Liquid Junction Potentials

A pH meter reads 7.02 instead of 7.00. The 0.02 pH error comes almost entirely from a few millivolts you never accounted for. Every electrochemical cell used for real measurement contains an interface where two dissimilar electrolyte solutions touch. Examples include a glass pH electrode, an ion-selective electrode in a hospital blood-gas analyzer, and a reference electrode dipped into seawater. At that interface, ions diffuse across the boundary at different rates. Charge separates, and a small but stubborn voltage appears: the liquid junction potential. It is not a mistake in the Nernst equation. It is an extra term the Nernst equation does not know about. It rides invisibly on top of every potentiometric measurement. This matters because Cell Potential and the Nernst Equation for Analysis rests on one key assumption. It assumes the measured cell voltage reflects only the thermodynamic activities of the analyte ion at the two electrodes. A junction potential of even 1–5 mV, left uncorrected, causes real errors. It translates into pH errors of 0.02–0.1 units. It can also cause relative concentration errors of several percent in ion-selective electrode work. These errors are routinely larger than the precision of the underlying Nernst response itself. Analytical chemists cannot eliminate junction potentials outright. But they can understand their origin well enough to minimize and stabilize them. That is exactly what a properly designed salt bridge does. The physical origin is beautifully simple: ions do not all move at the same speed. When a concentrated KCl solution meets a dilute HCl solution, H⁺ diffuses toward the dilute side faster than Cl⁻ diffuses the other way. This happens because the hydrated proton has an enormous mobility. This differential migration builds a thin dipole layer at the interface, a few nanometers thick. It forms a self-limiting electric field. This field decelerates the faster ion and accelerates the slower one. Both ions then cross the boundary at an equal net rate. The equilibrium voltage across that dipole layer is the liquid junction potential. Its size and sign depend entirely on the mobilities and charges of the ions present. This article derives the Henderson equation and its more transparent limiting form for a 1:1 electrolyte junction. It works three realistic numerical examples spanning a plain HCl/KCl junction, a biological buffer measurement, and a seawater reference-electrode problem. It explains why a saturated KCl (or a KNO₃) salt bridge is the standard fix. It closes by connecting junction potentials to the practical design of Reference Electrodes. It also links to the broader discipline of Potentiometry Fundamentals. There, controlling this "invisible" voltage separates a routine measurement from a systematic error. Otherwise, nobody notices that error until it is too late.

Key takeaways

A liquid junction potential (LJP) arises wherever two electrolyte solutions of different composition or concentration touch, because cations and anions diffuse across the boundary at different rates.
The size of the LJP is set by the mismatch in ionic mobilities (transference numbers); equal mobilities give zero junction potential.
The Henderson equation quantifies $E_j$ for a general junction; for a single 1:1 salt across a concentration gradient it reduces to a compact log expression in the cation/anion mobility ratio.
Typical junction potentials are 1–40 mV — small compared to a full cell voltage but often larger than the precision demanded of pH or ion-selective electrode measurements.
A saturated KCl salt bridge minimizes $E_j$ because K⁺ and Cl⁻ have nearly identical mobilities, and the huge KCl concentration swamps the mobility mismatch of whatever ions are actually in the sample.
Junction potentials cannot be calculated exactly for real, imperfectly-defined interfaces; they are managed by design (matched-mobility bridges, calibration bracketing) rather than corrected by formula in routine practice.

Key idea: A liquid junction potential is the steady-state Donnan-like voltage built by unequal ionic mobilities racing across a boundary, and it is minimized — never truly abolished — by a bridge electrolyte whose cation and anion move at nearly the same speed.

Where this lives today

Every commercial pH meter and every combination glass electrode relies on a KCl-filled reference junction whose residual potential is small enough to be absorbed into the two-point calibration against NIST-traceable buffers — the entire practical accuracy of pH measurement in clinical chemistry, environmental monitoring, and food science rests on keeping this junction potential both small and stable from calibration to sample. In clinical blood-gas analyzers, ion-selective electrodes for Na⁺, K⁺, Ca²⁺, and Cl⁻ measure physiological samples whose ionic strength differs from calibration standards; manufacturers explicitly design flowing or renewable liquid junctions (rather than static frits) so that $E_j$ resets to a reproducible value before every measurement, directly exploiting the mobility-matching physics derived here. Oceanographers measuring seawater pH must account for junction potentials that differ measurably between fresh-water and high-ionic-strength calibration buffers, which is why seawater pH scales (total scale, seawater scale) are defined with their own buffer sets rather than borrowing freshwater NIST buffers. In industrial process control, glass and ISE electrodes monitoring high-temperature or high-ionic-strength streams (paper pulping liquors, plating baths) require double-junction reference electrodes — an inner KCl fill isolated by a second bridge electrolyte compatible with the aggressive sample — precisely to prevent the sample from poisoning or altering the mobility-matched bridge chemistry. Even in modern solid-state and polymer-membrane ion sensors used in wearable and continuous glucose/electrolyte monitors, designers must still manage an analogous junction at the reference interface, showing that the century-old Henderson–Planck framework remains directly load-bearing in cutting-edge sensor engineering.

