all calibration standards and unknown samples, equating activity to molar concentration ($a_{F^-} = \gamma_{F^-} [F^-]$).
     3. Chelating Agent (CDTA or EDTA): Preferentially complexes polyvalent cations ($Fe^{3+}, Al^{3+}, Si^{4+}$) that strongly bind fluoride ($[FeF]^{2+}, [AlF_6]^{3-}$), releasing all bound fluoride into free, detectable $F^-$ form.

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Voltammetric Techniques and the Three-Electrode Cell

Unlike zero-current potentiometry, voltammetry is a dynamic method where a controlled potential waveform is impressed upon a micro working electrode while the resulting electrolysis current ($i$) is measured as a function of potential ($E$).

                    Potentiostat / Galvanostat
                    [ Voltage Generator / i-to-V ]
                     /            |            \
                    /             |             \
    [Counter/Auxiliary]      [Reference]       [Working Electrode]
       (Pt wire/coil)        (Ag/AgCl or SCE)   (Glassy C, Pt, Au, HMDE)
             |                    |                      |
             \                    |                      /
              +-------------------+---------------------+
              | Cell: Analyte + 0.1 M Supporting Salt   |
              +-----------------------------------------+

The Three-Electrode Architecture

In early two-electrode cells, current passing through the reference electrode induced concentration polarization and caused an uncompensated ohmic drop ($iR_{\text{cell}}$), changing the reference potential. Modern voltammetry isolates the circuit using three dedicated electrodes:

1. Working Electrode (WE): The micro-electrode (surface area $\sim 0.01-0.2\text{ cm}^2$) where the redox reaction of the analyte occurs. Substrates include glassy carbon (broad anodic and cathodic potential window), platinum, gold, boron-doped diamond (BDD), and mercury (Hanging Mercury Drop Electrode HMDE or Thin Mercury Film Electrode TMFE; exceptional high hydrogen overpotential allowing cathodic sweeps down to $-1.8\text{ V}$).
2. Reference Electrode (RE): ($Ag/AgCl$ or SCE) Positioned with its Luggin capillary tip placed immediately adjacent to the working electrode surface to minimize solution resistance. The potentiostat's high-input-impedance operational amplifier ensures zero net current flows through the reference electrode, preserving its thermodynamic half-cell potential.
3. Counter / Auxiliary Electrode (CE): A high-surface-area inert conductor (platinum coil or graphite rod) that completes the current circuit with the working electrode. The current flowing through the working electrode is balanced by an equal and opposite current at the counter electrode.

Role of the Supporting Electrolyte

Voltammetric solutions require a large excess ($0.1-1.0\text{ M}$) of an electrochemically inactive salt (e.g., $KNO_3, KCl, NaClO_4$, or tetraalkylammonium salts like $TBAPF_6$ for non-aqueous solvents):

1. Suppression of Migration Current: Eliminates electrostatic attraction/repulsion between charged ions and the charged electrode surface. The supporting electrolyte ions carry $>99.9\%$ of electrostatic charge migration, ensuring analyte mass transport occurs solely by diffusion (governed by Fick's laws).
2. Elimination of Ohmic $iR$ Drop: Decreases bulk solution electrical resistance ($R$).
3. Buffer and Ionic Strength Stabilization: Holds activity coefficients and double-layer capacitance constant.

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Cyclic Voltammetry (CV): Diagnostic Criteria of Reversibility

Cyclic Voltammetry applies a triangular potential excitation waveform to the working electrode, sweeping from an initial potential $E_1$ to a switching potential $E_2$ at a linear scan rate ($\nu = dE/dt$, typically $10-1000\text{ mV/s}$), then reversing the sweep back to $E_1$.

