Electroanalytical signals and chemical sensors

Potential, current and charge answer different questions

An electroanalytical response can depend on an equilibrium potential, an electron-transfer current, an integrated charge or another calibrated electrical quantity. Potential is an energy-per-charge difference; current is charge flow per time; integrated current gives charge. The response mechanism and reference should be named before treating a number as chemical amount.

An ion-selective electrode response can be related to ion activity under suitable conditions. Voltammetric or amperometric current can depend on reaction kinetics and transport as well as concentration. Coulometry uses transferred charge in a specified reaction relation. None of these method labels alone establishes selectivity or a complete amount measurement.

Keep the analyte, matrix, reference orientation, temperature, reaction and time basis explicit. The examples below use supplied hypothetical models, without operating conditions or claims of actual sensor performance.

An ideal potential slope is logarithmic in activity

Choose the fixed-reference convention E=E_offset+(RT/zF)ln a, where a is dimensionless activity, z the signed response charge number, R the molar gas constant, T temperature and F the Faraday constant. Potential differences in this convention are ΔE=(RT/zF)ln(a₂/a₁). The sign belongs to the stated reference orientation and response model.

At assigned 298.15 K, use R=8.31446261815324 J mol⁻¹ K⁻¹ and F=96485.33212331 C mol⁻¹ from current SI constants. The positive monovalent decade slope RT ln10/F is 0.05915935 V per decade. For z=+2 and activities 10⁻⁴ and 10⁻², the hundredfold increase gives ΔE=0.05915935 V.

Activity is not a concentration with units inside a logarithm. A relation such as a=γc/c° requires an activity coefficient γ and a specified standard concentration c°. Unknown γ, interfering responses or reference changes can break a concentration interpretation. The offset and reference are held fixed in this difference calculation.

A small selectivity coefficient can matter beside a large interferent activity

Use a restricted empirical equal-charge model for two positive monovalent ions: a_eff=a_A+K_AB a_B. The invented coefficient K_AB represents the interferent’s contribution under fixed stipulated conditions. This is not a general unequal-charge law or an actual electrode property.

Assign target activity a_A=1.00×10⁻⁵, interferent activity a_B=1.00×10⁻³ and K_AB=0.0200. Effective activity is 3.00×10⁻⁵, three times the target activity. In the same monovalent logarithmic convention at 298.15 K, the potential contribution relative to the target-only case is (RT/F)ln3=0.02822618 V.

The coefficient is small, but the interferent is a hundred times as active as the target. A response interpreted as target-only would overstate activity threefold in this model. A calibrated sensor needs evidence for relevant interferents and matrices; a nominally selective response need not be exclusive.

Transferred charge needs a named electron count and contribution

Faraday accounting relates charge Q to amount of transferred electrons through Q=Fn_e. Relating that to a chemical entity also requires the number of electrons per entity in the selected reaction. Background charge and side-reaction contributions should be distinguished rather than assigned automatically to the analyte.

An original inventory gives measured charge 0.120 C and independently applicable background 0.0200 C. Corrected charge is 0.100 C, corresponding to 1.03642697 μmol electrons. For a stipulated two-electron analyte reaction with all corrected charge assigned to it, transformed analyte amount would be 0.518213483 μmol.

Now give a separately stipulated conditional current efficiency η=0.900, meaning that fraction of the corrected charge drives the selected analyte reaction. Assigned analyte charge is ηQ, so transformed amount is ηQ/(2F)=0.466392135 μmol. Multiplication by this current fraction is different from dividing a recovered mass by a preparation recovery.

This is an amount transformed by the specified charge, not automatically the initial total analyte in an unknown sample. The background, selected electron stoichiometry and efficiency require evidence in a real coulometric procedure. Substituting an unexplained reaction factor can preserve neat arithmetic while changing the chemical quantity.

An ideal planar diffusion current decreases with time

For the specified Cottrell model, current magnitude is |I|=nFA_ec_bulk√D/√(πt). Here n is electrons per reacting entity, A_e electrode area, c_bulk the stipulated initial bulk concentration, D diffusivity and t elapsed time after the idealized change that depletes the reacting species at the surface. Oxidation/reduction sign is separate from the positive magnitude plotted here.

Assume a sufficiently large planar electrode, semi-infinite unstirred diffusion, excess supporting electrolyte and transport-limited surface depletion. Kinetic limits, capacitive current, convection, migration, finite-size effects and other reactions are excluded. The t=0 singular limit is not a finite plotted point or a physical prediction of infinite measured current.

Invent n=1, A_e=0.0100 cm², D=8.00×10⁻⁶ cm² s⁻¹ and c_bulk=2.00×10⁻⁶ mol cm⁻³. Consistent SI values are area 1.00×10⁻⁶ m², D=8.00×10⁻¹⁰ m² s⁻¹ and concentration 2.00 mol m⁻³. Either unit basis gives |I|=(3.07936627 μA√s)/√t.

At t=0.250, 1.00, 4.00 and 16.0 s, magnitudes are 6.15873255, 3.07936627, 1.53968314 and 0.769841568 μA. Each fourfold time increase halves current, so |I|√t is constant within this model. It is a decreasing transient, not a steady-current concentration calibration.

A useful response model still requires validation

Potential changes, interference coefficients, current transients and transferred charge constrain different chemical models. Similar responses can arise from different mixtures, transport conditions or reactions. The appropriate calibration and checks depend on the actual measurand and intended matrix.

Connect the observation to a defined chemical entity and basis, and evaluate relevant interferences, references, time behavior and uncertainty. The hypothetical calculations show what follows from supplied assumptions; they do not certify a sensor, electrolysis or clinical application.

Worked examples

Try it yourself

1. Keep the hundredfold activity increase and fixed-reference convention, but use z=−1. Find ΔE.

   Show the answer and reasoning

   −0.11831870 V.

   The signed response factor reverses the sign; its magnitude is twice the earlier z=+2 result.

2. Reduce interferent activity to 10⁻⁴ with the same target activity and K_AB. Find effective activity, apparent factor and potential shift.

   Show the answer and reasoning

   1.20×10⁻⁵, factor 1.20 and +0.004684311 V.

   The interferent contribution is 2.00×10⁻⁶, twenty percent of target activity; use (RT/F)ln1.20 in the stated convention.

3. For the corrected charge and efficiency above, select a one-electron reaction instead. Find transformed amount.

   Show the answer and reasoning

   0.932784269 μmol.

   The same assigned charge now corresponds to twice as many reacting entities as the two-electron model.

4. Within the same planar diffusion model, double bulk concentration and quadruple elapsed time together. What happens to current magnitude?

   Show the answer and reasoning

   It remains unchanged.

   Concentration supplies a factor two, while 1/√t supplies a factor one half; other model conditions remain stipulated.

What to carry forward

• Potential, current and integrated charge are distinct observables.
• A logarithmic response uses dimensionless activity and a reference convention.
• Check the interferent activity as well as its response coefficient.
• Relate charge to a specified reaction and assigned current contribution.
• A Cottrell transient is conditional and decreases with time.