Ask the tutor: How does the liquid junction potential change if the salt bridge itself has a concentration gradient along its length, rather than being uniform?

The physical picture

Picture two beakers of electrolyte connected by a narrow tube or porous frit: one side holds 0.10 M HCl, the other 0.010 M HCl. Concentration differs, so ions diffuse from high to low concentration, exactly as in Fick's first law. But H⁺ and Cl⁻ do not diffuse at the same rate. The proton's mobility in water, $u_{H^+} \approx 36.2 \times 10^{-8}\ \text{m}^2\text{s}^{-1}\text{V}^{-1}$ , is roughly 5 times that of chloride, $u_{Cl^-} \approx 7.9 \times 10^{-8}\ \text{m}^2\text{s}^{-1}\text{V}^{-1}$ (this unusually high proton mobility comes from the Grotthuss "hopping" mechanism, not ordinary hydrodynamic diffusion). H⁺ races ahead into the dilute region while Cl⁻ lags behind. The result is a minute charge separation: the dilute side becomes transiently positive (excess H⁺ arrived first), the concentrated side becomes transiently negative (excess Cl⁻ left behind). This separated charge creates a local electric field pointing from the dilute side back toward the concentrated side. That field decelerates H⁺ and accelerates Cl⁻. Within roughly a nanosecond, the system reaches a steady state where the electric field exactly compensates the mobility mismatch: H⁺ and Cl⁻ now cross the boundary together, at an equal net rate, preserving electroneutrality in the bulk. The voltage needed to enforce that equal crossing rate is the liquid junction potential, $E_j$ . Because the layer where this happens is only a few ion diameters thick, it behaves like a thin, self-adjusting capacitor sitting between two electroneutral bulk solutions. The bulk solutions on either side remain (to excellent approximation) electrically neutral and Nernstian; the entire departure from ideal cell behavior is concentrated in that thin dipole sheet. This is the same conceptual family as a Donnan potential across a semipermeable membrane, except here there is no membrane at all — just an unstirred, gradually mixing zone of free electrolyte.

Figure

Ion mobility mismatch creates a dipole layer at an HCl concentration junction

The magnitude of $E_j$ depends only on how badly mismatched the mobilities are and how large the concentration ratio is; it does not depend on the geometry of the tube or frit connecting the two solutions (to the extent that mixing there is purely diffusive and one-dimensional). This is why chemists can write a closed-form expression for $E_j$ at all — despite the interface being physically a smear rather than a sharp boundary.

Figure

Salt bridge strategy: swamp the junction with a near-equal-mobility electrolyte

The core ideas

For a junction between two solutions in which a single 1:1 electrolyte (cation charge $z_+ = +1$ , anion charge $z_- = -1$ ) varies only in concentration, the junction potential is given by the classical Henderson–Planck limiting expression:

E_j = \frac{u_+ - u_-}{u_+ + u_-}\cdot\frac{RT}{F}\ln\!\left(\frac{a_1}{a_2}\right)

where:

$u_+, u_-$ = ionic mobilities of the cation and anion (m² s⁻¹ V⁻¹), or equivalently their limiting molar conductivities $\lambda_+^{0}, \lambda_-^{0}$
$R$ = gas constant, $8.314\ \text{J mol}^{-1}\text{K}^{-1}$
$T$ = absolute temperature (K)
$F$ = Faraday constant, $96{,}485\ \text{C mol}^{-1}$
$a_1, a_2$ = activities of the electrolyte on side 1 (usually the more concentrated) and side 2

