Lesson 1 · 25 min

Potentiometry: reference and indicator electrodes

A voltmeter only measures differences. The same silver electrode reads differently against calomel and against silver–silver chloride; learn to convert between potential scales and to turn a reading into a concentration with the Nernst slope.

In this lesson

Explain why every potential measurement needs a reference electrode and write the cell potential $E = E_\text{ind} - E_\text{ref}$
Explain the potentials of silver–silver chloride and calomel electrodes from E° and chloride activity, and convert between reference scales
Distinguish electrodes of the first and second kind, redox electrodes and membrane electrodes, and find ion activities from readings with the Nernst slope
Calculate the Nernst slope at different temperatures and electron numbers

Put a silver wire into silver nitrate solution and connect a voltmeter — nothing happens. The meter needs two leads, and the moment the second lead touches the solution it becomes an electrode itself. The potential of a single electrode cannot be measured; what we measure is always the difference between two electrodes. So one key to potentiometry is a “far end” whose potential is stable, known and indifferent to the sample: the reference electrode.

Predict first

An electrode reads +0.500 V against the saturated calomel electrode (SCE). What does it read against a silver–silver chloride electrode in saturated KCl?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

+0.456 V +0.500 V (the reference does not matter) +0.544 V +0.741 V

Reference electrodes: the zero of the scale

The standard hydrogen electrode (SHE) defines zero, but with platinum black, 1 bar of hydrogen and $\ce{H+}$ at unit activity it is impractical for routine work. Labs use two “electrodes of the second kind” — a metal in contact with one of its insoluble salts, dipping in a solution of that salt’s anion:

silver–silver chloride: $\ce{AgCl(s)} + \ce{e-} \mathrel{\ce{<=>}} \ce{Ag(s)} + \ce{Cl-(aq)}$ , $E^\circ = +0.2223$ V;
calomel: $\ce{Hg2Cl2(s)} + \ce{2e-} \mathrel{\ce{<=>}} \ce{2Hg(l)} + \ce{2Cl-(aq)}$ , $E^\circ = +0.2681$ V.

Solids have unit activity, so the potential depends only on the $\ce{Cl-}$ activity. Filling the electrode with saturated KCl pins the chloride activity (at about 2.8) and the potential with it: +0.241 V for the saturated calomel electrode, +0.197 V for saturated Ag/AgCl. With 1 M or 0.1 M KCl the potentials shift accordingly — all of them follow from the Nernst equation and the activity coefficients of KCl.

The reference meets the sample through a salt bridge or porous ceramic junction. The boundary between two solutions develops a junction potential of a few millivolts; a concentrated KCl bridge minimises it because $\ce{K+}$ and $\ce{Cl-}$ move almost equally fast (see the conductometry lesson).

E_\text{cell} = E_\text{ind} - E_\text{ref}\qquad E(\ce{Ag|AgCl}) = E^\circ - \frac{RT}{F}\ln a_{\ce{Cl-}}

The indicator electrode goes to the meter’s positive terminal and the reference to the negative. The reference potential is fixed by the chloride activity of its filling solution.

$E_\text{ind}$: indicator electrode potential (follows the sample)
$E_\text{ref}$: reference electrode potential (constant)
$a_{\ce{Cl-}}$: chloride activity of the filling solution

Indicator electrodes and the Nernst slope

An indicator electrode has a potential that follows the activity of one ion in the sample:

first kind: a metal in a solution of its own ion, such as $\ce{Ag | Ag+}$ or $\ce{Cu | Cu^2+}$ ;
second kind: an Ag/AgCl wire, for instance, turned around to indicate $\ce{Cl-}$ ;
redox: an inert platinum wire responding to a ratio such as $[\ce{Fe^3+}]/[\ce{Fe^2+}]$ ;
membrane: the glass pH electrode and the ion-selective electrodes of the next lesson.

All obey the Nernst equation: each tenfold change in activity moves the potential by $(RT\ln 10)/(nF)$ . At 25 °C this Nernst slope is 59.16 mV (n = 1) or 29.58 mV (n = 2), and it is proportional to absolute temperature: 54.20 mV at 0 °C and 61.54 mV at 37 °C. That is why a pH meter must know the sample temperature.

E = E^\circ + \frac{RT\ln 10}{nF}\lg a\qquad \frac{RT\ln 10}{F} = 59.16\ \text{mV}\ (25\ ^\circ\text{C})

The Nernst equation in decadic form; the slope is proportional to T and inversely proportional to n.

$a$: activity of the measured ion (≈ concentration when dilute)
$n$: electrons in the electrode reaction
$R, T, F$: gas constant, absolute temperature, Faraday constant

E(cell) = E(indicator) − E(reference) · slope = (RT ln10)/(nF)

Indicator electrodes: a logarithmic response

Indicator electrode Reference electrode lg a(Ag⁺) · −2.0 Temperature · 25 °C

Slope (25 °C): 59.16 mV per decade
E°: 0.7996 V
E(SCE): 0.241 V

At this temperature (25 °C)
25 °C

Drag the handle to change the Ag⁺ activity: 59.16 mV per tenfold change. Switch to copper and the slope halves; raise the temperature to 60 °C and it becomes 66.1 mV. Now change the reference from SCE to Ag/AgCl — the whole line shifts by 44 mV while the slope stays the same.

Worked example

Silver-ion concentration from a cell reading

A silver wire and a saturated calomel electrode form a cell. What should it read with the silver in 0.0100 M $\ce{AgNO3}$ ? If an unknown solution reads +0.382 V, what is $[\ce{Ag+}]$ ? ( $E^\circ(\ce{Ag+/Ag}) = 0.7996$ V, 25 °C, activities taken as concentrations.)

Concept check

What is the Nernst slope of a $\ce{Cu | Cu^2+}$ indicator electrode at 37 °C?

59.16 mV per decade 61.54 mV per decade 29.58 mV per decade 30.77 mV per decade

Try the calculation

An Ag/AgCl reference is filled with 0.100 M KCl (mean activity coefficient 0.770). What is its potential against SHE? ( $E^\circ = 0.2223$ V, 25 °C)

Key takeaways

Only differences can be measured: $E_\text{cell} = E_\text{ind} - E_\text{ref}$ , and the reference must always be stated.
SCE (saturated KCl) is +0.241 V and Ag/AgCl (saturated KCl) +0.197 V against SHE at 25 °C, set by E° and the chloride activity.
The Nernst slope $(RT\ln10)/(nF)$ is 59.16/n mV at 25 °C and proportional to absolute temperature.
A 1 mV error in reading means about a 4 % error in the concentration of a monovalent ion.

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Data and references

D. C. Harris, Quantitative Chemical Analysis, 9th ed. (2016), Ch. 14 (fundamentals of electrochemistry) and Ch. 15 (electrodes and potentiometry).
A. J. Bard, L. R. Faulkner, Electrochemical Methods, 2nd ed. (Wiley, 2001), Ch. 2 and Appendix C (potentials of reference electrodes).
P. Vanýsek, “Electrochemical Series”, CRC Handbook of Chemistry and Physics, 91st ed. (2010): E°(AgCl/Ag) = 0.22233 V, E°(Hg₂Cl₂/Hg) = 0.26808 V.
R. A. Robinson, R. H. Stokes, Electrolyte Solutions, 2nd ed. (Butterworths, 1959), Appendix 8.10 (activity coefficients of KCl).