$\text{Forward Sweep: } O + ne^- \xrightarrow{\text{cathodic reduction}} R \quad (\text{producing cathodic peak current } i_{pc} \text{ at } E_{pc})$ $\text{Reverse Sweep: } R \xrightarrow{\text{anodic oxidation}} O + ne^- \quad (\text{producing anodic peak current } i_{pa} \text{ at } E_{pa})$

Diagnostic Criteria for an Electrochemically Reversible Redox Couple at $25^\circ\text{C}$

For a diffusion-controlled, reversible Nernstian electron transfer ($O + ne^- \rightleftharpoons R$):

1. Peak Potential Separation ($\Delta E_p$): $\Delta E_p = |E_{pa} - E_{pc}| = \frac{0.0592\text{ V}}{n} = \frac{59.2\text{ mV}}{n}$

   • For a $1$-electron transfer ($n = 1$): $\Delta E_p \approx 59\text{ mV}$ (experimentally $58-65\text{ mV}$ due to small residual uncompensated resistance).
   • For a $2$-electron transfer ($n = 2$): $\Delta E_p \approx 29.6\text{ mV} \approx 30\text{ mV}$.
   • $\Delta E_p$ is independent of scan rate ($\nu$) for a reversible system. If $\Delta E_p$ increases markedly as scan rate increases, the electron transfer kinetics are quasi-reversible or irreversible (governed by Butler-Volmer kinetics).

2. Formal Reduction Potential ($E^{0\prime}$): $E^{0\prime} = \frac{E_{pa} + E_{pc}}{2}$ The formal potential is located midway between the anodic and cathodic peak potentials.

3. Peak Current Ratio: $\left| \frac{i_{pa}}{i_{pc}} \right| = 1.0$ The ratio of anodic to cathodic peak heights is unity, confirming that the reduced intermediate $R$ is chemically stable and does not undergo follow-up chemical decay (such as EC mechanisms).

4. Scan Rate Dependence (The Randles-Ševčík Equation): At $298\text{ K}$ ($25^\circ\text{C}$), the peak current for a reversible process is governed by: $i_p = (2.69 \times 10^5) \, n^{3/2} \, A \, D^{1/2} \, C^* \, \nu^{1/2}$ where $i_p$ is peak current (Amperes), $n$ is number of transferred electrons, $A$ is electrode surface area ($\text{cm}^2$), $D$ is diffusion coefficient ($\text{cm}^2/\text{s}$), $C^*$ is bulk analyte concentration ($\text{mol/cm}^3$), and $\nu$ is potential scan rate ($\text{V/s}$).

   • A linear plot of $i_p$ versus $\nu^{1/2}$ passing through the origin is the gold-standard proof that the electrode process is controlled purely by diffusion rather than surface adsorption or catalytic cycles.

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Stripping Voltammetry and Heavy Metal Analysis

Stripping Voltammetry is the most sensitive electroanalytical technique available, routinely achieving detection limits in the sub-ppb ($10^{-9}-10^{-11}\text{ M}$) domain for heavy metals ($Pb^{2+}, Cd^{2+}, Cu^{2+}, Zn^{2+}, As^{3+}$).

Mechanism of Anodic Stripping Voltammetry (ASV)

ASV is a two-step hyphenated electrochemical process:

1. Preconcentration / Deposition Step (Cathodic Reduction):

   • The working electrode (typically a Hanging Mercury Drop Electrode HMDE or Thin Mercury Film Electrode TMFE on glassy carbon) is held at a constant cathodic deposition potential ($E_{\text{dep}}$) that is $200-400\text{ mV}$ more negative than the standard reduction potential of the analyte metal ions.
   • The solution is vigorously stirred (hydrodynamic transport) for a fixed period ($t_{\text{dep}} = 1-10\text{ minutes}$): $M^{n+}(aq) + ne^- + Hg(l) \xrightarrow{\text{electrochemical reduction}} M(Hg)\text{ (amalgam)}$
   • Because the volume of the mercury drop or film is orders of magnitude smaller than the sample solution ($V_{\text{Hg}} \ll V_{\text{soln}}$), the metal analyte is concentrated into the mercury by a factor of $100$ to $10,000$ times its initial bulk solution concentration.

2. Quiet / Rest Period:

   • Stirring is stopped for $15-30\text{ seconds}$ to allow the solution to become quiescent and the concentration gradient in the mercury drop to homogenize.