Reading: the log term is just the familiar Nernstian concentration-ratio factor; the prefactor $(u_+-u_-)/(u_++u_-)$ is a "leakage fraction" that says how much of that ideal concentration-cell voltage actually survives as a junction potential once you account for unequal ionic velocities. If the two ions moved at identical speed, $u_+ = u_-$ , the prefactor vanishes and $E_j = 0$ exactly — the concentration cell then behaves as if there were no junction at all, a case realized almost perfectly by KCl. For the more general Henderson equation, covering a junction between two solutions of arbitrary, possibly different, dissolved ions (mixed electrolytes on each side), the result generalizes to:

E_j = -\frac{RT}{F}\left(\frac{\sum_i z_i u_i (c_{i,2}-c_{i,1})}{\sum_i z_i^2 u_i (c_{i,2}-c_{i,1})}\right)\ln\!\left(\frac{\sum_i z_i^2 u_i c_{i,2}}{\sum_i z_i^2 u_i c_{i,1}}\right)

where:

the sums run over every ionic species $i$ present in either solution
$c_{i,1}, c_{i,2}$ = concentration of ion $i$ in solution 1 and solution 2 (mol L⁻¹), taken as zero if that ion is absent on one side
$z_i$ = signed charge number of ion $i$
$u_i$ = mobility of ion $i$

Reading: this unwieldy-looking expression is the same physics as the simple form, just bookkeeping every ion's contribution to both the net diffusive charge flux (numerator-like term) and the total conductance (denominator-like term); it reduces exactly to the compact 1:1 formula when only one salt is present at two concentrations.

Derivation AThe steady-state condition and the compact 1:1 formula

Goal: show that requiring equal net flux of cation and anion across the junction — the physical steady-state condition — forces the electric field, and hence $E_j$ , to take the value in the boxed 1:1 formula above.

Step 1 — Flux of each ion has a diffusive part and a migration part. In the thin junction region, each ion obeys the Nernst–Planck flux equation, which combines Fick diffusion with electrophoretic drift in the local field $\mathscr{E}(x)$ :

J_\pm = -D_\pm \frac{dc_\pm}{dx} \pm u_\pm c_\pm \mathscr{E}(x)

Here $D_\pm$ is the diffusion coefficient of each ion, linked to mobility by the Einstein relation $D_\pm = u_\pm RT/F$ (per mole of charge). The plus/minus sign on the migration term reflects that the field pushes cations and anions in opposite directions.

Step 2 — Electroneutrality forces equal net ionic flux. Because no net charge can accumulate once steady state is reached (the thin dipole layer's charge density is fixed, not growing), the cation and anion must cross the boundary together: the net flux of positive charge equals the net flux of negative charge, $J_+ = J_-$ (for a 1:1 salt where $c_+ = c_- = c$ everywhere by electroneutrality in bulk). Setting $J_+ = J_-$ :

-D_+\frac{dc}{dx} + u_+ c\,\mathscr{E} = -D_-\frac{dc}{dx} - u_- c\,\mathscr{E}

Step 3 — Solve for the local field. Rearranging to isolate $\mathscr{E}$ :

(u_+ + u_-)\,c\,\mathscr{E} = (D_+ - D_-)\frac{dc}{dx}

\mathscr{E}(x) = \frac{D_+ - D_-}{u_+ + u_-}\cdot\frac{1}{c}\frac{dc}{dx}

Substituting the Einstein relation $D_\pm = u_\pm RT/F$ :

\mathscr{E}(x) = \frac{RT}{F}\cdot\frac{u_+ - u_-}{u_+ + u_-}\cdot\frac{1}{c}\frac{dc}{dx}

Step 4 — Integrate the field across the junction to get the potential. The junction potential is $E_j = -\int_1^2 \mathscr{E}(x)\,dx$ , integrated from solution 1 (concentration $c_1$ ) to solution 2 (concentration $c_2$ ). Since $\frac{1}{c}\frac{dc}{dx}dx = d(\ln c)$ , this integral is elementary:

E_j = -\frac{RT}{F}\cdot\frac{u_+-u_-}{u_++u_-}\int_{c_1}^{c_2} d(\ln c) = \frac{RT}{F}\cdot\frac{u_+-u_-}{u_++u_-}\ln\!\left(\frac{c_1}{c_2}\right)

which is exactly the boxed 1:1 Henderson–Planck result (with activities replacing concentrations once non-ideality is included). Notice the self-consistency check: if $u_+ = u_-$ the field $\mathscr{E}(x)$ is zero everywhere in Step 3, so no junction potential can build up at all — confirming the limiting case stated above.

