Selectivity Coefficients and ISE Errors

Clinical blood-gas and electrolyte analyzers (measuring Na⁺, K⁺, Ca²⁺, Cl⁻, and pH simultaneously from a single small blood sample) are built entirely around selectivity-coefficient engineering: valinomycin-based K⁺ membranes are specifically chosen because their $K^{\text{pot}}_{\text{K,Na}}$ is small enough (typically $10^{-4}$–$10^{-5}$) to survive the roughly 30-to-1 Na⁺-to-K⁺ activity ratio in serum, a calculation directly analogous to Example 2 above but with a membrane engineered to pass the clinical tolerance rather than fail it. Environmental and agricultural laboratories use fluoride, nitrate, and ammonium ISEs in soil extracts and natural waters where sulfate, chloride, and other major ions are present at far higher concentration than the trace analyte — precisely the charge-mismatch detection-limit calculation of Example 3, which is why total ionic strength adjustment buffers (TISAB) are added: they suppress activity-coefficient variability and, in the fluoride case, complex interfering polyvalent cations that would otherwise distort the free fluoride activity itself, on top of any pure selectivity-coefficient interference. Industrial process control uses ISEs for real-time monitoring of electroplating baths, boiler water, and fermentation broths, where the sample matrix is often far more concentrated and far less "clean" than any calibration standard; selectivity-coefficient error budgets, computed exactly as in this article, are used to decide whether an in-line ISE reading can be trusted directly or must be periodically cross-checked against a reference method such as ion chromatography. Biomedical and materials research groups developing new solid-contact and all-solid-state ISEs (replacing the classical liquid internal filling solution with a conducting polymer or ionic-liquid layer) report selectivity coefficients using the same SSM/FIM protocols as a primary performance metric, because a new membrane chemistry is not considered viable for real-sample deployment — for instance wearable sweat-electrolyte sensors — until its selectivity coefficients against the dominant physiological interferents are quantified and shown to meet the same kind of error budget worked through above. This same selectivity-and-error-budget logic carries forward directly into Potentiometric Titrations, where an ISE used as an endpoint-detecting indicator electrode must likewise be checked for interference from other ions present in the titration matrix before its inflection point can be trusted as a true stoichiometric endpoint.

Ask the tutor: If my Ca²⁺ ISE has $K^{\text{pot}}_{\text{Ca,Mg}} = 0.015$ and my sample has both Mg²⁺ and Na⁺ present, how do I combine both interference terms correctly in the same detection-limit calculation?

An ion-selective membrane — glass, crystalline (LaF₃, AgCl/Ag₂S), or liquid ion-exchanger in PVC — works by establishing a phase-boundary potential that depends on the activity ratio of an ion between the bulk sample and the membrane surface, exactly as developed for the glass pH electrode. The Nernstian response assumes that only one ion, the "primary" ion $i$, participates in this exchange equilibrium. Real membranes are never perfectly selective: exchange sites (fixed anionic groups on a glass surface, or a charged ionophore in a PVC liquid membrane) have some finite affinity for other ions of the same or opposite sign, denoted $j$. When both $i$ and $j$ exchange onto the membrane simultaneously, the phase-boundary potential is no longer a function of $a_i$ alone — it becomes a function of an effective activity that blends $a_i$ and a scaled version of $a_j$. The scaling factor is the potentiometric selectivity coefficient $K^{\text{pot}}_{i,j}$. Physically it is a ratio of exchange-equilibrium constants and mobilities inside the membrane: how easily $j$ replaces $i$ at an exchange site, multiplied by how much more or less mobile $j$ is once inside the membrane phase, relative to $i$. It is not a thermodynamic constant in the strict sense — it depends on membrane composition, ionic strength, and (for imperfect membranes) even on the concentration ratio used to measure it, which is why IUPAC prescribes standardized measurement protocols rather than leaving it undefined. The consequence for the analyst is that every ISE reading is really a reading of an "apparent activity" $a_i^{\text{app}} = a_i + \sum_j K^{\text{pot}}_{i,j}\,a_j^{z_i/z_j}$, and everything downstream — calibration curves, detection limits, systematic bias in real samples — follows from taking this apparent activity seriously.

