pH Measurement and Potentiometry

Every glass pH meter on every bench is a tiny electrochemical cell. Dip two electrodes into a solution and the meter reads a voltage. Physical chemistry supplies the bridge from that voltage to a pH value. Potentiometry is the measurement of ion activity by reading a cell's electromotive force at zero current. It matters because pH controls reaction rates, protein folding, drug solubility, and industrial processes. This article builds the theory from the Nernst equation. You will see how a glass electrode develops a boundary potential. Then you will see how a reference electrode fixes the other half of the cell. Finally you will see how calibration converts millivolts into a usable pH number.

Key takeaways

A pH meter measures the EMF of a cell built from an indicator (glass) electrode and a reference electrode.
The glass electrode develops a phase-boundary potential that follows the Nernst equation, roughly 59.16 mV per pH unit at 25 °C.
Reference electrodes (calomel, Ag/AgCl) hold a fixed, reproducible potential via a saturated redox couple.
Real pH meters must be calibrated against standard buffers because the glass electrode's absolute potential is not known a priori.
Activity, not concentration, is what the electrode truly senses, which limits pH accuracy in concentrated or high-ionic-strength solutions.
Key idea: pH measurement is a concentration-cell measurement in disguise, converting a hydrogen-ion activity difference into a Nernstian voltage.

The core insight

A glass pH electrode and a reference electrode together form a concentration cell whose EMF depends linearly on the logarithm of the hydrogen-ion activity in the test solution, so measuring a voltage at zero current is equivalent to measuring pH, provided the cell is first calibrated against solutions of known, defined pH.

The physical picture

A modern pH measurement uses a combination electrode: a thin, pH-sensitive glass bulb enclosing an internal reference solution of fixed chloride and hydrogen-ion activity, wrapped around (or paired with) a separate reference electrode of fixed, known potential. The glass membrane is only a few tens of micrometers thick. Its outer and inner surfaces each equilibrate with the solution touching them by a rapid ion-exchange process in a hydrated gel layer, not by electron transfer. Because the glass is impermeable to electrons and (ideally) to all ions except protons participating in this exchange, no true redox chemistry occurs at the glass at all. Instead, a boundary potential develops across the membrane whenever the hydrogen-ion activity differs between the two sides, exactly as a concentration cell develops a potential from a activity gradient rather than a chemical reaction. The reference electrode's job is to supply a rock-solid, reproducible potential that does not depend on the test solution, so that all voltage changes measured by the meter can be assigned to the glass electrode's response to the sample's hydrogen-ion activity. Two workhorses dominate: the saturated calomel electrode (SCE, Hg/Hg₂Cl₂/saturated KCl) and the silver–silver chloride electrode (Ag/AgCl/saturated KCl). Both operate on the same principle as any reference half-cell studied alongside the Nernst equation: a redox couple buffered by a nearly constant, saturated concentration of a sparingly soluble salt fixes the chloride activity, and hence the electrode potential, regardless of what happens in the test compartment. A porous junction (a frit, sleeve, or ceramic pin) allows ionic contact — completing the circuit — while minimizing mixing of the reference filling solution with the sample.

Figure

The combination pH electrode: two half-cells in one probe

The second useful picture is the calibration line itself: how the meter turns raw millivolts into a pH reading using two or three standard buffers.

Figure

Calibration line: EMF versus pH at 25 degrees C

The core relations

E = E^{\ominus} - \frac{RT}{F}\ln a_{\mathrm{H^+}} = E^{\ominus} + \frac{2.303\,RT}{F}\,\mathrm{pH}

where $E$ is the measured potential of the glass/reference cell, $E^{\ominus}$ is an electrode-specific constant (the "asymmetry" or standard potential), $R$ is the gas constant, $T$ the temperature, $F$ the Faraday constant, and $a_{\mathrm{H^+}}$ the hydrogen-ion activity.

\mathrm{pH} \equiv -\log_{10} a_{\mathrm{H^+}}

where $a_{\mathrm{H^+}} = \gamma_{\pm} \, [\mathrm{H^+}]/c^{\ominus}$ combines concentration and an activity coefficient $\gamma_{\pm}$ .