Go deeper · Derivation B · 1 minWhy a saturated KCl bridge nearly zeroes two junction potentials in series

Derivation BWhy a saturated KCl bridge nearly zeroes two junction potentials in series

Goal: show quantitatively why inserting a concentrated KCl bridge between two arbitrary solutions produces a net series junction potential far smaller than the single direct junction would have been.

Step 1 — Set up the two-junction path. Suppose solution A (sample, arbitrary ions) is connected to solution B (reference fill) not directly, but through a bridge filled with saturated KCl (~4.2 M at room temperature). There are now two liquid junctions in series: A|bridge and bridge|B. The measured net contribution is the algebraic sum,

E_{j,\text{net}} = E_{j}(\text{A|bridge}) + E_{j}(\text{bridge|B})

Step 2 — Each junction is now dominated by the bridge composition. Because the bridge concentration (≈4.2 M) vastly exceeds typical sample/reference concentrations (10⁻⁴–10⁻¹ M), the diffusion of K⁺ and Cl⁻ out of the bridge dominates the ionic flux at each interface; the sample's own ions contribute negligibly to the net flux balance. Each junction can then be approximated with the compact 1:1 formula using KCl's own mobilities:

E_j(\text{each}) \approx \frac{u_{K^+}-u_{Cl^-}}{u_{K^+}+u_{Cl^-}}\cdot\frac{RT}{F}\ln\!\left(\frac{a_{\text{bridge}}}{a_{\text{outside}}}\right)

Step 3 — Evaluate the mobility mismatch prefactor for KCl. Using standard limiting ionic conductivities at 25 °C, $\lambda^0_{K^+} = 73.5\ \text{S cm}^2\text{mol}^{-1}$ and $\lambda^0_{Cl^-} = 76.3\ \text{S cm}^2\text{mol}^{-1}$ (proportional to mobility, since $\lambda^0 = zFu$ ):

\frac{u_{K^+}-u_{Cl^-}}{u_{K^+}+u_{Cl^-}} = \frac{73.5-76.3}{73.5+76.3} = \frac{-2.8}{149.8} = -0.0187

Step 4 — Compare to a mismatched pair, e.g., H⁺/Cl⁻. For HCl, $\lambda^0_{H^+} = 349.8$ , $\lambda^0_{Cl^-} = 76.3\ \text{S cm}^2\text{mol}^{-1}$ :

\frac{u_{H^+}-u_{Cl^-}}{u_{H^+}+u_{Cl^-}} = \frac{349.8-76.3}{349.8+76.3} = \frac{273.5}{426.1} = 0.642

Step 5 — Compare the two prefactors directly. The KCl prefactor (0.0187 in magnitude) is roughly 34 times smaller than the HCl/Cl⁻ prefactor (0.642). Since $E_j$ scales linearly with this prefactor for a fixed log-concentration-ratio term, replacing a direct HCl-type junction with two KCl-bridge junctions cuts the residual junction voltage by well over an order of magnitude — even though the log term itself is not tiny, because K⁺ and Cl⁻ so nearly cancel each other's contribution to the field. This is the quantitative reason "matched mobility" bridge salts (KCl, and to a lesser extent KNO₃ or NH₄NO₃ for junctions containing heavy metal ions) are the standard laboratory choice, and why the residual, uncorrectable $E_j$ in a well-designed reference electrode is typically only 1–2 mV rather than tens of millivolts.

Worked examples

Example 1 — Foundational: HCl concentration junction. A junction is formed between 0.10 M HCl (side 1) and 0.010 M HCl (side 2) at 25.00 °C, activity coefficients neglected for a first pass (activities ≈ concentrations). Using $\lambda^0_{H^+} = 349.8$ and $\lambda^0_{Cl^-} = 76.3\ \text{S cm}^2\text{mol}^{-1}$ from above, the mobility-ratio prefactor was already computed:

\frac{u_{H^+}-u_{Cl^-}}{u_{H^+}+u_{Cl^-}} = 0.642

The thermal voltage at 25.00 °C (298.15 K) is

\frac{RT}{F} = \frac{(8.314\ \text{J mol}^{-1}\text{K}^{-1})(298.15\ \text{K})}{96{,}485\ \text{C mol}^{-1}} = 0.025693\ \text{V} = 25.69\ \text{mV}

The log-ratio term, with $c_1/c_2 = 0.10/0.010 = 10.0$ :