3. Stripping Step (Anodic Sweep):

   • The working electrode potential is scanned anodically (positive direction) toward zero. As the potential reaches the formal oxidation potential of each amalgamated metal, the concentrated metal atoms are reoxidized and "stripped" back into solution: $M(Hg) \xrightarrow{\text{anodic oxidation}} M^{n+}(aq) + ne^- + Hg(l)$
   • Each metal generates a sharp current peak at its characteristic oxidation potential ($E_p$). The peak area (or peak height) is directly proportional to the original concentration of the metal in the sample.

Cathodic Preconcentration (Stirred)     -->     Quiescent Rest     -->     Anodic Stripping Sweep
  Apply E_dep (-1.2 V vs. SCE)                   Hold E_dep               Ramp Potential from -1.2 V to +0.2 V
  M^(n+) + ne- + Hg --> M(Hg) amalgam            Stirring OFF             M(Hg) --> M^(n+) + ne- (Sharp Peaks!)

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Coulometric Methods: Controlled-Potential vs. Coulometric Titrations

Coulometry measures the total quantity of electricity ($Q$, coulombs) required to convert an analyte completely into a different oxidation state. Governed by Faraday's First and Second Laws of Electrolysis, coulometric analysis requires no calibration curves, chemical standards, or indicator titrations:

$Q = \int_0^t i(t) \, dt = n \, F \, N = n \, F \, \frac{m}{M}$

where $F = 96485.3\text{ C/mol } e^-$, $n$ is stoichiometric electron transfer, $m$ is mass of analyte (grams), and $M$ is molar mass ($\text{g/mol}$).

Comparison of Coulometric Methodologies

The Coulometric Karl Fischer Titration for Trace Moisture

The coulometric Karl Fischer titration is the universal gold-standard method for determining trace moisture ($1\text{ ppm}$ to $1\%$) in pharmaceutical ingredients, organic solvents, and petrochemical fuels.

• Reaction Chemistry: Water reacts quantitatively with iodine, sulfur dioxide, and a base (imidazole or pyridine, denoted $B$) in methanol: $H_2O + I_2 + SO_2 + 3B + CH_3OH \longrightarrow 2BH^+I^- + BSO_4CH_3^-$ Stoichiometry: Exactly $1\text{ mole of } I_2$ consumes $1\text{ mole of } H_2O$.
• In-situ Coulometric Generation: Rather than adding liquid iodine titrant via burette, iodide ($I^-$) present in the electrolytic reagent is oxidized to iodine ($I_2$) at a platinum generator anode at a constant current $I$: $2I^- \xrightarrow{\text{anode}} I_2 + 2e^-$
• Because $n = 2$ electrons per molecule of water ($H_2O \equiv I_2 \equiv 2e^-$): $m_{H_2O} = \frac{Q \times M_{H_2O}}{n F} = \frac{(I \times t) \times 18.015\text{ g/mol}}{2 \times 96485\text{ C/mol}}$

Worked Example: A $2.500\text{ mL}$ aliquot of anhydrous methanol is analyzed for trace water contamination by coulometric Karl Fischer titration. A constant generator current of $64.32\text{ mA}$ ($0.06432\text{ A}$) operates for $180.0\text{ seconds}$ to reach the biamperometric endpoint. Calculate the water concentration in the methanol sample in milligrams per liter ($\text{mg/L}$).

• Step 1: Compute total electrical charge ($Q$): $Q = I \times t = 0.06432\text{ A} \times 180.0\text{ s} = 11.5776\text{ C}$
• Step 2: Calculate mass of water consumed: $m_{H_2O} = \frac{11.5776\text{ C} \times 18.015\text{ g/mol}}{2 \times 96485\text{ C/mol}} = \frac{208.5705}{192970}\text{ g} = 1.0808 \times 10^{-3}\text{ g} = 1.0808\text{ mg}$
• Step 3: Calculate concentration in $\text{mg/L}$: $\text{Concentration} = \frac{1.0808\text{ mg}}{2.500 \times 10^{-3}\text{ L}} = 432.3\text{ mg/L (or ppm)}$