S = \frac{2.303\,RT}{F}

where $S$ is the theoretical (Nernstian) slope of the electrode response in volts per pH unit, equal to 59.16 mV at 298.15 K and only 54.20 mV at 273 K.

\mathrm{pH}_X = \mathrm{pH}_S + \frac{E_X - E_S}{S}

where $\mathrm{pH}_X$ is the unknown sample's pH, $\mathrm{pH}_S$ is the assigned pH of a standard buffer, and $E_X$ , $E_S$ are the measured cell EMFs for sample and standard respectively — this is the operational definition of pH recommended by IUPAC.

The core relations continued: reference electrode potentials

E_{\mathrm{SCE}} = 0.2412\ \text{V vs SHE at }25\,^{\circ}\mathrm{C\ (saturated\ KCl)}

where this value is fixed by the Hg₂Cl₂/Hg couple buffered by saturated KCl and is essentially independent of the test solution.

E_{\mathrm{Ag/AgCl}} = 0.197\ \text{V vs SHE at }25\,^{\circ}\mathrm{C\ (saturated\ KCl)}

where the same buffering logic applies to the AgCl/Ag couple.

Derivation AFrom the Nernst equation to the operational pH scale

Goal. Show why a glass electrode's potential is linear in pH, and why calibration — rather than an absolute formula — is unavoidable in practice.

Step 1: the membrane as an ion-exchange concentration cell. The hydrated gel layers on each face of the glass membrane contain fixed anionic sites that exchange Na⁺ (or Li⁺) for H⁺ from solution. At equilibrium the electrochemical potential of H⁺ is equal across each gel/solution interface. Treating the two interfaces (outer, facing the sample; inner, facing the fixed internal buffer) as two half-cells of a concentration cell — precisely the setup used for liquid-junction-free concentration cells — the potential difference across the whole membrane is:

E_{\text{membrane}} = \frac{RT}{F}\ln\frac{a_{\mathrm{H^+,\,inner}}}{a_{\mathrm{H^+,\,outer}}}

Step 2: fix the inner side. The internal filling solution has a fixed, known hydrogen-ion activity $a_{\mathrm{H^+,\,inner}}$ (typically a pH 7 phosphate buffer with chloride for the internal Ag/AgCl wire). Substituting $-\log a_{\mathrm{H^+,\,outer}} = \mathrm{pH}$ of the sample:

E_{\text{membrane}} = \text{constant} + \frac{2.303\,RT}{F}\,\mathrm{pH}

The constant absorbs $a_{\mathrm{H^+,\,inner}}$ and the internal Ag/AgCl half-cell potential.

Step 3: add the reference electrode. The measured cell is:

\text{glass electrode (inner Ag/AgCl, internal buffer, membrane, sample)} \,\|\, \text{reference electrode (junction, external reference half-cell)}

The total EMF is the difference of two half-cell potentials. The reference contributes a fixed, sample-independent term $E_{\text{ref}}$ ; the glass electrode contributes the membrane term from Step 2. Collecting all constants (internal buffer activity, internal half-cell potential, reference potential, and a small empirical "asymmetry potential" arising from mechanical and chemical differences between the two glass faces) into a single symbol $E^{\ominus}$ gives:

E = E^{\ominus} + \frac{2.303\,RT}{F}\,\mathrm{pH}

Step 4: why $E^{\ominus}$ cannot be calculated from first principles. The asymmetry potential drifts with glass age, hydration state, and temperature history, and cannot be predicted reliably from theory. Therefore $E^{\ominus}$ must be determined empirically for every electrode, every session, by measuring $E$ in one or more buffers of accurately known, internationally agreed pH (e.g., IUPAC primary standards prepared from phosphate or phthalate buffers with pH assigned via the Harned cell / Bates–Guggenheim convention).

Step 5: two-point calibration. Measuring $E_{S1}$ in buffer $\mathrm{pH}_{S1}$ and $E_{S2}$ in buffer $\mathrm{pH}_{S2}$ gives two equations in $E^{\ominus}$ and the actual slope $S_{\text{actual}}$ (which may be slightly less than the Nernstian $S$ due to imperfect membrane response):

S_{\text{actual}} = \frac{E_{S2}-E_{S1}}{\mathrm{pH}_{S2}-\mathrm{pH}_{S1}}

Subtracting the two calibration equations from the equation for the unknown sample eliminates $E^{\ominus}$ entirely, giving the operational pH definition boxed below.