\ln(10.0) = 2.303

Assembling the compact 1:1 formula:

E_j = 0.642 \times 25.69\ \text{mV} \times 2.303 = 38.0\ \text{mV}

Interpretation: the dilute side (side 2) becomes positive relative to the concentrated side, because H⁺ (the faster ion) races ahead into the dilute solution, leaving it with a transient excess of positive charge that the steady-state field must maintain. This 38 mV is not negligible — it is comparable to a full Nernst decade (59.16 mV at 25 °C for a monovalent ion), so an uncorrected HCl|HCl junction of this kind would introduce an error equivalent to nearly two-thirds of a full activity decade. This is precisely the scenario a salt bridge is built to avoid.

Limiting-case check: if the two HCl concentrations were made equal ( $c_1 = c_2$ ), $\ln(c_1/c_2) \to \ln(1) = 0$ and $E_j \to 0$ regardless of the mobility mismatch — consistent with the physical expectation that no concentration gradient means no diffusion-driven charge separation at all.

Example 2 — Applied: a KCl salt bridge separating a physiological buffer from a reference fill. A combination pH electrode's reference half-cell contains saturated KCl (treated here as 4.2 M with activity coefficient corrections folded into an effective activity of $a_{\text{bridge}} = 3.5$ M), connected through a bridge to a blood-gas sample buffered at ionic strength corresponding to an effective KCl-equivalent activity of $a_{\text{sample}} \approx 0.15$ M (physiological saline background). Estimate the single bridge|sample junction potential. Using the KCl mobility prefactor from Step 3 above, $-0.0187$ , and

\ln\!\left(\frac{a_{\text{bridge}}}{a_{\text{sample}}}\right) = \ln\!\left(\frac{3.5}{0.15}\right) = \ln(23.3) = 3.150

E_j = (-0.0187)(25.69\ \text{mV})(3.150) = -1.51\ \text{mV}

Interpretation: even though the concentration ratio here (23.3-fold) is comparable in log-magnitude to Example 1's tenfold ratio, the near-cancellation of K⁺ and Cl⁻ mobilities keeps $|E_j|$ under 2 mV — about 25 times smaller than the unmitigated HCl case despite a larger concentration ratio. In pH terms, a stray 1.51 mV error corresponds to

\Delta \text{pH} = \frac{E_j}{59.16\ \text{mV}} = \frac{1.51}{59.16} = 0.026\ \text{pH units}

which is why calibration bracketing (running standard buffers with ionic strength close to the sample) is still recommended even with a KCl bridge — the residual 0.02–0.03 pH uncertainty is exactly this leftover junction contribution, and it is the practical floor on pH-meter accuracy that no amount of electronic precision can improve.

Example 3 — Exam-level: mixed-electrolyte Henderson calculation for a seawater reference junction. An oceanographic reference electrode with saturated KCl fill (treat as 4.0 M K⁺, 4.0 M Cl⁻ for this calculation) is inserted directly into seawater, approximated as containing only Na⁺ (0.45 M) and Cl⁻ (0.52 M) with charge balance made up by other unspecified minor ions (ignore for this estimate). Use the following limiting ionic conductivities (S cm² mol⁻¹, proportional to mobility since $u_i = \lambda_i^0/(z_i F)$ ): $\lambda^0_{K^+}=73.5$ , $\lambda^0_{Na^+}=50.1$ , $\lambda^0_{Cl^-}=76.3$ . Estimate $E_j$ across the direct KCl(bridge)|seawater junction using the full Henderson treatment.

Step 1 — Tabulate concentration differences. Side 1 = bridge (KCl fill), side 2 = seawater.

Ion	$z_i$	$c_{i,1}$ (mol/L)	$c_{i,2}$ (mol/L)	$c_{i,2}-c_{i,1}$
K⁺	+1	4.00	0	−4.00
Na⁺	+1	0	0.45	+0.45
Cl⁻	−1	4.00	0.52	−3.48

Step 2 — Numerator sum, $\sum z_i u_i (c_{i,2}-c_{i,1})$ . Using $u_i \propto \lambda_i^0$ (the proportionality constant $1/F$ cancels between numerator and denominator, so raw $\lambda^0$ values may be used directly):

\sum z_i \lambda_i^0 (c_{i,2}-c_{i,1}) = (+1)(73.5)(-4.00) + (+1)(50.1)(+0.45) + (-1)(76.3)(-3.48)