The equation at the bottom of that figure is the Nikolsky-Eisenman relation itself, and it is worth reading slowly before deriving it: the electrode measures a single effective activity that is a sum, not a product, of the primary and (scaled) interferent activities — meaning interference is additive on the activity scale, not multiplicative on the potential scale.

The full Nikolsky-Eisenman equation for a primary ion $i$ of charge $z_i$ in the presence of interferents $j$ of charge $z_j$:

where:

• $E$ is the measured cell potential (V), $L$ a constant absorbing the reference electrode, asymmetry, and standard potentials,
• $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$, $T$ the absolute temperature (K), $F = 96485\ \text{C mol}^{-1}$,
• $a_i$ the activity of the primary ion, $a_j$ the activity of each interferent,
• $z_i, z_j$ the signed charges of primary ion and interferent (same sign required for interference to occur at all — a cation-selective membrane is not perturbed by anions in this term),
• $K^{\text{pot}}_{i,j}$ the (dimensionless, ion-pair-specific) potentiometric selectivity coefficient.

Reading it: the electrode behaves as though it were seeing a single ion at "apparent activity" $a_i + \sum_j K^{\text{pot}}_{i,j} a_j^{z_i/z_j}$; the exponent $z_i/z_j$ is required so that the interferent term has the same dimensional weight in the exponential response as the primary-ion term — a doubly charged interferent competing with a singly charged primary ion must be raised to the ½ power. The systematic relative error caused by an interferent, expressed as a fractional error in the reported activity of $i$ (assuming a single interferent for clarity):

where:

• $\Delta a_i$ is the difference between the apparent (reported) activity and the true activity of $i$,
• all other symbols as above.

Reading it: this is the ratio of the two terms inside the logarithm's argument. It says the relative error is what matters, not the absolute size of $K^{\text{pot}}_{i,j}$ alone — a selectivity coefficient of $10^{-3}$ is harmless against an interferent at the same activity as the analyte, but ruinous against an interferent present at $10^4$ times the analyte's activity. The detection limit $a_i^{\text{DL}}$ — the lowest activity of $i$ at which the calibration curve is still Nernstian rather than dominated by interference — is IUPAC-defined as the activity at the intersection of the two asymptotes of the calibration curve, which in the single-interferent, equal-charge case reduces to:

where:

• $a_i^{\text{DL}}$ is the primary-ion activity at which the true term equals the interference term (the intersection of the low-concentration and high-concentration asymptotes on a semilog plot of $E$ vs. $\log a_i$).

Reading it: you cannot measure $a_i$ reliably below the level at which the interferent's contribution to the "apparent activity" equals the primary ion's own contribution — below this point, the electrode's response flattens and becomes governed entirely by the constant background of $a_j$.

Example 1 — Foundational: computing a selectivity coefficient from the separate-solution method (SSM). A calcium ISE (divalent primary ion, $z_{\text{Ca}} = +2$) is calibrated in a pure CaCl₂ solution and gives $E_1 = -18.4\ \text{mV}$ (vs. a fixed reference) at $a_{\text{Ca}} = 1.00\times10^{-2}$. In a separate pure NaCl solution ($z_{\text{Na}} = +1$) at $a_{\text{Na}} = 1.00\times10^{-1}$, the same electrode gives $E_2 = +42.6\ \text{mV}$. Find $K^{\text{pot}}_{\text{Ca,Na}}$ at 25.0 °C. The SSM protocol: measure $E$ in a solution of pure $i$ at activity $a_i$, then separately in a solution of pure $j$ at activity $a_j$, and use the Nikolsky-Eisenman equation applied to each solution separately (in the second solution, $a_i = 0$, so the electrode responds purely as if to $K^{\text{pot}}_{i,j}\,a_j^{z_i/z_j}$). For the pure-Ca solution:

For the pure-Na solution, treating the electrode as responding to the "equivalent Ca activity" produced by Na:

Subtracting the second equation from the first eliminates $L$:

At 25.0 °C, $RT/F = 0.025693\ \text{V}$, so $RT/(2F) = 0.012846\ \text{V} = 12.846\ \text{mV}$. Compute the left side:

Solve for the log term:

Exponentiate:

Now insert $a_{\text{Ca}} = 1.00\times10^{-2}$ and $a_{\text{Na}} = 1.00\times10^{-1}$, so $a_{\text{Na}}^2 = 1.00\times10^{-2}$:

So $K^{\text{pot}}_{\text{Ca,Na}} \approx 8.7\times10^{-3}$ (two sig figs, matching the precision of the millivolt readings). This is a physically reasonable value for a Ca-ISE with a phosphate-ester liquid ion-exchanger membrane, whose real published selectivity coefficients for Na⁺ interference typically fall in the $10^{-2}$–$10^{-3}$ range.

Limiting-case check: if the electrode were perfectly selective ($K^{\text{pot}}_{\text{Ca,Na}} \to 0$), $E_2$ would have to approach $-\infty$ for a fixed finite $a_{\text{Na}}$ — the pure-interferent solution would generate no measurable signal at all, which matches the physical intuition that a perfectly selective membrane is utterly blind to the interferent.

Example 2 — Applied: error budget for a K⁺ ISE in serum. A potassium ISE has manufacturer-quoted selectivity coefficient $K^{\text{pot}}_{\text{K,Na}} = 1.0\times10^{-3}$ (both monovalent, $z_{\text{K}} = z_{\text{Na}} = +1$, so the exponent $z_i/z_j = 1$). A patient's serum has $a_{\text{K}} = 4.20\times10^{-3}$ (physiological ~4.2 mM, assuming activity ≈ concentration for this estimate at physiological ionic strength with a fixed ionic-strength background already accounted for in calibration) and $a_{\text{Na}} = 1.40\times10^{-1}$ (140 mM sodium, a typical serum value). Find the absolute and relative error in the reported potassium activity, and check whether this error is clinically tolerable (clinical labs typically require better than ±2% accuracy on serum K⁺). The apparent activity the electrode reports:

Compute the interference term:

So:

The absolute error is $\Delta a_{\text{K}} = 1.40\times10^{-4}$, and the relative error is:

This 3.3% positive bias exceeds the ±2% clinical tolerance — meaning that a K⁺ ISE with this selectivity coefficient, used naively on undiluted serum, would systematically over-report potassium by a clinically significant margin at the low end of the normal range, which is exactly the scenario clinical instruments must guard against (real clinical analyzers use ionophores, typically valinomycin-based membranes, engineered for $K^{\text{pot}}_{\text{K,Na}} \sim 10^{-4}$ to $10^{-5}$, specifically to survive this test).

Limiting-case check: if $a_{\text{Na}}$ were reduced to zero (a hypothetical sodium-free sample), the interference term vanishes exactly and $a_{\text{K}}^{\text{app}} \to a_{\text{K}}$, recovering the ideal Nernstian reading — confirming the error scales linearly with the interferent activity as it must for two ions of equal charge.

Example 3 — Exam-level: detection limit with a charge mismatch, and back-calculating a required selectivity coefficient. A fluoride ISE (LaF₃ crystal membrane, $z_{\text{F}} = -1$) is to be used in groundwater containing a fixed background of sulfate interference, $a_{\text{SO}_4} = 5.0\times10^{-4}$ ($z_{\text{SO}_4} = -2$). A groundwater standards program requires the electrode to reliably measure fluoride down to $a_{\text{F}} = 1.0\times10^{-6}$ (roughly 0.019 mg/L, a low but environmentally relevant level). What is the maximum tolerable $K^{\text{pot}}_{\text{F,SO}_4}$ for the detection limit to sit at or below this required level? Then verify what detection limit results if the electrode actually has $K^{\text{pot}}_{\text{F,SO}_4} = 2.0\times10^{-4}$ (a realistic literature value for fluoride electrodes' generally excellent rejection of sulfate). First, note the exponent required: with $z_i = -1$ (fluoride) and $z_j = -2$ (sulfate), the Nikolsky-Eisenman exponent is

so the detection-limit formula becomes:

Part A — required selectivity coefficient. Require $a_{\text{F}}^{\text{DL}} \le 1.0\times10^{-6}$:

Compute the square root:

So:

The membrane must reject sulfate to better than $K^{\text{pot}}_{\text{F,SO}_4} = 4.5\times10^{-5}$ for the stated detection limit to be achievable in this matrix.

Part B — actual detection limit with the literature value. With $K^{\text{pot}}_{\text{F,SO}_4} = 2.0\times10^{-4}$:

This is roughly 4.5 times higher (worse) than the required $1.0\times10^{-6}$ — so the standard literature LaF₃ electrode, with this selectivity coefficient, would fail to meet the required low-level detection limit in this particular sulfate-rich groundwater; a fluoride-selective preconcentration step or an ionic-strength-adjustment buffer that also complexes or dilutes sulfate would be needed, or a lower-interference membrane batch must be sourced.

Limiting-case check: notice the half-power exponent means the detection limit depends only on the square root of the sulfate activity — a tenfold increase in sulfate activity worsens the detection limit by only about $\sqrt{10} \approx 3.16\times$, not tenfold, precisely because of the charge mismatch built into the Nikolsky-Eisenman exponent. If sulfate activity were instead raised to a full power (as it would be for an equal-charge interferent), the detection limit would degrade proportionally faster — confirming that the fractional exponent is doing real, checkable work in tempering the impact of a highly charged interferent.

The Nikolsky-Eisenman equation carries two names because it was built in two historical stages, separated by more than two decades and by the Cold War's division of electrochemical research communities. Boris Petrovich Nikolsky was a Soviet physical chemist working at Leningrad State University in the 1930s, at a time when the glass electrode's Nernstian pH response was already an established laboratory tool but its systematic deviations at high pH and in the presence of alkali-metal ions were not understood mechanistically. Nikolsky proposed, based on an ion-exchange model of the glass surface, that the glass membrane's potential should respond not to hydrogen ion activity alone but to a weighted sum of hydrogen and interfering alkali-metal-ion activities — introducing what is now recognized as the first selectivity-coefficient formalism, originally derived specifically for the H⁺/Na⁺ "alkaline error" of glass pH electrodes. His derivation, grounded in the same ion-exchange equilibrium logic used above, correctly predicted that the alkaline error should worsen at high pH (low H⁺ activity) and high Na⁺ activity — exactly the detection-limit crossover behavior this article derives in general form. The equation's generalization to arbitrary ion-selective membranes — crystalline, liquid-ion-exchanger, and neutral-carrier types alike, with arbitrary charges $z_i$ and $z_j$ — came from the American electrochemist James W. Ross and collaborators, alongside the broader systematization efforts of Garry A. Rechnitz and others through the 1960s, as the ISE field exploded beyond glass pH electrodes into calcium-, fluoride-, and later ammonia- and CO₂-gas-sensing electrodes. Ross's 1966 work on a practical liquid-membrane calcium electrode, and the accompanying formal extension of Nikolsky's exchange-equilibrium treatment to non-equal charges, produced the exponent $z_i/z_j$ in essentially the modern form. Eisenman's name entered through parallel theoretical work on glass and liquid-membrane ion selectivity mechanisms, particularly his systematic classification of alkali-metal cation selectivity sequences in glass electrodes, which gave the physical-chemical foundation for why particular membranes prefer particular ions — connecting the empirical selectivity coefficient to the underlying ion-exchange energetics.