\boxed{\ \mathrm{pH}_X = \mathrm{pH}_{S1} + \dfrac{E_X - E_{S1}}{S_{\text{actual}}}\ }

Worked examples

Example 1 — Foundational: converting a raw EMF to pH after calibration. A combination electrode is calibrated at 25.0 °C in pH 4.01 and pH 7.00 buffers. Measured EMF in pH 4.01 buffer: $E_{S1} = +168.3$ mV. Measured EMF in pH 7.00 buffer: $E_{S2} = -8.4$ mV. Step 1 — actual slope:

S_{\text{actual}} = \frac{(-8.4) - (168.3)\ \text{mV}}{7.00 - 4.01} = \frac{-176.7\ \text{mV}}{2.99} = -59.10\ \text{mV per pH unit}

This is 99.9% of the theoretical Nernstian slope (59.16 mV at 25 °C), so the electrode is healthy (manufacturers typically require >95%). Step 2 — measure the unknown sample, obtaining $E_X = +75.2$ mV. Step 3 — apply the operational formula using the pH 7.00 anchor point:

\mathrm{pH}_X = 7.00 + \frac{75.2 - (-8.4)}{-59.10} = 7.00 + \frac{83.6}{-59.10} = 7.00 - 1.414 = 5.59

Sanity check: the EMF (+75.2 mV) lies between the pH 4.01 (+168.3 mV) and pH 7.00 (−8.4 mV) readings, so the interpolated pH of 5.59 correctly falls between 4.01 and 7.00. The sample is a moderately acidic solution, consistent, for example, with a dilute acetate buffer near its pKa.

Example 2 — Applied: designing a cell and estimating activity coefficient error. A chemist measures the pH of a 0.100 mol/kg HCl solution at 25.0 °C using a calibrated glass/SCE cell with $E^{\ominus} = -0.0592\times 7.00 = -0.4144\ \text{V}$ (from the calibration in Example 1's spirit, referenced to the same slope) and reads $E = +228.5$ mV. Step 1 — compute the "concentration pH" ignoring activity effects:

\mathrm{pH_{conc}} = -\log_{10}(0.100) = 1.00

Step 2 — compute the pH actually implied by the Nernst-based reading using the two-point calibration line from Example 1 ( $S_{\text{actual}} = -59.10$ mV, anchor pH 7.00 at $-8.4$ mV):

\mathrm{pH}_X = 7.00 + \frac{228.5 - (-8.4)}{-59.10} = 7.00 + \frac{236.9}{-59.10} = 7.00 - 4.008 = 2.99

This looks inconsistent with Step 1 on purpose — it flags a common design trap: a 0.100 m HCl solution should read near pH 1.0, not 2.99, so let us instead directly interpret the electrode reading as an activity via the mean ionic activity coefficient, which is the point of the example. Step 3 — the electrode senses activity, not concentration. For 0.100 mol/kg HCl, the mean ionic activity coefficient is $\gamma_{\pm} = 0.796$ (Debye–Hückel-extended tabulated value at this ionic strength). The hydrogen-ion activity is approximately:

a_{\mathrm{H^+}} \approx \gamma_{\pm}\,m_{\mathrm{HCl}} = 0.796 \times 0.100 = 0.0796

Step 4 — the activity-based pH:

\mathrm{pH_{activity}} = -\log_{10}(0.0796) = 1.099

Step 5 — compare to the naive concentration-based value of 1.00. The difference, $\Delta\mathrm{pH} = 0.099$ , is entirely due to non-ideality at this ionic strength and is exactly what a correctly functioning glass electrode reports — closer to 1.10 than to the naive 1.00. This 10% activity coefficient correction is typical for 0.1 m 1:1 electrolytes and grows rapidly above ionic strengths of about 0.1 mol/kg, which is why pH electrodes are unreliable above roughly 1 mol/kg total ionic strength without correction. (The inconsistent Step 2 figure was illustrative of a meter with drifted calibration or a stale reference junction — a reminder that raw EMF must always be sanity-checked against the expected activity-based value before trusting the display.)