= -294.0 + 22.545 + 265.5 = -5.955

Step 3 — Denominator sum, $\sum z_i^2 u_i (c_{i,2}-c_{i,1})$ . Since $z_i^2 = 1$ for all these monovalent ions, this sum uses the same terms without the sign of $z_i$ applied twice (i.e., $z_i^2=+1$ throughout):

\sum z_i^2 \lambda_i^0(c_{i,2}-c_{i,1}) = (73.5)(-4.00) + (50.1)(+0.45) + (76.3)(-3.48)

= -294.0 + 22.545 - 265.5 = -536.96

Step 4 — The prefactor.

\frac{-5.955}{-536.96} = 0.01109

Step 5 — The log term, $\sum z_i^2 \lambda_i^0 c_{i,2}$ over $\sum z_i^2 \lambda_i^0 c_{i,1}$ :

\sum z_i^2\lambda_i^0 c_{i,2} = (73.5)(0) + (50.1)(0.45) + (76.3)(0.52) = 22.545 + 39.676 = 62.22

\sum z_i^2\lambda_i^0 c_{i,1} = (73.5)(4.00) + (50.1)(0) + (76.3)(4.00) = 294.0 + 305.2 = 599.2

\ln\!\left(\frac{62.22}{599.2}\right) = \ln(0.10384) = -2.265

Step 6 — Assemble.

E_j = -\frac{RT}{F}(0.01109)(-2.265) = -(25.69\ \text{mV})(0.01109)(-2.265) = +0.645\ \text{mV}

Interpretation: despite seawater being a complex multi-ion medium, the full Henderson treatment predicts a sub-millivolt junction potential (≈0.6 mV) at a direct KCl-bridge/seawater interface — consistent with the design intent of the saturated KCl bridge, and small enough that oceanographic pH and ISE measurements can proceed with only modest residual uncertainty. Limiting-case check: if seawater's Cl⁻ and Na⁺ concentrations were set equal to the bridge's own K⁺ and Cl⁻ (i.e., both sides identical), every $(c_{i,2}-c_{i,1})$ term would vanish, the numerator and denominator sums would both go to zero, and the log-ratio argument would go to $\ln(1)=0$ — correctly forcing $E_j \to 0$ for an identical-composition, no-gradient junction.

Limiting cases, scope & common traps

Situation	Behavior of $E_j$	Why
$u_+ = u_-$ exactly (idealized matched pair)	$E_j = 0$ for any concentration ratio	Prefactor in the 1:1 formula vanishes identically
$c_1 = c_2$ (no concentration gradient)	$E_j = 0$ regardless of mobility mismatch	$\ln(c_1/c_2)=\ln(1)=0$ ; no driving force for differential diffusion
Large mobility mismatch (e.g., H⁺ or OH⁻ present)	$E_j$ can reach tens of millivolts	H⁺/OH⁻ mobilities are 4–7× larger than typical ions (Grotthuss mechanism)
Saturated KCl bridge inserted between arbitrary solutions	$E_j$ reduced to roughly 1–2 mV, sometimes sub-mV	Bridge concentration dominates flux; K⁺/Cl⁻ mobilities nearly equal
Multivalent ions present (e.g., Ca²⁺, SO₄²⁻)	Henderson equation still applies but $z_i^2$ weighting changes the mixture; KCl bridge less effective	Charge-squared weighting in the denominator amplifies multivalent ion contributions
Very dilute solutions (activity coefficients far from 1)	Simple formula becomes unreliable; must use activities, not concentrations	Debye–Hückel corrections become significant below ~10⁻³ M
Flowing or stirred junctions, or a physically ill-defined frit	Henderson/Planck theory assumes a smooth, one-dimensional diffusion profile; real frits and flowing junctions deviate	Turbulent mixing or non-uniform pore geometry breaks the 1-D diffusion assumption used in the derivation

Common trapCommon misconception: "the Nernst equation is wrong" or "junction potentials can be calculated exactly for any real electrode"

Students sometimes conclude that a measured potentiometric error means the Nernst equation itself has failed. In fact the Nernst equation remains exactly correct for each electrode half-reaction; the junction potential is a separate, additive term arising from the physical liquid interface between the two half-cells, not a flaw in electrode thermodynamics. Equally common is the belief that the Henderson equation gives an exact, universally reliable number for $E_j$ in any real cell. It does not: Henderson theory assumes an idealized, one-dimensional, purely diffusive mixing zone with well-defined bulk concentrations on either side. Real frits, flowing junctions, and asymmetric geometries introduce additional, generally uncalculable contributions. In practice, chemists treat $E_j$ as minimized by good bridge design and calibration, not as a quantity to be computed and subtracted from every measurement — hence the universal reliance on matched-ionic-strength calibration standards in careful potentiometry.

The chemist behind it

The theory used throughout this article is named for Lawrence Joseph Henderson, a Harvard biochemist and physiologist better known for the Henderson–Hasselbalch buffer equation, whose 1908 analysis of diffusion potentials across membrane-like boundaries was later adapted and made rigorous for electrolyte junctions by Max Planck (1890) and, independently, by Peter Henderson (a distinct contemporary electrochemist, often conflated in casual usage with L. J. Henderson) in a series of papers refining the mixed-electrolyte treatment used in the general formula above. The lineage is genuinely tangled — the "Henderson equation" of electroanalytical chemistry is most directly due to the 1907 work of Peter Henderson himself, who solved the steady-state diffusion problem for arbitrary mixed electrolytes at a planar junction, building on Max Planck's earlier (1890) treatment of the simpler single-salt case that appears as the compact 1:1 formula derived above. Planck's derivation, rooted in the Nernst–Planck flux equation that also underlies membrane and semiconductor transport theory, remains the cleanest route to the result, which is why it is the derivation reproduced here.

Origins · 1 minMax Karl Ernst Ludwig Planck and how this was discovered

Max Karl Ernst Ludwig Planck

1858–1947 · German theoretical physicist working at the intersection of thermodynamics and early electrochemical transport theory before his famous 1900 work on blackbody radiation

Long before quantum theory made his name a household word, Planck spent the 1880s and 1890s working through problems in classical thermodynamics and electrochemistry, including the diffusion of dissociated electrolytes across a boundary — precisely the physics that gives rise to the liquid junction potential. His 1890 paper on the potential difference at the boundary between two electrolyte solutions solved the steady-state Nernst–Planck flux balance for a single binary salt, arriving at the compact logarithmic mobility-ratio formula that underlies all subsequent junction potential theory, including the more general mixed-electrolyte treatment. It is a quiet irony of the history of science that the same mathematical machinery — flux, gradient, and a self-consistent electric field — that Planck used to describe ions crossing a beaker boundary would, a decade later, echo through his treatment of energy quanta in a very different physical problem.

History

- 1888 — Walther Nernst publishes the theory of diffusion potentials and ionic mobility, laying the groundwork for treating ion transport electrochemically.

- 1890 — Max Planck derives the potential at a single-salt liquid junction, giving the compact logarithmic mobility-ratio formula.

- 1907 — Peter Henderson extends the treatment to arbitrary mixed electrolytes on both sides of the junction, producing the general Henderson equation.

- 1920s–1930s — Saturated KCl salt bridges become standard laboratory practice for reference electrodes, following empirical recognition that K⁺/Cl⁻ mobility matching minimizes residual junction voltage.

- 1950s–present — IUPAC operational pH conventions and NIST buffer standards are defined explicitly to keep residual junction potentials small and reproducible, formalizing salt-bridge practice into a metrological standard.

What exactly causes a liquid junction potential — is it a chemical reaction at the interface?

No chemical reaction is involved. A liquid junction potential arises purely from physical transport: when two electrolyte solutions of different composition or concentration are placed in contact, ions diffuse across the boundary driven by the concentration gradient, exactly as described by Fick's laws. Because different ions have different intrinsic mobilities (some move faster through water than others, largely due to their size, charge, and hydration shell), the faster ion initially outpaces the slower one across the boundary. This creates a tiny, transient charge separation — a thin dipole layer only a few ion diameters thick — whose resulting electric field then decelerates the fast ion and accelerates the slow one until they cross together at equal rates. The steady voltage sustaining that field is the junction potential. It is a kinetic and electrostatic phenomenon, not a redox or acid-base reaction.

Why doesn't the Nernst equation already include the junction potential?