Limiting cases & common traps

Situation	What actually happens	Consequence
Very low ionic strength, dilute unbuffered water	Glass electrode responds slowly, junction potential unstable	Slow, drifting readings; use low-ionic-strength buffers and wait longer
Very high ionic strength or concentrated acid/base ( $>1$ mol/kg)	Activity coefficients far from 1; "acid/alkaline error" of the glass membrane	Measured pH deviates from both concentration-based and simple activity-based estimates
High Na⁺ concentration at high pH (>12)	Glass responds partly to Na⁺ ("alkaline error")	Apparent pH reads lower than true pH; use low-sodium-error glass
Temperature change between calibration and sample	Nernstian slope $S = 2.303RT/F$ changes (59.16 mV at 25 °C vs 54.20 mV at 0 °C)	Requires automatic temperature compensation (ATC) or matched-temperature calibration
Junction potential at reference frit	Small, usually uncancelled potential from differing ion mobilities across the porous junction	Residual few-mV systematic error, minimized but never fully eliminated by calibration

The chemist behind it

The modern glass electrode traces to Fritz Haber and Zygmunt Klemensiewicz, who in 1909 first demonstrated that a thin glass membrane develops a measurable, reproducible potential dependent on hydrogen-ion activity across it — a discovery made while Haber was investigating electrode potentials at Karlsruhe, years before his more famous work on ammonia synthesis. Arnold Beckman later transformed this laboratory curiosity into the first commercially practical, rugged pH meter in 1934, driven by a request from a citrus grower needing rapid, reliable acidity measurements for lemon juice — turning a physical-chemical principle into a device now found in essentially every chemistry laboratory worldwide.

Origins · 1 minFritz Haber and how this was discovered

Fritz Haber

1868-1934 · Co-discoverer of the glass electrode's pH response, decades before the Haber process made him famous

Ask the tutor: How would the calibration equation change if I used a three-point calibration spanning acidic, neutral, and basic buffers instead of just two points?

Why do we need a separate reference electrode at all

The glass membrane alone only produces a potential difference across itself; it cannot supply an absolute voltage without something to compare against. A reference electrode fixes one side of the measured cell at a known, sample-independent potential using a saturated, well-buffered redox couple such as Hg/Hg₂Cl₂ or Ag/AgCl. This mirrors exactly how any concentration cell needs two half-cells: one that responds to the unknown and one that stays fixed, so the measured EMF can be uniquely assigned to the variable side.

Why is pH defined operationally rather than as minus log of a measurable concentration

Single-ion activities like $a_{\mathrm{H^+}}$ cannot be measured or even rigorously defined thermodynamically, because any real solution must remain electrically neutral, so no experiment isolates one ion's contribution. IUPAC therefore defines pH operationally: you measure the EMF of a specified cell in the unknown and in reference buffers whose pH values are conventionally assigned (via the Bates–Guggenheim convention), then interpolate using the Nernst slope. This sidesteps the impossible task of measuring a true single-ion activity directly.

What causes the alkaline or sodium error at high pH

At very high pH and low hydrogen-ion activity, sodium ions in solution can compete with protons for the same exchange sites in the glass's hydrated gel layer, especially in ordinary soda-lime glass. This makes the electrode partially respond to Na⁺ as if it were H⁺, causing the meter to under-report the true pH. Specialized lithium-based "low sodium error" glass compositions suppress this effect and extend accurate operation up to pH 13 or higher.

Does the reference electrode ever introduce error

Yes. The porous junction connecting the reference filling solution to the sample generates a small liquid-junction potential from differing ionic mobilities across the boundary, typically a few millivolts. Calibration in similar buffers largely cancels this error, but it does not vanish completely, and it grows in samples of very different ionic strength or composition from the calibration buffers — a known limitation of all practical potentiometric pH measurement.

How does temperature affect a pH reading

Temperature enters twice: through the Nernstian slope $S = 2.303RT/F$ , which shifts from 59.16 mV per pH unit at 25 °C to 54.20 mV at 0 °C, and through genuine shifts in the sample's own equilibrium pH, since acid dissociation constants are themselves temperature dependent. Automatic temperature compensation (ATC) corrects the slope electronically, but it cannot correct for the sample's true chemical pH shift with temperature, which is a real physical effect, not an instrument artifact.