The Nernst equation describes the equilibrium potential of a single electrode half-reaction as a function of ion activities at that electrode surface, derived from the thermodynamics of the electron-transfer (or ion-exchange) reaction itself. A liquid junction potential arises from an entirely different physical process — differential ionic diffusion at a boundary between two electrolyte solutions where no electron transfer or specific chemical equilibrium occurs. When you measure a real cell voltage with a voltmeter, you are actually measuring the sum of both electrode potentials (via the Nernst equation for each) plus any liquid junction potentials sitting between the two half-cells. The measured cell voltage is therefore $E_{\text{cell}} = E_{\text{cathode}} - E_{\text{anode}} + E_j$ , and the junction term must be estimated, minimized, or calibrated away separately.

Can a liquid junction potential ever be calculated exactly for a real electrode in the laboratory?

Not with full rigor for a genuinely realistic system. The Henderson equation assumes an idealized, one-dimensional, purely diffusive mixing zone with sharply defined bulk concentrations on either side and no convection, stirring, or irregular pore geometry. Real junctions — porous frits, ceramic pins, ground-glass sleeves, flowing capillaries — have complicated, often poorly characterized internal geometry, and the local mixing profile rarely matches the idealized one-dimensional diffusion picture exactly. For this reason, analytical chemists treat calculated $E_j$ values (like those in the worked examples here) as useful order-of-magnitude estimates and design guides, not as exact corrections to subtract from a measured cell voltage. The practical solution is to minimize $E_j$ by bridge design and to calibrate instruments using standards whose ionic composition closely matches the sample, so that residual junction potentials largely cancel between calibration and measurement.

Why does a junction between two different concentrations of the same salt, like HCl in Example 1, still give such a large junction potential compared to a KCl bridge?

The size of the junction potential is set by two multiplicative factors: how large the concentration ratio is (through the logarithm) and how badly mismatched the two ions' mobilities are (through the prefactor). HCl is an extreme case because H⁺ has an anomalously large mobility, roughly 4.6 times that of Cl⁻, due to the Grotthuss proton-hopping mechanism rather than ordinary hydrodynamic diffusion. This gives a large mobility-mismatch prefactor (≈0.64), so even though the concentration ratio in Example 1 was a modest tenfold, the resulting junction potential (about 38 mV) is large. By contrast, K⁺ and Cl⁻ happen to have almost identical mobilities in water, giving a prefactor of only about −0.019, so even a KCl junction with a much larger concentration ratio produces only a few millivolts.

How does a salt bridge physically "minimize" the junction potential rather than truly eliminating it?

A salt bridge works by ensuring that at each of its two ends, the dominant diffusive flux is that of the bridge's own ions (typically K⁺ and Cl⁻ from a concentrated or saturated fill solution) rather than the sample's or reference's ions, because the bridge concentration vastly exceeds the concentrations on either side. Since K⁺ and Cl⁻ have nearly equal mobilities, the junction potential generated at each bridge interface is intrinsically small, even though it is not exactly zero (their mobilities differ by about 4%, not 0%). Because there are now two small junctions in series rather than one large, uncontrolled junction between the sample and the reference fill directly, the net residual voltage is typically reduced from tens of millivolts to only one or two millivolts — a large practical improvement, though never a perfect cancellation.

Does temperature affect the liquid junction potential, and if so how?

Yes, in two distinct ways. First, the thermal voltage prefactor $RT/F$ scales linearly with absolute temperature, so at a fixed mobility mismatch and concentration ratio, $E_j$ grows proportionally with $T$ — roughly a 0.3% increase in $E_j$ per kelvin near room temperature, the same scaling that affects the ordinary Nernst slope. Second, and less predictably, ionic mobilities themselves are temperature-dependent (typically increasing with temperature due to lower solution viscosity), and different ions' mobilities do not necessarily change at exactly the same rate with temperature, so the mobility-mismatch prefactor itself can drift slightly with temperature. In precision potentiometry, both effects mean that calibration should be performed at, or very near, the temperature of the actual sample measurement.

Are there electrolytes even better matched than KCl for salt bridges?

KNO₃ and NH₄NO₃ are sometimes used instead of KCl, particularly when the sample contains ions that would form an insoluble precipitate with chloride (for example, silver, lead, or mercury-containing samples, where AgCl or PbCl₂ could clog the junction). Their cation/anion mobility pairs are also reasonably well matched, though generally not quite as close as K⁺/Cl⁻. For most routine aqueous potentiometry, however, saturated KCl remains the default choice because of its combination of near-ideal mobility matching, high solubility (allowing a large swamping concentration), and low cost, and because so much reference-electrode engineering and calibration-buffer chemistry has been built around it as the de facto standard.